Error Parsing the Output

Exception

Traceback (most recent call last):

  File "./prover.py", line 176, in prover_aprove
    time = conv_timeline(err.readline())

  File "./prover.py", line 81, in conv_timeline
    line = line.strip().split("\t")[1]

IndexError: list index out of range

Standard Out

KILLED







Termination w.r.t. Q of the following Term Rewriting System could not be shown:

Q restricted rewrite system:
The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))

a__2ndspos(0, Z) → rnil

a__2ndspos(s(N), cons(X, Z)) → a__2ndspos(s(mark(N)), cons2(X, mark(Z)))

a__2ndspos(s(N), cons2(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))

a__2ndsneg(0, Z) → rnil

a__2ndsneg(s(N), cons(X, Z)) → a__2ndsneg(s(mark(N)), cons2(X, mark(Z)))

a__2ndsneg(s(N), cons2(X, cons(Y, Z))) → rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z)))

a__pi(X) → a__2ndspos(mark(X), a__from(0))

a__plus(0, Y) → mark(Y)

a__plus(s(X), Y) → s(a__plus(mark(X), mark(Y)))

a__times(0, Y) → 0

a__times(s(X), Y) → a__plus(mark(Y), a__times(mark(X), mark(Y)))

a__square(X) → a__times(mark(X), mark(X))

mark(from(X)) → a__from(mark(X))

mark(2ndspos(X1, X2)) → a__2ndspos(mark(X1), mark(X2))

mark(2ndsneg(X1, X2)) → a__2ndsneg(mark(X1), mark(X2))

mark(pi(X)) → a__pi(mark(X))

mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))

mark(times(X1, X2)) → a__times(mark(X1), mark(X2))

mark(square(X)) → a__square(mark(X))

mark(0) → 0

mark(s(X)) → s(mark(X))

mark(posrecip(X)) → posrecip(mark(X))

mark(negrecip(X)) → negrecip(mark(X))

mark(nil) → nil

mark(cons(X1, X2)) → cons(mark(X1), X2)

mark(cons2(X1, X2)) → cons2(X1, mark(X2))

mark(rnil) → rnil

mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))

a__from(X) → from(X)

a__2ndspos(X1, X2) → 2ndspos(X1, X2)

a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)

a__pi(X) → pi(X)

a__plus(X1, X2) → plus(X1, X2)

a__times(X1, X2) → times(X1, X2)

a__square(X) → square(X)

Q is empty.


↳ QTRS
  ↳ DependencyPairsProof

Q restricted rewrite system:
The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))

a__2ndspos(0, Z) → rnil

a__2ndspos(s(N), cons(X, Z)) → a__2ndspos(s(mark(N)), cons2(X, mark(Z)))

a__2ndspos(s(N), cons2(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))

a__2ndsneg(0, Z) → rnil

a__2ndsneg(s(N), cons(X, Z)) → a__2ndsneg(s(mark(N)), cons2(X, mark(Z)))

a__2ndsneg(s(N), cons2(X, cons(Y, Z))) → rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z)))

a__pi(X) → a__2ndspos(mark(X), a__from(0))

a__plus(0, Y) → mark(Y)

a__plus(s(X), Y) → s(a__plus(mark(X), mark(Y)))

a__times(0, Y) → 0

a__times(s(X), Y) → a__plus(mark(Y), a__times(mark(X), mark(Y)))

a__square(X) → a__times(mark(X), mark(X))

mark(from(X)) → a__from(mark(X))

mark(2ndspos(X1, X2)) → a__2ndspos(mark(X1), mark(X2))

mark(2ndsneg(X1, X2)) → a__2ndsneg(mark(X1), mark(X2))

mark(pi(X)) → a__pi(mark(X))

mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))

mark(times(X1, X2)) → a__times(mark(X1), mark(X2))

mark(square(X)) → a__square(mark(X))

mark(0) → 0

mark(s(X)) → s(mark(X))

mark(posrecip(X)) → posrecip(mark(X))

mark(negrecip(X)) → negrecip(mark(X))

mark(nil) → nil

mark(cons(X1, X2)) → cons(mark(X1), X2)

mark(cons2(X1, X2)) → cons2(X1, mark(X2))

mark(rnil) → rnil

mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))

a__from(X) → from(X)

a__2ndspos(X1, X2) → 2ndspos(X1, X2)

a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)

a__pi(X) → pi(X)

a__plus(X1, X2) → plus(X1, X2)

a__times(X1, X2) → times(X1, X2)

a__square(X) → square(X)

Q is empty.

Using Dependency Pairs [1,15] we result in the following initial DP problem:
Q DP problem:
The TRS P consists of the following rules:

MARK(times(X1, X2)) → A__TIMES(mark(X1), mark(X2))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(N)

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(N)

A__2NDSNEG(s(N), cons(X, Z)) → A__2NDSNEG(s(mark(N)), cons2(X, mark(Z)))

MARK(cons(X1, X2)) → MARK(X1)

MARK(2ndspos(X1, X2)) → MARK(X2)

MARK(times(X1, X2)) → MARK(X2)

A__2NDSNEG(s(N), cons(X, Z)) → MARK(Z)

MARK(2ndsneg(X1, X2)) → MARK(X1)

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(Y)

MARK(rcons(X1, X2)) → MARK(X1)

A__TIMES(s(X), Y) → MARK(Y)

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(Z)

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → A__2NDSNEG(mark(N), mark(Z))

A__2NDSNEG(s(N), cons(X, Z)) → MARK(N)

MARK(square(X)) → MARK(X)

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(Y)

A__FROM(X) → MARK(X)

A__TIMES(s(X), Y) → A__PLUS(mark(Y), a__times(mark(X), mark(Y)))

MARK(plus(X1, X2)) → A__PLUS(mark(X1), mark(X2))

MARK(rcons(X1, X2)) → MARK(X2)

MARK(2ndsneg(X1, X2)) → A__2NDSNEG(mark(X1), mark(X2))

MARK(cons2(X1, X2)) → MARK(X2)

A__TIMES(s(X), Y) → A__TIMES(mark(X), mark(Y))

MARK(2ndspos(X1, X2)) → MARK(X1)

MARK(s(X)) → MARK(X)

A__PLUS(s(X), Y) → MARK(X)

MARK(pi(X)) → MARK(X)

MARK(times(X1, X2)) → MARK(X1)

MARK(plus(X1, X2)) → MARK(X2)

A__SQUARE(X) → MARK(X)

A__2NDSPOS(s(N), cons(X, Z)) → MARK(Z)

MARK(pi(X)) → A__PI(mark(X))

A__2NDSPOS(s(N), cons(X, Z)) → MARK(N)

MARK(negrecip(X)) → MARK(X)

A__PLUS(0, Y) → MARK(Y)

MARK(posrecip(X)) → MARK(X)

MARK(2ndsneg(X1, X2)) → MARK(X2)

A__PI(X) → A__2NDSPOS(mark(X), a__from(0))

MARK(from(X)) → MARK(X)

MARK(2ndspos(X1, X2)) → A__2NDSPOS(mark(X1), mark(X2))

A__PI(X) → A__FROM(0)

MARK(from(X)) → A__FROM(mark(X))

A__PLUS(s(X), Y) → A__PLUS(mark(X), mark(Y))

A__SQUARE(X) → A__TIMES(mark(X), mark(X))

MARK(plus(X1, X2)) → MARK(X1)

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(Z)

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → A__2NDSPOS(mark(N), mark(Z))

A__PI(X) → MARK(X)

MARK(square(X)) → A__SQUARE(mark(X))

A__PLUS(s(X), Y) → MARK(Y)

A__2NDSPOS(s(N), cons(X, Z)) → A__2NDSPOS(s(mark(N)), cons2(X, mark(Z)))

A__TIMES(s(X), Y) → MARK(X)

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))

a__2ndspos(0, Z) → rnil

a__2ndspos(s(N), cons(X, Z)) → a__2ndspos(s(mark(N)), cons2(X, mark(Z)))

a__2ndspos(s(N), cons2(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))

a__2ndsneg(0, Z) → rnil

a__2ndsneg(s(N), cons(X, Z)) → a__2ndsneg(s(mark(N)), cons2(X, mark(Z)))

a__2ndsneg(s(N), cons2(X, cons(Y, Z))) → rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z)))

a__pi(X) → a__2ndspos(mark(X), a__from(0))

a__plus(0, Y) → mark(Y)

a__plus(s(X), Y) → s(a__plus(mark(X), mark(Y)))

a__times(0, Y) → 0

a__times(s(X), Y) → a__plus(mark(Y), a__times(mark(X), mark(Y)))

a__square(X) → a__times(mark(X), mark(X))

mark(from(X)) → a__from(mark(X))

mark(2ndspos(X1, X2)) → a__2ndspos(mark(X1), mark(X2))

mark(2ndsneg(X1, X2)) → a__2ndsneg(mark(X1), mark(X2))

mark(pi(X)) → a__pi(mark(X))

mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))

mark(times(X1, X2)) → a__times(mark(X1), mark(X2))

mark(square(X)) → a__square(mark(X))

mark(0) → 0

mark(s(X)) → s(mark(X))

mark(posrecip(X)) → posrecip(mark(X))

mark(negrecip(X)) → negrecip(mark(X))

mark(nil) → nil

mark(cons(X1, X2)) → cons(mark(X1), X2)

mark(cons2(X1, X2)) → cons2(X1, mark(X2))

mark(rnil) → rnil

mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))

a__from(X) → from(X)

a__2ndspos(X1, X2) → 2ndspos(X1, X2)

a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)

a__pi(X) → pi(X)

a__plus(X1, X2) → plus(X1, X2)

a__times(X1, X2) → times(X1, X2)

a__square(X) → square(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ Narrowing

Q DP problem:
The TRS P consists of the following rules:

MARK(times(X1, X2)) → A__TIMES(mark(X1), mark(X2))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(N)

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(N)

A__2NDSNEG(s(N), cons(X, Z)) → A__2NDSNEG(s(mark(N)), cons2(X, mark(Z)))

MARK(cons(X1, X2)) → MARK(X1)

MARK(2ndspos(X1, X2)) → MARK(X2)

MARK(times(X1, X2)) → MARK(X2)

A__2NDSNEG(s(N), cons(X, Z)) → MARK(Z)

MARK(2ndsneg(X1, X2)) → MARK(X1)

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(Y)

MARK(rcons(X1, X2)) → MARK(X1)

A__TIMES(s(X), Y) → MARK(Y)

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(Z)

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → A__2NDSNEG(mark(N), mark(Z))

A__2NDSNEG(s(N), cons(X, Z)) → MARK(N)

MARK(square(X)) → MARK(X)

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(Y)

A__FROM(X) → MARK(X)

A__TIMES(s(X), Y) → A__PLUS(mark(Y), a__times(mark(X), mark(Y)))

MARK(plus(X1, X2)) → A__PLUS(mark(X1), mark(X2))

MARK(rcons(X1, X2)) → MARK(X2)

MARK(2ndsneg(X1, X2)) → A__2NDSNEG(mark(X1), mark(X2))

MARK(cons2(X1, X2)) → MARK(X2)

A__TIMES(s(X), Y) → A__TIMES(mark(X), mark(Y))

MARK(2ndspos(X1, X2)) → MARK(X1)

MARK(s(X)) → MARK(X)

A__PLUS(s(X), Y) → MARK(X)

MARK(pi(X)) → MARK(X)

MARK(times(X1, X2)) → MARK(X1)

MARK(plus(X1, X2)) → MARK(X2)

A__SQUARE(X) → MARK(X)

A__2NDSPOS(s(N), cons(X, Z)) → MARK(Z)

MARK(pi(X)) → A__PI(mark(X))

A__2NDSPOS(s(N), cons(X, Z)) → MARK(N)

MARK(negrecip(X)) → MARK(X)

A__PLUS(0, Y) → MARK(Y)

MARK(posrecip(X)) → MARK(X)

MARK(2ndsneg(X1, X2)) → MARK(X2)

A__PI(X) → A__2NDSPOS(mark(X), a__from(0))

MARK(from(X)) → MARK(X)

MARK(2ndspos(X1, X2)) → A__2NDSPOS(mark(X1), mark(X2))

A__PI(X) → A__FROM(0)

MARK(from(X)) → A__FROM(mark(X))

A__PLUS(s(X), Y) → A__PLUS(mark(X), mark(Y))

A__SQUARE(X) → A__TIMES(mark(X), mark(X))

MARK(plus(X1, X2)) → MARK(X1)

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(Z)

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → A__2NDSPOS(mark(N), mark(Z))

A__PI(X) → MARK(X)

MARK(square(X)) → A__SQUARE(mark(X))

A__PLUS(s(X), Y) → MARK(Y)

A__2NDSPOS(s(N), cons(X, Z)) → A__2NDSPOS(s(mark(N)), cons2(X, mark(Z)))

A__TIMES(s(X), Y) → MARK(X)

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))

a__2ndspos(0, Z) → rnil

a__2ndspos(s(N), cons(X, Z)) → a__2ndspos(s(mark(N)), cons2(X, mark(Z)))

a__2ndspos(s(N), cons2(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))

a__2ndsneg(0, Z) → rnil

a__2ndsneg(s(N), cons(X, Z)) → a__2ndsneg(s(mark(N)), cons2(X, mark(Z)))

a__2ndsneg(s(N), cons2(X, cons(Y, Z))) → rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z)))

a__pi(X) → a__2ndspos(mark(X), a__from(0))

a__plus(0, Y) → mark(Y)

a__plus(s(X), Y) → s(a__plus(mark(X), mark(Y)))

a__times(0, Y) → 0

a__times(s(X), Y) → a__plus(mark(Y), a__times(mark(X), mark(Y)))

a__square(X) → a__times(mark(X), mark(X))

mark(from(X)) → a__from(mark(X))

mark(2ndspos(X1, X2)) → a__2ndspos(mark(X1), mark(X2))

mark(2ndsneg(X1, X2)) → a__2ndsneg(mark(X1), mark(X2))

mark(pi(X)) → a__pi(mark(X))

mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))

mark(times(X1, X2)) → a__times(mark(X1), mark(X2))

mark(square(X)) → a__square(mark(X))

mark(0) → 0

mark(s(X)) → s(mark(X))

mark(posrecip(X)) → posrecip(mark(X))

mark(negrecip(X)) → negrecip(mark(X))

mark(nil) → nil

mark(cons(X1, X2)) → cons(mark(X1), X2)

mark(cons2(X1, X2)) → cons2(X1, mark(X2))

mark(rnil) → rnil

mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))

a__from(X) → from(X)

a__2ndspos(X1, X2) → 2ndspos(X1, X2)

a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)

a__pi(X) → pi(X)

a__plus(X1, X2) → plus(X1, X2)

a__times(X1, X2) → times(X1, X2)

a__square(X) → square(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule A__2NDSPOS(s(N), cons(X, Z)) → A__2NDSPOS(s(mark(N)), cons2(X, mark(Z))) at position [1,1] we obtained the following new rules:

A__2NDSPOS(s(y0), cons(y1, nil)) → A__2NDSPOS(s(mark(y0)), cons2(y1, nil))

A__2NDSPOS(s(y0), cons(y1, plus(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__plus(mark(x0), mark(x1))))

A__2NDSPOS(s(y0), cons(y1, posrecip(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, posrecip(mark(x0))))

A__2NDSPOS(s(y0), cons(y1, 2ndspos(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__2ndspos(mark(x0), mark(x1))))

A__2NDSPOS(s(y0), cons(y1, cons(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, cons(mark(x0), x1)))

A__2NDSPOS(s(y0), cons(y1, rcons(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, rcons(mark(x0), mark(x1))))

A__2NDSPOS(s(y0), cons(y1, negrecip(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, negrecip(mark(x0))))

A__2NDSPOS(s(y0), cons(y1, times(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__times(mark(x0), mark(x1))))

A__2NDSPOS(s(y0), cons(y1, cons2(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, cons2(x0, mark(x1))))

A__2NDSPOS(s(y0), cons(y1, s(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, s(mark(x0))))

A__2NDSPOS(s(y0), cons(y1, from(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__from(mark(x0))))

A__2NDSPOS(s(y0), cons(y1, 0)) → A__2NDSPOS(s(mark(y0)), cons2(y1, 0))

A__2NDSPOS(s(y0), cons(y1, 2ndsneg(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__2ndsneg(mark(x0), mark(x1))))

A__2NDSPOS(s(y0), cons(y1, rnil)) → A__2NDSPOS(s(mark(y0)), cons2(y1, rnil))

A__2NDSPOS(s(y0), cons(y1, pi(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__pi(mark(x0))))

A__2NDSPOS(s(y0), cons(y1, square(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__square(mark(x0))))



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ Narrowing
        ↳ QDP
          ↳ DependencyGraphProof

Q DP problem:
The TRS P consists of the following rules:

MARK(times(X1, X2)) → A__TIMES(mark(X1), mark(X2))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(N)

A__2NDSPOS(s(y0), cons(y1, posrecip(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, posrecip(mark(x0))))

A__2NDSPOS(s(y0), cons(y1, 2ndspos(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__2ndspos(mark(x0), mark(x1))))

A__2NDSNEG(s(N), cons(X, Z)) → A__2NDSNEG(s(mark(N)), cons2(X, mark(Z)))

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(N)

A__2NDSPOS(s(y0), cons(y1, rcons(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, rcons(mark(x0), mark(x1))))

MARK(cons(X1, X2)) → MARK(X1)

MARK(2ndspos(X1, X2)) → MARK(X2)

MARK(times(X1, X2)) → MARK(X2)

MARK(2ndsneg(X1, X2)) → MARK(X1)

A__2NDSNEG(s(N), cons(X, Z)) → MARK(Z)

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(Y)

MARK(rcons(X1, X2)) → MARK(X1)

A__TIMES(s(X), Y) → MARK(Y)

A__2NDSPOS(s(y0), cons(y1, plus(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__plus(mark(x0), mark(x1))))

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(Z)

MARK(square(X)) → MARK(X)

A__2NDSNEG(s(N), cons(X, Z)) → MARK(N)

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → A__2NDSNEG(mark(N), mark(Z))

A__2NDSPOS(s(y0), cons(y1, negrecip(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, negrecip(mark(x0))))

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(Y)

A__FROM(X) → MARK(X)

A__2NDSPOS(s(y0), cons(y1, from(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__from(mark(x0))))

A__TIMES(s(X), Y) → A__PLUS(mark(Y), a__times(mark(X), mark(Y)))

MARK(plus(X1, X2)) → A__PLUS(mark(X1), mark(X2))

MARK(rcons(X1, X2)) → MARK(X2)

MARK(2ndsneg(X1, X2)) → A__2NDSNEG(mark(X1), mark(X2))

MARK(cons2(X1, X2)) → MARK(X2)

A__TIMES(s(X), Y) → A__TIMES(mark(X), mark(Y))

MARK(s(X)) → MARK(X)

MARK(2ndspos(X1, X2)) → MARK(X1)

A__2NDSPOS(s(y0), cons(y1, cons(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, cons(mark(x0), x1)))

A__PLUS(s(X), Y) → MARK(X)

MARK(times(X1, X2)) → MARK(X1)

MARK(pi(X)) → MARK(X)

MARK(plus(X1, X2)) → MARK(X2)

A__2NDSPOS(s(y0), cons(y1, s(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, s(mark(x0))))

A__SQUARE(X) → MARK(X)

A__2NDSPOS(s(N), cons(X, Z)) → MARK(Z)

A__2NDSPOS(s(y0), cons(y1, 0)) → A__2NDSPOS(s(mark(y0)), cons2(y1, 0))

MARK(pi(X)) → A__PI(mark(X))

A__2NDSPOS(s(y0), cons(y1, 2ndsneg(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__2ndsneg(mark(x0), mark(x1))))

MARK(negrecip(X)) → MARK(X)

A__2NDSPOS(s(N), cons(X, Z)) → MARK(N)

A__PLUS(0, Y) → MARK(Y)

MARK(posrecip(X)) → MARK(X)

MARK(2ndsneg(X1, X2)) → MARK(X2)

A__2NDSPOS(s(y0), cons(y1, square(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__square(mark(x0))))

A__2NDSPOS(s(y0), cons(y1, nil)) → A__2NDSPOS(s(mark(y0)), cons2(y1, nil))

MARK(2ndspos(X1, X2)) → A__2NDSPOS(mark(X1), mark(X2))

MARK(from(X)) → MARK(X)

A__PI(X) → A__2NDSPOS(mark(X), a__from(0))

MARK(from(X)) → A__FROM(mark(X))

A__PI(X) → A__FROM(0)

A__PLUS(s(X), Y) → A__PLUS(mark(X), mark(Y))

MARK(plus(X1, X2)) → MARK(X1)

A__SQUARE(X) → A__TIMES(mark(X), mark(X))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(Z)

A__2NDSPOS(s(y0), cons(y1, times(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__times(mark(x0), mark(x1))))

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → A__2NDSPOS(mark(N), mark(Z))

A__2NDSPOS(s(y0), cons(y1, cons2(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, cons2(x0, mark(x1))))

MARK(square(X)) → A__SQUARE(mark(X))

A__PI(X) → MARK(X)

A__2NDSPOS(s(y0), cons(y1, rnil)) → A__2NDSPOS(s(mark(y0)), cons2(y1, rnil))

A__PLUS(s(X), Y) → MARK(Y)

A__2NDSPOS(s(y0), cons(y1, pi(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__pi(mark(x0))))

A__TIMES(s(X), Y) → MARK(X)

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))

a__2ndspos(0, Z) → rnil

a__2ndspos(s(N), cons(X, Z)) → a__2ndspos(s(mark(N)), cons2(X, mark(Z)))

a__2ndspos(s(N), cons2(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))

a__2ndsneg(0, Z) → rnil

a__2ndsneg(s(N), cons(X, Z)) → a__2ndsneg(s(mark(N)), cons2(X, mark(Z)))

a__2ndsneg(s(N), cons2(X, cons(Y, Z))) → rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z)))

a__pi(X) → a__2ndspos(mark(X), a__from(0))

a__plus(0, Y) → mark(Y)

a__plus(s(X), Y) → s(a__plus(mark(X), mark(Y)))

a__times(0, Y) → 0

a__times(s(X), Y) → a__plus(mark(Y), a__times(mark(X), mark(Y)))

a__square(X) → a__times(mark(X), mark(X))

mark(from(X)) → a__from(mark(X))

mark(2ndspos(X1, X2)) → a__2ndspos(mark(X1), mark(X2))

mark(2ndsneg(X1, X2)) → a__2ndsneg(mark(X1), mark(X2))

mark(pi(X)) → a__pi(mark(X))

mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))

mark(times(X1, X2)) → a__times(mark(X1), mark(X2))

mark(square(X)) → a__square(mark(X))

mark(0) → 0

mark(s(X)) → s(mark(X))

mark(posrecip(X)) → posrecip(mark(X))

mark(negrecip(X)) → negrecip(mark(X))

mark(nil) → nil

mark(cons(X1, X2)) → cons(mark(X1), X2)

mark(cons2(X1, X2)) → cons2(X1, mark(X2))

mark(rnil) → rnil

mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))

a__from(X) → from(X)

a__2ndspos(X1, X2) → 2ndspos(X1, X2)

a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)

a__pi(X) → pi(X)

a__plus(X1, X2) → plus(X1, X2)

a__times(X1, X2) → times(X1, X2)

a__square(X) → square(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 8 less nodes.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ Narrowing
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ QDP
              ↳ Narrowing

Q DP problem:
The TRS P consists of the following rules:

MARK(times(X1, X2)) → A__TIMES(mark(X1), mark(X2))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(N)

A__2NDSPOS(s(y0), cons(y1, 2ndspos(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__2ndspos(mark(x0), mark(x1))))

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(N)

A__2NDSNEG(s(N), cons(X, Z)) → A__2NDSNEG(s(mark(N)), cons2(X, mark(Z)))

MARK(cons(X1, X2)) → MARK(X1)

MARK(2ndspos(X1, X2)) → MARK(X2)

MARK(times(X1, X2)) → MARK(X2)

MARK(2ndsneg(X1, X2)) → MARK(X1)

A__2NDSNEG(s(N), cons(X, Z)) → MARK(Z)

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(Y)

MARK(rcons(X1, X2)) → MARK(X1)

A__TIMES(s(X), Y) → MARK(Y)

A__2NDSPOS(s(y0), cons(y1, plus(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__plus(mark(x0), mark(x1))))

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(Z)

MARK(square(X)) → MARK(X)

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → A__2NDSNEG(mark(N), mark(Z))

A__2NDSNEG(s(N), cons(X, Z)) → MARK(N)

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(Y)

A__FROM(X) → MARK(X)

A__2NDSPOS(s(y0), cons(y1, from(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__from(mark(x0))))

A__TIMES(s(X), Y) → A__PLUS(mark(Y), a__times(mark(X), mark(Y)))

MARK(plus(X1, X2)) → A__PLUS(mark(X1), mark(X2))

MARK(rcons(X1, X2)) → MARK(X2)

MARK(2ndsneg(X1, X2)) → A__2NDSNEG(mark(X1), mark(X2))

MARK(cons2(X1, X2)) → MARK(X2)

A__TIMES(s(X), Y) → A__TIMES(mark(X), mark(Y))

MARK(2ndspos(X1, X2)) → MARK(X1)

MARK(s(X)) → MARK(X)

A__2NDSPOS(s(y0), cons(y1, cons(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, cons(mark(x0), x1)))

A__PLUS(s(X), Y) → MARK(X)

MARK(times(X1, X2)) → MARK(X1)

MARK(pi(X)) → MARK(X)

MARK(plus(X1, X2)) → MARK(X2)

A__SQUARE(X) → MARK(X)

A__2NDSPOS(s(N), cons(X, Z)) → MARK(Z)

MARK(pi(X)) → A__PI(mark(X))

A__2NDSPOS(s(y0), cons(y1, 2ndsneg(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__2ndsneg(mark(x0), mark(x1))))

A__2NDSPOS(s(N), cons(X, Z)) → MARK(N)

MARK(negrecip(X)) → MARK(X)

A__PLUS(0, Y) → MARK(Y)

MARK(posrecip(X)) → MARK(X)

MARK(2ndsneg(X1, X2)) → MARK(X2)

A__2NDSPOS(s(y0), cons(y1, square(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__square(mark(x0))))

A__PI(X) → A__2NDSPOS(mark(X), a__from(0))

MARK(from(X)) → MARK(X)

MARK(2ndspos(X1, X2)) → A__2NDSPOS(mark(X1), mark(X2))

A__PI(X) → A__FROM(0)

MARK(from(X)) → A__FROM(mark(X))

A__PLUS(s(X), Y) → A__PLUS(mark(X), mark(Y))

MARK(plus(X1, X2)) → MARK(X1)

A__SQUARE(X) → A__TIMES(mark(X), mark(X))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(Z)

A__2NDSPOS(s(y0), cons(y1, times(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__times(mark(x0), mark(x1))))

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → A__2NDSPOS(mark(N), mark(Z))

A__PI(X) → MARK(X)

MARK(square(X)) → A__SQUARE(mark(X))

A__PLUS(s(X), Y) → MARK(Y)

A__2NDSPOS(s(y0), cons(y1, pi(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__pi(mark(x0))))

A__TIMES(s(X), Y) → MARK(X)

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))

a__2ndspos(0, Z) → rnil

a__2ndspos(s(N), cons(X, Z)) → a__2ndspos(s(mark(N)), cons2(X, mark(Z)))

a__2ndspos(s(N), cons2(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))

a__2ndsneg(0, Z) → rnil

a__2ndsneg(s(N), cons(X, Z)) → a__2ndsneg(s(mark(N)), cons2(X, mark(Z)))

a__2ndsneg(s(N), cons2(X, cons(Y, Z))) → rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z)))

a__pi(X) → a__2ndspos(mark(X), a__from(0))

a__plus(0, Y) → mark(Y)

a__plus(s(X), Y) → s(a__plus(mark(X), mark(Y)))

a__times(0, Y) → 0

a__times(s(X), Y) → a__plus(mark(Y), a__times(mark(X), mark(Y)))

a__square(X) → a__times(mark(X), mark(X))

mark(from(X)) → a__from(mark(X))

mark(2ndspos(X1, X2)) → a__2ndspos(mark(X1), mark(X2))

mark(2ndsneg(X1, X2)) → a__2ndsneg(mark(X1), mark(X2))

mark(pi(X)) → a__pi(mark(X))

mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))

mark(times(X1, X2)) → a__times(mark(X1), mark(X2))

mark(square(X)) → a__square(mark(X))

mark(0) → 0

mark(s(X)) → s(mark(X))

mark(posrecip(X)) → posrecip(mark(X))

mark(negrecip(X)) → negrecip(mark(X))

mark(nil) → nil

mark(cons(X1, X2)) → cons(mark(X1), X2)

mark(cons2(X1, X2)) → cons2(X1, mark(X2))

mark(rnil) → rnil

mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))

a__from(X) → from(X)

a__2ndspos(X1, X2) → 2ndspos(X1, X2)

a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)

a__pi(X) → pi(X)

a__plus(X1, X2) → plus(X1, X2)

a__times(X1, X2) → times(X1, X2)

a__square(X) → square(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule MARK(2ndspos(X1, X2)) → A__2NDSPOS(mark(X1), mark(X2)) at position [0] we obtained the following new rules:

MARK(2ndspos(square(x0), y1)) → A__2NDSPOS(a__square(mark(x0)), mark(y1))

MARK(2ndspos(from(x0), y1)) → A__2NDSPOS(a__from(mark(x0)), mark(y1))

MARK(2ndspos(s(x0), y1)) → A__2NDSPOS(s(mark(x0)), mark(y1))

MARK(2ndspos(2ndsneg(x0, x1), y1)) → A__2NDSPOS(a__2ndsneg(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(rcons(x0, x1), y1)) → A__2NDSPOS(rcons(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(rnil, y1)) → A__2NDSPOS(rnil, mark(y1))

MARK(2ndspos(posrecip(x0), y1)) → A__2NDSPOS(posrecip(mark(x0)), mark(y1))

MARK(2ndspos(pi(x0), y1)) → A__2NDSPOS(a__pi(mark(x0)), mark(y1))

MARK(2ndspos(nil, y1)) → A__2NDSPOS(nil, mark(y1))

MARK(2ndspos(cons(x0, x1), y1)) → A__2NDSPOS(cons(mark(x0), x1), mark(y1))

MARK(2ndspos(times(x0, x1), y1)) → A__2NDSPOS(a__times(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(2ndspos(x0, x1), y1)) → A__2NDSPOS(a__2ndspos(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(0, y1)) → A__2NDSPOS(0, mark(y1))

MARK(2ndspos(plus(x0, x1), y1)) → A__2NDSPOS(a__plus(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(cons2(x0, x1), y1)) → A__2NDSPOS(cons2(x0, mark(x1)), mark(y1))

MARK(2ndspos(negrecip(x0), y1)) → A__2NDSPOS(negrecip(mark(x0)), mark(y1))



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ Narrowing
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ QDP
              ↳ Narrowing
                ↳ QDP
                  ↳ DependencyGraphProof

Q DP problem:
The TRS P consists of the following rules:

MARK(times(X1, X2)) → A__TIMES(mark(X1), mark(X2))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(N)

MARK(2ndspos(square(x0), y1)) → A__2NDSPOS(a__square(mark(x0)), mark(y1))

A__2NDSPOS(s(y0), cons(y1, 2ndspos(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__2ndspos(mark(x0), mark(x1))))

A__2NDSNEG(s(N), cons(X, Z)) → A__2NDSNEG(s(mark(N)), cons2(X, mark(Z)))

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(N)

MARK(cons(X1, X2)) → MARK(X1)

MARK(2ndspos(X1, X2)) → MARK(X2)

MARK(times(X1, X2)) → MARK(X2)

MARK(2ndspos(cons(x0, x1), y1)) → A__2NDSPOS(cons(mark(x0), x1), mark(y1))

A__2NDSNEG(s(N), cons(X, Z)) → MARK(Z)

MARK(2ndsneg(X1, X2)) → MARK(X1)

MARK(2ndspos(cons2(x0, x1), y1)) → A__2NDSPOS(cons2(x0, mark(x1)), mark(y1))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(Y)

MARK(rcons(X1, X2)) → MARK(X1)

A__TIMES(s(X), Y) → MARK(Y)

MARK(2ndspos(s(x0), y1)) → A__2NDSPOS(s(mark(x0)), mark(y1))

MARK(2ndspos(from(x0), y1)) → A__2NDSPOS(a__from(mark(x0)), mark(y1))

A__2NDSPOS(s(y0), cons(y1, plus(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__plus(mark(x0), mark(x1))))

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(Z)

A__2NDSNEG(s(N), cons(X, Z)) → MARK(N)

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → A__2NDSNEG(mark(N), mark(Z))

MARK(square(X)) → MARK(X)

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(Y)

A__FROM(X) → MARK(X)

A__2NDSPOS(s(y0), cons(y1, from(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__from(mark(x0))))

MARK(2ndspos(times(x0, x1), y1)) → A__2NDSPOS(a__times(mark(x0), mark(x1)), mark(y1))

A__TIMES(s(X), Y) → A__PLUS(mark(Y), a__times(mark(X), mark(Y)))

MARK(2ndspos(0, y1)) → A__2NDSPOS(0, mark(y1))

MARK(plus(X1, X2)) → A__PLUS(mark(X1), mark(X2))

MARK(2ndspos(plus(x0, x1), y1)) → A__2NDSPOS(a__plus(mark(x0), mark(x1)), mark(y1))

MARK(rcons(X1, X2)) → MARK(X2)

MARK(2ndsneg(X1, X2)) → A__2NDSNEG(mark(X1), mark(X2))

MARK(2ndspos(negrecip(x0), y1)) → A__2NDSPOS(negrecip(mark(x0)), mark(y1))

MARK(cons2(X1, X2)) → MARK(X2)

A__TIMES(s(X), Y) → A__TIMES(mark(X), mark(Y))

MARK(s(X)) → MARK(X)

MARK(2ndspos(X1, X2)) → MARK(X1)

A__PLUS(s(X), Y) → MARK(X)

A__2NDSPOS(s(y0), cons(y1, cons(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, cons(mark(x0), x1)))

MARK(2ndspos(rcons(x0, x1), y1)) → A__2NDSPOS(rcons(mark(x0), mark(x1)), mark(y1))

MARK(pi(X)) → MARK(X)

MARK(times(X1, X2)) → MARK(X1)

MARK(plus(X1, X2)) → MARK(X2)

MARK(2ndspos(rnil, y1)) → A__2NDSPOS(rnil, mark(y1))

A__SQUARE(X) → MARK(X)

A__2NDSPOS(s(N), cons(X, Z)) → MARK(Z)

MARK(pi(X)) → A__PI(mark(X))

A__2NDSPOS(s(y0), cons(y1, 2ndsneg(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__2ndsneg(mark(x0), mark(x1))))

MARK(negrecip(X)) → MARK(X)

A__2NDSPOS(s(N), cons(X, Z)) → MARK(N)

A__PLUS(0, Y) → MARK(Y)

MARK(2ndspos(2ndspos(x0, x1), y1)) → A__2NDSPOS(a__2ndspos(mark(x0), mark(x1)), mark(y1))

MARK(posrecip(X)) → MARK(X)

MARK(2ndsneg(X1, X2)) → MARK(X2)

A__2NDSPOS(s(y0), cons(y1, square(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__square(mark(x0))))

MARK(from(X)) → MARK(X)

A__PI(X) → A__2NDSPOS(mark(X), a__from(0))

MARK(from(X)) → A__FROM(mark(X))

A__PI(X) → A__FROM(0)

MARK(2ndspos(2ndsneg(x0, x1), y1)) → A__2NDSPOS(a__2ndsneg(mark(x0), mark(x1)), mark(y1))

A__PLUS(s(X), Y) → A__PLUS(mark(X), mark(Y))

A__SQUARE(X) → A__TIMES(mark(X), mark(X))

MARK(plus(X1, X2)) → MARK(X1)

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(Z)

A__2NDSPOS(s(y0), cons(y1, times(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__times(mark(x0), mark(x1))))

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → A__2NDSPOS(mark(N), mark(Z))

MARK(2ndspos(posrecip(x0), y1)) → A__2NDSPOS(posrecip(mark(x0)), mark(y1))

MARK(2ndspos(nil, y1)) → A__2NDSPOS(nil, mark(y1))

MARK(2ndspos(pi(x0), y1)) → A__2NDSPOS(a__pi(mark(x0)), mark(y1))

MARK(square(X)) → A__SQUARE(mark(X))

A__PI(X) → MARK(X)

A__PLUS(s(X), Y) → MARK(Y)

A__TIMES(s(X), Y) → MARK(X)

A__2NDSPOS(s(y0), cons(y1, pi(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__pi(mark(x0))))

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))

a__2ndspos(0, Z) → rnil

a__2ndspos(s(N), cons(X, Z)) → a__2ndspos(s(mark(N)), cons2(X, mark(Z)))

a__2ndspos(s(N), cons2(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))

a__2ndsneg(0, Z) → rnil

a__2ndsneg(s(N), cons(X, Z)) → a__2ndsneg(s(mark(N)), cons2(X, mark(Z)))

a__2ndsneg(s(N), cons2(X, cons(Y, Z))) → rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z)))

a__pi(X) → a__2ndspos(mark(X), a__from(0))

a__plus(0, Y) → mark(Y)

a__plus(s(X), Y) → s(a__plus(mark(X), mark(Y)))

a__times(0, Y) → 0

a__times(s(X), Y) → a__plus(mark(Y), a__times(mark(X), mark(Y)))

a__square(X) → a__times(mark(X), mark(X))

mark(from(X)) → a__from(mark(X))

mark(2ndspos(X1, X2)) → a__2ndspos(mark(X1), mark(X2))

mark(2ndsneg(X1, X2)) → a__2ndsneg(mark(X1), mark(X2))

mark(pi(X)) → a__pi(mark(X))

mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))

mark(times(X1, X2)) → a__times(mark(X1), mark(X2))

mark(square(X)) → a__square(mark(X))

mark(0) → 0

mark(s(X)) → s(mark(X))

mark(posrecip(X)) → posrecip(mark(X))

mark(negrecip(X)) → negrecip(mark(X))

mark(nil) → nil

mark(cons(X1, X2)) → cons(mark(X1), X2)

mark(cons2(X1, X2)) → cons2(X1, mark(X2))

mark(rnil) → rnil

mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))

a__from(X) → from(X)

a__2ndspos(X1, X2) → 2ndspos(X1, X2)

a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)

a__pi(X) → pi(X)

a__plus(X1, X2) → plus(X1, X2)

a__times(X1, X2) → times(X1, X2)

a__square(X) → square(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 8 less nodes.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ Narrowing
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ QDP
              ↳ Narrowing
                ↳ QDP
                  ↳ DependencyGraphProof
                    ↳ QDP
                      ↳ Narrowing

Q DP problem:
The TRS P consists of the following rules:

MARK(times(X1, X2)) → A__TIMES(mark(X1), mark(X2))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(N)

MARK(2ndspos(square(x0), y1)) → A__2NDSPOS(a__square(mark(x0)), mark(y1))

A__2NDSPOS(s(y0), cons(y1, 2ndspos(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__2ndspos(mark(x0), mark(x1))))

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(N)

A__2NDSNEG(s(N), cons(X, Z)) → A__2NDSNEG(s(mark(N)), cons2(X, mark(Z)))

MARK(cons(X1, X2)) → MARK(X1)

MARK(2ndspos(X1, X2)) → MARK(X2)

MARK(times(X1, X2)) → MARK(X2)

A__2NDSNEG(s(N), cons(X, Z)) → MARK(Z)

MARK(2ndsneg(X1, X2)) → MARK(X1)

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(Y)

MARK(rcons(X1, X2)) → MARK(X1)

A__TIMES(s(X), Y) → MARK(Y)

MARK(2ndspos(from(x0), y1)) → A__2NDSPOS(a__from(mark(x0)), mark(y1))

MARK(2ndspos(s(x0), y1)) → A__2NDSPOS(s(mark(x0)), mark(y1))

A__2NDSPOS(s(y0), cons(y1, plus(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__plus(mark(x0), mark(x1))))

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(Z)

MARK(square(X)) → MARK(X)

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → A__2NDSNEG(mark(N), mark(Z))

A__2NDSNEG(s(N), cons(X, Z)) → MARK(N)

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(Y)

A__FROM(X) → MARK(X)

A__2NDSPOS(s(y0), cons(y1, from(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__from(mark(x0))))

MARK(2ndspos(times(x0, x1), y1)) → A__2NDSPOS(a__times(mark(x0), mark(x1)), mark(y1))

A__TIMES(s(X), Y) → A__PLUS(mark(Y), a__times(mark(X), mark(Y)))

MARK(plus(X1, X2)) → A__PLUS(mark(X1), mark(X2))

MARK(2ndspos(plus(x0, x1), y1)) → A__2NDSPOS(a__plus(mark(x0), mark(x1)), mark(y1))

MARK(rcons(X1, X2)) → MARK(X2)

MARK(2ndsneg(X1, X2)) → A__2NDSNEG(mark(X1), mark(X2))

MARK(cons2(X1, X2)) → MARK(X2)

A__TIMES(s(X), Y) → A__TIMES(mark(X), mark(Y))

MARK(2ndspos(X1, X2)) → MARK(X1)

MARK(s(X)) → MARK(X)

A__2NDSPOS(s(y0), cons(y1, cons(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, cons(mark(x0), x1)))

A__PLUS(s(X), Y) → MARK(X)

MARK(times(X1, X2)) → MARK(X1)

MARK(pi(X)) → MARK(X)

MARK(plus(X1, X2)) → MARK(X2)

A__SQUARE(X) → MARK(X)

A__2NDSPOS(s(N), cons(X, Z)) → MARK(Z)

MARK(pi(X)) → A__PI(mark(X))

A__2NDSPOS(s(y0), cons(y1, 2ndsneg(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__2ndsneg(mark(x0), mark(x1))))

A__2NDSPOS(s(N), cons(X, Z)) → MARK(N)

MARK(negrecip(X)) → MARK(X)

A__PLUS(0, Y) → MARK(Y)

MARK(2ndspos(2ndspos(x0, x1), y1)) → A__2NDSPOS(a__2ndspos(mark(x0), mark(x1)), mark(y1))

MARK(posrecip(X)) → MARK(X)

MARK(2ndsneg(X1, X2)) → MARK(X2)

A__2NDSPOS(s(y0), cons(y1, square(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__square(mark(x0))))

MARK(from(X)) → MARK(X)

A__PI(X) → A__2NDSPOS(mark(X), a__from(0))

A__PI(X) → A__FROM(0)

MARK(from(X)) → A__FROM(mark(X))

MARK(2ndspos(2ndsneg(x0, x1), y1)) → A__2NDSPOS(a__2ndsneg(mark(x0), mark(x1)), mark(y1))

A__PLUS(s(X), Y) → A__PLUS(mark(X), mark(Y))

MARK(plus(X1, X2)) → MARK(X1)

A__SQUARE(X) → A__TIMES(mark(X), mark(X))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(Z)

A__2NDSPOS(s(y0), cons(y1, times(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__times(mark(x0), mark(x1))))

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → A__2NDSPOS(mark(N), mark(Z))

A__PI(X) → MARK(X)

MARK(square(X)) → A__SQUARE(mark(X))

MARK(2ndspos(pi(x0), y1)) → A__2NDSPOS(a__pi(mark(x0)), mark(y1))

A__PLUS(s(X), Y) → MARK(Y)

A__TIMES(s(X), Y) → MARK(X)

A__2NDSPOS(s(y0), cons(y1, pi(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__pi(mark(x0))))

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))

a__2ndspos(0, Z) → rnil

a__2ndspos(s(N), cons(X, Z)) → a__2ndspos(s(mark(N)), cons2(X, mark(Z)))

a__2ndspos(s(N), cons2(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))

a__2ndsneg(0, Z) → rnil

a__2ndsneg(s(N), cons(X, Z)) → a__2ndsneg(s(mark(N)), cons2(X, mark(Z)))

a__2ndsneg(s(N), cons2(X, cons(Y, Z))) → rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z)))

a__pi(X) → a__2ndspos(mark(X), a__from(0))

a__plus(0, Y) → mark(Y)

a__plus(s(X), Y) → s(a__plus(mark(X), mark(Y)))

a__times(0, Y) → 0

a__times(s(X), Y) → a__plus(mark(Y), a__times(mark(X), mark(Y)))

a__square(X) → a__times(mark(X), mark(X))

mark(from(X)) → a__from(mark(X))

mark(2ndspos(X1, X2)) → a__2ndspos(mark(X1), mark(X2))

mark(2ndsneg(X1, X2)) → a__2ndsneg(mark(X1), mark(X2))

mark(pi(X)) → a__pi(mark(X))

mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))

mark(times(X1, X2)) → a__times(mark(X1), mark(X2))

mark(square(X)) → a__square(mark(X))

mark(0) → 0

mark(s(X)) → s(mark(X))

mark(posrecip(X)) → posrecip(mark(X))

mark(negrecip(X)) → negrecip(mark(X))

mark(nil) → nil

mark(cons(X1, X2)) → cons(mark(X1), X2)

mark(cons2(X1, X2)) → cons2(X1, mark(X2))

mark(rnil) → rnil

mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))

a__from(X) → from(X)

a__2ndspos(X1, X2) → 2ndspos(X1, X2)

a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)

a__pi(X) → pi(X)

a__plus(X1, X2) → plus(X1, X2)

a__times(X1, X2) → times(X1, X2)

a__square(X) → square(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule MARK(2ndsneg(X1, X2)) → A__2NDSNEG(mark(X1), mark(X2)) at position [0] we obtained the following new rules:

MARK(2ndsneg(rcons(x0, x1), y1)) → A__2NDSNEG(rcons(mark(x0), mark(x1)), mark(y1))

MARK(2ndsneg(cons(x0, x1), y1)) → A__2NDSNEG(cons(mark(x0), x1), mark(y1))

MARK(2ndsneg(square(x0), y1)) → A__2NDSNEG(a__square(mark(x0)), mark(y1))

MARK(2ndsneg(posrecip(x0), y1)) → A__2NDSNEG(posrecip(mark(x0)), mark(y1))

MARK(2ndsneg(2ndspos(x0, x1), y1)) → A__2NDSNEG(a__2ndspos(mark(x0), mark(x1)), mark(y1))

MARK(2ndsneg(plus(x0, x1), y1)) → A__2NDSNEG(a__plus(mark(x0), mark(x1)), mark(y1))

MARK(2ndsneg(from(x0), y1)) → A__2NDSNEG(a__from(mark(x0)), mark(y1))

MARK(2ndsneg(0, y1)) → A__2NDSNEG(0, mark(y1))

MARK(2ndsneg(cons2(x0, x1), y1)) → A__2NDSNEG(cons2(x0, mark(x1)), mark(y1))

MARK(2ndsneg(times(x0, x1), y1)) → A__2NDSNEG(a__times(mark(x0), mark(x1)), mark(y1))

MARK(2ndsneg(2ndsneg(x0, x1), y1)) → A__2NDSNEG(a__2ndsneg(mark(x0), mark(x1)), mark(y1))

MARK(2ndsneg(negrecip(x0), y1)) → A__2NDSNEG(negrecip(mark(x0)), mark(y1))

MARK(2ndsneg(s(x0), y1)) → A__2NDSNEG(s(mark(x0)), mark(y1))

MARK(2ndsneg(pi(x0), y1)) → A__2NDSNEG(a__pi(mark(x0)), mark(y1))

MARK(2ndsneg(rnil, y1)) → A__2NDSNEG(rnil, mark(y1))

MARK(2ndsneg(nil, y1)) → A__2NDSNEG(nil, mark(y1))



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ Narrowing
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ QDP
              ↳ Narrowing
                ↳ QDP
                  ↳ DependencyGraphProof
                    ↳ QDP
                      ↳ Narrowing
                        ↳ QDP
                          ↳ DependencyGraphProof

Q DP problem:
The TRS P consists of the following rules:

MARK(2ndsneg(cons(x0, x1), y1)) → A__2NDSNEG(cons(mark(x0), x1), mark(y1))

MARK(times(X1, X2)) → A__TIMES(mark(X1), mark(X2))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(N)

MARK(2ndsneg(square(x0), y1)) → A__2NDSNEG(a__square(mark(x0)), mark(y1))

MARK(2ndspos(square(x0), y1)) → A__2NDSPOS(a__square(mark(x0)), mark(y1))

A__2NDSPOS(s(y0), cons(y1, 2ndspos(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__2ndspos(mark(x0), mark(x1))))

A__2NDSNEG(s(N), cons(X, Z)) → A__2NDSNEG(s(mark(N)), cons2(X, mark(Z)))

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(N)

MARK(cons(X1, X2)) → MARK(X1)

MARK(2ndspos(X1, X2)) → MARK(X2)

MARK(times(X1, X2)) → MARK(X2)

MARK(2ndsneg(2ndsneg(x0, x1), y1)) → A__2NDSNEG(a__2ndsneg(mark(x0), mark(x1)), mark(y1))

MARK(2ndsneg(X1, X2)) → MARK(X1)

A__2NDSNEG(s(N), cons(X, Z)) → MARK(Z)

MARK(2ndsneg(nil, y1)) → A__2NDSNEG(nil, mark(y1))

MARK(2ndsneg(rnil, y1)) → A__2NDSNEG(rnil, mark(y1))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(Y)

MARK(rcons(X1, X2)) → MARK(X1)

A__TIMES(s(X), Y) → MARK(Y)

MARK(2ndsneg(2ndspos(x0, x1), y1)) → A__2NDSNEG(a__2ndspos(mark(x0), mark(x1)), mark(y1))

A__2NDSPOS(s(y0), cons(y1, plus(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__plus(mark(x0), mark(x1))))

MARK(2ndspos(s(x0), y1)) → A__2NDSPOS(s(mark(x0)), mark(y1))

MARK(2ndspos(from(x0), y1)) → A__2NDSPOS(a__from(mark(x0)), mark(y1))

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(Z)

A__2NDSNEG(s(N), cons(X, Z)) → MARK(N)

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → A__2NDSNEG(mark(N), mark(Z))

MARK(square(X)) → MARK(X)

MARK(2ndsneg(plus(x0, x1), y1)) → A__2NDSNEG(a__plus(mark(x0), mark(x1)), mark(y1))

MARK(2ndsneg(from(x0), y1)) → A__2NDSNEG(a__from(mark(x0)), mark(y1))

MARK(2ndsneg(0, y1)) → A__2NDSNEG(0, mark(y1))

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(Y)

A__FROM(X) → MARK(X)

MARK(2ndsneg(cons2(x0, x1), y1)) → A__2NDSNEG(cons2(x0, mark(x1)), mark(y1))

A__2NDSPOS(s(y0), cons(y1, from(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__from(mark(x0))))

MARK(2ndspos(times(x0, x1), y1)) → A__2NDSPOS(a__times(mark(x0), mark(x1)), mark(y1))

A__TIMES(s(X), Y) → A__PLUS(mark(Y), a__times(mark(X), mark(Y)))

MARK(plus(X1, X2)) → A__PLUS(mark(X1), mark(X2))

MARK(rcons(X1, X2)) → MARK(X2)

MARK(2ndspos(plus(x0, x1), y1)) → A__2NDSPOS(a__plus(mark(x0), mark(x1)), mark(y1))

MARK(cons2(X1, X2)) → MARK(X2)

MARK(2ndsneg(rcons(x0, x1), y1)) → A__2NDSNEG(rcons(mark(x0), mark(x1)), mark(y1))

MARK(2ndsneg(posrecip(x0), y1)) → A__2NDSNEG(posrecip(mark(x0)), mark(y1))

A__TIMES(s(X), Y) → A__TIMES(mark(X), mark(Y))

MARK(s(X)) → MARK(X)

MARK(2ndspos(X1, X2)) → MARK(X1)

A__PLUS(s(X), Y) → MARK(X)

A__2NDSPOS(s(y0), cons(y1, cons(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, cons(mark(x0), x1)))

MARK(pi(X)) → MARK(X)

MARK(times(X1, X2)) → MARK(X1)

MARK(plus(X1, X2)) → MARK(X2)

A__SQUARE(X) → MARK(X)

A__2NDSPOS(s(N), cons(X, Z)) → MARK(Z)

MARK(2ndsneg(times(x0, x1), y1)) → A__2NDSNEG(a__times(mark(x0), mark(x1)), mark(y1))

MARK(pi(X)) → A__PI(mark(X))

A__2NDSPOS(s(y0), cons(y1, 2ndsneg(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__2ndsneg(mark(x0), mark(x1))))

MARK(2ndsneg(negrecip(x0), y1)) → A__2NDSNEG(negrecip(mark(x0)), mark(y1))

MARK(negrecip(X)) → MARK(X)

A__2NDSPOS(s(N), cons(X, Z)) → MARK(N)

A__PLUS(0, Y) → MARK(Y)

MARK(2ndspos(2ndspos(x0, x1), y1)) → A__2NDSPOS(a__2ndspos(mark(x0), mark(x1)), mark(y1))

MARK(posrecip(X)) → MARK(X)

MARK(2ndsneg(pi(x0), y1)) → A__2NDSNEG(a__pi(mark(x0)), mark(y1))

MARK(2ndsneg(X1, X2)) → MARK(X2)

A__2NDSPOS(s(y0), cons(y1, square(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__square(mark(x0))))

A__PI(X) → A__2NDSPOS(mark(X), a__from(0))

MARK(from(X)) → MARK(X)

MARK(from(X)) → A__FROM(mark(X))

A__PI(X) → A__FROM(0)

MARK(2ndspos(2ndsneg(x0, x1), y1)) → A__2NDSPOS(a__2ndsneg(mark(x0), mark(x1)), mark(y1))

A__PLUS(s(X), Y) → A__PLUS(mark(X), mark(Y))

A__SQUARE(X) → A__TIMES(mark(X), mark(X))

MARK(plus(X1, X2)) → MARK(X1)

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(Z)

A__2NDSPOS(s(y0), cons(y1, times(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__times(mark(x0), mark(x1))))

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → A__2NDSPOS(mark(N), mark(Z))

MARK(2ndspos(pi(x0), y1)) → A__2NDSPOS(a__pi(mark(x0)), mark(y1))

MARK(square(X)) → A__SQUARE(mark(X))

A__PI(X) → MARK(X)

MARK(2ndsneg(s(x0), y1)) → A__2NDSNEG(s(mark(x0)), mark(y1))

A__PLUS(s(X), Y) → MARK(Y)

A__2NDSPOS(s(y0), cons(y1, pi(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__pi(mark(x0))))

A__TIMES(s(X), Y) → MARK(X)

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))

a__2ndspos(0, Z) → rnil

a__2ndspos(s(N), cons(X, Z)) → a__2ndspos(s(mark(N)), cons2(X, mark(Z)))

a__2ndspos(s(N), cons2(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))

a__2ndsneg(0, Z) → rnil

a__2ndsneg(s(N), cons(X, Z)) → a__2ndsneg(s(mark(N)), cons2(X, mark(Z)))

a__2ndsneg(s(N), cons2(X, cons(Y, Z))) → rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z)))

a__pi(X) → a__2ndspos(mark(X), a__from(0))

a__plus(0, Y) → mark(Y)

a__plus(s(X), Y) → s(a__plus(mark(X), mark(Y)))

a__times(0, Y) → 0

a__times(s(X), Y) → a__plus(mark(Y), a__times(mark(X), mark(Y)))

a__square(X) → a__times(mark(X), mark(X))

mark(from(X)) → a__from(mark(X))

mark(2ndspos(X1, X2)) → a__2ndspos(mark(X1), mark(X2))

mark(2ndsneg(X1, X2)) → a__2ndsneg(mark(X1), mark(X2))

mark(pi(X)) → a__pi(mark(X))

mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))

mark(times(X1, X2)) → a__times(mark(X1), mark(X2))

mark(square(X)) → a__square(mark(X))

mark(0) → 0

mark(s(X)) → s(mark(X))

mark(posrecip(X)) → posrecip(mark(X))

mark(negrecip(X)) → negrecip(mark(X))

mark(nil) → nil

mark(cons(X1, X2)) → cons(mark(X1), X2)

mark(cons2(X1, X2)) → cons2(X1, mark(X2))

mark(rnil) → rnil

mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))

a__from(X) → from(X)

a__2ndspos(X1, X2) → 2ndspos(X1, X2)

a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)

a__pi(X) → pi(X)

a__plus(X1, X2) → plus(X1, X2)

a__times(X1, X2) → times(X1, X2)

a__square(X) → square(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 8 less nodes.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ Narrowing
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ QDP
              ↳ Narrowing
                ↳ QDP
                  ↳ DependencyGraphProof
                    ↳ QDP
                      ↳ Narrowing
                        ↳ QDP
                          ↳ DependencyGraphProof
                            ↳ QDP
                              ↳ Narrowing

Q DP problem:
The TRS P consists of the following rules:

MARK(times(X1, X2)) → A__TIMES(mark(X1), mark(X2))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(N)

MARK(2ndsneg(square(x0), y1)) → A__2NDSNEG(a__square(mark(x0)), mark(y1))

MARK(2ndspos(square(x0), y1)) → A__2NDSPOS(a__square(mark(x0)), mark(y1))

A__2NDSPOS(s(y0), cons(y1, 2ndspos(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__2ndspos(mark(x0), mark(x1))))

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(N)

A__2NDSNEG(s(N), cons(X, Z)) → A__2NDSNEG(s(mark(N)), cons2(X, mark(Z)))

MARK(cons(X1, X2)) → MARK(X1)

MARK(2ndspos(X1, X2)) → MARK(X2)

MARK(times(X1, X2)) → MARK(X2)

MARK(2ndsneg(2ndsneg(x0, x1), y1)) → A__2NDSNEG(a__2ndsneg(mark(x0), mark(x1)), mark(y1))

A__2NDSNEG(s(N), cons(X, Z)) → MARK(Z)

MARK(2ndsneg(X1, X2)) → MARK(X1)

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(Y)

MARK(rcons(X1, X2)) → MARK(X1)

A__TIMES(s(X), Y) → MARK(Y)

MARK(2ndsneg(2ndspos(x0, x1), y1)) → A__2NDSNEG(a__2ndspos(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(from(x0), y1)) → A__2NDSPOS(a__from(mark(x0)), mark(y1))

MARK(2ndspos(s(x0), y1)) → A__2NDSPOS(s(mark(x0)), mark(y1))

A__2NDSPOS(s(y0), cons(y1, plus(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__plus(mark(x0), mark(x1))))

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(Z)

A__2NDSNEG(s(N), cons(X, Z)) → MARK(N)

MARK(square(X)) → MARK(X)

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → A__2NDSNEG(mark(N), mark(Z))

MARK(2ndsneg(plus(x0, x1), y1)) → A__2NDSNEG(a__plus(mark(x0), mark(x1)), mark(y1))

MARK(2ndsneg(from(x0), y1)) → A__2NDSNEG(a__from(mark(x0)), mark(y1))

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(Y)

A__FROM(X) → MARK(X)

A__2NDSPOS(s(y0), cons(y1, from(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__from(mark(x0))))

MARK(2ndspos(times(x0, x1), y1)) → A__2NDSPOS(a__times(mark(x0), mark(x1)), mark(y1))

A__TIMES(s(X), Y) → A__PLUS(mark(Y), a__times(mark(X), mark(Y)))

MARK(plus(X1, X2)) → A__PLUS(mark(X1), mark(X2))

MARK(2ndspos(plus(x0, x1), y1)) → A__2NDSPOS(a__plus(mark(x0), mark(x1)), mark(y1))

MARK(rcons(X1, X2)) → MARK(X2)

MARK(cons2(X1, X2)) → MARK(X2)

A__TIMES(s(X), Y) → A__TIMES(mark(X), mark(Y))

MARK(2ndspos(X1, X2)) → MARK(X1)

MARK(s(X)) → MARK(X)

A__PLUS(s(X), Y) → MARK(X)

A__2NDSPOS(s(y0), cons(y1, cons(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, cons(mark(x0), x1)))

MARK(times(X1, X2)) → MARK(X1)

MARK(pi(X)) → MARK(X)

MARK(plus(X1, X2)) → MARK(X2)

A__SQUARE(X) → MARK(X)

A__2NDSPOS(s(N), cons(X, Z)) → MARK(Z)

MARK(2ndsneg(times(x0, x1), y1)) → A__2NDSNEG(a__times(mark(x0), mark(x1)), mark(y1))

MARK(pi(X)) → A__PI(mark(X))

A__2NDSPOS(s(y0), cons(y1, 2ndsneg(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__2ndsneg(mark(x0), mark(x1))))

A__2NDSPOS(s(N), cons(X, Z)) → MARK(N)

MARK(negrecip(X)) → MARK(X)

A__PLUS(0, Y) → MARK(Y)

MARK(2ndspos(2ndspos(x0, x1), y1)) → A__2NDSPOS(a__2ndspos(mark(x0), mark(x1)), mark(y1))

MARK(posrecip(X)) → MARK(X)

MARK(2ndsneg(pi(x0), y1)) → A__2NDSNEG(a__pi(mark(x0)), mark(y1))

MARK(2ndsneg(X1, X2)) → MARK(X2)

A__2NDSPOS(s(y0), cons(y1, square(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__square(mark(x0))))

MARK(from(X)) → MARK(X)

A__PI(X) → A__2NDSPOS(mark(X), a__from(0))

A__PI(X) → A__FROM(0)

MARK(from(X)) → A__FROM(mark(X))

MARK(2ndspos(2ndsneg(x0, x1), y1)) → A__2NDSPOS(a__2ndsneg(mark(x0), mark(x1)), mark(y1))

A__PLUS(s(X), Y) → A__PLUS(mark(X), mark(Y))

A__SQUARE(X) → A__TIMES(mark(X), mark(X))

MARK(plus(X1, X2)) → MARK(X1)

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(Z)

A__2NDSPOS(s(y0), cons(y1, times(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__times(mark(x0), mark(x1))))

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → A__2NDSPOS(mark(N), mark(Z))

A__PI(X) → MARK(X)

MARK(square(X)) → A__SQUARE(mark(X))

MARK(2ndspos(pi(x0), y1)) → A__2NDSPOS(a__pi(mark(x0)), mark(y1))

MARK(2ndsneg(s(x0), y1)) → A__2NDSNEG(s(mark(x0)), mark(y1))

A__PLUS(s(X), Y) → MARK(Y)

A__TIMES(s(X), Y) → MARK(X)

A__2NDSPOS(s(y0), cons(y1, pi(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__pi(mark(x0))))

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))

a__2ndspos(0, Z) → rnil

a__2ndspos(s(N), cons(X, Z)) → a__2ndspos(s(mark(N)), cons2(X, mark(Z)))

a__2ndspos(s(N), cons2(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))

a__2ndsneg(0, Z) → rnil

a__2ndsneg(s(N), cons(X, Z)) → a__2ndsneg(s(mark(N)), cons2(X, mark(Z)))

a__2ndsneg(s(N), cons2(X, cons(Y, Z))) → rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z)))

a__pi(X) → a__2ndspos(mark(X), a__from(0))

a__plus(0, Y) → mark(Y)

a__plus(s(X), Y) → s(a__plus(mark(X), mark(Y)))

a__times(0, Y) → 0

a__times(s(X), Y) → a__plus(mark(Y), a__times(mark(X), mark(Y)))

a__square(X) → a__times(mark(X), mark(X))

mark(from(X)) → a__from(mark(X))

mark(2ndspos(X1, X2)) → a__2ndspos(mark(X1), mark(X2))

mark(2ndsneg(X1, X2)) → a__2ndsneg(mark(X1), mark(X2))

mark(pi(X)) → a__pi(mark(X))

mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))

mark(times(X1, X2)) → a__times(mark(X1), mark(X2))

mark(square(X)) → a__square(mark(X))

mark(0) → 0

mark(s(X)) → s(mark(X))

mark(posrecip(X)) → posrecip(mark(X))

mark(negrecip(X)) → negrecip(mark(X))

mark(nil) → nil

mark(cons(X1, X2)) → cons(mark(X1), X2)

mark(cons2(X1, X2)) → cons2(X1, mark(X2))

mark(rnil) → rnil

mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))

a__from(X) → from(X)

a__2ndspos(X1, X2) → 2ndspos(X1, X2)

a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)

a__pi(X) → pi(X)

a__plus(X1, X2) → plus(X1, X2)

a__times(X1, X2) → times(X1, X2)

a__square(X) → square(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule A__PI(X) → A__2NDSPOS(mark(X), a__from(0)) at position [0] we obtained the following new rules:

A__PI(rnil) → A__2NDSPOS(rnil, a__from(0))

A__PI(posrecip(x0)) → A__2NDSPOS(posrecip(mark(x0)), a__from(0))

A__PI(times(x0, x1)) → A__2NDSPOS(a__times(mark(x0), mark(x1)), a__from(0))

A__PI(cons2(x0, x1)) → A__2NDSPOS(cons2(x0, mark(x1)), a__from(0))

A__PI(pi(x0)) → A__2NDSPOS(a__pi(mark(x0)), a__from(0))

A__PI(from(x0)) → A__2NDSPOS(a__from(mark(x0)), a__from(0))

A__PI(s(x0)) → A__2NDSPOS(s(mark(x0)), a__from(0))

A__PI(cons(x0, x1)) → A__2NDSPOS(cons(mark(x0), x1), a__from(0))

A__PI(negrecip(x0)) → A__2NDSPOS(negrecip(mark(x0)), a__from(0))

A__PI(square(x0)) → A__2NDSPOS(a__square(mark(x0)), a__from(0))

A__PI(nil) → A__2NDSPOS(nil, a__from(0))

A__PI(rcons(x0, x1)) → A__2NDSPOS(rcons(mark(x0), mark(x1)), a__from(0))

A__PI(2ndspos(x0, x1)) → A__2NDSPOS(a__2ndspos(mark(x0), mark(x1)), a__from(0))

A__PI(plus(x0, x1)) → A__2NDSPOS(a__plus(mark(x0), mark(x1)), a__from(0))

A__PI(2ndsneg(x0, x1)) → A__2NDSPOS(a__2ndsneg(mark(x0), mark(x1)), a__from(0))

A__PI(0) → A__2NDSPOS(0, a__from(0))



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ Narrowing
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ QDP
              ↳ Narrowing
                ↳ QDP
                  ↳ DependencyGraphProof
                    ↳ QDP
                      ↳ Narrowing
                        ↳ QDP
                          ↳ DependencyGraphProof
                            ↳ QDP
                              ↳ Narrowing
                                ↳ QDP
                                  ↳ DependencyGraphProof

Q DP problem:
The TRS P consists of the following rules:

MARK(times(X1, X2)) → A__TIMES(mark(X1), mark(X2))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(N)

MARK(2ndspos(square(x0), y1)) → A__2NDSPOS(a__square(mark(x0)), mark(y1))

MARK(2ndsneg(square(x0), y1)) → A__2NDSNEG(a__square(mark(x0)), mark(y1))

A__PI(times(x0, x1)) → A__2NDSPOS(a__times(mark(x0), mark(x1)), a__from(0))

A__2NDSPOS(s(y0), cons(y1, 2ndspos(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__2ndspos(mark(x0), mark(x1))))

A__2NDSNEG(s(N), cons(X, Z)) → A__2NDSNEG(s(mark(N)), cons2(X, mark(Z)))

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(N)

A__PI(from(x0)) → A__2NDSPOS(a__from(mark(x0)), a__from(0))

MARK(cons(X1, X2)) → MARK(X1)

MARK(2ndspos(X1, X2)) → MARK(X2)

MARK(times(X1, X2)) → MARK(X2)

A__PI(square(x0)) → A__2NDSPOS(a__square(mark(x0)), a__from(0))

MARK(2ndsneg(2ndsneg(x0, x1), y1)) → A__2NDSNEG(a__2ndsneg(mark(x0), mark(x1)), mark(y1))

MARK(2ndsneg(X1, X2)) → MARK(X1)

A__2NDSNEG(s(N), cons(X, Z)) → MARK(Z)

A__PI(rcons(x0, x1)) → A__2NDSPOS(rcons(mark(x0), mark(x1)), a__from(0))

A__PI(2ndsneg(x0, x1)) → A__2NDSPOS(a__2ndsneg(mark(x0), mark(x1)), a__from(0))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(Y)

MARK(rcons(X1, X2)) → MARK(X1)

A__TIMES(s(X), Y) → MARK(Y)

MARK(2ndsneg(2ndspos(x0, x1), y1)) → A__2NDSNEG(a__2ndspos(mark(x0), mark(x1)), mark(y1))

A__2NDSPOS(s(y0), cons(y1, plus(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__plus(mark(x0), mark(x1))))

MARK(2ndspos(s(x0), y1)) → A__2NDSPOS(s(mark(x0)), mark(y1))

MARK(2ndspos(from(x0), y1)) → A__2NDSPOS(a__from(mark(x0)), mark(y1))

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(Z)

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → A__2NDSNEG(mark(N), mark(Z))

MARK(square(X)) → MARK(X)

A__2NDSNEG(s(N), cons(X, Z)) → MARK(N)

MARK(2ndsneg(plus(x0, x1), y1)) → A__2NDSNEG(a__plus(mark(x0), mark(x1)), mark(y1))

A__PI(s(x0)) → A__2NDSPOS(s(mark(x0)), a__from(0))

MARK(2ndsneg(from(x0), y1)) → A__2NDSNEG(a__from(mark(x0)), mark(y1))

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(Y)

A__FROM(X) → MARK(X)

A__2NDSPOS(s(y0), cons(y1, from(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__from(mark(x0))))

MARK(2ndspos(times(x0, x1), y1)) → A__2NDSPOS(a__times(mark(x0), mark(x1)), mark(y1))

A__TIMES(s(X), Y) → A__PLUS(mark(Y), a__times(mark(X), mark(Y)))

A__PI(2ndspos(x0, x1)) → A__2NDSPOS(a__2ndspos(mark(x0), mark(x1)), a__from(0))

MARK(plus(X1, X2)) → A__PLUS(mark(X1), mark(X2))

MARK(rcons(X1, X2)) → MARK(X2)

MARK(2ndspos(plus(x0, x1), y1)) → A__2NDSPOS(a__plus(mark(x0), mark(x1)), mark(y1))

A__PI(0) → A__2NDSPOS(0, a__from(0))

A__PI(plus(x0, x1)) → A__2NDSPOS(a__plus(mark(x0), mark(x1)), a__from(0))

MARK(cons2(X1, X2)) → MARK(X2)

A__TIMES(s(X), Y) → A__TIMES(mark(X), mark(Y))

MARK(s(X)) → MARK(X)

MARK(2ndspos(X1, X2)) → MARK(X1)

A__2NDSPOS(s(y0), cons(y1, cons(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, cons(mark(x0), x1)))

A__PLUS(s(X), Y) → MARK(X)

MARK(pi(X)) → MARK(X)

MARK(times(X1, X2)) → MARK(X1)

MARK(plus(X1, X2)) → MARK(X2)

A__PI(negrecip(x0)) → A__2NDSPOS(negrecip(mark(x0)), a__from(0))

A__SQUARE(X) → MARK(X)

A__2NDSPOS(s(N), cons(X, Z)) → MARK(Z)

MARK(pi(X)) → A__PI(mark(X))

MARK(2ndsneg(times(x0, x1), y1)) → A__2NDSNEG(a__times(mark(x0), mark(x1)), mark(y1))

A__2NDSPOS(s(y0), cons(y1, 2ndsneg(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__2ndsneg(mark(x0), mark(x1))))

MARK(negrecip(X)) → MARK(X)

A__2NDSPOS(s(N), cons(X, Z)) → MARK(N)

A__PI(nil) → A__2NDSPOS(nil, a__from(0))

A__PLUS(0, Y) → MARK(Y)

MARK(2ndspos(2ndspos(x0, x1), y1)) → A__2NDSPOS(a__2ndspos(mark(x0), mark(x1)), mark(y1))

MARK(posrecip(X)) → MARK(X)

MARK(2ndsneg(pi(x0), y1)) → A__2NDSNEG(a__pi(mark(x0)), mark(y1))

MARK(2ndsneg(X1, X2)) → MARK(X2)

A__2NDSPOS(s(y0), cons(y1, square(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__square(mark(x0))))

A__PI(rnil) → A__2NDSPOS(rnil, a__from(0))

MARK(from(X)) → MARK(X)

MARK(from(X)) → A__FROM(mark(X))

A__PI(X) → A__FROM(0)

A__PI(posrecip(x0)) → A__2NDSPOS(posrecip(mark(x0)), a__from(0))

MARK(2ndspos(2ndsneg(x0, x1), y1)) → A__2NDSPOS(a__2ndsneg(mark(x0), mark(x1)), mark(y1))

A__PLUS(s(X), Y) → A__PLUS(mark(X), mark(Y))

MARK(plus(X1, X2)) → MARK(X1)

A__SQUARE(X) → A__TIMES(mark(X), mark(X))

A__PI(pi(x0)) → A__2NDSPOS(a__pi(mark(x0)), a__from(0))

A__PI(cons2(x0, x1)) → A__2NDSPOS(cons2(x0, mark(x1)), a__from(0))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(Z)

A__PI(cons(x0, x1)) → A__2NDSPOS(cons(mark(x0), x1), a__from(0))

A__2NDSPOS(s(y0), cons(y1, times(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__times(mark(x0), mark(x1))))

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → A__2NDSPOS(mark(N), mark(Z))

MARK(2ndspos(pi(x0), y1)) → A__2NDSPOS(a__pi(mark(x0)), mark(y1))

MARK(square(X)) → A__SQUARE(mark(X))

A__PI(X) → MARK(X)

MARK(2ndsneg(s(x0), y1)) → A__2NDSNEG(s(mark(x0)), mark(y1))

A__PLUS(s(X), Y) → MARK(Y)

A__2NDSPOS(s(y0), cons(y1, pi(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__pi(mark(x0))))

A__TIMES(s(X), Y) → MARK(X)

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))

a__2ndspos(0, Z) → rnil

a__2ndspos(s(N), cons(X, Z)) → a__2ndspos(s(mark(N)), cons2(X, mark(Z)))

a__2ndspos(s(N), cons2(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))

a__2ndsneg(0, Z) → rnil

a__2ndsneg(s(N), cons(X, Z)) → a__2ndsneg(s(mark(N)), cons2(X, mark(Z)))

a__2ndsneg(s(N), cons2(X, cons(Y, Z))) → rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z)))

a__pi(X) → a__2ndspos(mark(X), a__from(0))

a__plus(0, Y) → mark(Y)

a__plus(s(X), Y) → s(a__plus(mark(X), mark(Y)))

a__times(0, Y) → 0

a__times(s(X), Y) → a__plus(mark(Y), a__times(mark(X), mark(Y)))

a__square(X) → a__times(mark(X), mark(X))

mark(from(X)) → a__from(mark(X))

mark(2ndspos(X1, X2)) → a__2ndspos(mark(X1), mark(X2))

mark(2ndsneg(X1, X2)) → a__2ndsneg(mark(X1), mark(X2))

mark(pi(X)) → a__pi(mark(X))

mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))

mark(times(X1, X2)) → a__times(mark(X1), mark(X2))

mark(square(X)) → a__square(mark(X))

mark(0) → 0

mark(s(X)) → s(mark(X))

mark(posrecip(X)) → posrecip(mark(X))

mark(negrecip(X)) → negrecip(mark(X))

mark(nil) → nil

mark(cons(X1, X2)) → cons(mark(X1), X2)

mark(cons2(X1, X2)) → cons2(X1, mark(X2))

mark(rnil) → rnil

mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))

a__from(X) → from(X)

a__2ndspos(X1, X2) → 2ndspos(X1, X2)

a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)

a__pi(X) → pi(X)

a__plus(X1, X2) → plus(X1, X2)

a__times(X1, X2) → times(X1, X2)

a__square(X) → square(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 8 less nodes.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ Narrowing
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ QDP
              ↳ Narrowing
                ↳ QDP
                  ↳ DependencyGraphProof
                    ↳ QDP
                      ↳ Narrowing
                        ↳ QDP
                          ↳ DependencyGraphProof
                            ↳ QDP
                              ↳ Narrowing
                                ↳ QDP
                                  ↳ DependencyGraphProof
                                    ↳ QDP
                                      ↳ Narrowing

Q DP problem:
The TRS P consists of the following rules:

MARK(times(X1, X2)) → A__TIMES(mark(X1), mark(X2))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(N)

MARK(2ndsneg(square(x0), y1)) → A__2NDSNEG(a__square(mark(x0)), mark(y1))

MARK(2ndspos(square(x0), y1)) → A__2NDSPOS(a__square(mark(x0)), mark(y1))

A__PI(times(x0, x1)) → A__2NDSPOS(a__times(mark(x0), mark(x1)), a__from(0))

A__2NDSPOS(s(y0), cons(y1, 2ndspos(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__2ndspos(mark(x0), mark(x1))))

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(N)

A__2NDSNEG(s(N), cons(X, Z)) → A__2NDSNEG(s(mark(N)), cons2(X, mark(Z)))

A__PI(from(x0)) → A__2NDSPOS(a__from(mark(x0)), a__from(0))

MARK(cons(X1, X2)) → MARK(X1)

MARK(2ndspos(X1, X2)) → MARK(X2)

MARK(times(X1, X2)) → MARK(X2)

A__PI(square(x0)) → A__2NDSPOS(a__square(mark(x0)), a__from(0))

MARK(2ndsneg(2ndsneg(x0, x1), y1)) → A__2NDSNEG(a__2ndsneg(mark(x0), mark(x1)), mark(y1))

A__2NDSNEG(s(N), cons(X, Z)) → MARK(Z)

MARK(2ndsneg(X1, X2)) → MARK(X1)

A__PI(2ndsneg(x0, x1)) → A__2NDSPOS(a__2ndsneg(mark(x0), mark(x1)), a__from(0))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(Y)

MARK(rcons(X1, X2)) → MARK(X1)

A__TIMES(s(X), Y) → MARK(Y)

MARK(2ndsneg(2ndspos(x0, x1), y1)) → A__2NDSNEG(a__2ndspos(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(from(x0), y1)) → A__2NDSPOS(a__from(mark(x0)), mark(y1))

MARK(2ndspos(s(x0), y1)) → A__2NDSPOS(s(mark(x0)), mark(y1))

A__2NDSPOS(s(y0), cons(y1, plus(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__plus(mark(x0), mark(x1))))

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(Z)

A__2NDSNEG(s(N), cons(X, Z)) → MARK(N)

MARK(square(X)) → MARK(X)

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → A__2NDSNEG(mark(N), mark(Z))

MARK(2ndsneg(plus(x0, x1), y1)) → A__2NDSNEG(a__plus(mark(x0), mark(x1)), mark(y1))

A__PI(s(x0)) → A__2NDSPOS(s(mark(x0)), a__from(0))

MARK(2ndsneg(from(x0), y1)) → A__2NDSNEG(a__from(mark(x0)), mark(y1))

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(Y)

A__FROM(X) → MARK(X)

A__2NDSPOS(s(y0), cons(y1, from(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__from(mark(x0))))

MARK(2ndspos(times(x0, x1), y1)) → A__2NDSPOS(a__times(mark(x0), mark(x1)), mark(y1))

A__TIMES(s(X), Y) → A__PLUS(mark(Y), a__times(mark(X), mark(Y)))

A__PI(2ndspos(x0, x1)) → A__2NDSPOS(a__2ndspos(mark(x0), mark(x1)), a__from(0))

MARK(plus(X1, X2)) → A__PLUS(mark(X1), mark(X2))

MARK(2ndspos(plus(x0, x1), y1)) → A__2NDSPOS(a__plus(mark(x0), mark(x1)), mark(y1))

MARK(rcons(X1, X2)) → MARK(X2)

A__PI(plus(x0, x1)) → A__2NDSPOS(a__plus(mark(x0), mark(x1)), a__from(0))

MARK(cons2(X1, X2)) → MARK(X2)

A__TIMES(s(X), Y) → A__TIMES(mark(X), mark(Y))

MARK(s(X)) → MARK(X)

MARK(2ndspos(X1, X2)) → MARK(X1)

A__PLUS(s(X), Y) → MARK(X)

A__2NDSPOS(s(y0), cons(y1, cons(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, cons(mark(x0), x1)))

MARK(times(X1, X2)) → MARK(X1)

MARK(pi(X)) → MARK(X)

MARK(plus(X1, X2)) → MARK(X2)

A__SQUARE(X) → MARK(X)

A__2NDSPOS(s(N), cons(X, Z)) → MARK(Z)

MARK(2ndsneg(times(x0, x1), y1)) → A__2NDSNEG(a__times(mark(x0), mark(x1)), mark(y1))

MARK(pi(X)) → A__PI(mark(X))

A__2NDSPOS(s(y0), cons(y1, 2ndsneg(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__2ndsneg(mark(x0), mark(x1))))

MARK(negrecip(X)) → MARK(X)

A__2NDSPOS(s(N), cons(X, Z)) → MARK(N)

A__PLUS(0, Y) → MARK(Y)

MARK(2ndspos(2ndspos(x0, x1), y1)) → A__2NDSPOS(a__2ndspos(mark(x0), mark(x1)), mark(y1))

MARK(posrecip(X)) → MARK(X)

MARK(2ndsneg(pi(x0), y1)) → A__2NDSNEG(a__pi(mark(x0)), mark(y1))

MARK(2ndsneg(X1, X2)) → MARK(X2)

A__2NDSPOS(s(y0), cons(y1, square(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__square(mark(x0))))

MARK(from(X)) → MARK(X)

A__PI(X) → A__FROM(0)

MARK(from(X)) → A__FROM(mark(X))

MARK(2ndspos(2ndsneg(x0, x1), y1)) → A__2NDSPOS(a__2ndsneg(mark(x0), mark(x1)), mark(y1))

A__PLUS(s(X), Y) → A__PLUS(mark(X), mark(Y))

A__SQUARE(X) → A__TIMES(mark(X), mark(X))

MARK(plus(X1, X2)) → MARK(X1)

A__PI(pi(x0)) → A__2NDSPOS(a__pi(mark(x0)), a__from(0))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(Z)

A__2NDSPOS(s(y0), cons(y1, times(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__times(mark(x0), mark(x1))))

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → A__2NDSPOS(mark(N), mark(Z))

MARK(square(X)) → A__SQUARE(mark(X))

A__PI(X) → MARK(X)

MARK(2ndspos(pi(x0), y1)) → A__2NDSPOS(a__pi(mark(x0)), mark(y1))

MARK(2ndsneg(s(x0), y1)) → A__2NDSNEG(s(mark(x0)), mark(y1))

A__PLUS(s(X), Y) → MARK(Y)

A__TIMES(s(X), Y) → MARK(X)

A__2NDSPOS(s(y0), cons(y1, pi(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__pi(mark(x0))))

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))

a__2ndspos(0, Z) → rnil

a__2ndspos(s(N), cons(X, Z)) → a__2ndspos(s(mark(N)), cons2(X, mark(Z)))

a__2ndspos(s(N), cons2(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))

a__2ndsneg(0, Z) → rnil

a__2ndsneg(s(N), cons(X, Z)) → a__2ndsneg(s(mark(N)), cons2(X, mark(Z)))

a__2ndsneg(s(N), cons2(X, cons(Y, Z))) → rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z)))

a__pi(X) → a__2ndspos(mark(X), a__from(0))

a__plus(0, Y) → mark(Y)

a__plus(s(X), Y) → s(a__plus(mark(X), mark(Y)))

a__times(0, Y) → 0

a__times(s(X), Y) → a__plus(mark(Y), a__times(mark(X), mark(Y)))

a__square(X) → a__times(mark(X), mark(X))

mark(from(X)) → a__from(mark(X))

mark(2ndspos(X1, X2)) → a__2ndspos(mark(X1), mark(X2))

mark(2ndsneg(X1, X2)) → a__2ndsneg(mark(X1), mark(X2))

mark(pi(X)) → a__pi(mark(X))

mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))

mark(times(X1, X2)) → a__times(mark(X1), mark(X2))

mark(square(X)) → a__square(mark(X))

mark(0) → 0

mark(s(X)) → s(mark(X))

mark(posrecip(X)) → posrecip(mark(X))

mark(negrecip(X)) → negrecip(mark(X))

mark(nil) → nil

mark(cons(X1, X2)) → cons(mark(X1), X2)

mark(cons2(X1, X2)) → cons2(X1, mark(X2))

mark(rnil) → rnil

mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))

a__from(X) → from(X)

a__2ndspos(X1, X2) → 2ndspos(X1, X2)

a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)

a__pi(X) → pi(X)

a__plus(X1, X2) → plus(X1, X2)

a__times(X1, X2) → times(X1, X2)

a__square(X) → square(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule MARK(plus(X1, X2)) → A__PLUS(mark(X1), mark(X2)) at position [0] we obtained the following new rules:

MARK(plus(0, y1)) → A__PLUS(0, mark(y1))

MARK(plus(2ndspos(x0, x1), y1)) → A__PLUS(a__2ndspos(mark(x0), mark(x1)), mark(y1))

MARK(plus(nil, y1)) → A__PLUS(nil, mark(y1))

MARK(plus(times(x0, x1), y1)) → A__PLUS(a__times(mark(x0), mark(x1)), mark(y1))

MARK(plus(from(x0), y1)) → A__PLUS(a__from(mark(x0)), mark(y1))

MARK(plus(plus(x0, x1), y1)) → A__PLUS(a__plus(mark(x0), mark(x1)), mark(y1))

MARK(plus(negrecip(x0), y1)) → A__PLUS(negrecip(mark(x0)), mark(y1))

MARK(plus(cons2(x0, x1), y1)) → A__PLUS(cons2(x0, mark(x1)), mark(y1))

MARK(plus(cons(x0, x1), y1)) → A__PLUS(cons(mark(x0), x1), mark(y1))

MARK(plus(2ndsneg(x0, x1), y1)) → A__PLUS(a__2ndsneg(mark(x0), mark(x1)), mark(y1))

MARK(plus(rnil, y1)) → A__PLUS(rnil, mark(y1))

MARK(plus(square(x0), y1)) → A__PLUS(a__square(mark(x0)), mark(y1))

MARK(plus(rcons(x0, x1), y1)) → A__PLUS(rcons(mark(x0), mark(x1)), mark(y1))

MARK(plus(s(x0), y1)) → A__PLUS(s(mark(x0)), mark(y1))

MARK(plus(pi(x0), y1)) → A__PLUS(a__pi(mark(x0)), mark(y1))

MARK(plus(posrecip(x0), y1)) → A__PLUS(posrecip(mark(x0)), mark(y1))



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ Narrowing
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ QDP
              ↳ Narrowing
                ↳ QDP
                  ↳ DependencyGraphProof
                    ↳ QDP
                      ↳ Narrowing
                        ↳ QDP
                          ↳ DependencyGraphProof
                            ↳ QDP
                              ↳ Narrowing
                                ↳ QDP
                                  ↳ DependencyGraphProof
                                    ↳ QDP
                                      ↳ Narrowing
                                        ↳ QDP
                                          ↳ DependencyGraphProof

Q DP problem:
The TRS P consists of the following rules:

MARK(2ndsneg(square(x0), y1)) → A__2NDSNEG(a__square(mark(x0)), mark(y1))

MARK(2ndspos(square(x0), y1)) → A__2NDSPOS(a__square(mark(x0)), mark(y1))

A__PI(times(x0, x1)) → A__2NDSPOS(a__times(mark(x0), mark(x1)), a__from(0))

A__2NDSPOS(s(y0), cons(y1, 2ndspos(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__2ndspos(mark(x0), mark(x1))))

A__2NDSNEG(s(N), cons(X, Z)) → A__2NDSNEG(s(mark(N)), cons2(X, mark(Z)))

MARK(2ndspos(X1, X2)) → MARK(X2)

MARK(plus(plus(x0, x1), y1)) → A__PLUS(a__plus(mark(x0), mark(x1)), mark(y1))

MARK(plus(negrecip(x0), y1)) → A__PLUS(negrecip(mark(x0)), mark(y1))

MARK(2ndsneg(2ndsneg(x0, x1), y1)) → A__2NDSNEG(a__2ndsneg(mark(x0), mark(x1)), mark(y1))

A__2NDSNEG(s(N), cons(X, Z)) → MARK(Z)

A__TIMES(s(X), Y) → MARK(Y)

MARK(2ndsneg(2ndspos(x0, x1), y1)) → A__2NDSNEG(a__2ndspos(mark(x0), mark(x1)), mark(y1))

A__2NDSPOS(s(y0), cons(y1, plus(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__plus(mark(x0), mark(x1))))

A__2NDSNEG(s(N), cons(X, Z)) → MARK(N)

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → A__2NDSNEG(mark(N), mark(Z))

MARK(2ndsneg(plus(x0, x1), y1)) → A__2NDSNEG(a__plus(mark(x0), mark(x1)), mark(y1))

MARK(2ndsneg(from(x0), y1)) → A__2NDSNEG(a__from(mark(x0)), mark(y1))

A__TIMES(s(X), Y) → A__PLUS(mark(Y), a__times(mark(X), mark(Y)))

A__PI(2ndspos(x0, x1)) → A__2NDSPOS(a__2ndspos(mark(x0), mark(x1)), a__from(0))

MARK(plus(square(x0), y1)) → A__PLUS(a__square(mark(x0)), mark(y1))

MARK(2ndspos(plus(x0, x1), y1)) → A__2NDSPOS(a__plus(mark(x0), mark(x1)), mark(y1))

MARK(plus(rcons(x0, x1), y1)) → A__PLUS(rcons(mark(x0), mark(x1)), mark(y1))

MARK(plus(s(x0), y1)) → A__PLUS(s(mark(x0)), mark(y1))

A__PI(plus(x0, x1)) → A__2NDSPOS(a__plus(mark(x0), mark(x1)), a__from(0))

MARK(plus(posrecip(x0), y1)) → A__PLUS(posrecip(mark(x0)), mark(y1))

MARK(cons2(X1, X2)) → MARK(X2)

MARK(s(X)) → MARK(X)

MARK(2ndspos(X1, X2)) → MARK(X1)

MARK(pi(X)) → MARK(X)

MARK(plus(X1, X2)) → MARK(X2)

A__2NDSPOS(s(N), cons(X, Z)) → MARK(Z)

MARK(pi(X)) → A__PI(mark(X))

MARK(plus(2ndsneg(x0, x1), y1)) → A__PLUS(a__2ndsneg(mark(x0), mark(x1)), mark(y1))

MARK(2ndsneg(X1, X2)) → MARK(X2)

A__PI(X) → A__FROM(0)

MARK(from(X)) → A__FROM(mark(X))

MARK(plus(X1, X2)) → MARK(X1)

MARK(plus(nil, y1)) → A__PLUS(nil, mark(y1))

A__2NDSPOS(s(y0), cons(y1, times(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__times(mark(x0), mark(x1))))

A__PI(X) → MARK(X)

MARK(plus(cons(x0, x1), y1)) → A__PLUS(cons(mark(x0), x1), mark(y1))

MARK(2ndsneg(s(x0), y1)) → A__2NDSNEG(s(mark(x0)), mark(y1))

A__PLUS(s(X), Y) → MARK(Y)

A__TIMES(s(X), Y) → MARK(X)

A__2NDSPOS(s(y0), cons(y1, pi(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__pi(mark(x0))))

MARK(times(X1, X2)) → A__TIMES(mark(X1), mark(X2))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(N)

MARK(plus(2ndspos(x0, x1), y1)) → A__PLUS(a__2ndspos(mark(x0), mark(x1)), mark(y1))

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(N)

A__PI(from(x0)) → A__2NDSPOS(a__from(mark(x0)), a__from(0))

MARK(cons(X1, X2)) → MARK(X1)

MARK(times(X1, X2)) → MARK(X2)

A__PI(square(x0)) → A__2NDSPOS(a__square(mark(x0)), a__from(0))

MARK(2ndsneg(X1, X2)) → MARK(X1)

A__PI(2ndsneg(x0, x1)) → A__2NDSPOS(a__2ndsneg(mark(x0), mark(x1)), a__from(0))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(Y)

MARK(plus(pi(x0), y1)) → A__PLUS(a__pi(mark(x0)), mark(y1))

MARK(rcons(X1, X2)) → MARK(X1)

MARK(2ndspos(from(x0), y1)) → A__2NDSPOS(a__from(mark(x0)), mark(y1))

MARK(2ndspos(s(x0), y1)) → A__2NDSPOS(s(mark(x0)), mark(y1))

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(Z)

MARK(square(X)) → MARK(X)

A__PI(s(x0)) → A__2NDSPOS(s(mark(x0)), a__from(0))

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(Y)

A__FROM(X) → MARK(X)

A__2NDSPOS(s(y0), cons(y1, from(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__from(mark(x0))))

MARK(2ndspos(times(x0, x1), y1)) → A__2NDSPOS(a__times(mark(x0), mark(x1)), mark(y1))

MARK(plus(rnil, y1)) → A__PLUS(rnil, mark(y1))

MARK(rcons(X1, X2)) → MARK(X2)

A__TIMES(s(X), Y) → A__TIMES(mark(X), mark(Y))

A__PLUS(s(X), Y) → MARK(X)

A__2NDSPOS(s(y0), cons(y1, cons(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, cons(mark(x0), x1)))

MARK(times(X1, X2)) → MARK(X1)

MARK(plus(times(x0, x1), y1)) → A__PLUS(a__times(mark(x0), mark(x1)), mark(y1))

A__SQUARE(X) → MARK(X)

MARK(2ndsneg(times(x0, x1), y1)) → A__2NDSNEG(a__times(mark(x0), mark(x1)), mark(y1))

A__2NDSPOS(s(y0), cons(y1, 2ndsneg(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__2ndsneg(mark(x0), mark(x1))))

MARK(negrecip(X)) → MARK(X)

A__2NDSPOS(s(N), cons(X, Z)) → MARK(N)

A__PLUS(0, Y) → MARK(Y)

MARK(2ndspos(2ndspos(x0, x1), y1)) → A__2NDSPOS(a__2ndspos(mark(x0), mark(x1)), mark(y1))

MARK(posrecip(X)) → MARK(X)

MARK(2ndsneg(pi(x0), y1)) → A__2NDSNEG(a__pi(mark(x0)), mark(y1))

A__2NDSPOS(s(y0), cons(y1, square(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__square(mark(x0))))

MARK(plus(0, y1)) → A__PLUS(0, mark(y1))

MARK(from(X)) → MARK(X)

MARK(2ndspos(2ndsneg(x0, x1), y1)) → A__2NDSPOS(a__2ndsneg(mark(x0), mark(x1)), mark(y1))

A__PLUS(s(X), Y) → A__PLUS(mark(X), mark(Y))

A__SQUARE(X) → A__TIMES(mark(X), mark(X))

A__PI(pi(x0)) → A__2NDSPOS(a__pi(mark(x0)), a__from(0))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(Z)

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → A__2NDSPOS(mark(N), mark(Z))

MARK(plus(from(x0), y1)) → A__PLUS(a__from(mark(x0)), mark(y1))

MARK(square(X)) → A__SQUARE(mark(X))

MARK(2ndspos(pi(x0), y1)) → A__2NDSPOS(a__pi(mark(x0)), mark(y1))

MARK(plus(cons2(x0, x1), y1)) → A__PLUS(cons2(x0, mark(x1)), mark(y1))

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))

a__2ndspos(0, Z) → rnil

a__2ndspos(s(N), cons(X, Z)) → a__2ndspos(s(mark(N)), cons2(X, mark(Z)))

a__2ndspos(s(N), cons2(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))

a__2ndsneg(0, Z) → rnil

a__2ndsneg(s(N), cons(X, Z)) → a__2ndsneg(s(mark(N)), cons2(X, mark(Z)))

a__2ndsneg(s(N), cons2(X, cons(Y, Z))) → rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z)))

a__pi(X) → a__2ndspos(mark(X), a__from(0))

a__plus(0, Y) → mark(Y)

a__plus(s(X), Y) → s(a__plus(mark(X), mark(Y)))

a__times(0, Y) → 0

a__times(s(X), Y) → a__plus(mark(Y), a__times(mark(X), mark(Y)))

a__square(X) → a__times(mark(X), mark(X))

mark(from(X)) → a__from(mark(X))

mark(2ndspos(X1, X2)) → a__2ndspos(mark(X1), mark(X2))

mark(2ndsneg(X1, X2)) → a__2ndsneg(mark(X1), mark(X2))

mark(pi(X)) → a__pi(mark(X))

mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))

mark(times(X1, X2)) → a__times(mark(X1), mark(X2))

mark(square(X)) → a__square(mark(X))

mark(0) → 0

mark(s(X)) → s(mark(X))

mark(posrecip(X)) → posrecip(mark(X))

mark(negrecip(X)) → negrecip(mark(X))

mark(nil) → nil

mark(cons(X1, X2)) → cons(mark(X1), X2)

mark(cons2(X1, X2)) → cons2(X1, mark(X2))

mark(rnil) → rnil

mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))

a__from(X) → from(X)

a__2ndspos(X1, X2) → 2ndspos(X1, X2)

a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)

a__pi(X) → pi(X)

a__plus(X1, X2) → plus(X1, X2)

a__times(X1, X2) → times(X1, X2)

a__square(X) → square(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 7 less nodes.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ Narrowing
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ QDP
              ↳ Narrowing
                ↳ QDP
                  ↳ DependencyGraphProof
                    ↳ QDP
                      ↳ Narrowing
                        ↳ QDP
                          ↳ DependencyGraphProof
                            ↳ QDP
                              ↳ Narrowing
                                ↳ QDP
                                  ↳ DependencyGraphProof
                                    ↳ QDP
                                      ↳ Narrowing
                                        ↳ QDP
                                          ↳ DependencyGraphProof
                                            ↳ QDP
                                              ↳ Narrowing

Q DP problem:
The TRS P consists of the following rules:

MARK(times(X1, X2)) → A__TIMES(mark(X1), mark(X2))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(N)

MARK(2ndsneg(square(x0), y1)) → A__2NDSNEG(a__square(mark(x0)), mark(y1))

MARK(2ndspos(square(x0), y1)) → A__2NDSPOS(a__square(mark(x0)), mark(y1))

A__PI(times(x0, x1)) → A__2NDSPOS(a__times(mark(x0), mark(x1)), a__from(0))

MARK(plus(2ndspos(x0, x1), y1)) → A__PLUS(a__2ndspos(mark(x0), mark(x1)), mark(y1))

A__2NDSPOS(s(y0), cons(y1, 2ndspos(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__2ndspos(mark(x0), mark(x1))))

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(N)

A__2NDSNEG(s(N), cons(X, Z)) → A__2NDSNEG(s(mark(N)), cons2(X, mark(Z)))

A__PI(from(x0)) → A__2NDSPOS(a__from(mark(x0)), a__from(0))

MARK(cons(X1, X2)) → MARK(X1)

MARK(2ndspos(X1, X2)) → MARK(X2)

MARK(times(X1, X2)) → MARK(X2)

A__PI(square(x0)) → A__2NDSPOS(a__square(mark(x0)), a__from(0))

MARK(plus(plus(x0, x1), y1)) → A__PLUS(a__plus(mark(x0), mark(x1)), mark(y1))

MARK(2ndsneg(2ndsneg(x0, x1), y1)) → A__2NDSNEG(a__2ndsneg(mark(x0), mark(x1)), mark(y1))

A__2NDSNEG(s(N), cons(X, Z)) → MARK(Z)

MARK(2ndsneg(X1, X2)) → MARK(X1)

A__PI(2ndsneg(x0, x1)) → A__2NDSPOS(a__2ndsneg(mark(x0), mark(x1)), a__from(0))

MARK(plus(pi(x0), y1)) → A__PLUS(a__pi(mark(x0)), mark(y1))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(Y)

MARK(rcons(X1, X2)) → MARK(X1)

A__TIMES(s(X), Y) → MARK(Y)

MARK(2ndsneg(2ndspos(x0, x1), y1)) → A__2NDSNEG(a__2ndspos(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(from(x0), y1)) → A__2NDSPOS(a__from(mark(x0)), mark(y1))

MARK(2ndspos(s(x0), y1)) → A__2NDSPOS(s(mark(x0)), mark(y1))

A__2NDSPOS(s(y0), cons(y1, plus(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__plus(mark(x0), mark(x1))))

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(Z)

A__2NDSNEG(s(N), cons(X, Z)) → MARK(N)

MARK(square(X)) → MARK(X)

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → A__2NDSNEG(mark(N), mark(Z))

MARK(2ndsneg(plus(x0, x1), y1)) → A__2NDSNEG(a__plus(mark(x0), mark(x1)), mark(y1))

A__PI(s(x0)) → A__2NDSPOS(s(mark(x0)), a__from(0))

MARK(2ndsneg(from(x0), y1)) → A__2NDSNEG(a__from(mark(x0)), mark(y1))

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(Y)

A__FROM(X) → MARK(X)

A__2NDSPOS(s(y0), cons(y1, from(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__from(mark(x0))))

MARK(2ndspos(times(x0, x1), y1)) → A__2NDSPOS(a__times(mark(x0), mark(x1)), mark(y1))

A__TIMES(s(X), Y) → A__PLUS(mark(Y), a__times(mark(X), mark(Y)))

A__PI(2ndspos(x0, x1)) → A__2NDSPOS(a__2ndspos(mark(x0), mark(x1)), a__from(0))

MARK(plus(square(x0), y1)) → A__PLUS(a__square(mark(x0)), mark(y1))

MARK(plus(s(x0), y1)) → A__PLUS(s(mark(x0)), mark(y1))

MARK(2ndspos(plus(x0, x1), y1)) → A__2NDSPOS(a__plus(mark(x0), mark(x1)), mark(y1))

MARK(rcons(X1, X2)) → MARK(X2)

A__PI(plus(x0, x1)) → A__2NDSPOS(a__plus(mark(x0), mark(x1)), a__from(0))

MARK(cons2(X1, X2)) → MARK(X2)

A__TIMES(s(X), Y) → A__TIMES(mark(X), mark(Y))

MARK(s(X)) → MARK(X)

MARK(2ndspos(X1, X2)) → MARK(X1)

A__PLUS(s(X), Y) → MARK(X)

A__2NDSPOS(s(y0), cons(y1, cons(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, cons(mark(x0), x1)))

MARK(times(X1, X2)) → MARK(X1)

MARK(pi(X)) → MARK(X)

MARK(plus(X1, X2)) → MARK(X2)

MARK(plus(times(x0, x1), y1)) → A__PLUS(a__times(mark(x0), mark(x1)), mark(y1))

A__SQUARE(X) → MARK(X)

A__2NDSPOS(s(N), cons(X, Z)) → MARK(Z)

MARK(2ndsneg(times(x0, x1), y1)) → A__2NDSNEG(a__times(mark(x0), mark(x1)), mark(y1))

MARK(pi(X)) → A__PI(mark(X))

A__2NDSPOS(s(y0), cons(y1, 2ndsneg(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__2ndsneg(mark(x0), mark(x1))))

MARK(negrecip(X)) → MARK(X)

A__2NDSPOS(s(N), cons(X, Z)) → MARK(N)

MARK(plus(2ndsneg(x0, x1), y1)) → A__PLUS(a__2ndsneg(mark(x0), mark(x1)), mark(y1))

A__PLUS(0, Y) → MARK(Y)

MARK(2ndspos(2ndspos(x0, x1), y1)) → A__2NDSPOS(a__2ndspos(mark(x0), mark(x1)), mark(y1))

MARK(posrecip(X)) → MARK(X)

MARK(2ndsneg(pi(x0), y1)) → A__2NDSNEG(a__pi(mark(x0)), mark(y1))

MARK(2ndsneg(X1, X2)) → MARK(X2)

A__2NDSPOS(s(y0), cons(y1, square(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__square(mark(x0))))

MARK(plus(0, y1)) → A__PLUS(0, mark(y1))

MARK(from(X)) → MARK(X)

A__PI(X) → A__FROM(0)

MARK(from(X)) → A__FROM(mark(X))

MARK(2ndspos(2ndsneg(x0, x1), y1)) → A__2NDSPOS(a__2ndsneg(mark(x0), mark(x1)), mark(y1))

A__PLUS(s(X), Y) → A__PLUS(mark(X), mark(Y))

A__SQUARE(X) → A__TIMES(mark(X), mark(X))

MARK(plus(X1, X2)) → MARK(X1)

A__PI(pi(x0)) → A__2NDSPOS(a__pi(mark(x0)), a__from(0))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(Z)

A__2NDSPOS(s(y0), cons(y1, times(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__times(mark(x0), mark(x1))))

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → A__2NDSPOS(mark(N), mark(Z))

MARK(plus(from(x0), y1)) → A__PLUS(a__from(mark(x0)), mark(y1))

MARK(square(X)) → A__SQUARE(mark(X))

A__PI(X) → MARK(X)

MARK(2ndspos(pi(x0), y1)) → A__2NDSPOS(a__pi(mark(x0)), mark(y1))

MARK(2ndsneg(s(x0), y1)) → A__2NDSNEG(s(mark(x0)), mark(y1))

A__PLUS(s(X), Y) → MARK(Y)

A__TIMES(s(X), Y) → MARK(X)

A__2NDSPOS(s(y0), cons(y1, pi(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__pi(mark(x0))))

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))

a__2ndspos(0, Z) → rnil

a__2ndspos(s(N), cons(X, Z)) → a__2ndspos(s(mark(N)), cons2(X, mark(Z)))

a__2ndspos(s(N), cons2(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))

a__2ndsneg(0, Z) → rnil

a__2ndsneg(s(N), cons(X, Z)) → a__2ndsneg(s(mark(N)), cons2(X, mark(Z)))

a__2ndsneg(s(N), cons2(X, cons(Y, Z))) → rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z)))

a__pi(X) → a__2ndspos(mark(X), a__from(0))

a__plus(0, Y) → mark(Y)

a__plus(s(X), Y) → s(a__plus(mark(X), mark(Y)))

a__times(0, Y) → 0

a__times(s(X), Y) → a__plus(mark(Y), a__times(mark(X), mark(Y)))

a__square(X) → a__times(mark(X), mark(X))

mark(from(X)) → a__from(mark(X))

mark(2ndspos(X1, X2)) → a__2ndspos(mark(X1), mark(X2))

mark(2ndsneg(X1, X2)) → a__2ndsneg(mark(X1), mark(X2))

mark(pi(X)) → a__pi(mark(X))

mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))

mark(times(X1, X2)) → a__times(mark(X1), mark(X2))

mark(square(X)) → a__square(mark(X))

mark(0) → 0

mark(s(X)) → s(mark(X))

mark(posrecip(X)) → posrecip(mark(X))

mark(negrecip(X)) → negrecip(mark(X))

mark(nil) → nil

mark(cons(X1, X2)) → cons(mark(X1), X2)

mark(cons2(X1, X2)) → cons2(X1, mark(X2))

mark(rnil) → rnil

mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))

a__from(X) → from(X)

a__2ndspos(X1, X2) → 2ndspos(X1, X2)

a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)

a__pi(X) → pi(X)

a__plus(X1, X2) → plus(X1, X2)

a__times(X1, X2) → times(X1, X2)

a__square(X) → square(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule MARK(times(X1, X2)) → A__TIMES(mark(X1), mark(X2)) at position [0] we obtained the following new rules:

MARK(times(square(x0), y1)) → A__TIMES(a__square(mark(x0)), mark(y1))

MARK(times(negrecip(x0), y1)) → A__TIMES(negrecip(mark(x0)), mark(y1))

MARK(times(from(x0), y1)) → A__TIMES(a__from(mark(x0)), mark(y1))

MARK(times(2ndsneg(x0, x1), y1)) → A__TIMES(a__2ndsneg(mark(x0), mark(x1)), mark(y1))

MARK(times(times(x0, x1), y1)) → A__TIMES(a__times(mark(x0), mark(x1)), mark(y1))

MARK(times(cons2(x0, x1), y1)) → A__TIMES(cons2(x0, mark(x1)), mark(y1))

MARK(times(plus(x0, x1), y1)) → A__TIMES(a__plus(mark(x0), mark(x1)), mark(y1))

MARK(times(rnil, y1)) → A__TIMES(rnil, mark(y1))

MARK(times(cons(x0, x1), y1)) → A__TIMES(cons(mark(x0), x1), mark(y1))

MARK(times(rcons(x0, x1), y1)) → A__TIMES(rcons(mark(x0), mark(x1)), mark(y1))

MARK(times(0, y1)) → A__TIMES(0, mark(y1))

MARK(times(s(x0), y1)) → A__TIMES(s(mark(x0)), mark(y1))

MARK(times(nil, y1)) → A__TIMES(nil, mark(y1))

MARK(times(posrecip(x0), y1)) → A__TIMES(posrecip(mark(x0)), mark(y1))

MARK(times(2ndspos(x0, x1), y1)) → A__TIMES(a__2ndspos(mark(x0), mark(x1)), mark(y1))

MARK(times(pi(x0), y1)) → A__TIMES(a__pi(mark(x0)), mark(y1))



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ Narrowing
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ QDP
              ↳ Narrowing
                ↳ QDP
                  ↳ DependencyGraphProof
                    ↳ QDP
                      ↳ Narrowing
                        ↳ QDP
                          ↳ DependencyGraphProof
                            ↳ QDP
                              ↳ Narrowing
                                ↳ QDP
                                  ↳ DependencyGraphProof
                                    ↳ QDP
                                      ↳ Narrowing
                                        ↳ QDP
                                          ↳ DependencyGraphProof
                                            ↳ QDP
                                              ↳ Narrowing
                                                ↳ QDP
                                                  ↳ DependencyGraphProof

Q DP problem:
The TRS P consists of the following rules:

MARK(2ndsneg(square(x0), y1)) → A__2NDSNEG(a__square(mark(x0)), mark(y1))

MARK(2ndspos(square(x0), y1)) → A__2NDSPOS(a__square(mark(x0)), mark(y1))

A__PI(times(x0, x1)) → A__2NDSPOS(a__times(mark(x0), mark(x1)), a__from(0))

A__2NDSPOS(s(y0), cons(y1, 2ndspos(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__2ndspos(mark(x0), mark(x1))))

A__2NDSNEG(s(N), cons(X, Z)) → A__2NDSNEG(s(mark(N)), cons2(X, mark(Z)))

MARK(2ndspos(X1, X2)) → MARK(X2)

MARK(times(times(x0, x1), y1)) → A__TIMES(a__times(mark(x0), mark(x1)), mark(y1))

MARK(plus(plus(x0, x1), y1)) → A__PLUS(a__plus(mark(x0), mark(x1)), mark(y1))

MARK(2ndsneg(2ndsneg(x0, x1), y1)) → A__2NDSNEG(a__2ndsneg(mark(x0), mark(x1)), mark(y1))

A__2NDSNEG(s(N), cons(X, Z)) → MARK(Z)

A__TIMES(s(X), Y) → MARK(Y)

MARK(2ndsneg(2ndspos(x0, x1), y1)) → A__2NDSNEG(a__2ndspos(mark(x0), mark(x1)), mark(y1))

A__2NDSPOS(s(y0), cons(y1, plus(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__plus(mark(x0), mark(x1))))

A__2NDSNEG(s(N), cons(X, Z)) → MARK(N)

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → A__2NDSNEG(mark(N), mark(Z))

MARK(2ndsneg(plus(x0, x1), y1)) → A__2NDSNEG(a__plus(mark(x0), mark(x1)), mark(y1))

MARK(times(2ndsneg(x0, x1), y1)) → A__TIMES(a__2ndsneg(mark(x0), mark(x1)), mark(y1))

MARK(2ndsneg(from(x0), y1)) → A__2NDSNEG(a__from(mark(x0)), mark(y1))

A__TIMES(s(X), Y) → A__PLUS(mark(Y), a__times(mark(X), mark(Y)))

A__PI(2ndspos(x0, x1)) → A__2NDSPOS(a__2ndspos(mark(x0), mark(x1)), a__from(0))

MARK(plus(square(x0), y1)) → A__PLUS(a__square(mark(x0)), mark(y1))

MARK(plus(s(x0), y1)) → A__PLUS(s(mark(x0)), mark(y1))

MARK(2ndspos(plus(x0, x1), y1)) → A__2NDSPOS(a__plus(mark(x0), mark(x1)), mark(y1))

A__PI(plus(x0, x1)) → A__2NDSPOS(a__plus(mark(x0), mark(x1)), a__from(0))

MARK(cons2(X1, X2)) → MARK(X2)

MARK(s(X)) → MARK(X)

MARK(2ndspos(X1, X2)) → MARK(X1)

MARK(pi(X)) → MARK(X)

MARK(plus(X1, X2)) → MARK(X2)

MARK(times(cons2(x0, x1), y1)) → A__TIMES(cons2(x0, mark(x1)), mark(y1))

A__2NDSPOS(s(N), cons(X, Z)) → MARK(Z)

MARK(times(cons(x0, x1), y1)) → A__TIMES(cons(mark(x0), x1), mark(y1))

MARK(pi(X)) → A__PI(mark(X))

MARK(times(0, y1)) → A__TIMES(0, mark(y1))

MARK(plus(2ndsneg(x0, x1), y1)) → A__PLUS(a__2ndsneg(mark(x0), mark(x1)), mark(y1))

MARK(2ndsneg(X1, X2)) → MARK(X2)

MARK(times(square(x0), y1)) → A__TIMES(a__square(mark(x0)), mark(y1))

A__PI(X) → A__FROM(0)

MARK(from(X)) → A__FROM(mark(X))

MARK(plus(X1, X2)) → MARK(X1)

MARK(times(from(x0), y1)) → A__TIMES(a__from(mark(x0)), mark(y1))

A__2NDSPOS(s(y0), cons(y1, times(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__times(mark(x0), mark(x1))))

A__PI(X) → MARK(X)

MARK(times(s(x0), y1)) → A__TIMES(s(mark(x0)), mark(y1))

MARK(2ndsneg(s(x0), y1)) → A__2NDSNEG(s(mark(x0)), mark(y1))

A__PLUS(s(X), Y) → MARK(Y)

A__TIMES(s(X), Y) → MARK(X)

A__2NDSPOS(s(y0), cons(y1, pi(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__pi(mark(x0))))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(N)

MARK(plus(2ndspos(x0, x1), y1)) → A__PLUS(a__2ndspos(mark(x0), mark(x1)), mark(y1))

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(N)

A__PI(from(x0)) → A__2NDSPOS(a__from(mark(x0)), a__from(0))

MARK(cons(X1, X2)) → MARK(X1)

MARK(times(X1, X2)) → MARK(X2)

A__PI(square(x0)) → A__2NDSPOS(a__square(mark(x0)), a__from(0))

MARK(times(rcons(x0, x1), y1)) → A__TIMES(rcons(mark(x0), mark(x1)), mark(y1))

MARK(2ndsneg(X1, X2)) → MARK(X1)

A__PI(2ndsneg(x0, x1)) → A__2NDSPOS(a__2ndsneg(mark(x0), mark(x1)), a__from(0))

MARK(times(pi(x0), y1)) → A__TIMES(a__pi(mark(x0)), mark(y1))

MARK(plus(pi(x0), y1)) → A__PLUS(a__pi(mark(x0)), mark(y1))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(Y)

MARK(rcons(X1, X2)) → MARK(X1)

MARK(2ndspos(from(x0), y1)) → A__2NDSPOS(a__from(mark(x0)), mark(y1))

MARK(2ndspos(s(x0), y1)) → A__2NDSPOS(s(mark(x0)), mark(y1))

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(Z)

MARK(square(X)) → MARK(X)

A__PI(s(x0)) → A__2NDSPOS(s(mark(x0)), a__from(0))

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(Y)

A__FROM(X) → MARK(X)

MARK(times(plus(x0, x1), y1)) → A__TIMES(a__plus(mark(x0), mark(x1)), mark(y1))

MARK(times(rnil, y1)) → A__TIMES(rnil, mark(y1))

A__2NDSPOS(s(y0), cons(y1, from(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__from(mark(x0))))

MARK(2ndspos(times(x0, x1), y1)) → A__2NDSPOS(a__times(mark(x0), mark(x1)), mark(y1))

MARK(times(posrecip(x0), y1)) → A__TIMES(posrecip(mark(x0)), mark(y1))

MARK(rcons(X1, X2)) → MARK(X2)

MARK(times(2ndspos(x0, x1), y1)) → A__TIMES(a__2ndspos(mark(x0), mark(x1)), mark(y1))

A__TIMES(s(X), Y) → A__TIMES(mark(X), mark(Y))

A__PLUS(s(X), Y) → MARK(X)

A__2NDSPOS(s(y0), cons(y1, cons(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, cons(mark(x0), x1)))

MARK(times(X1, X2)) → MARK(X1)

MARK(plus(times(x0, x1), y1)) → A__PLUS(a__times(mark(x0), mark(x1)), mark(y1))

A__SQUARE(X) → MARK(X)

MARK(2ndsneg(times(x0, x1), y1)) → A__2NDSNEG(a__times(mark(x0), mark(x1)), mark(y1))

A__2NDSPOS(s(y0), cons(y1, 2ndsneg(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__2ndsneg(mark(x0), mark(x1))))

MARK(negrecip(X)) → MARK(X)

A__2NDSPOS(s(N), cons(X, Z)) → MARK(N)

MARK(times(nil, y1)) → A__TIMES(nil, mark(y1))

A__PLUS(0, Y) → MARK(Y)

MARK(2ndspos(2ndspos(x0, x1), y1)) → A__2NDSPOS(a__2ndspos(mark(x0), mark(x1)), mark(y1))

MARK(posrecip(X)) → MARK(X)

MARK(2ndsneg(pi(x0), y1)) → A__2NDSNEG(a__pi(mark(x0)), mark(y1))

A__2NDSPOS(s(y0), cons(y1, square(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__square(mark(x0))))

MARK(plus(0, y1)) → A__PLUS(0, mark(y1))

MARK(from(X)) → MARK(X)

MARK(times(negrecip(x0), y1)) → A__TIMES(negrecip(mark(x0)), mark(y1))

MARK(2ndspos(2ndsneg(x0, x1), y1)) → A__2NDSPOS(a__2ndsneg(mark(x0), mark(x1)), mark(y1))

A__PLUS(s(X), Y) → A__PLUS(mark(X), mark(Y))

A__SQUARE(X) → A__TIMES(mark(X), mark(X))

A__PI(pi(x0)) → A__2NDSPOS(a__pi(mark(x0)), a__from(0))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(Z)

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → A__2NDSPOS(mark(N), mark(Z))

MARK(plus(from(x0), y1)) → A__PLUS(a__from(mark(x0)), mark(y1))

MARK(square(X)) → A__SQUARE(mark(X))

MARK(2ndspos(pi(x0), y1)) → A__2NDSPOS(a__pi(mark(x0)), mark(y1))

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))

a__2ndspos(0, Z) → rnil

a__2ndspos(s(N), cons(X, Z)) → a__2ndspos(s(mark(N)), cons2(X, mark(Z)))

a__2ndspos(s(N), cons2(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))

a__2ndsneg(0, Z) → rnil

a__2ndsneg(s(N), cons(X, Z)) → a__2ndsneg(s(mark(N)), cons2(X, mark(Z)))

a__2ndsneg(s(N), cons2(X, cons(Y, Z))) → rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z)))

a__pi(X) → a__2ndspos(mark(X), a__from(0))

a__plus(0, Y) → mark(Y)

a__plus(s(X), Y) → s(a__plus(mark(X), mark(Y)))

a__times(0, Y) → 0

a__times(s(X), Y) → a__plus(mark(Y), a__times(mark(X), mark(Y)))

a__square(X) → a__times(mark(X), mark(X))

mark(from(X)) → a__from(mark(X))

mark(2ndspos(X1, X2)) → a__2ndspos(mark(X1), mark(X2))

mark(2ndsneg(X1, X2)) → a__2ndsneg(mark(X1), mark(X2))

mark(pi(X)) → a__pi(mark(X))

mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))

mark(times(X1, X2)) → a__times(mark(X1), mark(X2))

mark(square(X)) → a__square(mark(X))

mark(0) → 0

mark(s(X)) → s(mark(X))

mark(posrecip(X)) → posrecip(mark(X))

mark(negrecip(X)) → negrecip(mark(X))

mark(nil) → nil

mark(cons(X1, X2)) → cons(mark(X1), X2)

mark(cons2(X1, X2)) → cons2(X1, mark(X2))

mark(rnil) → rnil

mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))

a__from(X) → from(X)

a__2ndspos(X1, X2) → 2ndspos(X1, X2)

a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)

a__pi(X) → pi(X)

a__plus(X1, X2) → plus(X1, X2)

a__times(X1, X2) → times(X1, X2)

a__square(X) → square(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 8 less nodes.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ Narrowing
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ QDP
              ↳ Narrowing
                ↳ QDP
                  ↳ DependencyGraphProof
                    ↳ QDP
                      ↳ Narrowing
                        ↳ QDP
                          ↳ DependencyGraphProof
                            ↳ QDP
                              ↳ Narrowing
                                ↳ QDP
                                  ↳ DependencyGraphProof
                                    ↳ QDP
                                      ↳ Narrowing
                                        ↳ QDP
                                          ↳ DependencyGraphProof
                                            ↳ QDP
                                              ↳ Narrowing
                                                ↳ QDP
                                                  ↳ DependencyGraphProof
                                                    ↳ QDP
                                                      ↳ Narrowing

Q DP problem:
The TRS P consists of the following rules:

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(N)

MARK(2ndsneg(square(x0), y1)) → A__2NDSNEG(a__square(mark(x0)), mark(y1))

MARK(2ndspos(square(x0), y1)) → A__2NDSPOS(a__square(mark(x0)), mark(y1))

A__PI(times(x0, x1)) → A__2NDSPOS(a__times(mark(x0), mark(x1)), a__from(0))

MARK(plus(2ndspos(x0, x1), y1)) → A__PLUS(a__2ndspos(mark(x0), mark(x1)), mark(y1))

A__2NDSPOS(s(y0), cons(y1, 2ndspos(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__2ndspos(mark(x0), mark(x1))))

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(N)

A__2NDSNEG(s(N), cons(X, Z)) → A__2NDSNEG(s(mark(N)), cons2(X, mark(Z)))

A__PI(from(x0)) → A__2NDSPOS(a__from(mark(x0)), a__from(0))

MARK(cons(X1, X2)) → MARK(X1)

MARK(times(times(x0, x1), y1)) → A__TIMES(a__times(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(X1, X2)) → MARK(X2)

MARK(times(X1, X2)) → MARK(X2)

A__PI(square(x0)) → A__2NDSPOS(a__square(mark(x0)), a__from(0))

MARK(plus(plus(x0, x1), y1)) → A__PLUS(a__plus(mark(x0), mark(x1)), mark(y1))

MARK(2ndsneg(2ndsneg(x0, x1), y1)) → A__2NDSNEG(a__2ndsneg(mark(x0), mark(x1)), mark(y1))

A__2NDSNEG(s(N), cons(X, Z)) → MARK(Z)

MARK(2ndsneg(X1, X2)) → MARK(X1)

A__PI(2ndsneg(x0, x1)) → A__2NDSPOS(a__2ndsneg(mark(x0), mark(x1)), a__from(0))

MARK(times(pi(x0), y1)) → A__TIMES(a__pi(mark(x0)), mark(y1))

MARK(plus(pi(x0), y1)) → A__PLUS(a__pi(mark(x0)), mark(y1))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(Y)

MARK(rcons(X1, X2)) → MARK(X1)

A__TIMES(s(X), Y) → MARK(Y)

MARK(2ndsneg(2ndspos(x0, x1), y1)) → A__2NDSNEG(a__2ndspos(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(from(x0), y1)) → A__2NDSPOS(a__from(mark(x0)), mark(y1))

MARK(2ndspos(s(x0), y1)) → A__2NDSPOS(s(mark(x0)), mark(y1))

A__2NDSPOS(s(y0), cons(y1, plus(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__plus(mark(x0), mark(x1))))

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(Z)

A__2NDSNEG(s(N), cons(X, Z)) → MARK(N)

MARK(square(X)) → MARK(X)

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → A__2NDSNEG(mark(N), mark(Z))

MARK(times(2ndsneg(x0, x1), y1)) → A__TIMES(a__2ndsneg(mark(x0), mark(x1)), mark(y1))

MARK(2ndsneg(plus(x0, x1), y1)) → A__2NDSNEG(a__plus(mark(x0), mark(x1)), mark(y1))

A__PI(s(x0)) → A__2NDSPOS(s(mark(x0)), a__from(0))

MARK(2ndsneg(from(x0), y1)) → A__2NDSNEG(a__from(mark(x0)), mark(y1))

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(Y)

A__FROM(X) → MARK(X)

MARK(times(plus(x0, x1), y1)) → A__TIMES(a__plus(mark(x0), mark(x1)), mark(y1))

A__2NDSPOS(s(y0), cons(y1, from(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__from(mark(x0))))

MARK(2ndspos(times(x0, x1), y1)) → A__2NDSPOS(a__times(mark(x0), mark(x1)), mark(y1))

A__TIMES(s(X), Y) → A__PLUS(mark(Y), a__times(mark(X), mark(Y)))

A__PI(2ndspos(x0, x1)) → A__2NDSPOS(a__2ndspos(mark(x0), mark(x1)), a__from(0))

MARK(plus(square(x0), y1)) → A__PLUS(a__square(mark(x0)), mark(y1))

MARK(plus(s(x0), y1)) → A__PLUS(s(mark(x0)), mark(y1))

MARK(2ndspos(plus(x0, x1), y1)) → A__2NDSPOS(a__plus(mark(x0), mark(x1)), mark(y1))

MARK(rcons(X1, X2)) → MARK(X2)

A__PI(plus(x0, x1)) → A__2NDSPOS(a__plus(mark(x0), mark(x1)), a__from(0))

MARK(times(2ndspos(x0, x1), y1)) → A__TIMES(a__2ndspos(mark(x0), mark(x1)), mark(y1))

MARK(cons2(X1, X2)) → MARK(X2)

A__TIMES(s(X), Y) → A__TIMES(mark(X), mark(Y))

MARK(s(X)) → MARK(X)

MARK(2ndspos(X1, X2)) → MARK(X1)

A__PLUS(s(X), Y) → MARK(X)

A__2NDSPOS(s(y0), cons(y1, cons(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, cons(mark(x0), x1)))

MARK(pi(X)) → MARK(X)

MARK(times(X1, X2)) → MARK(X1)

MARK(plus(X1, X2)) → MARK(X2)

MARK(plus(times(x0, x1), y1)) → A__PLUS(a__times(mark(x0), mark(x1)), mark(y1))

A__SQUARE(X) → MARK(X)

A__2NDSPOS(s(N), cons(X, Z)) → MARK(Z)

MARK(2ndsneg(times(x0, x1), y1)) → A__2NDSNEG(a__times(mark(x0), mark(x1)), mark(y1))

MARK(pi(X)) → A__PI(mark(X))

A__2NDSPOS(s(y0), cons(y1, 2ndsneg(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__2ndsneg(mark(x0), mark(x1))))

MARK(negrecip(X)) → MARK(X)

A__2NDSPOS(s(N), cons(X, Z)) → MARK(N)

MARK(plus(2ndsneg(x0, x1), y1)) → A__PLUS(a__2ndsneg(mark(x0), mark(x1)), mark(y1))

A__PLUS(0, Y) → MARK(Y)

MARK(2ndspos(2ndspos(x0, x1), y1)) → A__2NDSPOS(a__2ndspos(mark(x0), mark(x1)), mark(y1))

MARK(posrecip(X)) → MARK(X)

MARK(2ndsneg(pi(x0), y1)) → A__2NDSNEG(a__pi(mark(x0)), mark(y1))

MARK(2ndsneg(X1, X2)) → MARK(X2)

A__2NDSPOS(s(y0), cons(y1, square(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__square(mark(x0))))

MARK(plus(0, y1)) → A__PLUS(0, mark(y1))

MARK(times(square(x0), y1)) → A__TIMES(a__square(mark(x0)), mark(y1))

MARK(from(X)) → MARK(X)

A__PI(X) → A__FROM(0)

MARK(from(X)) → A__FROM(mark(X))

MARK(2ndspos(2ndsneg(x0, x1), y1)) → A__2NDSPOS(a__2ndsneg(mark(x0), mark(x1)), mark(y1))

A__PLUS(s(X), Y) → A__PLUS(mark(X), mark(Y))

MARK(times(from(x0), y1)) → A__TIMES(a__from(mark(x0)), mark(y1))

A__SQUARE(X) → A__TIMES(mark(X), mark(X))

MARK(plus(X1, X2)) → MARK(X1)

A__PI(pi(x0)) → A__2NDSPOS(a__pi(mark(x0)), a__from(0))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(Z)

A__2NDSPOS(s(y0), cons(y1, times(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__times(mark(x0), mark(x1))))

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → A__2NDSPOS(mark(N), mark(Z))

MARK(plus(from(x0), y1)) → A__PLUS(a__from(mark(x0)), mark(y1))

MARK(square(X)) → A__SQUARE(mark(X))

A__PI(X) → MARK(X)

MARK(2ndspos(pi(x0), y1)) → A__2NDSPOS(a__pi(mark(x0)), mark(y1))

MARK(times(s(x0), y1)) → A__TIMES(s(mark(x0)), mark(y1))

MARK(2ndsneg(s(x0), y1)) → A__2NDSNEG(s(mark(x0)), mark(y1))

A__PLUS(s(X), Y) → MARK(Y)

A__TIMES(s(X), Y) → MARK(X)

A__2NDSPOS(s(y0), cons(y1, pi(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__pi(mark(x0))))

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))

a__2ndspos(0, Z) → rnil

a__2ndspos(s(N), cons(X, Z)) → a__2ndspos(s(mark(N)), cons2(X, mark(Z)))

a__2ndspos(s(N), cons2(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))

a__2ndsneg(0, Z) → rnil

a__2ndsneg(s(N), cons(X, Z)) → a__2ndsneg(s(mark(N)), cons2(X, mark(Z)))

a__2ndsneg(s(N), cons2(X, cons(Y, Z))) → rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z)))

a__pi(X) → a__2ndspos(mark(X), a__from(0))

a__plus(0, Y) → mark(Y)

a__plus(s(X), Y) → s(a__plus(mark(X), mark(Y)))

a__times(0, Y) → 0

a__times(s(X), Y) → a__plus(mark(Y), a__times(mark(X), mark(Y)))

a__square(X) → a__times(mark(X), mark(X))

mark(from(X)) → a__from(mark(X))

mark(2ndspos(X1, X2)) → a__2ndspos(mark(X1), mark(X2))

mark(2ndsneg(X1, X2)) → a__2ndsneg(mark(X1), mark(X2))

mark(pi(X)) → a__pi(mark(X))

mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))

mark(times(X1, X2)) → a__times(mark(X1), mark(X2))

mark(square(X)) → a__square(mark(X))

mark(0) → 0

mark(s(X)) → s(mark(X))

mark(posrecip(X)) → posrecip(mark(X))

mark(negrecip(X)) → negrecip(mark(X))

mark(nil) → nil

mark(cons(X1, X2)) → cons(mark(X1), X2)

mark(cons2(X1, X2)) → cons2(X1, mark(X2))

mark(rnil) → rnil

mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))

a__from(X) → from(X)

a__2ndspos(X1, X2) → 2ndspos(X1, X2)

a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)

a__pi(X) → pi(X)

a__plus(X1, X2) → plus(X1, X2)

a__times(X1, X2) → times(X1, X2)

a__square(X) → square(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule MARK(2ndspos(square(x0), y1)) → A__2NDSPOS(a__square(mark(x0)), mark(y1)) at position [1] we obtained the following new rules:

MARK(2ndspos(square(y0), times(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(square(y0), plus(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(square(y0), rnil)) → A__2NDSPOS(a__square(mark(y0)), rnil)

MARK(2ndspos(square(y0), square(x0))) → A__2NDSPOS(a__square(mark(y0)), a__square(mark(x0)))

MARK(2ndspos(square(y0), 2ndsneg(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndspos(square(y0), nil)) → A__2NDSPOS(a__square(mark(y0)), nil)

MARK(2ndspos(square(y0), negrecip(x0))) → A__2NDSPOS(a__square(mark(y0)), negrecip(mark(x0)))

MARK(2ndspos(square(y0), 2ndspos(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndspos(square(y0), rcons(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), rcons(mark(x0), mark(x1)))

MARK(2ndspos(square(y0), 0)) → A__2NDSPOS(a__square(mark(y0)), 0)

MARK(2ndspos(square(y0), pi(x0))) → A__2NDSPOS(a__square(mark(y0)), a__pi(mark(x0)))

MARK(2ndspos(square(y0), cons2(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), cons2(x0, mark(x1)))

MARK(2ndspos(square(y0), s(x0))) → A__2NDSPOS(a__square(mark(y0)), s(mark(x0)))

MARK(2ndspos(square(y0), from(x0))) → A__2NDSPOS(a__square(mark(y0)), a__from(mark(x0)))

MARK(2ndspos(square(y0), cons(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), cons(mark(x0), x1))

MARK(2ndspos(square(y0), posrecip(x0))) → A__2NDSPOS(a__square(mark(y0)), posrecip(mark(x0)))



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ Narrowing
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ QDP
              ↳ Narrowing
                ↳ QDP
                  ↳ DependencyGraphProof
                    ↳ QDP
                      ↳ Narrowing
                        ↳ QDP
                          ↳ DependencyGraphProof
                            ↳ QDP
                              ↳ Narrowing
                                ↳ QDP
                                  ↳ DependencyGraphProof
                                    ↳ QDP
                                      ↳ Narrowing
                                        ↳ QDP
                                          ↳ DependencyGraphProof
                                            ↳ QDP
                                              ↳ Narrowing
                                                ↳ QDP
                                                  ↳ DependencyGraphProof
                                                    ↳ QDP
                                                      ↳ Narrowing
                                                        ↳ QDP
                                                          ↳ DependencyGraphProof

Q DP problem:
The TRS P consists of the following rules:

MARK(2ndspos(square(y0), plus(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(2ndsneg(square(x0), y1)) → A__2NDSNEG(a__square(mark(x0)), mark(y1))

A__2NDSPOS(s(y0), cons(y1, 2ndspos(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__2ndspos(mark(x0), mark(x1))))

A__PI(times(x0, x1)) → A__2NDSPOS(a__times(mark(x0), mark(x1)), a__from(0))

A__2NDSNEG(s(N), cons(X, Z)) → A__2NDSNEG(s(mark(N)), cons2(X, mark(Z)))

MARK(2ndspos(square(y0), nil)) → A__2NDSPOS(a__square(mark(y0)), nil)

MARK(2ndspos(X1, X2)) → MARK(X2)

MARK(times(times(x0, x1), y1)) → A__TIMES(a__times(mark(x0), mark(x1)), mark(y1))

MARK(plus(plus(x0, x1), y1)) → A__PLUS(a__plus(mark(x0), mark(x1)), mark(y1))

MARK(2ndsneg(2ndsneg(x0, x1), y1)) → A__2NDSNEG(a__2ndsneg(mark(x0), mark(x1)), mark(y1))

A__2NDSNEG(s(N), cons(X, Z)) → MARK(Z)

A__TIMES(s(X), Y) → MARK(Y)

MARK(2ndsneg(2ndspos(x0, x1), y1)) → A__2NDSNEG(a__2ndspos(mark(x0), mark(x1)), mark(y1))

A__2NDSPOS(s(y0), cons(y1, plus(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__plus(mark(x0), mark(x1))))

MARK(2ndspos(square(y0), 2ndsneg(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → A__2NDSNEG(mark(N), mark(Z))

A__2NDSNEG(s(N), cons(X, Z)) → MARK(N)

MARK(2ndsneg(plus(x0, x1), y1)) → A__2NDSNEG(a__plus(mark(x0), mark(x1)), mark(y1))

MARK(times(2ndsneg(x0, x1), y1)) → A__TIMES(a__2ndsneg(mark(x0), mark(x1)), mark(y1))

MARK(2ndsneg(from(x0), y1)) → A__2NDSNEG(a__from(mark(x0)), mark(y1))

A__TIMES(s(X), Y) → A__PLUS(mark(Y), a__times(mark(X), mark(Y)))

MARK(2ndspos(square(y0), cons2(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), cons2(x0, mark(x1)))

A__PI(2ndspos(x0, x1)) → A__2NDSPOS(a__2ndspos(mark(x0), mark(x1)), a__from(0))

MARK(plus(square(x0), y1)) → A__PLUS(a__square(mark(x0)), mark(y1))

MARK(2ndspos(plus(x0, x1), y1)) → A__2NDSPOS(a__plus(mark(x0), mark(x1)), mark(y1))

MARK(plus(s(x0), y1)) → A__PLUS(s(mark(x0)), mark(y1))

MARK(2ndspos(square(y0), cons(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), cons(mark(x0), x1))

A__PI(plus(x0, x1)) → A__2NDSPOS(a__plus(mark(x0), mark(x1)), a__from(0))

MARK(cons2(X1, X2)) → MARK(X2)

MARK(2ndspos(X1, X2)) → MARK(X1)

MARK(s(X)) → MARK(X)

MARK(pi(X)) → MARK(X)

MARK(plus(X1, X2)) → MARK(X2)

A__2NDSPOS(s(N), cons(X, Z)) → MARK(Z)

MARK(2ndspos(square(y0), 0)) → A__2NDSPOS(a__square(mark(y0)), 0)

MARK(pi(X)) → A__PI(mark(X))

MARK(plus(2ndsneg(x0, x1), y1)) → A__PLUS(a__2ndsneg(mark(x0), mark(x1)), mark(y1))

MARK(2ndsneg(X1, X2)) → MARK(X2)

MARK(2ndspos(square(y0), times(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(times(square(x0), y1)) → A__TIMES(a__square(mark(x0)), mark(y1))

MARK(2ndspos(square(y0), rnil)) → A__2NDSPOS(a__square(mark(y0)), rnil)

MARK(from(X)) → A__FROM(mark(X))

A__PI(X) → A__FROM(0)

MARK(plus(X1, X2)) → MARK(X1)

MARK(times(from(x0), y1)) → A__TIMES(a__from(mark(x0)), mark(y1))

A__2NDSPOS(s(y0), cons(y1, times(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__times(mark(x0), mark(x1))))

A__PI(X) → MARK(X)

MARK(times(s(x0), y1)) → A__TIMES(s(mark(x0)), mark(y1))

MARK(2ndsneg(s(x0), y1)) → A__2NDSNEG(s(mark(x0)), mark(y1))

A__PLUS(s(X), Y) → MARK(Y)

A__2NDSPOS(s(y0), cons(y1, pi(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__pi(mark(x0))))

A__TIMES(s(X), Y) → MARK(X)

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(N)

MARK(plus(2ndspos(x0, x1), y1)) → A__PLUS(a__2ndspos(mark(x0), mark(x1)), mark(y1))

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(N)

MARK(cons(X1, X2)) → MARK(X1)

A__PI(from(x0)) → A__2NDSPOS(a__from(mark(x0)), a__from(0))

MARK(2ndspos(square(y0), 2ndspos(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

MARK(times(X1, X2)) → MARK(X2)

MARK(2ndspos(square(y0), rcons(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), rcons(mark(x0), mark(x1)))

A__PI(square(x0)) → A__2NDSPOS(a__square(mark(x0)), a__from(0))

MARK(2ndsneg(X1, X2)) → MARK(X1)

A__PI(2ndsneg(x0, x1)) → A__2NDSPOS(a__2ndsneg(mark(x0), mark(x1)), a__from(0))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(Y)

MARK(plus(pi(x0), y1)) → A__PLUS(a__pi(mark(x0)), mark(y1))

MARK(times(pi(x0), y1)) → A__TIMES(a__pi(mark(x0)), mark(y1))

MARK(rcons(X1, X2)) → MARK(X1)

MARK(2ndspos(s(x0), y1)) → A__2NDSPOS(s(mark(x0)), mark(y1))

MARK(2ndspos(from(x0), y1)) → A__2NDSPOS(a__from(mark(x0)), mark(y1))

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(Z)

MARK(2ndspos(square(y0), negrecip(x0))) → A__2NDSPOS(a__square(mark(y0)), negrecip(mark(x0)))

MARK(square(X)) → MARK(X)

A__PI(s(x0)) → A__2NDSPOS(s(mark(x0)), a__from(0))

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(Y)

A__FROM(X) → MARK(X)

MARK(times(plus(x0, x1), y1)) → A__TIMES(a__plus(mark(x0), mark(x1)), mark(y1))

A__2NDSPOS(s(y0), cons(y1, from(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__from(mark(x0))))

MARK(2ndspos(times(x0, x1), y1)) → A__2NDSPOS(a__times(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(square(y0), pi(x0))) → A__2NDSPOS(a__square(mark(y0)), a__pi(mark(x0)))

MARK(2ndspos(square(y0), s(x0))) → A__2NDSPOS(a__square(mark(y0)), s(mark(x0)))

MARK(rcons(X1, X2)) → MARK(X2)

MARK(times(2ndspos(x0, x1), y1)) → A__TIMES(a__2ndspos(mark(x0), mark(x1)), mark(y1))

A__TIMES(s(X), Y) → A__TIMES(mark(X), mark(Y))

A__2NDSPOS(s(y0), cons(y1, cons(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, cons(mark(x0), x1)))

A__PLUS(s(X), Y) → MARK(X)

MARK(times(X1, X2)) → MARK(X1)

MARK(plus(times(x0, x1), y1)) → A__PLUS(a__times(mark(x0), mark(x1)), mark(y1))

A__SQUARE(X) → MARK(X)

MARK(2ndsneg(times(x0, x1), y1)) → A__2NDSNEG(a__times(mark(x0), mark(x1)), mark(y1))

A__2NDSPOS(s(y0), cons(y1, 2ndsneg(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__2ndsneg(mark(x0), mark(x1))))

A__2NDSPOS(s(N), cons(X, Z)) → MARK(N)

MARK(negrecip(X)) → MARK(X)

A__PLUS(0, Y) → MARK(Y)

MARK(2ndspos(2ndspos(x0, x1), y1)) → A__2NDSPOS(a__2ndspos(mark(x0), mark(x1)), mark(y1))

MARK(posrecip(X)) → MARK(X)

MARK(2ndspos(square(y0), from(x0))) → A__2NDSPOS(a__square(mark(y0)), a__from(mark(x0)))

MARK(2ndsneg(pi(x0), y1)) → A__2NDSNEG(a__pi(mark(x0)), mark(y1))

MARK(2ndspos(square(y0), posrecip(x0))) → A__2NDSPOS(a__square(mark(y0)), posrecip(mark(x0)))

A__2NDSPOS(s(y0), cons(y1, square(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__square(mark(x0))))

MARK(plus(0, y1)) → A__PLUS(0, mark(y1))

MARK(from(X)) → MARK(X)

MARK(2ndspos(2ndsneg(x0, x1), y1)) → A__2NDSPOS(a__2ndsneg(mark(x0), mark(x1)), mark(y1))

A__PLUS(s(X), Y) → A__PLUS(mark(X), mark(Y))

MARK(2ndspos(square(y0), square(x0))) → A__2NDSPOS(a__square(mark(y0)), a__square(mark(x0)))

A__SQUARE(X) → A__TIMES(mark(X), mark(X))

A__PI(pi(x0)) → A__2NDSPOS(a__pi(mark(x0)), a__from(0))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(Z)

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → A__2NDSPOS(mark(N), mark(Z))

MARK(plus(from(x0), y1)) → A__PLUS(a__from(mark(x0)), mark(y1))

MARK(2ndspos(pi(x0), y1)) → A__2NDSPOS(a__pi(mark(x0)), mark(y1))

MARK(square(X)) → A__SQUARE(mark(X))

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))

a__2ndspos(0, Z) → rnil

a__2ndspos(s(N), cons(X, Z)) → a__2ndspos(s(mark(N)), cons2(X, mark(Z)))

a__2ndspos(s(N), cons2(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))

a__2ndsneg(0, Z) → rnil

a__2ndsneg(s(N), cons(X, Z)) → a__2ndsneg(s(mark(N)), cons2(X, mark(Z)))

a__2ndsneg(s(N), cons2(X, cons(Y, Z))) → rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z)))

a__pi(X) → a__2ndspos(mark(X), a__from(0))

a__plus(0, Y) → mark(Y)

a__plus(s(X), Y) → s(a__plus(mark(X), mark(Y)))

a__times(0, Y) → 0

a__times(s(X), Y) → a__plus(mark(Y), a__times(mark(X), mark(Y)))

a__square(X) → a__times(mark(X), mark(X))

mark(from(X)) → a__from(mark(X))

mark(2ndspos(X1, X2)) → a__2ndspos(mark(X1), mark(X2))

mark(2ndsneg(X1, X2)) → a__2ndsneg(mark(X1), mark(X2))

mark(pi(X)) → a__pi(mark(X))

mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))

mark(times(X1, X2)) → a__times(mark(X1), mark(X2))

mark(square(X)) → a__square(mark(X))

mark(0) → 0

mark(s(X)) → s(mark(X))

mark(posrecip(X)) → posrecip(mark(X))

mark(negrecip(X)) → negrecip(mark(X))

mark(nil) → nil

mark(cons(X1, X2)) → cons(mark(X1), X2)

mark(cons2(X1, X2)) → cons2(X1, mark(X2))

mark(rnil) → rnil

mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))

a__from(X) → from(X)

a__2ndspos(X1, X2) → 2ndspos(X1, X2)

a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)

a__pi(X) → pi(X)

a__plus(X1, X2) → plus(X1, X2)

a__times(X1, X2) → times(X1, X2)

a__square(X) → square(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 7 less nodes.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ Narrowing
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ QDP
              ↳ Narrowing
                ↳ QDP
                  ↳ DependencyGraphProof
                    ↳ QDP
                      ↳ Narrowing
                        ↳ QDP
                          ↳ DependencyGraphProof
                            ↳ QDP
                              ↳ Narrowing
                                ↳ QDP
                                  ↳ DependencyGraphProof
                                    ↳ QDP
                                      ↳ Narrowing
                                        ↳ QDP
                                          ↳ DependencyGraphProof
                                            ↳ QDP
                                              ↳ Narrowing
                                                ↳ QDP
                                                  ↳ DependencyGraphProof
                                                    ↳ QDP
                                                      ↳ Narrowing
                                                        ↳ QDP
                                                          ↳ DependencyGraphProof
                                                            ↳ QDP
                                                              ↳ Narrowing

Q DP problem:
The TRS P consists of the following rules:

MARK(2ndsneg(square(x0), y1)) → A__2NDSNEG(a__square(mark(x0)), mark(y1))

MARK(2ndspos(square(y0), plus(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__plus(mark(x0), mark(x1)))

A__2NDSPOS(s(y0), cons(y1, 2ndspos(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__2ndspos(mark(x0), mark(x1))))

A__PI(times(x0, x1)) → A__2NDSPOS(a__times(mark(x0), mark(x1)), a__from(0))

A__2NDSNEG(s(N), cons(X, Z)) → A__2NDSNEG(s(mark(N)), cons2(X, mark(Z)))

MARK(2ndspos(X1, X2)) → MARK(X2)

MARK(times(times(x0, x1), y1)) → A__TIMES(a__times(mark(x0), mark(x1)), mark(y1))

MARK(plus(plus(x0, x1), y1)) → A__PLUS(a__plus(mark(x0), mark(x1)), mark(y1))

MARK(2ndsneg(2ndsneg(x0, x1), y1)) → A__2NDSNEG(a__2ndsneg(mark(x0), mark(x1)), mark(y1))

A__2NDSNEG(s(N), cons(X, Z)) → MARK(Z)

A__TIMES(s(X), Y) → MARK(Y)

MARK(2ndsneg(2ndspos(x0, x1), y1)) → A__2NDSNEG(a__2ndspos(mark(x0), mark(x1)), mark(y1))

A__2NDSPOS(s(y0), cons(y1, plus(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__plus(mark(x0), mark(x1))))

MARK(2ndspos(square(y0), 2ndsneg(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → A__2NDSNEG(mark(N), mark(Z))

A__2NDSNEG(s(N), cons(X, Z)) → MARK(N)

MARK(2ndsneg(plus(x0, x1), y1)) → A__2NDSNEG(a__plus(mark(x0), mark(x1)), mark(y1))

MARK(times(2ndsneg(x0, x1), y1)) → A__TIMES(a__2ndsneg(mark(x0), mark(x1)), mark(y1))

MARK(2ndsneg(from(x0), y1)) → A__2NDSNEG(a__from(mark(x0)), mark(y1))

A__TIMES(s(X), Y) → A__PLUS(mark(Y), a__times(mark(X), mark(Y)))

A__PI(2ndspos(x0, x1)) → A__2NDSPOS(a__2ndspos(mark(x0), mark(x1)), a__from(0))

MARK(2ndspos(square(y0), cons2(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), cons2(x0, mark(x1)))

MARK(plus(square(x0), y1)) → A__PLUS(a__square(mark(x0)), mark(y1))

MARK(2ndspos(plus(x0, x1), y1)) → A__2NDSPOS(a__plus(mark(x0), mark(x1)), mark(y1))

MARK(plus(s(x0), y1)) → A__PLUS(s(mark(x0)), mark(y1))

A__PI(plus(x0, x1)) → A__2NDSPOS(a__plus(mark(x0), mark(x1)), a__from(0))

MARK(2ndspos(square(y0), cons(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), cons(mark(x0), x1))

MARK(cons2(X1, X2)) → MARK(X2)

MARK(2ndspos(X1, X2)) → MARK(X1)

MARK(s(X)) → MARK(X)

MARK(pi(X)) → MARK(X)

MARK(plus(X1, X2)) → MARK(X2)

A__2NDSPOS(s(N), cons(X, Z)) → MARK(Z)

MARK(pi(X)) → A__PI(mark(X))

MARK(plus(2ndsneg(x0, x1), y1)) → A__PLUS(a__2ndsneg(mark(x0), mark(x1)), mark(y1))

MARK(2ndsneg(X1, X2)) → MARK(X2)

MARK(2ndspos(square(y0), times(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(times(square(x0), y1)) → A__TIMES(a__square(mark(x0)), mark(y1))

MARK(from(X)) → A__FROM(mark(X))

A__PI(X) → A__FROM(0)

MARK(plus(X1, X2)) → MARK(X1)

MARK(times(from(x0), y1)) → A__TIMES(a__from(mark(x0)), mark(y1))

A__2NDSPOS(s(y0), cons(y1, times(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__times(mark(x0), mark(x1))))

A__PI(X) → MARK(X)

MARK(times(s(x0), y1)) → A__TIMES(s(mark(x0)), mark(y1))

MARK(2ndsneg(s(x0), y1)) → A__2NDSNEG(s(mark(x0)), mark(y1))

A__PLUS(s(X), Y) → MARK(Y)

A__2NDSPOS(s(y0), cons(y1, pi(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__pi(mark(x0))))

A__TIMES(s(X), Y) → MARK(X)

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(N)

MARK(plus(2ndspos(x0, x1), y1)) → A__PLUS(a__2ndspos(mark(x0), mark(x1)), mark(y1))

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(N)

A__PI(from(x0)) → A__2NDSPOS(a__from(mark(x0)), a__from(0))

MARK(cons(X1, X2)) → MARK(X1)

MARK(2ndspos(square(y0), 2ndspos(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

MARK(times(X1, X2)) → MARK(X2)

A__PI(square(x0)) → A__2NDSPOS(a__square(mark(x0)), a__from(0))

MARK(2ndsneg(X1, X2)) → MARK(X1)

A__PI(2ndsneg(x0, x1)) → A__2NDSPOS(a__2ndsneg(mark(x0), mark(x1)), a__from(0))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(Y)

MARK(plus(pi(x0), y1)) → A__PLUS(a__pi(mark(x0)), mark(y1))

MARK(times(pi(x0), y1)) → A__TIMES(a__pi(mark(x0)), mark(y1))

MARK(rcons(X1, X2)) → MARK(X1)

MARK(2ndspos(s(x0), y1)) → A__2NDSPOS(s(mark(x0)), mark(y1))

MARK(2ndspos(from(x0), y1)) → A__2NDSPOS(a__from(mark(x0)), mark(y1))

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(Z)

MARK(square(X)) → MARK(X)

A__PI(s(x0)) → A__2NDSPOS(s(mark(x0)), a__from(0))

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(Y)

A__FROM(X) → MARK(X)

MARK(times(plus(x0, x1), y1)) → A__TIMES(a__plus(mark(x0), mark(x1)), mark(y1))

A__2NDSPOS(s(y0), cons(y1, from(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__from(mark(x0))))

MARK(2ndspos(times(x0, x1), y1)) → A__2NDSPOS(a__times(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(square(y0), pi(x0))) → A__2NDSPOS(a__square(mark(y0)), a__pi(mark(x0)))

MARK(rcons(X1, X2)) → MARK(X2)

MARK(times(2ndspos(x0, x1), y1)) → A__TIMES(a__2ndspos(mark(x0), mark(x1)), mark(y1))

A__TIMES(s(X), Y) → A__TIMES(mark(X), mark(Y))

A__2NDSPOS(s(y0), cons(y1, cons(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, cons(mark(x0), x1)))

A__PLUS(s(X), Y) → MARK(X)

MARK(times(X1, X2)) → MARK(X1)

MARK(plus(times(x0, x1), y1)) → A__PLUS(a__times(mark(x0), mark(x1)), mark(y1))

A__SQUARE(X) → MARK(X)

MARK(2ndsneg(times(x0, x1), y1)) → A__2NDSNEG(a__times(mark(x0), mark(x1)), mark(y1))

A__2NDSPOS(s(y0), cons(y1, 2ndsneg(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__2ndsneg(mark(x0), mark(x1))))

A__2NDSPOS(s(N), cons(X, Z)) → MARK(N)

MARK(negrecip(X)) → MARK(X)

A__PLUS(0, Y) → MARK(Y)

MARK(2ndspos(2ndspos(x0, x1), y1)) → A__2NDSPOS(a__2ndspos(mark(x0), mark(x1)), mark(y1))

MARK(posrecip(X)) → MARK(X)

MARK(2ndsneg(pi(x0), y1)) → A__2NDSNEG(a__pi(mark(x0)), mark(y1))

MARK(2ndspos(square(y0), from(x0))) → A__2NDSPOS(a__square(mark(y0)), a__from(mark(x0)))

A__2NDSPOS(s(y0), cons(y1, square(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__square(mark(x0))))

MARK(plus(0, y1)) → A__PLUS(0, mark(y1))

MARK(from(X)) → MARK(X)

MARK(2ndspos(2ndsneg(x0, x1), y1)) → A__2NDSPOS(a__2ndsneg(mark(x0), mark(x1)), mark(y1))

A__PLUS(s(X), Y) → A__PLUS(mark(X), mark(Y))

A__SQUARE(X) → A__TIMES(mark(X), mark(X))

MARK(2ndspos(square(y0), square(x0))) → A__2NDSPOS(a__square(mark(y0)), a__square(mark(x0)))

A__PI(pi(x0)) → A__2NDSPOS(a__pi(mark(x0)), a__from(0))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(Z)

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → A__2NDSPOS(mark(N), mark(Z))

MARK(plus(from(x0), y1)) → A__PLUS(a__from(mark(x0)), mark(y1))

MARK(2ndspos(pi(x0), y1)) → A__2NDSPOS(a__pi(mark(x0)), mark(y1))

MARK(square(X)) → A__SQUARE(mark(X))

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))

a__2ndspos(0, Z) → rnil

a__2ndspos(s(N), cons(X, Z)) → a__2ndspos(s(mark(N)), cons2(X, mark(Z)))

a__2ndspos(s(N), cons2(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))

a__2ndsneg(0, Z) → rnil

a__2ndsneg(s(N), cons(X, Z)) → a__2ndsneg(s(mark(N)), cons2(X, mark(Z)))

a__2ndsneg(s(N), cons2(X, cons(Y, Z))) → rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z)))

a__pi(X) → a__2ndspos(mark(X), a__from(0))

a__plus(0, Y) → mark(Y)

a__plus(s(X), Y) → s(a__plus(mark(X), mark(Y)))

a__times(0, Y) → 0

a__times(s(X), Y) → a__plus(mark(Y), a__times(mark(X), mark(Y)))

a__square(X) → a__times(mark(X), mark(X))

mark(from(X)) → a__from(mark(X))

mark(2ndspos(X1, X2)) → a__2ndspos(mark(X1), mark(X2))

mark(2ndsneg(X1, X2)) → a__2ndsneg(mark(X1), mark(X2))

mark(pi(X)) → a__pi(mark(X))

mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))

mark(times(X1, X2)) → a__times(mark(X1), mark(X2))

mark(square(X)) → a__square(mark(X))

mark(0) → 0

mark(s(X)) → s(mark(X))

mark(posrecip(X)) → posrecip(mark(X))

mark(negrecip(X)) → negrecip(mark(X))

mark(nil) → nil

mark(cons(X1, X2)) → cons(mark(X1), X2)

mark(cons2(X1, X2)) → cons2(X1, mark(X2))

mark(rnil) → rnil

mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))

a__from(X) → from(X)

a__2ndspos(X1, X2) → 2ndspos(X1, X2)

a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)

a__pi(X) → pi(X)

a__plus(X1, X2) → plus(X1, X2)

a__times(X1, X2) → times(X1, X2)

a__square(X) → square(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule MARK(2ndspos(from(x0), y1)) → A__2NDSPOS(a__from(mark(x0)), mark(y1)) at position [1] we obtained the following new rules:

MARK(2ndspos(from(y0), posrecip(x0))) → A__2NDSPOS(a__from(mark(y0)), posrecip(mark(x0)))

MARK(2ndspos(from(y0), nil)) → A__2NDSPOS(a__from(mark(y0)), nil)

MARK(2ndspos(from(y0), pi(x0))) → A__2NDSPOS(a__from(mark(y0)), a__pi(mark(x0)))

MARK(2ndspos(from(y0), from(x0))) → A__2NDSPOS(a__from(mark(y0)), a__from(mark(x0)))

MARK(2ndspos(from(y0), rcons(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), rcons(mark(x0), mark(x1)))

MARK(2ndspos(from(y0), cons2(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), cons2(x0, mark(x1)))

MARK(2ndspos(from(y0), s(x0))) → A__2NDSPOS(a__from(mark(y0)), s(mark(x0)))

MARK(2ndspos(from(y0), plus(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(from(y0), 2ndsneg(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndspos(from(y0), cons(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), cons(mark(x0), x1))

MARK(2ndspos(from(y0), negrecip(x0))) → A__2NDSPOS(a__from(mark(y0)), negrecip(mark(x0)))

MARK(2ndspos(from(y0), square(x0))) → A__2NDSPOS(a__from(mark(y0)), a__square(mark(x0)))

MARK(2ndspos(from(y0), 0)) → A__2NDSPOS(a__from(mark(y0)), 0)

MARK(2ndspos(from(y0), times(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(from(y0), rnil)) → A__2NDSPOS(a__from(mark(y0)), rnil)

MARK(2ndspos(from(y0), 2ndspos(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), a__2ndspos(mark(x0), mark(x1)))



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ Narrowing
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ QDP
              ↳ Narrowing
                ↳ QDP
                  ↳ DependencyGraphProof
                    ↳ QDP
                      ↳ Narrowing
                        ↳ QDP
                          ↳ DependencyGraphProof
                            ↳ QDP
                              ↳ Narrowing
                                ↳ QDP
                                  ↳ DependencyGraphProof
                                    ↳ QDP
                                      ↳ Narrowing
                                        ↳ QDP
                                          ↳ DependencyGraphProof
                                            ↳ QDP
                                              ↳ Narrowing
                                                ↳ QDP
                                                  ↳ DependencyGraphProof
                                                    ↳ QDP
                                                      ↳ Narrowing
                                                        ↳ QDP
                                                          ↳ DependencyGraphProof
                                                            ↳ QDP
                                                              ↳ Narrowing
                                                                ↳ QDP
                                                                  ↳ DependencyGraphProof

Q DP problem:
The TRS P consists of the following rules:

MARK(2ndspos(square(y0), plus(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(2ndsneg(square(x0), y1)) → A__2NDSNEG(a__square(mark(x0)), mark(y1))

A__PI(times(x0, x1)) → A__2NDSPOS(a__times(mark(x0), mark(x1)), a__from(0))

A__2NDSPOS(s(y0), cons(y1, 2ndspos(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__2ndspos(mark(x0), mark(x1))))

A__2NDSNEG(s(N), cons(X, Z)) → A__2NDSNEG(s(mark(N)), cons2(X, mark(Z)))

MARK(times(times(x0, x1), y1)) → A__TIMES(a__times(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(X1, X2)) → MARK(X2)

MARK(2ndspos(from(y0), cons2(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), cons2(x0, mark(x1)))

MARK(plus(plus(x0, x1), y1)) → A__PLUS(a__plus(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(from(y0), 2ndsneg(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndspos(from(y0), s(x0))) → A__2NDSPOS(a__from(mark(y0)), s(mark(x0)))

MARK(2ndsneg(2ndsneg(x0, x1), y1)) → A__2NDSNEG(a__2ndsneg(mark(x0), mark(x1)), mark(y1))

A__2NDSNEG(s(N), cons(X, Z)) → MARK(Z)

MARK(2ndspos(from(y0), 0)) → A__2NDSPOS(a__from(mark(y0)), 0)

A__TIMES(s(X), Y) → MARK(Y)

MARK(2ndsneg(2ndspos(x0, x1), y1)) → A__2NDSNEG(a__2ndspos(mark(x0), mark(x1)), mark(y1))

A__2NDSPOS(s(y0), cons(y1, plus(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__plus(mark(x0), mark(x1))))

MARK(2ndspos(from(y0), pi(x0))) → A__2NDSPOS(a__from(mark(y0)), a__pi(mark(x0)))

MARK(2ndspos(square(y0), 2ndsneg(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

A__2NDSNEG(s(N), cons(X, Z)) → MARK(N)

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → A__2NDSNEG(mark(N), mark(Z))

MARK(times(2ndsneg(x0, x1), y1)) → A__TIMES(a__2ndsneg(mark(x0), mark(x1)), mark(y1))

MARK(2ndsneg(plus(x0, x1), y1)) → A__2NDSNEG(a__plus(mark(x0), mark(x1)), mark(y1))

MARK(2ndsneg(from(x0), y1)) → A__2NDSNEG(a__from(mark(x0)), mark(y1))

MARK(2ndspos(from(y0), from(x0))) → A__2NDSPOS(a__from(mark(y0)), a__from(mark(x0)))

A__TIMES(s(X), Y) → A__PLUS(mark(Y), a__times(mark(X), mark(Y)))

MARK(2ndspos(square(y0), cons2(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), cons2(x0, mark(x1)))

A__PI(2ndspos(x0, x1)) → A__2NDSPOS(a__2ndspos(mark(x0), mark(x1)), a__from(0))

MARK(plus(square(x0), y1)) → A__PLUS(a__square(mark(x0)), mark(y1))

MARK(plus(s(x0), y1)) → A__PLUS(s(mark(x0)), mark(y1))

MARK(2ndspos(plus(x0, x1), y1)) → A__2NDSPOS(a__plus(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(square(y0), cons(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), cons(mark(x0), x1))

A__PI(plus(x0, x1)) → A__2NDSPOS(a__plus(mark(x0), mark(x1)), a__from(0))

MARK(cons2(X1, X2)) → MARK(X2)

MARK(s(X)) → MARK(X)

MARK(2ndspos(X1, X2)) → MARK(X1)

MARK(pi(X)) → MARK(X)

MARK(plus(X1, X2)) → MARK(X2)

MARK(2ndspos(from(y0), rcons(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), rcons(mark(x0), mark(x1)))

A__2NDSPOS(s(N), cons(X, Z)) → MARK(Z)

MARK(pi(X)) → A__PI(mark(X))

MARK(plus(2ndsneg(x0, x1), y1)) → A__PLUS(a__2ndsneg(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(from(y0), negrecip(x0))) → A__2NDSPOS(a__from(mark(y0)), negrecip(mark(x0)))

MARK(2ndsneg(X1, X2)) → MARK(X2)

MARK(2ndspos(square(y0), times(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(times(square(x0), y1)) → A__TIMES(a__square(mark(x0)), mark(y1))

A__PI(X) → A__FROM(0)

MARK(from(X)) → A__FROM(mark(X))

MARK(times(from(x0), y1)) → A__TIMES(a__from(mark(x0)), mark(y1))

MARK(plus(X1, X2)) → MARK(X1)

A__2NDSPOS(s(y0), cons(y1, times(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__times(mark(x0), mark(x1))))

A__PI(X) → MARK(X)

MARK(times(s(x0), y1)) → A__TIMES(s(mark(x0)), mark(y1))

MARK(2ndspos(from(y0), cons(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), cons(mark(x0), x1))

MARK(2ndsneg(s(x0), y1)) → A__2NDSNEG(s(mark(x0)), mark(y1))

A__PLUS(s(X), Y) → MARK(Y)

A__TIMES(s(X), Y) → MARK(X)

A__2NDSPOS(s(y0), cons(y1, pi(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__pi(mark(x0))))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(N)

MARK(plus(2ndspos(x0, x1), y1)) → A__PLUS(a__2ndspos(mark(x0), mark(x1)), mark(y1))

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(N)

MARK(cons(X1, X2)) → MARK(X1)

A__PI(from(x0)) → A__2NDSPOS(a__from(mark(x0)), a__from(0))

MARK(2ndspos(square(y0), 2ndspos(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

MARK(times(X1, X2)) → MARK(X2)

A__PI(square(x0)) → A__2NDSPOS(a__square(mark(x0)), a__from(0))

MARK(2ndsneg(X1, X2)) → MARK(X1)

A__PI(2ndsneg(x0, x1)) → A__2NDSPOS(a__2ndsneg(mark(x0), mark(x1)), a__from(0))

MARK(2ndspos(from(y0), rnil)) → A__2NDSPOS(a__from(mark(y0)), rnil)

MARK(times(pi(x0), y1)) → A__TIMES(a__pi(mark(x0)), mark(y1))

MARK(plus(pi(x0), y1)) → A__PLUS(a__pi(mark(x0)), mark(y1))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(Y)

MARK(rcons(X1, X2)) → MARK(X1)

MARK(2ndspos(s(x0), y1)) → A__2NDSPOS(s(mark(x0)), mark(y1))

MARK(2ndspos(from(y0), nil)) → A__2NDSPOS(a__from(mark(y0)), nil)

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(Z)

MARK(square(X)) → MARK(X)

A__PI(s(x0)) → A__2NDSPOS(s(mark(x0)), a__from(0))

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(Y)

A__FROM(X) → MARK(X)

MARK(times(plus(x0, x1), y1)) → A__TIMES(a__plus(mark(x0), mark(x1)), mark(y1))

A__2NDSPOS(s(y0), cons(y1, from(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__from(mark(x0))))

MARK(2ndspos(times(x0, x1), y1)) → A__2NDSPOS(a__times(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(square(y0), pi(x0))) → A__2NDSPOS(a__square(mark(y0)), a__pi(mark(x0)))

MARK(rcons(X1, X2)) → MARK(X2)

MARK(times(2ndspos(x0, x1), y1)) → A__TIMES(a__2ndspos(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(from(y0), posrecip(x0))) → A__2NDSPOS(a__from(mark(y0)), posrecip(mark(x0)))

A__TIMES(s(X), Y) → A__TIMES(mark(X), mark(Y))

A__PLUS(s(X), Y) → MARK(X)

A__2NDSPOS(s(y0), cons(y1, cons(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, cons(mark(x0), x1)))

MARK(times(X1, X2)) → MARK(X1)

MARK(plus(times(x0, x1), y1)) → A__PLUS(a__times(mark(x0), mark(x1)), mark(y1))

A__SQUARE(X) → MARK(X)

MARK(2ndsneg(times(x0, x1), y1)) → A__2NDSNEG(a__times(mark(x0), mark(x1)), mark(y1))

A__2NDSPOS(s(y0), cons(y1, 2ndsneg(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__2ndsneg(mark(x0), mark(x1))))

MARK(negrecip(X)) → MARK(X)

A__2NDSPOS(s(N), cons(X, Z)) → MARK(N)

A__PLUS(0, Y) → MARK(Y)

MARK(2ndspos(2ndspos(x0, x1), y1)) → A__2NDSPOS(a__2ndspos(mark(x0), mark(x1)), mark(y1))

MARK(posrecip(X)) → MARK(X)

MARK(2ndspos(from(y0), square(x0))) → A__2NDSPOS(a__from(mark(y0)), a__square(mark(x0)))

MARK(2ndspos(square(y0), from(x0))) → A__2NDSPOS(a__square(mark(y0)), a__from(mark(x0)))

MARK(2ndsneg(pi(x0), y1)) → A__2NDSNEG(a__pi(mark(x0)), mark(y1))

MARK(2ndspos(from(y0), times(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), a__times(mark(x0), mark(x1)))

A__2NDSPOS(s(y0), cons(y1, square(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__square(mark(x0))))

MARK(plus(0, y1)) → A__PLUS(0, mark(y1))

MARK(from(X)) → MARK(X)

MARK(2ndspos(2ndsneg(x0, x1), y1)) → A__2NDSPOS(a__2ndsneg(mark(x0), mark(x1)), mark(y1))

A__PLUS(s(X), Y) → A__PLUS(mark(X), mark(Y))

MARK(2ndspos(square(y0), square(x0))) → A__2NDSPOS(a__square(mark(y0)), a__square(mark(x0)))

A__SQUARE(X) → A__TIMES(mark(X), mark(X))

A__PI(pi(x0)) → A__2NDSPOS(a__pi(mark(x0)), a__from(0))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(Z)

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → A__2NDSPOS(mark(N), mark(Z))

MARK(plus(from(x0), y1)) → A__PLUS(a__from(mark(x0)), mark(y1))

MARK(square(X)) → A__SQUARE(mark(X))

MARK(2ndspos(pi(x0), y1)) → A__2NDSPOS(a__pi(mark(x0)), mark(y1))

MARK(2ndspos(from(y0), plus(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(from(y0), 2ndspos(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))

a__2ndspos(0, Z) → rnil

a__2ndspos(s(N), cons(X, Z)) → a__2ndspos(s(mark(N)), cons2(X, mark(Z)))

a__2ndspos(s(N), cons2(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))

a__2ndsneg(0, Z) → rnil

a__2ndsneg(s(N), cons(X, Z)) → a__2ndsneg(s(mark(N)), cons2(X, mark(Z)))

a__2ndsneg(s(N), cons2(X, cons(Y, Z))) → rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z)))

a__pi(X) → a__2ndspos(mark(X), a__from(0))

a__plus(0, Y) → mark(Y)

a__plus(s(X), Y) → s(a__plus(mark(X), mark(Y)))

a__times(0, Y) → 0

a__times(s(X), Y) → a__plus(mark(Y), a__times(mark(X), mark(Y)))

a__square(X) → a__times(mark(X), mark(X))

mark(from(X)) → a__from(mark(X))

mark(2ndspos(X1, X2)) → a__2ndspos(mark(X1), mark(X2))

mark(2ndsneg(X1, X2)) → a__2ndsneg(mark(X1), mark(X2))

mark(pi(X)) → a__pi(mark(X))

mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))

mark(times(X1, X2)) → a__times(mark(X1), mark(X2))

mark(square(X)) → a__square(mark(X))

mark(0) → 0

mark(s(X)) → s(mark(X))

mark(posrecip(X)) → posrecip(mark(X))

mark(negrecip(X)) → negrecip(mark(X))

mark(nil) → nil

mark(cons(X1, X2)) → cons(mark(X1), X2)

mark(cons2(X1, X2)) → cons2(X1, mark(X2))

mark(rnil) → rnil

mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))

a__from(X) → from(X)

a__2ndspos(X1, X2) → 2ndspos(X1, X2)

a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)

a__pi(X) → pi(X)

a__plus(X1, X2) → plus(X1, X2)

a__times(X1, X2) → times(X1, X2)

a__square(X) → square(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 7 less nodes.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ Narrowing
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ QDP
              ↳ Narrowing
                ↳ QDP
                  ↳ DependencyGraphProof
                    ↳ QDP
                      ↳ Narrowing
                        ↳ QDP
                          ↳ DependencyGraphProof
                            ↳ QDP
                              ↳ Narrowing
                                ↳ QDP
                                  ↳ DependencyGraphProof
                                    ↳ QDP
                                      ↳ Narrowing
                                        ↳ QDP
                                          ↳ DependencyGraphProof
                                            ↳ QDP
                                              ↳ Narrowing
                                                ↳ QDP
                                                  ↳ DependencyGraphProof
                                                    ↳ QDP
                                                      ↳ Narrowing
                                                        ↳ QDP
                                                          ↳ DependencyGraphProof
                                                            ↳ QDP
                                                              ↳ Narrowing
                                                                ↳ QDP
                                                                  ↳ DependencyGraphProof
                                                                    ↳ QDP
                                                                      ↳ Narrowing

Q DP problem:
The TRS P consists of the following rules:

MARK(2ndsneg(square(x0), y1)) → A__2NDSNEG(a__square(mark(x0)), mark(y1))

MARK(2ndspos(square(y0), plus(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__plus(mark(x0), mark(x1)))

A__PI(times(x0, x1)) → A__2NDSPOS(a__times(mark(x0), mark(x1)), a__from(0))

A__2NDSPOS(s(y0), cons(y1, 2ndspos(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__2ndspos(mark(x0), mark(x1))))

A__2NDSNEG(s(N), cons(X, Z)) → A__2NDSNEG(s(mark(N)), cons2(X, mark(Z)))

MARK(2ndspos(X1, X2)) → MARK(X2)

MARK(times(times(x0, x1), y1)) → A__TIMES(a__times(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(from(y0), cons2(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), cons2(x0, mark(x1)))

MARK(plus(plus(x0, x1), y1)) → A__PLUS(a__plus(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(from(y0), 2ndsneg(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndsneg(2ndsneg(x0, x1), y1)) → A__2NDSNEG(a__2ndsneg(mark(x0), mark(x1)), mark(y1))

A__2NDSNEG(s(N), cons(X, Z)) → MARK(Z)

A__TIMES(s(X), Y) → MARK(Y)

MARK(2ndsneg(2ndspos(x0, x1), y1)) → A__2NDSNEG(a__2ndspos(mark(x0), mark(x1)), mark(y1))

A__2NDSPOS(s(y0), cons(y1, plus(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__plus(mark(x0), mark(x1))))

MARK(2ndspos(square(y0), 2ndsneg(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndspos(from(y0), pi(x0))) → A__2NDSPOS(a__from(mark(y0)), a__pi(mark(x0)))

A__2NDSNEG(s(N), cons(X, Z)) → MARK(N)

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → A__2NDSNEG(mark(N), mark(Z))

MARK(2ndsneg(plus(x0, x1), y1)) → A__2NDSNEG(a__plus(mark(x0), mark(x1)), mark(y1))

MARK(times(2ndsneg(x0, x1), y1)) → A__TIMES(a__2ndsneg(mark(x0), mark(x1)), mark(y1))

MARK(2ndsneg(from(x0), y1)) → A__2NDSNEG(a__from(mark(x0)), mark(y1))

MARK(2ndspos(from(y0), from(x0))) → A__2NDSPOS(a__from(mark(y0)), a__from(mark(x0)))

A__TIMES(s(X), Y) → A__PLUS(mark(Y), a__times(mark(X), mark(Y)))

A__PI(2ndspos(x0, x1)) → A__2NDSPOS(a__2ndspos(mark(x0), mark(x1)), a__from(0))

MARK(2ndspos(square(y0), cons2(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), cons2(x0, mark(x1)))

MARK(plus(square(x0), y1)) → A__PLUS(a__square(mark(x0)), mark(y1))

MARK(2ndspos(plus(x0, x1), y1)) → A__2NDSPOS(a__plus(mark(x0), mark(x1)), mark(y1))

MARK(plus(s(x0), y1)) → A__PLUS(s(mark(x0)), mark(y1))

A__PI(plus(x0, x1)) → A__2NDSPOS(a__plus(mark(x0), mark(x1)), a__from(0))

MARK(2ndspos(square(y0), cons(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), cons(mark(x0), x1))

MARK(cons2(X1, X2)) → MARK(X2)

MARK(2ndspos(X1, X2)) → MARK(X1)

MARK(s(X)) → MARK(X)

MARK(pi(X)) → MARK(X)

MARK(plus(X1, X2)) → MARK(X2)

A__2NDSPOS(s(N), cons(X, Z)) → MARK(Z)

MARK(pi(X)) → A__PI(mark(X))

MARK(plus(2ndsneg(x0, x1), y1)) → A__PLUS(a__2ndsneg(mark(x0), mark(x1)), mark(y1))

MARK(2ndsneg(X1, X2)) → MARK(X2)

MARK(2ndspos(square(y0), times(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(times(square(x0), y1)) → A__TIMES(a__square(mark(x0)), mark(y1))

A__PI(X) → A__FROM(0)

MARK(from(X)) → A__FROM(mark(X))

MARK(plus(X1, X2)) → MARK(X1)

MARK(times(from(x0), y1)) → A__TIMES(a__from(mark(x0)), mark(y1))

A__2NDSPOS(s(y0), cons(y1, times(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__times(mark(x0), mark(x1))))

A__PI(X) → MARK(X)

MARK(times(s(x0), y1)) → A__TIMES(s(mark(x0)), mark(y1))

MARK(2ndspos(from(y0), cons(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), cons(mark(x0), x1))

MARK(2ndsneg(s(x0), y1)) → A__2NDSNEG(s(mark(x0)), mark(y1))

A__PLUS(s(X), Y) → MARK(Y)

A__2NDSPOS(s(y0), cons(y1, pi(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__pi(mark(x0))))

A__TIMES(s(X), Y) → MARK(X)

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(N)

MARK(plus(2ndspos(x0, x1), y1)) → A__PLUS(a__2ndspos(mark(x0), mark(x1)), mark(y1))

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(N)

A__PI(from(x0)) → A__2NDSPOS(a__from(mark(x0)), a__from(0))

MARK(cons(X1, X2)) → MARK(X1)

MARK(2ndspos(square(y0), 2ndspos(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

MARK(times(X1, X2)) → MARK(X2)

A__PI(square(x0)) → A__2NDSPOS(a__square(mark(x0)), a__from(0))

MARK(2ndsneg(X1, X2)) → MARK(X1)

A__PI(2ndsneg(x0, x1)) → A__2NDSPOS(a__2ndsneg(mark(x0), mark(x1)), a__from(0))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(Y)

MARK(plus(pi(x0), y1)) → A__PLUS(a__pi(mark(x0)), mark(y1))

MARK(times(pi(x0), y1)) → A__TIMES(a__pi(mark(x0)), mark(y1))

MARK(rcons(X1, X2)) → MARK(X1)

MARK(2ndspos(s(x0), y1)) → A__2NDSPOS(s(mark(x0)), mark(y1))

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(Z)

MARK(square(X)) → MARK(X)

A__PI(s(x0)) → A__2NDSPOS(s(mark(x0)), a__from(0))

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(Y)

A__FROM(X) → MARK(X)

MARK(times(plus(x0, x1), y1)) → A__TIMES(a__plus(mark(x0), mark(x1)), mark(y1))

A__2NDSPOS(s(y0), cons(y1, from(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__from(mark(x0))))

MARK(2ndspos(times(x0, x1), y1)) → A__2NDSPOS(a__times(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(square(y0), pi(x0))) → A__2NDSPOS(a__square(mark(y0)), a__pi(mark(x0)))

MARK(rcons(X1, X2)) → MARK(X2)

MARK(times(2ndspos(x0, x1), y1)) → A__TIMES(a__2ndspos(mark(x0), mark(x1)), mark(y1))

A__TIMES(s(X), Y) → A__TIMES(mark(X), mark(Y))

A__2NDSPOS(s(y0), cons(y1, cons(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, cons(mark(x0), x1)))

A__PLUS(s(X), Y) → MARK(X)

MARK(times(X1, X2)) → MARK(X1)

MARK(plus(times(x0, x1), y1)) → A__PLUS(a__times(mark(x0), mark(x1)), mark(y1))

A__SQUARE(X) → MARK(X)

MARK(2ndsneg(times(x0, x1), y1)) → A__2NDSNEG(a__times(mark(x0), mark(x1)), mark(y1))

A__2NDSPOS(s(y0), cons(y1, 2ndsneg(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__2ndsneg(mark(x0), mark(x1))))

A__2NDSPOS(s(N), cons(X, Z)) → MARK(N)

MARK(negrecip(X)) → MARK(X)

A__PLUS(0, Y) → MARK(Y)

MARK(2ndspos(2ndspos(x0, x1), y1)) → A__2NDSPOS(a__2ndspos(mark(x0), mark(x1)), mark(y1))

MARK(posrecip(X)) → MARK(X)

MARK(2ndspos(from(y0), square(x0))) → A__2NDSPOS(a__from(mark(y0)), a__square(mark(x0)))

MARK(2ndsneg(pi(x0), y1)) → A__2NDSNEG(a__pi(mark(x0)), mark(y1))

MARK(2ndspos(square(y0), from(x0))) → A__2NDSPOS(a__square(mark(y0)), a__from(mark(x0)))

MARK(2ndspos(from(y0), times(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), a__times(mark(x0), mark(x1)))

A__2NDSPOS(s(y0), cons(y1, square(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__square(mark(x0))))

MARK(plus(0, y1)) → A__PLUS(0, mark(y1))

MARK(from(X)) → MARK(X)

MARK(2ndspos(2ndsneg(x0, x1), y1)) → A__2NDSPOS(a__2ndsneg(mark(x0), mark(x1)), mark(y1))

A__PLUS(s(X), Y) → A__PLUS(mark(X), mark(Y))

A__SQUARE(X) → A__TIMES(mark(X), mark(X))

MARK(2ndspos(square(y0), square(x0))) → A__2NDSPOS(a__square(mark(y0)), a__square(mark(x0)))

A__PI(pi(x0)) → A__2NDSPOS(a__pi(mark(x0)), a__from(0))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(Z)

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → A__2NDSPOS(mark(N), mark(Z))

MARK(plus(from(x0), y1)) → A__PLUS(a__from(mark(x0)), mark(y1))

MARK(2ndspos(pi(x0), y1)) → A__2NDSPOS(a__pi(mark(x0)), mark(y1))

MARK(square(X)) → A__SQUARE(mark(X))

MARK(2ndspos(from(y0), plus(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(from(y0), 2ndspos(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))

a__2ndspos(0, Z) → rnil

a__2ndspos(s(N), cons(X, Z)) → a__2ndspos(s(mark(N)), cons2(X, mark(Z)))

a__2ndspos(s(N), cons2(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))

a__2ndsneg(0, Z) → rnil

a__2ndsneg(s(N), cons(X, Z)) → a__2ndsneg(s(mark(N)), cons2(X, mark(Z)))

a__2ndsneg(s(N), cons2(X, cons(Y, Z))) → rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z)))

a__pi(X) → a__2ndspos(mark(X), a__from(0))

a__plus(0, Y) → mark(Y)

a__plus(s(X), Y) → s(a__plus(mark(X), mark(Y)))

a__times(0, Y) → 0

a__times(s(X), Y) → a__plus(mark(Y), a__times(mark(X), mark(Y)))

a__square(X) → a__times(mark(X), mark(X))

mark(from(X)) → a__from(mark(X))

mark(2ndspos(X1, X2)) → a__2ndspos(mark(X1), mark(X2))

mark(2ndsneg(X1, X2)) → a__2ndsneg(mark(X1), mark(X2))

mark(pi(X)) → a__pi(mark(X))

mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))

mark(times(X1, X2)) → a__times(mark(X1), mark(X2))

mark(square(X)) → a__square(mark(X))

mark(0) → 0

mark(s(X)) → s(mark(X))

mark(posrecip(X)) → posrecip(mark(X))

mark(negrecip(X)) → negrecip(mark(X))

mark(nil) → nil

mark(cons(X1, X2)) → cons(mark(X1), X2)

mark(cons2(X1, X2)) → cons2(X1, mark(X2))

mark(rnil) → rnil

mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))

a__from(X) → from(X)

a__2ndspos(X1, X2) → 2ndspos(X1, X2)

a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)

a__pi(X) → pi(X)

a__plus(X1, X2) → plus(X1, X2)

a__times(X1, X2) → times(X1, X2)

a__square(X) → square(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule MARK(2ndspos(s(x0), y1)) → A__2NDSPOS(s(mark(x0)), mark(y1)) at position [1] we obtained the following new rules:

MARK(2ndspos(s(y0), rnil)) → A__2NDSPOS(s(mark(y0)), rnil)

MARK(2ndspos(s(y0), cons2(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(x0, mark(x1)))

MARK(2ndspos(s(y0), plus(x0, x1))) → A__2NDSPOS(s(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(s(y0), negrecip(x0))) → A__2NDSPOS(s(mark(y0)), negrecip(mark(x0)))

MARK(2ndspos(s(y0), 2ndspos(x0, x1))) → A__2NDSPOS(s(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndspos(s(y0), cons(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons(mark(x0), x1))

MARK(2ndspos(s(y0), from(x0))) → A__2NDSPOS(s(mark(y0)), a__from(mark(x0)))

MARK(2ndspos(s(y0), times(x0, x1))) → A__2NDSPOS(s(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(s(y0), posrecip(x0))) → A__2NDSPOS(s(mark(y0)), posrecip(mark(x0)))

MARK(2ndspos(s(y0), 2ndsneg(x0, x1))) → A__2NDSPOS(s(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndspos(s(y0), pi(x0))) → A__2NDSPOS(s(mark(y0)), a__pi(mark(x0)))

MARK(2ndspos(s(y0), 0)) → A__2NDSPOS(s(mark(y0)), 0)

MARK(2ndspos(s(y0), rcons(x0, x1))) → A__2NDSPOS(s(mark(y0)), rcons(mark(x0), mark(x1)))

MARK(2ndspos(s(y0), square(x0))) → A__2NDSPOS(s(mark(y0)), a__square(mark(x0)))

MARK(2ndspos(s(y0), s(x0))) → A__2NDSPOS(s(mark(y0)), s(mark(x0)))

MARK(2ndspos(s(y0), nil)) → A__2NDSPOS(s(mark(y0)), nil)



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ Narrowing
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ QDP
              ↳ Narrowing
                ↳ QDP
                  ↳ DependencyGraphProof
                    ↳ QDP
                      ↳ Narrowing
                        ↳ QDP
                          ↳ DependencyGraphProof
                            ↳ QDP
                              ↳ Narrowing
                                ↳ QDP
                                  ↳ DependencyGraphProof
                                    ↳ QDP
                                      ↳ Narrowing
                                        ↳ QDP
                                          ↳ DependencyGraphProof
                                            ↳ QDP
                                              ↳ Narrowing
                                                ↳ QDP
                                                  ↳ DependencyGraphProof
                                                    ↳ QDP
                                                      ↳ Narrowing
                                                        ↳ QDP
                                                          ↳ DependencyGraphProof
                                                            ↳ QDP
                                                              ↳ Narrowing
                                                                ↳ QDP
                                                                  ↳ DependencyGraphProof
                                                                    ↳ QDP
                                                                      ↳ Narrowing
                                                                        ↳ QDP
                                                                          ↳ DependencyGraphProof

Q DP problem:
The TRS P consists of the following rules:

MARK(2ndspos(square(y0), plus(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(2ndsneg(square(x0), y1)) → A__2NDSNEG(a__square(mark(x0)), mark(y1))

A__2NDSPOS(s(y0), cons(y1, 2ndspos(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__2ndspos(mark(x0), mark(x1))))

A__PI(times(x0, x1)) → A__2NDSPOS(a__times(mark(x0), mark(x1)), a__from(0))

A__2NDSNEG(s(N), cons(X, Z)) → A__2NDSNEG(s(mark(N)), cons2(X, mark(Z)))

MARK(2ndspos(s(y0), negrecip(x0))) → A__2NDSPOS(s(mark(y0)), negrecip(mark(x0)))

MARK(times(times(x0, x1), y1)) → A__TIMES(a__times(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(X1, X2)) → MARK(X2)

MARK(2ndspos(from(y0), cons2(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), cons2(x0, mark(x1)))

MARK(plus(plus(x0, x1), y1)) → A__PLUS(a__plus(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(s(y0), from(x0))) → A__2NDSPOS(s(mark(y0)), a__from(mark(x0)))

MARK(2ndspos(from(y0), 2ndsneg(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndsneg(2ndsneg(x0, x1), y1)) → A__2NDSNEG(a__2ndsneg(mark(x0), mark(x1)), mark(y1))

A__2NDSNEG(s(N), cons(X, Z)) → MARK(Z)

MARK(2ndspos(s(y0), s(x0))) → A__2NDSPOS(s(mark(y0)), s(mark(x0)))

MARK(2ndspos(s(y0), rnil)) → A__2NDSPOS(s(mark(y0)), rnil)

A__TIMES(s(X), Y) → MARK(Y)

MARK(2ndsneg(2ndspos(x0, x1), y1)) → A__2NDSNEG(a__2ndspos(mark(x0), mark(x1)), mark(y1))

A__2NDSPOS(s(y0), cons(y1, plus(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__plus(mark(x0), mark(x1))))

MARK(2ndspos(from(y0), pi(x0))) → A__2NDSPOS(a__from(mark(y0)), a__pi(mark(x0)))

MARK(2ndspos(square(y0), 2ndsneg(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → A__2NDSNEG(mark(N), mark(Z))

A__2NDSNEG(s(N), cons(X, Z)) → MARK(N)

MARK(times(2ndsneg(x0, x1), y1)) → A__TIMES(a__2ndsneg(mark(x0), mark(x1)), mark(y1))

MARK(2ndsneg(plus(x0, x1), y1)) → A__2NDSNEG(a__plus(mark(x0), mark(x1)), mark(y1))

MARK(2ndsneg(from(x0), y1)) → A__2NDSNEG(a__from(mark(x0)), mark(y1))

MARK(2ndspos(from(y0), from(x0))) → A__2NDSPOS(a__from(mark(y0)), a__from(mark(x0)))

MARK(2ndspos(s(y0), pi(x0))) → A__2NDSPOS(s(mark(y0)), a__pi(mark(x0)))

A__TIMES(s(X), Y) → A__PLUS(mark(Y), a__times(mark(X), mark(Y)))

MARK(2ndspos(square(y0), cons2(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), cons2(x0, mark(x1)))

A__PI(2ndspos(x0, x1)) → A__2NDSPOS(a__2ndspos(mark(x0), mark(x1)), a__from(0))

MARK(plus(square(x0), y1)) → A__PLUS(a__square(mark(x0)), mark(y1))

MARK(plus(s(x0), y1)) → A__PLUS(s(mark(x0)), mark(y1))

MARK(2ndspos(plus(x0, x1), y1)) → A__2NDSPOS(a__plus(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(square(y0), cons(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), cons(mark(x0), x1))

A__PI(plus(x0, x1)) → A__2NDSPOS(a__plus(mark(x0), mark(x1)), a__from(0))

MARK(cons2(X1, X2)) → MARK(X2)

MARK(s(X)) → MARK(X)

MARK(2ndspos(X1, X2)) → MARK(X1)

MARK(pi(X)) → MARK(X)

MARK(plus(X1, X2)) → MARK(X2)

A__2NDSPOS(s(N), cons(X, Z)) → MARK(Z)

MARK(pi(X)) → A__PI(mark(X))

MARK(plus(2ndsneg(x0, x1), y1)) → A__PLUS(a__2ndsneg(mark(x0), mark(x1)), mark(y1))

MARK(2ndsneg(X1, X2)) → MARK(X2)

MARK(2ndspos(square(y0), times(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(times(square(x0), y1)) → A__TIMES(a__square(mark(x0)), mark(y1))

MARK(2ndspos(s(y0), plus(x0, x1))) → A__2NDSPOS(s(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(from(X)) → A__FROM(mark(X))

A__PI(X) → A__FROM(0)

MARK(times(from(x0), y1)) → A__TIMES(a__from(mark(x0)), mark(y1))

MARK(plus(X1, X2)) → MARK(X1)

A__2NDSPOS(s(y0), cons(y1, times(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__times(mark(x0), mark(x1))))

MARK(2ndspos(s(y0), cons(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons(mark(x0), x1))

A__PI(X) → MARK(X)

MARK(times(s(x0), y1)) → A__TIMES(s(mark(x0)), mark(y1))

MARK(2ndsneg(s(x0), y1)) → A__2NDSNEG(s(mark(x0)), mark(y1))

MARK(2ndspos(from(y0), cons(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), cons(mark(x0), x1))

MARK(2ndspos(s(y0), rcons(x0, x1))) → A__2NDSPOS(s(mark(y0)), rcons(mark(x0), mark(x1)))

A__PLUS(s(X), Y) → MARK(Y)

A__TIMES(s(X), Y) → MARK(X)

A__2NDSPOS(s(y0), cons(y1, pi(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__pi(mark(x0))))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(N)

MARK(plus(2ndspos(x0, x1), y1)) → A__PLUS(a__2ndspos(mark(x0), mark(x1)), mark(y1))

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(N)

MARK(cons(X1, X2)) → MARK(X1)

A__PI(from(x0)) → A__2NDSPOS(a__from(mark(x0)), a__from(0))

MARK(2ndspos(square(y0), 2ndspos(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndspos(s(y0), 2ndspos(x0, x1))) → A__2NDSPOS(s(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

MARK(times(X1, X2)) → MARK(X2)

A__PI(square(x0)) → A__2NDSPOS(a__square(mark(x0)), a__from(0))

MARK(2ndspos(s(y0), times(x0, x1))) → A__2NDSPOS(s(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndsneg(X1, X2)) → MARK(X1)

MARK(2ndspos(s(y0), 2ndsneg(x0, x1))) → A__2NDSPOS(s(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

A__PI(2ndsneg(x0, x1)) → A__2NDSPOS(a__2ndsneg(mark(x0), mark(x1)), a__from(0))

MARK(times(pi(x0), y1)) → A__TIMES(a__pi(mark(x0)), mark(y1))

MARK(plus(pi(x0), y1)) → A__PLUS(a__pi(mark(x0)), mark(y1))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(Y)

MARK(2ndspos(s(y0), nil)) → A__2NDSPOS(s(mark(y0)), nil)

MARK(rcons(X1, X2)) → MARK(X1)

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(Z)

MARK(square(X)) → MARK(X)

A__PI(s(x0)) → A__2NDSPOS(s(mark(x0)), a__from(0))

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(Y)

A__FROM(X) → MARK(X)

MARK(times(plus(x0, x1), y1)) → A__TIMES(a__plus(mark(x0), mark(x1)), mark(y1))

A__2NDSPOS(s(y0), cons(y1, from(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__from(mark(x0))))

MARK(2ndspos(times(x0, x1), y1)) → A__2NDSPOS(a__times(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(square(y0), pi(x0))) → A__2NDSPOS(a__square(mark(y0)), a__pi(mark(x0)))

MARK(2ndspos(s(y0), square(x0))) → A__2NDSPOS(s(mark(y0)), a__square(mark(x0)))

MARK(rcons(X1, X2)) → MARK(X2)

MARK(times(2ndspos(x0, x1), y1)) → A__TIMES(a__2ndspos(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(s(y0), cons2(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(x0, mark(x1)))

A__TIMES(s(X), Y) → A__TIMES(mark(X), mark(Y))

A__PLUS(s(X), Y) → MARK(X)

A__2NDSPOS(s(y0), cons(y1, cons(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, cons(mark(x0), x1)))

MARK(times(X1, X2)) → MARK(X1)

MARK(plus(times(x0, x1), y1)) → A__PLUS(a__times(mark(x0), mark(x1)), mark(y1))

A__SQUARE(X) → MARK(X)

MARK(2ndsneg(times(x0, x1), y1)) → A__2NDSNEG(a__times(mark(x0), mark(x1)), mark(y1))

A__2NDSPOS(s(y0), cons(y1, 2ndsneg(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__2ndsneg(mark(x0), mark(x1))))

MARK(negrecip(X)) → MARK(X)

A__2NDSPOS(s(N), cons(X, Z)) → MARK(N)

MARK(2ndspos(s(y0), posrecip(x0))) → A__2NDSPOS(s(mark(y0)), posrecip(mark(x0)))

A__PLUS(0, Y) → MARK(Y)

MARK(2ndspos(2ndspos(x0, x1), y1)) → A__2NDSPOS(a__2ndspos(mark(x0), mark(x1)), mark(y1))

MARK(posrecip(X)) → MARK(X)

MARK(2ndspos(square(y0), from(x0))) → A__2NDSPOS(a__square(mark(y0)), a__from(mark(x0)))

MARK(2ndsneg(pi(x0), y1)) → A__2NDSNEG(a__pi(mark(x0)), mark(y1))

MARK(2ndspos(from(y0), square(x0))) → A__2NDSPOS(a__from(mark(y0)), a__square(mark(x0)))

MARK(2ndspos(from(y0), times(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), a__times(mark(x0), mark(x1)))

A__2NDSPOS(s(y0), cons(y1, square(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__square(mark(x0))))

MARK(plus(0, y1)) → A__PLUS(0, mark(y1))

MARK(from(X)) → MARK(X)

MARK(2ndspos(2ndsneg(x0, x1), y1)) → A__2NDSPOS(a__2ndsneg(mark(x0), mark(x1)), mark(y1))

A__PLUS(s(X), Y) → A__PLUS(mark(X), mark(Y))

MARK(2ndspos(square(y0), square(x0))) → A__2NDSPOS(a__square(mark(y0)), a__square(mark(x0)))

A__SQUARE(X) → A__TIMES(mark(X), mark(X))

A__PI(pi(x0)) → A__2NDSPOS(a__pi(mark(x0)), a__from(0))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(Z)

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → A__2NDSPOS(mark(N), mark(Z))

MARK(plus(from(x0), y1)) → A__PLUS(a__from(mark(x0)), mark(y1))

MARK(square(X)) → A__SQUARE(mark(X))

MARK(2ndspos(pi(x0), y1)) → A__2NDSPOS(a__pi(mark(x0)), mark(y1))

MARK(2ndspos(from(y0), plus(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(s(y0), 0)) → A__2NDSPOS(s(mark(y0)), 0)

MARK(2ndspos(from(y0), 2ndspos(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))

a__2ndspos(0, Z) → rnil

a__2ndspos(s(N), cons(X, Z)) → a__2ndspos(s(mark(N)), cons2(X, mark(Z)))

a__2ndspos(s(N), cons2(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))

a__2ndsneg(0, Z) → rnil

a__2ndsneg(s(N), cons(X, Z)) → a__2ndsneg(s(mark(N)), cons2(X, mark(Z)))

a__2ndsneg(s(N), cons2(X, cons(Y, Z))) → rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z)))

a__pi(X) → a__2ndspos(mark(X), a__from(0))

a__plus(0, Y) → mark(Y)

a__plus(s(X), Y) → s(a__plus(mark(X), mark(Y)))

a__times(0, Y) → 0

a__times(s(X), Y) → a__plus(mark(Y), a__times(mark(X), mark(Y)))

a__square(X) → a__times(mark(X), mark(X))

mark(from(X)) → a__from(mark(X))

mark(2ndspos(X1, X2)) → a__2ndspos(mark(X1), mark(X2))

mark(2ndsneg(X1, X2)) → a__2ndsneg(mark(X1), mark(X2))

mark(pi(X)) → a__pi(mark(X))

mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))

mark(times(X1, X2)) → a__times(mark(X1), mark(X2))

mark(square(X)) → a__square(mark(X))

mark(0) → 0

mark(s(X)) → s(mark(X))

mark(posrecip(X)) → posrecip(mark(X))

mark(negrecip(X)) → negrecip(mark(X))

mark(nil) → nil

mark(cons(X1, X2)) → cons(mark(X1), X2)

mark(cons2(X1, X2)) → cons2(X1, mark(X2))

mark(rnil) → rnil

mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))

a__from(X) → from(X)

a__2ndspos(X1, X2) → 2ndspos(X1, X2)

a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)

a__pi(X) → pi(X)

a__plus(X1, X2) → plus(X1, X2)

a__times(X1, X2) → times(X1, X2)

a__square(X) → square(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 7 less nodes.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ Narrowing
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ QDP
              ↳ Narrowing
                ↳ QDP
                  ↳ DependencyGraphProof
                    ↳ QDP
                      ↳ Narrowing
                        ↳ QDP
                          ↳ DependencyGraphProof
                            ↳ QDP
                              ↳ Narrowing
                                ↳ QDP
                                  ↳ DependencyGraphProof
                                    ↳ QDP
                                      ↳ Narrowing
                                        ↳ QDP
                                          ↳ DependencyGraphProof
                                            ↳ QDP
                                              ↳ Narrowing
                                                ↳ QDP
                                                  ↳ DependencyGraphProof
                                                    ↳ QDP
                                                      ↳ Narrowing
                                                        ↳ QDP
                                                          ↳ DependencyGraphProof
                                                            ↳ QDP
                                                              ↳ Narrowing
                                                                ↳ QDP
                                                                  ↳ DependencyGraphProof
                                                                    ↳ QDP
                                                                      ↳ Narrowing
                                                                        ↳ QDP
                                                                          ↳ DependencyGraphProof
                                                                            ↳ QDP
                                                                              ↳ Narrowing

Q DP problem:
The TRS P consists of the following rules:

MARK(2ndsneg(square(x0), y1)) → A__2NDSNEG(a__square(mark(x0)), mark(y1))

MARK(2ndspos(square(y0), plus(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__plus(mark(x0), mark(x1)))

A__PI(times(x0, x1)) → A__2NDSPOS(a__times(mark(x0), mark(x1)), a__from(0))

A__2NDSPOS(s(y0), cons(y1, 2ndspos(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__2ndspos(mark(x0), mark(x1))))

A__2NDSNEG(s(N), cons(X, Z)) → A__2NDSNEG(s(mark(N)), cons2(X, mark(Z)))

MARK(2ndspos(X1, X2)) → MARK(X2)

MARK(times(times(x0, x1), y1)) → A__TIMES(a__times(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(from(y0), cons2(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), cons2(x0, mark(x1)))

MARK(plus(plus(x0, x1), y1)) → A__PLUS(a__plus(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(s(y0), from(x0))) → A__2NDSPOS(s(mark(y0)), a__from(mark(x0)))

MARK(2ndspos(from(y0), 2ndsneg(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndsneg(2ndsneg(x0, x1), y1)) → A__2NDSNEG(a__2ndsneg(mark(x0), mark(x1)), mark(y1))

A__2NDSNEG(s(N), cons(X, Z)) → MARK(Z)

A__TIMES(s(X), Y) → MARK(Y)

MARK(2ndsneg(2ndspos(x0, x1), y1)) → A__2NDSNEG(a__2ndspos(mark(x0), mark(x1)), mark(y1))

A__2NDSPOS(s(y0), cons(y1, plus(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__plus(mark(x0), mark(x1))))

MARK(2ndspos(square(y0), 2ndsneg(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndspos(from(y0), pi(x0))) → A__2NDSPOS(a__from(mark(y0)), a__pi(mark(x0)))

A__2NDSNEG(s(N), cons(X, Z)) → MARK(N)

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → A__2NDSNEG(mark(N), mark(Z))

MARK(2ndsneg(plus(x0, x1), y1)) → A__2NDSNEG(a__plus(mark(x0), mark(x1)), mark(y1))

MARK(times(2ndsneg(x0, x1), y1)) → A__TIMES(a__2ndsneg(mark(x0), mark(x1)), mark(y1))

MARK(2ndsneg(from(x0), y1)) → A__2NDSNEG(a__from(mark(x0)), mark(y1))

MARK(2ndspos(from(y0), from(x0))) → A__2NDSPOS(a__from(mark(y0)), a__from(mark(x0)))

MARK(2ndspos(s(y0), pi(x0))) → A__2NDSPOS(s(mark(y0)), a__pi(mark(x0)))

A__TIMES(s(X), Y) → A__PLUS(mark(Y), a__times(mark(X), mark(Y)))

A__PI(2ndspos(x0, x1)) → A__2NDSPOS(a__2ndspos(mark(x0), mark(x1)), a__from(0))

MARK(2ndspos(square(y0), cons2(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), cons2(x0, mark(x1)))

MARK(plus(square(x0), y1)) → A__PLUS(a__square(mark(x0)), mark(y1))

MARK(2ndspos(plus(x0, x1), y1)) → A__2NDSPOS(a__plus(mark(x0), mark(x1)), mark(y1))

MARK(plus(s(x0), y1)) → A__PLUS(s(mark(x0)), mark(y1))

A__PI(plus(x0, x1)) → A__2NDSPOS(a__plus(mark(x0), mark(x1)), a__from(0))

MARK(2ndspos(square(y0), cons(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), cons(mark(x0), x1))

MARK(cons2(X1, X2)) → MARK(X2)

MARK(2ndspos(X1, X2)) → MARK(X1)

MARK(s(X)) → MARK(X)

MARK(pi(X)) → MARK(X)

MARK(plus(X1, X2)) → MARK(X2)

A__2NDSPOS(s(N), cons(X, Z)) → MARK(Z)

MARK(pi(X)) → A__PI(mark(X))

MARK(plus(2ndsneg(x0, x1), y1)) → A__PLUS(a__2ndsneg(mark(x0), mark(x1)), mark(y1))

MARK(2ndsneg(X1, X2)) → MARK(X2)

MARK(2ndspos(square(y0), times(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(times(square(x0), y1)) → A__TIMES(a__square(mark(x0)), mark(y1))

A__PI(X) → A__FROM(0)

MARK(2ndspos(s(y0), plus(x0, x1))) → A__2NDSPOS(s(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(from(X)) → A__FROM(mark(X))

MARK(plus(X1, X2)) → MARK(X1)

MARK(times(from(x0), y1)) → A__TIMES(a__from(mark(x0)), mark(y1))

A__2NDSPOS(s(y0), cons(y1, times(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__times(mark(x0), mark(x1))))

MARK(2ndspos(s(y0), cons(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons(mark(x0), x1))

A__PI(X) → MARK(X)

MARK(times(s(x0), y1)) → A__TIMES(s(mark(x0)), mark(y1))

MARK(2ndspos(from(y0), cons(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), cons(mark(x0), x1))

MARK(2ndsneg(s(x0), y1)) → A__2NDSNEG(s(mark(x0)), mark(y1))

A__PLUS(s(X), Y) → MARK(Y)

A__TIMES(s(X), Y) → MARK(X)

A__2NDSPOS(s(y0), cons(y1, pi(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__pi(mark(x0))))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(N)

MARK(plus(2ndspos(x0, x1), y1)) → A__PLUS(a__2ndspos(mark(x0), mark(x1)), mark(y1))

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(N)

A__PI(from(x0)) → A__2NDSPOS(a__from(mark(x0)), a__from(0))

MARK(cons(X1, X2)) → MARK(X1)

MARK(2ndspos(square(y0), 2ndspos(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndspos(s(y0), 2ndspos(x0, x1))) → A__2NDSPOS(s(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

MARK(times(X1, X2)) → MARK(X2)

A__PI(square(x0)) → A__2NDSPOS(a__square(mark(x0)), a__from(0))

MARK(2ndspos(s(y0), times(x0, x1))) → A__2NDSPOS(s(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndsneg(X1, X2)) → MARK(X1)

MARK(2ndspos(s(y0), 2ndsneg(x0, x1))) → A__2NDSPOS(s(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

A__PI(2ndsneg(x0, x1)) → A__2NDSPOS(a__2ndsneg(mark(x0), mark(x1)), a__from(0))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(Y)

MARK(plus(pi(x0), y1)) → A__PLUS(a__pi(mark(x0)), mark(y1))

MARK(times(pi(x0), y1)) → A__TIMES(a__pi(mark(x0)), mark(y1))

MARK(rcons(X1, X2)) → MARK(X1)

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(Z)

MARK(square(X)) → MARK(X)

A__PI(s(x0)) → A__2NDSPOS(s(mark(x0)), a__from(0))

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(Y)

A__FROM(X) → MARK(X)

MARK(times(plus(x0, x1), y1)) → A__TIMES(a__plus(mark(x0), mark(x1)), mark(y1))

A__2NDSPOS(s(y0), cons(y1, from(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__from(mark(x0))))

MARK(2ndspos(times(x0, x1), y1)) → A__2NDSPOS(a__times(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(square(y0), pi(x0))) → A__2NDSPOS(a__square(mark(y0)), a__pi(mark(x0)))

MARK(2ndspos(s(y0), square(x0))) → A__2NDSPOS(s(mark(y0)), a__square(mark(x0)))

MARK(rcons(X1, X2)) → MARK(X2)

MARK(times(2ndspos(x0, x1), y1)) → A__TIMES(a__2ndspos(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(s(y0), cons2(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(x0, mark(x1)))

A__TIMES(s(X), Y) → A__TIMES(mark(X), mark(Y))

A__PLUS(s(X), Y) → MARK(X)

A__2NDSPOS(s(y0), cons(y1, cons(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, cons(mark(x0), x1)))

MARK(times(X1, X2)) → MARK(X1)

MARK(plus(times(x0, x1), y1)) → A__PLUS(a__times(mark(x0), mark(x1)), mark(y1))

A__SQUARE(X) → MARK(X)

MARK(2ndsneg(times(x0, x1), y1)) → A__2NDSNEG(a__times(mark(x0), mark(x1)), mark(y1))

A__2NDSPOS(s(y0), cons(y1, 2ndsneg(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__2ndsneg(mark(x0), mark(x1))))

A__2NDSPOS(s(N), cons(X, Z)) → MARK(N)

MARK(negrecip(X)) → MARK(X)

A__PLUS(0, Y) → MARK(Y)

MARK(2ndspos(2ndspos(x0, x1), y1)) → A__2NDSPOS(a__2ndspos(mark(x0), mark(x1)), mark(y1))

MARK(posrecip(X)) → MARK(X)

MARK(2ndspos(from(y0), square(x0))) → A__2NDSPOS(a__from(mark(y0)), a__square(mark(x0)))

MARK(2ndsneg(pi(x0), y1)) → A__2NDSNEG(a__pi(mark(x0)), mark(y1))

MARK(2ndspos(square(y0), from(x0))) → A__2NDSPOS(a__square(mark(y0)), a__from(mark(x0)))

MARK(2ndspos(from(y0), times(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), a__times(mark(x0), mark(x1)))

A__2NDSPOS(s(y0), cons(y1, square(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__square(mark(x0))))

MARK(plus(0, y1)) → A__PLUS(0, mark(y1))

MARK(from(X)) → MARK(X)

MARK(2ndspos(2ndsneg(x0, x1), y1)) → A__2NDSPOS(a__2ndsneg(mark(x0), mark(x1)), mark(y1))

A__PLUS(s(X), Y) → A__PLUS(mark(X), mark(Y))

A__SQUARE(X) → A__TIMES(mark(X), mark(X))

MARK(2ndspos(square(y0), square(x0))) → A__2NDSPOS(a__square(mark(y0)), a__square(mark(x0)))

A__PI(pi(x0)) → A__2NDSPOS(a__pi(mark(x0)), a__from(0))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(Z)

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → A__2NDSPOS(mark(N), mark(Z))

MARK(plus(from(x0), y1)) → A__PLUS(a__from(mark(x0)), mark(y1))

MARK(2ndspos(pi(x0), y1)) → A__2NDSPOS(a__pi(mark(x0)), mark(y1))

MARK(square(X)) → A__SQUARE(mark(X))

MARK(2ndspos(from(y0), plus(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(from(y0), 2ndspos(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))

a__2ndspos(0, Z) → rnil

a__2ndspos(s(N), cons(X, Z)) → a__2ndspos(s(mark(N)), cons2(X, mark(Z)))

a__2ndspos(s(N), cons2(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))

a__2ndsneg(0, Z) → rnil

a__2ndsneg(s(N), cons(X, Z)) → a__2ndsneg(s(mark(N)), cons2(X, mark(Z)))

a__2ndsneg(s(N), cons2(X, cons(Y, Z))) → rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z)))

a__pi(X) → a__2ndspos(mark(X), a__from(0))

a__plus(0, Y) → mark(Y)

a__plus(s(X), Y) → s(a__plus(mark(X), mark(Y)))

a__times(0, Y) → 0

a__times(s(X), Y) → a__plus(mark(Y), a__times(mark(X), mark(Y)))

a__square(X) → a__times(mark(X), mark(X))

mark(from(X)) → a__from(mark(X))

mark(2ndspos(X1, X2)) → a__2ndspos(mark(X1), mark(X2))

mark(2ndsneg(X1, X2)) → a__2ndsneg(mark(X1), mark(X2))

mark(pi(X)) → a__pi(mark(X))

mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))

mark(times(X1, X2)) → a__times(mark(X1), mark(X2))

mark(square(X)) → a__square(mark(X))

mark(0) → 0

mark(s(X)) → s(mark(X))

mark(posrecip(X)) → posrecip(mark(X))

mark(negrecip(X)) → negrecip(mark(X))

mark(nil) → nil

mark(cons(X1, X2)) → cons(mark(X1), X2)

mark(cons2(X1, X2)) → cons2(X1, mark(X2))

mark(rnil) → rnil

mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))

a__from(X) → from(X)

a__2ndspos(X1, X2) → 2ndspos(X1, X2)

a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)

a__pi(X) → pi(X)

a__plus(X1, X2) → plus(X1, X2)

a__times(X1, X2) → times(X1, X2)

a__square(X) → square(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule MARK(2ndspos(2ndsneg(x0, x1), y1)) → A__2NDSPOS(a__2ndsneg(mark(x0), mark(x1)), mark(y1)) at position [1] we obtained the following new rules:

MARK(2ndspos(2ndsneg(y0, y1), 0)) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), 0)

MARK(2ndspos(2ndsneg(y0, y1), negrecip(x0))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), negrecip(mark(x0)))

MARK(2ndspos(2ndsneg(y0, y1), rcons(x0, x1))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), rcons(mark(x0), mark(x1)))

MARK(2ndspos(2ndsneg(y0, y1), 2ndsneg(x0, x1))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndspos(2ndsneg(y0, y1), from(x0))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndspos(2ndsneg(y0, y1), nil)) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), nil)

MARK(2ndspos(2ndsneg(y0, y1), pi(x0))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__pi(mark(x0)))

MARK(2ndspos(2ndsneg(y0, y1), s(x0))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), s(mark(x0)))

MARK(2ndspos(2ndsneg(y0, y1), plus(x0, x1))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(2ndsneg(y0, y1), times(x0, x1))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(2ndsneg(y0, y1), cons2(x0, x1))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(2ndspos(2ndsneg(y0, y1), square(x0))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndspos(2ndsneg(y0, y1), 2ndspos(x0, x1))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndspos(2ndsneg(y0, y1), posrecip(x0))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), posrecip(mark(x0)))

MARK(2ndspos(2ndsneg(y0, y1), rnil)) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), rnil)

MARK(2ndspos(2ndsneg(y0, y1), cons(x0, x1))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), cons(mark(x0), x1))



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ Narrowing
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ QDP
              ↳ Narrowing
                ↳ QDP
                  ↳ DependencyGraphProof
                    ↳ QDP
                      ↳ Narrowing
                        ↳ QDP
                          ↳ DependencyGraphProof
                            ↳ QDP
                              ↳ Narrowing
                                ↳ QDP
                                  ↳ DependencyGraphProof
                                    ↳ QDP
                                      ↳ Narrowing
                                        ↳ QDP
                                          ↳ DependencyGraphProof
                                            ↳ QDP
                                              ↳ Narrowing
                                                ↳ QDP
                                                  ↳ DependencyGraphProof
                                                    ↳ QDP
                                                      ↳ Narrowing
                                                        ↳ QDP
                                                          ↳ DependencyGraphProof
                                                            ↳ QDP
                                                              ↳ Narrowing
                                                                ↳ QDP
                                                                  ↳ DependencyGraphProof
                                                                    ↳ QDP
                                                                      ↳ Narrowing
                                                                        ↳ QDP
                                                                          ↳ DependencyGraphProof
                                                                            ↳ QDP
                                                                              ↳ Narrowing
                                                                                ↳ QDP
                                                                                  ↳ DependencyGraphProof

Q DP problem:
The TRS P consists of the following rules:

MARK(2ndspos(square(y0), plus(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(2ndsneg(square(x0), y1)) → A__2NDSNEG(a__square(mark(x0)), mark(y1))

A__2NDSPOS(s(y0), cons(y1, 2ndspos(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__2ndspos(mark(x0), mark(x1))))

A__PI(times(x0, x1)) → A__2NDSPOS(a__times(mark(x0), mark(x1)), a__from(0))

A__2NDSNEG(s(N), cons(X, Z)) → A__2NDSNEG(s(mark(N)), cons2(X, mark(Z)))

MARK(times(times(x0, x1), y1)) → A__TIMES(a__times(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(X1, X2)) → MARK(X2)

MARK(2ndspos(from(y0), cons2(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), cons2(x0, mark(x1)))

MARK(plus(plus(x0, x1), y1)) → A__PLUS(a__plus(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(2ndsneg(y0, y1), from(x0))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndspos(from(y0), 2ndsneg(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndspos(s(y0), from(x0))) → A__2NDSPOS(s(mark(y0)), a__from(mark(x0)))

MARK(2ndspos(2ndsneg(y0, y1), pi(x0))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__pi(mark(x0)))

MARK(2ndsneg(2ndsneg(x0, x1), y1)) → A__2NDSNEG(a__2ndsneg(mark(x0), mark(x1)), mark(y1))

A__2NDSNEG(s(N), cons(X, Z)) → MARK(Z)

A__TIMES(s(X), Y) → MARK(Y)

MARK(2ndsneg(2ndspos(x0, x1), y1)) → A__2NDSNEG(a__2ndspos(mark(x0), mark(x1)), mark(y1))

A__2NDSPOS(s(y0), cons(y1, plus(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__plus(mark(x0), mark(x1))))

MARK(2ndspos(2ndsneg(y0, y1), square(x0))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndspos(from(y0), pi(x0))) → A__2NDSPOS(a__from(mark(y0)), a__pi(mark(x0)))

MARK(2ndspos(square(y0), 2ndsneg(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → A__2NDSNEG(mark(N), mark(Z))

A__2NDSNEG(s(N), cons(X, Z)) → MARK(N)

MARK(times(2ndsneg(x0, x1), y1)) → A__TIMES(a__2ndsneg(mark(x0), mark(x1)), mark(y1))

MARK(2ndsneg(plus(x0, x1), y1)) → A__2NDSNEG(a__plus(mark(x0), mark(x1)), mark(y1))

MARK(2ndsneg(from(x0), y1)) → A__2NDSNEG(a__from(mark(x0)), mark(y1))

MARK(2ndspos(from(y0), from(x0))) → A__2NDSPOS(a__from(mark(y0)), a__from(mark(x0)))

A__TIMES(s(X), Y) → A__PLUS(mark(Y), a__times(mark(X), mark(Y)))

MARK(2ndspos(s(y0), pi(x0))) → A__2NDSPOS(s(mark(y0)), a__pi(mark(x0)))

MARK(2ndspos(square(y0), cons2(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), cons2(x0, mark(x1)))

A__PI(2ndspos(x0, x1)) → A__2NDSPOS(a__2ndspos(mark(x0), mark(x1)), a__from(0))

MARK(plus(square(x0), y1)) → A__PLUS(a__square(mark(x0)), mark(y1))

MARK(plus(s(x0), y1)) → A__PLUS(s(mark(x0)), mark(y1))

MARK(2ndspos(plus(x0, x1), y1)) → A__2NDSPOS(a__plus(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(square(y0), cons(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), cons(mark(x0), x1))

A__PI(plus(x0, x1)) → A__2NDSPOS(a__plus(mark(x0), mark(x1)), a__from(0))

MARK(cons2(X1, X2)) → MARK(X2)

MARK(s(X)) → MARK(X)

MARK(2ndspos(X1, X2)) → MARK(X1)

MARK(pi(X)) → MARK(X)

MARK(plus(X1, X2)) → MARK(X2)

A__2NDSPOS(s(N), cons(X, Z)) → MARK(Z)

MARK(pi(X)) → A__PI(mark(X))

MARK(plus(2ndsneg(x0, x1), y1)) → A__PLUS(a__2ndsneg(mark(x0), mark(x1)), mark(y1))

MARK(2ndsneg(X1, X2)) → MARK(X2)

MARK(2ndspos(square(y0), times(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(2ndsneg(y0, y1), cons2(x0, x1))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(times(square(x0), y1)) → A__TIMES(a__square(mark(x0)), mark(y1))

MARK(from(X)) → A__FROM(mark(X))

MARK(2ndspos(s(y0), plus(x0, x1))) → A__2NDSPOS(s(mark(y0)), a__plus(mark(x0), mark(x1)))

A__PI(X) → A__FROM(0)

MARK(times(from(x0), y1)) → A__TIMES(a__from(mark(x0)), mark(y1))

MARK(plus(X1, X2)) → MARK(X1)

A__2NDSPOS(s(y0), cons(y1, times(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__times(mark(x0), mark(x1))))

A__PI(X) → MARK(X)

MARK(2ndspos(s(y0), cons(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons(mark(x0), x1))

MARK(2ndspos(2ndsneg(y0, y1), posrecip(x0))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), posrecip(mark(x0)))

MARK(times(s(x0), y1)) → A__TIMES(s(mark(x0)), mark(y1))

MARK(2ndsneg(s(x0), y1)) → A__2NDSNEG(s(mark(x0)), mark(y1))

MARK(2ndspos(from(y0), cons(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), cons(mark(x0), x1))

A__PLUS(s(X), Y) → MARK(Y)

A__2NDSPOS(s(y0), cons(y1, pi(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__pi(mark(x0))))

A__TIMES(s(X), Y) → MARK(X)

MARK(2ndspos(2ndsneg(y0, y1), rnil)) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), rnil)

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(N)

MARK(plus(2ndspos(x0, x1), y1)) → A__PLUS(a__2ndspos(mark(x0), mark(x1)), mark(y1))

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(N)

MARK(cons(X1, X2)) → MARK(X1)

A__PI(from(x0)) → A__2NDSPOS(a__from(mark(x0)), a__from(0))

MARK(2ndspos(2ndsneg(y0, y1), 0)) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), 0)

MARK(2ndspos(2ndsneg(y0, y1), rcons(x0, x1))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), rcons(mark(x0), mark(x1)))

MARK(2ndspos(square(y0), 2ndspos(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

MARK(times(X1, X2)) → MARK(X2)

MARK(2ndspos(s(y0), 2ndspos(x0, x1))) → A__2NDSPOS(s(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndspos(2ndsneg(y0, y1), 2ndsneg(x0, x1))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__2ndsneg(mark(x0), mark(x1)))

A__PI(square(x0)) → A__2NDSPOS(a__square(mark(x0)), a__from(0))

MARK(2ndspos(s(y0), times(x0, x1))) → A__2NDSPOS(s(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(2ndsneg(y0, y1), nil)) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), nil)

MARK(2ndsneg(X1, X2)) → MARK(X1)

MARK(2ndspos(2ndsneg(y0, y1), s(x0))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), s(mark(x0)))

MARK(2ndspos(s(y0), 2ndsneg(x0, x1))) → A__2NDSPOS(s(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndspos(2ndsneg(y0, y1), plus(x0, x1))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

A__PI(2ndsneg(x0, x1)) → A__2NDSPOS(a__2ndsneg(mark(x0), mark(x1)), a__from(0))

MARK(times(pi(x0), y1)) → A__TIMES(a__pi(mark(x0)), mark(y1))

MARK(plus(pi(x0), y1)) → A__PLUS(a__pi(mark(x0)), mark(y1))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(Y)

MARK(rcons(X1, X2)) → MARK(X1)

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(Z)

MARK(square(X)) → MARK(X)

MARK(2ndspos(2ndsneg(y0, y1), 2ndspos(x0, x1))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__2ndspos(mark(x0), mark(x1)))

A__PI(s(x0)) → A__2NDSPOS(s(mark(x0)), a__from(0))

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(Y)

A__FROM(X) → MARK(X)

MARK(times(plus(x0, x1), y1)) → A__TIMES(a__plus(mark(x0), mark(x1)), mark(y1))

A__2NDSPOS(s(y0), cons(y1, from(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__from(mark(x0))))

MARK(2ndspos(times(x0, x1), y1)) → A__2NDSPOS(a__times(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(square(y0), pi(x0))) → A__2NDSPOS(a__square(mark(y0)), a__pi(mark(x0)))

MARK(rcons(X1, X2)) → MARK(X2)

MARK(2ndspos(s(y0), square(x0))) → A__2NDSPOS(s(mark(y0)), a__square(mark(x0)))

MARK(times(2ndspos(x0, x1), y1)) → A__TIMES(a__2ndspos(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(s(y0), cons2(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(x0, mark(x1)))

A__TIMES(s(X), Y) → A__TIMES(mark(X), mark(Y))

A__2NDSPOS(s(y0), cons(y1, cons(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, cons(mark(x0), x1)))

A__PLUS(s(X), Y) → MARK(X)

MARK(times(X1, X2)) → MARK(X1)

MARK(2ndspos(2ndsneg(y0, y1), negrecip(x0))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), negrecip(mark(x0)))

MARK(plus(times(x0, x1), y1)) → A__PLUS(a__times(mark(x0), mark(x1)), mark(y1))

A__SQUARE(X) → MARK(X)

MARK(2ndsneg(times(x0, x1), y1)) → A__2NDSNEG(a__times(mark(x0), mark(x1)), mark(y1))

A__2NDSPOS(s(y0), cons(y1, 2ndsneg(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__2ndsneg(mark(x0), mark(x1))))

MARK(negrecip(X)) → MARK(X)

A__2NDSPOS(s(N), cons(X, Z)) → MARK(N)

A__PLUS(0, Y) → MARK(Y)

MARK(2ndspos(2ndspos(x0, x1), y1)) → A__2NDSPOS(a__2ndspos(mark(x0), mark(x1)), mark(y1))

MARK(posrecip(X)) → MARK(X)

MARK(2ndspos(square(y0), from(x0))) → A__2NDSPOS(a__square(mark(y0)), a__from(mark(x0)))

MARK(2ndsneg(pi(x0), y1)) → A__2NDSNEG(a__pi(mark(x0)), mark(y1))

MARK(2ndspos(from(y0), square(x0))) → A__2NDSPOS(a__from(mark(y0)), a__square(mark(x0)))

MARK(2ndspos(from(y0), times(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), a__times(mark(x0), mark(x1)))

A__2NDSPOS(s(y0), cons(y1, square(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__square(mark(x0))))

MARK(2ndspos(2ndsneg(y0, y1), times(x0, x1))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(plus(0, y1)) → A__PLUS(0, mark(y1))

MARK(from(X)) → MARK(X)

A__PLUS(s(X), Y) → A__PLUS(mark(X), mark(Y))

MARK(2ndspos(square(y0), square(x0))) → A__2NDSPOS(a__square(mark(y0)), a__square(mark(x0)))

A__SQUARE(X) → A__TIMES(mark(X), mark(X))

A__PI(pi(x0)) → A__2NDSPOS(a__pi(mark(x0)), a__from(0))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(Z)

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → A__2NDSPOS(mark(N), mark(Z))

MARK(plus(from(x0), y1)) → A__PLUS(a__from(mark(x0)), mark(y1))

MARK(square(X)) → A__SQUARE(mark(X))

MARK(2ndspos(pi(x0), y1)) → A__2NDSPOS(a__pi(mark(x0)), mark(y1))

MARK(2ndspos(from(y0), plus(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(from(y0), 2ndspos(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndspos(2ndsneg(y0, y1), cons(x0, x1))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), cons(mark(x0), x1))

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))

a__2ndspos(0, Z) → rnil

a__2ndspos(s(N), cons(X, Z)) → a__2ndspos(s(mark(N)), cons2(X, mark(Z)))

a__2ndspos(s(N), cons2(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))

a__2ndsneg(0, Z) → rnil

a__2ndsneg(s(N), cons(X, Z)) → a__2ndsneg(s(mark(N)), cons2(X, mark(Z)))

a__2ndsneg(s(N), cons2(X, cons(Y, Z))) → rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z)))

a__pi(X) → a__2ndspos(mark(X), a__from(0))

a__plus(0, Y) → mark(Y)

a__plus(s(X), Y) → s(a__plus(mark(X), mark(Y)))

a__times(0, Y) → 0

a__times(s(X), Y) → a__plus(mark(Y), a__times(mark(X), mark(Y)))

a__square(X) → a__times(mark(X), mark(X))

mark(from(X)) → a__from(mark(X))

mark(2ndspos(X1, X2)) → a__2ndspos(mark(X1), mark(X2))

mark(2ndsneg(X1, X2)) → a__2ndsneg(mark(X1), mark(X2))

mark(pi(X)) → a__pi(mark(X))

mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))

mark(times(X1, X2)) → a__times(mark(X1), mark(X2))

mark(square(X)) → a__square(mark(X))

mark(0) → 0

mark(s(X)) → s(mark(X))

mark(posrecip(X)) → posrecip(mark(X))

mark(negrecip(X)) → negrecip(mark(X))

mark(nil) → nil

mark(cons(X1, X2)) → cons(mark(X1), X2)

mark(cons2(X1, X2)) → cons2(X1, mark(X2))

mark(rnil) → rnil

mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))

a__from(X) → from(X)

a__2ndspos(X1, X2) → 2ndspos(X1, X2)

a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)

a__pi(X) → pi(X)

a__plus(X1, X2) → plus(X1, X2)

a__times(X1, X2) → times(X1, X2)

a__square(X) → square(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 7 less nodes.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ Narrowing
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ QDP
              ↳ Narrowing
                ↳ QDP
                  ↳ DependencyGraphProof
                    ↳ QDP
                      ↳ Narrowing
                        ↳ QDP
                          ↳ DependencyGraphProof
                            ↳ QDP
                              ↳ Narrowing
                                ↳ QDP
                                  ↳ DependencyGraphProof
                                    ↳ QDP
                                      ↳ Narrowing
                                        ↳ QDP
                                          ↳ DependencyGraphProof
                                            ↳ QDP
                                              ↳ Narrowing
                                                ↳ QDP
                                                  ↳ DependencyGraphProof
                                                    ↳ QDP
                                                      ↳ Narrowing
                                                        ↳ QDP
                                                          ↳ DependencyGraphProof
                                                            ↳ QDP
                                                              ↳ Narrowing
                                                                ↳ QDP
                                                                  ↳ DependencyGraphProof
                                                                    ↳ QDP
                                                                      ↳ Narrowing
                                                                        ↳ QDP
                                                                          ↳ DependencyGraphProof
                                                                            ↳ QDP
                                                                              ↳ Narrowing
                                                                                ↳ QDP
                                                                                  ↳ DependencyGraphProof
                                                                                    ↳ QDP
                                                                                      ↳ Narrowing

Q DP problem:
The TRS P consists of the following rules:

MARK(2ndsneg(square(x0), y1)) → A__2NDSNEG(a__square(mark(x0)), mark(y1))

MARK(2ndspos(square(y0), plus(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__plus(mark(x0), mark(x1)))

A__PI(times(x0, x1)) → A__2NDSPOS(a__times(mark(x0), mark(x1)), a__from(0))

A__2NDSPOS(s(y0), cons(y1, 2ndspos(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__2ndspos(mark(x0), mark(x1))))

A__2NDSNEG(s(N), cons(X, Z)) → A__2NDSNEG(s(mark(N)), cons2(X, mark(Z)))

MARK(2ndspos(X1, X2)) → MARK(X2)

MARK(times(times(x0, x1), y1)) → A__TIMES(a__times(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(from(y0), cons2(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), cons2(x0, mark(x1)))

MARK(plus(plus(x0, x1), y1)) → A__PLUS(a__plus(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(2ndsneg(y0, y1), from(x0))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndspos(s(y0), from(x0))) → A__2NDSPOS(s(mark(y0)), a__from(mark(x0)))

MARK(2ndspos(from(y0), 2ndsneg(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndspos(2ndsneg(y0, y1), pi(x0))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__pi(mark(x0)))

MARK(2ndsneg(2ndsneg(x0, x1), y1)) → A__2NDSNEG(a__2ndsneg(mark(x0), mark(x1)), mark(y1))

A__2NDSNEG(s(N), cons(X, Z)) → MARK(Z)

A__TIMES(s(X), Y) → MARK(Y)

MARK(2ndsneg(2ndspos(x0, x1), y1)) → A__2NDSNEG(a__2ndspos(mark(x0), mark(x1)), mark(y1))

A__2NDSPOS(s(y0), cons(y1, plus(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__plus(mark(x0), mark(x1))))

MARK(2ndspos(2ndsneg(y0, y1), square(x0))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndspos(square(y0), 2ndsneg(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndspos(from(y0), pi(x0))) → A__2NDSPOS(a__from(mark(y0)), a__pi(mark(x0)))

A__2NDSNEG(s(N), cons(X, Z)) → MARK(N)

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → A__2NDSNEG(mark(N), mark(Z))

MARK(2ndsneg(plus(x0, x1), y1)) → A__2NDSNEG(a__plus(mark(x0), mark(x1)), mark(y1))

MARK(times(2ndsneg(x0, x1), y1)) → A__TIMES(a__2ndsneg(mark(x0), mark(x1)), mark(y1))

MARK(2ndsneg(from(x0), y1)) → A__2NDSNEG(a__from(mark(x0)), mark(y1))

MARK(2ndspos(from(y0), from(x0))) → A__2NDSPOS(a__from(mark(y0)), a__from(mark(x0)))

MARK(2ndspos(s(y0), pi(x0))) → A__2NDSPOS(s(mark(y0)), a__pi(mark(x0)))

A__TIMES(s(X), Y) → A__PLUS(mark(Y), a__times(mark(X), mark(Y)))

A__PI(2ndspos(x0, x1)) → A__2NDSPOS(a__2ndspos(mark(x0), mark(x1)), a__from(0))

MARK(2ndspos(square(y0), cons2(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), cons2(x0, mark(x1)))

MARK(plus(square(x0), y1)) → A__PLUS(a__square(mark(x0)), mark(y1))

MARK(2ndspos(plus(x0, x1), y1)) → A__2NDSPOS(a__plus(mark(x0), mark(x1)), mark(y1))

MARK(plus(s(x0), y1)) → A__PLUS(s(mark(x0)), mark(y1))

A__PI(plus(x0, x1)) → A__2NDSPOS(a__plus(mark(x0), mark(x1)), a__from(0))

MARK(2ndspos(square(y0), cons(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), cons(mark(x0), x1))

MARK(cons2(X1, X2)) → MARK(X2)

MARK(2ndspos(X1, X2)) → MARK(X1)

MARK(s(X)) → MARK(X)

MARK(pi(X)) → MARK(X)

MARK(plus(X1, X2)) → MARK(X2)

A__2NDSPOS(s(N), cons(X, Z)) → MARK(Z)

MARK(pi(X)) → A__PI(mark(X))

MARK(plus(2ndsneg(x0, x1), y1)) → A__PLUS(a__2ndsneg(mark(x0), mark(x1)), mark(y1))

MARK(2ndsneg(X1, X2)) → MARK(X2)

MARK(2ndspos(square(y0), times(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(2ndsneg(y0, y1), cons2(x0, x1))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(times(square(x0), y1)) → A__TIMES(a__square(mark(x0)), mark(y1))

A__PI(X) → A__FROM(0)

MARK(2ndspos(s(y0), plus(x0, x1))) → A__2NDSPOS(s(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(from(X)) → A__FROM(mark(X))

MARK(plus(X1, X2)) → MARK(X1)

MARK(times(from(x0), y1)) → A__TIMES(a__from(mark(x0)), mark(y1))

A__2NDSPOS(s(y0), cons(y1, times(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__times(mark(x0), mark(x1))))

MARK(2ndspos(s(y0), cons(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons(mark(x0), x1))

A__PI(X) → MARK(X)

MARK(times(s(x0), y1)) → A__TIMES(s(mark(x0)), mark(y1))

MARK(2ndsneg(s(x0), y1)) → A__2NDSNEG(s(mark(x0)), mark(y1))

MARK(2ndspos(from(y0), cons(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), cons(mark(x0), x1))

A__PLUS(s(X), Y) → MARK(Y)

A__TIMES(s(X), Y) → MARK(X)

A__2NDSPOS(s(y0), cons(y1, pi(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__pi(mark(x0))))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(N)

MARK(plus(2ndspos(x0, x1), y1)) → A__PLUS(a__2ndspos(mark(x0), mark(x1)), mark(y1))

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(N)

A__PI(from(x0)) → A__2NDSPOS(a__from(mark(x0)), a__from(0))

MARK(cons(X1, X2)) → MARK(X1)

MARK(2ndspos(square(y0), 2ndspos(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndspos(s(y0), 2ndspos(x0, x1))) → A__2NDSPOS(s(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

MARK(times(X1, X2)) → MARK(X2)

A__PI(square(x0)) → A__2NDSPOS(a__square(mark(x0)), a__from(0))

MARK(2ndspos(2ndsneg(y0, y1), 2ndsneg(x0, x1))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndspos(s(y0), times(x0, x1))) → A__2NDSPOS(s(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndsneg(X1, X2)) → MARK(X1)

MARK(2ndspos(s(y0), 2ndsneg(x0, x1))) → A__2NDSPOS(s(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndspos(2ndsneg(y0, y1), plus(x0, x1))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

A__PI(2ndsneg(x0, x1)) → A__2NDSPOS(a__2ndsneg(mark(x0), mark(x1)), a__from(0))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(Y)

MARK(plus(pi(x0), y1)) → A__PLUS(a__pi(mark(x0)), mark(y1))

MARK(times(pi(x0), y1)) → A__TIMES(a__pi(mark(x0)), mark(y1))

MARK(rcons(X1, X2)) → MARK(X1)

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(Z)

MARK(square(X)) → MARK(X)

MARK(2ndspos(2ndsneg(y0, y1), 2ndspos(x0, x1))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__2ndspos(mark(x0), mark(x1)))

A__PI(s(x0)) → A__2NDSPOS(s(mark(x0)), a__from(0))

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(Y)

A__FROM(X) → MARK(X)

MARK(times(plus(x0, x1), y1)) → A__TIMES(a__plus(mark(x0), mark(x1)), mark(y1))

A__2NDSPOS(s(y0), cons(y1, from(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__from(mark(x0))))

MARK(2ndspos(times(x0, x1), y1)) → A__2NDSPOS(a__times(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(square(y0), pi(x0))) → A__2NDSPOS(a__square(mark(y0)), a__pi(mark(x0)))

MARK(2ndspos(s(y0), square(x0))) → A__2NDSPOS(s(mark(y0)), a__square(mark(x0)))

MARK(rcons(X1, X2)) → MARK(X2)

MARK(times(2ndspos(x0, x1), y1)) → A__TIMES(a__2ndspos(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(s(y0), cons2(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(x0, mark(x1)))

A__TIMES(s(X), Y) → A__TIMES(mark(X), mark(Y))

A__PLUS(s(X), Y) → MARK(X)

A__2NDSPOS(s(y0), cons(y1, cons(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, cons(mark(x0), x1)))

MARK(times(X1, X2)) → MARK(X1)

MARK(plus(times(x0, x1), y1)) → A__PLUS(a__times(mark(x0), mark(x1)), mark(y1))

A__SQUARE(X) → MARK(X)

MARK(2ndsneg(times(x0, x1), y1)) → A__2NDSNEG(a__times(mark(x0), mark(x1)), mark(y1))

A__2NDSPOS(s(y0), cons(y1, 2ndsneg(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__2ndsneg(mark(x0), mark(x1))))

A__2NDSPOS(s(N), cons(X, Z)) → MARK(N)

MARK(negrecip(X)) → MARK(X)

A__PLUS(0, Y) → MARK(Y)

MARK(2ndspos(2ndspos(x0, x1), y1)) → A__2NDSPOS(a__2ndspos(mark(x0), mark(x1)), mark(y1))

MARK(posrecip(X)) → MARK(X)

MARK(2ndspos(from(y0), square(x0))) → A__2NDSPOS(a__from(mark(y0)), a__square(mark(x0)))

MARK(2ndsneg(pi(x0), y1)) → A__2NDSNEG(a__pi(mark(x0)), mark(y1))

MARK(2ndspos(square(y0), from(x0))) → A__2NDSPOS(a__square(mark(y0)), a__from(mark(x0)))

MARK(2ndspos(from(y0), times(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), a__times(mark(x0), mark(x1)))

A__2NDSPOS(s(y0), cons(y1, square(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__square(mark(x0))))

MARK(2ndspos(2ndsneg(y0, y1), times(x0, x1))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(plus(0, y1)) → A__PLUS(0, mark(y1))

MARK(from(X)) → MARK(X)

A__PLUS(s(X), Y) → A__PLUS(mark(X), mark(Y))

A__SQUARE(X) → A__TIMES(mark(X), mark(X))

MARK(2ndspos(square(y0), square(x0))) → A__2NDSPOS(a__square(mark(y0)), a__square(mark(x0)))

A__PI(pi(x0)) → A__2NDSPOS(a__pi(mark(x0)), a__from(0))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(Z)

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → A__2NDSPOS(mark(N), mark(Z))

MARK(plus(from(x0), y1)) → A__PLUS(a__from(mark(x0)), mark(y1))

MARK(2ndspos(pi(x0), y1)) → A__2NDSPOS(a__pi(mark(x0)), mark(y1))

MARK(square(X)) → A__SQUARE(mark(X))

MARK(2ndspos(from(y0), plus(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(from(y0), 2ndspos(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndspos(2ndsneg(y0, y1), cons(x0, x1))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), cons(mark(x0), x1))

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))

a__2ndspos(0, Z) → rnil

a__2ndspos(s(N), cons(X, Z)) → a__2ndspos(s(mark(N)), cons2(X, mark(Z)))

a__2ndspos(s(N), cons2(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))

a__2ndsneg(0, Z) → rnil

a__2ndsneg(s(N), cons(X, Z)) → a__2ndsneg(s(mark(N)), cons2(X, mark(Z)))

a__2ndsneg(s(N), cons2(X, cons(Y, Z))) → rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z)))

a__pi(X) → a__2ndspos(mark(X), a__from(0))

a__plus(0, Y) → mark(Y)

a__plus(s(X), Y) → s(a__plus(mark(X), mark(Y)))

a__times(0, Y) → 0

a__times(s(X), Y) → a__plus(mark(Y), a__times(mark(X), mark(Y)))

a__square(X) → a__times(mark(X), mark(X))

mark(from(X)) → a__from(mark(X))

mark(2ndspos(X1, X2)) → a__2ndspos(mark(X1), mark(X2))

mark(2ndsneg(X1, X2)) → a__2ndsneg(mark(X1), mark(X2))

mark(pi(X)) → a__pi(mark(X))

mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))

mark(times(X1, X2)) → a__times(mark(X1), mark(X2))

mark(square(X)) → a__square(mark(X))

mark(0) → 0

mark(s(X)) → s(mark(X))

mark(posrecip(X)) → posrecip(mark(X))

mark(negrecip(X)) → negrecip(mark(X))

mark(nil) → nil

mark(cons(X1, X2)) → cons(mark(X1), X2)

mark(cons2(X1, X2)) → cons2(X1, mark(X2))

mark(rnil) → rnil

mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))

a__from(X) → from(X)

a__2ndspos(X1, X2) → 2ndspos(X1, X2)

a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)

a__pi(X) → pi(X)

a__plus(X1, X2) → plus(X1, X2)

a__times(X1, X2) → times(X1, X2)

a__square(X) → square(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule MARK(2ndspos(pi(x0), y1)) → A__2NDSPOS(a__pi(mark(x0)), mark(y1)) at position [1] we obtained the following new rules:

MARK(2ndspos(pi(y0), negrecip(x0))) → A__2NDSPOS(a__pi(mark(y0)), negrecip(mark(x0)))

MARK(2ndspos(pi(y0), plus(x0, x1))) → A__2NDSPOS(a__pi(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(pi(y0), posrecip(x0))) → A__2NDSPOS(a__pi(mark(y0)), posrecip(mark(x0)))

MARK(2ndspos(pi(y0), rnil)) → A__2NDSPOS(a__pi(mark(y0)), rnil)

MARK(2ndspos(pi(y0), cons(x0, x1))) → A__2NDSPOS(a__pi(mark(y0)), cons(mark(x0), x1))

MARK(2ndspos(pi(y0), from(x0))) → A__2NDSPOS(a__pi(mark(y0)), a__from(mark(x0)))

MARK(2ndspos(pi(y0), cons2(x0, x1))) → A__2NDSPOS(a__pi(mark(y0)), cons2(x0, mark(x1)))

MARK(2ndspos(pi(y0), times(x0, x1))) → A__2NDSPOS(a__pi(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(pi(y0), pi(x0))) → A__2NDSPOS(a__pi(mark(y0)), a__pi(mark(x0)))

MARK(2ndspos(pi(y0), nil)) → A__2NDSPOS(a__pi(mark(y0)), nil)

MARK(2ndspos(pi(y0), 2ndsneg(x0, x1))) → A__2NDSPOS(a__pi(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndspos(pi(y0), square(x0))) → A__2NDSPOS(a__pi(mark(y0)), a__square(mark(x0)))

MARK(2ndspos(pi(y0), rcons(x0, x1))) → A__2NDSPOS(a__pi(mark(y0)), rcons(mark(x0), mark(x1)))

MARK(2ndspos(pi(y0), s(x0))) → A__2NDSPOS(a__pi(mark(y0)), s(mark(x0)))

MARK(2ndspos(pi(y0), 2ndspos(x0, x1))) → A__2NDSPOS(a__pi(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndspos(pi(y0), 0)) → A__2NDSPOS(a__pi(mark(y0)), 0)



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ Narrowing
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ QDP
              ↳ Narrowing
                ↳ QDP
                  ↳ DependencyGraphProof
                    ↳ QDP
                      ↳ Narrowing
                        ↳ QDP
                          ↳ DependencyGraphProof
                            ↳ QDP
                              ↳ Narrowing
                                ↳ QDP
                                  ↳ DependencyGraphProof
                                    ↳ QDP
                                      ↳ Narrowing
                                        ↳ QDP
                                          ↳ DependencyGraphProof
                                            ↳ QDP
                                              ↳ Narrowing
                                                ↳ QDP
                                                  ↳ DependencyGraphProof
                                                    ↳ QDP
                                                      ↳ Narrowing
                                                        ↳ QDP
                                                          ↳ DependencyGraphProof
                                                            ↳ QDP
                                                              ↳ Narrowing
                                                                ↳ QDP
                                                                  ↳ DependencyGraphProof
                                                                    ↳ QDP
                                                                      ↳ Narrowing
                                                                        ↳ QDP
                                                                          ↳ DependencyGraphProof
                                                                            ↳ QDP
                                                                              ↳ Narrowing
                                                                                ↳ QDP
                                                                                  ↳ DependencyGraphProof
                                                                                    ↳ QDP
                                                                                      ↳ Narrowing
                                                                                        ↳ QDP
                                                                                          ↳ DependencyGraphProof

Q DP problem:
The TRS P consists of the following rules:

MARK(2ndspos(square(y0), plus(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(2ndsneg(square(x0), y1)) → A__2NDSNEG(a__square(mark(x0)), mark(y1))

MARK(2ndspos(pi(y0), posrecip(x0))) → A__2NDSPOS(a__pi(mark(y0)), posrecip(mark(x0)))

A__2NDSPOS(s(y0), cons(y1, 2ndspos(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__2ndspos(mark(x0), mark(x1))))

A__PI(times(x0, x1)) → A__2NDSPOS(a__times(mark(x0), mark(x1)), a__from(0))

A__2NDSNEG(s(N), cons(X, Z)) → A__2NDSNEG(s(mark(N)), cons2(X, mark(Z)))

MARK(2ndspos(pi(y0), cons2(x0, x1))) → A__2NDSPOS(a__pi(mark(y0)), cons2(x0, mark(x1)))

MARK(times(times(x0, x1), y1)) → A__TIMES(a__times(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(X1, X2)) → MARK(X2)

MARK(2ndspos(from(y0), cons2(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), cons2(x0, mark(x1)))

MARK(plus(plus(x0, x1), y1)) → A__PLUS(a__plus(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(from(y0), 2ndsneg(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndspos(s(y0), from(x0))) → A__2NDSPOS(s(mark(y0)), a__from(mark(x0)))

MARK(2ndspos(2ndsneg(y0, y1), from(x0))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndspos(pi(y0), 2ndsneg(x0, x1))) → A__2NDSPOS(a__pi(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndsneg(2ndsneg(x0, x1), y1)) → A__2NDSNEG(a__2ndsneg(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(2ndsneg(y0, y1), pi(x0))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__pi(mark(x0)))

A__2NDSNEG(s(N), cons(X, Z)) → MARK(Z)

MARK(2ndspos(pi(y0), negrecip(x0))) → A__2NDSPOS(a__pi(mark(y0)), negrecip(mark(x0)))

A__TIMES(s(X), Y) → MARK(Y)

MARK(2ndsneg(2ndspos(x0, x1), y1)) → A__2NDSNEG(a__2ndspos(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(pi(y0), rnil)) → A__2NDSPOS(a__pi(mark(y0)), rnil)

A__2NDSPOS(s(y0), cons(y1, plus(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__plus(mark(x0), mark(x1))))

MARK(2ndspos(pi(y0), from(x0))) → A__2NDSPOS(a__pi(mark(y0)), a__from(mark(x0)))

MARK(2ndspos(from(y0), pi(x0))) → A__2NDSPOS(a__from(mark(y0)), a__pi(mark(x0)))

MARK(2ndspos(square(y0), 2ndsneg(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndspos(2ndsneg(y0, y1), square(x0))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__square(mark(x0)))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → A__2NDSNEG(mark(N), mark(Z))

A__2NDSNEG(s(N), cons(X, Z)) → MARK(N)

MARK(times(2ndsneg(x0, x1), y1)) → A__TIMES(a__2ndsneg(mark(x0), mark(x1)), mark(y1))

MARK(2ndsneg(plus(x0, x1), y1)) → A__2NDSNEG(a__plus(mark(x0), mark(x1)), mark(y1))

MARK(2ndsneg(from(x0), y1)) → A__2NDSNEG(a__from(mark(x0)), mark(y1))

MARK(2ndspos(from(y0), from(x0))) → A__2NDSPOS(a__from(mark(y0)), a__from(mark(x0)))

MARK(2ndspos(pi(y0), pi(x0))) → A__2NDSPOS(a__pi(mark(y0)), a__pi(mark(x0)))

MARK(2ndspos(pi(y0), s(x0))) → A__2NDSPOS(a__pi(mark(y0)), s(mark(x0)))

A__TIMES(s(X), Y) → A__PLUS(mark(Y), a__times(mark(X), mark(Y)))

MARK(2ndspos(s(y0), pi(x0))) → A__2NDSPOS(s(mark(y0)), a__pi(mark(x0)))

MARK(2ndspos(square(y0), cons2(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), cons2(x0, mark(x1)))

A__PI(2ndspos(x0, x1)) → A__2NDSPOS(a__2ndspos(mark(x0), mark(x1)), a__from(0))

MARK(plus(square(x0), y1)) → A__PLUS(a__square(mark(x0)), mark(y1))

MARK(2ndspos(pi(y0), 0)) → A__2NDSPOS(a__pi(mark(y0)), 0)

MARK(plus(s(x0), y1)) → A__PLUS(s(mark(x0)), mark(y1))

MARK(2ndspos(plus(x0, x1), y1)) → A__2NDSPOS(a__plus(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(square(y0), cons(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), cons(mark(x0), x1))

A__PI(plus(x0, x1)) → A__2NDSPOS(a__plus(mark(x0), mark(x1)), a__from(0))

MARK(cons2(X1, X2)) → MARK(X2)

MARK(s(X)) → MARK(X)

MARK(2ndspos(X1, X2)) → MARK(X1)

MARK(pi(X)) → MARK(X)

MARK(plus(X1, X2)) → MARK(X2)

A__2NDSPOS(s(N), cons(X, Z)) → MARK(Z)

MARK(pi(X)) → A__PI(mark(X))

MARK(plus(2ndsneg(x0, x1), y1)) → A__PLUS(a__2ndsneg(mark(x0), mark(x1)), mark(y1))

MARK(2ndsneg(X1, X2)) → MARK(X2)

MARK(2ndspos(square(y0), times(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(2ndsneg(y0, y1), cons2(x0, x1))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(times(square(x0), y1)) → A__TIMES(a__square(mark(x0)), mark(y1))

MARK(from(X)) → A__FROM(mark(X))

MARK(2ndspos(s(y0), plus(x0, x1))) → A__2NDSPOS(s(mark(y0)), a__plus(mark(x0), mark(x1)))

A__PI(X) → A__FROM(0)

MARK(2ndspos(pi(y0), cons(x0, x1))) → A__2NDSPOS(a__pi(mark(y0)), cons(mark(x0), x1))

MARK(times(from(x0), y1)) → A__TIMES(a__from(mark(x0)), mark(y1))

MARK(plus(X1, X2)) → MARK(X1)

A__2NDSPOS(s(y0), cons(y1, times(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__times(mark(x0), mark(x1))))

MARK(2ndspos(pi(y0), nil)) → A__2NDSPOS(a__pi(mark(y0)), nil)

A__PI(X) → MARK(X)

MARK(2ndspos(s(y0), cons(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons(mark(x0), x1))

MARK(times(s(x0), y1)) → A__TIMES(s(mark(x0)), mark(y1))

MARK(2ndspos(pi(y0), 2ndspos(x0, x1))) → A__2NDSPOS(a__pi(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndspos(from(y0), cons(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), cons(mark(x0), x1))

MARK(2ndsneg(s(x0), y1)) → A__2NDSNEG(s(mark(x0)), mark(y1))

A__PLUS(s(X), Y) → MARK(Y)

A__2NDSPOS(s(y0), cons(y1, pi(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__pi(mark(x0))))

A__TIMES(s(X), Y) → MARK(X)

MARK(2ndspos(pi(y0), plus(x0, x1))) → A__2NDSPOS(a__pi(mark(y0)), a__plus(mark(x0), mark(x1)))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(N)

MARK(plus(2ndspos(x0, x1), y1)) → A__PLUS(a__2ndspos(mark(x0), mark(x1)), mark(y1))

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(N)

MARK(cons(X1, X2)) → MARK(X1)

A__PI(from(x0)) → A__2NDSPOS(a__from(mark(x0)), a__from(0))

MARK(2ndspos(square(y0), 2ndspos(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

MARK(times(X1, X2)) → MARK(X2)

MARK(2ndspos(s(y0), 2ndspos(x0, x1))) → A__2NDSPOS(s(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndspos(2ndsneg(y0, y1), 2ndsneg(x0, x1))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__2ndsneg(mark(x0), mark(x1)))

A__PI(square(x0)) → A__2NDSPOS(a__square(mark(x0)), a__from(0))

MARK(2ndspos(s(y0), times(x0, x1))) → A__2NDSPOS(s(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndsneg(X1, X2)) → MARK(X1)

MARK(2ndspos(s(y0), 2ndsneg(x0, x1))) → A__2NDSPOS(s(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndspos(2ndsneg(y0, y1), plus(x0, x1))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

A__PI(2ndsneg(x0, x1)) → A__2NDSPOS(a__2ndsneg(mark(x0), mark(x1)), a__from(0))

MARK(times(pi(x0), y1)) → A__TIMES(a__pi(mark(x0)), mark(y1))

MARK(plus(pi(x0), y1)) → A__PLUS(a__pi(mark(x0)), mark(y1))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(Y)

MARK(rcons(X1, X2)) → MARK(X1)

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(Z)

MARK(square(X)) → MARK(X)

MARK(2ndspos(2ndsneg(y0, y1), 2ndspos(x0, x1))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndspos(pi(y0), times(x0, x1))) → A__2NDSPOS(a__pi(mark(y0)), a__times(mark(x0), mark(x1)))

A__PI(s(x0)) → A__2NDSPOS(s(mark(x0)), a__from(0))

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(Y)

A__FROM(X) → MARK(X)

MARK(times(plus(x0, x1), y1)) → A__TIMES(a__plus(mark(x0), mark(x1)), mark(y1))

A__2NDSPOS(s(y0), cons(y1, from(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__from(mark(x0))))

MARK(2ndspos(times(x0, x1), y1)) → A__2NDSPOS(a__times(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(pi(y0), square(x0))) → A__2NDSPOS(a__pi(mark(y0)), a__square(mark(x0)))

MARK(2ndspos(square(y0), pi(x0))) → A__2NDSPOS(a__square(mark(y0)), a__pi(mark(x0)))

MARK(rcons(X1, X2)) → MARK(X2)

MARK(2ndspos(s(y0), square(x0))) → A__2NDSPOS(s(mark(y0)), a__square(mark(x0)))

MARK(times(2ndspos(x0, x1), y1)) → A__TIMES(a__2ndspos(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(s(y0), cons2(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(x0, mark(x1)))

A__TIMES(s(X), Y) → A__TIMES(mark(X), mark(Y))

A__2NDSPOS(s(y0), cons(y1, cons(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, cons(mark(x0), x1)))

A__PLUS(s(X), Y) → MARK(X)

MARK(times(X1, X2)) → MARK(X1)

MARK(plus(times(x0, x1), y1)) → A__PLUS(a__times(mark(x0), mark(x1)), mark(y1))

A__SQUARE(X) → MARK(X)

MARK(2ndsneg(times(x0, x1), y1)) → A__2NDSNEG(a__times(mark(x0), mark(x1)), mark(y1))

A__2NDSPOS(s(y0), cons(y1, 2ndsneg(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__2ndsneg(mark(x0), mark(x1))))

MARK(negrecip(X)) → MARK(X)

A__2NDSPOS(s(N), cons(X, Z)) → MARK(N)

A__PLUS(0, Y) → MARK(Y)

MARK(2ndspos(2ndspos(x0, x1), y1)) → A__2NDSPOS(a__2ndspos(mark(x0), mark(x1)), mark(y1))

MARK(posrecip(X)) → MARK(X)

MARK(2ndspos(square(y0), from(x0))) → A__2NDSPOS(a__square(mark(y0)), a__from(mark(x0)))

MARK(2ndsneg(pi(x0), y1)) → A__2NDSNEG(a__pi(mark(x0)), mark(y1))

MARK(2ndspos(from(y0), square(x0))) → A__2NDSPOS(a__from(mark(y0)), a__square(mark(x0)))

MARK(2ndspos(from(y0), times(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), a__times(mark(x0), mark(x1)))

A__2NDSPOS(s(y0), cons(y1, square(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__square(mark(x0))))

MARK(2ndspos(2ndsneg(y0, y1), times(x0, x1))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(plus(0, y1)) → A__PLUS(0, mark(y1))

MARK(from(X)) → MARK(X)

A__PLUS(s(X), Y) → A__PLUS(mark(X), mark(Y))

MARK(2ndspos(square(y0), square(x0))) → A__2NDSPOS(a__square(mark(y0)), a__square(mark(x0)))

A__SQUARE(X) → A__TIMES(mark(X), mark(X))

A__PI(pi(x0)) → A__2NDSPOS(a__pi(mark(x0)), a__from(0))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(Z)

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → A__2NDSPOS(mark(N), mark(Z))

MARK(plus(from(x0), y1)) → A__PLUS(a__from(mark(x0)), mark(y1))

MARK(square(X)) → A__SQUARE(mark(X))

MARK(2ndspos(from(y0), plus(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(pi(y0), rcons(x0, x1))) → A__2NDSPOS(a__pi(mark(y0)), rcons(mark(x0), mark(x1)))

MARK(2ndspos(from(y0), 2ndspos(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndspos(2ndsneg(y0, y1), cons(x0, x1))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), cons(mark(x0), x1))

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))

a__2ndspos(0, Z) → rnil

a__2ndspos(s(N), cons(X, Z)) → a__2ndspos(s(mark(N)), cons2(X, mark(Z)))

a__2ndspos(s(N), cons2(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))

a__2ndsneg(0, Z) → rnil

a__2ndsneg(s(N), cons(X, Z)) → a__2ndsneg(s(mark(N)), cons2(X, mark(Z)))

a__2ndsneg(s(N), cons2(X, cons(Y, Z))) → rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z)))

a__pi(X) → a__2ndspos(mark(X), a__from(0))

a__plus(0, Y) → mark(Y)

a__plus(s(X), Y) → s(a__plus(mark(X), mark(Y)))

a__times(0, Y) → 0

a__times(s(X), Y) → a__plus(mark(Y), a__times(mark(X), mark(Y)))

a__square(X) → a__times(mark(X), mark(X))

mark(from(X)) → a__from(mark(X))

mark(2ndspos(X1, X2)) → a__2ndspos(mark(X1), mark(X2))

mark(2ndsneg(X1, X2)) → a__2ndsneg(mark(X1), mark(X2))

mark(pi(X)) → a__pi(mark(X))

mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))

mark(times(X1, X2)) → a__times(mark(X1), mark(X2))

mark(square(X)) → a__square(mark(X))

mark(0) → 0

mark(s(X)) → s(mark(X))

mark(posrecip(X)) → posrecip(mark(X))

mark(negrecip(X)) → negrecip(mark(X))

mark(nil) → nil

mark(cons(X1, X2)) → cons(mark(X1), X2)

mark(cons2(X1, X2)) → cons2(X1, mark(X2))

mark(rnil) → rnil

mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))

a__from(X) → from(X)

a__2ndspos(X1, X2) → 2ndspos(X1, X2)

a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)

a__pi(X) → pi(X)

a__plus(X1, X2) → plus(X1, X2)

a__times(X1, X2) → times(X1, X2)

a__square(X) → square(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 7 less nodes.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ Narrowing
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ QDP
              ↳ Narrowing
                ↳ QDP
                  ↳ DependencyGraphProof
                    ↳ QDP
                      ↳ Narrowing
                        ↳ QDP
                          ↳ DependencyGraphProof
                            ↳ QDP
                              ↳ Narrowing
                                ↳ QDP
                                  ↳ DependencyGraphProof
                                    ↳ QDP
                                      ↳ Narrowing
                                        ↳ QDP
                                          ↳ DependencyGraphProof
                                            ↳ QDP
                                              ↳ Narrowing
                                                ↳ QDP
                                                  ↳ DependencyGraphProof
                                                    ↳ QDP
                                                      ↳ Narrowing
                                                        ↳ QDP
                                                          ↳ DependencyGraphProof
                                                            ↳ QDP
                                                              ↳ Narrowing
                                                                ↳ QDP
                                                                  ↳ DependencyGraphProof
                                                                    ↳ QDP
                                                                      ↳ Narrowing
                                                                        ↳ QDP
                                                                          ↳ DependencyGraphProof
                                                                            ↳ QDP
                                                                              ↳ Narrowing
                                                                                ↳ QDP
                                                                                  ↳ DependencyGraphProof
                                                                                    ↳ QDP
                                                                                      ↳ Narrowing
                                                                                        ↳ QDP
                                                                                          ↳ DependencyGraphProof
                                                                                            ↳ QDP
                                                                                              ↳ Narrowing

Q DP problem:
The TRS P consists of the following rules:

MARK(2ndsneg(square(x0), y1)) → A__2NDSNEG(a__square(mark(x0)), mark(y1))

MARK(2ndspos(square(y0), plus(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__plus(mark(x0), mark(x1)))

A__PI(times(x0, x1)) → A__2NDSPOS(a__times(mark(x0), mark(x1)), a__from(0))

A__2NDSPOS(s(y0), cons(y1, 2ndspos(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__2ndspos(mark(x0), mark(x1))))

A__2NDSNEG(s(N), cons(X, Z)) → A__2NDSNEG(s(mark(N)), cons2(X, mark(Z)))

MARK(2ndspos(pi(y0), cons2(x0, x1))) → A__2NDSPOS(a__pi(mark(y0)), cons2(x0, mark(x1)))

MARK(2ndspos(X1, X2)) → MARK(X2)

MARK(times(times(x0, x1), y1)) → A__TIMES(a__times(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(from(y0), cons2(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), cons2(x0, mark(x1)))

MARK(plus(plus(x0, x1), y1)) → A__PLUS(a__plus(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(2ndsneg(y0, y1), from(x0))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndspos(s(y0), from(x0))) → A__2NDSPOS(s(mark(y0)), a__from(mark(x0)))

MARK(2ndspos(from(y0), 2ndsneg(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndspos(pi(y0), 2ndsneg(x0, x1))) → A__2NDSPOS(a__pi(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndspos(2ndsneg(y0, y1), pi(x0))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__pi(mark(x0)))

MARK(2ndsneg(2ndsneg(x0, x1), y1)) → A__2NDSNEG(a__2ndsneg(mark(x0), mark(x1)), mark(y1))

A__2NDSNEG(s(N), cons(X, Z)) → MARK(Z)

A__TIMES(s(X), Y) → MARK(Y)

MARK(2ndsneg(2ndspos(x0, x1), y1)) → A__2NDSNEG(a__2ndspos(mark(x0), mark(x1)), mark(y1))

A__2NDSPOS(s(y0), cons(y1, plus(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__plus(mark(x0), mark(x1))))

MARK(2ndspos(pi(y0), from(x0))) → A__2NDSPOS(a__pi(mark(y0)), a__from(mark(x0)))

MARK(2ndspos(2ndsneg(y0, y1), square(x0))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndspos(square(y0), 2ndsneg(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndspos(from(y0), pi(x0))) → A__2NDSPOS(a__from(mark(y0)), a__pi(mark(x0)))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → A__2NDSNEG(mark(N), mark(Z))

A__2NDSNEG(s(N), cons(X, Z)) → MARK(N)

MARK(2ndsneg(plus(x0, x1), y1)) → A__2NDSNEG(a__plus(mark(x0), mark(x1)), mark(y1))

MARK(times(2ndsneg(x0, x1), y1)) → A__TIMES(a__2ndsneg(mark(x0), mark(x1)), mark(y1))

MARK(2ndsneg(from(x0), y1)) → A__2NDSNEG(a__from(mark(x0)), mark(y1))

MARK(2ndspos(from(y0), from(x0))) → A__2NDSPOS(a__from(mark(y0)), a__from(mark(x0)))

MARK(2ndspos(pi(y0), pi(x0))) → A__2NDSPOS(a__pi(mark(y0)), a__pi(mark(x0)))

MARK(2ndspos(s(y0), pi(x0))) → A__2NDSPOS(s(mark(y0)), a__pi(mark(x0)))

A__TIMES(s(X), Y) → A__PLUS(mark(Y), a__times(mark(X), mark(Y)))

A__PI(2ndspos(x0, x1)) → A__2NDSPOS(a__2ndspos(mark(x0), mark(x1)), a__from(0))

MARK(2ndspos(square(y0), cons2(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), cons2(x0, mark(x1)))

MARK(plus(square(x0), y1)) → A__PLUS(a__square(mark(x0)), mark(y1))

MARK(2ndspos(plus(x0, x1), y1)) → A__2NDSPOS(a__plus(mark(x0), mark(x1)), mark(y1))

MARK(plus(s(x0), y1)) → A__PLUS(s(mark(x0)), mark(y1))

A__PI(plus(x0, x1)) → A__2NDSPOS(a__plus(mark(x0), mark(x1)), a__from(0))

MARK(2ndspos(square(y0), cons(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), cons(mark(x0), x1))

MARK(cons2(X1, X2)) → MARK(X2)

MARK(2ndspos(X1, X2)) → MARK(X1)

MARK(s(X)) → MARK(X)

MARK(pi(X)) → MARK(X)

MARK(plus(X1, X2)) → MARK(X2)

A__2NDSPOS(s(N), cons(X, Z)) → MARK(Z)

MARK(pi(X)) → A__PI(mark(X))

MARK(plus(2ndsneg(x0, x1), y1)) → A__PLUS(a__2ndsneg(mark(x0), mark(x1)), mark(y1))

MARK(2ndsneg(X1, X2)) → MARK(X2)

MARK(2ndspos(square(y0), times(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(2ndsneg(y0, y1), cons2(x0, x1))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(times(square(x0), y1)) → A__TIMES(a__square(mark(x0)), mark(y1))

A__PI(X) → A__FROM(0)

MARK(2ndspos(s(y0), plus(x0, x1))) → A__2NDSPOS(s(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(from(X)) → A__FROM(mark(X))

MARK(2ndspos(pi(y0), cons(x0, x1))) → A__2NDSPOS(a__pi(mark(y0)), cons(mark(x0), x1))

MARK(plus(X1, X2)) → MARK(X1)

MARK(times(from(x0), y1)) → A__TIMES(a__from(mark(x0)), mark(y1))

A__2NDSPOS(s(y0), cons(y1, times(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__times(mark(x0), mark(x1))))

MARK(2ndspos(s(y0), cons(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons(mark(x0), x1))

A__PI(X) → MARK(X)

MARK(times(s(x0), y1)) → A__TIMES(s(mark(x0)), mark(y1))

MARK(2ndspos(pi(y0), 2ndspos(x0, x1))) → A__2NDSPOS(a__pi(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndsneg(s(x0), y1)) → A__2NDSNEG(s(mark(x0)), mark(y1))

MARK(2ndspos(from(y0), cons(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), cons(mark(x0), x1))

A__PLUS(s(X), Y) → MARK(Y)

A__TIMES(s(X), Y) → MARK(X)

A__2NDSPOS(s(y0), cons(y1, pi(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__pi(mark(x0))))

MARK(2ndspos(pi(y0), plus(x0, x1))) → A__2NDSPOS(a__pi(mark(y0)), a__plus(mark(x0), mark(x1)))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(N)

MARK(plus(2ndspos(x0, x1), y1)) → A__PLUS(a__2ndspos(mark(x0), mark(x1)), mark(y1))

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(N)

A__PI(from(x0)) → A__2NDSPOS(a__from(mark(x0)), a__from(0))

MARK(cons(X1, X2)) → MARK(X1)

MARK(2ndspos(square(y0), 2ndspos(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndspos(s(y0), 2ndspos(x0, x1))) → A__2NDSPOS(s(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

MARK(times(X1, X2)) → MARK(X2)

A__PI(square(x0)) → A__2NDSPOS(a__square(mark(x0)), a__from(0))

MARK(2ndspos(2ndsneg(y0, y1), 2ndsneg(x0, x1))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndspos(s(y0), times(x0, x1))) → A__2NDSPOS(s(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndsneg(X1, X2)) → MARK(X1)

MARK(2ndspos(s(y0), 2ndsneg(x0, x1))) → A__2NDSPOS(s(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndspos(2ndsneg(y0, y1), plus(x0, x1))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

A__PI(2ndsneg(x0, x1)) → A__2NDSPOS(a__2ndsneg(mark(x0), mark(x1)), a__from(0))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(Y)

MARK(plus(pi(x0), y1)) → A__PLUS(a__pi(mark(x0)), mark(y1))

MARK(times(pi(x0), y1)) → A__TIMES(a__pi(mark(x0)), mark(y1))

MARK(rcons(X1, X2)) → MARK(X1)

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(Z)

MARK(square(X)) → MARK(X)

MARK(2ndspos(2ndsneg(y0, y1), 2ndspos(x0, x1))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndspos(pi(y0), times(x0, x1))) → A__2NDSPOS(a__pi(mark(y0)), a__times(mark(x0), mark(x1)))

A__PI(s(x0)) → A__2NDSPOS(s(mark(x0)), a__from(0))

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(Y)

A__FROM(X) → MARK(X)

MARK(times(plus(x0, x1), y1)) → A__TIMES(a__plus(mark(x0), mark(x1)), mark(y1))

A__2NDSPOS(s(y0), cons(y1, from(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__from(mark(x0))))

MARK(2ndspos(times(x0, x1), y1)) → A__2NDSPOS(a__times(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(pi(y0), square(x0))) → A__2NDSPOS(a__pi(mark(y0)), a__square(mark(x0)))

MARK(2ndspos(square(y0), pi(x0))) → A__2NDSPOS(a__square(mark(y0)), a__pi(mark(x0)))

MARK(2ndspos(s(y0), square(x0))) → A__2NDSPOS(s(mark(y0)), a__square(mark(x0)))

MARK(rcons(X1, X2)) → MARK(X2)

MARK(times(2ndspos(x0, x1), y1)) → A__TIMES(a__2ndspos(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(s(y0), cons2(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(x0, mark(x1)))

A__TIMES(s(X), Y) → A__TIMES(mark(X), mark(Y))

A__PLUS(s(X), Y) → MARK(X)

A__2NDSPOS(s(y0), cons(y1, cons(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, cons(mark(x0), x1)))

MARK(times(X1, X2)) → MARK(X1)

MARK(plus(times(x0, x1), y1)) → A__PLUS(a__times(mark(x0), mark(x1)), mark(y1))

A__SQUARE(X) → MARK(X)

MARK(2ndsneg(times(x0, x1), y1)) → A__2NDSNEG(a__times(mark(x0), mark(x1)), mark(y1))

A__2NDSPOS(s(y0), cons(y1, 2ndsneg(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__2ndsneg(mark(x0), mark(x1))))

A__2NDSPOS(s(N), cons(X, Z)) → MARK(N)

MARK(negrecip(X)) → MARK(X)

A__PLUS(0, Y) → MARK(Y)

MARK(2ndspos(2ndspos(x0, x1), y1)) → A__2NDSPOS(a__2ndspos(mark(x0), mark(x1)), mark(y1))

MARK(posrecip(X)) → MARK(X)

MARK(2ndsneg(pi(x0), y1)) → A__2NDSNEG(a__pi(mark(x0)), mark(y1))

MARK(2ndspos(square(y0), from(x0))) → A__2NDSPOS(a__square(mark(y0)), a__from(mark(x0)))

MARK(2ndspos(from(y0), square(x0))) → A__2NDSPOS(a__from(mark(y0)), a__square(mark(x0)))

MARK(2ndspos(from(y0), times(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), a__times(mark(x0), mark(x1)))

A__2NDSPOS(s(y0), cons(y1, square(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__square(mark(x0))))

MARK(2ndspos(2ndsneg(y0, y1), times(x0, x1))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(plus(0, y1)) → A__PLUS(0, mark(y1))

MARK(from(X)) → MARK(X)

A__PLUS(s(X), Y) → A__PLUS(mark(X), mark(Y))

A__SQUARE(X) → A__TIMES(mark(X), mark(X))

MARK(2ndspos(square(y0), square(x0))) → A__2NDSPOS(a__square(mark(y0)), a__square(mark(x0)))

A__PI(pi(x0)) → A__2NDSPOS(a__pi(mark(x0)), a__from(0))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(Z)

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → A__2NDSPOS(mark(N), mark(Z))

MARK(plus(from(x0), y1)) → A__PLUS(a__from(mark(x0)), mark(y1))

MARK(square(X)) → A__SQUARE(mark(X))

MARK(2ndspos(from(y0), plus(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(from(y0), 2ndspos(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndspos(2ndsneg(y0, y1), cons(x0, x1))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), cons(mark(x0), x1))

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))

a__2ndspos(0, Z) → rnil

a__2ndspos(s(N), cons(X, Z)) → a__2ndspos(s(mark(N)), cons2(X, mark(Z)))

a__2ndspos(s(N), cons2(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))

a__2ndsneg(0, Z) → rnil

a__2ndsneg(s(N), cons(X, Z)) → a__2ndsneg(s(mark(N)), cons2(X, mark(Z)))

a__2ndsneg(s(N), cons2(X, cons(Y, Z))) → rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z)))

a__pi(X) → a__2ndspos(mark(X), a__from(0))

a__plus(0, Y) → mark(Y)

a__plus(s(X), Y) → s(a__plus(mark(X), mark(Y)))

a__times(0, Y) → 0

a__times(s(X), Y) → a__plus(mark(Y), a__times(mark(X), mark(Y)))

a__square(X) → a__times(mark(X), mark(X))

mark(from(X)) → a__from(mark(X))

mark(2ndspos(X1, X2)) → a__2ndspos(mark(X1), mark(X2))

mark(2ndsneg(X1, X2)) → a__2ndsneg(mark(X1), mark(X2))

mark(pi(X)) → a__pi(mark(X))

mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))

mark(times(X1, X2)) → a__times(mark(X1), mark(X2))

mark(square(X)) → a__square(mark(X))

mark(0) → 0

mark(s(X)) → s(mark(X))

mark(posrecip(X)) → posrecip(mark(X))

mark(negrecip(X)) → negrecip(mark(X))

mark(nil) → nil

mark(cons(X1, X2)) → cons(mark(X1), X2)

mark(cons2(X1, X2)) → cons2(X1, mark(X2))

mark(rnil) → rnil

mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))

a__from(X) → from(X)

a__2ndspos(X1, X2) → 2ndspos(X1, X2)

a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)

a__pi(X) → pi(X)

a__plus(X1, X2) → plus(X1, X2)

a__times(X1, X2) → times(X1, X2)

a__square(X) → square(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule MARK(2ndspos(times(x0, x1), y1)) → A__2NDSPOS(a__times(mark(x0), mark(x1)), mark(y1)) at position [1] we obtained the following new rules:

MARK(2ndspos(times(y0, y1), rnil)) → A__2NDSPOS(a__times(mark(y0), mark(y1)), rnil)

MARK(2ndspos(times(y0, y1), posrecip(x0))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), posrecip(mark(x0)))

MARK(2ndspos(times(y0, y1), cons(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), cons(mark(x0), x1))

MARK(2ndspos(times(y0, y1), plus(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(times(y0, y1), s(x0))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), s(mark(x0)))

MARK(2ndspos(times(y0, y1), cons2(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(2ndspos(times(y0, y1), negrecip(x0))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), negrecip(mark(x0)))

MARK(2ndspos(times(y0, y1), 0)) → A__2NDSPOS(a__times(mark(y0), mark(y1)), 0)

MARK(2ndspos(times(y0, y1), from(x0))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndspos(times(y0, y1), 2ndsneg(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndspos(times(y0, y1), square(x0))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndspos(times(y0, y1), pi(x0))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__pi(mark(x0)))

MARK(2ndspos(times(y0, y1), rcons(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), rcons(mark(x0), mark(x1)))

MARK(2ndspos(times(y0, y1), 2ndspos(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndspos(times(y0, y1), times(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(times(y0, y1), nil)) → A__2NDSPOS(a__times(mark(y0), mark(y1)), nil)



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ Narrowing
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ QDP
              ↳ Narrowing
                ↳ QDP
                  ↳ DependencyGraphProof
                    ↳ QDP
                      ↳ Narrowing
                        ↳ QDP
                          ↳ DependencyGraphProof
                            ↳ QDP
                              ↳ Narrowing
                                ↳ QDP
                                  ↳ DependencyGraphProof
                                    ↳ QDP
                                      ↳ Narrowing
                                        ↳ QDP
                                          ↳ DependencyGraphProof
                                            ↳ QDP
                                              ↳ Narrowing
                                                ↳ QDP
                                                  ↳ DependencyGraphProof
                                                    ↳ QDP
                                                      ↳ Narrowing
                                                        ↳ QDP
                                                          ↳ DependencyGraphProof
                                                            ↳ QDP
                                                              ↳ Narrowing
                                                                ↳ QDP
                                                                  ↳ DependencyGraphProof
                                                                    ↳ QDP
                                                                      ↳ Narrowing
                                                                        ↳ QDP
                                                                          ↳ DependencyGraphProof
                                                                            ↳ QDP
                                                                              ↳ Narrowing
                                                                                ↳ QDP
                                                                                  ↳ DependencyGraphProof
                                                                                    ↳ QDP
                                                                                      ↳ Narrowing
                                                                                        ↳ QDP
                                                                                          ↳ DependencyGraphProof
                                                                                            ↳ QDP
                                                                                              ↳ Narrowing
                                                                                                ↳ QDP
                                                                                                  ↳ DependencyGraphProof

Q DP problem:
The TRS P consists of the following rules:

MARK(2ndspos(square(y0), plus(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(2ndsneg(square(x0), y1)) → A__2NDSNEG(a__square(mark(x0)), mark(y1))

A__2NDSPOS(s(y0), cons(y1, 2ndspos(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__2ndspos(mark(x0), mark(x1))))

A__PI(times(x0, x1)) → A__2NDSPOS(a__times(mark(x0), mark(x1)), a__from(0))

A__2NDSNEG(s(N), cons(X, Z)) → A__2NDSNEG(s(mark(N)), cons2(X, mark(Z)))

MARK(times(times(x0, x1), y1)) → A__TIMES(a__times(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(X1, X2)) → MARK(X2)

MARK(2ndspos(pi(y0), cons2(x0, x1))) → A__2NDSPOS(a__pi(mark(y0)), cons2(x0, mark(x1)))

MARK(2ndspos(from(y0), cons2(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), cons2(x0, mark(x1)))

MARK(plus(plus(x0, x1), y1)) → A__PLUS(a__plus(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(from(y0), 2ndsneg(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndspos(s(y0), from(x0))) → A__2NDSPOS(s(mark(y0)), a__from(mark(x0)))

MARK(2ndspos(2ndsneg(y0, y1), from(x0))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndsneg(2ndsneg(x0, x1), y1)) → A__2NDSNEG(a__2ndsneg(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(2ndsneg(y0, y1), pi(x0))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__pi(mark(x0)))

MARK(2ndspos(pi(y0), 2ndsneg(x0, x1))) → A__2NDSPOS(a__pi(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

A__2NDSNEG(s(N), cons(X, Z)) → MARK(Z)

A__TIMES(s(X), Y) → MARK(Y)

MARK(2ndsneg(2ndspos(x0, x1), y1)) → A__2NDSNEG(a__2ndspos(mark(x0), mark(x1)), mark(y1))

A__2NDSPOS(s(y0), cons(y1, plus(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__plus(mark(x0), mark(x1))))

MARK(2ndspos(from(y0), pi(x0))) → A__2NDSPOS(a__from(mark(y0)), a__pi(mark(x0)))

MARK(2ndspos(square(y0), 2ndsneg(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndspos(2ndsneg(y0, y1), square(x0))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndspos(pi(y0), from(x0))) → A__2NDSPOS(a__pi(mark(y0)), a__from(mark(x0)))

A__2NDSNEG(s(N), cons(X, Z)) → MARK(N)

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → A__2NDSNEG(mark(N), mark(Z))

MARK(times(2ndsneg(x0, x1), y1)) → A__TIMES(a__2ndsneg(mark(x0), mark(x1)), mark(y1))

MARK(2ndsneg(plus(x0, x1), y1)) → A__2NDSNEG(a__plus(mark(x0), mark(x1)), mark(y1))

MARK(2ndsneg(from(x0), y1)) → A__2NDSNEG(a__from(mark(x0)), mark(y1))

MARK(2ndspos(from(y0), from(x0))) → A__2NDSPOS(a__from(mark(y0)), a__from(mark(x0)))

MARK(2ndspos(times(y0, y1), rcons(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), rcons(mark(x0), mark(x1)))

MARK(2ndspos(pi(y0), pi(x0))) → A__2NDSPOS(a__pi(mark(y0)), a__pi(mark(x0)))

MARK(2ndspos(times(y0, y1), times(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

A__TIMES(s(X), Y) → A__PLUS(mark(Y), a__times(mark(X), mark(Y)))

MARK(2ndspos(s(y0), pi(x0))) → A__2NDSPOS(s(mark(y0)), a__pi(mark(x0)))

MARK(2ndspos(square(y0), cons2(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), cons2(x0, mark(x1)))

A__PI(2ndspos(x0, x1)) → A__2NDSPOS(a__2ndspos(mark(x0), mark(x1)), a__from(0))

MARK(plus(square(x0), y1)) → A__PLUS(a__square(mark(x0)), mark(y1))

MARK(plus(s(x0), y1)) → A__PLUS(s(mark(x0)), mark(y1))

MARK(2ndspos(plus(x0, x1), y1)) → A__2NDSPOS(a__plus(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(square(y0), cons(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), cons(mark(x0), x1))

A__PI(plus(x0, x1)) → A__2NDSPOS(a__plus(mark(x0), mark(x1)), a__from(0))

MARK(cons2(X1, X2)) → MARK(X2)

MARK(2ndspos(times(y0, y1), rnil)) → A__2NDSPOS(a__times(mark(y0), mark(y1)), rnil)

MARK(s(X)) → MARK(X)

MARK(2ndspos(X1, X2)) → MARK(X1)

MARK(pi(X)) → MARK(X)

MARK(plus(X1, X2)) → MARK(X2)

MARK(2ndspos(times(y0, y1), posrecip(x0))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), posrecip(mark(x0)))

A__2NDSPOS(s(N), cons(X, Z)) → MARK(Z)

MARK(pi(X)) → A__PI(mark(X))

MARK(plus(2ndsneg(x0, x1), y1)) → A__PLUS(a__2ndsneg(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(times(y0, y1), cons2(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(2ndspos(times(y0, y1), 0)) → A__2NDSPOS(a__times(mark(y0), mark(y1)), 0)

MARK(2ndsneg(X1, X2)) → MARK(X2)

MARK(2ndspos(times(y0, y1), square(x0))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndspos(square(y0), times(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(2ndsneg(y0, y1), cons2(x0, x1))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(times(square(x0), y1)) → A__TIMES(a__square(mark(x0)), mark(y1))

MARK(from(X)) → A__FROM(mark(X))

MARK(2ndspos(s(y0), plus(x0, x1))) → A__2NDSPOS(s(mark(y0)), a__plus(mark(x0), mark(x1)))

A__PI(X) → A__FROM(0)

MARK(times(from(x0), y1)) → A__TIMES(a__from(mark(x0)), mark(y1))

MARK(plus(X1, X2)) → MARK(X1)

MARK(2ndspos(pi(y0), cons(x0, x1))) → A__2NDSPOS(a__pi(mark(y0)), cons(mark(x0), x1))

A__2NDSPOS(s(y0), cons(y1, times(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__times(mark(x0), mark(x1))))

A__PI(X) → MARK(X)

MARK(2ndspos(s(y0), cons(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons(mark(x0), x1))

MARK(times(s(x0), y1)) → A__TIMES(s(mark(x0)), mark(y1))

MARK(2ndspos(pi(y0), 2ndspos(x0, x1))) → A__2NDSPOS(a__pi(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndspos(from(y0), cons(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), cons(mark(x0), x1))

MARK(2ndsneg(s(x0), y1)) → A__2NDSNEG(s(mark(x0)), mark(y1))

A__PLUS(s(X), Y) → MARK(Y)

A__2NDSPOS(s(y0), cons(y1, pi(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__pi(mark(x0))))

A__TIMES(s(X), Y) → MARK(X)

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(N)

MARK(2ndspos(pi(y0), plus(x0, x1))) → A__2NDSPOS(a__pi(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(plus(2ndspos(x0, x1), y1)) → A__PLUS(a__2ndspos(mark(x0), mark(x1)), mark(y1))

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(N)

MARK(cons(X1, X2)) → MARK(X1)

A__PI(from(x0)) → A__2NDSPOS(a__from(mark(x0)), a__from(0))

MARK(2ndspos(square(y0), 2ndspos(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndspos(times(y0, y1), cons(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), cons(mark(x0), x1))

MARK(times(X1, X2)) → MARK(X2)

MARK(2ndspos(s(y0), 2ndspos(x0, x1))) → A__2NDSPOS(s(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndspos(2ndsneg(y0, y1), 2ndsneg(x0, x1))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__2ndsneg(mark(x0), mark(x1)))

A__PI(square(x0)) → A__2NDSPOS(a__square(mark(x0)), a__from(0))

MARK(2ndspos(times(y0, y1), plus(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(times(y0, y1), s(x0))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), s(mark(x0)))

MARK(2ndspos(s(y0), times(x0, x1))) → A__2NDSPOS(s(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndsneg(X1, X2)) → MARK(X1)

MARK(2ndspos(s(y0), 2ndsneg(x0, x1))) → A__2NDSPOS(s(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndspos(times(y0, y1), negrecip(x0))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), negrecip(mark(x0)))

MARK(2ndspos(2ndsneg(y0, y1), plus(x0, x1))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

A__PI(2ndsneg(x0, x1)) → A__2NDSPOS(a__2ndsneg(mark(x0), mark(x1)), a__from(0))

MARK(times(pi(x0), y1)) → A__TIMES(a__pi(mark(x0)), mark(y1))

MARK(plus(pi(x0), y1)) → A__PLUS(a__pi(mark(x0)), mark(y1))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(Y)

MARK(rcons(X1, X2)) → MARK(X1)

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(Z)

MARK(square(X)) → MARK(X)

MARK(2ndspos(2ndsneg(y0, y1), 2ndspos(x0, x1))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndspos(times(y0, y1), pi(x0))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__pi(mark(x0)))

MARK(2ndspos(pi(y0), times(x0, x1))) → A__2NDSPOS(a__pi(mark(y0)), a__times(mark(x0), mark(x1)))

A__PI(s(x0)) → A__2NDSPOS(s(mark(x0)), a__from(0))

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(Y)

A__FROM(X) → MARK(X)

MARK(times(plus(x0, x1), y1)) → A__TIMES(a__plus(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(times(y0, y1), 2ndspos(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__2ndspos(mark(x0), mark(x1)))

A__2NDSPOS(s(y0), cons(y1, from(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__from(mark(x0))))

MARK(2ndspos(square(y0), pi(x0))) → A__2NDSPOS(a__square(mark(y0)), a__pi(mark(x0)))

MARK(2ndspos(pi(y0), square(x0))) → A__2NDSPOS(a__pi(mark(y0)), a__square(mark(x0)))

MARK(2ndspos(times(y0, y1), nil)) → A__2NDSPOS(a__times(mark(y0), mark(y1)), nil)

MARK(rcons(X1, X2)) → MARK(X2)

MARK(2ndspos(s(y0), square(x0))) → A__2NDSPOS(s(mark(y0)), a__square(mark(x0)))

MARK(times(2ndspos(x0, x1), y1)) → A__TIMES(a__2ndspos(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(s(y0), cons2(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(x0, mark(x1)))

A__TIMES(s(X), Y) → A__TIMES(mark(X), mark(Y))

A__2NDSPOS(s(y0), cons(y1, cons(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, cons(mark(x0), x1)))

A__PLUS(s(X), Y) → MARK(X)

MARK(times(X1, X2)) → MARK(X1)

MARK(plus(times(x0, x1), y1)) → A__PLUS(a__times(mark(x0), mark(x1)), mark(y1))

A__SQUARE(X) → MARK(X)

MARK(2ndsneg(times(x0, x1), y1)) → A__2NDSNEG(a__times(mark(x0), mark(x1)), mark(y1))

A__2NDSPOS(s(y0), cons(y1, 2ndsneg(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__2ndsneg(mark(x0), mark(x1))))

MARK(negrecip(X)) → MARK(X)

A__2NDSPOS(s(N), cons(X, Z)) → MARK(N)

A__PLUS(0, Y) → MARK(Y)

MARK(2ndspos(2ndspos(x0, x1), y1)) → A__2NDSPOS(a__2ndspos(mark(x0), mark(x1)), mark(y1))

MARK(posrecip(X)) → MARK(X)

MARK(2ndspos(from(y0), square(x0))) → A__2NDSPOS(a__from(mark(y0)), a__square(mark(x0)))

MARK(2ndspos(square(y0), from(x0))) → A__2NDSPOS(a__square(mark(y0)), a__from(mark(x0)))

MARK(2ndsneg(pi(x0), y1)) → A__2NDSNEG(a__pi(mark(x0)), mark(y1))

MARK(2ndspos(times(y0, y1), from(x0))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndspos(from(y0), times(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), a__times(mark(x0), mark(x1)))

A__2NDSPOS(s(y0), cons(y1, square(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__square(mark(x0))))

MARK(2ndspos(times(y0, y1), 2ndsneg(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndspos(2ndsneg(y0, y1), times(x0, x1))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(plus(0, y1)) → A__PLUS(0, mark(y1))

MARK(from(X)) → MARK(X)

A__PLUS(s(X), Y) → A__PLUS(mark(X), mark(Y))

MARK(2ndspos(square(y0), square(x0))) → A__2NDSPOS(a__square(mark(y0)), a__square(mark(x0)))

A__SQUARE(X) → A__TIMES(mark(X), mark(X))

A__PI(pi(x0)) → A__2NDSPOS(a__pi(mark(x0)), a__from(0))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(Z)

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → A__2NDSPOS(mark(N), mark(Z))

MARK(plus(from(x0), y1)) → A__PLUS(a__from(mark(x0)), mark(y1))

MARK(square(X)) → A__SQUARE(mark(X))

MARK(2ndspos(from(y0), plus(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(from(y0), 2ndspos(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndspos(2ndsneg(y0, y1), cons(x0, x1))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), cons(mark(x0), x1))

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))

a__2ndspos(0, Z) → rnil

a__2ndspos(s(N), cons(X, Z)) → a__2ndspos(s(mark(N)), cons2(X, mark(Z)))

a__2ndspos(s(N), cons2(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))

a__2ndsneg(0, Z) → rnil

a__2ndsneg(s(N), cons(X, Z)) → a__2ndsneg(s(mark(N)), cons2(X, mark(Z)))

a__2ndsneg(s(N), cons2(X, cons(Y, Z))) → rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z)))

a__pi(X) → a__2ndspos(mark(X), a__from(0))

a__plus(0, Y) → mark(Y)

a__plus(s(X), Y) → s(a__plus(mark(X), mark(Y)))

a__times(0, Y) → 0

a__times(s(X), Y) → a__plus(mark(Y), a__times(mark(X), mark(Y)))

a__square(X) → a__times(mark(X), mark(X))

mark(from(X)) → a__from(mark(X))

mark(2ndspos(X1, X2)) → a__2ndspos(mark(X1), mark(X2))

mark(2ndsneg(X1, X2)) → a__2ndsneg(mark(X1), mark(X2))

mark(pi(X)) → a__pi(mark(X))

mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))

mark(times(X1, X2)) → a__times(mark(X1), mark(X2))

mark(square(X)) → a__square(mark(X))

mark(0) → 0

mark(s(X)) → s(mark(X))

mark(posrecip(X)) → posrecip(mark(X))

mark(negrecip(X)) → negrecip(mark(X))

mark(nil) → nil

mark(cons(X1, X2)) → cons(mark(X1), X2)

mark(cons2(X1, X2)) → cons2(X1, mark(X2))

mark(rnil) → rnil

mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))

a__from(X) → from(X)

a__2ndspos(X1, X2) → 2ndspos(X1, X2)

a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)

a__pi(X) → pi(X)

a__plus(X1, X2) → plus(X1, X2)

a__times(X1, X2) → times(X1, X2)

a__square(X) → square(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 7 less nodes.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ Narrowing
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ QDP
              ↳ Narrowing
                ↳ QDP
                  ↳ DependencyGraphProof
                    ↳ QDP
                      ↳ Narrowing
                        ↳ QDP
                          ↳ DependencyGraphProof
                            ↳ QDP
                              ↳ Narrowing
                                ↳ QDP
                                  ↳ DependencyGraphProof
                                    ↳ QDP
                                      ↳ Narrowing
                                        ↳ QDP
                                          ↳ DependencyGraphProof
                                            ↳ QDP
                                              ↳ Narrowing
                                                ↳ QDP
                                                  ↳ DependencyGraphProof
                                                    ↳ QDP
                                                      ↳ Narrowing
                                                        ↳ QDP
                                                          ↳ DependencyGraphProof
                                                            ↳ QDP
                                                              ↳ Narrowing
                                                                ↳ QDP
                                                                  ↳ DependencyGraphProof
                                                                    ↳ QDP
                                                                      ↳ Narrowing
                                                                        ↳ QDP
                                                                          ↳ DependencyGraphProof
                                                                            ↳ QDP
                                                                              ↳ Narrowing
                                                                                ↳ QDP
                                                                                  ↳ DependencyGraphProof
                                                                                    ↳ QDP
                                                                                      ↳ Narrowing
                                                                                        ↳ QDP
                                                                                          ↳ DependencyGraphProof
                                                                                            ↳ QDP
                                                                                              ↳ Narrowing
                                                                                                ↳ QDP
                                                                                                  ↳ DependencyGraphProof
                                                                                                    ↳ QDP
                                                                                                      ↳ Narrowing

Q DP problem:
The TRS P consists of the following rules:

MARK(2ndsneg(square(x0), y1)) → A__2NDSNEG(a__square(mark(x0)), mark(y1))

MARK(2ndspos(square(y0), plus(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__plus(mark(x0), mark(x1)))

A__PI(times(x0, x1)) → A__2NDSPOS(a__times(mark(x0), mark(x1)), a__from(0))

A__2NDSPOS(s(y0), cons(y1, 2ndspos(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__2ndspos(mark(x0), mark(x1))))

A__2NDSNEG(s(N), cons(X, Z)) → A__2NDSNEG(s(mark(N)), cons2(X, mark(Z)))

MARK(2ndspos(pi(y0), cons2(x0, x1))) → A__2NDSPOS(a__pi(mark(y0)), cons2(x0, mark(x1)))

MARK(2ndspos(X1, X2)) → MARK(X2)

MARK(times(times(x0, x1), y1)) → A__TIMES(a__times(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(from(y0), cons2(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), cons2(x0, mark(x1)))

MARK(plus(plus(x0, x1), y1)) → A__PLUS(a__plus(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(2ndsneg(y0, y1), from(x0))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndspos(s(y0), from(x0))) → A__2NDSPOS(s(mark(y0)), a__from(mark(x0)))

MARK(2ndspos(from(y0), 2ndsneg(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndspos(pi(y0), 2ndsneg(x0, x1))) → A__2NDSPOS(a__pi(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndspos(2ndsneg(y0, y1), pi(x0))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__pi(mark(x0)))

MARK(2ndsneg(2ndsneg(x0, x1), y1)) → A__2NDSNEG(a__2ndsneg(mark(x0), mark(x1)), mark(y1))

A__2NDSNEG(s(N), cons(X, Z)) → MARK(Z)

A__TIMES(s(X), Y) → MARK(Y)

MARK(2ndsneg(2ndspos(x0, x1), y1)) → A__2NDSNEG(a__2ndspos(mark(x0), mark(x1)), mark(y1))

A__2NDSPOS(s(y0), cons(y1, plus(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__plus(mark(x0), mark(x1))))

MARK(2ndspos(pi(y0), from(x0))) → A__2NDSPOS(a__pi(mark(y0)), a__from(mark(x0)))

MARK(2ndspos(2ndsneg(y0, y1), square(x0))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndspos(square(y0), 2ndsneg(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndspos(from(y0), pi(x0))) → A__2NDSPOS(a__from(mark(y0)), a__pi(mark(x0)))

A__2NDSNEG(s(N), cons(X, Z)) → MARK(N)

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → A__2NDSNEG(mark(N), mark(Z))

MARK(2ndsneg(plus(x0, x1), y1)) → A__2NDSNEG(a__plus(mark(x0), mark(x1)), mark(y1))

MARK(times(2ndsneg(x0, x1), y1)) → A__TIMES(a__2ndsneg(mark(x0), mark(x1)), mark(y1))

MARK(2ndsneg(from(x0), y1)) → A__2NDSNEG(a__from(mark(x0)), mark(y1))

MARK(2ndspos(from(y0), from(x0))) → A__2NDSPOS(a__from(mark(y0)), a__from(mark(x0)))

MARK(2ndspos(pi(y0), pi(x0))) → A__2NDSPOS(a__pi(mark(y0)), a__pi(mark(x0)))

MARK(2ndspos(times(y0, y1), times(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(s(y0), pi(x0))) → A__2NDSPOS(s(mark(y0)), a__pi(mark(x0)))

A__TIMES(s(X), Y) → A__PLUS(mark(Y), a__times(mark(X), mark(Y)))

A__PI(2ndspos(x0, x1)) → A__2NDSPOS(a__2ndspos(mark(x0), mark(x1)), a__from(0))

MARK(2ndspos(square(y0), cons2(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), cons2(x0, mark(x1)))

MARK(plus(square(x0), y1)) → A__PLUS(a__square(mark(x0)), mark(y1))

MARK(2ndspos(plus(x0, x1), y1)) → A__2NDSPOS(a__plus(mark(x0), mark(x1)), mark(y1))

MARK(plus(s(x0), y1)) → A__PLUS(s(mark(x0)), mark(y1))

A__PI(plus(x0, x1)) → A__2NDSPOS(a__plus(mark(x0), mark(x1)), a__from(0))

MARK(2ndspos(square(y0), cons(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), cons(mark(x0), x1))

MARK(cons2(X1, X2)) → MARK(X2)

MARK(2ndspos(X1, X2)) → MARK(X1)

MARK(s(X)) → MARK(X)

MARK(pi(X)) → MARK(X)

MARK(plus(X1, X2)) → MARK(X2)

A__2NDSPOS(s(N), cons(X, Z)) → MARK(Z)

MARK(pi(X)) → A__PI(mark(X))

MARK(plus(2ndsneg(x0, x1), y1)) → A__PLUS(a__2ndsneg(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(times(y0, y1), cons2(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(2ndsneg(X1, X2)) → MARK(X2)

MARK(2ndspos(times(y0, y1), square(x0))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndspos(square(y0), times(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(2ndsneg(y0, y1), cons2(x0, x1))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(times(square(x0), y1)) → A__TIMES(a__square(mark(x0)), mark(y1))

A__PI(X) → A__FROM(0)

MARK(2ndspos(s(y0), plus(x0, x1))) → A__2NDSPOS(s(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(from(X)) → A__FROM(mark(X))

MARK(2ndspos(pi(y0), cons(x0, x1))) → A__2NDSPOS(a__pi(mark(y0)), cons(mark(x0), x1))

MARK(plus(X1, X2)) → MARK(X1)

MARK(times(from(x0), y1)) → A__TIMES(a__from(mark(x0)), mark(y1))

A__2NDSPOS(s(y0), cons(y1, times(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__times(mark(x0), mark(x1))))

MARK(2ndspos(s(y0), cons(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons(mark(x0), x1))

A__PI(X) → MARK(X)

MARK(times(s(x0), y1)) → A__TIMES(s(mark(x0)), mark(y1))

MARK(2ndspos(pi(y0), 2ndspos(x0, x1))) → A__2NDSPOS(a__pi(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndsneg(s(x0), y1)) → A__2NDSNEG(s(mark(x0)), mark(y1))

MARK(2ndspos(from(y0), cons(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), cons(mark(x0), x1))

A__PLUS(s(X), Y) → MARK(Y)

A__TIMES(s(X), Y) → MARK(X)

A__2NDSPOS(s(y0), cons(y1, pi(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__pi(mark(x0))))

MARK(2ndspos(pi(y0), plus(x0, x1))) → A__2NDSPOS(a__pi(mark(y0)), a__plus(mark(x0), mark(x1)))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(N)

MARK(plus(2ndspos(x0, x1), y1)) → A__PLUS(a__2ndspos(mark(x0), mark(x1)), mark(y1))

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(N)

A__PI(from(x0)) → A__2NDSPOS(a__from(mark(x0)), a__from(0))

MARK(cons(X1, X2)) → MARK(X1)

MARK(2ndspos(square(y0), 2ndspos(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndspos(times(y0, y1), cons(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), cons(mark(x0), x1))

MARK(2ndspos(s(y0), 2ndspos(x0, x1))) → A__2NDSPOS(s(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

MARK(times(X1, X2)) → MARK(X2)

A__PI(square(x0)) → A__2NDSPOS(a__square(mark(x0)), a__from(0))

MARK(2ndspos(2ndsneg(y0, y1), 2ndsneg(x0, x1))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndspos(times(y0, y1), plus(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(s(y0), times(x0, x1))) → A__2NDSPOS(s(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndsneg(X1, X2)) → MARK(X1)

MARK(2ndspos(s(y0), 2ndsneg(x0, x1))) → A__2NDSPOS(s(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndspos(2ndsneg(y0, y1), plus(x0, x1))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

A__PI(2ndsneg(x0, x1)) → A__2NDSPOS(a__2ndsneg(mark(x0), mark(x1)), a__from(0))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(Y)

MARK(plus(pi(x0), y1)) → A__PLUS(a__pi(mark(x0)), mark(y1))

MARK(times(pi(x0), y1)) → A__TIMES(a__pi(mark(x0)), mark(y1))

MARK(rcons(X1, X2)) → MARK(X1)

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(Z)

MARK(square(X)) → MARK(X)

MARK(2ndspos(2ndsneg(y0, y1), 2ndspos(x0, x1))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndspos(times(y0, y1), pi(x0))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__pi(mark(x0)))

MARK(2ndspos(pi(y0), times(x0, x1))) → A__2NDSPOS(a__pi(mark(y0)), a__times(mark(x0), mark(x1)))

A__PI(s(x0)) → A__2NDSPOS(s(mark(x0)), a__from(0))

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(Y)

A__FROM(X) → MARK(X)

MARK(times(plus(x0, x1), y1)) → A__TIMES(a__plus(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(times(y0, y1), 2ndspos(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__2ndspos(mark(x0), mark(x1)))

A__2NDSPOS(s(y0), cons(y1, from(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__from(mark(x0))))

MARK(2ndspos(pi(y0), square(x0))) → A__2NDSPOS(a__pi(mark(y0)), a__square(mark(x0)))

MARK(2ndspos(square(y0), pi(x0))) → A__2NDSPOS(a__square(mark(y0)), a__pi(mark(x0)))

MARK(2ndspos(s(y0), square(x0))) → A__2NDSPOS(s(mark(y0)), a__square(mark(x0)))

MARK(rcons(X1, X2)) → MARK(X2)

MARK(times(2ndspos(x0, x1), y1)) → A__TIMES(a__2ndspos(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(s(y0), cons2(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(x0, mark(x1)))

A__TIMES(s(X), Y) → A__TIMES(mark(X), mark(Y))

A__PLUS(s(X), Y) → MARK(X)

A__2NDSPOS(s(y0), cons(y1, cons(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, cons(mark(x0), x1)))

MARK(times(X1, X2)) → MARK(X1)

MARK(plus(times(x0, x1), y1)) → A__PLUS(a__times(mark(x0), mark(x1)), mark(y1))

A__SQUARE(X) → MARK(X)

MARK(2ndsneg(times(x0, x1), y1)) → A__2NDSNEG(a__times(mark(x0), mark(x1)), mark(y1))

A__2NDSPOS(s(y0), cons(y1, 2ndsneg(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__2ndsneg(mark(x0), mark(x1))))

A__2NDSPOS(s(N), cons(X, Z)) → MARK(N)

MARK(negrecip(X)) → MARK(X)

A__PLUS(0, Y) → MARK(Y)

MARK(2ndspos(2ndspos(x0, x1), y1)) → A__2NDSPOS(a__2ndspos(mark(x0), mark(x1)), mark(y1))

MARK(posrecip(X)) → MARK(X)

MARK(2ndsneg(pi(x0), y1)) → A__2NDSNEG(a__pi(mark(x0)), mark(y1))

MARK(2ndspos(square(y0), from(x0))) → A__2NDSPOS(a__square(mark(y0)), a__from(mark(x0)))

MARK(2ndspos(from(y0), square(x0))) → A__2NDSPOS(a__from(mark(y0)), a__square(mark(x0)))

MARK(2ndspos(times(y0, y1), from(x0))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndspos(from(y0), times(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), a__times(mark(x0), mark(x1)))

A__2NDSPOS(s(y0), cons(y1, square(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__square(mark(x0))))

MARK(2ndspos(times(y0, y1), 2ndsneg(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndspos(2ndsneg(y0, y1), times(x0, x1))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(plus(0, y1)) → A__PLUS(0, mark(y1))

MARK(from(X)) → MARK(X)

A__PLUS(s(X), Y) → A__PLUS(mark(X), mark(Y))

A__SQUARE(X) → A__TIMES(mark(X), mark(X))

MARK(2ndspos(square(y0), square(x0))) → A__2NDSPOS(a__square(mark(y0)), a__square(mark(x0)))

A__PI(pi(x0)) → A__2NDSPOS(a__pi(mark(x0)), a__from(0))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(Z)

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → A__2NDSPOS(mark(N), mark(Z))

MARK(plus(from(x0), y1)) → A__PLUS(a__from(mark(x0)), mark(y1))

MARK(square(X)) → A__SQUARE(mark(X))

MARK(2ndspos(from(y0), plus(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(from(y0), 2ndspos(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndspos(2ndsneg(y0, y1), cons(x0, x1))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), cons(mark(x0), x1))

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))

a__2ndspos(0, Z) → rnil

a__2ndspos(s(N), cons(X, Z)) → a__2ndspos(s(mark(N)), cons2(X, mark(Z)))

a__2ndspos(s(N), cons2(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))

a__2ndsneg(0, Z) → rnil

a__2ndsneg(s(N), cons(X, Z)) → a__2ndsneg(s(mark(N)), cons2(X, mark(Z)))

a__2ndsneg(s(N), cons2(X, cons(Y, Z))) → rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z)))

a__pi(X) → a__2ndspos(mark(X), a__from(0))

a__plus(0, Y) → mark(Y)

a__plus(s(X), Y) → s(a__plus(mark(X), mark(Y)))

a__times(0, Y) → 0

a__times(s(X), Y) → a__plus(mark(Y), a__times(mark(X), mark(Y)))

a__square(X) → a__times(mark(X), mark(X))

mark(from(X)) → a__from(mark(X))

mark(2ndspos(X1, X2)) → a__2ndspos(mark(X1), mark(X2))

mark(2ndsneg(X1, X2)) → a__2ndsneg(mark(X1), mark(X2))

mark(pi(X)) → a__pi(mark(X))

mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))

mark(times(X1, X2)) → a__times(mark(X1), mark(X2))

mark(square(X)) → a__square(mark(X))

mark(0) → 0

mark(s(X)) → s(mark(X))

mark(posrecip(X)) → posrecip(mark(X))

mark(negrecip(X)) → negrecip(mark(X))

mark(nil) → nil

mark(cons(X1, X2)) → cons(mark(X1), X2)

mark(cons2(X1, X2)) → cons2(X1, mark(X2))

mark(rnil) → rnil

mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))

a__from(X) → from(X)

a__2ndspos(X1, X2) → 2ndspos(X1, X2)

a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)

a__pi(X) → pi(X)

a__plus(X1, X2) → plus(X1, X2)

a__times(X1, X2) → times(X1, X2)

a__square(X) → square(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule MARK(2ndspos(2ndspos(x0, x1), y1)) → A__2NDSPOS(a__2ndspos(mark(x0), mark(x1)), mark(y1)) at position [1] we obtained the following new rules:

MARK(2ndspos(2ndspos(y0, y1), negrecip(x0))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), negrecip(mark(x0)))

MARK(2ndspos(2ndspos(y0, y1), square(x0))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndspos(2ndspos(y0, y1), s(x0))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), s(mark(x0)))

MARK(2ndspos(2ndspos(y0, y1), posrecip(x0))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), posrecip(mark(x0)))

MARK(2ndspos(2ndspos(y0, y1), 0)) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), 0)

MARK(2ndspos(2ndspos(y0, y1), cons2(x0, x1))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(2ndspos(2ndspos(y0, y1), 2ndspos(x0, x1))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndspos(2ndspos(y0, y1), 2ndsneg(x0, x1))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndspos(2ndspos(y0, y1), cons(x0, x1))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), cons(mark(x0), x1))

MARK(2ndspos(2ndspos(y0, y1), times(x0, x1))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(2ndspos(y0, y1), rcons(x0, x1))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), rcons(mark(x0), mark(x1)))

MARK(2ndspos(2ndspos(y0, y1), nil)) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), nil)

MARK(2ndspos(2ndspos(y0, y1), rnil)) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), rnil)

MARK(2ndspos(2ndspos(y0, y1), pi(x0))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), a__pi(mark(x0)))

MARK(2ndspos(2ndspos(y0, y1), plus(x0, x1))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(2ndspos(y0, y1), from(x0))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), a__from(mark(x0)))



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ Narrowing
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ QDP
              ↳ Narrowing
                ↳ QDP
                  ↳ DependencyGraphProof
                    ↳ QDP
                      ↳ Narrowing
                        ↳ QDP
                          ↳ DependencyGraphProof
                            ↳ QDP
                              ↳ Narrowing
                                ↳ QDP
                                  ↳ DependencyGraphProof
                                    ↳ QDP
                                      ↳ Narrowing
                                        ↳ QDP
                                          ↳ DependencyGraphProof
                                            ↳ QDP
                                              ↳ Narrowing
                                                ↳ QDP
                                                  ↳ DependencyGraphProof
                                                    ↳ QDP
                                                      ↳ Narrowing
                                                        ↳ QDP
                                                          ↳ DependencyGraphProof
                                                            ↳ QDP
                                                              ↳ Narrowing
                                                                ↳ QDP
                                                                  ↳ DependencyGraphProof
                                                                    ↳ QDP
                                                                      ↳ Narrowing
                                                                        ↳ QDP
                                                                          ↳ DependencyGraphProof
                                                                            ↳ QDP
                                                                              ↳ Narrowing
                                                                                ↳ QDP
                                                                                  ↳ DependencyGraphProof
                                                                                    ↳ QDP
                                                                                      ↳ Narrowing
                                                                                        ↳ QDP
                                                                                          ↳ DependencyGraphProof
                                                                                            ↳ QDP
                                                                                              ↳ Narrowing
                                                                                                ↳ QDP
                                                                                                  ↳ DependencyGraphProof
                                                                                                    ↳ QDP
                                                                                                      ↳ Narrowing
                                                                                                        ↳ QDP
                                                                                                          ↳ DependencyGraphProof

Q DP problem:
The TRS P consists of the following rules:

MARK(2ndspos(square(y0), plus(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(2ndsneg(square(x0), y1)) → A__2NDSNEG(a__square(mark(x0)), mark(y1))

A__2NDSPOS(s(y0), cons(y1, 2ndspos(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__2ndspos(mark(x0), mark(x1))))

A__PI(times(x0, x1)) → A__2NDSPOS(a__times(mark(x0), mark(x1)), a__from(0))

A__2NDSNEG(s(N), cons(X, Z)) → A__2NDSNEG(s(mark(N)), cons2(X, mark(Z)))

MARK(2ndspos(2ndspos(y0, y1), negrecip(x0))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), negrecip(mark(x0)))

MARK(times(times(x0, x1), y1)) → A__TIMES(a__times(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(X1, X2)) → MARK(X2)

MARK(2ndspos(pi(y0), cons2(x0, x1))) → A__2NDSPOS(a__pi(mark(y0)), cons2(x0, mark(x1)))

MARK(2ndspos(from(y0), cons2(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), cons2(x0, mark(x1)))

MARK(plus(plus(x0, x1), y1)) → A__PLUS(a__plus(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(from(y0), 2ndsneg(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndspos(s(y0), from(x0))) → A__2NDSPOS(s(mark(y0)), a__from(mark(x0)))

MARK(2ndspos(2ndsneg(y0, y1), from(x0))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndsneg(2ndsneg(x0, x1), y1)) → A__2NDSNEG(a__2ndsneg(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(2ndsneg(y0, y1), pi(x0))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__pi(mark(x0)))

MARK(2ndspos(pi(y0), 2ndsneg(x0, x1))) → A__2NDSPOS(a__pi(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

A__2NDSNEG(s(N), cons(X, Z)) → MARK(Z)

A__TIMES(s(X), Y) → MARK(Y)

MARK(2ndsneg(2ndspos(x0, x1), y1)) → A__2NDSNEG(a__2ndspos(mark(x0), mark(x1)), mark(y1))

A__2NDSPOS(s(y0), cons(y1, plus(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__plus(mark(x0), mark(x1))))

MARK(2ndspos(from(y0), pi(x0))) → A__2NDSPOS(a__from(mark(y0)), a__pi(mark(x0)))

MARK(2ndspos(square(y0), 2ndsneg(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndspos(2ndsneg(y0, y1), square(x0))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndspos(pi(y0), from(x0))) → A__2NDSPOS(a__pi(mark(y0)), a__from(mark(x0)))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → A__2NDSNEG(mark(N), mark(Z))

A__2NDSNEG(s(N), cons(X, Z)) → MARK(N)

MARK(times(2ndsneg(x0, x1), y1)) → A__TIMES(a__2ndsneg(mark(x0), mark(x1)), mark(y1))

MARK(2ndsneg(plus(x0, x1), y1)) → A__2NDSNEG(a__plus(mark(x0), mark(x1)), mark(y1))

MARK(2ndsneg(from(x0), y1)) → A__2NDSNEG(a__from(mark(x0)), mark(y1))

MARK(2ndspos(from(y0), from(x0))) → A__2NDSPOS(a__from(mark(y0)), a__from(mark(x0)))

MARK(2ndspos(pi(y0), pi(x0))) → A__2NDSPOS(a__pi(mark(y0)), a__pi(mark(x0)))

MARK(2ndspos(times(y0, y1), times(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(2ndspos(y0, y1), rcons(x0, x1))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), rcons(mark(x0), mark(x1)))

A__TIMES(s(X), Y) → A__PLUS(mark(Y), a__times(mark(X), mark(Y)))

MARK(2ndspos(s(y0), pi(x0))) → A__2NDSPOS(s(mark(y0)), a__pi(mark(x0)))

MARK(2ndspos(square(y0), cons2(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), cons2(x0, mark(x1)))

A__PI(2ndspos(x0, x1)) → A__2NDSPOS(a__2ndspos(mark(x0), mark(x1)), a__from(0))

MARK(plus(square(x0), y1)) → A__PLUS(a__square(mark(x0)), mark(y1))

MARK(plus(s(x0), y1)) → A__PLUS(s(mark(x0)), mark(y1))

MARK(2ndspos(plus(x0, x1), y1)) → A__2NDSPOS(a__plus(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(square(y0), cons(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), cons(mark(x0), x1))

A__PI(plus(x0, x1)) → A__2NDSPOS(a__plus(mark(x0), mark(x1)), a__from(0))

MARK(cons2(X1, X2)) → MARK(X2)

MARK(s(X)) → MARK(X)

MARK(2ndspos(X1, X2)) → MARK(X1)

MARK(pi(X)) → MARK(X)

MARK(plus(X1, X2)) → MARK(X2)

A__2NDSPOS(s(N), cons(X, Z)) → MARK(Z)

MARK(pi(X)) → A__PI(mark(X))

MARK(plus(2ndsneg(x0, x1), y1)) → A__PLUS(a__2ndsneg(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(2ndspos(y0, y1), 0)) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), 0)

MARK(2ndspos(times(y0, y1), cons2(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(2ndspos(2ndspos(y0, y1), cons2(x0, x1))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(2ndsneg(X1, X2)) → MARK(X2)

MARK(2ndspos(square(y0), times(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(times(y0, y1), square(x0))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndspos(2ndsneg(y0, y1), cons2(x0, x1))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(times(square(x0), y1)) → A__TIMES(a__square(mark(x0)), mark(y1))

MARK(from(X)) → A__FROM(mark(X))

MARK(2ndspos(s(y0), plus(x0, x1))) → A__2NDSPOS(s(mark(y0)), a__plus(mark(x0), mark(x1)))

A__PI(X) → A__FROM(0)

MARK(times(from(x0), y1)) → A__TIMES(a__from(mark(x0)), mark(y1))

MARK(plus(X1, X2)) → MARK(X1)

MARK(2ndspos(pi(y0), cons(x0, x1))) → A__2NDSPOS(a__pi(mark(y0)), cons(mark(x0), x1))

A__2NDSPOS(s(y0), cons(y1, times(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__times(mark(x0), mark(x1))))

A__PI(X) → MARK(X)

MARK(2ndspos(s(y0), cons(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons(mark(x0), x1))

MARK(times(s(x0), y1)) → A__TIMES(s(mark(x0)), mark(y1))

MARK(2ndspos(2ndspos(y0, y1), pi(x0))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), a__pi(mark(x0)))

MARK(2ndspos(pi(y0), 2ndspos(x0, x1))) → A__2NDSPOS(a__pi(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndspos(from(y0), cons(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), cons(mark(x0), x1))

MARK(2ndsneg(s(x0), y1)) → A__2NDSNEG(s(mark(x0)), mark(y1))

MARK(2ndspos(2ndspos(y0, y1), plus(x0, x1))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

A__PLUS(s(X), Y) → MARK(Y)

A__2NDSPOS(s(y0), cons(y1, pi(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__pi(mark(x0))))

A__TIMES(s(X), Y) → MARK(X)

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(N)

MARK(2ndspos(pi(y0), plus(x0, x1))) → A__2NDSPOS(a__pi(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(plus(2ndspos(x0, x1), y1)) → A__PLUS(a__2ndspos(mark(x0), mark(x1)), mark(y1))

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(N)

MARK(cons(X1, X2)) → MARK(X1)

A__PI(from(x0)) → A__2NDSPOS(a__from(mark(x0)), a__from(0))

MARK(2ndspos(2ndspos(y0, y1), square(x0))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndspos(square(y0), 2ndspos(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndspos(2ndspos(y0, y1), s(x0))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), s(mark(x0)))

MARK(times(X1, X2)) → MARK(X2)

MARK(2ndspos(s(y0), 2ndspos(x0, x1))) → A__2NDSPOS(s(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndspos(times(y0, y1), cons(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), cons(mark(x0), x1))

MARK(2ndspos(2ndsneg(y0, y1), 2ndsneg(x0, x1))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__2ndsneg(mark(x0), mark(x1)))

A__PI(square(x0)) → A__2NDSPOS(a__square(mark(x0)), a__from(0))

MARK(2ndspos(times(y0, y1), plus(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(s(y0), times(x0, x1))) → A__2NDSPOS(s(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndsneg(X1, X2)) → MARK(X1)

MARK(2ndspos(s(y0), 2ndsneg(x0, x1))) → A__2NDSPOS(s(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndspos(2ndsneg(y0, y1), plus(x0, x1))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

A__PI(2ndsneg(x0, x1)) → A__2NDSPOS(a__2ndsneg(mark(x0), mark(x1)), a__from(0))

MARK(times(pi(x0), y1)) → A__TIMES(a__pi(mark(x0)), mark(y1))

MARK(plus(pi(x0), y1)) → A__PLUS(a__pi(mark(x0)), mark(y1))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(Y)

MARK(2ndspos(2ndspos(y0, y1), 2ndspos(x0, x1))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), a__2ndspos(mark(x0), mark(x1)))

MARK(rcons(X1, X2)) → MARK(X1)

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(Z)

MARK(square(X)) → MARK(X)

MARK(2ndspos(2ndspos(y0, y1), 2ndsneg(x0, x1))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndspos(2ndsneg(y0, y1), 2ndspos(x0, x1))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndspos(pi(y0), times(x0, x1))) → A__2NDSPOS(a__pi(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(times(y0, y1), pi(x0))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__pi(mark(x0)))

A__PI(s(x0)) → A__2NDSPOS(s(mark(x0)), a__from(0))

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(Y)

MARK(2ndspos(2ndspos(y0, y1), cons(x0, x1))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), cons(mark(x0), x1))

A__FROM(X) → MARK(X)

MARK(times(plus(x0, x1), y1)) → A__TIMES(a__plus(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(2ndspos(y0, y1), times(x0, x1))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

A__2NDSPOS(s(y0), cons(y1, from(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__from(mark(x0))))

MARK(2ndspos(times(y0, y1), 2ndspos(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndspos(square(y0), pi(x0))) → A__2NDSPOS(a__square(mark(y0)), a__pi(mark(x0)))

MARK(2ndspos(pi(y0), square(x0))) → A__2NDSPOS(a__pi(mark(y0)), a__square(mark(x0)))

MARK(2ndspos(2ndspos(y0, y1), rnil)) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), rnil)

MARK(rcons(X1, X2)) → MARK(X2)

MARK(2ndspos(s(y0), square(x0))) → A__2NDSPOS(s(mark(y0)), a__square(mark(x0)))

MARK(times(2ndspos(x0, x1), y1)) → A__TIMES(a__2ndspos(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(s(y0), cons2(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(x0, mark(x1)))

A__TIMES(s(X), Y) → A__TIMES(mark(X), mark(Y))

A__2NDSPOS(s(y0), cons(y1, cons(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, cons(mark(x0), x1)))

A__PLUS(s(X), Y) → MARK(X)

MARK(times(X1, X2)) → MARK(X1)

MARK(plus(times(x0, x1), y1)) → A__PLUS(a__times(mark(x0), mark(x1)), mark(y1))

A__SQUARE(X) → MARK(X)

MARK(2ndsneg(times(x0, x1), y1)) → A__2NDSNEG(a__times(mark(x0), mark(x1)), mark(y1))

A__2NDSPOS(s(y0), cons(y1, 2ndsneg(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__2ndsneg(mark(x0), mark(x1))))

MARK(negrecip(X)) → MARK(X)

A__2NDSPOS(s(N), cons(X, Z)) → MARK(N)

MARK(2ndspos(2ndspos(y0, y1), posrecip(x0))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), posrecip(mark(x0)))

A__PLUS(0, Y) → MARK(Y)

MARK(posrecip(X)) → MARK(X)

MARK(2ndspos(from(y0), square(x0))) → A__2NDSPOS(a__from(mark(y0)), a__square(mark(x0)))

MARK(2ndspos(square(y0), from(x0))) → A__2NDSPOS(a__square(mark(y0)), a__from(mark(x0)))

MARK(2ndsneg(pi(x0), y1)) → A__2NDSNEG(a__pi(mark(x0)), mark(y1))

MARK(2ndspos(from(y0), times(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(times(y0, y1), from(x0))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__from(mark(x0)))

A__2NDSPOS(s(y0), cons(y1, square(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__square(mark(x0))))

MARK(2ndspos(2ndsneg(y0, y1), times(x0, x1))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(times(y0, y1), 2ndsneg(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__2ndsneg(mark(x0), mark(x1)))

MARK(plus(0, y1)) → A__PLUS(0, mark(y1))

MARK(from(X)) → MARK(X)

A__PLUS(s(X), Y) → A__PLUS(mark(X), mark(Y))

MARK(2ndspos(square(y0), square(x0))) → A__2NDSPOS(a__square(mark(y0)), a__square(mark(x0)))

A__SQUARE(X) → A__TIMES(mark(X), mark(X))

A__PI(pi(x0)) → A__2NDSPOS(a__pi(mark(x0)), a__from(0))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(Z)

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → A__2NDSPOS(mark(N), mark(Z))

MARK(plus(from(x0), y1)) → A__PLUS(a__from(mark(x0)), mark(y1))

MARK(square(X)) → A__SQUARE(mark(X))

MARK(2ndspos(from(y0), plus(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(2ndspos(y0, y1), nil)) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), nil)

MARK(2ndspos(2ndspos(y0, y1), from(x0))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndspos(from(y0), 2ndspos(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndspos(2ndsneg(y0, y1), cons(x0, x1))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), cons(mark(x0), x1))

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))

a__2ndspos(0, Z) → rnil

a__2ndspos(s(N), cons(X, Z)) → a__2ndspos(s(mark(N)), cons2(X, mark(Z)))

a__2ndspos(s(N), cons2(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))

a__2ndsneg(0, Z) → rnil

a__2ndsneg(s(N), cons(X, Z)) → a__2ndsneg(s(mark(N)), cons2(X, mark(Z)))

a__2ndsneg(s(N), cons2(X, cons(Y, Z))) → rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z)))

a__pi(X) → a__2ndspos(mark(X), a__from(0))

a__plus(0, Y) → mark(Y)

a__plus(s(X), Y) → s(a__plus(mark(X), mark(Y)))

a__times(0, Y) → 0

a__times(s(X), Y) → a__plus(mark(Y), a__times(mark(X), mark(Y)))

a__square(X) → a__times(mark(X), mark(X))

mark(from(X)) → a__from(mark(X))

mark(2ndspos(X1, X2)) → a__2ndspos(mark(X1), mark(X2))

mark(2ndsneg(X1, X2)) → a__2ndsneg(mark(X1), mark(X2))

mark(pi(X)) → a__pi(mark(X))

mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))

mark(times(X1, X2)) → a__times(mark(X1), mark(X2))

mark(square(X)) → a__square(mark(X))

mark(0) → 0

mark(s(X)) → s(mark(X))

mark(posrecip(X)) → posrecip(mark(X))

mark(negrecip(X)) → negrecip(mark(X))

mark(nil) → nil

mark(cons(X1, X2)) → cons(mark(X1), X2)

mark(cons2(X1, X2)) → cons2(X1, mark(X2))

mark(rnil) → rnil

mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))

a__from(X) → from(X)

a__2ndspos(X1, X2) → 2ndspos(X1, X2)

a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)

a__pi(X) → pi(X)

a__plus(X1, X2) → plus(X1, X2)

a__times(X1, X2) → times(X1, X2)

a__square(X) → square(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 7 less nodes.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ Narrowing
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ QDP
              ↳ Narrowing
                ↳ QDP
                  ↳ DependencyGraphProof
                    ↳ QDP
                      ↳ Narrowing
                        ↳ QDP
                          ↳ DependencyGraphProof
                            ↳ QDP
                              ↳ Narrowing
                                ↳ QDP
                                  ↳ DependencyGraphProof
                                    ↳ QDP
                                      ↳ Narrowing
                                        ↳ QDP
                                          ↳ DependencyGraphProof
                                            ↳ QDP
                                              ↳ Narrowing
                                                ↳ QDP
                                                  ↳ DependencyGraphProof
                                                    ↳ QDP
                                                      ↳ Narrowing
                                                        ↳ QDP
                                                          ↳ DependencyGraphProof
                                                            ↳ QDP
                                                              ↳ Narrowing
                                                                ↳ QDP
                                                                  ↳ DependencyGraphProof
                                                                    ↳ QDP
                                                                      ↳ Narrowing
                                                                        ↳ QDP
                                                                          ↳ DependencyGraphProof
                                                                            ↳ QDP
                                                                              ↳ Narrowing
                                                                                ↳ QDP
                                                                                  ↳ DependencyGraphProof
                                                                                    ↳ QDP
                                                                                      ↳ Narrowing
                                                                                        ↳ QDP
                                                                                          ↳ DependencyGraphProof
                                                                                            ↳ QDP
                                                                                              ↳ Narrowing
                                                                                                ↳ QDP
                                                                                                  ↳ DependencyGraphProof
                                                                                                    ↳ QDP
                                                                                                      ↳ Narrowing
                                                                                                        ↳ QDP
                                                                                                          ↳ DependencyGraphProof
                                                                                                            ↳ QDP
                                                                                                              ↳ Narrowing

Q DP problem:
The TRS P consists of the following rules:

MARK(2ndsneg(square(x0), y1)) → A__2NDSNEG(a__square(mark(x0)), mark(y1))

MARK(2ndspos(square(y0), plus(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__plus(mark(x0), mark(x1)))

A__PI(times(x0, x1)) → A__2NDSPOS(a__times(mark(x0), mark(x1)), a__from(0))

A__2NDSPOS(s(y0), cons(y1, 2ndspos(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__2ndspos(mark(x0), mark(x1))))

A__2NDSNEG(s(N), cons(X, Z)) → A__2NDSNEG(s(mark(N)), cons2(X, mark(Z)))

MARK(2ndspos(pi(y0), cons2(x0, x1))) → A__2NDSPOS(a__pi(mark(y0)), cons2(x0, mark(x1)))

MARK(2ndspos(X1, X2)) → MARK(X2)

MARK(times(times(x0, x1), y1)) → A__TIMES(a__times(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(from(y0), cons2(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), cons2(x0, mark(x1)))

MARK(plus(plus(x0, x1), y1)) → A__PLUS(a__plus(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(2ndsneg(y0, y1), from(x0))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndspos(s(y0), from(x0))) → A__2NDSPOS(s(mark(y0)), a__from(mark(x0)))

MARK(2ndspos(from(y0), 2ndsneg(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndspos(pi(y0), 2ndsneg(x0, x1))) → A__2NDSPOS(a__pi(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndspos(2ndsneg(y0, y1), pi(x0))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__pi(mark(x0)))

MARK(2ndsneg(2ndsneg(x0, x1), y1)) → A__2NDSNEG(a__2ndsneg(mark(x0), mark(x1)), mark(y1))

A__2NDSNEG(s(N), cons(X, Z)) → MARK(Z)

A__TIMES(s(X), Y) → MARK(Y)

MARK(2ndsneg(2ndspos(x0, x1), y1)) → A__2NDSNEG(a__2ndspos(mark(x0), mark(x1)), mark(y1))

A__2NDSPOS(s(y0), cons(y1, plus(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__plus(mark(x0), mark(x1))))

MARK(2ndspos(pi(y0), from(x0))) → A__2NDSPOS(a__pi(mark(y0)), a__from(mark(x0)))

MARK(2ndspos(2ndsneg(y0, y1), square(x0))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndspos(square(y0), 2ndsneg(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndspos(from(y0), pi(x0))) → A__2NDSPOS(a__from(mark(y0)), a__pi(mark(x0)))

A__2NDSNEG(s(N), cons(X, Z)) → MARK(N)

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → A__2NDSNEG(mark(N), mark(Z))

MARK(2ndsneg(plus(x0, x1), y1)) → A__2NDSNEG(a__plus(mark(x0), mark(x1)), mark(y1))

MARK(times(2ndsneg(x0, x1), y1)) → A__TIMES(a__2ndsneg(mark(x0), mark(x1)), mark(y1))

MARK(2ndsneg(from(x0), y1)) → A__2NDSNEG(a__from(mark(x0)), mark(y1))

MARK(2ndspos(from(y0), from(x0))) → A__2NDSPOS(a__from(mark(y0)), a__from(mark(x0)))

MARK(2ndspos(pi(y0), pi(x0))) → A__2NDSPOS(a__pi(mark(y0)), a__pi(mark(x0)))

MARK(2ndspos(times(y0, y1), times(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(s(y0), pi(x0))) → A__2NDSPOS(s(mark(y0)), a__pi(mark(x0)))

A__TIMES(s(X), Y) → A__PLUS(mark(Y), a__times(mark(X), mark(Y)))

A__PI(2ndspos(x0, x1)) → A__2NDSPOS(a__2ndspos(mark(x0), mark(x1)), a__from(0))

MARK(2ndspos(square(y0), cons2(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), cons2(x0, mark(x1)))

MARK(plus(square(x0), y1)) → A__PLUS(a__square(mark(x0)), mark(y1))

MARK(2ndspos(plus(x0, x1), y1)) → A__2NDSPOS(a__plus(mark(x0), mark(x1)), mark(y1))

MARK(plus(s(x0), y1)) → A__PLUS(s(mark(x0)), mark(y1))

A__PI(plus(x0, x1)) → A__2NDSPOS(a__plus(mark(x0), mark(x1)), a__from(0))

MARK(2ndspos(square(y0), cons(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), cons(mark(x0), x1))

MARK(cons2(X1, X2)) → MARK(X2)

MARK(2ndspos(X1, X2)) → MARK(X1)

MARK(s(X)) → MARK(X)

MARK(pi(X)) → MARK(X)

MARK(plus(X1, X2)) → MARK(X2)

A__2NDSPOS(s(N), cons(X, Z)) → MARK(Z)

MARK(pi(X)) → A__PI(mark(X))

MARK(plus(2ndsneg(x0, x1), y1)) → A__PLUS(a__2ndsneg(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(times(y0, y1), cons2(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(2ndspos(2ndspos(y0, y1), cons2(x0, x1))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(2ndsneg(X1, X2)) → MARK(X2)

MARK(2ndspos(times(y0, y1), square(x0))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndspos(square(y0), times(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(2ndsneg(y0, y1), cons2(x0, x1))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(times(square(x0), y1)) → A__TIMES(a__square(mark(x0)), mark(y1))

A__PI(X) → A__FROM(0)

MARK(2ndspos(s(y0), plus(x0, x1))) → A__2NDSPOS(s(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(from(X)) → A__FROM(mark(X))

MARK(2ndspos(pi(y0), cons(x0, x1))) → A__2NDSPOS(a__pi(mark(y0)), cons(mark(x0), x1))

MARK(plus(X1, X2)) → MARK(X1)

MARK(times(from(x0), y1)) → A__TIMES(a__from(mark(x0)), mark(y1))

A__2NDSPOS(s(y0), cons(y1, times(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__times(mark(x0), mark(x1))))

MARK(2ndspos(s(y0), cons(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons(mark(x0), x1))

A__PI(X) → MARK(X)

MARK(times(s(x0), y1)) → A__TIMES(s(mark(x0)), mark(y1))

MARK(2ndspos(2ndspos(y0, y1), pi(x0))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), a__pi(mark(x0)))

MARK(2ndspos(pi(y0), 2ndspos(x0, x1))) → A__2NDSPOS(a__pi(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndsneg(s(x0), y1)) → A__2NDSNEG(s(mark(x0)), mark(y1))

MARK(2ndspos(from(y0), cons(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), cons(mark(x0), x1))

MARK(2ndspos(2ndspos(y0, y1), plus(x0, x1))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

A__PLUS(s(X), Y) → MARK(Y)

A__TIMES(s(X), Y) → MARK(X)

A__2NDSPOS(s(y0), cons(y1, pi(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__pi(mark(x0))))

MARK(2ndspos(pi(y0), plus(x0, x1))) → A__2NDSPOS(a__pi(mark(y0)), a__plus(mark(x0), mark(x1)))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(N)

MARK(plus(2ndspos(x0, x1), y1)) → A__PLUS(a__2ndspos(mark(x0), mark(x1)), mark(y1))

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(N)

A__PI(from(x0)) → A__2NDSPOS(a__from(mark(x0)), a__from(0))

MARK(cons(X1, X2)) → MARK(X1)

MARK(2ndspos(2ndspos(y0, y1), square(x0))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndspos(square(y0), 2ndspos(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndspos(times(y0, y1), cons(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), cons(mark(x0), x1))

MARK(2ndspos(s(y0), 2ndspos(x0, x1))) → A__2NDSPOS(s(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

MARK(times(X1, X2)) → MARK(X2)

A__PI(square(x0)) → A__2NDSPOS(a__square(mark(x0)), a__from(0))

MARK(2ndspos(2ndsneg(y0, y1), 2ndsneg(x0, x1))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndspos(times(y0, y1), plus(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(s(y0), times(x0, x1))) → A__2NDSPOS(s(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndsneg(X1, X2)) → MARK(X1)

MARK(2ndspos(s(y0), 2ndsneg(x0, x1))) → A__2NDSPOS(s(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndspos(2ndsneg(y0, y1), plus(x0, x1))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

A__PI(2ndsneg(x0, x1)) → A__2NDSPOS(a__2ndsneg(mark(x0), mark(x1)), a__from(0))

MARK(plus(pi(x0), y1)) → A__PLUS(a__pi(mark(x0)), mark(y1))

MARK(times(pi(x0), y1)) → A__TIMES(a__pi(mark(x0)), mark(y1))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(Y)

MARK(2ndspos(2ndspos(y0, y1), 2ndspos(x0, x1))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), a__2ndspos(mark(x0), mark(x1)))

MARK(rcons(X1, X2)) → MARK(X1)

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(Z)

MARK(square(X)) → MARK(X)

MARK(2ndspos(2ndspos(y0, y1), 2ndsneg(x0, x1))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndspos(2ndsneg(y0, y1), 2ndspos(x0, x1))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndspos(times(y0, y1), pi(x0))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__pi(mark(x0)))

MARK(2ndspos(pi(y0), times(x0, x1))) → A__2NDSPOS(a__pi(mark(y0)), a__times(mark(x0), mark(x1)))

A__PI(s(x0)) → A__2NDSPOS(s(mark(x0)), a__from(0))

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(Y)

MARK(2ndspos(2ndspos(y0, y1), cons(x0, x1))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), cons(mark(x0), x1))

A__FROM(X) → MARK(X)

MARK(times(plus(x0, x1), y1)) → A__TIMES(a__plus(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(2ndspos(y0, y1), times(x0, x1))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(times(y0, y1), 2ndspos(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__2ndspos(mark(x0), mark(x1)))

A__2NDSPOS(s(y0), cons(y1, from(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__from(mark(x0))))

MARK(2ndspos(pi(y0), square(x0))) → A__2NDSPOS(a__pi(mark(y0)), a__square(mark(x0)))

MARK(2ndspos(square(y0), pi(x0))) → A__2NDSPOS(a__square(mark(y0)), a__pi(mark(x0)))

MARK(2ndspos(s(y0), square(x0))) → A__2NDSPOS(s(mark(y0)), a__square(mark(x0)))

MARK(rcons(X1, X2)) → MARK(X2)

MARK(times(2ndspos(x0, x1), y1)) → A__TIMES(a__2ndspos(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(s(y0), cons2(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(x0, mark(x1)))

A__TIMES(s(X), Y) → A__TIMES(mark(X), mark(Y))

A__PLUS(s(X), Y) → MARK(X)

A__2NDSPOS(s(y0), cons(y1, cons(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, cons(mark(x0), x1)))

MARK(times(X1, X2)) → MARK(X1)

MARK(plus(times(x0, x1), y1)) → A__PLUS(a__times(mark(x0), mark(x1)), mark(y1))

A__SQUARE(X) → MARK(X)

MARK(2ndsneg(times(x0, x1), y1)) → A__2NDSNEG(a__times(mark(x0), mark(x1)), mark(y1))

A__2NDSPOS(s(y0), cons(y1, 2ndsneg(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__2ndsneg(mark(x0), mark(x1))))

A__2NDSPOS(s(N), cons(X, Z)) → MARK(N)

MARK(negrecip(X)) → MARK(X)

A__PLUS(0, Y) → MARK(Y)

MARK(posrecip(X)) → MARK(X)

MARK(2ndsneg(pi(x0), y1)) → A__2NDSNEG(a__pi(mark(x0)), mark(y1))

MARK(2ndspos(square(y0), from(x0))) → A__2NDSPOS(a__square(mark(y0)), a__from(mark(x0)))

MARK(2ndspos(from(y0), square(x0))) → A__2NDSPOS(a__from(mark(y0)), a__square(mark(x0)))

MARK(2ndspos(times(y0, y1), from(x0))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndspos(from(y0), times(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), a__times(mark(x0), mark(x1)))

A__2NDSPOS(s(y0), cons(y1, square(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__square(mark(x0))))

MARK(2ndspos(times(y0, y1), 2ndsneg(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndspos(2ndsneg(y0, y1), times(x0, x1))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(plus(0, y1)) → A__PLUS(0, mark(y1))

MARK(from(X)) → MARK(X)

A__PLUS(s(X), Y) → A__PLUS(mark(X), mark(Y))

A__SQUARE(X) → A__TIMES(mark(X), mark(X))

MARK(2ndspos(square(y0), square(x0))) → A__2NDSPOS(a__square(mark(y0)), a__square(mark(x0)))

A__PI(pi(x0)) → A__2NDSPOS(a__pi(mark(x0)), a__from(0))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(Z)

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → A__2NDSPOS(mark(N), mark(Z))

MARK(plus(from(x0), y1)) → A__PLUS(a__from(mark(x0)), mark(y1))

MARK(square(X)) → A__SQUARE(mark(X))

MARK(2ndspos(from(y0), plus(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(2ndspos(y0, y1), from(x0))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndspos(from(y0), 2ndspos(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndspos(2ndsneg(y0, y1), cons(x0, x1))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), cons(mark(x0), x1))

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))

a__2ndspos(0, Z) → rnil

a__2ndspos(s(N), cons(X, Z)) → a__2ndspos(s(mark(N)), cons2(X, mark(Z)))

a__2ndspos(s(N), cons2(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))

a__2ndsneg(0, Z) → rnil

a__2ndsneg(s(N), cons(X, Z)) → a__2ndsneg(s(mark(N)), cons2(X, mark(Z)))

a__2ndsneg(s(N), cons2(X, cons(Y, Z))) → rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z)))

a__pi(X) → a__2ndspos(mark(X), a__from(0))

a__plus(0, Y) → mark(Y)

a__plus(s(X), Y) → s(a__plus(mark(X), mark(Y)))

a__times(0, Y) → 0

a__times(s(X), Y) → a__plus(mark(Y), a__times(mark(X), mark(Y)))

a__square(X) → a__times(mark(X), mark(X))

mark(from(X)) → a__from(mark(X))

mark(2ndspos(X1, X2)) → a__2ndspos(mark(X1), mark(X2))

mark(2ndsneg(X1, X2)) → a__2ndsneg(mark(X1), mark(X2))

mark(pi(X)) → a__pi(mark(X))

mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))

mark(times(X1, X2)) → a__times(mark(X1), mark(X2))

mark(square(X)) → a__square(mark(X))

mark(0) → 0

mark(s(X)) → s(mark(X))

mark(posrecip(X)) → posrecip(mark(X))

mark(negrecip(X)) → negrecip(mark(X))

mark(nil) → nil

mark(cons(X1, X2)) → cons(mark(X1), X2)

mark(cons2(X1, X2)) → cons2(X1, mark(X2))

mark(rnil) → rnil

mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))

a__from(X) → from(X)

a__2ndspos(X1, X2) → 2ndspos(X1, X2)

a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)

a__pi(X) → pi(X)

a__plus(X1, X2) → plus(X1, X2)

a__times(X1, X2) → times(X1, X2)

a__square(X) → square(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule MARK(2ndspos(plus(x0, x1), y1)) → A__2NDSPOS(a__plus(mark(x0), mark(x1)), mark(y1)) at position [1] we obtained the following new rules:

MARK(2ndspos(plus(y0, y1), pi(x0))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__pi(mark(x0)))

MARK(2ndspos(plus(y0, y1), square(x0))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndspos(plus(y0, y1), rnil)) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), rnil)

MARK(2ndspos(plus(y0, y1), cons2(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(2ndspos(plus(y0, y1), times(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(plus(y0, y1), cons(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), cons(mark(x0), x1))

MARK(2ndspos(plus(y0, y1), posrecip(x0))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), posrecip(mark(x0)))

MARK(2ndspos(plus(y0, y1), 2ndsneg(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndspos(plus(y0, y1), s(x0))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), s(mark(x0)))

MARK(2ndspos(plus(y0, y1), nil)) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), nil)

MARK(2ndspos(plus(y0, y1), 0)) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), 0)

MARK(2ndspos(plus(y0, y1), rcons(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), rcons(mark(x0), mark(x1)))

MARK(2ndspos(plus(y0, y1), from(x0))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndspos(plus(y0, y1), plus(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(plus(y0, y1), 2ndspos(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndspos(plus(y0, y1), negrecip(x0))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), negrecip(mark(x0)))



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ Narrowing
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ QDP
              ↳ Narrowing
                ↳ QDP
                  ↳ DependencyGraphProof
                    ↳ QDP
                      ↳ Narrowing
                        ↳ QDP
                          ↳ DependencyGraphProof
                            ↳ QDP
                              ↳ Narrowing
                                ↳ QDP
                                  ↳ DependencyGraphProof
                                    ↳ QDP
                                      ↳ Narrowing
                                        ↳ QDP
                                          ↳ DependencyGraphProof
                                            ↳ QDP
                                              ↳ Narrowing
                                                ↳ QDP
                                                  ↳ DependencyGraphProof
                                                    ↳ QDP
                                                      ↳ Narrowing
                                                        ↳ QDP
                                                          ↳ DependencyGraphProof
                                                            ↳ QDP
                                                              ↳ Narrowing
                                                                ↳ QDP
                                                                  ↳ DependencyGraphProof
                                                                    ↳ QDP
                                                                      ↳ Narrowing
                                                                        ↳ QDP
                                                                          ↳ DependencyGraphProof
                                                                            ↳ QDP
                                                                              ↳ Narrowing
                                                                                ↳ QDP
                                                                                  ↳ DependencyGraphProof
                                                                                    ↳ QDP
                                                                                      ↳ Narrowing
                                                                                        ↳ QDP
                                                                                          ↳ DependencyGraphProof
                                                                                            ↳ QDP
                                                                                              ↳ Narrowing
                                                                                                ↳ QDP
                                                                                                  ↳ DependencyGraphProof
                                                                                                    ↳ QDP
                                                                                                      ↳ Narrowing
                                                                                                        ↳ QDP
                                                                                                          ↳ DependencyGraphProof
                                                                                                            ↳ QDP
                                                                                                              ↳ Narrowing
                                                                                                                ↳ QDP
                                                                                                                  ↳ DependencyGraphProof

Q DP problem:
The TRS P consists of the following rules:

MARK(2ndspos(plus(y0, y1), square(x0))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndspos(square(y0), plus(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(2ndsneg(square(x0), y1)) → A__2NDSNEG(a__square(mark(x0)), mark(y1))

MARK(2ndspos(plus(y0, y1), rnil)) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), rnil)

A__2NDSPOS(s(y0), cons(y1, 2ndspos(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__2ndspos(mark(x0), mark(x1))))

A__PI(times(x0, x1)) → A__2NDSPOS(a__times(mark(x0), mark(x1)), a__from(0))

A__2NDSNEG(s(N), cons(X, Z)) → A__2NDSNEG(s(mark(N)), cons2(X, mark(Z)))

MARK(times(times(x0, x1), y1)) → A__TIMES(a__times(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(X1, X2)) → MARK(X2)

MARK(2ndspos(pi(y0), cons2(x0, x1))) → A__2NDSPOS(a__pi(mark(y0)), cons2(x0, mark(x1)))

MARK(2ndspos(from(y0), cons2(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), cons2(x0, mark(x1)))

MARK(plus(plus(x0, x1), y1)) → A__PLUS(a__plus(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(from(y0), 2ndsneg(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndspos(s(y0), from(x0))) → A__2NDSPOS(s(mark(y0)), a__from(mark(x0)))

MARK(2ndspos(2ndsneg(y0, y1), from(x0))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndsneg(2ndsneg(x0, x1), y1)) → A__2NDSNEG(a__2ndsneg(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(2ndsneg(y0, y1), pi(x0))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__pi(mark(x0)))

MARK(2ndspos(pi(y0), 2ndsneg(x0, x1))) → A__2NDSPOS(a__pi(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

A__2NDSNEG(s(N), cons(X, Z)) → MARK(Z)

MARK(2ndspos(plus(y0, y1), posrecip(x0))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), posrecip(mark(x0)))

A__TIMES(s(X), Y) → MARK(Y)

MARK(2ndsneg(2ndspos(x0, x1), y1)) → A__2NDSNEG(a__2ndspos(mark(x0), mark(x1)), mark(y1))

A__2NDSPOS(s(y0), cons(y1, plus(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__plus(mark(x0), mark(x1))))

MARK(2ndspos(from(y0), pi(x0))) → A__2NDSPOS(a__from(mark(y0)), a__pi(mark(x0)))

MARK(2ndspos(square(y0), 2ndsneg(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndspos(2ndsneg(y0, y1), square(x0))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndspos(pi(y0), from(x0))) → A__2NDSPOS(a__pi(mark(y0)), a__from(mark(x0)))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → A__2NDSNEG(mark(N), mark(Z))

A__2NDSNEG(s(N), cons(X, Z)) → MARK(N)

MARK(times(2ndsneg(x0, x1), y1)) → A__TIMES(a__2ndsneg(mark(x0), mark(x1)), mark(y1))

MARK(2ndsneg(plus(x0, x1), y1)) → A__2NDSNEG(a__plus(mark(x0), mark(x1)), mark(y1))

MARK(2ndsneg(from(x0), y1)) → A__2NDSNEG(a__from(mark(x0)), mark(y1))

MARK(2ndspos(from(y0), from(x0))) → A__2NDSPOS(a__from(mark(y0)), a__from(mark(x0)))

MARK(2ndspos(pi(y0), pi(x0))) → A__2NDSPOS(a__pi(mark(y0)), a__pi(mark(x0)))

MARK(2ndspos(times(y0, y1), times(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

A__TIMES(s(X), Y) → A__PLUS(mark(Y), a__times(mark(X), mark(Y)))

MARK(2ndspos(s(y0), pi(x0))) → A__2NDSPOS(s(mark(y0)), a__pi(mark(x0)))

MARK(2ndspos(square(y0), cons2(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), cons2(x0, mark(x1)))

A__PI(2ndspos(x0, x1)) → A__2NDSPOS(a__2ndspos(mark(x0), mark(x1)), a__from(0))

MARK(plus(square(x0), y1)) → A__PLUS(a__square(mark(x0)), mark(y1))

MARK(plus(s(x0), y1)) → A__PLUS(s(mark(x0)), mark(y1))

MARK(2ndspos(square(y0), cons(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), cons(mark(x0), x1))

A__PI(plus(x0, x1)) → A__2NDSPOS(a__plus(mark(x0), mark(x1)), a__from(0))

MARK(cons2(X1, X2)) → MARK(X2)

MARK(s(X)) → MARK(X)

MARK(2ndspos(X1, X2)) → MARK(X1)

MARK(2ndspos(plus(y0, y1), cons2(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(pi(X)) → MARK(X)

MARK(plus(X1, X2)) → MARK(X2)

A__2NDSPOS(s(N), cons(X, Z)) → MARK(Z)

MARK(pi(X)) → A__PI(mark(X))

MARK(2ndspos(plus(y0, y1), cons(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), cons(mark(x0), x1))

MARK(plus(2ndsneg(x0, x1), y1)) → A__PLUS(a__2ndsneg(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(times(y0, y1), cons2(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(2ndspos(2ndspos(y0, y1), cons2(x0, x1))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(2ndsneg(X1, X2)) → MARK(X2)

MARK(2ndspos(square(y0), times(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(times(y0, y1), square(x0))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndspos(2ndsneg(y0, y1), cons2(x0, x1))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(times(square(x0), y1)) → A__TIMES(a__square(mark(x0)), mark(y1))

MARK(2ndspos(plus(y0, y1), s(x0))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), s(mark(x0)))

MARK(from(X)) → A__FROM(mark(X))

MARK(2ndspos(s(y0), plus(x0, x1))) → A__2NDSPOS(s(mark(y0)), a__plus(mark(x0), mark(x1)))

A__PI(X) → A__FROM(0)

MARK(2ndspos(plus(y0, y1), nil)) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), nil)

MARK(times(from(x0), y1)) → A__TIMES(a__from(mark(x0)), mark(y1))

MARK(plus(X1, X2)) → MARK(X1)

MARK(2ndspos(pi(y0), cons(x0, x1))) → A__2NDSPOS(a__pi(mark(y0)), cons(mark(x0), x1))

A__2NDSPOS(s(y0), cons(y1, times(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__times(mark(x0), mark(x1))))

A__PI(X) → MARK(X)

MARK(2ndspos(s(y0), cons(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons(mark(x0), x1))

MARK(2ndspos(plus(y0, y1), rcons(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), rcons(mark(x0), mark(x1)))

MARK(times(s(x0), y1)) → A__TIMES(s(mark(x0)), mark(y1))

MARK(2ndspos(pi(y0), 2ndspos(x0, x1))) → A__2NDSPOS(a__pi(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndspos(2ndspos(y0, y1), pi(x0))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), a__pi(mark(x0)))

MARK(2ndspos(from(y0), cons(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), cons(mark(x0), x1))

MARK(2ndsneg(s(x0), y1)) → A__2NDSNEG(s(mark(x0)), mark(y1))

MARK(2ndspos(plus(y0, y1), 2ndspos(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndspos(2ndspos(y0, y1), plus(x0, x1))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

A__PLUS(s(X), Y) → MARK(Y)

A__2NDSPOS(s(y0), cons(y1, pi(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__pi(mark(x0))))

A__TIMES(s(X), Y) → MARK(X)

MARK(2ndspos(plus(y0, y1), negrecip(x0))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), negrecip(mark(x0)))

MARK(2ndspos(plus(y0, y1), pi(x0))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__pi(mark(x0)))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(N)

MARK(2ndspos(pi(y0), plus(x0, x1))) → A__2NDSPOS(a__pi(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(plus(2ndspos(x0, x1), y1)) → A__PLUS(a__2ndspos(mark(x0), mark(x1)), mark(y1))

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(N)

MARK(cons(X1, X2)) → MARK(X1)

A__PI(from(x0)) → A__2NDSPOS(a__from(mark(x0)), a__from(0))

MARK(2ndspos(square(y0), 2ndspos(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndspos(2ndspos(y0, y1), square(x0))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(times(X1, X2)) → MARK(X2)

MARK(2ndspos(s(y0), 2ndspos(x0, x1))) → A__2NDSPOS(s(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndspos(times(y0, y1), cons(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), cons(mark(x0), x1))

MARK(2ndspos(2ndsneg(y0, y1), 2ndsneg(x0, x1))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__2ndsneg(mark(x0), mark(x1)))

A__PI(square(x0)) → A__2NDSPOS(a__square(mark(x0)), a__from(0))

MARK(2ndspos(plus(y0, y1), times(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(times(y0, y1), plus(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(s(y0), times(x0, x1))) → A__2NDSPOS(s(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndsneg(X1, X2)) → MARK(X1)

MARK(2ndspos(s(y0), 2ndsneg(x0, x1))) → A__2NDSPOS(s(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndspos(plus(y0, y1), 2ndsneg(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndspos(2ndsneg(y0, y1), plus(x0, x1))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

A__PI(2ndsneg(x0, x1)) → A__2NDSPOS(a__2ndsneg(mark(x0), mark(x1)), a__from(0))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(Y)

MARK(times(pi(x0), y1)) → A__TIMES(a__pi(mark(x0)), mark(y1))

MARK(plus(pi(x0), y1)) → A__PLUS(a__pi(mark(x0)), mark(y1))

MARK(2ndspos(2ndspos(y0, y1), 2ndspos(x0, x1))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), a__2ndspos(mark(x0), mark(x1)))

MARK(rcons(X1, X2)) → MARK(X1)

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(Z)

MARK(square(X)) → MARK(X)

MARK(2ndspos(2ndsneg(y0, y1), 2ndspos(x0, x1))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndspos(2ndspos(y0, y1), 2ndsneg(x0, x1))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndspos(plus(y0, y1), 0)) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), 0)

MARK(2ndspos(pi(y0), times(x0, x1))) → A__2NDSPOS(a__pi(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(times(y0, y1), pi(x0))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__pi(mark(x0)))

A__PI(s(x0)) → A__2NDSPOS(s(mark(x0)), a__from(0))

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(Y)

A__FROM(X) → MARK(X)

MARK(2ndspos(2ndspos(y0, y1), cons(x0, x1))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), cons(mark(x0), x1))

MARK(times(plus(x0, x1), y1)) → A__TIMES(a__plus(mark(x0), mark(x1)), mark(y1))

A__2NDSPOS(s(y0), cons(y1, from(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__from(mark(x0))))

MARK(2ndspos(times(y0, y1), 2ndspos(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndspos(2ndspos(y0, y1), times(x0, x1))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(square(y0), pi(x0))) → A__2NDSPOS(a__square(mark(y0)), a__pi(mark(x0)))

MARK(2ndspos(pi(y0), square(x0))) → A__2NDSPOS(a__pi(mark(y0)), a__square(mark(x0)))

MARK(rcons(X1, X2)) → MARK(X2)

MARK(2ndspos(s(y0), square(x0))) → A__2NDSPOS(s(mark(y0)), a__square(mark(x0)))

MARK(times(2ndspos(x0, x1), y1)) → A__TIMES(a__2ndspos(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(s(y0), cons2(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(x0, mark(x1)))

A__TIMES(s(X), Y) → A__TIMES(mark(X), mark(Y))

A__2NDSPOS(s(y0), cons(y1, cons(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, cons(mark(x0), x1)))

A__PLUS(s(X), Y) → MARK(X)

MARK(times(X1, X2)) → MARK(X1)

MARK(plus(times(x0, x1), y1)) → A__PLUS(a__times(mark(x0), mark(x1)), mark(y1))

A__SQUARE(X) → MARK(X)

MARK(2ndsneg(times(x0, x1), y1)) → A__2NDSNEG(a__times(mark(x0), mark(x1)), mark(y1))

A__2NDSPOS(s(y0), cons(y1, 2ndsneg(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__2ndsneg(mark(x0), mark(x1))))

MARK(negrecip(X)) → MARK(X)

A__2NDSPOS(s(N), cons(X, Z)) → MARK(N)

A__PLUS(0, Y) → MARK(Y)

MARK(posrecip(X)) → MARK(X)

MARK(2ndspos(from(y0), square(x0))) → A__2NDSPOS(a__from(mark(y0)), a__square(mark(x0)))

MARK(2ndspos(square(y0), from(x0))) → A__2NDSPOS(a__square(mark(y0)), a__from(mark(x0)))

MARK(2ndsneg(pi(x0), y1)) → A__2NDSNEG(a__pi(mark(x0)), mark(y1))

MARK(2ndspos(from(y0), times(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(times(y0, y1), from(x0))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__from(mark(x0)))

A__2NDSPOS(s(y0), cons(y1, square(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__square(mark(x0))))

MARK(2ndspos(2ndsneg(y0, y1), times(x0, x1))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(times(y0, y1), 2ndsneg(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__2ndsneg(mark(x0), mark(x1)))

MARK(plus(0, y1)) → A__PLUS(0, mark(y1))

MARK(from(X)) → MARK(X)

A__PLUS(s(X), Y) → A__PLUS(mark(X), mark(Y))

MARK(2ndspos(square(y0), square(x0))) → A__2NDSPOS(a__square(mark(y0)), a__square(mark(x0)))

A__SQUARE(X) → A__TIMES(mark(X), mark(X))

A__PI(pi(x0)) → A__2NDSPOS(a__pi(mark(x0)), a__from(0))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(Z)

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → A__2NDSPOS(mark(N), mark(Z))

MARK(plus(from(x0), y1)) → A__PLUS(a__from(mark(x0)), mark(y1))

MARK(square(X)) → A__SQUARE(mark(X))

MARK(2ndspos(plus(y0, y1), from(x0))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndspos(from(y0), plus(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(plus(y0, y1), plus(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(from(y0), 2ndspos(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndspos(2ndspos(y0, y1), from(x0))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndspos(2ndsneg(y0, y1), cons(x0, x1))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), cons(mark(x0), x1))

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))

a__2ndspos(0, Z) → rnil

a__2ndspos(s(N), cons(X, Z)) → a__2ndspos(s(mark(N)), cons2(X, mark(Z)))

a__2ndspos(s(N), cons2(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))

a__2ndsneg(0, Z) → rnil

a__2ndsneg(s(N), cons(X, Z)) → a__2ndsneg(s(mark(N)), cons2(X, mark(Z)))

a__2ndsneg(s(N), cons2(X, cons(Y, Z))) → rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z)))

a__pi(X) → a__2ndspos(mark(X), a__from(0))

a__plus(0, Y) → mark(Y)

a__plus(s(X), Y) → s(a__plus(mark(X), mark(Y)))

a__times(0, Y) → 0

a__times(s(X), Y) → a__plus(mark(Y), a__times(mark(X), mark(Y)))

a__square(X) → a__times(mark(X), mark(X))

mark(from(X)) → a__from(mark(X))

mark(2ndspos(X1, X2)) → a__2ndspos(mark(X1), mark(X2))

mark(2ndsneg(X1, X2)) → a__2ndsneg(mark(X1), mark(X2))

mark(pi(X)) → a__pi(mark(X))

mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))

mark(times(X1, X2)) → a__times(mark(X1), mark(X2))

mark(square(X)) → a__square(mark(X))

mark(0) → 0

mark(s(X)) → s(mark(X))

mark(posrecip(X)) → posrecip(mark(X))

mark(negrecip(X)) → negrecip(mark(X))

mark(nil) → nil

mark(cons(X1, X2)) → cons(mark(X1), X2)

mark(cons2(X1, X2)) → cons2(X1, mark(X2))

mark(rnil) → rnil

mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))

a__from(X) → from(X)

a__2ndspos(X1, X2) → 2ndspos(X1, X2)

a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)

a__pi(X) → pi(X)

a__plus(X1, X2) → plus(X1, X2)

a__times(X1, X2) → times(X1, X2)

a__square(X) → square(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 7 less nodes.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ Narrowing
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ QDP
              ↳ Narrowing
                ↳ QDP
                  ↳ DependencyGraphProof
                    ↳ QDP
                      ↳ Narrowing
                        ↳ QDP
                          ↳ DependencyGraphProof
                            ↳ QDP
                              ↳ Narrowing
                                ↳ QDP
                                  ↳ DependencyGraphProof
                                    ↳ QDP
                                      ↳ Narrowing
                                        ↳ QDP
                                          ↳ DependencyGraphProof
                                            ↳ QDP
                                              ↳ Narrowing
                                                ↳ QDP
                                                  ↳ DependencyGraphProof
                                                    ↳ QDP
                                                      ↳ Narrowing
                                                        ↳ QDP
                                                          ↳ DependencyGraphProof
                                                            ↳ QDP
                                                              ↳ Narrowing
                                                                ↳ QDP
                                                                  ↳ DependencyGraphProof
                                                                    ↳ QDP
                                                                      ↳ Narrowing
                                                                        ↳ QDP
                                                                          ↳ DependencyGraphProof
                                                                            ↳ QDP
                                                                              ↳ Narrowing
                                                                                ↳ QDP
                                                                                  ↳ DependencyGraphProof
                                                                                    ↳ QDP
                                                                                      ↳ Narrowing
                                                                                        ↳ QDP
                                                                                          ↳ DependencyGraphProof
                                                                                            ↳ QDP
                                                                                              ↳ Narrowing
                                                                                                ↳ QDP
                                                                                                  ↳ DependencyGraphProof
                                                                                                    ↳ QDP
                                                                                                      ↳ Narrowing
                                                                                                        ↳ QDP
                                                                                                          ↳ DependencyGraphProof
                                                                                                            ↳ QDP
                                                                                                              ↳ Narrowing
                                                                                                                ↳ QDP
                                                                                                                  ↳ DependencyGraphProof
                                                                                                                    ↳ QDP
                                                                                                                      ↳ Narrowing

Q DP problem:
The TRS P consists of the following rules:

MARK(2ndspos(plus(y0, y1), square(x0))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndsneg(square(x0), y1)) → A__2NDSNEG(a__square(mark(x0)), mark(y1))

MARK(2ndspos(square(y0), plus(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__plus(mark(x0), mark(x1)))

A__PI(times(x0, x1)) → A__2NDSPOS(a__times(mark(x0), mark(x1)), a__from(0))

A__2NDSPOS(s(y0), cons(y1, 2ndspos(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__2ndspos(mark(x0), mark(x1))))

A__2NDSNEG(s(N), cons(X, Z)) → A__2NDSNEG(s(mark(N)), cons2(X, mark(Z)))

MARK(2ndspos(pi(y0), cons2(x0, x1))) → A__2NDSPOS(a__pi(mark(y0)), cons2(x0, mark(x1)))

MARK(2ndspos(X1, X2)) → MARK(X2)

MARK(times(times(x0, x1), y1)) → A__TIMES(a__times(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(from(y0), cons2(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), cons2(x0, mark(x1)))

MARK(plus(plus(x0, x1), y1)) → A__PLUS(a__plus(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(2ndsneg(y0, y1), from(x0))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndspos(s(y0), from(x0))) → A__2NDSPOS(s(mark(y0)), a__from(mark(x0)))

MARK(2ndspos(from(y0), 2ndsneg(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndspos(pi(y0), 2ndsneg(x0, x1))) → A__2NDSPOS(a__pi(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndspos(2ndsneg(y0, y1), pi(x0))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__pi(mark(x0)))

MARK(2ndsneg(2ndsneg(x0, x1), y1)) → A__2NDSNEG(a__2ndsneg(mark(x0), mark(x1)), mark(y1))

A__2NDSNEG(s(N), cons(X, Z)) → MARK(Z)

A__TIMES(s(X), Y) → MARK(Y)

MARK(2ndsneg(2ndspos(x0, x1), y1)) → A__2NDSNEG(a__2ndspos(mark(x0), mark(x1)), mark(y1))

A__2NDSPOS(s(y0), cons(y1, plus(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__plus(mark(x0), mark(x1))))

MARK(2ndspos(pi(y0), from(x0))) → A__2NDSPOS(a__pi(mark(y0)), a__from(mark(x0)))

MARK(2ndspos(2ndsneg(y0, y1), square(x0))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndspos(square(y0), 2ndsneg(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndspos(from(y0), pi(x0))) → A__2NDSPOS(a__from(mark(y0)), a__pi(mark(x0)))

A__2NDSNEG(s(N), cons(X, Z)) → MARK(N)

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → A__2NDSNEG(mark(N), mark(Z))

MARK(2ndsneg(plus(x0, x1), y1)) → A__2NDSNEG(a__plus(mark(x0), mark(x1)), mark(y1))

MARK(times(2ndsneg(x0, x1), y1)) → A__TIMES(a__2ndsneg(mark(x0), mark(x1)), mark(y1))

MARK(2ndsneg(from(x0), y1)) → A__2NDSNEG(a__from(mark(x0)), mark(y1))

MARK(2ndspos(from(y0), from(x0))) → A__2NDSPOS(a__from(mark(y0)), a__from(mark(x0)))

MARK(2ndspos(pi(y0), pi(x0))) → A__2NDSPOS(a__pi(mark(y0)), a__pi(mark(x0)))

MARK(2ndspos(times(y0, y1), times(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(s(y0), pi(x0))) → A__2NDSPOS(s(mark(y0)), a__pi(mark(x0)))

A__TIMES(s(X), Y) → A__PLUS(mark(Y), a__times(mark(X), mark(Y)))

A__PI(2ndspos(x0, x1)) → A__2NDSPOS(a__2ndspos(mark(x0), mark(x1)), a__from(0))

MARK(2ndspos(square(y0), cons2(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), cons2(x0, mark(x1)))

MARK(plus(square(x0), y1)) → A__PLUS(a__square(mark(x0)), mark(y1))

MARK(plus(s(x0), y1)) → A__PLUS(s(mark(x0)), mark(y1))

A__PI(plus(x0, x1)) → A__2NDSPOS(a__plus(mark(x0), mark(x1)), a__from(0))

MARK(2ndspos(square(y0), cons(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), cons(mark(x0), x1))

MARK(cons2(X1, X2)) → MARK(X2)

MARK(2ndspos(X1, X2)) → MARK(X1)

MARK(s(X)) → MARK(X)

MARK(2ndspos(plus(y0, y1), cons2(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(pi(X)) → MARK(X)

MARK(plus(X1, X2)) → MARK(X2)

A__2NDSPOS(s(N), cons(X, Z)) → MARK(Z)

MARK(pi(X)) → A__PI(mark(X))

MARK(2ndspos(plus(y0, y1), cons(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), cons(mark(x0), x1))

MARK(plus(2ndsneg(x0, x1), y1)) → A__PLUS(a__2ndsneg(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(times(y0, y1), cons2(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(2ndspos(2ndspos(y0, y1), cons2(x0, x1))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(2ndsneg(X1, X2)) → MARK(X2)

MARK(2ndspos(times(y0, y1), square(x0))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndspos(square(y0), times(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(2ndsneg(y0, y1), cons2(x0, x1))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(times(square(x0), y1)) → A__TIMES(a__square(mark(x0)), mark(y1))

A__PI(X) → A__FROM(0)

MARK(2ndspos(s(y0), plus(x0, x1))) → A__2NDSPOS(s(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(from(X)) → A__FROM(mark(X))

MARK(2ndspos(pi(y0), cons(x0, x1))) → A__2NDSPOS(a__pi(mark(y0)), cons(mark(x0), x1))

MARK(plus(X1, X2)) → MARK(X1)

MARK(times(from(x0), y1)) → A__TIMES(a__from(mark(x0)), mark(y1))

A__2NDSPOS(s(y0), cons(y1, times(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__times(mark(x0), mark(x1))))

MARK(2ndspos(s(y0), cons(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons(mark(x0), x1))

A__PI(X) → MARK(X)

MARK(times(s(x0), y1)) → A__TIMES(s(mark(x0)), mark(y1))

MARK(2ndspos(2ndspos(y0, y1), pi(x0))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), a__pi(mark(x0)))

MARK(2ndspos(pi(y0), 2ndspos(x0, x1))) → A__2NDSPOS(a__pi(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndsneg(s(x0), y1)) → A__2NDSNEG(s(mark(x0)), mark(y1))

MARK(2ndspos(from(y0), cons(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), cons(mark(x0), x1))

MARK(2ndspos(plus(y0, y1), 2ndspos(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndspos(2ndspos(y0, y1), plus(x0, x1))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

A__PLUS(s(X), Y) → MARK(Y)

A__TIMES(s(X), Y) → MARK(X)

A__2NDSPOS(s(y0), cons(y1, pi(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__pi(mark(x0))))

MARK(2ndspos(plus(y0, y1), pi(x0))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__pi(mark(x0)))

MARK(2ndspos(pi(y0), plus(x0, x1))) → A__2NDSPOS(a__pi(mark(y0)), a__plus(mark(x0), mark(x1)))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(N)

MARK(plus(2ndspos(x0, x1), y1)) → A__PLUS(a__2ndspos(mark(x0), mark(x1)), mark(y1))

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(N)

A__PI(from(x0)) → A__2NDSPOS(a__from(mark(x0)), a__from(0))

MARK(cons(X1, X2)) → MARK(X1)

MARK(2ndspos(2ndspos(y0, y1), square(x0))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndspos(square(y0), 2ndspos(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndspos(times(y0, y1), cons(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), cons(mark(x0), x1))

MARK(2ndspos(s(y0), 2ndspos(x0, x1))) → A__2NDSPOS(s(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

MARK(times(X1, X2)) → MARK(X2)

A__PI(square(x0)) → A__2NDSPOS(a__square(mark(x0)), a__from(0))

MARK(2ndspos(2ndsneg(y0, y1), 2ndsneg(x0, x1))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndspos(plus(y0, y1), times(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(times(y0, y1), plus(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(s(y0), times(x0, x1))) → A__2NDSPOS(s(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndsneg(X1, X2)) → MARK(X1)

MARK(2ndspos(s(y0), 2ndsneg(x0, x1))) → A__2NDSPOS(s(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndspos(plus(y0, y1), 2ndsneg(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndspos(2ndsneg(y0, y1), plus(x0, x1))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

A__PI(2ndsneg(x0, x1)) → A__2NDSPOS(a__2ndsneg(mark(x0), mark(x1)), a__from(0))

MARK(plus(pi(x0), y1)) → A__PLUS(a__pi(mark(x0)), mark(y1))

MARK(times(pi(x0), y1)) → A__TIMES(a__pi(mark(x0)), mark(y1))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(Y)

MARK(2ndspos(2ndspos(y0, y1), 2ndspos(x0, x1))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), a__2ndspos(mark(x0), mark(x1)))

MARK(rcons(X1, X2)) → MARK(X1)

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(Z)

MARK(square(X)) → MARK(X)

MARK(2ndspos(2ndspos(y0, y1), 2ndsneg(x0, x1))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndspos(2ndsneg(y0, y1), 2ndspos(x0, x1))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndspos(times(y0, y1), pi(x0))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__pi(mark(x0)))

MARK(2ndspos(pi(y0), times(x0, x1))) → A__2NDSPOS(a__pi(mark(y0)), a__times(mark(x0), mark(x1)))

A__PI(s(x0)) → A__2NDSPOS(s(mark(x0)), a__from(0))

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(Y)

MARK(2ndspos(2ndspos(y0, y1), cons(x0, x1))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), cons(mark(x0), x1))

A__FROM(X) → MARK(X)

MARK(times(plus(x0, x1), y1)) → A__TIMES(a__plus(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(2ndspos(y0, y1), times(x0, x1))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(times(y0, y1), 2ndspos(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__2ndspos(mark(x0), mark(x1)))

A__2NDSPOS(s(y0), cons(y1, from(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__from(mark(x0))))

MARK(2ndspos(pi(y0), square(x0))) → A__2NDSPOS(a__pi(mark(y0)), a__square(mark(x0)))

MARK(2ndspos(square(y0), pi(x0))) → A__2NDSPOS(a__square(mark(y0)), a__pi(mark(x0)))

MARK(2ndspos(s(y0), square(x0))) → A__2NDSPOS(s(mark(y0)), a__square(mark(x0)))

MARK(rcons(X1, X2)) → MARK(X2)

MARK(times(2ndspos(x0, x1), y1)) → A__TIMES(a__2ndspos(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(s(y0), cons2(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(x0, mark(x1)))

A__TIMES(s(X), Y) → A__TIMES(mark(X), mark(Y))

A__PLUS(s(X), Y) → MARK(X)

A__2NDSPOS(s(y0), cons(y1, cons(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, cons(mark(x0), x1)))

MARK(times(X1, X2)) → MARK(X1)

MARK(plus(times(x0, x1), y1)) → A__PLUS(a__times(mark(x0), mark(x1)), mark(y1))

A__SQUARE(X) → MARK(X)

MARK(2ndsneg(times(x0, x1), y1)) → A__2NDSNEG(a__times(mark(x0), mark(x1)), mark(y1))

A__2NDSPOS(s(y0), cons(y1, 2ndsneg(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__2ndsneg(mark(x0), mark(x1))))

A__2NDSPOS(s(N), cons(X, Z)) → MARK(N)

MARK(negrecip(X)) → MARK(X)

A__PLUS(0, Y) → MARK(Y)

MARK(posrecip(X)) → MARK(X)

MARK(2ndsneg(pi(x0), y1)) → A__2NDSNEG(a__pi(mark(x0)), mark(y1))

MARK(2ndspos(square(y0), from(x0))) → A__2NDSPOS(a__square(mark(y0)), a__from(mark(x0)))

MARK(2ndspos(from(y0), square(x0))) → A__2NDSPOS(a__from(mark(y0)), a__square(mark(x0)))

MARK(2ndspos(times(y0, y1), from(x0))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndspos(from(y0), times(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), a__times(mark(x0), mark(x1)))

A__2NDSPOS(s(y0), cons(y1, square(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__square(mark(x0))))

MARK(2ndspos(times(y0, y1), 2ndsneg(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndspos(2ndsneg(y0, y1), times(x0, x1))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(plus(0, y1)) → A__PLUS(0, mark(y1))

MARK(from(X)) → MARK(X)

A__PLUS(s(X), Y) → A__PLUS(mark(X), mark(Y))

A__SQUARE(X) → A__TIMES(mark(X), mark(X))

MARK(2ndspos(square(y0), square(x0))) → A__2NDSPOS(a__square(mark(y0)), a__square(mark(x0)))

A__PI(pi(x0)) → A__2NDSPOS(a__pi(mark(x0)), a__from(0))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(Z)

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → A__2NDSPOS(mark(N), mark(Z))

MARK(plus(from(x0), y1)) → A__PLUS(a__from(mark(x0)), mark(y1))

MARK(square(X)) → A__SQUARE(mark(X))

MARK(2ndspos(plus(y0, y1), from(x0))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndspos(from(y0), plus(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(plus(y0, y1), plus(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(2ndspos(y0, y1), from(x0))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndspos(from(y0), 2ndspos(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndspos(2ndsneg(y0, y1), cons(x0, x1))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), cons(mark(x0), x1))

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))

a__2ndspos(0, Z) → rnil

a__2ndspos(s(N), cons(X, Z)) → a__2ndspos(s(mark(N)), cons2(X, mark(Z)))

a__2ndspos(s(N), cons2(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))

a__2ndsneg(0, Z) → rnil

a__2ndsneg(s(N), cons(X, Z)) → a__2ndsneg(s(mark(N)), cons2(X, mark(Z)))

a__2ndsneg(s(N), cons2(X, cons(Y, Z))) → rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z)))

a__pi(X) → a__2ndspos(mark(X), a__from(0))

a__plus(0, Y) → mark(Y)

a__plus(s(X), Y) → s(a__plus(mark(X), mark(Y)))

a__times(0, Y) → 0

a__times(s(X), Y) → a__plus(mark(Y), a__times(mark(X), mark(Y)))

a__square(X) → a__times(mark(X), mark(X))

mark(from(X)) → a__from(mark(X))

mark(2ndspos(X1, X2)) → a__2ndspos(mark(X1), mark(X2))

mark(2ndsneg(X1, X2)) → a__2ndsneg(mark(X1), mark(X2))

mark(pi(X)) → a__pi(mark(X))

mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))

mark(times(X1, X2)) → a__times(mark(X1), mark(X2))

mark(square(X)) → a__square(mark(X))

mark(0) → 0

mark(s(X)) → s(mark(X))

mark(posrecip(X)) → posrecip(mark(X))

mark(negrecip(X)) → negrecip(mark(X))

mark(nil) → nil

mark(cons(X1, X2)) → cons(mark(X1), X2)

mark(cons2(X1, X2)) → cons2(X1, mark(X2))

mark(rnil) → rnil

mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))

a__from(X) → from(X)

a__2ndspos(X1, X2) → 2ndspos(X1, X2)

a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)

a__pi(X) → pi(X)

a__plus(X1, X2) → plus(X1, X2)

a__times(X1, X2) → times(X1, X2)

a__square(X) → square(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule MARK(2ndsneg(square(x0), y1)) → A__2NDSNEG(a__square(mark(x0)), mark(y1)) at position [1] we obtained the following new rules:

MARK(2ndsneg(square(y0), cons2(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), cons2(x0, mark(x1)))

MARK(2ndsneg(square(y0), 0)) → A__2NDSNEG(a__square(mark(y0)), 0)

MARK(2ndsneg(square(y0), rcons(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), rcons(mark(x0), mark(x1)))

MARK(2ndsneg(square(y0), plus(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(2ndsneg(square(y0), negrecip(x0))) → A__2NDSNEG(a__square(mark(y0)), negrecip(mark(x0)))

MARK(2ndsneg(square(y0), nil)) → A__2NDSNEG(a__square(mark(y0)), nil)

MARK(2ndsneg(square(y0), s(x0))) → A__2NDSNEG(a__square(mark(y0)), s(mark(x0)))

MARK(2ndsneg(square(y0), pi(x0))) → A__2NDSNEG(a__square(mark(y0)), a__pi(mark(x0)))

MARK(2ndsneg(square(y0), posrecip(x0))) → A__2NDSNEG(a__square(mark(y0)), posrecip(mark(x0)))

MARK(2ndsneg(square(y0), 2ndsneg(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndsneg(square(y0), rnil)) → A__2NDSNEG(a__square(mark(y0)), rnil)

MARK(2ndsneg(square(y0), cons(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), cons(mark(x0), x1))

MARK(2ndsneg(square(y0), times(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndsneg(square(y0), 2ndspos(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndsneg(square(y0), square(x0))) → A__2NDSNEG(a__square(mark(y0)), a__square(mark(x0)))

MARK(2ndsneg(square(y0), from(x0))) → A__2NDSNEG(a__square(mark(y0)), a__from(mark(x0)))



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ Narrowing
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ QDP
              ↳ Narrowing
                ↳ QDP
                  ↳ DependencyGraphProof
                    ↳ QDP
                      ↳ Narrowing
                        ↳ QDP
                          ↳ DependencyGraphProof
                            ↳ QDP
                              ↳ Narrowing
                                ↳ QDP
                                  ↳ DependencyGraphProof
                                    ↳ QDP
                                      ↳ Narrowing
                                        ↳ QDP
                                          ↳ DependencyGraphProof
                                            ↳ QDP
                                              ↳ Narrowing
                                                ↳ QDP
                                                  ↳ DependencyGraphProof
                                                    ↳ QDP
                                                      ↳ Narrowing
                                                        ↳ QDP
                                                          ↳ DependencyGraphProof
                                                            ↳ QDP
                                                              ↳ Narrowing
                                                                ↳ QDP
                                                                  ↳ DependencyGraphProof
                                                                    ↳ QDP
                                                                      ↳ Narrowing
                                                                        ↳ QDP
                                                                          ↳ DependencyGraphProof
                                                                            ↳ QDP
                                                                              ↳ Narrowing
                                                                                ↳ QDP
                                                                                  ↳ DependencyGraphProof
                                                                                    ↳ QDP
                                                                                      ↳ Narrowing
                                                                                        ↳ QDP
                                                                                          ↳ DependencyGraphProof
                                                                                            ↳ QDP
                                                                                              ↳ Narrowing
                                                                                                ↳ QDP
                                                                                                  ↳ DependencyGraphProof
                                                                                                    ↳ QDP
                                                                                                      ↳ Narrowing
                                                                                                        ↳ QDP
                                                                                                          ↳ DependencyGraphProof
                                                                                                            ↳ QDP
                                                                                                              ↳ Narrowing
                                                                                                                ↳ QDP
                                                                                                                  ↳ DependencyGraphProof
                                                                                                                    ↳ QDP
                                                                                                                      ↳ Narrowing
                                                                                                                        ↳ QDP
                                                                                                                          ↳ DependencyGraphProof

Q DP problem:
The TRS P consists of the following rules:

MARK(2ndspos(square(y0), plus(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(plus(y0, y1), square(x0))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__square(mark(x0)))

A__2NDSPOS(s(y0), cons(y1, 2ndspos(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__2ndspos(mark(x0), mark(x1))))

A__PI(times(x0, x1)) → A__2NDSPOS(a__times(mark(x0), mark(x1)), a__from(0))

A__2NDSNEG(s(N), cons(X, Z)) → A__2NDSNEG(s(mark(N)), cons2(X, mark(Z)))

MARK(2ndsneg(square(y0), s(x0))) → A__2NDSNEG(a__square(mark(y0)), s(mark(x0)))

MARK(times(times(x0, x1), y1)) → A__TIMES(a__times(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(X1, X2)) → MARK(X2)

MARK(2ndspos(pi(y0), cons2(x0, x1))) → A__2NDSPOS(a__pi(mark(y0)), cons2(x0, mark(x1)))

MARK(2ndspos(from(y0), cons2(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), cons2(x0, mark(x1)))

MARK(plus(plus(x0, x1), y1)) → A__PLUS(a__plus(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(from(y0), 2ndsneg(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndspos(s(y0), from(x0))) → A__2NDSPOS(s(mark(y0)), a__from(mark(x0)))

MARK(2ndspos(2ndsneg(y0, y1), from(x0))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndsneg(2ndsneg(x0, x1), y1)) → A__2NDSNEG(a__2ndsneg(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(2ndsneg(y0, y1), pi(x0))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__pi(mark(x0)))

MARK(2ndspos(pi(y0), 2ndsneg(x0, x1))) → A__2NDSPOS(a__pi(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndsneg(square(y0), 2ndspos(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

A__2NDSNEG(s(N), cons(X, Z)) → MARK(Z)

A__TIMES(s(X), Y) → MARK(Y)

MARK(2ndsneg(square(y0), 0)) → A__2NDSNEG(a__square(mark(y0)), 0)

MARK(2ndsneg(2ndspos(x0, x1), y1)) → A__2NDSNEG(a__2ndspos(mark(x0), mark(x1)), mark(y1))

A__2NDSPOS(s(y0), cons(y1, plus(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__plus(mark(x0), mark(x1))))

MARK(2ndsneg(square(y0), negrecip(x0))) → A__2NDSNEG(a__square(mark(y0)), negrecip(mark(x0)))

MARK(2ndspos(from(y0), pi(x0))) → A__2NDSPOS(a__from(mark(y0)), a__pi(mark(x0)))

MARK(2ndspos(square(y0), 2ndsneg(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndspos(2ndsneg(y0, y1), square(x0))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndspos(pi(y0), from(x0))) → A__2NDSPOS(a__pi(mark(y0)), a__from(mark(x0)))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → A__2NDSNEG(mark(N), mark(Z))

A__2NDSNEG(s(N), cons(X, Z)) → MARK(N)

MARK(times(2ndsneg(x0, x1), y1)) → A__TIMES(a__2ndsneg(mark(x0), mark(x1)), mark(y1))

MARK(2ndsneg(plus(x0, x1), y1)) → A__2NDSNEG(a__plus(mark(x0), mark(x1)), mark(y1))

MARK(2ndsneg(square(y0), posrecip(x0))) → A__2NDSNEG(a__square(mark(y0)), posrecip(mark(x0)))

MARK(2ndsneg(from(x0), y1)) → A__2NDSNEG(a__from(mark(x0)), mark(y1))

MARK(2ndspos(from(y0), from(x0))) → A__2NDSPOS(a__from(mark(y0)), a__from(mark(x0)))

MARK(2ndspos(pi(y0), pi(x0))) → A__2NDSPOS(a__pi(mark(y0)), a__pi(mark(x0)))

MARK(2ndsneg(square(y0), 2ndsneg(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndsneg(square(y0), rnil)) → A__2NDSNEG(a__square(mark(y0)), rnil)

MARK(2ndspos(times(y0, y1), times(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

A__TIMES(s(X), Y) → A__PLUS(mark(Y), a__times(mark(X), mark(Y)))

MARK(2ndspos(s(y0), pi(x0))) → A__2NDSPOS(s(mark(y0)), a__pi(mark(x0)))

MARK(2ndspos(square(y0), cons2(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), cons2(x0, mark(x1)))

A__PI(2ndspos(x0, x1)) → A__2NDSPOS(a__2ndspos(mark(x0), mark(x1)), a__from(0))

MARK(plus(square(x0), y1)) → A__PLUS(a__square(mark(x0)), mark(y1))

MARK(plus(s(x0), y1)) → A__PLUS(s(mark(x0)), mark(y1))

MARK(2ndspos(square(y0), cons(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), cons(mark(x0), x1))

A__PI(plus(x0, x1)) → A__2NDSPOS(a__plus(mark(x0), mark(x1)), a__from(0))

MARK(cons2(X1, X2)) → MARK(X2)

MARK(2ndsneg(square(y0), cons2(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), cons2(x0, mark(x1)))

MARK(2ndsneg(square(y0), plus(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(2ndsneg(square(y0), rcons(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), rcons(mark(x0), mark(x1)))

MARK(s(X)) → MARK(X)

MARK(2ndspos(X1, X2)) → MARK(X1)

MARK(2ndspos(plus(y0, y1), cons2(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(pi(X)) → MARK(X)

MARK(plus(X1, X2)) → MARK(X2)

A__2NDSPOS(s(N), cons(X, Z)) → MARK(Z)

MARK(pi(X)) → A__PI(mark(X))

MARK(2ndsneg(square(y0), times(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(plus(y0, y1), cons(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), cons(mark(x0), x1))

MARK(plus(2ndsneg(x0, x1), y1)) → A__PLUS(a__2ndsneg(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(times(y0, y1), cons2(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(2ndspos(2ndspos(y0, y1), cons2(x0, x1))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(2ndsneg(X1, X2)) → MARK(X2)

MARK(2ndsneg(square(y0), from(x0))) → A__2NDSNEG(a__square(mark(y0)), a__from(mark(x0)))

MARK(2ndspos(square(y0), times(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(times(y0, y1), square(x0))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndspos(2ndsneg(y0, y1), cons2(x0, x1))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(times(square(x0), y1)) → A__TIMES(a__square(mark(x0)), mark(y1))

MARK(from(X)) → A__FROM(mark(X))

MARK(2ndspos(s(y0), plus(x0, x1))) → A__2NDSPOS(s(mark(y0)), a__plus(mark(x0), mark(x1)))

A__PI(X) → A__FROM(0)

MARK(times(from(x0), y1)) → A__TIMES(a__from(mark(x0)), mark(y1))

MARK(plus(X1, X2)) → MARK(X1)

MARK(2ndspos(pi(y0), cons(x0, x1))) → A__2NDSPOS(a__pi(mark(y0)), cons(mark(x0), x1))

MARK(2ndsneg(square(y0), nil)) → A__2NDSNEG(a__square(mark(y0)), nil)

A__2NDSPOS(s(y0), cons(y1, times(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__times(mark(x0), mark(x1))))

A__PI(X) → MARK(X)

MARK(2ndspos(s(y0), cons(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons(mark(x0), x1))

MARK(times(s(x0), y1)) → A__TIMES(s(mark(x0)), mark(y1))

MARK(2ndspos(pi(y0), 2ndspos(x0, x1))) → A__2NDSPOS(a__pi(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndspos(2ndspos(y0, y1), pi(x0))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), a__pi(mark(x0)))

MARK(2ndspos(from(y0), cons(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), cons(mark(x0), x1))

MARK(2ndsneg(s(x0), y1)) → A__2NDSNEG(s(mark(x0)), mark(y1))

MARK(2ndspos(2ndspos(y0, y1), plus(x0, x1))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(plus(y0, y1), 2ndspos(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__2ndspos(mark(x0), mark(x1)))

A__PLUS(s(X), Y) → MARK(Y)

A__2NDSPOS(s(y0), cons(y1, pi(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__pi(mark(x0))))

A__TIMES(s(X), Y) → MARK(X)

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(N)

MARK(2ndspos(pi(y0), plus(x0, x1))) → A__2NDSPOS(a__pi(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(plus(y0, y1), pi(x0))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__pi(mark(x0)))

MARK(plus(2ndspos(x0, x1), y1)) → A__PLUS(a__2ndspos(mark(x0), mark(x1)), mark(y1))

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(N)

MARK(cons(X1, X2)) → MARK(X1)

A__PI(from(x0)) → A__2NDSPOS(a__from(mark(x0)), a__from(0))

MARK(2ndspos(square(y0), 2ndspos(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndspos(2ndspos(y0, y1), square(x0))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(times(X1, X2)) → MARK(X2)

MARK(2ndspos(s(y0), 2ndspos(x0, x1))) → A__2NDSPOS(s(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndspos(times(y0, y1), cons(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), cons(mark(x0), x1))

MARK(2ndspos(2ndsneg(y0, y1), 2ndsneg(x0, x1))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__2ndsneg(mark(x0), mark(x1)))

A__PI(square(x0)) → A__2NDSPOS(a__square(mark(x0)), a__from(0))

MARK(2ndsneg(square(y0), cons(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), cons(mark(x0), x1))

MARK(2ndspos(times(y0, y1), plus(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(plus(y0, y1), times(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(s(y0), times(x0, x1))) → A__2NDSPOS(s(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndsneg(X1, X2)) → MARK(X1)

MARK(2ndspos(s(y0), 2ndsneg(x0, x1))) → A__2NDSPOS(s(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndsneg(square(y0), square(x0))) → A__2NDSNEG(a__square(mark(y0)), a__square(mark(x0)))

MARK(2ndspos(2ndsneg(y0, y1), plus(x0, x1))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(plus(y0, y1), 2ndsneg(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__2ndsneg(mark(x0), mark(x1)))

A__PI(2ndsneg(x0, x1)) → A__2NDSPOS(a__2ndsneg(mark(x0), mark(x1)), a__from(0))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(Y)

MARK(times(pi(x0), y1)) → A__TIMES(a__pi(mark(x0)), mark(y1))

MARK(plus(pi(x0), y1)) → A__PLUS(a__pi(mark(x0)), mark(y1))

MARK(2ndspos(2ndspos(y0, y1), 2ndspos(x0, x1))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), a__2ndspos(mark(x0), mark(x1)))

MARK(rcons(X1, X2)) → MARK(X1)

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(Z)

MARK(2ndsneg(square(y0), pi(x0))) → A__2NDSNEG(a__square(mark(y0)), a__pi(mark(x0)))

MARK(square(X)) → MARK(X)

MARK(2ndspos(2ndsneg(y0, y1), 2ndspos(x0, x1))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndspos(2ndspos(y0, y1), 2ndsneg(x0, x1))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndspos(pi(y0), times(x0, x1))) → A__2NDSPOS(a__pi(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(times(y0, y1), pi(x0))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__pi(mark(x0)))

A__PI(s(x0)) → A__2NDSPOS(s(mark(x0)), a__from(0))

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(Y)

A__FROM(X) → MARK(X)

MARK(2ndspos(2ndspos(y0, y1), cons(x0, x1))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), cons(mark(x0), x1))

MARK(times(plus(x0, x1), y1)) → A__TIMES(a__plus(mark(x0), mark(x1)), mark(y1))

A__2NDSPOS(s(y0), cons(y1, from(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__from(mark(x0))))

MARK(2ndspos(times(y0, y1), 2ndspos(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndspos(2ndspos(y0, y1), times(x0, x1))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(square(y0), pi(x0))) → A__2NDSPOS(a__square(mark(y0)), a__pi(mark(x0)))

MARK(2ndspos(pi(y0), square(x0))) → A__2NDSPOS(a__pi(mark(y0)), a__square(mark(x0)))

MARK(rcons(X1, X2)) → MARK(X2)

MARK(2ndspos(s(y0), square(x0))) → A__2NDSPOS(s(mark(y0)), a__square(mark(x0)))

MARK(times(2ndspos(x0, x1), y1)) → A__TIMES(a__2ndspos(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(s(y0), cons2(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(x0, mark(x1)))

A__TIMES(s(X), Y) → A__TIMES(mark(X), mark(Y))

A__2NDSPOS(s(y0), cons(y1, cons(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, cons(mark(x0), x1)))

A__PLUS(s(X), Y) → MARK(X)

MARK(times(X1, X2)) → MARK(X1)

MARK(plus(times(x0, x1), y1)) → A__PLUS(a__times(mark(x0), mark(x1)), mark(y1))

A__SQUARE(X) → MARK(X)

MARK(2ndsneg(times(x0, x1), y1)) → A__2NDSNEG(a__times(mark(x0), mark(x1)), mark(y1))

A__2NDSPOS(s(y0), cons(y1, 2ndsneg(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__2ndsneg(mark(x0), mark(x1))))

MARK(negrecip(X)) → MARK(X)

A__2NDSPOS(s(N), cons(X, Z)) → MARK(N)

A__PLUS(0, Y) → MARK(Y)

MARK(posrecip(X)) → MARK(X)

MARK(2ndspos(from(y0), square(x0))) → A__2NDSPOS(a__from(mark(y0)), a__square(mark(x0)))

MARK(2ndspos(square(y0), from(x0))) → A__2NDSPOS(a__square(mark(y0)), a__from(mark(x0)))

MARK(2ndsneg(pi(x0), y1)) → A__2NDSNEG(a__pi(mark(x0)), mark(y1))

MARK(2ndspos(from(y0), times(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(times(y0, y1), from(x0))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__from(mark(x0)))

A__2NDSPOS(s(y0), cons(y1, square(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__square(mark(x0))))

MARK(2ndspos(2ndsneg(y0, y1), times(x0, x1))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(times(y0, y1), 2ndsneg(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__2ndsneg(mark(x0), mark(x1)))

MARK(plus(0, y1)) → A__PLUS(0, mark(y1))

MARK(from(X)) → MARK(X)

A__PLUS(s(X), Y) → A__PLUS(mark(X), mark(Y))

MARK(2ndspos(square(y0), square(x0))) → A__2NDSPOS(a__square(mark(y0)), a__square(mark(x0)))

A__SQUARE(X) → A__TIMES(mark(X), mark(X))

A__PI(pi(x0)) → A__2NDSPOS(a__pi(mark(x0)), a__from(0))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(Z)

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → A__2NDSPOS(mark(N), mark(Z))

MARK(plus(from(x0), y1)) → A__PLUS(a__from(mark(x0)), mark(y1))

MARK(square(X)) → A__SQUARE(mark(X))

MARK(2ndspos(from(y0), plus(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(plus(y0, y1), from(x0))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndspos(plus(y0, y1), plus(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(from(y0), 2ndspos(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndspos(2ndspos(y0, y1), from(x0))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndspos(2ndsneg(y0, y1), cons(x0, x1))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), cons(mark(x0), x1))

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))

a__2ndspos(0, Z) → rnil

a__2ndspos(s(N), cons(X, Z)) → a__2ndspos(s(mark(N)), cons2(X, mark(Z)))

a__2ndspos(s(N), cons2(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))

a__2ndsneg(0, Z) → rnil

a__2ndsneg(s(N), cons(X, Z)) → a__2ndsneg(s(mark(N)), cons2(X, mark(Z)))

a__2ndsneg(s(N), cons2(X, cons(Y, Z))) → rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z)))

a__pi(X) → a__2ndspos(mark(X), a__from(0))

a__plus(0, Y) → mark(Y)

a__plus(s(X), Y) → s(a__plus(mark(X), mark(Y)))

a__times(0, Y) → 0

a__times(s(X), Y) → a__plus(mark(Y), a__times(mark(X), mark(Y)))

a__square(X) → a__times(mark(X), mark(X))

mark(from(X)) → a__from(mark(X))

mark(2ndspos(X1, X2)) → a__2ndspos(mark(X1), mark(X2))

mark(2ndsneg(X1, X2)) → a__2ndsneg(mark(X1), mark(X2))

mark(pi(X)) → a__pi(mark(X))

mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))

mark(times(X1, X2)) → a__times(mark(X1), mark(X2))

mark(square(X)) → a__square(mark(X))

mark(0) → 0

mark(s(X)) → s(mark(X))

mark(posrecip(X)) → posrecip(mark(X))

mark(negrecip(X)) → negrecip(mark(X))

mark(nil) → nil

mark(cons(X1, X2)) → cons(mark(X1), X2)

mark(cons2(X1, X2)) → cons2(X1, mark(X2))

mark(rnil) → rnil

mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))

a__from(X) → from(X)

a__2ndspos(X1, X2) → 2ndspos(X1, X2)

a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)

a__pi(X) → pi(X)

a__plus(X1, X2) → plus(X1, X2)

a__times(X1, X2) → times(X1, X2)

a__square(X) → square(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 7 less nodes.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ Narrowing
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ QDP
              ↳ Narrowing
                ↳ QDP
                  ↳ DependencyGraphProof
                    ↳ QDP
                      ↳ Narrowing
                        ↳ QDP
                          ↳ DependencyGraphProof
                            ↳ QDP
                              ↳ Narrowing
                                ↳ QDP
                                  ↳ DependencyGraphProof
                                    ↳ QDP
                                      ↳ Narrowing
                                        ↳ QDP
                                          ↳ DependencyGraphProof
                                            ↳ QDP
                                              ↳ Narrowing
                                                ↳ QDP
                                                  ↳ DependencyGraphProof
                                                    ↳ QDP
                                                      ↳ Narrowing
                                                        ↳ QDP
                                                          ↳ DependencyGraphProof
                                                            ↳ QDP
                                                              ↳ Narrowing
                                                                ↳ QDP
                                                                  ↳ DependencyGraphProof
                                                                    ↳ QDP
                                                                      ↳ Narrowing
                                                                        ↳ QDP
                                                                          ↳ DependencyGraphProof
                                                                            ↳ QDP
                                                                              ↳ Narrowing
                                                                                ↳ QDP
                                                                                  ↳ DependencyGraphProof
                                                                                    ↳ QDP
                                                                                      ↳ Narrowing
                                                                                        ↳ QDP
                                                                                          ↳ DependencyGraphProof
                                                                                            ↳ QDP
                                                                                              ↳ Narrowing
                                                                                                ↳ QDP
                                                                                                  ↳ DependencyGraphProof
                                                                                                    ↳ QDP
                                                                                                      ↳ Narrowing
                                                                                                        ↳ QDP
                                                                                                          ↳ DependencyGraphProof
                                                                                                            ↳ QDP
                                                                                                              ↳ Narrowing
                                                                                                                ↳ QDP
                                                                                                                  ↳ DependencyGraphProof
                                                                                                                    ↳ QDP
                                                                                                                      ↳ Narrowing
                                                                                                                        ↳ QDP
                                                                                                                          ↳ DependencyGraphProof
                                                                                                                            ↳ QDP
                                                                                                                              ↳ Narrowing

Q DP problem:
The TRS P consists of the following rules:

MARK(2ndspos(plus(y0, y1), square(x0))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndspos(square(y0), plus(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__plus(mark(x0), mark(x1)))

A__PI(times(x0, x1)) → A__2NDSPOS(a__times(mark(x0), mark(x1)), a__from(0))

A__2NDSPOS(s(y0), cons(y1, 2ndspos(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__2ndspos(mark(x0), mark(x1))))

A__2NDSNEG(s(N), cons(X, Z)) → A__2NDSNEG(s(mark(N)), cons2(X, mark(Z)))

MARK(2ndspos(pi(y0), cons2(x0, x1))) → A__2NDSPOS(a__pi(mark(y0)), cons2(x0, mark(x1)))

MARK(2ndspos(X1, X2)) → MARK(X2)

MARK(times(times(x0, x1), y1)) → A__TIMES(a__times(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(from(y0), cons2(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), cons2(x0, mark(x1)))

MARK(plus(plus(x0, x1), y1)) → A__PLUS(a__plus(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(2ndsneg(y0, y1), from(x0))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndspos(s(y0), from(x0))) → A__2NDSPOS(s(mark(y0)), a__from(mark(x0)))

MARK(2ndspos(from(y0), 2ndsneg(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndspos(pi(y0), 2ndsneg(x0, x1))) → A__2NDSPOS(a__pi(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndspos(2ndsneg(y0, y1), pi(x0))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__pi(mark(x0)))

MARK(2ndsneg(2ndsneg(x0, x1), y1)) → A__2NDSNEG(a__2ndsneg(mark(x0), mark(x1)), mark(y1))

MARK(2ndsneg(square(y0), 2ndspos(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

A__2NDSNEG(s(N), cons(X, Z)) → MARK(Z)

A__TIMES(s(X), Y) → MARK(Y)

MARK(2ndsneg(2ndspos(x0, x1), y1)) → A__2NDSNEG(a__2ndspos(mark(x0), mark(x1)), mark(y1))

A__2NDSPOS(s(y0), cons(y1, plus(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__plus(mark(x0), mark(x1))))

MARK(2ndspos(pi(y0), from(x0))) → A__2NDSPOS(a__pi(mark(y0)), a__from(mark(x0)))

MARK(2ndspos(2ndsneg(y0, y1), square(x0))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndspos(square(y0), 2ndsneg(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndspos(from(y0), pi(x0))) → A__2NDSPOS(a__from(mark(y0)), a__pi(mark(x0)))

A__2NDSNEG(s(N), cons(X, Z)) → MARK(N)

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → A__2NDSNEG(mark(N), mark(Z))

MARK(2ndsneg(plus(x0, x1), y1)) → A__2NDSNEG(a__plus(mark(x0), mark(x1)), mark(y1))

MARK(times(2ndsneg(x0, x1), y1)) → A__TIMES(a__2ndsneg(mark(x0), mark(x1)), mark(y1))

MARK(2ndsneg(from(x0), y1)) → A__2NDSNEG(a__from(mark(x0)), mark(y1))

MARK(2ndspos(from(y0), from(x0))) → A__2NDSPOS(a__from(mark(y0)), a__from(mark(x0)))

MARK(2ndspos(pi(y0), pi(x0))) → A__2NDSPOS(a__pi(mark(y0)), a__pi(mark(x0)))

MARK(2ndsneg(square(y0), 2ndsneg(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndspos(times(y0, y1), times(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(s(y0), pi(x0))) → A__2NDSPOS(s(mark(y0)), a__pi(mark(x0)))

A__TIMES(s(X), Y) → A__PLUS(mark(Y), a__times(mark(X), mark(Y)))

A__PI(2ndspos(x0, x1)) → A__2NDSPOS(a__2ndspos(mark(x0), mark(x1)), a__from(0))

MARK(2ndspos(square(y0), cons2(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), cons2(x0, mark(x1)))

MARK(plus(square(x0), y1)) → A__PLUS(a__square(mark(x0)), mark(y1))

MARK(plus(s(x0), y1)) → A__PLUS(s(mark(x0)), mark(y1))

A__PI(plus(x0, x1)) → A__2NDSPOS(a__plus(mark(x0), mark(x1)), a__from(0))

MARK(2ndspos(square(y0), cons(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), cons(mark(x0), x1))

MARK(cons2(X1, X2)) → MARK(X2)

MARK(2ndsneg(square(y0), cons2(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), cons2(x0, mark(x1)))

MARK(2ndsneg(square(y0), plus(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(X1, X2)) → MARK(X1)

MARK(s(X)) → MARK(X)

MARK(2ndspos(plus(y0, y1), cons2(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(pi(X)) → MARK(X)

MARK(plus(X1, X2)) → MARK(X2)

A__2NDSPOS(s(N), cons(X, Z)) → MARK(Z)

MARK(pi(X)) → A__PI(mark(X))

MARK(2ndsneg(square(y0), times(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(plus(y0, y1), cons(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), cons(mark(x0), x1))

MARK(plus(2ndsneg(x0, x1), y1)) → A__PLUS(a__2ndsneg(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(times(y0, y1), cons2(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(2ndspos(2ndspos(y0, y1), cons2(x0, x1))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(2ndsneg(X1, X2)) → MARK(X2)

MARK(2ndsneg(square(y0), from(x0))) → A__2NDSNEG(a__square(mark(y0)), a__from(mark(x0)))

MARK(2ndspos(times(y0, y1), square(x0))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndspos(square(y0), times(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(2ndsneg(y0, y1), cons2(x0, x1))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(times(square(x0), y1)) → A__TIMES(a__square(mark(x0)), mark(y1))

A__PI(X) → A__FROM(0)

MARK(from(X)) → A__FROM(mark(X))

MARK(2ndspos(s(y0), plus(x0, x1))) → A__2NDSPOS(s(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(pi(y0), cons(x0, x1))) → A__2NDSPOS(a__pi(mark(y0)), cons(mark(x0), x1))

MARK(plus(X1, X2)) → MARK(X1)

MARK(times(from(x0), y1)) → A__TIMES(a__from(mark(x0)), mark(y1))

A__2NDSPOS(s(y0), cons(y1, times(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__times(mark(x0), mark(x1))))

MARK(2ndspos(s(y0), cons(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons(mark(x0), x1))

A__PI(X) → MARK(X)

MARK(times(s(x0), y1)) → A__TIMES(s(mark(x0)), mark(y1))

MARK(2ndspos(2ndspos(y0, y1), pi(x0))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), a__pi(mark(x0)))

MARK(2ndspos(pi(y0), 2ndspos(x0, x1))) → A__2NDSPOS(a__pi(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndsneg(s(x0), y1)) → A__2NDSNEG(s(mark(x0)), mark(y1))

MARK(2ndspos(from(y0), cons(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), cons(mark(x0), x1))

MARK(2ndspos(plus(y0, y1), 2ndspos(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndspos(2ndspos(y0, y1), plus(x0, x1))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

A__PLUS(s(X), Y) → MARK(Y)

A__TIMES(s(X), Y) → MARK(X)

A__2NDSPOS(s(y0), cons(y1, pi(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__pi(mark(x0))))

MARK(2ndspos(plus(y0, y1), pi(x0))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__pi(mark(x0)))

MARK(2ndspos(pi(y0), plus(x0, x1))) → A__2NDSPOS(a__pi(mark(y0)), a__plus(mark(x0), mark(x1)))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(N)

MARK(plus(2ndspos(x0, x1), y1)) → A__PLUS(a__2ndspos(mark(x0), mark(x1)), mark(y1))

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(N)

A__PI(from(x0)) → A__2NDSPOS(a__from(mark(x0)), a__from(0))

MARK(cons(X1, X2)) → MARK(X1)

MARK(2ndspos(2ndspos(y0, y1), square(x0))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndspos(square(y0), 2ndspos(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndspos(times(y0, y1), cons(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), cons(mark(x0), x1))

MARK(2ndspos(s(y0), 2ndspos(x0, x1))) → A__2NDSPOS(s(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

MARK(times(X1, X2)) → MARK(X2)

A__PI(square(x0)) → A__2NDSPOS(a__square(mark(x0)), a__from(0))

MARK(2ndspos(2ndsneg(y0, y1), 2ndsneg(x0, x1))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndsneg(square(y0), cons(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), cons(mark(x0), x1))

MARK(2ndspos(plus(y0, y1), times(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(times(y0, y1), plus(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(s(y0), times(x0, x1))) → A__2NDSPOS(s(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndsneg(X1, X2)) → MARK(X1)

MARK(2ndspos(s(y0), 2ndsneg(x0, x1))) → A__2NDSPOS(s(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndsneg(square(y0), square(x0))) → A__2NDSNEG(a__square(mark(y0)), a__square(mark(x0)))

MARK(2ndspos(plus(y0, y1), 2ndsneg(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndspos(2ndsneg(y0, y1), plus(x0, x1))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

A__PI(2ndsneg(x0, x1)) → A__2NDSPOS(a__2ndsneg(mark(x0), mark(x1)), a__from(0))

MARK(plus(pi(x0), y1)) → A__PLUS(a__pi(mark(x0)), mark(y1))

MARK(times(pi(x0), y1)) → A__TIMES(a__pi(mark(x0)), mark(y1))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(Y)

MARK(2ndspos(2ndspos(y0, y1), 2ndspos(x0, x1))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), a__2ndspos(mark(x0), mark(x1)))

MARK(rcons(X1, X2)) → MARK(X1)

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(Z)

MARK(2ndsneg(square(y0), pi(x0))) → A__2NDSNEG(a__square(mark(y0)), a__pi(mark(x0)))

MARK(square(X)) → MARK(X)

MARK(2ndspos(2ndspos(y0, y1), 2ndsneg(x0, x1))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndspos(2ndsneg(y0, y1), 2ndspos(x0, x1))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndspos(times(y0, y1), pi(x0))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__pi(mark(x0)))

MARK(2ndspos(pi(y0), times(x0, x1))) → A__2NDSPOS(a__pi(mark(y0)), a__times(mark(x0), mark(x1)))

A__PI(s(x0)) → A__2NDSPOS(s(mark(x0)), a__from(0))

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(Y)

MARK(2ndspos(2ndspos(y0, y1), cons(x0, x1))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), cons(mark(x0), x1))

A__FROM(X) → MARK(X)

MARK(times(plus(x0, x1), y1)) → A__TIMES(a__plus(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(2ndspos(y0, y1), times(x0, x1))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(times(y0, y1), 2ndspos(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__2ndspos(mark(x0), mark(x1)))

A__2NDSPOS(s(y0), cons(y1, from(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__from(mark(x0))))

MARK(2ndspos(pi(y0), square(x0))) → A__2NDSPOS(a__pi(mark(y0)), a__square(mark(x0)))

MARK(2ndspos(square(y0), pi(x0))) → A__2NDSPOS(a__square(mark(y0)), a__pi(mark(x0)))

MARK(2ndspos(s(y0), square(x0))) → A__2NDSPOS(s(mark(y0)), a__square(mark(x0)))

MARK(rcons(X1, X2)) → MARK(X2)

MARK(times(2ndspos(x0, x1), y1)) → A__TIMES(a__2ndspos(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(s(y0), cons2(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(x0, mark(x1)))

A__TIMES(s(X), Y) → A__TIMES(mark(X), mark(Y))

A__PLUS(s(X), Y) → MARK(X)

A__2NDSPOS(s(y0), cons(y1, cons(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, cons(mark(x0), x1)))

MARK(times(X1, X2)) → MARK(X1)

MARK(plus(times(x0, x1), y1)) → A__PLUS(a__times(mark(x0), mark(x1)), mark(y1))

A__SQUARE(X) → MARK(X)

MARK(2ndsneg(times(x0, x1), y1)) → A__2NDSNEG(a__times(mark(x0), mark(x1)), mark(y1))

A__2NDSPOS(s(y0), cons(y1, 2ndsneg(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__2ndsneg(mark(x0), mark(x1))))

A__2NDSPOS(s(N), cons(X, Z)) → MARK(N)

MARK(negrecip(X)) → MARK(X)

A__PLUS(0, Y) → MARK(Y)

MARK(posrecip(X)) → MARK(X)

MARK(2ndsneg(pi(x0), y1)) → A__2NDSNEG(a__pi(mark(x0)), mark(y1))

MARK(2ndspos(square(y0), from(x0))) → A__2NDSPOS(a__square(mark(y0)), a__from(mark(x0)))

MARK(2ndspos(from(y0), square(x0))) → A__2NDSPOS(a__from(mark(y0)), a__square(mark(x0)))

MARK(2ndspos(times(y0, y1), from(x0))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndspos(from(y0), times(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), a__times(mark(x0), mark(x1)))

A__2NDSPOS(s(y0), cons(y1, square(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__square(mark(x0))))

MARK(2ndspos(times(y0, y1), 2ndsneg(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndspos(2ndsneg(y0, y1), times(x0, x1))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(plus(0, y1)) → A__PLUS(0, mark(y1))

MARK(from(X)) → MARK(X)

A__PLUS(s(X), Y) → A__PLUS(mark(X), mark(Y))

A__SQUARE(X) → A__TIMES(mark(X), mark(X))

MARK(2ndspos(square(y0), square(x0))) → A__2NDSPOS(a__square(mark(y0)), a__square(mark(x0)))

A__PI(pi(x0)) → A__2NDSPOS(a__pi(mark(x0)), a__from(0))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(Z)

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → A__2NDSPOS(mark(N), mark(Z))

MARK(plus(from(x0), y1)) → A__PLUS(a__from(mark(x0)), mark(y1))

MARK(square(X)) → A__SQUARE(mark(X))

MARK(2ndspos(plus(y0, y1), from(x0))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndspos(from(y0), plus(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(plus(y0, y1), plus(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(2ndspos(y0, y1), from(x0))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndspos(from(y0), 2ndspos(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndspos(2ndsneg(y0, y1), cons(x0, x1))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), cons(mark(x0), x1))

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))

a__2ndspos(0, Z) → rnil

a__2ndspos(s(N), cons(X, Z)) → a__2ndspos(s(mark(N)), cons2(X, mark(Z)))

a__2ndspos(s(N), cons2(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))

a__2ndsneg(0, Z) → rnil

a__2ndsneg(s(N), cons(X, Z)) → a__2ndsneg(s(mark(N)), cons2(X, mark(Z)))

a__2ndsneg(s(N), cons2(X, cons(Y, Z))) → rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z)))

a__pi(X) → a__2ndspos(mark(X), a__from(0))

a__plus(0, Y) → mark(Y)

a__plus(s(X), Y) → s(a__plus(mark(X), mark(Y)))

a__times(0, Y) → 0

a__times(s(X), Y) → a__plus(mark(Y), a__times(mark(X), mark(Y)))

a__square(X) → a__times(mark(X), mark(X))

mark(from(X)) → a__from(mark(X))

mark(2ndspos(X1, X2)) → a__2ndspos(mark(X1), mark(X2))

mark(2ndsneg(X1, X2)) → a__2ndsneg(mark(X1), mark(X2))

mark(pi(X)) → a__pi(mark(X))

mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))

mark(times(X1, X2)) → a__times(mark(X1), mark(X2))

mark(square(X)) → a__square(mark(X))

mark(0) → 0

mark(s(X)) → s(mark(X))

mark(posrecip(X)) → posrecip(mark(X))

mark(negrecip(X)) → negrecip(mark(X))

mark(nil) → nil

mark(cons(X1, X2)) → cons(mark(X1), X2)

mark(cons2(X1, X2)) → cons2(X1, mark(X2))

mark(rnil) → rnil

mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))

a__from(X) → from(X)

a__2ndspos(X1, X2) → 2ndspos(X1, X2)

a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)

a__pi(X) → pi(X)

a__plus(X1, X2) → plus(X1, X2)

a__times(X1, X2) → times(X1, X2)

a__square(X) → square(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule MARK(2ndsneg(2ndspos(x0, x1), y1)) → A__2NDSNEG(a__2ndspos(mark(x0), mark(x1)), mark(y1)) at position [1] we obtained the following new rules:

MARK(2ndsneg(2ndspos(y0, y1), times(x0, x1))) → A__2NDSNEG(a__2ndspos(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndsneg(2ndspos(y0, y1), 0)) → A__2NDSNEG(a__2ndspos(mark(y0), mark(y1)), 0)

MARK(2ndsneg(2ndspos(y0, y1), rcons(x0, x1))) → A__2NDSNEG(a__2ndspos(mark(y0), mark(y1)), rcons(mark(x0), mark(x1)))

MARK(2ndsneg(2ndspos(y0, y1), 2ndsneg(x0, x1))) → A__2NDSNEG(a__2ndspos(mark(y0), mark(y1)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndsneg(2ndspos(y0, y1), s(x0))) → A__2NDSNEG(a__2ndspos(mark(y0), mark(y1)), s(mark(x0)))

MARK(2ndsneg(2ndspos(y0, y1), 2ndspos(x0, x1))) → A__2NDSNEG(a__2ndspos(mark(y0), mark(y1)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndsneg(2ndspos(y0, y1), from(x0))) → A__2NDSNEG(a__2ndspos(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndsneg(2ndspos(y0, y1), square(x0))) → A__2NDSNEG(a__2ndspos(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndsneg(2ndspos(y0, y1), cons2(x0, x1))) → A__2NDSNEG(a__2ndspos(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(2ndsneg(2ndspos(y0, y1), nil)) → A__2NDSNEG(a__2ndspos(mark(y0), mark(y1)), nil)

MARK(2ndsneg(2ndspos(y0, y1), pi(x0))) → A__2NDSNEG(a__2ndspos(mark(y0), mark(y1)), a__pi(mark(x0)))

MARK(2ndsneg(2ndspos(y0, y1), negrecip(x0))) → A__2NDSNEG(a__2ndspos(mark(y0), mark(y1)), negrecip(mark(x0)))

MARK(2ndsneg(2ndspos(y0, y1), rnil)) → A__2NDSNEG(a__2ndspos(mark(y0), mark(y1)), rnil)

MARK(2ndsneg(2ndspos(y0, y1), posrecip(x0))) → A__2NDSNEG(a__2ndspos(mark(y0), mark(y1)), posrecip(mark(x0)))

MARK(2ndsneg(2ndspos(y0, y1), plus(x0, x1))) → A__2NDSNEG(a__2ndspos(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(2ndsneg(2ndspos(y0, y1), cons(x0, x1))) → A__2NDSNEG(a__2ndspos(mark(y0), mark(y1)), cons(mark(x0), x1))



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ Narrowing
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ QDP
              ↳ Narrowing
                ↳ QDP
                  ↳ DependencyGraphProof
                    ↳ QDP
                      ↳ Narrowing
                        ↳ QDP
                          ↳ DependencyGraphProof
                            ↳ QDP
                              ↳ Narrowing
                                ↳ QDP
                                  ↳ DependencyGraphProof
                                    ↳ QDP
                                      ↳ Narrowing
                                        ↳ QDP
                                          ↳ DependencyGraphProof
                                            ↳ QDP
                                              ↳ Narrowing
                                                ↳ QDP
                                                  ↳ DependencyGraphProof
                                                    ↳ QDP
                                                      ↳ Narrowing
                                                        ↳ QDP
                                                          ↳ DependencyGraphProof
                                                            ↳ QDP
                                                              ↳ Narrowing
                                                                ↳ QDP
                                                                  ↳ DependencyGraphProof
                                                                    ↳ QDP
                                                                      ↳ Narrowing
                                                                        ↳ QDP
                                                                          ↳ DependencyGraphProof
                                                                            ↳ QDP
                                                                              ↳ Narrowing
                                                                                ↳ QDP
                                                                                  ↳ DependencyGraphProof
                                                                                    ↳ QDP
                                                                                      ↳ Narrowing
                                                                                        ↳ QDP
                                                                                          ↳ DependencyGraphProof
                                                                                            ↳ QDP
                                                                                              ↳ Narrowing
                                                                                                ↳ QDP
                                                                                                  ↳ DependencyGraphProof
                                                                                                    ↳ QDP
                                                                                                      ↳ Narrowing
                                                                                                        ↳ QDP
                                                                                                          ↳ DependencyGraphProof
                                                                                                            ↳ QDP
                                                                                                              ↳ Narrowing
                                                                                                                ↳ QDP
                                                                                                                  ↳ DependencyGraphProof
                                                                                                                    ↳ QDP
                                                                                                                      ↳ Narrowing
                                                                                                                        ↳ QDP
                                                                                                                          ↳ DependencyGraphProof
                                                                                                                            ↳ QDP
                                                                                                                              ↳ Narrowing
                                                                                                                                ↳ QDP
                                                                                                                                  ↳ DependencyGraphProof

Q DP problem:
The TRS P consists of the following rules:

MARK(2ndspos(square(y0), plus(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(plus(y0, y1), square(x0))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__square(mark(x0)))

A__2NDSPOS(s(y0), cons(y1, 2ndspos(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__2ndspos(mark(x0), mark(x1))))

A__PI(times(x0, x1)) → A__2NDSPOS(a__times(mark(x0), mark(x1)), a__from(0))

A__2NDSNEG(s(N), cons(X, Z)) → A__2NDSNEG(s(mark(N)), cons2(X, mark(Z)))

MARK(times(times(x0, x1), y1)) → A__TIMES(a__times(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(X1, X2)) → MARK(X2)

MARK(2ndspos(pi(y0), cons2(x0, x1))) → A__2NDSPOS(a__pi(mark(y0)), cons2(x0, mark(x1)))

MARK(2ndsneg(2ndspos(y0, y1), 2ndsneg(x0, x1))) → A__2NDSNEG(a__2ndspos(mark(y0), mark(y1)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndspos(from(y0), cons2(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), cons2(x0, mark(x1)))

MARK(2ndsneg(2ndspos(y0, y1), 2ndspos(x0, x1))) → A__2NDSNEG(a__2ndspos(mark(y0), mark(y1)), a__2ndspos(mark(x0), mark(x1)))

MARK(plus(plus(x0, x1), y1)) → A__PLUS(a__plus(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(from(y0), 2ndsneg(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndspos(s(y0), from(x0))) → A__2NDSPOS(s(mark(y0)), a__from(mark(x0)))

MARK(2ndspos(2ndsneg(y0, y1), from(x0))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndsneg(2ndsneg(x0, x1), y1)) → A__2NDSNEG(a__2ndsneg(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(2ndsneg(y0, y1), pi(x0))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__pi(mark(x0)))

MARK(2ndspos(pi(y0), 2ndsneg(x0, x1))) → A__2NDSPOS(a__pi(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndsneg(2ndspos(y0, y1), square(x0))) → A__2NDSNEG(a__2ndspos(mark(y0), mark(y1)), a__square(mark(x0)))

A__2NDSNEG(s(N), cons(X, Z)) → MARK(Z)

MARK(2ndsneg(square(y0), 2ndspos(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndsneg(2ndspos(y0, y1), pi(x0))) → A__2NDSNEG(a__2ndspos(mark(y0), mark(y1)), a__pi(mark(x0)))

A__TIMES(s(X), Y) → MARK(Y)

A__2NDSPOS(s(y0), cons(y1, plus(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__plus(mark(x0), mark(x1))))

MARK(2ndspos(from(y0), pi(x0))) → A__2NDSPOS(a__from(mark(y0)), a__pi(mark(x0)))

MARK(2ndspos(square(y0), 2ndsneg(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndspos(2ndsneg(y0, y1), square(x0))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndspos(pi(y0), from(x0))) → A__2NDSPOS(a__pi(mark(y0)), a__from(mark(x0)))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → A__2NDSNEG(mark(N), mark(Z))

A__2NDSNEG(s(N), cons(X, Z)) → MARK(N)

MARK(times(2ndsneg(x0, x1), y1)) → A__TIMES(a__2ndsneg(mark(x0), mark(x1)), mark(y1))

MARK(2ndsneg(plus(x0, x1), y1)) → A__2NDSNEG(a__plus(mark(x0), mark(x1)), mark(y1))

MARK(2ndsneg(from(x0), y1)) → A__2NDSNEG(a__from(mark(x0)), mark(y1))

MARK(2ndspos(from(y0), from(x0))) → A__2NDSPOS(a__from(mark(y0)), a__from(mark(x0)))

MARK(2ndspos(pi(y0), pi(x0))) → A__2NDSPOS(a__pi(mark(y0)), a__pi(mark(x0)))

MARK(2ndsneg(square(y0), 2ndsneg(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndspos(times(y0, y1), times(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

A__TIMES(s(X), Y) → A__PLUS(mark(Y), a__times(mark(X), mark(Y)))

MARK(2ndspos(s(y0), pi(x0))) → A__2NDSPOS(s(mark(y0)), a__pi(mark(x0)))

MARK(2ndspos(square(y0), cons2(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), cons2(x0, mark(x1)))

A__PI(2ndspos(x0, x1)) → A__2NDSPOS(a__2ndspos(mark(x0), mark(x1)), a__from(0))

MARK(plus(square(x0), y1)) → A__PLUS(a__square(mark(x0)), mark(y1))

MARK(plus(s(x0), y1)) → A__PLUS(s(mark(x0)), mark(y1))

MARK(2ndspos(square(y0), cons(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), cons(mark(x0), x1))

A__PI(plus(x0, x1)) → A__2NDSPOS(a__plus(mark(x0), mark(x1)), a__from(0))

MARK(cons2(X1, X2)) → MARK(X2)

MARK(2ndsneg(2ndspos(y0, y1), times(x0, x1))) → A__2NDSNEG(a__2ndspos(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndsneg(square(y0), cons2(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), cons2(x0, mark(x1)))

MARK(s(X)) → MARK(X)

MARK(2ndspos(X1, X2)) → MARK(X1)

MARK(2ndsneg(square(y0), plus(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(plus(y0, y1), cons2(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(pi(X)) → MARK(X)

MARK(plus(X1, X2)) → MARK(X2)

A__2NDSPOS(s(N), cons(X, Z)) → MARK(Z)

MARK(pi(X)) → A__PI(mark(X))

MARK(2ndspos(plus(y0, y1), cons(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), cons(mark(x0), x1))

MARK(2ndsneg(square(y0), times(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(plus(2ndsneg(x0, x1), y1)) → A__PLUS(a__2ndsneg(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(times(y0, y1), cons2(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(2ndspos(2ndspos(y0, y1), cons2(x0, x1))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(2ndsneg(X1, X2)) → MARK(X2)

MARK(2ndsneg(square(y0), from(x0))) → A__2NDSNEG(a__square(mark(y0)), a__from(mark(x0)))

MARK(2ndspos(square(y0), times(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(times(y0, y1), square(x0))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndspos(2ndsneg(y0, y1), cons2(x0, x1))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(times(square(x0), y1)) → A__TIMES(a__square(mark(x0)), mark(y1))

MARK(2ndspos(s(y0), plus(x0, x1))) → A__2NDSPOS(s(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(from(X)) → A__FROM(mark(X))

A__PI(X) → A__FROM(0)

MARK(times(from(x0), y1)) → A__TIMES(a__from(mark(x0)), mark(y1))

MARK(plus(X1, X2)) → MARK(X1)

MARK(2ndspos(pi(y0), cons(x0, x1))) → A__2NDSPOS(a__pi(mark(y0)), cons(mark(x0), x1))

A__2NDSPOS(s(y0), cons(y1, times(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__times(mark(x0), mark(x1))))

A__PI(X) → MARK(X)

MARK(2ndspos(s(y0), cons(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons(mark(x0), x1))

MARK(2ndsneg(2ndspos(y0, y1), plus(x0, x1))) → A__2NDSNEG(a__2ndspos(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(2ndsneg(2ndspos(y0, y1), posrecip(x0))) → A__2NDSNEG(a__2ndspos(mark(y0), mark(y1)), posrecip(mark(x0)))

MARK(times(s(x0), y1)) → A__TIMES(s(mark(x0)), mark(y1))

MARK(2ndspos(pi(y0), 2ndspos(x0, x1))) → A__2NDSPOS(a__pi(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndspos(2ndspos(y0, y1), pi(x0))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), a__pi(mark(x0)))

MARK(2ndspos(from(y0), cons(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), cons(mark(x0), x1))

MARK(2ndsneg(s(x0), y1)) → A__2NDSNEG(s(mark(x0)), mark(y1))

MARK(2ndsneg(2ndspos(y0, y1), cons(x0, x1))) → A__2NDSNEG(a__2ndspos(mark(y0), mark(y1)), cons(mark(x0), x1))

MARK(2ndspos(2ndspos(y0, y1), plus(x0, x1))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(plus(y0, y1), 2ndspos(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__2ndspos(mark(x0), mark(x1)))

A__PLUS(s(X), Y) → MARK(Y)

A__2NDSPOS(s(y0), cons(y1, pi(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__pi(mark(x0))))

A__TIMES(s(X), Y) → MARK(X)

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(N)

MARK(2ndspos(pi(y0), plus(x0, x1))) → A__2NDSPOS(a__pi(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(plus(y0, y1), pi(x0))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__pi(mark(x0)))

MARK(2ndsneg(2ndspos(y0, y1), rcons(x0, x1))) → A__2NDSNEG(a__2ndspos(mark(y0), mark(y1)), rcons(mark(x0), mark(x1)))

MARK(plus(2ndspos(x0, x1), y1)) → A__PLUS(a__2ndspos(mark(x0), mark(x1)), mark(y1))

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(N)

MARK(cons(X1, X2)) → MARK(X1)

A__PI(from(x0)) → A__2NDSPOS(a__from(mark(x0)), a__from(0))

MARK(2ndsneg(2ndspos(y0, y1), s(x0))) → A__2NDSNEG(a__2ndspos(mark(y0), mark(y1)), s(mark(x0)))

MARK(2ndspos(square(y0), 2ndspos(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndspos(2ndspos(y0, y1), square(x0))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(times(X1, X2)) → MARK(X2)

MARK(2ndspos(s(y0), 2ndspos(x0, x1))) → A__2NDSPOS(s(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndspos(times(y0, y1), cons(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), cons(mark(x0), x1))

MARK(2ndspos(2ndsneg(y0, y1), 2ndsneg(x0, x1))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__2ndsneg(mark(x0), mark(x1)))

A__PI(square(x0)) → A__2NDSPOS(a__square(mark(x0)), a__from(0))

MARK(2ndsneg(2ndspos(y0, y1), from(x0))) → A__2NDSNEG(a__2ndspos(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndspos(times(y0, y1), plus(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(plus(y0, y1), times(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndsneg(square(y0), cons(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), cons(mark(x0), x1))

MARK(2ndspos(s(y0), times(x0, x1))) → A__2NDSPOS(s(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndsneg(X1, X2)) → MARK(X1)

MARK(2ndsneg(2ndspos(y0, y1), cons2(x0, x1))) → A__2NDSNEG(a__2ndspos(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(2ndspos(s(y0), 2ndsneg(x0, x1))) → A__2NDSPOS(s(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndsneg(square(y0), square(x0))) → A__2NDSNEG(a__square(mark(y0)), a__square(mark(x0)))

MARK(2ndsneg(2ndspos(y0, y1), nil)) → A__2NDSNEG(a__2ndspos(mark(y0), mark(y1)), nil)

MARK(2ndspos(2ndsneg(y0, y1), plus(x0, x1))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(plus(y0, y1), 2ndsneg(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__2ndsneg(mark(x0), mark(x1)))

A__PI(2ndsneg(x0, x1)) → A__2NDSPOS(a__2ndsneg(mark(x0), mark(x1)), a__from(0))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(Y)

MARK(times(pi(x0), y1)) → A__TIMES(a__pi(mark(x0)), mark(y1))

MARK(plus(pi(x0), y1)) → A__PLUS(a__pi(mark(x0)), mark(y1))

MARK(2ndspos(2ndspos(y0, y1), 2ndspos(x0, x1))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), a__2ndspos(mark(x0), mark(x1)))

MARK(rcons(X1, X2)) → MARK(X1)

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(Z)

MARK(square(X)) → MARK(X)

MARK(2ndsneg(square(y0), pi(x0))) → A__2NDSNEG(a__square(mark(y0)), a__pi(mark(x0)))

MARK(2ndspos(2ndsneg(y0, y1), 2ndspos(x0, x1))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndspos(2ndspos(y0, y1), 2ndsneg(x0, x1))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndspos(pi(y0), times(x0, x1))) → A__2NDSPOS(a__pi(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(times(y0, y1), pi(x0))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__pi(mark(x0)))

A__PI(s(x0)) → A__2NDSPOS(s(mark(x0)), a__from(0))

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(Y)

A__FROM(X) → MARK(X)

MARK(2ndspos(2ndspos(y0, y1), cons(x0, x1))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), cons(mark(x0), x1))

MARK(times(plus(x0, x1), y1)) → A__TIMES(a__plus(mark(x0), mark(x1)), mark(y1))

A__2NDSPOS(s(y0), cons(y1, from(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__from(mark(x0))))

MARK(2ndspos(times(y0, y1), 2ndspos(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndspos(2ndspos(y0, y1), times(x0, x1))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(square(y0), pi(x0))) → A__2NDSPOS(a__square(mark(y0)), a__pi(mark(x0)))

MARK(2ndspos(pi(y0), square(x0))) → A__2NDSPOS(a__pi(mark(y0)), a__square(mark(x0)))

MARK(rcons(X1, X2)) → MARK(X2)

MARK(2ndspos(s(y0), square(x0))) → A__2NDSPOS(s(mark(y0)), a__square(mark(x0)))

MARK(times(2ndspos(x0, x1), y1)) → A__TIMES(a__2ndspos(mark(x0), mark(x1)), mark(y1))

MARK(2ndsneg(2ndspos(y0, y1), 0)) → A__2NDSNEG(a__2ndspos(mark(y0), mark(y1)), 0)

MARK(2ndspos(s(y0), cons2(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(x0, mark(x1)))

A__TIMES(s(X), Y) → A__TIMES(mark(X), mark(Y))

A__2NDSPOS(s(y0), cons(y1, cons(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, cons(mark(x0), x1)))

A__PLUS(s(X), Y) → MARK(X)

MARK(times(X1, X2)) → MARK(X1)

MARK(plus(times(x0, x1), y1)) → A__PLUS(a__times(mark(x0), mark(x1)), mark(y1))

A__SQUARE(X) → MARK(X)

MARK(2ndsneg(times(x0, x1), y1)) → A__2NDSNEG(a__times(mark(x0), mark(x1)), mark(y1))

A__2NDSPOS(s(y0), cons(y1, 2ndsneg(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__2ndsneg(mark(x0), mark(x1))))

MARK(negrecip(X)) → MARK(X)

A__2NDSPOS(s(N), cons(X, Z)) → MARK(N)

A__PLUS(0, Y) → MARK(Y)

MARK(posrecip(X)) → MARK(X)

MARK(2ndspos(from(y0), square(x0))) → A__2NDSPOS(a__from(mark(y0)), a__square(mark(x0)))

MARK(2ndspos(square(y0), from(x0))) → A__2NDSPOS(a__square(mark(y0)), a__from(mark(x0)))

MARK(2ndsneg(pi(x0), y1)) → A__2NDSNEG(a__pi(mark(x0)), mark(y1))

MARK(2ndspos(from(y0), times(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(times(y0, y1), from(x0))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__from(mark(x0)))

A__2NDSPOS(s(y0), cons(y1, square(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__square(mark(x0))))

MARK(2ndspos(2ndsneg(y0, y1), times(x0, x1))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(times(y0, y1), 2ndsneg(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__2ndsneg(mark(x0), mark(x1)))

MARK(plus(0, y1)) → A__PLUS(0, mark(y1))

MARK(from(X)) → MARK(X)

MARK(2ndsneg(2ndspos(y0, y1), negrecip(x0))) → A__2NDSNEG(a__2ndspos(mark(y0), mark(y1)), negrecip(mark(x0)))

A__PLUS(s(X), Y) → A__PLUS(mark(X), mark(Y))

MARK(2ndspos(square(y0), square(x0))) → A__2NDSPOS(a__square(mark(y0)), a__square(mark(x0)))

A__SQUARE(X) → A__TIMES(mark(X), mark(X))

A__PI(pi(x0)) → A__2NDSPOS(a__pi(mark(x0)), a__from(0))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(Z)

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → A__2NDSPOS(mark(N), mark(Z))

MARK(plus(from(x0), y1)) → A__PLUS(a__from(mark(x0)), mark(y1))

MARK(square(X)) → A__SQUARE(mark(X))

MARK(2ndspos(from(y0), plus(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(plus(y0, y1), from(x0))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndsneg(2ndspos(y0, y1), rnil)) → A__2NDSNEG(a__2ndspos(mark(y0), mark(y1)), rnil)

MARK(2ndspos(plus(y0, y1), plus(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(from(y0), 2ndspos(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndspos(2ndspos(y0, y1), from(x0))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndspos(2ndsneg(y0, y1), cons(x0, x1))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), cons(mark(x0), x1))

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))

a__2ndspos(0, Z) → rnil

a__2ndspos(s(N), cons(X, Z)) → a__2ndspos(s(mark(N)), cons2(X, mark(Z)))

a__2ndspos(s(N), cons2(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))

a__2ndsneg(0, Z) → rnil

a__2ndsneg(s(N), cons(X, Z)) → a__2ndsneg(s(mark(N)), cons2(X, mark(Z)))

a__2ndsneg(s(N), cons2(X, cons(Y, Z))) → rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z)))

a__pi(X) → a__2ndspos(mark(X), a__from(0))

a__plus(0, Y) → mark(Y)

a__plus(s(X), Y) → s(a__plus(mark(X), mark(Y)))

a__times(0, Y) → 0

a__times(s(X), Y) → a__plus(mark(Y), a__times(mark(X), mark(Y)))

a__square(X) → a__times(mark(X), mark(X))

mark(from(X)) → a__from(mark(X))

mark(2ndspos(X1, X2)) → a__2ndspos(mark(X1), mark(X2))

mark(2ndsneg(X1, X2)) → a__2ndsneg(mark(X1), mark(X2))

mark(pi(X)) → a__pi(mark(X))

mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))

mark(times(X1, X2)) → a__times(mark(X1), mark(X2))

mark(square(X)) → a__square(mark(X))

mark(0) → 0

mark(s(X)) → s(mark(X))

mark(posrecip(X)) → posrecip(mark(X))

mark(negrecip(X)) → negrecip(mark(X))

mark(nil) → nil

mark(cons(X1, X2)) → cons(mark(X1), X2)

mark(cons2(X1, X2)) → cons2(X1, mark(X2))

mark(rnil) → rnil

mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))

a__from(X) → from(X)

a__2ndspos(X1, X2) → 2ndspos(X1, X2)

a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)

a__pi(X) → pi(X)

a__plus(X1, X2) → plus(X1, X2)

a__times(X1, X2) → times(X1, X2)

a__square(X) → square(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 7 less nodes.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ Narrowing
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ QDP
              ↳ Narrowing
                ↳ QDP
                  ↳ DependencyGraphProof
                    ↳ QDP
                      ↳ Narrowing
                        ↳ QDP
                          ↳ DependencyGraphProof
                            ↳ QDP
                              ↳ Narrowing
                                ↳ QDP
                                  ↳ DependencyGraphProof
                                    ↳ QDP
                                      ↳ Narrowing
                                        ↳ QDP
                                          ↳ DependencyGraphProof
                                            ↳ QDP
                                              ↳ Narrowing
                                                ↳ QDP
                                                  ↳ DependencyGraphProof
                                                    ↳ QDP
                                                      ↳ Narrowing
                                                        ↳ QDP
                                                          ↳ DependencyGraphProof
                                                            ↳ QDP
                                                              ↳ Narrowing
                                                                ↳ QDP
                                                                  ↳ DependencyGraphProof
                                                                    ↳ QDP
                                                                      ↳ Narrowing
                                                                        ↳ QDP
                                                                          ↳ DependencyGraphProof
                                                                            ↳ QDP
                                                                              ↳ Narrowing
                                                                                ↳ QDP
                                                                                  ↳ DependencyGraphProof
                                                                                    ↳ QDP
                                                                                      ↳ Narrowing
                                                                                        ↳ QDP
                                                                                          ↳ DependencyGraphProof
                                                                                            ↳ QDP
                                                                                              ↳ Narrowing
                                                                                                ↳ QDP
                                                                                                  ↳ DependencyGraphProof
                                                                                                    ↳ QDP
                                                                                                      ↳ Narrowing
                                                                                                        ↳ QDP
                                                                                                          ↳ DependencyGraphProof
                                                                                                            ↳ QDP
                                                                                                              ↳ Narrowing
                                                                                                                ↳ QDP
                                                                                                                  ↳ DependencyGraphProof
                                                                                                                    ↳ QDP
                                                                                                                      ↳ Narrowing
                                                                                                                        ↳ QDP
                                                                                                                          ↳ DependencyGraphProof
                                                                                                                            ↳ QDP
                                                                                                                              ↳ Narrowing
                                                                                                                                ↳ QDP
                                                                                                                                  ↳ DependencyGraphProof
                                                                                                                                    ↳ QDP
                                                                                                                                      ↳ Narrowing

Q DP problem:
The TRS P consists of the following rules:

MARK(2ndspos(plus(y0, y1), square(x0))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndspos(square(y0), plus(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__plus(mark(x0), mark(x1)))

A__PI(times(x0, x1)) → A__2NDSPOS(a__times(mark(x0), mark(x1)), a__from(0))

A__2NDSPOS(s(y0), cons(y1, 2ndspos(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__2ndspos(mark(x0), mark(x1))))

A__2NDSNEG(s(N), cons(X, Z)) → A__2NDSNEG(s(mark(N)), cons2(X, mark(Z)))

MARK(2ndspos(pi(y0), cons2(x0, x1))) → A__2NDSPOS(a__pi(mark(y0)), cons2(x0, mark(x1)))

MARK(2ndspos(X1, X2)) → MARK(X2)

MARK(times(times(x0, x1), y1)) → A__TIMES(a__times(mark(x0), mark(x1)), mark(y1))

MARK(2ndsneg(2ndspos(y0, y1), 2ndsneg(x0, x1))) → A__2NDSNEG(a__2ndspos(mark(y0), mark(y1)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndspos(from(y0), cons2(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), cons2(x0, mark(x1)))

MARK(2ndsneg(2ndspos(y0, y1), 2ndspos(x0, x1))) → A__2NDSNEG(a__2ndspos(mark(y0), mark(y1)), a__2ndspos(mark(x0), mark(x1)))

MARK(plus(plus(x0, x1), y1)) → A__PLUS(a__plus(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(2ndsneg(y0, y1), from(x0))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndspos(s(y0), from(x0))) → A__2NDSPOS(s(mark(y0)), a__from(mark(x0)))

MARK(2ndspos(from(y0), 2ndsneg(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndspos(pi(y0), 2ndsneg(x0, x1))) → A__2NDSPOS(a__pi(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndspos(2ndsneg(y0, y1), pi(x0))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__pi(mark(x0)))

MARK(2ndsneg(2ndsneg(x0, x1), y1)) → A__2NDSNEG(a__2ndsneg(mark(x0), mark(x1)), mark(y1))

MARK(2ndsneg(2ndspos(y0, y1), square(x0))) → A__2NDSNEG(a__2ndspos(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndsneg(square(y0), 2ndspos(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

A__2NDSNEG(s(N), cons(X, Z)) → MARK(Z)

MARK(2ndsneg(2ndspos(y0, y1), pi(x0))) → A__2NDSNEG(a__2ndspos(mark(y0), mark(y1)), a__pi(mark(x0)))

A__TIMES(s(X), Y) → MARK(Y)

A__2NDSPOS(s(y0), cons(y1, plus(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__plus(mark(x0), mark(x1))))

MARK(2ndspos(pi(y0), from(x0))) → A__2NDSPOS(a__pi(mark(y0)), a__from(mark(x0)))

MARK(2ndspos(2ndsneg(y0, y1), square(x0))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndspos(square(y0), 2ndsneg(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndspos(from(y0), pi(x0))) → A__2NDSPOS(a__from(mark(y0)), a__pi(mark(x0)))

A__2NDSNEG(s(N), cons(X, Z)) → MARK(N)

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → A__2NDSNEG(mark(N), mark(Z))

MARK(2ndsneg(plus(x0, x1), y1)) → A__2NDSNEG(a__plus(mark(x0), mark(x1)), mark(y1))

MARK(times(2ndsneg(x0, x1), y1)) → A__TIMES(a__2ndsneg(mark(x0), mark(x1)), mark(y1))

MARK(2ndsneg(from(x0), y1)) → A__2NDSNEG(a__from(mark(x0)), mark(y1))

MARK(2ndspos(from(y0), from(x0))) → A__2NDSPOS(a__from(mark(y0)), a__from(mark(x0)))

MARK(2ndspos(pi(y0), pi(x0))) → A__2NDSPOS(a__pi(mark(y0)), a__pi(mark(x0)))

MARK(2ndsneg(square(y0), 2ndsneg(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndspos(times(y0, y1), times(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(s(y0), pi(x0))) → A__2NDSPOS(s(mark(y0)), a__pi(mark(x0)))

A__TIMES(s(X), Y) → A__PLUS(mark(Y), a__times(mark(X), mark(Y)))

A__PI(2ndspos(x0, x1)) → A__2NDSPOS(a__2ndspos(mark(x0), mark(x1)), a__from(0))

MARK(2ndspos(square(y0), cons2(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), cons2(x0, mark(x1)))

MARK(plus(square(x0), y1)) → A__PLUS(a__square(mark(x0)), mark(y1))

MARK(plus(s(x0), y1)) → A__PLUS(s(mark(x0)), mark(y1))

A__PI(plus(x0, x1)) → A__2NDSPOS(a__plus(mark(x0), mark(x1)), a__from(0))

MARK(2ndspos(square(y0), cons(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), cons(mark(x0), x1))

MARK(cons2(X1, X2)) → MARK(X2)

MARK(2ndsneg(2ndspos(y0, y1), times(x0, x1))) → A__2NDSNEG(a__2ndspos(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndsneg(square(y0), cons2(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), cons2(x0, mark(x1)))

MARK(2ndsneg(square(y0), plus(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(X1, X2)) → MARK(X1)

MARK(s(X)) → MARK(X)

MARK(2ndspos(plus(y0, y1), cons2(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(pi(X)) → MARK(X)

MARK(plus(X1, X2)) → MARK(X2)

A__2NDSPOS(s(N), cons(X, Z)) → MARK(Z)

MARK(pi(X)) → A__PI(mark(X))

MARK(2ndsneg(square(y0), times(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(plus(y0, y1), cons(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), cons(mark(x0), x1))

MARK(plus(2ndsneg(x0, x1), y1)) → A__PLUS(a__2ndsneg(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(times(y0, y1), cons2(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(2ndspos(2ndspos(y0, y1), cons2(x0, x1))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(2ndsneg(X1, X2)) → MARK(X2)

MARK(2ndsneg(square(y0), from(x0))) → A__2NDSNEG(a__square(mark(y0)), a__from(mark(x0)))

MARK(2ndspos(times(y0, y1), square(x0))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndspos(square(y0), times(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(2ndsneg(y0, y1), cons2(x0, x1))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(times(square(x0), y1)) → A__TIMES(a__square(mark(x0)), mark(y1))

A__PI(X) → A__FROM(0)

MARK(from(X)) → A__FROM(mark(X))

MARK(2ndspos(s(y0), plus(x0, x1))) → A__2NDSPOS(s(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(pi(y0), cons(x0, x1))) → A__2NDSPOS(a__pi(mark(y0)), cons(mark(x0), x1))

MARK(plus(X1, X2)) → MARK(X1)

MARK(times(from(x0), y1)) → A__TIMES(a__from(mark(x0)), mark(y1))

A__2NDSPOS(s(y0), cons(y1, times(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__times(mark(x0), mark(x1))))

MARK(2ndspos(s(y0), cons(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons(mark(x0), x1))

A__PI(X) → MARK(X)

MARK(2ndsneg(2ndspos(y0, y1), plus(x0, x1))) → A__2NDSNEG(a__2ndspos(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(times(s(x0), y1)) → A__TIMES(s(mark(x0)), mark(y1))

MARK(2ndspos(2ndspos(y0, y1), pi(x0))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), a__pi(mark(x0)))

MARK(2ndspos(pi(y0), 2ndspos(x0, x1))) → A__2NDSPOS(a__pi(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndsneg(s(x0), y1)) → A__2NDSNEG(s(mark(x0)), mark(y1))

MARK(2ndspos(from(y0), cons(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), cons(mark(x0), x1))

MARK(2ndsneg(2ndspos(y0, y1), cons(x0, x1))) → A__2NDSNEG(a__2ndspos(mark(y0), mark(y1)), cons(mark(x0), x1))

MARK(2ndspos(plus(y0, y1), 2ndspos(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndspos(2ndspos(y0, y1), plus(x0, x1))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

A__PLUS(s(X), Y) → MARK(Y)

A__TIMES(s(X), Y) → MARK(X)

A__2NDSPOS(s(y0), cons(y1, pi(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__pi(mark(x0))))

MARK(2ndspos(plus(y0, y1), pi(x0))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__pi(mark(x0)))

MARK(2ndspos(pi(y0), plus(x0, x1))) → A__2NDSPOS(a__pi(mark(y0)), a__plus(mark(x0), mark(x1)))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(N)

MARK(plus(2ndspos(x0, x1), y1)) → A__PLUS(a__2ndspos(mark(x0), mark(x1)), mark(y1))

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(N)

A__PI(from(x0)) → A__2NDSPOS(a__from(mark(x0)), a__from(0))

MARK(cons(X1, X2)) → MARK(X1)

MARK(2ndspos(2ndspos(y0, y1), square(x0))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndspos(square(y0), 2ndspos(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndspos(times(y0, y1), cons(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), cons(mark(x0), x1))

MARK(times(X1, X2)) → MARK(X2)

MARK(2ndspos(s(y0), 2ndspos(x0, x1))) → A__2NDSPOS(s(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

A__PI(square(x0)) → A__2NDSPOS(a__square(mark(x0)), a__from(0))

MARK(2ndspos(2ndsneg(y0, y1), 2ndsneg(x0, x1))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndsneg(2ndspos(y0, y1), from(x0))) → A__2NDSNEG(a__2ndspos(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndsneg(square(y0), cons(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), cons(mark(x0), x1))

MARK(2ndspos(plus(y0, y1), times(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(times(y0, y1), plus(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(s(y0), times(x0, x1))) → A__2NDSPOS(s(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndsneg(X1, X2)) → MARK(X1)

MARK(2ndsneg(2ndspos(y0, y1), cons2(x0, x1))) → A__2NDSNEG(a__2ndspos(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(2ndspos(s(y0), 2ndsneg(x0, x1))) → A__2NDSPOS(s(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndsneg(square(y0), square(x0))) → A__2NDSNEG(a__square(mark(y0)), a__square(mark(x0)))

MARK(2ndspos(plus(y0, y1), 2ndsneg(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndspos(2ndsneg(y0, y1), plus(x0, x1))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

A__PI(2ndsneg(x0, x1)) → A__2NDSPOS(a__2ndsneg(mark(x0), mark(x1)), a__from(0))

MARK(plus(pi(x0), y1)) → A__PLUS(a__pi(mark(x0)), mark(y1))

MARK(times(pi(x0), y1)) → A__TIMES(a__pi(mark(x0)), mark(y1))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(Y)

MARK(2ndspos(2ndspos(y0, y1), 2ndspos(x0, x1))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), a__2ndspos(mark(x0), mark(x1)))

MARK(rcons(X1, X2)) → MARK(X1)

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(Z)

MARK(2ndsneg(square(y0), pi(x0))) → A__2NDSNEG(a__square(mark(y0)), a__pi(mark(x0)))

MARK(square(X)) → MARK(X)

MARK(2ndspos(2ndspos(y0, y1), 2ndsneg(x0, x1))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndspos(2ndsneg(y0, y1), 2ndspos(x0, x1))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndspos(times(y0, y1), pi(x0))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__pi(mark(x0)))

MARK(2ndspos(pi(y0), times(x0, x1))) → A__2NDSPOS(a__pi(mark(y0)), a__times(mark(x0), mark(x1)))

A__PI(s(x0)) → A__2NDSPOS(s(mark(x0)), a__from(0))

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(Y)

MARK(2ndspos(2ndspos(y0, y1), cons(x0, x1))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), cons(mark(x0), x1))

A__FROM(X) → MARK(X)

MARK(times(plus(x0, x1), y1)) → A__TIMES(a__plus(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(2ndspos(y0, y1), times(x0, x1))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(times(y0, y1), 2ndspos(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__2ndspos(mark(x0), mark(x1)))

A__2NDSPOS(s(y0), cons(y1, from(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__from(mark(x0))))

MARK(2ndspos(pi(y0), square(x0))) → A__2NDSPOS(a__pi(mark(y0)), a__square(mark(x0)))

MARK(2ndspos(square(y0), pi(x0))) → A__2NDSPOS(a__square(mark(y0)), a__pi(mark(x0)))

MARK(2ndspos(s(y0), square(x0))) → A__2NDSPOS(s(mark(y0)), a__square(mark(x0)))

MARK(rcons(X1, X2)) → MARK(X2)

MARK(times(2ndspos(x0, x1), y1)) → A__TIMES(a__2ndspos(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(s(y0), cons2(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(x0, mark(x1)))

A__TIMES(s(X), Y) → A__TIMES(mark(X), mark(Y))

A__PLUS(s(X), Y) → MARK(X)

A__2NDSPOS(s(y0), cons(y1, cons(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, cons(mark(x0), x1)))

MARK(times(X1, X2)) → MARK(X1)

MARK(plus(times(x0, x1), y1)) → A__PLUS(a__times(mark(x0), mark(x1)), mark(y1))

A__SQUARE(X) → MARK(X)

MARK(2ndsneg(times(x0, x1), y1)) → A__2NDSNEG(a__times(mark(x0), mark(x1)), mark(y1))

A__2NDSPOS(s(y0), cons(y1, 2ndsneg(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__2ndsneg(mark(x0), mark(x1))))

A__2NDSPOS(s(N), cons(X, Z)) → MARK(N)

MARK(negrecip(X)) → MARK(X)

A__PLUS(0, Y) → MARK(Y)

MARK(posrecip(X)) → MARK(X)

MARK(2ndsneg(pi(x0), y1)) → A__2NDSNEG(a__pi(mark(x0)), mark(y1))

MARK(2ndspos(square(y0), from(x0))) → A__2NDSPOS(a__square(mark(y0)), a__from(mark(x0)))

MARK(2ndspos(from(y0), square(x0))) → A__2NDSPOS(a__from(mark(y0)), a__square(mark(x0)))

MARK(2ndspos(times(y0, y1), from(x0))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndspos(from(y0), times(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), a__times(mark(x0), mark(x1)))

A__2NDSPOS(s(y0), cons(y1, square(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__square(mark(x0))))

MARK(2ndspos(times(y0, y1), 2ndsneg(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndspos(2ndsneg(y0, y1), times(x0, x1))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(plus(0, y1)) → A__PLUS(0, mark(y1))

MARK(from(X)) → MARK(X)

A__PLUS(s(X), Y) → A__PLUS(mark(X), mark(Y))

A__SQUARE(X) → A__TIMES(mark(X), mark(X))

MARK(2ndspos(square(y0), square(x0))) → A__2NDSPOS(a__square(mark(y0)), a__square(mark(x0)))

A__PI(pi(x0)) → A__2NDSPOS(a__pi(mark(x0)), a__from(0))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(Z)

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → A__2NDSPOS(mark(N), mark(Z))

MARK(plus(from(x0), y1)) → A__PLUS(a__from(mark(x0)), mark(y1))

MARK(square(X)) → A__SQUARE(mark(X))

MARK(2ndspos(plus(y0, y1), from(x0))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndspos(from(y0), plus(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(plus(y0, y1), plus(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(2ndspos(y0, y1), from(x0))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndspos(from(y0), 2ndspos(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndspos(2ndsneg(y0, y1), cons(x0, x1))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), cons(mark(x0), x1))

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))

a__2ndspos(0, Z) → rnil

a__2ndspos(s(N), cons(X, Z)) → a__2ndspos(s(mark(N)), cons2(X, mark(Z)))

a__2ndspos(s(N), cons2(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))

a__2ndsneg(0, Z) → rnil

a__2ndsneg(s(N), cons(X, Z)) → a__2ndsneg(s(mark(N)), cons2(X, mark(Z)))

a__2ndsneg(s(N), cons2(X, cons(Y, Z))) → rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z)))

a__pi(X) → a__2ndspos(mark(X), a__from(0))

a__plus(0, Y) → mark(Y)

a__plus(s(X), Y) → s(a__plus(mark(X), mark(Y)))

a__times(0, Y) → 0

a__times(s(X), Y) → a__plus(mark(Y), a__times(mark(X), mark(Y)))

a__square(X) → a__times(mark(X), mark(X))

mark(from(X)) → a__from(mark(X))

mark(2ndspos(X1, X2)) → a__2ndspos(mark(X1), mark(X2))

mark(2ndsneg(X1, X2)) → a__2ndsneg(mark(X1), mark(X2))

mark(pi(X)) → a__pi(mark(X))

mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))

mark(times(X1, X2)) → a__times(mark(X1), mark(X2))

mark(square(X)) → a__square(mark(X))

mark(0) → 0

mark(s(X)) → s(mark(X))

mark(posrecip(X)) → posrecip(mark(X))

mark(negrecip(X)) → negrecip(mark(X))

mark(nil) → nil

mark(cons(X1, X2)) → cons(mark(X1), X2)

mark(cons2(X1, X2)) → cons2(X1, mark(X2))

mark(rnil) → rnil

mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))

a__from(X) → from(X)

a__2ndspos(X1, X2) → 2ndspos(X1, X2)

a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)

a__pi(X) → pi(X)

a__plus(X1, X2) → plus(X1, X2)

a__times(X1, X2) → times(X1, X2)

a__square(X) → square(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule MARK(2ndsneg(plus(x0, x1), y1)) → A__2NDSNEG(a__plus(mark(x0), mark(x1)), mark(y1)) at position [1] we obtained the following new rules:

MARK(2ndsneg(plus(y0, y1), rcons(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), rcons(mark(x0), mark(x1)))

MARK(2ndsneg(plus(y0, y1), cons2(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(2ndsneg(plus(y0, y1), square(x0))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndsneg(plus(y0, y1), from(x0))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndsneg(plus(y0, y1), 0)) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), 0)

MARK(2ndsneg(plus(y0, y1), 2ndsneg(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndsneg(plus(y0, y1), rnil)) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), rnil)

MARK(2ndsneg(plus(y0, y1), nil)) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), nil)

MARK(2ndsneg(plus(y0, y1), s(x0))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), s(mark(x0)))

MARK(2ndsneg(plus(y0, y1), plus(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(2ndsneg(plus(y0, y1), posrecip(x0))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), posrecip(mark(x0)))

MARK(2ndsneg(plus(y0, y1), 2ndspos(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndsneg(plus(y0, y1), times(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndsneg(plus(y0, y1), cons(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), cons(mark(x0), x1))

MARK(2ndsneg(plus(y0, y1), negrecip(x0))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), negrecip(mark(x0)))

MARK(2ndsneg(plus(y0, y1), pi(x0))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__pi(mark(x0)))



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ Narrowing
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ QDP
              ↳ Narrowing
                ↳ QDP
                  ↳ DependencyGraphProof
                    ↳ QDP
                      ↳ Narrowing
                        ↳ QDP
                          ↳ DependencyGraphProof
                            ↳ QDP
                              ↳ Narrowing
                                ↳ QDP
                                  ↳ DependencyGraphProof
                                    ↳ QDP
                                      ↳ Narrowing
                                        ↳ QDP
                                          ↳ DependencyGraphProof
                                            ↳ QDP
                                              ↳ Narrowing
                                                ↳ QDP
                                                  ↳ DependencyGraphProof
                                                    ↳ QDP
                                                      ↳ Narrowing
                                                        ↳ QDP
                                                          ↳ DependencyGraphProof
                                                            ↳ QDP
                                                              ↳ Narrowing
                                                                ↳ QDP
                                                                  ↳ DependencyGraphProof
                                                                    ↳ QDP
                                                                      ↳ Narrowing
                                                                        ↳ QDP
                                                                          ↳ DependencyGraphProof
                                                                            ↳ QDP
                                                                              ↳ Narrowing
                                                                                ↳ QDP
                                                                                  ↳ DependencyGraphProof
                                                                                    ↳ QDP
                                                                                      ↳ Narrowing
                                                                                        ↳ QDP
                                                                                          ↳ DependencyGraphProof
                                                                                            ↳ QDP
                                                                                              ↳ Narrowing
                                                                                                ↳ QDP
                                                                                                  ↳ DependencyGraphProof
                                                                                                    ↳ QDP
                                                                                                      ↳ Narrowing
                                                                                                        ↳ QDP
                                                                                                          ↳ DependencyGraphProof
                                                                                                            ↳ QDP
                                                                                                              ↳ Narrowing
                                                                                                                ↳ QDP
                                                                                                                  ↳ DependencyGraphProof
                                                                                                                    ↳ QDP
                                                                                                                      ↳ Narrowing
                                                                                                                        ↳ QDP
                                                                                                                          ↳ DependencyGraphProof
                                                                                                                            ↳ QDP
                                                                                                                              ↳ Narrowing
                                                                                                                                ↳ QDP
                                                                                                                                  ↳ DependencyGraphProof
                                                                                                                                    ↳ QDP
                                                                                                                                      ↳ Narrowing
                                                                                                                                        ↳ QDP
                                                                                                                                          ↳ DependencyGraphProof

Q DP problem:
The TRS P consists of the following rules:

MARK(2ndspos(square(y0), plus(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(plus(y0, y1), square(x0))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__square(mark(x0)))

A__2NDSPOS(s(y0), cons(y1, 2ndspos(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__2ndspos(mark(x0), mark(x1))))

A__PI(times(x0, x1)) → A__2NDSPOS(a__times(mark(x0), mark(x1)), a__from(0))

A__2NDSNEG(s(N), cons(X, Z)) → A__2NDSNEG(s(mark(N)), cons2(X, mark(Z)))

MARK(times(times(x0, x1), y1)) → A__TIMES(a__times(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(X1, X2)) → MARK(X2)

MARK(2ndspos(pi(y0), cons2(x0, x1))) → A__2NDSPOS(a__pi(mark(y0)), cons2(x0, mark(x1)))

MARK(2ndsneg(2ndspos(y0, y1), 2ndsneg(x0, x1))) → A__2NDSNEG(a__2ndspos(mark(y0), mark(y1)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndspos(from(y0), cons2(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), cons2(x0, mark(x1)))

MARK(2ndsneg(2ndspos(y0, y1), 2ndspos(x0, x1))) → A__2NDSNEG(a__2ndspos(mark(y0), mark(y1)), a__2ndspos(mark(x0), mark(x1)))

MARK(plus(plus(x0, x1), y1)) → A__PLUS(a__plus(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(from(y0), 2ndsneg(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndspos(s(y0), from(x0))) → A__2NDSPOS(s(mark(y0)), a__from(mark(x0)))

MARK(2ndspos(2ndsneg(y0, y1), from(x0))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndsneg(2ndsneg(x0, x1), y1)) → A__2NDSNEG(a__2ndsneg(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(2ndsneg(y0, y1), pi(x0))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__pi(mark(x0)))

MARK(2ndspos(pi(y0), 2ndsneg(x0, x1))) → A__2NDSPOS(a__pi(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

A__2NDSNEG(s(N), cons(X, Z)) → MARK(Z)

MARK(2ndsneg(square(y0), 2ndspos(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndsneg(2ndspos(y0, y1), square(x0))) → A__2NDSNEG(a__2ndspos(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndsneg(2ndspos(y0, y1), pi(x0))) → A__2NDSNEG(a__2ndspos(mark(y0), mark(y1)), a__pi(mark(x0)))

A__TIMES(s(X), Y) → MARK(Y)

A__2NDSPOS(s(y0), cons(y1, plus(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__plus(mark(x0), mark(x1))))

MARK(2ndsneg(plus(y0, y1), nil)) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), nil)

MARK(2ndspos(from(y0), pi(x0))) → A__2NDSPOS(a__from(mark(y0)), a__pi(mark(x0)))

MARK(2ndspos(square(y0), 2ndsneg(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndspos(2ndsneg(y0, y1), square(x0))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndspos(pi(y0), from(x0))) → A__2NDSPOS(a__pi(mark(y0)), a__from(mark(x0)))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → A__2NDSNEG(mark(N), mark(Z))

A__2NDSNEG(s(N), cons(X, Z)) → MARK(N)

MARK(times(2ndsneg(x0, x1), y1)) → A__TIMES(a__2ndsneg(mark(x0), mark(x1)), mark(y1))

MARK(2ndsneg(from(x0), y1)) → A__2NDSNEG(a__from(mark(x0)), mark(y1))

MARK(2ndspos(from(y0), from(x0))) → A__2NDSPOS(a__from(mark(y0)), a__from(mark(x0)))

MARK(2ndspos(pi(y0), pi(x0))) → A__2NDSPOS(a__pi(mark(y0)), a__pi(mark(x0)))

MARK(2ndsneg(square(y0), 2ndsneg(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndspos(times(y0, y1), times(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

A__TIMES(s(X), Y) → A__PLUS(mark(Y), a__times(mark(X), mark(Y)))

MARK(2ndspos(s(y0), pi(x0))) → A__2NDSPOS(s(mark(y0)), a__pi(mark(x0)))

MARK(2ndspos(square(y0), cons2(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), cons2(x0, mark(x1)))

A__PI(2ndspos(x0, x1)) → A__2NDSPOS(a__2ndspos(mark(x0), mark(x1)), a__from(0))

MARK(plus(square(x0), y1)) → A__PLUS(a__square(mark(x0)), mark(y1))

MARK(plus(s(x0), y1)) → A__PLUS(s(mark(x0)), mark(y1))

MARK(2ndspos(square(y0), cons(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), cons(mark(x0), x1))

A__PI(plus(x0, x1)) → A__2NDSPOS(a__plus(mark(x0), mark(x1)), a__from(0))

MARK(cons2(X1, X2)) → MARK(X2)

MARK(2ndsneg(square(y0), cons2(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), cons2(x0, mark(x1)))

MARK(2ndsneg(2ndspos(y0, y1), times(x0, x1))) → A__2NDSNEG(a__2ndspos(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndsneg(plus(y0, y1), square(x0))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(s(X)) → MARK(X)

MARK(2ndspos(X1, X2)) → MARK(X1)

MARK(2ndsneg(square(y0), plus(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(plus(y0, y1), cons2(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(pi(X)) → MARK(X)

MARK(plus(X1, X2)) → MARK(X2)

A__2NDSPOS(s(N), cons(X, Z)) → MARK(Z)

MARK(pi(X)) → A__PI(mark(X))

MARK(2ndsneg(plus(y0, y1), 0)) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), 0)

MARK(2ndspos(plus(y0, y1), cons(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), cons(mark(x0), x1))

MARK(2ndsneg(square(y0), times(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(plus(2ndsneg(x0, x1), y1)) → A__PLUS(a__2ndsneg(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(times(y0, y1), cons2(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(2ndspos(2ndspos(y0, y1), cons2(x0, x1))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(2ndsneg(X1, X2)) → MARK(X2)

MARK(2ndsneg(square(y0), from(x0))) → A__2NDSNEG(a__square(mark(y0)), a__from(mark(x0)))

MARK(2ndspos(square(y0), times(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(times(y0, y1), square(x0))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndspos(2ndsneg(y0, y1), cons2(x0, x1))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(times(square(x0), y1)) → A__TIMES(a__square(mark(x0)), mark(y1))

MARK(2ndspos(s(y0), plus(x0, x1))) → A__2NDSPOS(s(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(from(X)) → A__FROM(mark(X))

A__PI(X) → A__FROM(0)

MARK(times(from(x0), y1)) → A__TIMES(a__from(mark(x0)), mark(y1))

MARK(plus(X1, X2)) → MARK(X1)

MARK(2ndspos(pi(y0), cons(x0, x1))) → A__2NDSPOS(a__pi(mark(y0)), cons(mark(x0), x1))

A__2NDSPOS(s(y0), cons(y1, times(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__times(mark(x0), mark(x1))))

A__PI(X) → MARK(X)

MARK(2ndspos(s(y0), cons(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons(mark(x0), x1))

MARK(2ndsneg(plus(y0, y1), posrecip(x0))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), posrecip(mark(x0)))

MARK(2ndsneg(plus(y0, y1), 2ndspos(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndsneg(2ndspos(y0, y1), plus(x0, x1))) → A__2NDSNEG(a__2ndspos(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(times(s(x0), y1)) → A__TIMES(s(mark(x0)), mark(y1))

MARK(2ndspos(pi(y0), 2ndspos(x0, x1))) → A__2NDSPOS(a__pi(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndspos(2ndspos(y0, y1), pi(x0))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), a__pi(mark(x0)))

MARK(2ndspos(from(y0), cons(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), cons(mark(x0), x1))

MARK(2ndsneg(s(x0), y1)) → A__2NDSNEG(s(mark(x0)), mark(y1))

MARK(2ndsneg(plus(y0, y1), cons(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), cons(mark(x0), x1))

MARK(2ndspos(2ndspos(y0, y1), plus(x0, x1))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(plus(y0, y1), 2ndspos(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndsneg(2ndspos(y0, y1), cons(x0, x1))) → A__2NDSNEG(a__2ndspos(mark(y0), mark(y1)), cons(mark(x0), x1))

A__PLUS(s(X), Y) → MARK(Y)

A__2NDSPOS(s(y0), cons(y1, pi(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__pi(mark(x0))))

A__TIMES(s(X), Y) → MARK(X)

MARK(2ndsneg(plus(y0, y1), pi(x0))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__pi(mark(x0)))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(N)

MARK(2ndspos(pi(y0), plus(x0, x1))) → A__2NDSPOS(a__pi(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(plus(y0, y1), pi(x0))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__pi(mark(x0)))

MARK(2ndsneg(plus(y0, y1), rcons(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), rcons(mark(x0), mark(x1)))

MARK(plus(2ndspos(x0, x1), y1)) → A__PLUS(a__2ndspos(mark(x0), mark(x1)), mark(y1))

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(N)

MARK(cons(X1, X2)) → MARK(X1)

A__PI(from(x0)) → A__2NDSPOS(a__from(mark(x0)), a__from(0))

MARK(2ndsneg(plus(y0, y1), from(x0))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndspos(square(y0), 2ndspos(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndspos(2ndspos(y0, y1), square(x0))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndspos(s(y0), 2ndspos(x0, x1))) → A__2NDSPOS(s(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

MARK(times(X1, X2)) → MARK(X2)

MARK(2ndspos(times(y0, y1), cons(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), cons(mark(x0), x1))

MARK(2ndspos(2ndsneg(y0, y1), 2ndsneg(x0, x1))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__2ndsneg(mark(x0), mark(x1)))

A__PI(square(x0)) → A__2NDSPOS(a__square(mark(x0)), a__from(0))

MARK(2ndspos(times(y0, y1), plus(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(plus(y0, y1), times(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndsneg(square(y0), cons(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), cons(mark(x0), x1))

MARK(2ndsneg(2ndspos(y0, y1), from(x0))) → A__2NDSNEG(a__2ndspos(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndspos(s(y0), times(x0, x1))) → A__2NDSPOS(s(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndsneg(X1, X2)) → MARK(X1)

MARK(2ndspos(s(y0), 2ndsneg(x0, x1))) → A__2NDSPOS(s(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndsneg(2ndspos(y0, y1), cons2(x0, x1))) → A__2NDSNEG(a__2ndspos(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(2ndsneg(square(y0), square(x0))) → A__2NDSNEG(a__square(mark(y0)), a__square(mark(x0)))

MARK(2ndspos(2ndsneg(y0, y1), plus(x0, x1))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(plus(y0, y1), 2ndsneg(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__2ndsneg(mark(x0), mark(x1)))

A__PI(2ndsneg(x0, x1)) → A__2NDSPOS(a__2ndsneg(mark(x0), mark(x1)), a__from(0))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(Y)

MARK(times(pi(x0), y1)) → A__TIMES(a__pi(mark(x0)), mark(y1))

MARK(plus(pi(x0), y1)) → A__PLUS(a__pi(mark(x0)), mark(y1))

MARK(2ndspos(2ndspos(y0, y1), 2ndspos(x0, x1))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndsneg(plus(y0, y1), rnil)) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), rnil)

MARK(rcons(X1, X2)) → MARK(X1)

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(Z)

MARK(square(X)) → MARK(X)

MARK(2ndsneg(square(y0), pi(x0))) → A__2NDSNEG(a__square(mark(y0)), a__pi(mark(x0)))

MARK(2ndspos(2ndsneg(y0, y1), 2ndspos(x0, x1))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndspos(2ndspos(y0, y1), 2ndsneg(x0, x1))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndspos(pi(y0), times(x0, x1))) → A__2NDSPOS(a__pi(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(times(y0, y1), pi(x0))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__pi(mark(x0)))

A__PI(s(x0)) → A__2NDSPOS(s(mark(x0)), a__from(0))

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(Y)

A__FROM(X) → MARK(X)

MARK(2ndspos(2ndspos(y0, y1), cons(x0, x1))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), cons(mark(x0), x1))

MARK(2ndsneg(plus(y0, y1), s(x0))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), s(mark(x0)))

MARK(times(plus(x0, x1), y1)) → A__TIMES(a__plus(mark(x0), mark(x1)), mark(y1))

A__2NDSPOS(s(y0), cons(y1, from(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__from(mark(x0))))

MARK(2ndspos(times(y0, y1), 2ndspos(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndspos(2ndspos(y0, y1), times(x0, x1))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(square(y0), pi(x0))) → A__2NDSPOS(a__square(mark(y0)), a__pi(mark(x0)))

MARK(2ndspos(pi(y0), square(x0))) → A__2NDSPOS(a__pi(mark(y0)), a__square(mark(x0)))

MARK(rcons(X1, X2)) → MARK(X2)

MARK(2ndspos(s(y0), square(x0))) → A__2NDSPOS(s(mark(y0)), a__square(mark(x0)))

MARK(times(2ndspos(x0, x1), y1)) → A__TIMES(a__2ndspos(mark(x0), mark(x1)), mark(y1))

MARK(2ndsneg(plus(y0, y1), negrecip(x0))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), negrecip(mark(x0)))

MARK(2ndspos(s(y0), cons2(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(x0, mark(x1)))

MARK(2ndsneg(plus(y0, y1), cons2(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), cons2(x0, mark(x1)))

A__TIMES(s(X), Y) → A__TIMES(mark(X), mark(Y))

A__2NDSPOS(s(y0), cons(y1, cons(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, cons(mark(x0), x1)))

A__PLUS(s(X), Y) → MARK(X)

MARK(times(X1, X2)) → MARK(X1)

MARK(plus(times(x0, x1), y1)) → A__PLUS(a__times(mark(x0), mark(x1)), mark(y1))

A__SQUARE(X) → MARK(X)

MARK(2ndsneg(times(x0, x1), y1)) → A__2NDSNEG(a__times(mark(x0), mark(x1)), mark(y1))

A__2NDSPOS(s(y0), cons(y1, 2ndsneg(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__2ndsneg(mark(x0), mark(x1))))

MARK(negrecip(X)) → MARK(X)

A__2NDSPOS(s(N), cons(X, Z)) → MARK(N)

MARK(2ndsneg(plus(y0, y1), 2ndsneg(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__2ndsneg(mark(x0), mark(x1)))

A__PLUS(0, Y) → MARK(Y)

MARK(posrecip(X)) → MARK(X)

MARK(2ndspos(from(y0), square(x0))) → A__2NDSPOS(a__from(mark(y0)), a__square(mark(x0)))

MARK(2ndspos(square(y0), from(x0))) → A__2NDSPOS(a__square(mark(y0)), a__from(mark(x0)))

MARK(2ndsneg(pi(x0), y1)) → A__2NDSNEG(a__pi(mark(x0)), mark(y1))

MARK(2ndspos(from(y0), times(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(times(y0, y1), from(x0))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__from(mark(x0)))

A__2NDSPOS(s(y0), cons(y1, square(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__square(mark(x0))))

MARK(2ndspos(2ndsneg(y0, y1), times(x0, x1))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(times(y0, y1), 2ndsneg(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__2ndsneg(mark(x0), mark(x1)))

MARK(plus(0, y1)) → A__PLUS(0, mark(y1))

MARK(from(X)) → MARK(X)

A__PLUS(s(X), Y) → A__PLUS(mark(X), mark(Y))

MARK(2ndspos(square(y0), square(x0))) → A__2NDSPOS(a__square(mark(y0)), a__square(mark(x0)))

A__SQUARE(X) → A__TIMES(mark(X), mark(X))

A__PI(pi(x0)) → A__2NDSPOS(a__pi(mark(x0)), a__from(0))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(Z)

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → A__2NDSPOS(mark(N), mark(Z))

MARK(2ndsneg(plus(y0, y1), plus(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(plus(from(x0), y1)) → A__PLUS(a__from(mark(x0)), mark(y1))

MARK(square(X)) → A__SQUARE(mark(X))

MARK(2ndspos(from(y0), plus(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(plus(y0, y1), from(x0))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndsneg(plus(y0, y1), times(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(plus(y0, y1), plus(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(from(y0), 2ndspos(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndspos(2ndspos(y0, y1), from(x0))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndspos(2ndsneg(y0, y1), cons(x0, x1))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), cons(mark(x0), x1))

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))

a__2ndspos(0, Z) → rnil

a__2ndspos(s(N), cons(X, Z)) → a__2ndspos(s(mark(N)), cons2(X, mark(Z)))

a__2ndspos(s(N), cons2(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))

a__2ndsneg(0, Z) → rnil

a__2ndsneg(s(N), cons(X, Z)) → a__2ndsneg(s(mark(N)), cons2(X, mark(Z)))

a__2ndsneg(s(N), cons2(X, cons(Y, Z))) → rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z)))

a__pi(X) → a__2ndspos(mark(X), a__from(0))

a__plus(0, Y) → mark(Y)

a__plus(s(X), Y) → s(a__plus(mark(X), mark(Y)))

a__times(0, Y) → 0

a__times(s(X), Y) → a__plus(mark(Y), a__times(mark(X), mark(Y)))

a__square(X) → a__times(mark(X), mark(X))

mark(from(X)) → a__from(mark(X))

mark(2ndspos(X1, X2)) → a__2ndspos(mark(X1), mark(X2))

mark(2ndsneg(X1, X2)) → a__2ndsneg(mark(X1), mark(X2))

mark(pi(X)) → a__pi(mark(X))

mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))

mark(times(X1, X2)) → a__times(mark(X1), mark(X2))

mark(square(X)) → a__square(mark(X))

mark(0) → 0

mark(s(X)) → s(mark(X))

mark(posrecip(X)) → posrecip(mark(X))

mark(negrecip(X)) → negrecip(mark(X))

mark(nil) → nil

mark(cons(X1, X2)) → cons(mark(X1), X2)

mark(cons2(X1, X2)) → cons2(X1, mark(X2))

mark(rnil) → rnil

mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))

a__from(X) → from(X)

a__2ndspos(X1, X2) → 2ndspos(X1, X2)

a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)

a__pi(X) → pi(X)

a__plus(X1, X2) → plus(X1, X2)

a__times(X1, X2) → times(X1, X2)

a__square(X) → square(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 7 less nodes.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ Narrowing
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ QDP
              ↳ Narrowing
                ↳ QDP
                  ↳ DependencyGraphProof
                    ↳ QDP
                      ↳ Narrowing
                        ↳ QDP
                          ↳ DependencyGraphProof
                            ↳ QDP
                              ↳ Narrowing
                                ↳ QDP
                                  ↳ DependencyGraphProof
                                    ↳ QDP
                                      ↳ Narrowing
                                        ↳ QDP
                                          ↳ DependencyGraphProof
                                            ↳ QDP
                                              ↳ Narrowing
                                                ↳ QDP
                                                  ↳ DependencyGraphProof
                                                    ↳ QDP
                                                      ↳ Narrowing
                                                        ↳ QDP
                                                          ↳ DependencyGraphProof
                                                            ↳ QDP
                                                              ↳ Narrowing
                                                                ↳ QDP
                                                                  ↳ DependencyGraphProof
                                                                    ↳ QDP
                                                                      ↳ Narrowing
                                                                        ↳ QDP
                                                                          ↳ DependencyGraphProof
                                                                            ↳ QDP
                                                                              ↳ Narrowing
                                                                                ↳ QDP
                                                                                  ↳ DependencyGraphProof
                                                                                    ↳ QDP
                                                                                      ↳ Narrowing
                                                                                        ↳ QDP
                                                                                          ↳ DependencyGraphProof
                                                                                            ↳ QDP
                                                                                              ↳ Narrowing
                                                                                                ↳ QDP
                                                                                                  ↳ DependencyGraphProof
                                                                                                    ↳ QDP
                                                                                                      ↳ Narrowing
                                                                                                        ↳ QDP
                                                                                                          ↳ DependencyGraphProof
                                                                                                            ↳ QDP
                                                                                                              ↳ Narrowing
                                                                                                                ↳ QDP
                                                                                                                  ↳ DependencyGraphProof
                                                                                                                    ↳ QDP
                                                                                                                      ↳ Narrowing
                                                                                                                        ↳ QDP
                                                                                                                          ↳ DependencyGraphProof
                                                                                                                            ↳ QDP
                                                                                                                              ↳ Narrowing
                                                                                                                                ↳ QDP
                                                                                                                                  ↳ DependencyGraphProof
                                                                                                                                    ↳ QDP
                                                                                                                                      ↳ Narrowing
                                                                                                                                        ↳ QDP
                                                                                                                                          ↳ DependencyGraphProof
                                                                                                                                            ↳ QDP
                                                                                                                                              ↳ Narrowing

Q DP problem:
The TRS P consists of the following rules:

MARK(2ndspos(plus(y0, y1), square(x0))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndspos(square(y0), plus(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__plus(mark(x0), mark(x1)))

A__PI(times(x0, x1)) → A__2NDSPOS(a__times(mark(x0), mark(x1)), a__from(0))

A__2NDSPOS(s(y0), cons(y1, 2ndspos(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__2ndspos(mark(x0), mark(x1))))

A__2NDSNEG(s(N), cons(X, Z)) → A__2NDSNEG(s(mark(N)), cons2(X, mark(Z)))

MARK(2ndspos(pi(y0), cons2(x0, x1))) → A__2NDSPOS(a__pi(mark(y0)), cons2(x0, mark(x1)))

MARK(2ndspos(X1, X2)) → MARK(X2)

MARK(times(times(x0, x1), y1)) → A__TIMES(a__times(mark(x0), mark(x1)), mark(y1))

MARK(2ndsneg(2ndspos(y0, y1), 2ndsneg(x0, x1))) → A__2NDSNEG(a__2ndspos(mark(y0), mark(y1)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndspos(from(y0), cons2(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), cons2(x0, mark(x1)))

MARK(2ndsneg(2ndspos(y0, y1), 2ndspos(x0, x1))) → A__2NDSNEG(a__2ndspos(mark(y0), mark(y1)), a__2ndspos(mark(x0), mark(x1)))

MARK(plus(plus(x0, x1), y1)) → A__PLUS(a__plus(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(2ndsneg(y0, y1), from(x0))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndspos(s(y0), from(x0))) → A__2NDSPOS(s(mark(y0)), a__from(mark(x0)))

MARK(2ndspos(from(y0), 2ndsneg(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndspos(pi(y0), 2ndsneg(x0, x1))) → A__2NDSPOS(a__pi(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndspos(2ndsneg(y0, y1), pi(x0))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__pi(mark(x0)))

MARK(2ndsneg(2ndsneg(x0, x1), y1)) → A__2NDSNEG(a__2ndsneg(mark(x0), mark(x1)), mark(y1))

MARK(2ndsneg(2ndspos(y0, y1), square(x0))) → A__2NDSNEG(a__2ndspos(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndsneg(square(y0), 2ndspos(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

A__2NDSNEG(s(N), cons(X, Z)) → MARK(Z)

MARK(2ndsneg(2ndspos(y0, y1), pi(x0))) → A__2NDSNEG(a__2ndspos(mark(y0), mark(y1)), a__pi(mark(x0)))

A__TIMES(s(X), Y) → MARK(Y)

A__2NDSPOS(s(y0), cons(y1, plus(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__plus(mark(x0), mark(x1))))

MARK(2ndspos(pi(y0), from(x0))) → A__2NDSPOS(a__pi(mark(y0)), a__from(mark(x0)))

MARK(2ndspos(2ndsneg(y0, y1), square(x0))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndspos(square(y0), 2ndsneg(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndspos(from(y0), pi(x0))) → A__2NDSPOS(a__from(mark(y0)), a__pi(mark(x0)))

A__2NDSNEG(s(N), cons(X, Z)) → MARK(N)

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → A__2NDSNEG(mark(N), mark(Z))

MARK(times(2ndsneg(x0, x1), y1)) → A__TIMES(a__2ndsneg(mark(x0), mark(x1)), mark(y1))

MARK(2ndsneg(from(x0), y1)) → A__2NDSNEG(a__from(mark(x0)), mark(y1))

MARK(2ndspos(from(y0), from(x0))) → A__2NDSPOS(a__from(mark(y0)), a__from(mark(x0)))

MARK(2ndspos(pi(y0), pi(x0))) → A__2NDSPOS(a__pi(mark(y0)), a__pi(mark(x0)))

MARK(2ndsneg(square(y0), 2ndsneg(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndspos(times(y0, y1), times(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(s(y0), pi(x0))) → A__2NDSPOS(s(mark(y0)), a__pi(mark(x0)))

A__TIMES(s(X), Y) → A__PLUS(mark(Y), a__times(mark(X), mark(Y)))

A__PI(2ndspos(x0, x1)) → A__2NDSPOS(a__2ndspos(mark(x0), mark(x1)), a__from(0))

MARK(2ndspos(square(y0), cons2(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), cons2(x0, mark(x1)))

MARK(plus(square(x0), y1)) → A__PLUS(a__square(mark(x0)), mark(y1))

MARK(plus(s(x0), y1)) → A__PLUS(s(mark(x0)), mark(y1))

A__PI(plus(x0, x1)) → A__2NDSPOS(a__plus(mark(x0), mark(x1)), a__from(0))

MARK(2ndspos(square(y0), cons(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), cons(mark(x0), x1))

MARK(cons2(X1, X2)) → MARK(X2)

MARK(2ndsneg(2ndspos(y0, y1), times(x0, x1))) → A__2NDSNEG(a__2ndspos(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndsneg(square(y0), cons2(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), cons2(x0, mark(x1)))

MARK(2ndsneg(plus(y0, y1), square(x0))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndsneg(square(y0), plus(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(X1, X2)) → MARK(X1)

MARK(s(X)) → MARK(X)

MARK(2ndspos(plus(y0, y1), cons2(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(pi(X)) → MARK(X)

MARK(plus(X1, X2)) → MARK(X2)

A__2NDSPOS(s(N), cons(X, Z)) → MARK(Z)

MARK(pi(X)) → A__PI(mark(X))

MARK(2ndsneg(square(y0), times(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(plus(y0, y1), cons(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), cons(mark(x0), x1))

MARK(plus(2ndsneg(x0, x1), y1)) → A__PLUS(a__2ndsneg(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(times(y0, y1), cons2(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(2ndspos(2ndspos(y0, y1), cons2(x0, x1))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(2ndsneg(X1, X2)) → MARK(X2)

MARK(2ndsneg(square(y0), from(x0))) → A__2NDSNEG(a__square(mark(y0)), a__from(mark(x0)))

MARK(2ndspos(times(y0, y1), square(x0))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndspos(square(y0), times(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(2ndsneg(y0, y1), cons2(x0, x1))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(times(square(x0), y1)) → A__TIMES(a__square(mark(x0)), mark(y1))

A__PI(X) → A__FROM(0)

MARK(from(X)) → A__FROM(mark(X))

MARK(2ndspos(s(y0), plus(x0, x1))) → A__2NDSPOS(s(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(pi(y0), cons(x0, x1))) → A__2NDSPOS(a__pi(mark(y0)), cons(mark(x0), x1))

MARK(plus(X1, X2)) → MARK(X1)

MARK(times(from(x0), y1)) → A__TIMES(a__from(mark(x0)), mark(y1))

A__2NDSPOS(s(y0), cons(y1, times(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__times(mark(x0), mark(x1))))

A__PI(X) → MARK(X)

MARK(2ndspos(s(y0), cons(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons(mark(x0), x1))

MARK(2ndsneg(plus(y0, y1), 2ndspos(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndsneg(2ndspos(y0, y1), plus(x0, x1))) → A__2NDSNEG(a__2ndspos(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(times(s(x0), y1)) → A__TIMES(s(mark(x0)), mark(y1))

MARK(2ndspos(2ndspos(y0, y1), pi(x0))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), a__pi(mark(x0)))

MARK(2ndspos(pi(y0), 2ndspos(x0, x1))) → A__2NDSPOS(a__pi(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndsneg(s(x0), y1)) → A__2NDSNEG(s(mark(x0)), mark(y1))

MARK(2ndspos(from(y0), cons(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), cons(mark(x0), x1))

MARK(2ndsneg(plus(y0, y1), cons(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), cons(mark(x0), x1))

MARK(2ndsneg(2ndspos(y0, y1), cons(x0, x1))) → A__2NDSNEG(a__2ndspos(mark(y0), mark(y1)), cons(mark(x0), x1))

MARK(2ndspos(plus(y0, y1), 2ndspos(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndspos(2ndspos(y0, y1), plus(x0, x1))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

A__PLUS(s(X), Y) → MARK(Y)

A__TIMES(s(X), Y) → MARK(X)

A__2NDSPOS(s(y0), cons(y1, pi(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__pi(mark(x0))))

MARK(2ndsneg(plus(y0, y1), pi(x0))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__pi(mark(x0)))

MARK(2ndspos(plus(y0, y1), pi(x0))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__pi(mark(x0)))

MARK(2ndspos(pi(y0), plus(x0, x1))) → A__2NDSPOS(a__pi(mark(y0)), a__plus(mark(x0), mark(x1)))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(N)

MARK(plus(2ndspos(x0, x1), y1)) → A__PLUS(a__2ndspos(mark(x0), mark(x1)), mark(y1))

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(N)

A__PI(from(x0)) → A__2NDSPOS(a__from(mark(x0)), a__from(0))

MARK(cons(X1, X2)) → MARK(X1)

MARK(2ndsneg(plus(y0, y1), from(x0))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndspos(2ndspos(y0, y1), square(x0))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndspos(square(y0), 2ndspos(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndspos(times(y0, y1), cons(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), cons(mark(x0), x1))

MARK(times(X1, X2)) → MARK(X2)

MARK(2ndspos(s(y0), 2ndspos(x0, x1))) → A__2NDSPOS(s(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

A__PI(square(x0)) → A__2NDSPOS(a__square(mark(x0)), a__from(0))

MARK(2ndspos(2ndsneg(y0, y1), 2ndsneg(x0, x1))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndsneg(2ndspos(y0, y1), from(x0))) → A__2NDSNEG(a__2ndspos(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndsneg(square(y0), cons(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), cons(mark(x0), x1))

MARK(2ndspos(plus(y0, y1), times(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(times(y0, y1), plus(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(s(y0), times(x0, x1))) → A__2NDSPOS(s(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndsneg(X1, X2)) → MARK(X1)

MARK(2ndsneg(2ndspos(y0, y1), cons2(x0, x1))) → A__2NDSNEG(a__2ndspos(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(2ndspos(s(y0), 2ndsneg(x0, x1))) → A__2NDSPOS(s(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndsneg(square(y0), square(x0))) → A__2NDSNEG(a__square(mark(y0)), a__square(mark(x0)))

MARK(2ndspos(plus(y0, y1), 2ndsneg(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndspos(2ndsneg(y0, y1), plus(x0, x1))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

A__PI(2ndsneg(x0, x1)) → A__2NDSPOS(a__2ndsneg(mark(x0), mark(x1)), a__from(0))

MARK(plus(pi(x0), y1)) → A__PLUS(a__pi(mark(x0)), mark(y1))

MARK(times(pi(x0), y1)) → A__TIMES(a__pi(mark(x0)), mark(y1))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(Y)

MARK(2ndspos(2ndspos(y0, y1), 2ndspos(x0, x1))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), a__2ndspos(mark(x0), mark(x1)))

MARK(rcons(X1, X2)) → MARK(X1)

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(Z)

MARK(2ndsneg(square(y0), pi(x0))) → A__2NDSNEG(a__square(mark(y0)), a__pi(mark(x0)))

MARK(square(X)) → MARK(X)

MARK(2ndspos(2ndspos(y0, y1), 2ndsneg(x0, x1))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndspos(2ndsneg(y0, y1), 2ndspos(x0, x1))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndspos(times(y0, y1), pi(x0))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__pi(mark(x0)))

MARK(2ndspos(pi(y0), times(x0, x1))) → A__2NDSPOS(a__pi(mark(y0)), a__times(mark(x0), mark(x1)))

A__PI(s(x0)) → A__2NDSPOS(s(mark(x0)), a__from(0))

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(Y)

MARK(2ndspos(2ndspos(y0, y1), cons(x0, x1))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), cons(mark(x0), x1))

A__FROM(X) → MARK(X)

MARK(times(plus(x0, x1), y1)) → A__TIMES(a__plus(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(2ndspos(y0, y1), times(x0, x1))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(times(y0, y1), 2ndspos(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__2ndspos(mark(x0), mark(x1)))

A__2NDSPOS(s(y0), cons(y1, from(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__from(mark(x0))))

MARK(2ndspos(pi(y0), square(x0))) → A__2NDSPOS(a__pi(mark(y0)), a__square(mark(x0)))

MARK(2ndspos(square(y0), pi(x0))) → A__2NDSPOS(a__square(mark(y0)), a__pi(mark(x0)))

MARK(2ndspos(s(y0), square(x0))) → A__2NDSPOS(s(mark(y0)), a__square(mark(x0)))

MARK(rcons(X1, X2)) → MARK(X2)

MARK(times(2ndspos(x0, x1), y1)) → A__TIMES(a__2ndspos(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(s(y0), cons2(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(x0, mark(x1)))

MARK(2ndsneg(plus(y0, y1), cons2(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), cons2(x0, mark(x1)))

A__TIMES(s(X), Y) → A__TIMES(mark(X), mark(Y))

A__PLUS(s(X), Y) → MARK(X)

A__2NDSPOS(s(y0), cons(y1, cons(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, cons(mark(x0), x1)))

MARK(times(X1, X2)) → MARK(X1)

MARK(plus(times(x0, x1), y1)) → A__PLUS(a__times(mark(x0), mark(x1)), mark(y1))

A__SQUARE(X) → MARK(X)

MARK(2ndsneg(times(x0, x1), y1)) → A__2NDSNEG(a__times(mark(x0), mark(x1)), mark(y1))

A__2NDSPOS(s(y0), cons(y1, 2ndsneg(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__2ndsneg(mark(x0), mark(x1))))

A__2NDSPOS(s(N), cons(X, Z)) → MARK(N)

MARK(negrecip(X)) → MARK(X)

MARK(2ndsneg(plus(y0, y1), 2ndsneg(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__2ndsneg(mark(x0), mark(x1)))

A__PLUS(0, Y) → MARK(Y)

MARK(posrecip(X)) → MARK(X)

MARK(2ndsneg(pi(x0), y1)) → A__2NDSNEG(a__pi(mark(x0)), mark(y1))

MARK(2ndspos(square(y0), from(x0))) → A__2NDSPOS(a__square(mark(y0)), a__from(mark(x0)))

MARK(2ndspos(from(y0), square(x0))) → A__2NDSPOS(a__from(mark(y0)), a__square(mark(x0)))

MARK(2ndspos(times(y0, y1), from(x0))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndspos(from(y0), times(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), a__times(mark(x0), mark(x1)))

A__2NDSPOS(s(y0), cons(y1, square(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__square(mark(x0))))

MARK(2ndspos(times(y0, y1), 2ndsneg(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndspos(2ndsneg(y0, y1), times(x0, x1))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(plus(0, y1)) → A__PLUS(0, mark(y1))

MARK(from(X)) → MARK(X)

A__PLUS(s(X), Y) → A__PLUS(mark(X), mark(Y))

A__SQUARE(X) → A__TIMES(mark(X), mark(X))

MARK(2ndspos(square(y0), square(x0))) → A__2NDSPOS(a__square(mark(y0)), a__square(mark(x0)))

A__PI(pi(x0)) → A__2NDSPOS(a__pi(mark(x0)), a__from(0))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(Z)

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → A__2NDSPOS(mark(N), mark(Z))

MARK(2ndsneg(plus(y0, y1), plus(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(plus(from(x0), y1)) → A__PLUS(a__from(mark(x0)), mark(y1))

MARK(square(X)) → A__SQUARE(mark(X))

MARK(2ndspos(plus(y0, y1), from(x0))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndspos(from(y0), plus(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(2ndsneg(plus(y0, y1), times(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(plus(y0, y1), plus(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(2ndspos(y0, y1), from(x0))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndspos(from(y0), 2ndspos(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndspos(2ndsneg(y0, y1), cons(x0, x1))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), cons(mark(x0), x1))

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))

a__2ndspos(0, Z) → rnil

a__2ndspos(s(N), cons(X, Z)) → a__2ndspos(s(mark(N)), cons2(X, mark(Z)))

a__2ndspos(s(N), cons2(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))

a__2ndsneg(0, Z) → rnil

a__2ndsneg(s(N), cons(X, Z)) → a__2ndsneg(s(mark(N)), cons2(X, mark(Z)))

a__2ndsneg(s(N), cons2(X, cons(Y, Z))) → rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z)))

a__pi(X) → a__2ndspos(mark(X), a__from(0))

a__plus(0, Y) → mark(Y)

a__plus(s(X), Y) → s(a__plus(mark(X), mark(Y)))

a__times(0, Y) → 0

a__times(s(X), Y) → a__plus(mark(Y), a__times(mark(X), mark(Y)))

a__square(X) → a__times(mark(X), mark(X))

mark(from(X)) → a__from(mark(X))

mark(2ndspos(X1, X2)) → a__2ndspos(mark(X1), mark(X2))

mark(2ndsneg(X1, X2)) → a__2ndsneg(mark(X1), mark(X2))

mark(pi(X)) → a__pi(mark(X))

mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))

mark(times(X1, X2)) → a__times(mark(X1), mark(X2))

mark(square(X)) → a__square(mark(X))

mark(0) → 0

mark(s(X)) → s(mark(X))

mark(posrecip(X)) → posrecip(mark(X))

mark(negrecip(X)) → negrecip(mark(X))

mark(nil) → nil

mark(cons(X1, X2)) → cons(mark(X1), X2)

mark(cons2(X1, X2)) → cons2(X1, mark(X2))

mark(rnil) → rnil

mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))

a__from(X) → from(X)

a__2ndspos(X1, X2) → 2ndspos(X1, X2)

a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)

a__pi(X) → pi(X)

a__plus(X1, X2) → plus(X1, X2)

a__times(X1, X2) → times(X1, X2)

a__square(X) → square(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule MARK(2ndsneg(from(x0), y1)) → A__2NDSNEG(a__from(mark(x0)), mark(y1)) at position [1] we obtained the following new rules:

MARK(2ndsneg(from(y0), rnil)) → A__2NDSNEG(a__from(mark(y0)), rnil)

MARK(2ndsneg(from(y0), nil)) → A__2NDSNEG(a__from(mark(y0)), nil)

MARK(2ndsneg(from(y0), s(x0))) → A__2NDSNEG(a__from(mark(y0)), s(mark(x0)))

MARK(2ndsneg(from(y0), plus(x0, x1))) → A__2NDSNEG(a__from(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(2ndsneg(from(y0), 0)) → A__2NDSNEG(a__from(mark(y0)), 0)

MARK(2ndsneg(from(y0), pi(x0))) → A__2NDSNEG(a__from(mark(y0)), a__pi(mark(x0)))

MARK(2ndsneg(from(y0), from(x0))) → A__2NDSNEG(a__from(mark(y0)), a__from(mark(x0)))

MARK(2ndsneg(from(y0), negrecip(x0))) → A__2NDSNEG(a__from(mark(y0)), negrecip(mark(x0)))

MARK(2ndsneg(from(y0), cons2(x0, x1))) → A__2NDSNEG(a__from(mark(y0)), cons2(x0, mark(x1)))

MARK(2ndsneg(from(y0), cons(x0, x1))) → A__2NDSNEG(a__from(mark(y0)), cons(mark(x0), x1))

MARK(2ndsneg(from(y0), 2ndspos(x0, x1))) → A__2NDSNEG(a__from(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndsneg(from(y0), rcons(x0, x1))) → A__2NDSNEG(a__from(mark(y0)), rcons(mark(x0), mark(x1)))

MARK(2ndsneg(from(y0), times(x0, x1))) → A__2NDSNEG(a__from(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndsneg(from(y0), posrecip(x0))) → A__2NDSNEG(a__from(mark(y0)), posrecip(mark(x0)))

MARK(2ndsneg(from(y0), square(x0))) → A__2NDSNEG(a__from(mark(y0)), a__square(mark(x0)))

MARK(2ndsneg(from(y0), 2ndsneg(x0, x1))) → A__2NDSNEG(a__from(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ Narrowing
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ QDP
              ↳ Narrowing
                ↳ QDP
                  ↳ DependencyGraphProof
                    ↳ QDP
                      ↳ Narrowing
                        ↳ QDP
                          ↳ DependencyGraphProof
                            ↳ QDP
                              ↳ Narrowing
                                ↳ QDP
                                  ↳ DependencyGraphProof
                                    ↳ QDP
                                      ↳ Narrowing
                                        ↳ QDP
                                          ↳ DependencyGraphProof
                                            ↳ QDP
                                              ↳ Narrowing
                                                ↳ QDP
                                                  ↳ DependencyGraphProof
                                                    ↳ QDP
                                                      ↳ Narrowing
                                                        ↳ QDP
                                                          ↳ DependencyGraphProof
                                                            ↳ QDP
                                                              ↳ Narrowing
                                                                ↳ QDP
                                                                  ↳ DependencyGraphProof
                                                                    ↳ QDP
                                                                      ↳ Narrowing
                                                                        ↳ QDP
                                                                          ↳ DependencyGraphProof
                                                                            ↳ QDP
                                                                              ↳ Narrowing
                                                                                ↳ QDP
                                                                                  ↳ DependencyGraphProof
                                                                                    ↳ QDP
                                                                                      ↳ Narrowing
                                                                                        ↳ QDP
                                                                                          ↳ DependencyGraphProof
                                                                                            ↳ QDP
                                                                                              ↳ Narrowing
                                                                                                ↳ QDP
                                                                                                  ↳ DependencyGraphProof
                                                                                                    ↳ QDP
                                                                                                      ↳ Narrowing
                                                                                                        ↳ QDP
                                                                                                          ↳ DependencyGraphProof
                                                                                                            ↳ QDP
                                                                                                              ↳ Narrowing
                                                                                                                ↳ QDP
                                                                                                                  ↳ DependencyGraphProof
                                                                                                                    ↳ QDP
                                                                                                                      ↳ Narrowing
                                                                                                                        ↳ QDP
                                                                                                                          ↳ DependencyGraphProof
                                                                                                                            ↳ QDP
                                                                                                                              ↳ Narrowing
                                                                                                                                ↳ QDP
                                                                                                                                  ↳ DependencyGraphProof
                                                                                                                                    ↳ QDP
                                                                                                                                      ↳ Narrowing
                                                                                                                                        ↳ QDP
                                                                                                                                          ↳ DependencyGraphProof
                                                                                                                                            ↳ QDP
                                                                                                                                              ↳ Narrowing
                                                                                                                                                ↳ QDP
                                                                                                                                                  ↳ DependencyGraphProof

Q DP problem:
The TRS P consists of the following rules:

MARK(times(times(x0, x1), y1)) → A__TIMES(a__times(mark(x0), mark(x1)), mark(y1))

MARK(2ndsneg(2ndspos(y0, y1), 2ndsneg(x0, x1))) → A__2NDSNEG(a__2ndspos(mark(y0), mark(y1)), a__2ndsneg(mark(x0), mark(x1)))

MARK(plus(plus(x0, x1), y1)) → A__PLUS(a__plus(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(2ndsneg(y0, y1), from(x0))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndspos(from(y0), 2ndsneg(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndsneg(2ndsneg(x0, x1), y1)) → A__2NDSNEG(a__2ndsneg(mark(x0), mark(x1)), mark(y1))

MARK(2ndsneg(2ndspos(y0, y1), square(x0))) → A__2NDSNEG(a__2ndspos(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndsneg(square(y0), 2ndspos(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

A__2NDSNEG(s(N), cons(X, Z)) → MARK(Z)

MARK(2ndsneg(2ndspos(y0, y1), pi(x0))) → A__2NDSNEG(a__2ndspos(mark(y0), mark(y1)), a__pi(mark(x0)))

A__TIMES(s(X), Y) → MARK(Y)

MARK(2ndspos(2ndsneg(y0, y1), square(x0))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndspos(square(y0), 2ndsneg(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

A__2NDSNEG(s(N), cons(X, Z)) → MARK(N)

MARK(2ndsneg(from(y0), nil)) → A__2NDSNEG(a__from(mark(y0)), nil)

MARK(times(2ndsneg(x0, x1), y1)) → A__TIMES(a__2ndsneg(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(pi(y0), pi(x0))) → A__2NDSPOS(a__pi(mark(y0)), a__pi(mark(x0)))

MARK(2ndsneg(square(y0), 2ndsneg(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndsneg(from(y0), cons(x0, x1))) → A__2NDSNEG(a__from(mark(y0)), cons(mark(x0), x1))

MARK(2ndspos(s(y0), pi(x0))) → A__2NDSPOS(s(mark(y0)), a__pi(mark(x0)))

MARK(plus(square(x0), y1)) → A__PLUS(a__square(mark(x0)), mark(y1))

MARK(cons2(X1, X2)) → MARK(X2)

MARK(2ndsneg(2ndspos(y0, y1), times(x0, x1))) → A__2NDSNEG(a__2ndspos(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndsneg(square(y0), cons2(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), cons2(x0, mark(x1)))

MARK(2ndspos(X1, X2)) → MARK(X1)

MARK(pi(X)) → MARK(X)

MARK(2ndsneg(from(y0), s(x0))) → A__2NDSNEG(a__from(mark(y0)), s(mark(x0)))

MARK(pi(X)) → A__PI(mark(X))

MARK(2ndsneg(square(y0), times(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(plus(y0, y1), cons(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), cons(mark(x0), x1))

MARK(plus(2ndsneg(x0, x1), y1)) → A__PLUS(a__2ndsneg(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(2ndspos(y0, y1), cons2(x0, x1))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(2ndsneg(X1, X2)) → MARK(X2)

MARK(2ndspos(times(y0, y1), square(x0))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndspos(square(y0), times(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(times(square(x0), y1)) → A__TIMES(a__square(mark(x0)), mark(y1))

MARK(2ndspos(pi(y0), cons(x0, x1))) → A__2NDSPOS(a__pi(mark(y0)), cons(mark(x0), x1))

A__PI(X) → MARK(X)

MARK(times(s(x0), y1)) → A__TIMES(s(mark(x0)), mark(y1))

MARK(2ndspos(plus(y0, y1), 2ndspos(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndspos(2ndspos(y0, y1), plus(x0, x1))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

A__PLUS(s(X), Y) → MARK(Y)

MARK(2ndsneg(plus(y0, y1), pi(x0))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__pi(mark(x0)))

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(N)

MARK(2ndspos(2ndspos(y0, y1), square(x0))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndspos(square(y0), 2ndspos(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndspos(times(y0, y1), cons(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), cons(mark(x0), x1))

MARK(2ndspos(s(y0), 2ndspos(x0, x1))) → A__2NDSPOS(s(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndspos(2ndsneg(y0, y1), 2ndsneg(x0, x1))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndsneg(from(y0), 2ndspos(x0, x1))) → A__2NDSNEG(a__from(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndsneg(2ndspos(y0, y1), from(x0))) → A__2NDSNEG(a__2ndspos(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndspos(plus(y0, y1), times(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(times(y0, y1), plus(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(2ndsneg(2ndspos(y0, y1), cons2(x0, x1))) → A__2NDSNEG(a__2ndspos(mark(y0), mark(y1)), cons2(x0, mark(x1)))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(Y)

MARK(rcons(X1, X2)) → MARK(X1)

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(Z)

MARK(square(X)) → MARK(X)

MARK(2ndspos(times(y0, y1), pi(x0))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__pi(mark(x0)))

MARK(2ndspos(pi(y0), times(x0, x1))) → A__2NDSPOS(a__pi(mark(y0)), a__times(mark(x0), mark(x1)))

A__PI(s(x0)) → A__2NDSPOS(s(mark(x0)), a__from(0))

MARK(times(plus(x0, x1), y1)) → A__TIMES(a__plus(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(2ndspos(y0, y1), times(x0, x1))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(times(y0, y1), 2ndspos(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__2ndspos(mark(x0), mark(x1)))

A__2NDSPOS(s(y0), cons(y1, from(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__from(mark(x0))))

MARK(2ndsneg(from(y0), cons2(x0, x1))) → A__2NDSNEG(a__from(mark(y0)), cons2(x0, mark(x1)))

MARK(2ndspos(s(y0), square(x0))) → A__2NDSPOS(s(mark(y0)), a__square(mark(x0)))

MARK(times(2ndspos(x0, x1), y1)) → A__TIMES(a__2ndspos(mark(x0), mark(x1)), mark(y1))

MARK(2ndsneg(from(y0), 2ndsneg(x0, x1))) → A__2NDSNEG(a__from(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndspos(s(y0), cons2(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(x0, mark(x1)))

MARK(2ndsneg(plus(y0, y1), cons2(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), cons2(x0, mark(x1)))

A__PLUS(s(X), Y) → MARK(X)

MARK(times(X1, X2)) → MARK(X1)

A__SQUARE(X) → MARK(X)

MARK(2ndsneg(times(x0, x1), y1)) → A__2NDSNEG(a__times(mark(x0), mark(x1)), mark(y1))

MARK(negrecip(X)) → MARK(X)

MARK(2ndspos(times(y0, y1), from(x0))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndspos(from(y0), times(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(times(y0, y1), 2ndsneg(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndspos(2ndsneg(y0, y1), times(x0, x1))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(from(X)) → MARK(X)

MARK(2ndsneg(from(y0), rnil)) → A__2NDSNEG(a__from(mark(y0)), rnil)

A__SQUARE(X) → A__TIMES(mark(X), mark(X))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(Z)

MARK(2ndsneg(plus(y0, y1), plus(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(2ndsneg(from(y0), negrecip(x0))) → A__2NDSNEG(a__from(mark(y0)), negrecip(mark(x0)))

MARK(2ndspos(plus(y0, y1), from(x0))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndspos(from(y0), plus(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(2ndspos(y0, y1), from(x0))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndspos(from(y0), 2ndspos(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndspos(plus(y0, y1), square(x0))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndspos(square(y0), plus(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__plus(mark(x0), mark(x1)))

A__PI(times(x0, x1)) → A__2NDSPOS(a__times(mark(x0), mark(x1)), a__from(0))

A__2NDSPOS(s(y0), cons(y1, 2ndspos(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__2ndspos(mark(x0), mark(x1))))

A__2NDSNEG(s(N), cons(X, Z)) → A__2NDSNEG(s(mark(N)), cons2(X, mark(Z)))

MARK(2ndspos(pi(y0), cons2(x0, x1))) → A__2NDSPOS(a__pi(mark(y0)), cons2(x0, mark(x1)))

MARK(2ndspos(X1, X2)) → MARK(X2)

MARK(2ndspos(from(y0), cons2(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), cons2(x0, mark(x1)))

MARK(2ndsneg(from(y0), pi(x0))) → A__2NDSNEG(a__from(mark(y0)), a__pi(mark(x0)))

MARK(2ndsneg(2ndspos(y0, y1), 2ndspos(x0, x1))) → A__2NDSNEG(a__2ndspos(mark(y0), mark(y1)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndspos(s(y0), from(x0))) → A__2NDSPOS(s(mark(y0)), a__from(mark(x0)))

MARK(2ndspos(pi(y0), 2ndsneg(x0, x1))) → A__2NDSPOS(a__pi(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndspos(2ndsneg(y0, y1), pi(x0))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__pi(mark(x0)))

A__2NDSPOS(s(y0), cons(y1, plus(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__plus(mark(x0), mark(x1))))

MARK(2ndspos(pi(y0), from(x0))) → A__2NDSPOS(a__pi(mark(y0)), a__from(mark(x0)))

MARK(2ndspos(from(y0), pi(x0))) → A__2NDSPOS(a__from(mark(y0)), a__pi(mark(x0)))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → A__2NDSNEG(mark(N), mark(Z))

MARK(2ndspos(from(y0), from(x0))) → A__2NDSPOS(a__from(mark(y0)), a__from(mark(x0)))

MARK(2ndspos(times(y0, y1), times(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

A__TIMES(s(X), Y) → A__PLUS(mark(Y), a__times(mark(X), mark(Y)))

A__PI(2ndspos(x0, x1)) → A__2NDSPOS(a__2ndspos(mark(x0), mark(x1)), a__from(0))

MARK(2ndspos(square(y0), cons2(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), cons2(x0, mark(x1)))

MARK(2ndsneg(from(y0), posrecip(x0))) → A__2NDSNEG(a__from(mark(y0)), posrecip(mark(x0)))

MARK(plus(s(x0), y1)) → A__PLUS(s(mark(x0)), mark(y1))

A__PI(plus(x0, x1)) → A__2NDSPOS(a__plus(mark(x0), mark(x1)), a__from(0))

MARK(2ndspos(square(y0), cons(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), cons(mark(x0), x1))

MARK(2ndsneg(plus(y0, y1), square(x0))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndsneg(square(y0), plus(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(s(X)) → MARK(X)

MARK(2ndspos(plus(y0, y1), cons2(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(plus(X1, X2)) → MARK(X2)

A__2NDSPOS(s(N), cons(X, Z)) → MARK(Z)

MARK(2ndsneg(from(y0), times(x0, x1))) → A__2NDSNEG(a__from(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(times(y0, y1), cons2(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(2ndsneg(from(y0), square(x0))) → A__2NDSNEG(a__from(mark(y0)), a__square(mark(x0)))

MARK(2ndsneg(square(y0), from(x0))) → A__2NDSNEG(a__square(mark(y0)), a__from(mark(x0)))

MARK(2ndspos(2ndsneg(y0, y1), cons2(x0, x1))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), cons2(x0, mark(x1)))

A__PI(X) → A__FROM(0)

MARK(from(X)) → A__FROM(mark(X))

MARK(2ndspos(s(y0), plus(x0, x1))) → A__2NDSPOS(s(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(plus(X1, X2)) → MARK(X1)

MARK(times(from(x0), y1)) → A__TIMES(a__from(mark(x0)), mark(y1))

A__2NDSPOS(s(y0), cons(y1, times(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__times(mark(x0), mark(x1))))

MARK(2ndspos(s(y0), cons(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons(mark(x0), x1))

MARK(2ndsneg(plus(y0, y1), 2ndspos(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndsneg(2ndspos(y0, y1), plus(x0, x1))) → A__2NDSNEG(a__2ndspos(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(2ndspos(y0, y1), pi(x0))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), a__pi(mark(x0)))

MARK(2ndspos(pi(y0), 2ndspos(x0, x1))) → A__2NDSPOS(a__pi(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndsneg(s(x0), y1)) → A__2NDSNEG(s(mark(x0)), mark(y1))

MARK(2ndspos(from(y0), cons(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), cons(mark(x0), x1))

MARK(2ndsneg(plus(y0, y1), cons(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), cons(mark(x0), x1))

MARK(2ndsneg(2ndspos(y0, y1), cons(x0, x1))) → A__2NDSNEG(a__2ndspos(mark(y0), mark(y1)), cons(mark(x0), x1))

A__TIMES(s(X), Y) → MARK(X)

A__2NDSPOS(s(y0), cons(y1, pi(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__pi(mark(x0))))

MARK(2ndspos(plus(y0, y1), pi(x0))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__pi(mark(x0)))

MARK(2ndspos(pi(y0), plus(x0, x1))) → A__2NDSPOS(a__pi(mark(y0)), a__plus(mark(x0), mark(x1)))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(N)

MARK(plus(2ndspos(x0, x1), y1)) → A__PLUS(a__2ndspos(mark(x0), mark(x1)), mark(y1))

A__PI(from(x0)) → A__2NDSPOS(a__from(mark(x0)), a__from(0))

MARK(cons(X1, X2)) → MARK(X1)

MARK(2ndsneg(plus(y0, y1), from(x0))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndsneg(from(y0), plus(x0, x1))) → A__2NDSNEG(a__from(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(times(X1, X2)) → MARK(X2)

MARK(2ndsneg(from(y0), 0)) → A__2NDSNEG(a__from(mark(y0)), 0)

A__PI(square(x0)) → A__2NDSPOS(a__square(mark(x0)), a__from(0))

MARK(2ndsneg(square(y0), cons(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), cons(mark(x0), x1))

MARK(2ndspos(s(y0), times(x0, x1))) → A__2NDSPOS(s(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndsneg(X1, X2)) → MARK(X1)

MARK(2ndspos(s(y0), 2ndsneg(x0, x1))) → A__2NDSPOS(s(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndsneg(square(y0), square(x0))) → A__2NDSNEG(a__square(mark(y0)), a__square(mark(x0)))

MARK(2ndspos(plus(y0, y1), 2ndsneg(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndspos(2ndsneg(y0, y1), plus(x0, x1))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

A__PI(2ndsneg(x0, x1)) → A__2NDSPOS(a__2ndsneg(mark(x0), mark(x1)), a__from(0))

MARK(plus(pi(x0), y1)) → A__PLUS(a__pi(mark(x0)), mark(y1))

MARK(times(pi(x0), y1)) → A__TIMES(a__pi(mark(x0)), mark(y1))

MARK(2ndspos(2ndspos(y0, y1), 2ndspos(x0, x1))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndsneg(square(y0), pi(x0))) → A__2NDSNEG(a__square(mark(y0)), a__pi(mark(x0)))

MARK(2ndspos(2ndspos(y0, y1), 2ndsneg(x0, x1))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndspos(2ndsneg(y0, y1), 2ndspos(x0, x1))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__2ndspos(mark(x0), mark(x1)))

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(Y)

MARK(2ndspos(2ndspos(y0, y1), cons(x0, x1))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), cons(mark(x0), x1))

A__FROM(X) → MARK(X)

MARK(2ndsneg(from(y0), rcons(x0, x1))) → A__2NDSNEG(a__from(mark(y0)), rcons(mark(x0), mark(x1)))

MARK(2ndspos(pi(y0), square(x0))) → A__2NDSPOS(a__pi(mark(y0)), a__square(mark(x0)))

MARK(2ndspos(square(y0), pi(x0))) → A__2NDSPOS(a__square(mark(y0)), a__pi(mark(x0)))

MARK(rcons(X1, X2)) → MARK(X2)

A__TIMES(s(X), Y) → A__TIMES(mark(X), mark(Y))

A__2NDSPOS(s(y0), cons(y1, cons(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, cons(mark(x0), x1)))

MARK(plus(times(x0, x1), y1)) → A__PLUS(a__times(mark(x0), mark(x1)), mark(y1))

MARK(2ndsneg(from(y0), from(x0))) → A__2NDSNEG(a__from(mark(y0)), a__from(mark(x0)))

A__2NDSPOS(s(y0), cons(y1, 2ndsneg(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__2ndsneg(mark(x0), mark(x1))))

A__2NDSPOS(s(N), cons(X, Z)) → MARK(N)

MARK(2ndsneg(plus(y0, y1), 2ndsneg(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__2ndsneg(mark(x0), mark(x1)))

A__PLUS(0, Y) → MARK(Y)

MARK(posrecip(X)) → MARK(X)

MARK(2ndsneg(pi(x0), y1)) → A__2NDSNEG(a__pi(mark(x0)), mark(y1))

MARK(2ndspos(square(y0), from(x0))) → A__2NDSPOS(a__square(mark(y0)), a__from(mark(x0)))

MARK(2ndspos(from(y0), square(x0))) → A__2NDSPOS(a__from(mark(y0)), a__square(mark(x0)))

A__2NDSPOS(s(y0), cons(y1, square(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__square(mark(x0))))

MARK(plus(0, y1)) → A__PLUS(0, mark(y1))

A__PLUS(s(X), Y) → A__PLUS(mark(X), mark(Y))

MARK(2ndspos(square(y0), square(x0))) → A__2NDSPOS(a__square(mark(y0)), a__square(mark(x0)))

A__PI(pi(x0)) → A__2NDSPOS(a__pi(mark(x0)), a__from(0))

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → A__2NDSPOS(mark(N), mark(Z))

MARK(plus(from(x0), y1)) → A__PLUS(a__from(mark(x0)), mark(y1))

MARK(square(X)) → A__SQUARE(mark(X))

MARK(2ndsneg(plus(y0, y1), times(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(plus(y0, y1), plus(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(2ndsneg(y0, y1), cons(x0, x1))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), cons(mark(x0), x1))

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))

a__2ndspos(0, Z) → rnil

a__2ndspos(s(N), cons(X, Z)) → a__2ndspos(s(mark(N)), cons2(X, mark(Z)))

a__2ndspos(s(N), cons2(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))

a__2ndsneg(0, Z) → rnil

a__2ndsneg(s(N), cons(X, Z)) → a__2ndsneg(s(mark(N)), cons2(X, mark(Z)))

a__2ndsneg(s(N), cons2(X, cons(Y, Z))) → rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z)))

a__pi(X) → a__2ndspos(mark(X), a__from(0))

a__plus(0, Y) → mark(Y)

a__plus(s(X), Y) → s(a__plus(mark(X), mark(Y)))

a__times(0, Y) → 0

a__times(s(X), Y) → a__plus(mark(Y), a__times(mark(X), mark(Y)))

a__square(X) → a__times(mark(X), mark(X))

mark(from(X)) → a__from(mark(X))

mark(2ndspos(X1, X2)) → a__2ndspos(mark(X1), mark(X2))

mark(2ndsneg(X1, X2)) → a__2ndsneg(mark(X1), mark(X2))

mark(pi(X)) → a__pi(mark(X))

mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))

mark(times(X1, X2)) → a__times(mark(X1), mark(X2))

mark(square(X)) → a__square(mark(X))

mark(0) → 0

mark(s(X)) → s(mark(X))

mark(posrecip(X)) → posrecip(mark(X))

mark(negrecip(X)) → negrecip(mark(X))

mark(nil) → nil

mark(cons(X1, X2)) → cons(mark(X1), X2)

mark(cons2(X1, X2)) → cons2(X1, mark(X2))

mark(rnil) → rnil

mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))

a__from(X) → from(X)

a__2ndspos(X1, X2) → 2ndspos(X1, X2)

a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)

a__pi(X) → pi(X)

a__plus(X1, X2) → plus(X1, X2)

a__times(X1, X2) → times(X1, X2)

a__square(X) → square(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 7 less nodes.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ Narrowing
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ QDP
              ↳ Narrowing
                ↳ QDP
                  ↳ DependencyGraphProof
                    ↳ QDP
                      ↳ Narrowing
                        ↳ QDP
                          ↳ DependencyGraphProof
                            ↳ QDP
                              ↳ Narrowing
                                ↳ QDP
                                  ↳ DependencyGraphProof
                                    ↳ QDP
                                      ↳ Narrowing
                                        ↳ QDP
                                          ↳ DependencyGraphProof
                                            ↳ QDP
                                              ↳ Narrowing
                                                ↳ QDP
                                                  ↳ DependencyGraphProof
                                                    ↳ QDP
                                                      ↳ Narrowing
                                                        ↳ QDP
                                                          ↳ DependencyGraphProof
                                                            ↳ QDP
                                                              ↳ Narrowing
                                                                ↳ QDP
                                                                  ↳ DependencyGraphProof
                                                                    ↳ QDP
                                                                      ↳ Narrowing
                                                                        ↳ QDP
                                                                          ↳ DependencyGraphProof
                                                                            ↳ QDP
                                                                              ↳ Narrowing
                                                                                ↳ QDP
                                                                                  ↳ DependencyGraphProof
                                                                                    ↳ QDP
                                                                                      ↳ Narrowing
                                                                                        ↳ QDP
                                                                                          ↳ DependencyGraphProof
                                                                                            ↳ QDP
                                                                                              ↳ Narrowing
                                                                                                ↳ QDP
                                                                                                  ↳ DependencyGraphProof
                                                                                                    ↳ QDP
                                                                                                      ↳ Narrowing
                                                                                                        ↳ QDP
                                                                                                          ↳ DependencyGraphProof
                                                                                                            ↳ QDP
                                                                                                              ↳ Narrowing
                                                                                                                ↳ QDP
                                                                                                                  ↳ DependencyGraphProof
                                                                                                                    ↳ QDP
                                                                                                                      ↳ Narrowing
                                                                                                                        ↳ QDP
                                                                                                                          ↳ DependencyGraphProof
                                                                                                                            ↳ QDP
                                                                                                                              ↳ Narrowing
                                                                                                                                ↳ QDP
                                                                                                                                  ↳ DependencyGraphProof
                                                                                                                                    ↳ QDP
                                                                                                                                      ↳ Narrowing
                                                                                                                                        ↳ QDP
                                                                                                                                          ↳ DependencyGraphProof
                                                                                                                                            ↳ QDP
                                                                                                                                              ↳ Narrowing
                                                                                                                                                ↳ QDP
                                                                                                                                                  ↳ DependencyGraphProof
                                                                                                                                                    ↳ QDP
                                                                                                                                                      ↳ Narrowing

Q DP problem:
The TRS P consists of the following rules:

MARK(2ndspos(plus(y0, y1), square(x0))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndspos(square(y0), plus(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__plus(mark(x0), mark(x1)))

A__PI(times(x0, x1)) → A__2NDSPOS(a__times(mark(x0), mark(x1)), a__from(0))

A__2NDSPOS(s(y0), cons(y1, 2ndspos(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__2ndspos(mark(x0), mark(x1))))

A__2NDSNEG(s(N), cons(X, Z)) → A__2NDSNEG(s(mark(N)), cons2(X, mark(Z)))

MARK(2ndspos(pi(y0), cons2(x0, x1))) → A__2NDSPOS(a__pi(mark(y0)), cons2(x0, mark(x1)))

MARK(2ndspos(X1, X2)) → MARK(X2)

MARK(times(times(x0, x1), y1)) → A__TIMES(a__times(mark(x0), mark(x1)), mark(y1))

MARK(2ndsneg(2ndspos(y0, y1), 2ndsneg(x0, x1))) → A__2NDSNEG(a__2ndspos(mark(y0), mark(y1)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndspos(from(y0), cons2(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), cons2(x0, mark(x1)))

MARK(2ndsneg(from(y0), pi(x0))) → A__2NDSNEG(a__from(mark(y0)), a__pi(mark(x0)))

MARK(2ndsneg(2ndspos(y0, y1), 2ndspos(x0, x1))) → A__2NDSNEG(a__2ndspos(mark(y0), mark(y1)), a__2ndspos(mark(x0), mark(x1)))

MARK(plus(plus(x0, x1), y1)) → A__PLUS(a__plus(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(2ndsneg(y0, y1), from(x0))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndspos(from(y0), 2ndsneg(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndspos(s(y0), from(x0))) → A__2NDSPOS(s(mark(y0)), a__from(mark(x0)))

MARK(2ndspos(pi(y0), 2ndsneg(x0, x1))) → A__2NDSPOS(a__pi(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndspos(2ndsneg(y0, y1), pi(x0))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__pi(mark(x0)))

MARK(2ndsneg(2ndsneg(x0, x1), y1)) → A__2NDSNEG(a__2ndsneg(mark(x0), mark(x1)), mark(y1))

MARK(2ndsneg(2ndspos(y0, y1), square(x0))) → A__2NDSNEG(a__2ndspos(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndsneg(square(y0), 2ndspos(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

A__2NDSNEG(s(N), cons(X, Z)) → MARK(Z)

MARK(2ndsneg(2ndspos(y0, y1), pi(x0))) → A__2NDSNEG(a__2ndspos(mark(y0), mark(y1)), a__pi(mark(x0)))

A__TIMES(s(X), Y) → MARK(Y)

A__2NDSPOS(s(y0), cons(y1, plus(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__plus(mark(x0), mark(x1))))

MARK(2ndspos(pi(y0), from(x0))) → A__2NDSPOS(a__pi(mark(y0)), a__from(mark(x0)))

MARK(2ndspos(2ndsneg(y0, y1), square(x0))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndspos(square(y0), 2ndsneg(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndspos(from(y0), pi(x0))) → A__2NDSPOS(a__from(mark(y0)), a__pi(mark(x0)))

A__2NDSNEG(s(N), cons(X, Z)) → MARK(N)

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → A__2NDSNEG(mark(N), mark(Z))

MARK(times(2ndsneg(x0, x1), y1)) → A__TIMES(a__2ndsneg(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(from(y0), from(x0))) → A__2NDSPOS(a__from(mark(y0)), a__from(mark(x0)))

MARK(2ndspos(pi(y0), pi(x0))) → A__2NDSPOS(a__pi(mark(y0)), a__pi(mark(x0)))

MARK(2ndsneg(square(y0), 2ndsneg(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndsneg(from(y0), cons(x0, x1))) → A__2NDSNEG(a__from(mark(y0)), cons(mark(x0), x1))

MARK(2ndspos(times(y0, y1), times(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(s(y0), pi(x0))) → A__2NDSPOS(s(mark(y0)), a__pi(mark(x0)))

A__TIMES(s(X), Y) → A__PLUS(mark(Y), a__times(mark(X), mark(Y)))

A__PI(2ndspos(x0, x1)) → A__2NDSPOS(a__2ndspos(mark(x0), mark(x1)), a__from(0))

MARK(2ndspos(square(y0), cons2(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), cons2(x0, mark(x1)))

MARK(plus(square(x0), y1)) → A__PLUS(a__square(mark(x0)), mark(y1))

MARK(plus(s(x0), y1)) → A__PLUS(s(mark(x0)), mark(y1))

A__PI(plus(x0, x1)) → A__2NDSPOS(a__plus(mark(x0), mark(x1)), a__from(0))

MARK(2ndspos(square(y0), cons(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), cons(mark(x0), x1))

MARK(cons2(X1, X2)) → MARK(X2)

MARK(2ndsneg(2ndspos(y0, y1), times(x0, x1))) → A__2NDSNEG(a__2ndspos(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndsneg(square(y0), cons2(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), cons2(x0, mark(x1)))

MARK(2ndsneg(plus(y0, y1), square(x0))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndsneg(square(y0), plus(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(X1, X2)) → MARK(X1)

MARK(s(X)) → MARK(X)

MARK(2ndspos(plus(y0, y1), cons2(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(pi(X)) → MARK(X)

MARK(plus(X1, X2)) → MARK(X2)

A__2NDSPOS(s(N), cons(X, Z)) → MARK(Z)

MARK(pi(X)) → A__PI(mark(X))

MARK(2ndsneg(from(y0), times(x0, x1))) → A__2NDSNEG(a__from(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndsneg(square(y0), times(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(plus(y0, y1), cons(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), cons(mark(x0), x1))

MARK(plus(2ndsneg(x0, x1), y1)) → A__PLUS(a__2ndsneg(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(times(y0, y1), cons2(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(2ndspos(2ndspos(y0, y1), cons2(x0, x1))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(2ndsneg(X1, X2)) → MARK(X2)

MARK(2ndsneg(from(y0), square(x0))) → A__2NDSNEG(a__from(mark(y0)), a__square(mark(x0)))

MARK(2ndsneg(square(y0), from(x0))) → A__2NDSNEG(a__square(mark(y0)), a__from(mark(x0)))

MARK(2ndspos(times(y0, y1), square(x0))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndspos(square(y0), times(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(2ndsneg(y0, y1), cons2(x0, x1))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(times(square(x0), y1)) → A__TIMES(a__square(mark(x0)), mark(y1))

A__PI(X) → A__FROM(0)

MARK(from(X)) → A__FROM(mark(X))

MARK(2ndspos(s(y0), plus(x0, x1))) → A__2NDSPOS(s(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(pi(y0), cons(x0, x1))) → A__2NDSPOS(a__pi(mark(y0)), cons(mark(x0), x1))

MARK(plus(X1, X2)) → MARK(X1)

MARK(times(from(x0), y1)) → A__TIMES(a__from(mark(x0)), mark(y1))

A__2NDSPOS(s(y0), cons(y1, times(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__times(mark(x0), mark(x1))))

A__PI(X) → MARK(X)

MARK(2ndspos(s(y0), cons(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons(mark(x0), x1))

MARK(2ndsneg(plus(y0, y1), 2ndspos(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndsneg(2ndspos(y0, y1), plus(x0, x1))) → A__2NDSNEG(a__2ndspos(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(times(s(x0), y1)) → A__TIMES(s(mark(x0)), mark(y1))

MARK(2ndspos(2ndspos(y0, y1), pi(x0))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), a__pi(mark(x0)))

MARK(2ndspos(pi(y0), 2ndspos(x0, x1))) → A__2NDSPOS(a__pi(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndsneg(s(x0), y1)) → A__2NDSNEG(s(mark(x0)), mark(y1))

MARK(2ndspos(from(y0), cons(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), cons(mark(x0), x1))

MARK(2ndsneg(plus(y0, y1), cons(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), cons(mark(x0), x1))

MARK(2ndsneg(2ndspos(y0, y1), cons(x0, x1))) → A__2NDSNEG(a__2ndspos(mark(y0), mark(y1)), cons(mark(x0), x1))

MARK(2ndspos(plus(y0, y1), 2ndspos(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndspos(2ndspos(y0, y1), plus(x0, x1))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

A__PLUS(s(X), Y) → MARK(Y)

A__TIMES(s(X), Y) → MARK(X)

A__2NDSPOS(s(y0), cons(y1, pi(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__pi(mark(x0))))

MARK(2ndsneg(plus(y0, y1), pi(x0))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__pi(mark(x0)))

MARK(2ndspos(plus(y0, y1), pi(x0))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__pi(mark(x0)))

MARK(2ndspos(pi(y0), plus(x0, x1))) → A__2NDSPOS(a__pi(mark(y0)), a__plus(mark(x0), mark(x1)))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(N)

MARK(plus(2ndspos(x0, x1), y1)) → A__PLUS(a__2ndspos(mark(x0), mark(x1)), mark(y1))

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(N)

A__PI(from(x0)) → A__2NDSPOS(a__from(mark(x0)), a__from(0))

MARK(cons(X1, X2)) → MARK(X1)

MARK(2ndsneg(from(y0), plus(x0, x1))) → A__2NDSNEG(a__from(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(2ndsneg(plus(y0, y1), from(x0))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndspos(2ndspos(y0, y1), square(x0))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndspos(square(y0), 2ndspos(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndspos(times(y0, y1), cons(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), cons(mark(x0), x1))

MARK(times(X1, X2)) → MARK(X2)

MARK(2ndspos(s(y0), 2ndspos(x0, x1))) → A__2NDSPOS(s(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

A__PI(square(x0)) → A__2NDSPOS(a__square(mark(x0)), a__from(0))

MARK(2ndspos(2ndsneg(y0, y1), 2ndsneg(x0, x1))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndsneg(from(y0), 2ndspos(x0, x1))) → A__2NDSNEG(a__from(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndsneg(2ndspos(y0, y1), from(x0))) → A__2NDSNEG(a__2ndspos(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndsneg(square(y0), cons(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), cons(mark(x0), x1))

MARK(2ndspos(plus(y0, y1), times(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(times(y0, y1), plus(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(s(y0), times(x0, x1))) → A__2NDSPOS(s(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndsneg(X1, X2)) → MARK(X1)

MARK(2ndsneg(2ndspos(y0, y1), cons2(x0, x1))) → A__2NDSNEG(a__2ndspos(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(2ndspos(s(y0), 2ndsneg(x0, x1))) → A__2NDSPOS(s(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndsneg(square(y0), square(x0))) → A__2NDSNEG(a__square(mark(y0)), a__square(mark(x0)))

MARK(2ndspos(plus(y0, y1), 2ndsneg(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndspos(2ndsneg(y0, y1), plus(x0, x1))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

A__PI(2ndsneg(x0, x1)) → A__2NDSPOS(a__2ndsneg(mark(x0), mark(x1)), a__from(0))

MARK(plus(pi(x0), y1)) → A__PLUS(a__pi(mark(x0)), mark(y1))

MARK(times(pi(x0), y1)) → A__TIMES(a__pi(mark(x0)), mark(y1))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(Y)

MARK(2ndspos(2ndspos(y0, y1), 2ndspos(x0, x1))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), a__2ndspos(mark(x0), mark(x1)))

MARK(rcons(X1, X2)) → MARK(X1)

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(Z)

MARK(2ndsneg(square(y0), pi(x0))) → A__2NDSNEG(a__square(mark(y0)), a__pi(mark(x0)))

MARK(square(X)) → MARK(X)

MARK(2ndspos(2ndspos(y0, y1), 2ndsneg(x0, x1))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndspos(2ndsneg(y0, y1), 2ndspos(x0, x1))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndspos(times(y0, y1), pi(x0))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__pi(mark(x0)))

MARK(2ndspos(pi(y0), times(x0, x1))) → A__2NDSPOS(a__pi(mark(y0)), a__times(mark(x0), mark(x1)))

A__PI(s(x0)) → A__2NDSPOS(s(mark(x0)), a__from(0))

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(Y)

MARK(2ndspos(2ndspos(y0, y1), cons(x0, x1))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), cons(mark(x0), x1))

A__FROM(X) → MARK(X)

MARK(times(plus(x0, x1), y1)) → A__TIMES(a__plus(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(2ndspos(y0, y1), times(x0, x1))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(times(y0, y1), 2ndspos(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__2ndspos(mark(x0), mark(x1)))

A__2NDSPOS(s(y0), cons(y1, from(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__from(mark(x0))))

MARK(2ndsneg(from(y0), cons2(x0, x1))) → A__2NDSNEG(a__from(mark(y0)), cons2(x0, mark(x1)))

MARK(2ndspos(pi(y0), square(x0))) → A__2NDSPOS(a__pi(mark(y0)), a__square(mark(x0)))

MARK(2ndspos(square(y0), pi(x0))) → A__2NDSPOS(a__square(mark(y0)), a__pi(mark(x0)))

MARK(2ndspos(s(y0), square(x0))) → A__2NDSPOS(s(mark(y0)), a__square(mark(x0)))

MARK(rcons(X1, X2)) → MARK(X2)

MARK(times(2ndspos(x0, x1), y1)) → A__TIMES(a__2ndspos(mark(x0), mark(x1)), mark(y1))

MARK(2ndsneg(from(y0), 2ndsneg(x0, x1))) → A__2NDSNEG(a__from(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndspos(s(y0), cons2(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(x0, mark(x1)))

MARK(2ndsneg(plus(y0, y1), cons2(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), cons2(x0, mark(x1)))

A__TIMES(s(X), Y) → A__TIMES(mark(X), mark(Y))

A__PLUS(s(X), Y) → MARK(X)

A__2NDSPOS(s(y0), cons(y1, cons(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, cons(mark(x0), x1)))

MARK(times(X1, X2)) → MARK(X1)

MARK(plus(times(x0, x1), y1)) → A__PLUS(a__times(mark(x0), mark(x1)), mark(y1))

A__SQUARE(X) → MARK(X)

MARK(2ndsneg(from(y0), from(x0))) → A__2NDSNEG(a__from(mark(y0)), a__from(mark(x0)))

MARK(2ndsneg(times(x0, x1), y1)) → A__2NDSNEG(a__times(mark(x0), mark(x1)), mark(y1))

A__2NDSPOS(s(y0), cons(y1, 2ndsneg(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__2ndsneg(mark(x0), mark(x1))))

A__2NDSPOS(s(N), cons(X, Z)) → MARK(N)

MARK(negrecip(X)) → MARK(X)

MARK(2ndsneg(plus(y0, y1), 2ndsneg(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__2ndsneg(mark(x0), mark(x1)))

A__PLUS(0, Y) → MARK(Y)

MARK(posrecip(X)) → MARK(X)

MARK(2ndsneg(pi(x0), y1)) → A__2NDSNEG(a__pi(mark(x0)), mark(y1))

MARK(2ndspos(square(y0), from(x0))) → A__2NDSPOS(a__square(mark(y0)), a__from(mark(x0)))

MARK(2ndspos(from(y0), square(x0))) → A__2NDSPOS(a__from(mark(y0)), a__square(mark(x0)))

MARK(2ndspos(times(y0, y1), from(x0))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndspos(from(y0), times(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), a__times(mark(x0), mark(x1)))

A__2NDSPOS(s(y0), cons(y1, square(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__square(mark(x0))))

MARK(2ndspos(times(y0, y1), 2ndsneg(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndspos(2ndsneg(y0, y1), times(x0, x1))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(plus(0, y1)) → A__PLUS(0, mark(y1))

MARK(from(X)) → MARK(X)

A__PLUS(s(X), Y) → A__PLUS(mark(X), mark(Y))

A__SQUARE(X) → A__TIMES(mark(X), mark(X))

MARK(2ndspos(square(y0), square(x0))) → A__2NDSPOS(a__square(mark(y0)), a__square(mark(x0)))

A__PI(pi(x0)) → A__2NDSPOS(a__pi(mark(x0)), a__from(0))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(Z)

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → A__2NDSPOS(mark(N), mark(Z))

MARK(2ndsneg(plus(y0, y1), plus(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(plus(from(x0), y1)) → A__PLUS(a__from(mark(x0)), mark(y1))

MARK(square(X)) → A__SQUARE(mark(X))

MARK(2ndspos(plus(y0, y1), from(x0))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndspos(from(y0), plus(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(2ndsneg(plus(y0, y1), times(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(plus(y0, y1), plus(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(2ndspos(y0, y1), from(x0))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndspos(from(y0), 2ndspos(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndspos(2ndsneg(y0, y1), cons(x0, x1))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), cons(mark(x0), x1))

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))

a__2ndspos(0, Z) → rnil

a__2ndspos(s(N), cons(X, Z)) → a__2ndspos(s(mark(N)), cons2(X, mark(Z)))

a__2ndspos(s(N), cons2(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))

a__2ndsneg(0, Z) → rnil

a__2ndsneg(s(N), cons(X, Z)) → a__2ndsneg(s(mark(N)), cons2(X, mark(Z)))

a__2ndsneg(s(N), cons2(X, cons(Y, Z))) → rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z)))

a__pi(X) → a__2ndspos(mark(X), a__from(0))

a__plus(0, Y) → mark(Y)

a__plus(s(X), Y) → s(a__plus(mark(X), mark(Y)))

a__times(0, Y) → 0

a__times(s(X), Y) → a__plus(mark(Y), a__times(mark(X), mark(Y)))

a__square(X) → a__times(mark(X), mark(X))

mark(from(X)) → a__from(mark(X))

mark(2ndspos(X1, X2)) → a__2ndspos(mark(X1), mark(X2))

mark(2ndsneg(X1, X2)) → a__2ndsneg(mark(X1), mark(X2))

mark(pi(X)) → a__pi(mark(X))

mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))

mark(times(X1, X2)) → a__times(mark(X1), mark(X2))

mark(square(X)) → a__square(mark(X))

mark(0) → 0

mark(s(X)) → s(mark(X))

mark(posrecip(X)) → posrecip(mark(X))

mark(negrecip(X)) → negrecip(mark(X))

mark(nil) → nil

mark(cons(X1, X2)) → cons(mark(X1), X2)

mark(cons2(X1, X2)) → cons2(X1, mark(X2))

mark(rnil) → rnil

mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))

a__from(X) → from(X)

a__2ndspos(X1, X2) → 2ndspos(X1, X2)

a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)

a__pi(X) → pi(X)

a__plus(X1, X2) → plus(X1, X2)

a__times(X1, X2) → times(X1, X2)

a__square(X) → square(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule MARK(2ndsneg(times(x0, x1), y1)) → A__2NDSNEG(a__times(mark(x0), mark(x1)), mark(y1)) at position [1] we obtained the following new rules:

MARK(2ndsneg(times(y0, y1), 2ndspos(x0, x1))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndsneg(times(y0, y1), rnil)) → A__2NDSNEG(a__times(mark(y0), mark(y1)), rnil)

MARK(2ndsneg(times(y0, y1), square(x0))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndsneg(times(y0, y1), from(x0))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndsneg(times(y0, y1), s(x0))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), s(mark(x0)))

MARK(2ndsneg(times(y0, y1), times(x0, x1))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndsneg(times(y0, y1), nil)) → A__2NDSNEG(a__times(mark(y0), mark(y1)), nil)

MARK(2ndsneg(times(y0, y1), posrecip(x0))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), posrecip(mark(x0)))

MARK(2ndsneg(times(y0, y1), pi(x0))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__pi(mark(x0)))

MARK(2ndsneg(times(y0, y1), 0)) → A__2NDSNEG(a__times(mark(y0), mark(y1)), 0)

MARK(2ndsneg(times(y0, y1), plus(x0, x1))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(2ndsneg(times(y0, y1), negrecip(x0))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), negrecip(mark(x0)))

MARK(2ndsneg(times(y0, y1), 2ndsneg(x0, x1))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndsneg(times(y0, y1), rcons(x0, x1))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), rcons(mark(x0), mark(x1)))

MARK(2ndsneg(times(y0, y1), cons(x0, x1))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), cons(mark(x0), x1))

MARK(2ndsneg(times(y0, y1), cons2(x0, x1))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), cons2(x0, mark(x1)))



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ Narrowing
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ QDP
              ↳ Narrowing
                ↳ QDP
                  ↳ DependencyGraphProof
                    ↳ QDP
                      ↳ Narrowing
                        ↳ QDP
                          ↳ DependencyGraphProof
                            ↳ QDP
                              ↳ Narrowing
                                ↳ QDP
                                  ↳ DependencyGraphProof
                                    ↳ QDP
                                      ↳ Narrowing
                                        ↳ QDP
                                          ↳ DependencyGraphProof
                                            ↳ QDP
                                              ↳ Narrowing
                                                ↳ QDP
                                                  ↳ DependencyGraphProof
                                                    ↳ QDP
                                                      ↳ Narrowing
                                                        ↳ QDP
                                                          ↳ DependencyGraphProof
                                                            ↳ QDP
                                                              ↳ Narrowing
                                                                ↳ QDP
                                                                  ↳ DependencyGraphProof
                                                                    ↳ QDP
                                                                      ↳ Narrowing
                                                                        ↳ QDP
                                                                          ↳ DependencyGraphProof
                                                                            ↳ QDP
                                                                              ↳ Narrowing
                                                                                ↳ QDP
                                                                                  ↳ DependencyGraphProof
                                                                                    ↳ QDP
                                                                                      ↳ Narrowing
                                                                                        ↳ QDP
                                                                                          ↳ DependencyGraphProof
                                                                                            ↳ QDP
                                                                                              ↳ Narrowing
                                                                                                ↳ QDP
                                                                                                  ↳ DependencyGraphProof
                                                                                                    ↳ QDP
                                                                                                      ↳ Narrowing
                                                                                                        ↳ QDP
                                                                                                          ↳ DependencyGraphProof
                                                                                                            ↳ QDP
                                                                                                              ↳ Narrowing
                                                                                                                ↳ QDP
                                                                                                                  ↳ DependencyGraphProof
                                                                                                                    ↳ QDP
                                                                                                                      ↳ Narrowing
                                                                                                                        ↳ QDP
                                                                                                                          ↳ DependencyGraphProof
                                                                                                                            ↳ QDP
                                                                                                                              ↳ Narrowing
                                                                                                                                ↳ QDP
                                                                                                                                  ↳ DependencyGraphProof
                                                                                                                                    ↳ QDP
                                                                                                                                      ↳ Narrowing
                                                                                                                                        ↳ QDP
                                                                                                                                          ↳ DependencyGraphProof
                                                                                                                                            ↳ QDP
                                                                                                                                              ↳ Narrowing
                                                                                                                                                ↳ QDP
                                                                                                                                                  ↳ DependencyGraphProof
                                                                                                                                                    ↳ QDP
                                                                                                                                                      ↳ Narrowing
                                                                                                                                                        ↳ QDP
                                                                                                                                                          ↳ DependencyGraphProof

Q DP problem:
The TRS P consists of the following rules:

MARK(times(times(x0, x1), y1)) → A__TIMES(a__times(mark(x0), mark(x1)), mark(y1))

MARK(2ndsneg(2ndspos(y0, y1), 2ndsneg(x0, x1))) → A__2NDSNEG(a__2ndspos(mark(y0), mark(y1)), a__2ndsneg(mark(x0), mark(x1)))

MARK(plus(plus(x0, x1), y1)) → A__PLUS(a__plus(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(from(y0), 2ndsneg(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndspos(2ndsneg(y0, y1), from(x0))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndsneg(2ndsneg(x0, x1), y1)) → A__2NDSNEG(a__2ndsneg(mark(x0), mark(x1)), mark(y1))

A__2NDSNEG(s(N), cons(X, Z)) → MARK(Z)

MARK(2ndsneg(square(y0), 2ndspos(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndsneg(2ndspos(y0, y1), square(x0))) → A__2NDSNEG(a__2ndspos(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndsneg(2ndspos(y0, y1), pi(x0))) → A__2NDSNEG(a__2ndspos(mark(y0), mark(y1)), a__pi(mark(x0)))

A__TIMES(s(X), Y) → MARK(Y)

MARK(2ndspos(square(y0), 2ndsneg(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndspos(2ndsneg(y0, y1), square(x0))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__square(mark(x0)))

A__2NDSNEG(s(N), cons(X, Z)) → MARK(N)

MARK(times(2ndsneg(x0, x1), y1)) → A__TIMES(a__2ndsneg(mark(x0), mark(x1)), mark(y1))

MARK(2ndsneg(times(y0, y1), times(x0, x1))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(pi(y0), pi(x0))) → A__2NDSPOS(a__pi(mark(y0)), a__pi(mark(x0)))

MARK(2ndsneg(square(y0), 2ndsneg(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndsneg(from(y0), cons(x0, x1))) → A__2NDSNEG(a__from(mark(y0)), cons(mark(x0), x1))

MARK(2ndspos(s(y0), pi(x0))) → A__2NDSPOS(s(mark(y0)), a__pi(mark(x0)))

MARK(plus(square(x0), y1)) → A__PLUS(a__square(mark(x0)), mark(y1))

MARK(2ndsneg(times(y0, y1), cons2(x0, x1))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(cons2(X1, X2)) → MARK(X2)

MARK(2ndsneg(times(y0, y1), 2ndspos(x0, x1))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndsneg(square(y0), cons2(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), cons2(x0, mark(x1)))

MARK(2ndsneg(2ndspos(y0, y1), times(x0, x1))) → A__2NDSNEG(a__2ndspos(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(X1, X2)) → MARK(X1)

MARK(pi(X)) → MARK(X)

MARK(pi(X)) → A__PI(mark(X))

MARK(2ndsneg(times(y0, y1), square(x0))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndspos(plus(y0, y1), cons(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), cons(mark(x0), x1))

MARK(2ndsneg(square(y0), times(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(plus(2ndsneg(x0, x1), y1)) → A__PLUS(a__2ndsneg(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(2ndspos(y0, y1), cons2(x0, x1))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(2ndsneg(X1, X2)) → MARK(X2)

MARK(2ndspos(square(y0), times(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(times(y0, y1), square(x0))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(times(square(x0), y1)) → A__TIMES(a__square(mark(x0)), mark(y1))

MARK(2ndspos(pi(y0), cons(x0, x1))) → A__2NDSPOS(a__pi(mark(y0)), cons(mark(x0), x1))

A__PI(X) → MARK(X)

MARK(times(s(x0), y1)) → A__TIMES(s(mark(x0)), mark(y1))

MARK(2ndspos(2ndspos(y0, y1), plus(x0, x1))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(plus(y0, y1), 2ndspos(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__2ndspos(mark(x0), mark(x1)))

A__PLUS(s(X), Y) → MARK(Y)

MARK(2ndsneg(plus(y0, y1), pi(x0))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__pi(mark(x0)))

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(N)

MARK(2ndspos(square(y0), 2ndspos(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndspos(2ndspos(y0, y1), square(x0))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndspos(s(y0), 2ndspos(x0, x1))) → A__2NDSPOS(s(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndspos(times(y0, y1), cons(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), cons(mark(x0), x1))

MARK(2ndspos(2ndsneg(y0, y1), 2ndsneg(x0, x1))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndspos(times(y0, y1), plus(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(plus(y0, y1), times(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndsneg(2ndspos(y0, y1), from(x0))) → A__2NDSNEG(a__2ndspos(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndsneg(from(y0), 2ndspos(x0, x1))) → A__2NDSNEG(a__from(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndsneg(2ndspos(y0, y1), cons2(x0, x1))) → A__2NDSNEG(a__2ndspos(mark(y0), mark(y1)), cons2(x0, mark(x1)))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(Y)

MARK(rcons(X1, X2)) → MARK(X1)

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(Z)

MARK(square(X)) → MARK(X)

MARK(2ndspos(pi(y0), times(x0, x1))) → A__2NDSPOS(a__pi(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(times(y0, y1), pi(x0))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__pi(mark(x0)))

A__PI(s(x0)) → A__2NDSPOS(s(mark(x0)), a__from(0))

MARK(times(plus(x0, x1), y1)) → A__TIMES(a__plus(mark(x0), mark(x1)), mark(y1))

A__2NDSPOS(s(y0), cons(y1, from(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__from(mark(x0))))

MARK(2ndspos(times(y0, y1), 2ndspos(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndspos(2ndspos(y0, y1), times(x0, x1))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndsneg(times(y0, y1), 0)) → A__2NDSNEG(a__times(mark(y0), mark(y1)), 0)

MARK(2ndsneg(from(y0), cons2(x0, x1))) → A__2NDSNEG(a__from(mark(y0)), cons2(x0, mark(x1)))

MARK(2ndspos(s(y0), square(x0))) → A__2NDSPOS(s(mark(y0)), a__square(mark(x0)))

MARK(times(2ndspos(x0, x1), y1)) → A__TIMES(a__2ndspos(mark(x0), mark(x1)), mark(y1))

MARK(2ndsneg(from(y0), 2ndsneg(x0, x1))) → A__2NDSNEG(a__from(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndspos(s(y0), cons2(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(x0, mark(x1)))

MARK(2ndsneg(plus(y0, y1), cons2(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), cons2(x0, mark(x1)))

A__PLUS(s(X), Y) → MARK(X)

MARK(times(X1, X2)) → MARK(X1)

A__SQUARE(X) → MARK(X)

MARK(negrecip(X)) → MARK(X)

MARK(2ndspos(from(y0), times(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(times(y0, y1), from(x0))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndspos(2ndsneg(y0, y1), times(x0, x1))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(times(y0, y1), 2ndsneg(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__2ndsneg(mark(x0), mark(x1)))

MARK(from(X)) → MARK(X)

A__SQUARE(X) → A__TIMES(mark(X), mark(X))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(Z)

MARK(2ndsneg(times(y0, y1), nil)) → A__2NDSNEG(a__times(mark(y0), mark(y1)), nil)

MARK(2ndsneg(plus(y0, y1), plus(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(from(y0), plus(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(plus(y0, y1), from(x0))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndsneg(times(y0, y1), negrecip(x0))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), negrecip(mark(x0)))

MARK(2ndsneg(times(y0, y1), 2ndsneg(x0, x1))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndsneg(times(y0, y1), cons(x0, x1))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), cons(mark(x0), x1))

MARK(2ndspos(from(y0), 2ndspos(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndspos(2ndspos(y0, y1), from(x0))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndspos(square(y0), plus(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(plus(y0, y1), square(x0))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__square(mark(x0)))

A__2NDSPOS(s(y0), cons(y1, 2ndspos(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__2ndspos(mark(x0), mark(x1))))

A__PI(times(x0, x1)) → A__2NDSPOS(a__times(mark(x0), mark(x1)), a__from(0))

A__2NDSNEG(s(N), cons(X, Z)) → A__2NDSNEG(s(mark(N)), cons2(X, mark(Z)))

MARK(2ndspos(X1, X2)) → MARK(X2)

MARK(2ndspos(pi(y0), cons2(x0, x1))) → A__2NDSPOS(a__pi(mark(y0)), cons2(x0, mark(x1)))

MARK(2ndspos(from(y0), cons2(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), cons2(x0, mark(x1)))

MARK(2ndsneg(from(y0), pi(x0))) → A__2NDSNEG(a__from(mark(y0)), a__pi(mark(x0)))

MARK(2ndsneg(2ndspos(y0, y1), 2ndspos(x0, x1))) → A__2NDSNEG(a__2ndspos(mark(y0), mark(y1)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndspos(s(y0), from(x0))) → A__2NDSPOS(s(mark(y0)), a__from(mark(x0)))

MARK(2ndspos(2ndsneg(y0, y1), pi(x0))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__pi(mark(x0)))

MARK(2ndspos(pi(y0), 2ndsneg(x0, x1))) → A__2NDSPOS(a__pi(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

A__2NDSPOS(s(y0), cons(y1, plus(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__plus(mark(x0), mark(x1))))

MARK(2ndspos(from(y0), pi(x0))) → A__2NDSPOS(a__from(mark(y0)), a__pi(mark(x0)))

MARK(2ndspos(pi(y0), from(x0))) → A__2NDSPOS(a__pi(mark(y0)), a__from(mark(x0)))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → A__2NDSNEG(mark(N), mark(Z))

MARK(2ndspos(from(y0), from(x0))) → A__2NDSPOS(a__from(mark(y0)), a__from(mark(x0)))

MARK(2ndspos(times(y0, y1), times(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

A__TIMES(s(X), Y) → A__PLUS(mark(Y), a__times(mark(X), mark(Y)))

MARK(2ndspos(square(y0), cons2(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), cons2(x0, mark(x1)))

A__PI(2ndspos(x0, x1)) → A__2NDSPOS(a__2ndspos(mark(x0), mark(x1)), a__from(0))

MARK(plus(s(x0), y1)) → A__PLUS(s(mark(x0)), mark(y1))

MARK(2ndspos(square(y0), cons(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), cons(mark(x0), x1))

A__PI(plus(x0, x1)) → A__2NDSPOS(a__plus(mark(x0), mark(x1)), a__from(0))

MARK(s(X)) → MARK(X)

MARK(2ndsneg(square(y0), plus(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(2ndsneg(plus(y0, y1), square(x0))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndspos(plus(y0, y1), cons2(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(plus(X1, X2)) → MARK(X2)

A__2NDSPOS(s(N), cons(X, Z)) → MARK(Z)

MARK(2ndsneg(times(y0, y1), from(x0))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndsneg(from(y0), times(x0, x1))) → A__2NDSNEG(a__from(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(times(y0, y1), cons2(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(2ndsneg(square(y0), from(x0))) → A__2NDSNEG(a__square(mark(y0)), a__from(mark(x0)))

MARK(2ndsneg(from(y0), square(x0))) → A__2NDSNEG(a__from(mark(y0)), a__square(mark(x0)))

MARK(2ndspos(2ndsneg(y0, y1), cons2(x0, x1))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(2ndspos(s(y0), plus(x0, x1))) → A__2NDSPOS(s(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(from(X)) → A__FROM(mark(X))

A__PI(X) → A__FROM(0)

MARK(times(from(x0), y1)) → A__TIMES(a__from(mark(x0)), mark(y1))

MARK(plus(X1, X2)) → MARK(X1)

A__2NDSPOS(s(y0), cons(y1, times(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__times(mark(x0), mark(x1))))

MARK(2ndspos(s(y0), cons(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons(mark(x0), x1))

MARK(2ndsneg(2ndspos(y0, y1), plus(x0, x1))) → A__2NDSNEG(a__2ndspos(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(2ndsneg(plus(y0, y1), 2ndspos(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndspos(pi(y0), 2ndspos(x0, x1))) → A__2NDSPOS(a__pi(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndspos(2ndspos(y0, y1), pi(x0))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), a__pi(mark(x0)))

MARK(2ndspos(from(y0), cons(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), cons(mark(x0), x1))

MARK(2ndsneg(s(x0), y1)) → A__2NDSNEG(s(mark(x0)), mark(y1))

MARK(2ndsneg(2ndspos(y0, y1), cons(x0, x1))) → A__2NDSNEG(a__2ndspos(mark(y0), mark(y1)), cons(mark(x0), x1))

MARK(2ndsneg(plus(y0, y1), cons(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), cons(mark(x0), x1))

A__2NDSPOS(s(y0), cons(y1, pi(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__pi(mark(x0))))

A__TIMES(s(X), Y) → MARK(X)

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(N)

MARK(2ndspos(pi(y0), plus(x0, x1))) → A__2NDSPOS(a__pi(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(plus(y0, y1), pi(x0))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__pi(mark(x0)))

MARK(plus(2ndspos(x0, x1), y1)) → A__PLUS(a__2ndspos(mark(x0), mark(x1)), mark(y1))

MARK(cons(X1, X2)) → MARK(X1)

A__PI(from(x0)) → A__2NDSPOS(a__from(mark(x0)), a__from(0))

MARK(2ndsneg(times(y0, y1), rnil)) → A__2NDSNEG(a__times(mark(y0), mark(y1)), rnil)

MARK(2ndsneg(plus(y0, y1), from(x0))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndsneg(from(y0), plus(x0, x1))) → A__2NDSNEG(a__from(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(times(X1, X2)) → MARK(X2)

A__PI(square(x0)) → A__2NDSPOS(a__square(mark(x0)), a__from(0))

MARK(2ndsneg(square(y0), cons(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), cons(mark(x0), x1))

MARK(2ndspos(s(y0), times(x0, x1))) → A__2NDSPOS(s(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndsneg(X1, X2)) → MARK(X1)

MARK(2ndspos(s(y0), 2ndsneg(x0, x1))) → A__2NDSPOS(s(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndsneg(square(y0), square(x0))) → A__2NDSNEG(a__square(mark(y0)), a__square(mark(x0)))

MARK(2ndspos(2ndsneg(y0, y1), plus(x0, x1))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(plus(y0, y1), 2ndsneg(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__2ndsneg(mark(x0), mark(x1)))

A__PI(2ndsneg(x0, x1)) → A__2NDSPOS(a__2ndsneg(mark(x0), mark(x1)), a__from(0))

MARK(times(pi(x0), y1)) → A__TIMES(a__pi(mark(x0)), mark(y1))

MARK(plus(pi(x0), y1)) → A__PLUS(a__pi(mark(x0)), mark(y1))

MARK(2ndspos(2ndspos(y0, y1), 2ndspos(x0, x1))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndsneg(square(y0), pi(x0))) → A__2NDSNEG(a__square(mark(y0)), a__pi(mark(x0)))

MARK(2ndspos(2ndsneg(y0, y1), 2ndspos(x0, x1))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndspos(2ndspos(y0, y1), 2ndsneg(x0, x1))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), a__2ndsneg(mark(x0), mark(x1)))

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(Y)

A__FROM(X) → MARK(X)

MARK(2ndspos(2ndspos(y0, y1), cons(x0, x1))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), cons(mark(x0), x1))

MARK(2ndspos(square(y0), pi(x0))) → A__2NDSPOS(a__square(mark(y0)), a__pi(mark(x0)))

MARK(2ndspos(pi(y0), square(x0))) → A__2NDSPOS(a__pi(mark(y0)), a__square(mark(x0)))

MARK(2ndsneg(times(y0, y1), rcons(x0, x1))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), rcons(mark(x0), mark(x1)))

MARK(rcons(X1, X2)) → MARK(X2)

A__TIMES(s(X), Y) → A__TIMES(mark(X), mark(Y))

A__2NDSPOS(s(y0), cons(y1, cons(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, cons(mark(x0), x1)))

MARK(plus(times(x0, x1), y1)) → A__PLUS(a__times(mark(x0), mark(x1)), mark(y1))

MARK(2ndsneg(from(y0), from(x0))) → A__2NDSNEG(a__from(mark(y0)), a__from(mark(x0)))

A__2NDSPOS(s(y0), cons(y1, 2ndsneg(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__2ndsneg(mark(x0), mark(x1))))

A__2NDSPOS(s(N), cons(X, Z)) → MARK(N)

MARK(2ndsneg(plus(y0, y1), 2ndsneg(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__2ndsneg(mark(x0), mark(x1)))

A__PLUS(0, Y) → MARK(Y)

MARK(posrecip(X)) → MARK(X)

MARK(2ndspos(from(y0), square(x0))) → A__2NDSPOS(a__from(mark(y0)), a__square(mark(x0)))

MARK(2ndspos(square(y0), from(x0))) → A__2NDSPOS(a__square(mark(y0)), a__from(mark(x0)))

MARK(2ndsneg(pi(x0), y1)) → A__2NDSNEG(a__pi(mark(x0)), mark(y1))

A__2NDSPOS(s(y0), cons(y1, square(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__square(mark(x0))))

MARK(plus(0, y1)) → A__PLUS(0, mark(y1))

MARK(2ndsneg(times(y0, y1), s(x0))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), s(mark(x0)))

A__PLUS(s(X), Y) → A__PLUS(mark(X), mark(Y))

MARK(2ndspos(square(y0), square(x0))) → A__2NDSPOS(a__square(mark(y0)), a__square(mark(x0)))

A__PI(pi(x0)) → A__2NDSPOS(a__pi(mark(x0)), a__from(0))

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → A__2NDSPOS(mark(N), mark(Z))

MARK(2ndsneg(times(y0, y1), posrecip(x0))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), posrecip(mark(x0)))

MARK(plus(from(x0), y1)) → A__PLUS(a__from(mark(x0)), mark(y1))

MARK(2ndsneg(times(y0, y1), pi(x0))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__pi(mark(x0)))

MARK(square(X)) → A__SQUARE(mark(X))

MARK(2ndsneg(times(y0, y1), plus(x0, x1))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(2ndsneg(plus(y0, y1), times(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(plus(y0, y1), plus(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(2ndsneg(y0, y1), cons(x0, x1))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), cons(mark(x0), x1))

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))

a__2ndspos(0, Z) → rnil

a__2ndspos(s(N), cons(X, Z)) → a__2ndspos(s(mark(N)), cons2(X, mark(Z)))

a__2ndspos(s(N), cons2(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))

a__2ndsneg(0, Z) → rnil

a__2ndsneg(s(N), cons(X, Z)) → a__2ndsneg(s(mark(N)), cons2(X, mark(Z)))

a__2ndsneg(s(N), cons2(X, cons(Y, Z))) → rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z)))

a__pi(X) → a__2ndspos(mark(X), a__from(0))

a__plus(0, Y) → mark(Y)

a__plus(s(X), Y) → s(a__plus(mark(X), mark(Y)))

a__times(0, Y) → 0

a__times(s(X), Y) → a__plus(mark(Y), a__times(mark(X), mark(Y)))

a__square(X) → a__times(mark(X), mark(X))

mark(from(X)) → a__from(mark(X))

mark(2ndspos(X1, X2)) → a__2ndspos(mark(X1), mark(X2))

mark(2ndsneg(X1, X2)) → a__2ndsneg(mark(X1), mark(X2))

mark(pi(X)) → a__pi(mark(X))

mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))

mark(times(X1, X2)) → a__times(mark(X1), mark(X2))

mark(square(X)) → a__square(mark(X))

mark(0) → 0

mark(s(X)) → s(mark(X))

mark(posrecip(X)) → posrecip(mark(X))

mark(negrecip(X)) → negrecip(mark(X))

mark(nil) → nil

mark(cons(X1, X2)) → cons(mark(X1), X2)

mark(cons2(X1, X2)) → cons2(X1, mark(X2))

mark(rnil) → rnil

mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))

a__from(X) → from(X)

a__2ndspos(X1, X2) → 2ndspos(X1, X2)

a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)

a__pi(X) → pi(X)

a__plus(X1, X2) → plus(X1, X2)

a__times(X1, X2) → times(X1, X2)

a__square(X) → square(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 7 less nodes.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ Narrowing
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ QDP
              ↳ Narrowing
                ↳ QDP
                  ↳ DependencyGraphProof
                    ↳ QDP
                      ↳ Narrowing
                        ↳ QDP
                          ↳ DependencyGraphProof
                            ↳ QDP
                              ↳ Narrowing
                                ↳ QDP
                                  ↳ DependencyGraphProof
                                    ↳ QDP
                                      ↳ Narrowing
                                        ↳ QDP
                                          ↳ DependencyGraphProof
                                            ↳ QDP
                                              ↳ Narrowing
                                                ↳ QDP
                                                  ↳ DependencyGraphProof
                                                    ↳ QDP
                                                      ↳ Narrowing
                                                        ↳ QDP
                                                          ↳ DependencyGraphProof
                                                            ↳ QDP
                                                              ↳ Narrowing
                                                                ↳ QDP
                                                                  ↳ DependencyGraphProof
                                                                    ↳ QDP
                                                                      ↳ Narrowing
                                                                        ↳ QDP
                                                                          ↳ DependencyGraphProof
                                                                            ↳ QDP
                                                                              ↳ Narrowing
                                                                                ↳ QDP
                                                                                  ↳ DependencyGraphProof
                                                                                    ↳ QDP
                                                                                      ↳ Narrowing
                                                                                        ↳ QDP
                                                                                          ↳ DependencyGraphProof
                                                                                            ↳ QDP
                                                                                              ↳ Narrowing
                                                                                                ↳ QDP
                                                                                                  ↳ DependencyGraphProof
                                                                                                    ↳ QDP
                                                                                                      ↳ Narrowing
                                                                                                        ↳ QDP
                                                                                                          ↳ DependencyGraphProof
                                                                                                            ↳ QDP
                                                                                                              ↳ Narrowing
                                                                                                                ↳ QDP
                                                                                                                  ↳ DependencyGraphProof
                                                                                                                    ↳ QDP
                                                                                                                      ↳ Narrowing
                                                                                                                        ↳ QDP
                                                                                                                          ↳ DependencyGraphProof
                                                                                                                            ↳ QDP
                                                                                                                              ↳ Narrowing
                                                                                                                                ↳ QDP
                                                                                                                                  ↳ DependencyGraphProof
                                                                                                                                    ↳ QDP
                                                                                                                                      ↳ Narrowing
                                                                                                                                        ↳ QDP
                                                                                                                                          ↳ DependencyGraphProof
                                                                                                                                            ↳ QDP
                                                                                                                                              ↳ Narrowing
                                                                                                                                                ↳ QDP
                                                                                                                                                  ↳ DependencyGraphProof
                                                                                                                                                    ↳ QDP
                                                                                                                                                      ↳ Narrowing
                                                                                                                                                        ↳ QDP
                                                                                                                                                          ↳ DependencyGraphProof
                                                                                                                                                            ↳ QDP
                                                                                                                                                              ↳ Narrowing

Q DP problem:
The TRS P consists of the following rules:

MARK(times(times(x0, x1), y1)) → A__TIMES(a__times(mark(x0), mark(x1)), mark(y1))

MARK(2ndsneg(2ndspos(y0, y1), 2ndsneg(x0, x1))) → A__2NDSNEG(a__2ndspos(mark(y0), mark(y1)), a__2ndsneg(mark(x0), mark(x1)))

MARK(plus(plus(x0, x1), y1)) → A__PLUS(a__plus(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(from(y0), 2ndsneg(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndspos(2ndsneg(y0, y1), from(x0))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndsneg(2ndsneg(x0, x1), y1)) → A__2NDSNEG(a__2ndsneg(mark(x0), mark(x1)), mark(y1))

A__2NDSNEG(s(N), cons(X, Z)) → MARK(Z)

MARK(2ndsneg(square(y0), 2ndspos(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndsneg(2ndspos(y0, y1), square(x0))) → A__2NDSNEG(a__2ndspos(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndsneg(2ndspos(y0, y1), pi(x0))) → A__2NDSNEG(a__2ndspos(mark(y0), mark(y1)), a__pi(mark(x0)))

A__TIMES(s(X), Y) → MARK(Y)

MARK(2ndspos(square(y0), 2ndsneg(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndspos(2ndsneg(y0, y1), square(x0))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__square(mark(x0)))

A__2NDSNEG(s(N), cons(X, Z)) → MARK(N)

MARK(times(2ndsneg(x0, x1), y1)) → A__TIMES(a__2ndsneg(mark(x0), mark(x1)), mark(y1))

MARK(2ndsneg(times(y0, y1), times(x0, x1))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(pi(y0), pi(x0))) → A__2NDSPOS(a__pi(mark(y0)), a__pi(mark(x0)))

MARK(2ndsneg(square(y0), 2ndsneg(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndsneg(from(y0), cons(x0, x1))) → A__2NDSNEG(a__from(mark(y0)), cons(mark(x0), x1))

MARK(2ndspos(s(y0), pi(x0))) → A__2NDSPOS(s(mark(y0)), a__pi(mark(x0)))

MARK(plus(square(x0), y1)) → A__PLUS(a__square(mark(x0)), mark(y1))

MARK(2ndsneg(times(y0, y1), cons2(x0, x1))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(cons2(X1, X2)) → MARK(X2)

MARK(2ndsneg(square(y0), cons2(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), cons2(x0, mark(x1)))

MARK(2ndsneg(2ndspos(y0, y1), times(x0, x1))) → A__2NDSNEG(a__2ndspos(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndsneg(times(y0, y1), 2ndspos(x0, x1))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndspos(X1, X2)) → MARK(X1)

MARK(pi(X)) → MARK(X)

MARK(pi(X)) → A__PI(mark(X))

MARK(2ndspos(plus(y0, y1), cons(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), cons(mark(x0), x1))

MARK(2ndsneg(square(y0), times(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndsneg(times(y0, y1), square(x0))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(plus(2ndsneg(x0, x1), y1)) → A__PLUS(a__2ndsneg(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(2ndspos(y0, y1), cons2(x0, x1))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(2ndsneg(X1, X2)) → MARK(X2)

MARK(2ndspos(square(y0), times(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(times(y0, y1), square(x0))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(times(square(x0), y1)) → A__TIMES(a__square(mark(x0)), mark(y1))

MARK(2ndspos(pi(y0), cons(x0, x1))) → A__2NDSPOS(a__pi(mark(y0)), cons(mark(x0), x1))

A__PI(X) → MARK(X)

MARK(times(s(x0), y1)) → A__TIMES(s(mark(x0)), mark(y1))

MARK(2ndspos(2ndspos(y0, y1), plus(x0, x1))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(plus(y0, y1), 2ndspos(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__2ndspos(mark(x0), mark(x1)))

A__PLUS(s(X), Y) → MARK(Y)

MARK(2ndsneg(plus(y0, y1), pi(x0))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__pi(mark(x0)))

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(N)

MARK(2ndspos(square(y0), 2ndspos(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndspos(2ndspos(y0, y1), square(x0))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndspos(s(y0), 2ndspos(x0, x1))) → A__2NDSPOS(s(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndspos(times(y0, y1), cons(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), cons(mark(x0), x1))

MARK(2ndspos(2ndsneg(y0, y1), 2ndsneg(x0, x1))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndspos(times(y0, y1), plus(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(plus(y0, y1), times(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndsneg(2ndspos(y0, y1), from(x0))) → A__2NDSNEG(a__2ndspos(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndsneg(from(y0), 2ndspos(x0, x1))) → A__2NDSNEG(a__from(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndsneg(2ndspos(y0, y1), cons2(x0, x1))) → A__2NDSNEG(a__2ndspos(mark(y0), mark(y1)), cons2(x0, mark(x1)))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(Y)

MARK(rcons(X1, X2)) → MARK(X1)

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(Z)

MARK(square(X)) → MARK(X)

MARK(2ndspos(pi(y0), times(x0, x1))) → A__2NDSPOS(a__pi(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(times(y0, y1), pi(x0))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__pi(mark(x0)))

A__PI(s(x0)) → A__2NDSPOS(s(mark(x0)), a__from(0))

MARK(times(plus(x0, x1), y1)) → A__TIMES(a__plus(mark(x0), mark(x1)), mark(y1))

A__2NDSPOS(s(y0), cons(y1, from(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__from(mark(x0))))

MARK(2ndspos(times(y0, y1), 2ndspos(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndspos(2ndspos(y0, y1), times(x0, x1))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndsneg(from(y0), cons2(x0, x1))) → A__2NDSNEG(a__from(mark(y0)), cons2(x0, mark(x1)))

MARK(2ndspos(s(y0), square(x0))) → A__2NDSPOS(s(mark(y0)), a__square(mark(x0)))

MARK(times(2ndspos(x0, x1), y1)) → A__TIMES(a__2ndspos(mark(x0), mark(x1)), mark(y1))

MARK(2ndsneg(from(y0), 2ndsneg(x0, x1))) → A__2NDSNEG(a__from(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndspos(s(y0), cons2(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(x0, mark(x1)))

MARK(2ndsneg(plus(y0, y1), cons2(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), cons2(x0, mark(x1)))

A__PLUS(s(X), Y) → MARK(X)

MARK(times(X1, X2)) → MARK(X1)

A__SQUARE(X) → MARK(X)

MARK(negrecip(X)) → MARK(X)

MARK(2ndspos(from(y0), times(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(times(y0, y1), from(x0))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndspos(2ndsneg(y0, y1), times(x0, x1))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(times(y0, y1), 2ndsneg(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__2ndsneg(mark(x0), mark(x1)))

MARK(from(X)) → MARK(X)

A__SQUARE(X) → A__TIMES(mark(X), mark(X))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(Z)

MARK(2ndsneg(plus(y0, y1), plus(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(from(y0), plus(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(plus(y0, y1), from(x0))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndsneg(times(y0, y1), 2ndsneg(x0, x1))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndsneg(times(y0, y1), cons(x0, x1))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), cons(mark(x0), x1))

MARK(2ndspos(from(y0), 2ndspos(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndspos(2ndspos(y0, y1), from(x0))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndspos(square(y0), plus(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(plus(y0, y1), square(x0))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__square(mark(x0)))

A__2NDSPOS(s(y0), cons(y1, 2ndspos(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__2ndspos(mark(x0), mark(x1))))

A__PI(times(x0, x1)) → A__2NDSPOS(a__times(mark(x0), mark(x1)), a__from(0))

A__2NDSNEG(s(N), cons(X, Z)) → A__2NDSNEG(s(mark(N)), cons2(X, mark(Z)))

MARK(2ndspos(X1, X2)) → MARK(X2)

MARK(2ndspos(pi(y0), cons2(x0, x1))) → A__2NDSPOS(a__pi(mark(y0)), cons2(x0, mark(x1)))

MARK(2ndspos(from(y0), cons2(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), cons2(x0, mark(x1)))

MARK(2ndsneg(from(y0), pi(x0))) → A__2NDSNEG(a__from(mark(y0)), a__pi(mark(x0)))

MARK(2ndsneg(2ndspos(y0, y1), 2ndspos(x0, x1))) → A__2NDSNEG(a__2ndspos(mark(y0), mark(y1)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndspos(s(y0), from(x0))) → A__2NDSPOS(s(mark(y0)), a__from(mark(x0)))

MARK(2ndspos(2ndsneg(y0, y1), pi(x0))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__pi(mark(x0)))

MARK(2ndspos(pi(y0), 2ndsneg(x0, x1))) → A__2NDSPOS(a__pi(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

A__2NDSPOS(s(y0), cons(y1, plus(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__plus(mark(x0), mark(x1))))

MARK(2ndspos(from(y0), pi(x0))) → A__2NDSPOS(a__from(mark(y0)), a__pi(mark(x0)))

MARK(2ndspos(pi(y0), from(x0))) → A__2NDSPOS(a__pi(mark(y0)), a__from(mark(x0)))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → A__2NDSNEG(mark(N), mark(Z))

MARK(2ndspos(from(y0), from(x0))) → A__2NDSPOS(a__from(mark(y0)), a__from(mark(x0)))

MARK(2ndspos(times(y0, y1), times(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

A__TIMES(s(X), Y) → A__PLUS(mark(Y), a__times(mark(X), mark(Y)))

A__PI(2ndspos(x0, x1)) → A__2NDSPOS(a__2ndspos(mark(x0), mark(x1)), a__from(0))

MARK(2ndspos(square(y0), cons2(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), cons2(x0, mark(x1)))

MARK(plus(s(x0), y1)) → A__PLUS(s(mark(x0)), mark(y1))

A__PI(plus(x0, x1)) → A__2NDSPOS(a__plus(mark(x0), mark(x1)), a__from(0))

MARK(2ndspos(square(y0), cons(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), cons(mark(x0), x1))

MARK(s(X)) → MARK(X)

MARK(2ndsneg(square(y0), plus(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(2ndsneg(plus(y0, y1), square(x0))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndspos(plus(y0, y1), cons2(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(plus(X1, X2)) → MARK(X2)

A__2NDSPOS(s(N), cons(X, Z)) → MARK(Z)

MARK(2ndsneg(from(y0), times(x0, x1))) → A__2NDSNEG(a__from(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndsneg(times(y0, y1), from(x0))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndspos(times(y0, y1), cons2(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(2ndsneg(square(y0), from(x0))) → A__2NDSNEG(a__square(mark(y0)), a__from(mark(x0)))

MARK(2ndsneg(from(y0), square(x0))) → A__2NDSNEG(a__from(mark(y0)), a__square(mark(x0)))

MARK(2ndspos(2ndsneg(y0, y1), cons2(x0, x1))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(2ndspos(s(y0), plus(x0, x1))) → A__2NDSPOS(s(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(from(X)) → A__FROM(mark(X))

A__PI(X) → A__FROM(0)

MARK(plus(X1, X2)) → MARK(X1)

MARK(times(from(x0), y1)) → A__TIMES(a__from(mark(x0)), mark(y1))

A__2NDSPOS(s(y0), cons(y1, times(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__times(mark(x0), mark(x1))))

MARK(2ndspos(s(y0), cons(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons(mark(x0), x1))

MARK(2ndsneg(2ndspos(y0, y1), plus(x0, x1))) → A__2NDSNEG(a__2ndspos(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(2ndsneg(plus(y0, y1), 2ndspos(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndspos(pi(y0), 2ndspos(x0, x1))) → A__2NDSPOS(a__pi(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndspos(2ndspos(y0, y1), pi(x0))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), a__pi(mark(x0)))

MARK(2ndspos(from(y0), cons(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), cons(mark(x0), x1))

MARK(2ndsneg(s(x0), y1)) → A__2NDSNEG(s(mark(x0)), mark(y1))

MARK(2ndsneg(2ndspos(y0, y1), cons(x0, x1))) → A__2NDSNEG(a__2ndspos(mark(y0), mark(y1)), cons(mark(x0), x1))

MARK(2ndsneg(plus(y0, y1), cons(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), cons(mark(x0), x1))

A__2NDSPOS(s(y0), cons(y1, pi(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__pi(mark(x0))))

A__TIMES(s(X), Y) → MARK(X)

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(N)

MARK(2ndspos(pi(y0), plus(x0, x1))) → A__2NDSPOS(a__pi(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(plus(y0, y1), pi(x0))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__pi(mark(x0)))

MARK(plus(2ndspos(x0, x1), y1)) → A__PLUS(a__2ndspos(mark(x0), mark(x1)), mark(y1))

A__PI(from(x0)) → A__2NDSPOS(a__from(mark(x0)), a__from(0))

MARK(cons(X1, X2)) → MARK(X1)

MARK(2ndsneg(plus(y0, y1), from(x0))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndsneg(from(y0), plus(x0, x1))) → A__2NDSNEG(a__from(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(times(X1, X2)) → MARK(X2)

A__PI(square(x0)) → A__2NDSPOS(a__square(mark(x0)), a__from(0))

MARK(2ndsneg(square(y0), cons(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), cons(mark(x0), x1))

MARK(2ndspos(s(y0), times(x0, x1))) → A__2NDSPOS(s(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndsneg(X1, X2)) → MARK(X1)

MARK(2ndspos(s(y0), 2ndsneg(x0, x1))) → A__2NDSPOS(s(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndsneg(square(y0), square(x0))) → A__2NDSNEG(a__square(mark(y0)), a__square(mark(x0)))

MARK(2ndspos(2ndsneg(y0, y1), plus(x0, x1))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(plus(y0, y1), 2ndsneg(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__2ndsneg(mark(x0), mark(x1)))

A__PI(2ndsneg(x0, x1)) → A__2NDSPOS(a__2ndsneg(mark(x0), mark(x1)), a__from(0))

MARK(times(pi(x0), y1)) → A__TIMES(a__pi(mark(x0)), mark(y1))

MARK(plus(pi(x0), y1)) → A__PLUS(a__pi(mark(x0)), mark(y1))

MARK(2ndspos(2ndspos(y0, y1), 2ndspos(x0, x1))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndsneg(square(y0), pi(x0))) → A__2NDSNEG(a__square(mark(y0)), a__pi(mark(x0)))

MARK(2ndspos(2ndsneg(y0, y1), 2ndspos(x0, x1))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndspos(2ndspos(y0, y1), 2ndsneg(x0, x1))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), a__2ndsneg(mark(x0), mark(x1)))

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(Y)

A__FROM(X) → MARK(X)

MARK(2ndspos(2ndspos(y0, y1), cons(x0, x1))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), cons(mark(x0), x1))

MARK(2ndspos(square(y0), pi(x0))) → A__2NDSPOS(a__square(mark(y0)), a__pi(mark(x0)))

MARK(2ndspos(pi(y0), square(x0))) → A__2NDSPOS(a__pi(mark(y0)), a__square(mark(x0)))

MARK(rcons(X1, X2)) → MARK(X2)

A__TIMES(s(X), Y) → A__TIMES(mark(X), mark(Y))

A__2NDSPOS(s(y0), cons(y1, cons(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, cons(mark(x0), x1)))

MARK(plus(times(x0, x1), y1)) → A__PLUS(a__times(mark(x0), mark(x1)), mark(y1))

MARK(2ndsneg(from(y0), from(x0))) → A__2NDSNEG(a__from(mark(y0)), a__from(mark(x0)))

A__2NDSPOS(s(y0), cons(y1, 2ndsneg(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__2ndsneg(mark(x0), mark(x1))))

A__2NDSPOS(s(N), cons(X, Z)) → MARK(N)

MARK(2ndsneg(plus(y0, y1), 2ndsneg(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__2ndsneg(mark(x0), mark(x1)))

A__PLUS(0, Y) → MARK(Y)

MARK(posrecip(X)) → MARK(X)

MARK(2ndspos(from(y0), square(x0))) → A__2NDSPOS(a__from(mark(y0)), a__square(mark(x0)))

MARK(2ndspos(square(y0), from(x0))) → A__2NDSPOS(a__square(mark(y0)), a__from(mark(x0)))

MARK(2ndsneg(pi(x0), y1)) → A__2NDSNEG(a__pi(mark(x0)), mark(y1))

A__2NDSPOS(s(y0), cons(y1, square(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__square(mark(x0))))

MARK(plus(0, y1)) → A__PLUS(0, mark(y1))

A__PLUS(s(X), Y) → A__PLUS(mark(X), mark(Y))

MARK(2ndspos(square(y0), square(x0))) → A__2NDSPOS(a__square(mark(y0)), a__square(mark(x0)))

A__PI(pi(x0)) → A__2NDSPOS(a__pi(mark(x0)), a__from(0))

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → A__2NDSPOS(mark(N), mark(Z))

MARK(plus(from(x0), y1)) → A__PLUS(a__from(mark(x0)), mark(y1))

MARK(square(X)) → A__SQUARE(mark(X))

MARK(2ndsneg(times(y0, y1), pi(x0))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__pi(mark(x0)))

MARK(2ndsneg(plus(y0, y1), times(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndsneg(times(y0, y1), plus(x0, x1))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(plus(y0, y1), plus(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(2ndsneg(y0, y1), cons(x0, x1))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), cons(mark(x0), x1))

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))

a__2ndspos(0, Z) → rnil

a__2ndspos(s(N), cons(X, Z)) → a__2ndspos(s(mark(N)), cons2(X, mark(Z)))

a__2ndspos(s(N), cons2(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))

a__2ndsneg(0, Z) → rnil

a__2ndsneg(s(N), cons(X, Z)) → a__2ndsneg(s(mark(N)), cons2(X, mark(Z)))

a__2ndsneg(s(N), cons2(X, cons(Y, Z))) → rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z)))

a__pi(X) → a__2ndspos(mark(X), a__from(0))

a__plus(0, Y) → mark(Y)

a__plus(s(X), Y) → s(a__plus(mark(X), mark(Y)))

a__times(0, Y) → 0

a__times(s(X), Y) → a__plus(mark(Y), a__times(mark(X), mark(Y)))

a__square(X) → a__times(mark(X), mark(X))

mark(from(X)) → a__from(mark(X))

mark(2ndspos(X1, X2)) → a__2ndspos(mark(X1), mark(X2))

mark(2ndsneg(X1, X2)) → a__2ndsneg(mark(X1), mark(X2))

mark(pi(X)) → a__pi(mark(X))

mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))

mark(times(X1, X2)) → a__times(mark(X1), mark(X2))

mark(square(X)) → a__square(mark(X))

mark(0) → 0

mark(s(X)) → s(mark(X))

mark(posrecip(X)) → posrecip(mark(X))

mark(negrecip(X)) → negrecip(mark(X))

mark(nil) → nil

mark(cons(X1, X2)) → cons(mark(X1), X2)

mark(cons2(X1, X2)) → cons2(X1, mark(X2))

mark(rnil) → rnil

mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))

a__from(X) → from(X)

a__2ndspos(X1, X2) → 2ndspos(X1, X2)

a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)

a__pi(X) → pi(X)

a__plus(X1, X2) → plus(X1, X2)

a__times(X1, X2) → times(X1, X2)

a__square(X) → square(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule MARK(2ndsneg(2ndsneg(x0, x1), y1)) → A__2NDSNEG(a__2ndsneg(mark(x0), mark(x1)), mark(y1)) at position [1] we obtained the following new rules:

MARK(2ndsneg(2ndsneg(y0, y1), rnil)) → A__2NDSNEG(a__2ndsneg(mark(y0), mark(y1)), rnil)

MARK(2ndsneg(2ndsneg(y0, y1), 2ndspos(x0, x1))) → A__2NDSNEG(a__2ndsneg(mark(y0), mark(y1)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndsneg(2ndsneg(y0, y1), nil)) → A__2NDSNEG(a__2ndsneg(mark(y0), mark(y1)), nil)

MARK(2ndsneg(2ndsneg(y0, y1), plus(x0, x1))) → A__2NDSNEG(a__2ndsneg(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(2ndsneg(2ndsneg(y0, y1), posrecip(x0))) → A__2NDSNEG(a__2ndsneg(mark(y0), mark(y1)), posrecip(mark(x0)))

MARK(2ndsneg(2ndsneg(y0, y1), negrecip(x0))) → A__2NDSNEG(a__2ndsneg(mark(y0), mark(y1)), negrecip(mark(x0)))

MARK(2ndsneg(2ndsneg(y0, y1), 2ndsneg(x0, x1))) → A__2NDSNEG(a__2ndsneg(mark(y0), mark(y1)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndsneg(2ndsneg(y0, y1), s(x0))) → A__2NDSNEG(a__2ndsneg(mark(y0), mark(y1)), s(mark(x0)))

MARK(2ndsneg(2ndsneg(y0, y1), pi(x0))) → A__2NDSNEG(a__2ndsneg(mark(y0), mark(y1)), a__pi(mark(x0)))

MARK(2ndsneg(2ndsneg(y0, y1), square(x0))) → A__2NDSNEG(a__2ndsneg(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndsneg(2ndsneg(y0, y1), rcons(x0, x1))) → A__2NDSNEG(a__2ndsneg(mark(y0), mark(y1)), rcons(mark(x0), mark(x1)))

MARK(2ndsneg(2ndsneg(y0, y1), 0)) → A__2NDSNEG(a__2ndsneg(mark(y0), mark(y1)), 0)

MARK(2ndsneg(2ndsneg(y0, y1), cons2(x0, x1))) → A__2NDSNEG(a__2ndsneg(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(2ndsneg(2ndsneg(y0, y1), times(x0, x1))) → A__2NDSNEG(a__2ndsneg(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndsneg(2ndsneg(y0, y1), cons(x0, x1))) → A__2NDSNEG(a__2ndsneg(mark(y0), mark(y1)), cons(mark(x0), x1))

MARK(2ndsneg(2ndsneg(y0, y1), from(x0))) → A__2NDSNEG(a__2ndsneg(mark(y0), mark(y1)), a__from(mark(x0)))



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ Narrowing
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ QDP
              ↳ Narrowing
                ↳ QDP
                  ↳ DependencyGraphProof
                    ↳ QDP
                      ↳ Narrowing
                        ↳ QDP
                          ↳ DependencyGraphProof
                            ↳ QDP
                              ↳ Narrowing
                                ↳ QDP
                                  ↳ DependencyGraphProof
                                    ↳ QDP
                                      ↳ Narrowing
                                        ↳ QDP
                                          ↳ DependencyGraphProof
                                            ↳ QDP
                                              ↳ Narrowing
                                                ↳ QDP
                                                  ↳ DependencyGraphProof
                                                    ↳ QDP
                                                      ↳ Narrowing
                                                        ↳ QDP
                                                          ↳ DependencyGraphProof
                                                            ↳ QDP
                                                              ↳ Narrowing
                                                                ↳ QDP
                                                                  ↳ DependencyGraphProof
                                                                    ↳ QDP
                                                                      ↳ Narrowing
                                                                        ↳ QDP
                                                                          ↳ DependencyGraphProof
                                                                            ↳ QDP
                                                                              ↳ Narrowing
                                                                                ↳ QDP
                                                                                  ↳ DependencyGraphProof
                                                                                    ↳ QDP
                                                                                      ↳ Narrowing
                                                                                        ↳ QDP
                                                                                          ↳ DependencyGraphProof
                                                                                            ↳ QDP
                                                                                              ↳ Narrowing
                                                                                                ↳ QDP
                                                                                                  ↳ DependencyGraphProof
                                                                                                    ↳ QDP
                                                                                                      ↳ Narrowing
                                                                                                        ↳ QDP
                                                                                                          ↳ DependencyGraphProof
                                                                                                            ↳ QDP
                                                                                                              ↳ Narrowing
                                                                                                                ↳ QDP
                                                                                                                  ↳ DependencyGraphProof
                                                                                                                    ↳ QDP
                                                                                                                      ↳ Narrowing
                                                                                                                        ↳ QDP
                                                                                                                          ↳ DependencyGraphProof
                                                                                                                            ↳ QDP
                                                                                                                              ↳ Narrowing
                                                                                                                                ↳ QDP
                                                                                                                                  ↳ DependencyGraphProof
                                                                                                                                    ↳ QDP
                                                                                                                                      ↳ Narrowing
                                                                                                                                        ↳ QDP
                                                                                                                                          ↳ DependencyGraphProof
                                                                                                                                            ↳ QDP
                                                                                                                                              ↳ Narrowing
                                                                                                                                                ↳ QDP
                                                                                                                                                  ↳ DependencyGraphProof
                                                                                                                                                    ↳ QDP
                                                                                                                                                      ↳ Narrowing
                                                                                                                                                        ↳ QDP
                                                                                                                                                          ↳ DependencyGraphProof
                                                                                                                                                            ↳ QDP
                                                                                                                                                              ↳ Narrowing
                                                                                                                                                                ↳ QDP
                                                                                                                                                                  ↳ DependencyGraphProof

Q DP problem:
The TRS P consists of the following rules:

MARK(times(times(x0, x1), y1)) → A__TIMES(a__times(mark(x0), mark(x1)), mark(y1))

MARK(2ndsneg(2ndsneg(y0, y1), 2ndspos(x0, x1))) → A__2NDSNEG(a__2ndsneg(mark(y0), mark(y1)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndsneg(2ndspos(y0, y1), 2ndsneg(x0, x1))) → A__2NDSNEG(a__2ndspos(mark(y0), mark(y1)), a__2ndsneg(mark(x0), mark(x1)))

MARK(plus(plus(x0, x1), y1)) → A__PLUS(a__plus(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(2ndsneg(y0, y1), from(x0))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndspos(from(y0), 2ndsneg(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndsneg(2ndspos(y0, y1), square(x0))) → A__2NDSNEG(a__2ndspos(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndsneg(square(y0), 2ndspos(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

A__2NDSNEG(s(N), cons(X, Z)) → MARK(Z)

MARK(2ndsneg(2ndspos(y0, y1), pi(x0))) → A__2NDSNEG(a__2ndspos(mark(y0), mark(y1)), a__pi(mark(x0)))

A__TIMES(s(X), Y) → MARK(Y)

MARK(2ndspos(2ndsneg(y0, y1), square(x0))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndspos(square(y0), 2ndsneg(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

A__2NDSNEG(s(N), cons(X, Z)) → MARK(N)

MARK(times(2ndsneg(x0, x1), y1)) → A__TIMES(a__2ndsneg(mark(x0), mark(x1)), mark(y1))

MARK(2ndsneg(times(y0, y1), times(x0, x1))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(pi(y0), pi(x0))) → A__2NDSPOS(a__pi(mark(y0)), a__pi(mark(x0)))

MARK(2ndsneg(2ndsneg(y0, y1), square(x0))) → A__2NDSNEG(a__2ndsneg(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndsneg(square(y0), 2ndsneg(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndsneg(from(y0), cons(x0, x1))) → A__2NDSNEG(a__from(mark(y0)), cons(mark(x0), x1))

MARK(2ndspos(s(y0), pi(x0))) → A__2NDSPOS(s(mark(y0)), a__pi(mark(x0)))

MARK(plus(square(x0), y1)) → A__PLUS(a__square(mark(x0)), mark(y1))

MARK(2ndsneg(times(y0, y1), cons2(x0, x1))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(cons2(X1, X2)) → MARK(X2)

MARK(2ndsneg(times(y0, y1), 2ndspos(x0, x1))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndsneg(2ndspos(y0, y1), times(x0, x1))) → A__2NDSNEG(a__2ndspos(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndsneg(square(y0), cons2(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), cons2(x0, mark(x1)))

MARK(2ndspos(X1, X2)) → MARK(X1)

MARK(pi(X)) → MARK(X)

MARK(pi(X)) → A__PI(mark(X))

MARK(2ndsneg(times(y0, y1), square(x0))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndsneg(square(y0), times(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(plus(y0, y1), cons(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), cons(mark(x0), x1))

MARK(plus(2ndsneg(x0, x1), y1)) → A__PLUS(a__2ndsneg(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(2ndspos(y0, y1), cons2(x0, x1))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(2ndsneg(X1, X2)) → MARK(X2)

MARK(2ndsneg(2ndsneg(y0, y1), posrecip(x0))) → A__2NDSNEG(a__2ndsneg(mark(y0), mark(y1)), posrecip(mark(x0)))

MARK(2ndspos(times(y0, y1), square(x0))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndspos(square(y0), times(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(times(square(x0), y1)) → A__TIMES(a__square(mark(x0)), mark(y1))

MARK(2ndspos(pi(y0), cons(x0, x1))) → A__2NDSPOS(a__pi(mark(y0)), cons(mark(x0), x1))

A__PI(X) → MARK(X)

MARK(2ndsneg(2ndsneg(y0, y1), rcons(x0, x1))) → A__2NDSNEG(a__2ndsneg(mark(y0), mark(y1)), rcons(mark(x0), mark(x1)))

MARK(times(s(x0), y1)) → A__TIMES(s(mark(x0)), mark(y1))

MARK(2ndspos(plus(y0, y1), 2ndspos(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndspos(2ndspos(y0, y1), plus(x0, x1))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

A__PLUS(s(X), Y) → MARK(Y)

MARK(2ndsneg(plus(y0, y1), pi(x0))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__pi(mark(x0)))

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(N)

MARK(2ndspos(2ndspos(y0, y1), square(x0))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndspos(square(y0), 2ndspos(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndspos(times(y0, y1), cons(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), cons(mark(x0), x1))

MARK(2ndspos(s(y0), 2ndspos(x0, x1))) → A__2NDSPOS(s(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndspos(2ndsneg(y0, y1), 2ndsneg(x0, x1))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndsneg(from(y0), 2ndspos(x0, x1))) → A__2NDSNEG(a__from(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndsneg(2ndspos(y0, y1), from(x0))) → A__2NDSNEG(a__2ndspos(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndspos(plus(y0, y1), times(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(times(y0, y1), plus(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(2ndsneg(2ndspos(y0, y1), cons2(x0, x1))) → A__2NDSNEG(a__2ndspos(mark(y0), mark(y1)), cons2(x0, mark(x1)))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(Y)

MARK(rcons(X1, X2)) → MARK(X1)

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(Z)

MARK(square(X)) → MARK(X)

MARK(2ndspos(times(y0, y1), pi(x0))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__pi(mark(x0)))

MARK(2ndspos(pi(y0), times(x0, x1))) → A__2NDSPOS(a__pi(mark(y0)), a__times(mark(x0), mark(x1)))

A__PI(s(x0)) → A__2NDSPOS(s(mark(x0)), a__from(0))

MARK(2ndsneg(2ndsneg(y0, y1), pi(x0))) → A__2NDSNEG(a__2ndsneg(mark(y0), mark(y1)), a__pi(mark(x0)))

MARK(times(plus(x0, x1), y1)) → A__TIMES(a__plus(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(2ndspos(y0, y1), times(x0, x1))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(times(y0, y1), 2ndspos(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__2ndspos(mark(x0), mark(x1)))

A__2NDSPOS(s(y0), cons(y1, from(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__from(mark(x0))))

MARK(2ndsneg(from(y0), cons2(x0, x1))) → A__2NDSNEG(a__from(mark(y0)), cons2(x0, mark(x1)))

MARK(2ndspos(s(y0), square(x0))) → A__2NDSPOS(s(mark(y0)), a__square(mark(x0)))

MARK(times(2ndspos(x0, x1), y1)) → A__TIMES(a__2ndspos(mark(x0), mark(x1)), mark(y1))

MARK(2ndsneg(2ndsneg(y0, y1), from(x0))) → A__2NDSNEG(a__2ndsneg(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndsneg(from(y0), 2ndsneg(x0, x1))) → A__2NDSNEG(a__from(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndspos(s(y0), cons2(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(x0, mark(x1)))

MARK(2ndsneg(plus(y0, y1), cons2(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), cons2(x0, mark(x1)))

A__PLUS(s(X), Y) → MARK(X)

MARK(times(X1, X2)) → MARK(X1)

A__SQUARE(X) → MARK(X)

MARK(negrecip(X)) → MARK(X)

MARK(2ndspos(times(y0, y1), from(x0))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndspos(from(y0), times(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(times(y0, y1), 2ndsneg(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndspos(2ndsneg(y0, y1), times(x0, x1))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(from(X)) → MARK(X)

A__SQUARE(X) → A__TIMES(mark(X), mark(X))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(Z)

MARK(2ndsneg(plus(y0, y1), plus(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(plus(y0, y1), from(x0))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndspos(from(y0), plus(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(2ndsneg(2ndsneg(y0, y1), 0)) → A__2NDSNEG(a__2ndsneg(mark(y0), mark(y1)), 0)

MARK(2ndsneg(2ndsneg(y0, y1), times(x0, x1))) → A__2NDSNEG(a__2ndsneg(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndsneg(times(y0, y1), 2ndsneg(x0, x1))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndsneg(times(y0, y1), cons(x0, x1))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), cons(mark(x0), x1))

MARK(2ndsneg(2ndsneg(y0, y1), cons(x0, x1))) → A__2NDSNEG(a__2ndsneg(mark(y0), mark(y1)), cons(mark(x0), x1))

MARK(2ndspos(2ndspos(y0, y1), from(x0))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndspos(from(y0), 2ndspos(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndspos(plus(y0, y1), square(x0))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndspos(square(y0), plus(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(2ndsneg(2ndsneg(y0, y1), rnil)) → A__2NDSNEG(a__2ndsneg(mark(y0), mark(y1)), rnil)

A__PI(times(x0, x1)) → A__2NDSPOS(a__times(mark(x0), mark(x1)), a__from(0))

A__2NDSPOS(s(y0), cons(y1, 2ndspos(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__2ndspos(mark(x0), mark(x1))))

A__2NDSNEG(s(N), cons(X, Z)) → A__2NDSNEG(s(mark(N)), cons2(X, mark(Z)))

MARK(2ndspos(pi(y0), cons2(x0, x1))) → A__2NDSPOS(a__pi(mark(y0)), cons2(x0, mark(x1)))

MARK(2ndspos(X1, X2)) → MARK(X2)

MARK(2ndspos(from(y0), cons2(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), cons2(x0, mark(x1)))

MARK(2ndsneg(from(y0), pi(x0))) → A__2NDSNEG(a__from(mark(y0)), a__pi(mark(x0)))

MARK(2ndsneg(2ndspos(y0, y1), 2ndspos(x0, x1))) → A__2NDSNEG(a__2ndspos(mark(y0), mark(y1)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndspos(s(y0), from(x0))) → A__2NDSPOS(s(mark(y0)), a__from(mark(x0)))

MARK(2ndspos(pi(y0), 2ndsneg(x0, x1))) → A__2NDSPOS(a__pi(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndspos(2ndsneg(y0, y1), pi(x0))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__pi(mark(x0)))

A__2NDSPOS(s(y0), cons(y1, plus(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__plus(mark(x0), mark(x1))))

MARK(2ndspos(pi(y0), from(x0))) → A__2NDSPOS(a__pi(mark(y0)), a__from(mark(x0)))

MARK(2ndspos(from(y0), pi(x0))) → A__2NDSPOS(a__from(mark(y0)), a__pi(mark(x0)))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → A__2NDSNEG(mark(N), mark(Z))

MARK(2ndspos(from(y0), from(x0))) → A__2NDSPOS(a__from(mark(y0)), a__from(mark(x0)))

MARK(2ndspos(times(y0, y1), times(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

A__TIMES(s(X), Y) → A__PLUS(mark(Y), a__times(mark(X), mark(Y)))

MARK(2ndspos(square(y0), cons2(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), cons2(x0, mark(x1)))

A__PI(2ndspos(x0, x1)) → A__2NDSPOS(a__2ndspos(mark(x0), mark(x1)), a__from(0))

MARK(plus(s(x0), y1)) → A__PLUS(s(mark(x0)), mark(y1))

MARK(2ndspos(square(y0), cons(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), cons(mark(x0), x1))

A__PI(plus(x0, x1)) → A__2NDSPOS(a__plus(mark(x0), mark(x1)), a__from(0))

MARK(2ndsneg(plus(y0, y1), square(x0))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndsneg(square(y0), plus(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(s(X)) → MARK(X)

MARK(2ndspos(plus(y0, y1), cons2(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(plus(X1, X2)) → MARK(X2)

A__2NDSPOS(s(N), cons(X, Z)) → MARK(Z)

MARK(2ndsneg(times(y0, y1), from(x0))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndsneg(from(y0), times(x0, x1))) → A__2NDSNEG(a__from(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(times(y0, y1), cons2(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(2ndsneg(from(y0), square(x0))) → A__2NDSNEG(a__from(mark(y0)), a__square(mark(x0)))

MARK(2ndsneg(square(y0), from(x0))) → A__2NDSNEG(a__square(mark(y0)), a__from(mark(x0)))

MARK(2ndspos(2ndsneg(y0, y1), cons2(x0, x1))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), cons2(x0, mark(x1)))

A__PI(X) → A__FROM(0)

MARK(from(X)) → A__FROM(mark(X))

MARK(2ndspos(s(y0), plus(x0, x1))) → A__2NDSPOS(s(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(times(from(x0), y1)) → A__TIMES(a__from(mark(x0)), mark(y1))

MARK(plus(X1, X2)) → MARK(X1)

A__2NDSPOS(s(y0), cons(y1, times(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__times(mark(x0), mark(x1))))

MARK(2ndsneg(2ndsneg(y0, y1), 2ndsneg(x0, x1))) → A__2NDSNEG(a__2ndsneg(mark(y0), mark(y1)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndspos(s(y0), cons(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons(mark(x0), x1))

MARK(2ndsneg(plus(y0, y1), 2ndspos(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndsneg(2ndspos(y0, y1), plus(x0, x1))) → A__2NDSNEG(a__2ndspos(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(2ndspos(y0, y1), pi(x0))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), a__pi(mark(x0)))

MARK(2ndspos(pi(y0), 2ndspos(x0, x1))) → A__2NDSPOS(a__pi(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndsneg(s(x0), y1)) → A__2NDSNEG(s(mark(x0)), mark(y1))

MARK(2ndspos(from(y0), cons(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), cons(mark(x0), x1))

MARK(2ndsneg(plus(y0, y1), cons(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), cons(mark(x0), x1))

MARK(2ndsneg(2ndspos(y0, y1), cons(x0, x1))) → A__2NDSNEG(a__2ndspos(mark(y0), mark(y1)), cons(mark(x0), x1))

A__TIMES(s(X), Y) → MARK(X)

A__2NDSPOS(s(y0), cons(y1, pi(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__pi(mark(x0))))

MARK(2ndspos(plus(y0, y1), pi(x0))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__pi(mark(x0)))

MARK(2ndspos(pi(y0), plus(x0, x1))) → A__2NDSPOS(a__pi(mark(y0)), a__plus(mark(x0), mark(x1)))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(N)

MARK(plus(2ndspos(x0, x1), y1)) → A__PLUS(a__2ndspos(mark(x0), mark(x1)), mark(y1))

MARK(cons(X1, X2)) → MARK(X1)

A__PI(from(x0)) → A__2NDSPOS(a__from(mark(x0)), a__from(0))

MARK(2ndsneg(from(y0), plus(x0, x1))) → A__2NDSNEG(a__from(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(2ndsneg(plus(y0, y1), from(x0))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndsneg(2ndsneg(y0, y1), nil)) → A__2NDSNEG(a__2ndsneg(mark(y0), mark(y1)), nil)

MARK(times(X1, X2)) → MARK(X2)

A__PI(square(x0)) → A__2NDSPOS(a__square(mark(x0)), a__from(0))

MARK(2ndsneg(square(y0), cons(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), cons(mark(x0), x1))

MARK(2ndspos(s(y0), times(x0, x1))) → A__2NDSPOS(s(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndsneg(X1, X2)) → MARK(X1)

MARK(2ndspos(s(y0), 2ndsneg(x0, x1))) → A__2NDSPOS(s(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndsneg(square(y0), square(x0))) → A__2NDSNEG(a__square(mark(y0)), a__square(mark(x0)))

MARK(2ndspos(plus(y0, y1), 2ndsneg(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndspos(2ndsneg(y0, y1), plus(x0, x1))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

A__PI(2ndsneg(x0, x1)) → A__2NDSPOS(a__2ndsneg(mark(x0), mark(x1)), a__from(0))

MARK(plus(pi(x0), y1)) → A__PLUS(a__pi(mark(x0)), mark(y1))

MARK(times(pi(x0), y1)) → A__TIMES(a__pi(mark(x0)), mark(y1))

MARK(2ndspos(2ndspos(y0, y1), 2ndspos(x0, x1))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndsneg(square(y0), pi(x0))) → A__2NDSNEG(a__square(mark(y0)), a__pi(mark(x0)))

MARK(2ndspos(2ndspos(y0, y1), 2ndsneg(x0, x1))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndspos(2ndsneg(y0, y1), 2ndspos(x0, x1))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndsneg(2ndsneg(y0, y1), negrecip(x0))) → A__2NDSNEG(a__2ndsneg(mark(y0), mark(y1)), negrecip(mark(x0)))

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(Y)

MARK(2ndsneg(2ndsneg(y0, y1), s(x0))) → A__2NDSNEG(a__2ndsneg(mark(y0), mark(y1)), s(mark(x0)))

MARK(2ndspos(2ndspos(y0, y1), cons(x0, x1))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), cons(mark(x0), x1))

A__FROM(X) → MARK(X)

MARK(2ndspos(pi(y0), square(x0))) → A__2NDSPOS(a__pi(mark(y0)), a__square(mark(x0)))

MARK(2ndspos(square(y0), pi(x0))) → A__2NDSPOS(a__square(mark(y0)), a__pi(mark(x0)))

MARK(rcons(X1, X2)) → MARK(X2)

A__TIMES(s(X), Y) → A__TIMES(mark(X), mark(Y))

A__2NDSPOS(s(y0), cons(y1, cons(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, cons(mark(x0), x1)))

MARK(plus(times(x0, x1), y1)) → A__PLUS(a__times(mark(x0), mark(x1)), mark(y1))

MARK(2ndsneg(from(y0), from(x0))) → A__2NDSNEG(a__from(mark(y0)), a__from(mark(x0)))

A__2NDSPOS(s(y0), cons(y1, 2ndsneg(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__2ndsneg(mark(x0), mark(x1))))

A__2NDSPOS(s(N), cons(X, Z)) → MARK(N)

MARK(2ndsneg(2ndsneg(y0, y1), plus(x0, x1))) → A__2NDSNEG(a__2ndsneg(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(2ndsneg(plus(y0, y1), 2ndsneg(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__2ndsneg(mark(x0), mark(x1)))

A__PLUS(0, Y) → MARK(Y)

MARK(posrecip(X)) → MARK(X)

MARK(2ndsneg(pi(x0), y1)) → A__2NDSNEG(a__pi(mark(x0)), mark(y1))

MARK(2ndspos(square(y0), from(x0))) → A__2NDSPOS(a__square(mark(y0)), a__from(mark(x0)))

MARK(2ndspos(from(y0), square(x0))) → A__2NDSPOS(a__from(mark(y0)), a__square(mark(x0)))

A__2NDSPOS(s(y0), cons(y1, square(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__square(mark(x0))))

MARK(plus(0, y1)) → A__PLUS(0, mark(y1))

A__PLUS(s(X), Y) → A__PLUS(mark(X), mark(Y))

MARK(2ndspos(square(y0), square(x0))) → A__2NDSPOS(a__square(mark(y0)), a__square(mark(x0)))

A__PI(pi(x0)) → A__2NDSPOS(a__pi(mark(x0)), a__from(0))

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → A__2NDSPOS(mark(N), mark(Z))

MARK(plus(from(x0), y1)) → A__PLUS(a__from(mark(x0)), mark(y1))

MARK(2ndsneg(times(y0, y1), pi(x0))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__pi(mark(x0)))

MARK(square(X)) → A__SQUARE(mark(X))

MARK(2ndsneg(times(y0, y1), plus(x0, x1))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(2ndsneg(plus(y0, y1), times(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(plus(y0, y1), plus(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(2ndsneg(2ndsneg(y0, y1), cons2(x0, x1))) → A__2NDSNEG(a__2ndsneg(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(2ndspos(2ndsneg(y0, y1), cons(x0, x1))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), cons(mark(x0), x1))

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))

a__2ndspos(0, Z) → rnil

a__2ndspos(s(N), cons(X, Z)) → a__2ndspos(s(mark(N)), cons2(X, mark(Z)))

a__2ndspos(s(N), cons2(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))

a__2ndsneg(0, Z) → rnil

a__2ndsneg(s(N), cons(X, Z)) → a__2ndsneg(s(mark(N)), cons2(X, mark(Z)))

a__2ndsneg(s(N), cons2(X, cons(Y, Z))) → rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z)))

a__pi(X) → a__2ndspos(mark(X), a__from(0))

a__plus(0, Y) → mark(Y)

a__plus(s(X), Y) → s(a__plus(mark(X), mark(Y)))

a__times(0, Y) → 0

a__times(s(X), Y) → a__plus(mark(Y), a__times(mark(X), mark(Y)))

a__square(X) → a__times(mark(X), mark(X))

mark(from(X)) → a__from(mark(X))

mark(2ndspos(X1, X2)) → a__2ndspos(mark(X1), mark(X2))

mark(2ndsneg(X1, X2)) → a__2ndsneg(mark(X1), mark(X2))

mark(pi(X)) → a__pi(mark(X))

mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))

mark(times(X1, X2)) → a__times(mark(X1), mark(X2))

mark(square(X)) → a__square(mark(X))

mark(0) → 0

mark(s(X)) → s(mark(X))

mark(posrecip(X)) → posrecip(mark(X))

mark(negrecip(X)) → negrecip(mark(X))

mark(nil) → nil

mark(cons(X1, X2)) → cons(mark(X1), X2)

mark(cons2(X1, X2)) → cons2(X1, mark(X2))

mark(rnil) → rnil

mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))

a__from(X) → from(X)

a__2ndspos(X1, X2) → 2ndspos(X1, X2)

a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)

a__pi(X) → pi(X)

a__plus(X1, X2) → plus(X1, X2)

a__times(X1, X2) → times(X1, X2)

a__square(X) → square(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 7 less nodes.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ Narrowing
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ QDP
              ↳ Narrowing
                ↳ QDP
                  ↳ DependencyGraphProof
                    ↳ QDP
                      ↳ Narrowing
                        ↳ QDP
                          ↳ DependencyGraphProof
                            ↳ QDP
                              ↳ Narrowing
                                ↳ QDP
                                  ↳ DependencyGraphProof
                                    ↳ QDP
                                      ↳ Narrowing
                                        ↳ QDP
                                          ↳ DependencyGraphProof
                                            ↳ QDP
                                              ↳ Narrowing
                                                ↳ QDP
                                                  ↳ DependencyGraphProof
                                                    ↳ QDP
                                                      ↳ Narrowing
                                                        ↳ QDP
                                                          ↳ DependencyGraphProof
                                                            ↳ QDP
                                                              ↳ Narrowing
                                                                ↳ QDP
                                                                  ↳ DependencyGraphProof
                                                                    ↳ QDP
                                                                      ↳ Narrowing
                                                                        ↳ QDP
                                                                          ↳ DependencyGraphProof
                                                                            ↳ QDP
                                                                              ↳ Narrowing
                                                                                ↳ QDP
                                                                                  ↳ DependencyGraphProof
                                                                                    ↳ QDP
                                                                                      ↳ Narrowing
                                                                                        ↳ QDP
                                                                                          ↳ DependencyGraphProof
                                                                                            ↳ QDP
                                                                                              ↳ Narrowing
                                                                                                ↳ QDP
                                                                                                  ↳ DependencyGraphProof
                                                                                                    ↳ QDP
                                                                                                      ↳ Narrowing
                                                                                                        ↳ QDP
                                                                                                          ↳ DependencyGraphProof
                                                                                                            ↳ QDP
                                                                                                              ↳ Narrowing
                                                                                                                ↳ QDP
                                                                                                                  ↳ DependencyGraphProof
                                                                                                                    ↳ QDP
                                                                                                                      ↳ Narrowing
                                                                                                                        ↳ QDP
                                                                                                                          ↳ DependencyGraphProof
                                                                                                                            ↳ QDP
                                                                                                                              ↳ Narrowing
                                                                                                                                ↳ QDP
                                                                                                                                  ↳ DependencyGraphProof
                                                                                                                                    ↳ QDP
                                                                                                                                      ↳ Narrowing
                                                                                                                                        ↳ QDP
                                                                                                                                          ↳ DependencyGraphProof
                                                                                                                                            ↳ QDP
                                                                                                                                              ↳ Narrowing
                                                                                                                                                ↳ QDP
                                                                                                                                                  ↳ DependencyGraphProof
                                                                                                                                                    ↳ QDP
                                                                                                                                                      ↳ Narrowing
                                                                                                                                                        ↳ QDP
                                                                                                                                                          ↳ DependencyGraphProof
                                                                                                                                                            ↳ QDP
                                                                                                                                                              ↳ Narrowing
                                                                                                                                                                ↳ QDP
                                                                                                                                                                  ↳ DependencyGraphProof
                                                                                                                                                                    ↳ QDP
                                                                                                                                                                      ↳ Narrowing

Q DP problem:
The TRS P consists of the following rules:

MARK(times(times(x0, x1), y1)) → A__TIMES(a__times(mark(x0), mark(x1)), mark(y1))

MARK(2ndsneg(2ndspos(y0, y1), 2ndsneg(x0, x1))) → A__2NDSNEG(a__2ndspos(mark(y0), mark(y1)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndsneg(2ndsneg(y0, y1), 2ndspos(x0, x1))) → A__2NDSNEG(a__2ndsneg(mark(y0), mark(y1)), a__2ndspos(mark(x0), mark(x1)))

MARK(plus(plus(x0, x1), y1)) → A__PLUS(a__plus(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(from(y0), 2ndsneg(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndspos(2ndsneg(y0, y1), from(x0))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__from(mark(x0)))

A__2NDSNEG(s(N), cons(X, Z)) → MARK(Z)

MARK(2ndsneg(square(y0), 2ndspos(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndsneg(2ndspos(y0, y1), square(x0))) → A__2NDSNEG(a__2ndspos(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndsneg(2ndspos(y0, y1), pi(x0))) → A__2NDSNEG(a__2ndspos(mark(y0), mark(y1)), a__pi(mark(x0)))

A__TIMES(s(X), Y) → MARK(Y)

MARK(2ndspos(square(y0), 2ndsneg(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndspos(2ndsneg(y0, y1), square(x0))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__square(mark(x0)))

A__2NDSNEG(s(N), cons(X, Z)) → MARK(N)

MARK(times(2ndsneg(x0, x1), y1)) → A__TIMES(a__2ndsneg(mark(x0), mark(x1)), mark(y1))

MARK(2ndsneg(times(y0, y1), times(x0, x1))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(pi(y0), pi(x0))) → A__2NDSPOS(a__pi(mark(y0)), a__pi(mark(x0)))

MARK(2ndsneg(2ndsneg(y0, y1), square(x0))) → A__2NDSNEG(a__2ndsneg(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndsneg(square(y0), 2ndsneg(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndsneg(from(y0), cons(x0, x1))) → A__2NDSNEG(a__from(mark(y0)), cons(mark(x0), x1))

MARK(2ndspos(s(y0), pi(x0))) → A__2NDSPOS(s(mark(y0)), a__pi(mark(x0)))

MARK(plus(square(x0), y1)) → A__PLUS(a__square(mark(x0)), mark(y1))

MARK(2ndsneg(times(y0, y1), cons2(x0, x1))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(cons2(X1, X2)) → MARK(X2)

MARK(2ndsneg(square(y0), cons2(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), cons2(x0, mark(x1)))

MARK(2ndsneg(2ndspos(y0, y1), times(x0, x1))) → A__2NDSNEG(a__2ndspos(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndsneg(times(y0, y1), 2ndspos(x0, x1))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndspos(X1, X2)) → MARK(X1)

MARK(pi(X)) → MARK(X)

MARK(pi(X)) → A__PI(mark(X))

MARK(2ndspos(plus(y0, y1), cons(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), cons(mark(x0), x1))

MARK(2ndsneg(square(y0), times(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndsneg(times(y0, y1), square(x0))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(plus(2ndsneg(x0, x1), y1)) → A__PLUS(a__2ndsneg(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(2ndspos(y0, y1), cons2(x0, x1))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(2ndsneg(X1, X2)) → MARK(X2)

MARK(2ndspos(square(y0), times(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(times(y0, y1), square(x0))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(times(square(x0), y1)) → A__TIMES(a__square(mark(x0)), mark(y1))

MARK(2ndspos(pi(y0), cons(x0, x1))) → A__2NDSPOS(a__pi(mark(y0)), cons(mark(x0), x1))

A__PI(X) → MARK(X)

MARK(times(s(x0), y1)) → A__TIMES(s(mark(x0)), mark(y1))

MARK(2ndspos(2ndspos(y0, y1), plus(x0, x1))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(plus(y0, y1), 2ndspos(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__2ndspos(mark(x0), mark(x1)))

A__PLUS(s(X), Y) → MARK(Y)

MARK(2ndsneg(plus(y0, y1), pi(x0))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__pi(mark(x0)))

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(N)

MARK(2ndspos(square(y0), 2ndspos(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndspos(2ndspos(y0, y1), square(x0))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndspos(s(y0), 2ndspos(x0, x1))) → A__2NDSPOS(s(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndspos(times(y0, y1), cons(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), cons(mark(x0), x1))

MARK(2ndspos(2ndsneg(y0, y1), 2ndsneg(x0, x1))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndspos(times(y0, y1), plus(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(plus(y0, y1), times(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndsneg(2ndspos(y0, y1), from(x0))) → A__2NDSNEG(a__2ndspos(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndsneg(from(y0), 2ndspos(x0, x1))) → A__2NDSNEG(a__from(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndsneg(2ndspos(y0, y1), cons2(x0, x1))) → A__2NDSNEG(a__2ndspos(mark(y0), mark(y1)), cons2(x0, mark(x1)))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(Y)

MARK(rcons(X1, X2)) → MARK(X1)

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(Z)

MARK(square(X)) → MARK(X)

MARK(2ndspos(pi(y0), times(x0, x1))) → A__2NDSPOS(a__pi(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(times(y0, y1), pi(x0))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__pi(mark(x0)))

A__PI(s(x0)) → A__2NDSPOS(s(mark(x0)), a__from(0))

MARK(2ndsneg(2ndsneg(y0, y1), pi(x0))) → A__2NDSNEG(a__2ndsneg(mark(y0), mark(y1)), a__pi(mark(x0)))

MARK(times(plus(x0, x1), y1)) → A__TIMES(a__plus(mark(x0), mark(x1)), mark(y1))

A__2NDSPOS(s(y0), cons(y1, from(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__from(mark(x0))))

MARK(2ndspos(times(y0, y1), 2ndspos(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndspos(2ndspos(y0, y1), times(x0, x1))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndsneg(from(y0), cons2(x0, x1))) → A__2NDSNEG(a__from(mark(y0)), cons2(x0, mark(x1)))

MARK(2ndspos(s(y0), square(x0))) → A__2NDSPOS(s(mark(y0)), a__square(mark(x0)))

MARK(times(2ndspos(x0, x1), y1)) → A__TIMES(a__2ndspos(mark(x0), mark(x1)), mark(y1))

MARK(2ndsneg(2ndsneg(y0, y1), from(x0))) → A__2NDSNEG(a__2ndsneg(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndsneg(from(y0), 2ndsneg(x0, x1))) → A__2NDSNEG(a__from(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndspos(s(y0), cons2(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(x0, mark(x1)))

MARK(2ndsneg(plus(y0, y1), cons2(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), cons2(x0, mark(x1)))

A__PLUS(s(X), Y) → MARK(X)

MARK(times(X1, X2)) → MARK(X1)

A__SQUARE(X) → MARK(X)

MARK(negrecip(X)) → MARK(X)

MARK(2ndspos(from(y0), times(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(times(y0, y1), from(x0))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndspos(2ndsneg(y0, y1), times(x0, x1))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(times(y0, y1), 2ndsneg(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__2ndsneg(mark(x0), mark(x1)))

MARK(from(X)) → MARK(X)

A__SQUARE(X) → A__TIMES(mark(X), mark(X))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(Z)

MARK(2ndsneg(plus(y0, y1), plus(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(from(y0), plus(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(plus(y0, y1), from(x0))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndsneg(2ndsneg(y0, y1), times(x0, x1))) → A__2NDSNEG(a__2ndsneg(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndsneg(times(y0, y1), 2ndsneg(x0, x1))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndsneg(times(y0, y1), cons(x0, x1))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), cons(mark(x0), x1))

MARK(2ndsneg(2ndsneg(y0, y1), cons(x0, x1))) → A__2NDSNEG(a__2ndsneg(mark(y0), mark(y1)), cons(mark(x0), x1))

MARK(2ndspos(from(y0), 2ndspos(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndspos(2ndspos(y0, y1), from(x0))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndspos(square(y0), plus(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(plus(y0, y1), square(x0))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__square(mark(x0)))

A__PI(times(x0, x1)) → A__2NDSPOS(a__times(mark(x0), mark(x1)), a__from(0))

A__2NDSPOS(s(y0), cons(y1, 2ndspos(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__2ndspos(mark(x0), mark(x1))))

A__2NDSNEG(s(N), cons(X, Z)) → A__2NDSNEG(s(mark(N)), cons2(X, mark(Z)))

MARK(2ndspos(X1, X2)) → MARK(X2)

MARK(2ndspos(pi(y0), cons2(x0, x1))) → A__2NDSPOS(a__pi(mark(y0)), cons2(x0, mark(x1)))

MARK(2ndspos(from(y0), cons2(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), cons2(x0, mark(x1)))

MARK(2ndsneg(from(y0), pi(x0))) → A__2NDSNEG(a__from(mark(y0)), a__pi(mark(x0)))

MARK(2ndsneg(2ndspos(y0, y1), 2ndspos(x0, x1))) → A__2NDSNEG(a__2ndspos(mark(y0), mark(y1)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndspos(s(y0), from(x0))) → A__2NDSPOS(s(mark(y0)), a__from(mark(x0)))

MARK(2ndspos(2ndsneg(y0, y1), pi(x0))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__pi(mark(x0)))

MARK(2ndspos(pi(y0), 2ndsneg(x0, x1))) → A__2NDSPOS(a__pi(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

A__2NDSPOS(s(y0), cons(y1, plus(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__plus(mark(x0), mark(x1))))

MARK(2ndspos(from(y0), pi(x0))) → A__2NDSPOS(a__from(mark(y0)), a__pi(mark(x0)))

MARK(2ndspos(pi(y0), from(x0))) → A__2NDSPOS(a__pi(mark(y0)), a__from(mark(x0)))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → A__2NDSNEG(mark(N), mark(Z))

MARK(2ndspos(from(y0), from(x0))) → A__2NDSPOS(a__from(mark(y0)), a__from(mark(x0)))

MARK(2ndspos(times(y0, y1), times(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

A__TIMES(s(X), Y) → A__PLUS(mark(Y), a__times(mark(X), mark(Y)))

A__PI(2ndspos(x0, x1)) → A__2NDSPOS(a__2ndspos(mark(x0), mark(x1)), a__from(0))

MARK(2ndspos(square(y0), cons2(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), cons2(x0, mark(x1)))

MARK(plus(s(x0), y1)) → A__PLUS(s(mark(x0)), mark(y1))

A__PI(plus(x0, x1)) → A__2NDSPOS(a__plus(mark(x0), mark(x1)), a__from(0))

MARK(2ndspos(square(y0), cons(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), cons(mark(x0), x1))

MARK(s(X)) → MARK(X)

MARK(2ndsneg(square(y0), plus(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(2ndsneg(plus(y0, y1), square(x0))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndspos(plus(y0, y1), cons2(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(plus(X1, X2)) → MARK(X2)

A__2NDSPOS(s(N), cons(X, Z)) → MARK(Z)

MARK(2ndsneg(from(y0), times(x0, x1))) → A__2NDSNEG(a__from(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndsneg(times(y0, y1), from(x0))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndspos(times(y0, y1), cons2(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(2ndsneg(square(y0), from(x0))) → A__2NDSNEG(a__square(mark(y0)), a__from(mark(x0)))

MARK(2ndsneg(from(y0), square(x0))) → A__2NDSNEG(a__from(mark(y0)), a__square(mark(x0)))

MARK(2ndspos(2ndsneg(y0, y1), cons2(x0, x1))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), cons2(x0, mark(x1)))

A__PI(X) → A__FROM(0)

MARK(2ndspos(s(y0), plus(x0, x1))) → A__2NDSPOS(s(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(from(X)) → A__FROM(mark(X))

MARK(plus(X1, X2)) → MARK(X1)

MARK(times(from(x0), y1)) → A__TIMES(a__from(mark(x0)), mark(y1))

A__2NDSPOS(s(y0), cons(y1, times(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__times(mark(x0), mark(x1))))

MARK(2ndsneg(2ndsneg(y0, y1), 2ndsneg(x0, x1))) → A__2NDSNEG(a__2ndsneg(mark(y0), mark(y1)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndspos(s(y0), cons(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons(mark(x0), x1))

MARK(2ndsneg(2ndspos(y0, y1), plus(x0, x1))) → A__2NDSNEG(a__2ndspos(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(2ndsneg(plus(y0, y1), 2ndspos(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndspos(pi(y0), 2ndspos(x0, x1))) → A__2NDSPOS(a__pi(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndspos(2ndspos(y0, y1), pi(x0))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), a__pi(mark(x0)))

MARK(2ndspos(from(y0), cons(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), cons(mark(x0), x1))

MARK(2ndsneg(s(x0), y1)) → A__2NDSNEG(s(mark(x0)), mark(y1))

MARK(2ndsneg(2ndspos(y0, y1), cons(x0, x1))) → A__2NDSNEG(a__2ndspos(mark(y0), mark(y1)), cons(mark(x0), x1))

MARK(2ndsneg(plus(y0, y1), cons(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), cons(mark(x0), x1))

A__2NDSPOS(s(y0), cons(y1, pi(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__pi(mark(x0))))

A__TIMES(s(X), Y) → MARK(X)

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(N)

MARK(2ndspos(pi(y0), plus(x0, x1))) → A__2NDSPOS(a__pi(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(plus(y0, y1), pi(x0))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__pi(mark(x0)))

MARK(plus(2ndspos(x0, x1), y1)) → A__PLUS(a__2ndspos(mark(x0), mark(x1)), mark(y1))

A__PI(from(x0)) → A__2NDSPOS(a__from(mark(x0)), a__from(0))

MARK(cons(X1, X2)) → MARK(X1)

MARK(2ndsneg(plus(y0, y1), from(x0))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndsneg(from(y0), plus(x0, x1))) → A__2NDSNEG(a__from(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(times(X1, X2)) → MARK(X2)

A__PI(square(x0)) → A__2NDSPOS(a__square(mark(x0)), a__from(0))

MARK(2ndsneg(square(y0), cons(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), cons(mark(x0), x1))

MARK(2ndspos(s(y0), times(x0, x1))) → A__2NDSPOS(s(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndsneg(X1, X2)) → MARK(X1)

MARK(2ndspos(s(y0), 2ndsneg(x0, x1))) → A__2NDSPOS(s(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndsneg(square(y0), square(x0))) → A__2NDSNEG(a__square(mark(y0)), a__square(mark(x0)))

MARK(2ndspos(2ndsneg(y0, y1), plus(x0, x1))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(plus(y0, y1), 2ndsneg(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__2ndsneg(mark(x0), mark(x1)))

A__PI(2ndsneg(x0, x1)) → A__2NDSPOS(a__2ndsneg(mark(x0), mark(x1)), a__from(0))

MARK(times(pi(x0), y1)) → A__TIMES(a__pi(mark(x0)), mark(y1))

MARK(plus(pi(x0), y1)) → A__PLUS(a__pi(mark(x0)), mark(y1))

MARK(2ndspos(2ndspos(y0, y1), 2ndspos(x0, x1))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndsneg(square(y0), pi(x0))) → A__2NDSNEG(a__square(mark(y0)), a__pi(mark(x0)))

MARK(2ndspos(2ndsneg(y0, y1), 2ndspos(x0, x1))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndspos(2ndspos(y0, y1), 2ndsneg(x0, x1))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), a__2ndsneg(mark(x0), mark(x1)))

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(Y)

A__FROM(X) → MARK(X)

MARK(2ndspos(2ndspos(y0, y1), cons(x0, x1))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), cons(mark(x0), x1))

MARK(2ndspos(square(y0), pi(x0))) → A__2NDSPOS(a__square(mark(y0)), a__pi(mark(x0)))

MARK(2ndspos(pi(y0), square(x0))) → A__2NDSPOS(a__pi(mark(y0)), a__square(mark(x0)))

MARK(rcons(X1, X2)) → MARK(X2)

A__TIMES(s(X), Y) → A__TIMES(mark(X), mark(Y))

A__2NDSPOS(s(y0), cons(y1, cons(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, cons(mark(x0), x1)))

MARK(plus(times(x0, x1), y1)) → A__PLUS(a__times(mark(x0), mark(x1)), mark(y1))

MARK(2ndsneg(from(y0), from(x0))) → A__2NDSNEG(a__from(mark(y0)), a__from(mark(x0)))

A__2NDSPOS(s(y0), cons(y1, 2ndsneg(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__2ndsneg(mark(x0), mark(x1))))

A__2NDSPOS(s(N), cons(X, Z)) → MARK(N)

MARK(2ndsneg(plus(y0, y1), 2ndsneg(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndsneg(2ndsneg(y0, y1), plus(x0, x1))) → A__2NDSNEG(a__2ndsneg(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

A__PLUS(0, Y) → MARK(Y)

MARK(posrecip(X)) → MARK(X)

MARK(2ndspos(from(y0), square(x0))) → A__2NDSPOS(a__from(mark(y0)), a__square(mark(x0)))

MARK(2ndspos(square(y0), from(x0))) → A__2NDSPOS(a__square(mark(y0)), a__from(mark(x0)))

MARK(2ndsneg(pi(x0), y1)) → A__2NDSNEG(a__pi(mark(x0)), mark(y1))

A__2NDSPOS(s(y0), cons(y1, square(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__square(mark(x0))))

MARK(plus(0, y1)) → A__PLUS(0, mark(y1))

A__PLUS(s(X), Y) → A__PLUS(mark(X), mark(Y))

MARK(2ndspos(square(y0), square(x0))) → A__2NDSPOS(a__square(mark(y0)), a__square(mark(x0)))

A__PI(pi(x0)) → A__2NDSPOS(a__pi(mark(x0)), a__from(0))

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → A__2NDSPOS(mark(N), mark(Z))

MARK(plus(from(x0), y1)) → A__PLUS(a__from(mark(x0)), mark(y1))

MARK(square(X)) → A__SQUARE(mark(X))

MARK(2ndsneg(times(y0, y1), pi(x0))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__pi(mark(x0)))

MARK(2ndsneg(plus(y0, y1), times(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndsneg(times(y0, y1), plus(x0, x1))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(plus(y0, y1), plus(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(2ndsneg(2ndsneg(y0, y1), cons2(x0, x1))) → A__2NDSNEG(a__2ndsneg(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(2ndspos(2ndsneg(y0, y1), cons(x0, x1))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), cons(mark(x0), x1))

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))

a__2ndspos(0, Z) → rnil

a__2ndspos(s(N), cons(X, Z)) → a__2ndspos(s(mark(N)), cons2(X, mark(Z)))

a__2ndspos(s(N), cons2(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))

a__2ndsneg(0, Z) → rnil

a__2ndsneg(s(N), cons(X, Z)) → a__2ndsneg(s(mark(N)), cons2(X, mark(Z)))

a__2ndsneg(s(N), cons2(X, cons(Y, Z))) → rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z)))

a__pi(X) → a__2ndspos(mark(X), a__from(0))

a__plus(0, Y) → mark(Y)

a__plus(s(X), Y) → s(a__plus(mark(X), mark(Y)))

a__times(0, Y) → 0

a__times(s(X), Y) → a__plus(mark(Y), a__times(mark(X), mark(Y)))

a__square(X) → a__times(mark(X), mark(X))

mark(from(X)) → a__from(mark(X))

mark(2ndspos(X1, X2)) → a__2ndspos(mark(X1), mark(X2))

mark(2ndsneg(X1, X2)) → a__2ndsneg(mark(X1), mark(X2))

mark(pi(X)) → a__pi(mark(X))

mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))

mark(times(X1, X2)) → a__times(mark(X1), mark(X2))

mark(square(X)) → a__square(mark(X))

mark(0) → 0

mark(s(X)) → s(mark(X))

mark(posrecip(X)) → posrecip(mark(X))

mark(negrecip(X)) → negrecip(mark(X))

mark(nil) → nil

mark(cons(X1, X2)) → cons(mark(X1), X2)

mark(cons2(X1, X2)) → cons2(X1, mark(X2))

mark(rnil) → rnil

mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))

a__from(X) → from(X)

a__2ndspos(X1, X2) → 2ndspos(X1, X2)

a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)

a__pi(X) → pi(X)

a__plus(X1, X2) → plus(X1, X2)

a__times(X1, X2) → times(X1, X2)

a__square(X) → square(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule MARK(2ndsneg(s(x0), y1)) → A__2NDSNEG(s(mark(x0)), mark(y1)) at position [1] we obtained the following new rules:

MARK(2ndsneg(s(y0), s(x0))) → A__2NDSNEG(s(mark(y0)), s(mark(x0)))

MARK(2ndsneg(s(y0), cons(x0, x1))) → A__2NDSNEG(s(mark(y0)), cons(mark(x0), x1))

MARK(2ndsneg(s(y0), rcons(x0, x1))) → A__2NDSNEG(s(mark(y0)), rcons(mark(x0), mark(x1)))

MARK(2ndsneg(s(y0), cons2(x0, x1))) → A__2NDSNEG(s(mark(y0)), cons2(x0, mark(x1)))

MARK(2ndsneg(s(y0), times(x0, x1))) → A__2NDSNEG(s(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndsneg(s(y0), square(x0))) → A__2NDSNEG(s(mark(y0)), a__square(mark(x0)))

MARK(2ndsneg(s(y0), rnil)) → A__2NDSNEG(s(mark(y0)), rnil)

MARK(2ndsneg(s(y0), 0)) → A__2NDSNEG(s(mark(y0)), 0)

MARK(2ndsneg(s(y0), from(x0))) → A__2NDSNEG(s(mark(y0)), a__from(mark(x0)))

MARK(2ndsneg(s(y0), 2ndspos(x0, x1))) → A__2NDSNEG(s(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndsneg(s(y0), 2ndsneg(x0, x1))) → A__2NDSNEG(s(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndsneg(s(y0), plus(x0, x1))) → A__2NDSNEG(s(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(2ndsneg(s(y0), negrecip(x0))) → A__2NDSNEG(s(mark(y0)), negrecip(mark(x0)))

MARK(2ndsneg(s(y0), posrecip(x0))) → A__2NDSNEG(s(mark(y0)), posrecip(mark(x0)))

MARK(2ndsneg(s(y0), pi(x0))) → A__2NDSNEG(s(mark(y0)), a__pi(mark(x0)))

MARK(2ndsneg(s(y0), nil)) → A__2NDSNEG(s(mark(y0)), nil)



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ Narrowing
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ QDP
              ↳ Narrowing
                ↳ QDP
                  ↳ DependencyGraphProof
                    ↳ QDP
                      ↳ Narrowing
                        ↳ QDP
                          ↳ DependencyGraphProof
                            ↳ QDP
                              ↳ Narrowing
                                ↳ QDP
                                  ↳ DependencyGraphProof
                                    ↳ QDP
                                      ↳ Narrowing
                                        ↳ QDP
                                          ↳ DependencyGraphProof
                                            ↳ QDP
                                              ↳ Narrowing
                                                ↳ QDP
                                                  ↳ DependencyGraphProof
                                                    ↳ QDP
                                                      ↳ Narrowing
                                                        ↳ QDP
                                                          ↳ DependencyGraphProof
                                                            ↳ QDP
                                                              ↳ Narrowing
                                                                ↳ QDP
                                                                  ↳ DependencyGraphProof
                                                                    ↳ QDP
                                                                      ↳ Narrowing
                                                                        ↳ QDP
                                                                          ↳ DependencyGraphProof
                                                                            ↳ QDP
                                                                              ↳ Narrowing
                                                                                ↳ QDP
                                                                                  ↳ DependencyGraphProof
                                                                                    ↳ QDP
                                                                                      ↳ Narrowing
                                                                                        ↳ QDP
                                                                                          ↳ DependencyGraphProof
                                                                                            ↳ QDP
                                                                                              ↳ Narrowing
                                                                                                ↳ QDP
                                                                                                  ↳ DependencyGraphProof
                                                                                                    ↳ QDP
                                                                                                      ↳ Narrowing
                                                                                                        ↳ QDP
                                                                                                          ↳ DependencyGraphProof
                                                                                                            ↳ QDP
                                                                                                              ↳ Narrowing
                                                                                                                ↳ QDP
                                                                                                                  ↳ DependencyGraphProof
                                                                                                                    ↳ QDP
                                                                                                                      ↳ Narrowing
                                                                                                                        ↳ QDP
                                                                                                                          ↳ DependencyGraphProof
                                                                                                                            ↳ QDP
                                                                                                                              ↳ Narrowing
                                                                                                                                ↳ QDP
                                                                                                                                  ↳ DependencyGraphProof
                                                                                                                                    ↳ QDP
                                                                                                                                      ↳ Narrowing
                                                                                                                                        ↳ QDP
                                                                                                                                          ↳ DependencyGraphProof
                                                                                                                                            ↳ QDP
                                                                                                                                              ↳ Narrowing
                                                                                                                                                ↳ QDP
                                                                                                                                                  ↳ DependencyGraphProof
                                                                                                                                                    ↳ QDP
                                                                                                                                                      ↳ Narrowing
                                                                                                                                                        ↳ QDP
                                                                                                                                                          ↳ DependencyGraphProof
                                                                                                                                                            ↳ QDP
                                                                                                                                                              ↳ Narrowing
                                                                                                                                                                ↳ QDP
                                                                                                                                                                  ↳ DependencyGraphProof
                                                                                                                                                                    ↳ QDP
                                                                                                                                                                      ↳ Narrowing
                                                                                                                                                                        ↳ QDP
                                                                                                                                                                          ↳ DependencyGraphProof

Q DP problem:
The TRS P consists of the following rules:

MARK(2ndsneg(s(y0), rcons(x0, x1))) → A__2NDSNEG(s(mark(y0)), rcons(mark(x0), mark(x1)))

MARK(times(times(x0, x1), y1)) → A__TIMES(a__times(mark(x0), mark(x1)), mark(y1))

MARK(2ndsneg(2ndsneg(y0, y1), 2ndspos(x0, x1))) → A__2NDSNEG(a__2ndsneg(mark(y0), mark(y1)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndsneg(2ndspos(y0, y1), 2ndsneg(x0, x1))) → A__2NDSNEG(a__2ndspos(mark(y0), mark(y1)), a__2ndsneg(mark(x0), mark(x1)))

MARK(plus(plus(x0, x1), y1)) → A__PLUS(a__plus(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(2ndsneg(y0, y1), from(x0))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndspos(from(y0), 2ndsneg(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndsneg(2ndspos(y0, y1), square(x0))) → A__2NDSNEG(a__2ndspos(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndsneg(square(y0), 2ndspos(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

A__2NDSNEG(s(N), cons(X, Z)) → MARK(Z)

MARK(2ndsneg(s(y0), pi(x0))) → A__2NDSNEG(s(mark(y0)), a__pi(mark(x0)))

MARK(2ndsneg(2ndspos(y0, y1), pi(x0))) → A__2NDSNEG(a__2ndspos(mark(y0), mark(y1)), a__pi(mark(x0)))

A__TIMES(s(X), Y) → MARK(Y)

MARK(2ndspos(2ndsneg(y0, y1), square(x0))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndspos(square(y0), 2ndsneg(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

A__2NDSNEG(s(N), cons(X, Z)) → MARK(N)

MARK(times(2ndsneg(x0, x1), y1)) → A__TIMES(a__2ndsneg(mark(x0), mark(x1)), mark(y1))

MARK(2ndsneg(times(y0, y1), times(x0, x1))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(pi(y0), pi(x0))) → A__2NDSPOS(a__pi(mark(y0)), a__pi(mark(x0)))

MARK(2ndsneg(square(y0), 2ndsneg(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndsneg(2ndsneg(y0, y1), square(x0))) → A__2NDSNEG(a__2ndsneg(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndsneg(from(y0), cons(x0, x1))) → A__2NDSNEG(a__from(mark(y0)), cons(mark(x0), x1))

MARK(2ndspos(s(y0), pi(x0))) → A__2NDSPOS(s(mark(y0)), a__pi(mark(x0)))

MARK(plus(square(x0), y1)) → A__PLUS(a__square(mark(x0)), mark(y1))

MARK(2ndsneg(times(y0, y1), cons2(x0, x1))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(cons2(X1, X2)) → MARK(X2)

MARK(2ndsneg(times(y0, y1), 2ndspos(x0, x1))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndsneg(2ndspos(y0, y1), times(x0, x1))) → A__2NDSNEG(a__2ndspos(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndsneg(square(y0), cons2(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), cons2(x0, mark(x1)))

MARK(2ndspos(X1, X2)) → MARK(X1)

MARK(pi(X)) → MARK(X)

MARK(2ndsneg(s(y0), from(x0))) → A__2NDSNEG(s(mark(y0)), a__from(mark(x0)))

MARK(pi(X)) → A__PI(mark(X))

MARK(2ndsneg(times(y0, y1), square(x0))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndsneg(square(y0), times(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(plus(y0, y1), cons(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), cons(mark(x0), x1))

MARK(plus(2ndsneg(x0, x1), y1)) → A__PLUS(a__2ndsneg(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(2ndspos(y0, y1), cons2(x0, x1))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(2ndsneg(X1, X2)) → MARK(X2)

MARK(2ndspos(times(y0, y1), square(x0))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndspos(square(y0), times(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(times(square(x0), y1)) → A__TIMES(a__square(mark(x0)), mark(y1))

MARK(2ndspos(pi(y0), cons(x0, x1))) → A__2NDSPOS(a__pi(mark(y0)), cons(mark(x0), x1))

A__PI(X) → MARK(X)

MARK(times(s(x0), y1)) → A__TIMES(s(mark(x0)), mark(y1))

MARK(2ndspos(plus(y0, y1), 2ndspos(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndspos(2ndspos(y0, y1), plus(x0, x1))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

A__PLUS(s(X), Y) → MARK(Y)

MARK(2ndsneg(s(y0), nil)) → A__2NDSNEG(s(mark(y0)), nil)

MARK(2ndsneg(plus(y0, y1), pi(x0))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__pi(mark(x0)))

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(N)

MARK(2ndspos(2ndspos(y0, y1), square(x0))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndspos(square(y0), 2ndspos(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndspos(times(y0, y1), cons(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), cons(mark(x0), x1))

MARK(2ndspos(s(y0), 2ndspos(x0, x1))) → A__2NDSPOS(s(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndspos(2ndsneg(y0, y1), 2ndsneg(x0, x1))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndsneg(from(y0), 2ndspos(x0, x1))) → A__2NDSNEG(a__from(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndsneg(2ndspos(y0, y1), from(x0))) → A__2NDSNEG(a__2ndspos(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndspos(plus(y0, y1), times(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(times(y0, y1), plus(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(2ndsneg(s(y0), negrecip(x0))) → A__2NDSNEG(s(mark(y0)), negrecip(mark(x0)))

MARK(2ndsneg(2ndspos(y0, y1), cons2(x0, x1))) → A__2NDSNEG(a__2ndspos(mark(y0), mark(y1)), cons2(x0, mark(x1)))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(Y)

MARK(rcons(X1, X2)) → MARK(X1)

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(Z)

MARK(square(X)) → MARK(X)

MARK(2ndspos(times(y0, y1), pi(x0))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__pi(mark(x0)))

MARK(2ndspos(pi(y0), times(x0, x1))) → A__2NDSPOS(a__pi(mark(y0)), a__times(mark(x0), mark(x1)))

A__PI(s(x0)) → A__2NDSPOS(s(mark(x0)), a__from(0))

MARK(2ndsneg(2ndsneg(y0, y1), pi(x0))) → A__2NDSNEG(a__2ndsneg(mark(y0), mark(y1)), a__pi(mark(x0)))

MARK(times(plus(x0, x1), y1)) → A__TIMES(a__plus(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(2ndspos(y0, y1), times(x0, x1))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(times(y0, y1), 2ndspos(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__2ndspos(mark(x0), mark(x1)))

A__2NDSPOS(s(y0), cons(y1, from(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__from(mark(x0))))

MARK(2ndsneg(from(y0), cons2(x0, x1))) → A__2NDSNEG(a__from(mark(y0)), cons2(x0, mark(x1)))

MARK(2ndspos(s(y0), square(x0))) → A__2NDSPOS(s(mark(y0)), a__square(mark(x0)))

MARK(times(2ndspos(x0, x1), y1)) → A__TIMES(a__2ndspos(mark(x0), mark(x1)), mark(y1))

MARK(2ndsneg(from(y0), 2ndsneg(x0, x1))) → A__2NDSNEG(a__from(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndsneg(2ndsneg(y0, y1), from(x0))) → A__2NDSNEG(a__2ndsneg(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndspos(s(y0), cons2(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(x0, mark(x1)))

MARK(2ndsneg(plus(y0, y1), cons2(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), cons2(x0, mark(x1)))

A__PLUS(s(X), Y) → MARK(X)

MARK(times(X1, X2)) → MARK(X1)

A__SQUARE(X) → MARK(X)

MARK(negrecip(X)) → MARK(X)

MARK(2ndspos(times(y0, y1), from(x0))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndspos(from(y0), times(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(times(y0, y1), 2ndsneg(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndspos(2ndsneg(y0, y1), times(x0, x1))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(from(X)) → MARK(X)

A__SQUARE(X) → A__TIMES(mark(X), mark(X))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(Z)

MARK(2ndsneg(s(y0), 0)) → A__2NDSNEG(s(mark(y0)), 0)

MARK(2ndsneg(plus(y0, y1), plus(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(plus(y0, y1), from(x0))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndspos(from(y0), plus(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(2ndsneg(times(y0, y1), 2ndsneg(x0, x1))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndsneg(2ndsneg(y0, y1), times(x0, x1))) → A__2NDSNEG(a__2ndsneg(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndsneg(times(y0, y1), cons(x0, x1))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), cons(mark(x0), x1))

MARK(2ndsneg(2ndsneg(y0, y1), cons(x0, x1))) → A__2NDSNEG(a__2ndsneg(mark(y0), mark(y1)), cons(mark(x0), x1))

MARK(2ndspos(2ndspos(y0, y1), from(x0))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndspos(from(y0), 2ndspos(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndspos(plus(y0, y1), square(x0))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndspos(square(y0), plus(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__plus(mark(x0), mark(x1)))

A__2NDSPOS(s(y0), cons(y1, 2ndspos(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__2ndspos(mark(x0), mark(x1))))

A__PI(times(x0, x1)) → A__2NDSPOS(a__times(mark(x0), mark(x1)), a__from(0))

A__2NDSNEG(s(N), cons(X, Z)) → A__2NDSNEG(s(mark(N)), cons2(X, mark(Z)))

MARK(2ndsneg(s(y0), square(x0))) → A__2NDSNEG(s(mark(y0)), a__square(mark(x0)))

MARK(2ndspos(pi(y0), cons2(x0, x1))) → A__2NDSPOS(a__pi(mark(y0)), cons2(x0, mark(x1)))

MARK(2ndspos(X1, X2)) → MARK(X2)

MARK(2ndspos(from(y0), cons2(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), cons2(x0, mark(x1)))

MARK(2ndsneg(from(y0), pi(x0))) → A__2NDSNEG(a__from(mark(y0)), a__pi(mark(x0)))

MARK(2ndsneg(2ndspos(y0, y1), 2ndspos(x0, x1))) → A__2NDSNEG(a__2ndspos(mark(y0), mark(y1)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndspos(s(y0), from(x0))) → A__2NDSPOS(s(mark(y0)), a__from(mark(x0)))

MARK(2ndspos(pi(y0), 2ndsneg(x0, x1))) → A__2NDSPOS(a__pi(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndspos(2ndsneg(y0, y1), pi(x0))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__pi(mark(x0)))

A__2NDSPOS(s(y0), cons(y1, plus(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__plus(mark(x0), mark(x1))))

MARK(2ndspos(pi(y0), from(x0))) → A__2NDSPOS(a__pi(mark(y0)), a__from(mark(x0)))

MARK(2ndspos(from(y0), pi(x0))) → A__2NDSPOS(a__from(mark(y0)), a__pi(mark(x0)))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → A__2NDSNEG(mark(N), mark(Z))

MARK(2ndspos(from(y0), from(x0))) → A__2NDSPOS(a__from(mark(y0)), a__from(mark(x0)))

MARK(2ndspos(times(y0, y1), times(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

A__TIMES(s(X), Y) → A__PLUS(mark(Y), a__times(mark(X), mark(Y)))

MARK(2ndspos(square(y0), cons2(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), cons2(x0, mark(x1)))

A__PI(2ndspos(x0, x1)) → A__2NDSPOS(a__2ndspos(mark(x0), mark(x1)), a__from(0))

MARK(plus(s(x0), y1)) → A__PLUS(s(mark(x0)), mark(y1))

MARK(2ndspos(square(y0), cons(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), cons(mark(x0), x1))

A__PI(plus(x0, x1)) → A__2NDSPOS(a__plus(mark(x0), mark(x1)), a__from(0))

MARK(2ndsneg(plus(y0, y1), square(x0))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndsneg(square(y0), plus(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(s(X)) → MARK(X)

MARK(2ndspos(plus(y0, y1), cons2(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(2ndsneg(s(y0), rnil)) → A__2NDSNEG(s(mark(y0)), rnil)

MARK(plus(X1, X2)) → MARK(X2)

A__2NDSPOS(s(N), cons(X, Z)) → MARK(Z)

MARK(2ndsneg(times(y0, y1), from(x0))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndsneg(from(y0), times(x0, x1))) → A__2NDSNEG(a__from(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(times(y0, y1), cons2(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(2ndsneg(from(y0), square(x0))) → A__2NDSNEG(a__from(mark(y0)), a__square(mark(x0)))

MARK(2ndsneg(square(y0), from(x0))) → A__2NDSNEG(a__square(mark(y0)), a__from(mark(x0)))

MARK(2ndspos(2ndsneg(y0, y1), cons2(x0, x1))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(2ndsneg(s(y0), cons(x0, x1))) → A__2NDSNEG(s(mark(y0)), cons(mark(x0), x1))

MARK(from(X)) → A__FROM(mark(X))

MARK(2ndspos(s(y0), plus(x0, x1))) → A__2NDSPOS(s(mark(y0)), a__plus(mark(x0), mark(x1)))

A__PI(X) → A__FROM(0)

MARK(times(from(x0), y1)) → A__TIMES(a__from(mark(x0)), mark(y1))

MARK(plus(X1, X2)) → MARK(X1)

A__2NDSPOS(s(y0), cons(y1, times(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__times(mark(x0), mark(x1))))

MARK(2ndsneg(2ndsneg(y0, y1), 2ndsneg(x0, x1))) → A__2NDSNEG(a__2ndsneg(mark(y0), mark(y1)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndspos(s(y0), cons(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons(mark(x0), x1))

MARK(2ndsneg(plus(y0, y1), 2ndspos(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndsneg(2ndspos(y0, y1), plus(x0, x1))) → A__2NDSNEG(a__2ndspos(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(2ndspos(y0, y1), pi(x0))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), a__pi(mark(x0)))

MARK(2ndspos(pi(y0), 2ndspos(x0, x1))) → A__2NDSPOS(a__pi(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndspos(from(y0), cons(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), cons(mark(x0), x1))

MARK(2ndsneg(plus(y0, y1), cons(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), cons(mark(x0), x1))

MARK(2ndsneg(2ndspos(y0, y1), cons(x0, x1))) → A__2NDSNEG(a__2ndspos(mark(y0), mark(y1)), cons(mark(x0), x1))

A__TIMES(s(X), Y) → MARK(X)

A__2NDSPOS(s(y0), cons(y1, pi(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__pi(mark(x0))))

MARK(2ndsneg(s(y0), s(x0))) → A__2NDSNEG(s(mark(y0)), s(mark(x0)))

MARK(2ndspos(plus(y0, y1), pi(x0))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__pi(mark(x0)))

MARK(2ndspos(pi(y0), plus(x0, x1))) → A__2NDSPOS(a__pi(mark(y0)), a__plus(mark(x0), mark(x1)))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(N)

MARK(2ndsneg(s(y0), cons2(x0, x1))) → A__2NDSNEG(s(mark(y0)), cons2(x0, mark(x1)))

MARK(plus(2ndspos(x0, x1), y1)) → A__PLUS(a__2ndspos(mark(x0), mark(x1)), mark(y1))

MARK(cons(X1, X2)) → MARK(X1)

A__PI(from(x0)) → A__2NDSPOS(a__from(mark(x0)), a__from(0))

MARK(2ndsneg(s(y0), 2ndspos(x0, x1))) → A__2NDSNEG(s(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndsneg(from(y0), plus(x0, x1))) → A__2NDSNEG(a__from(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(2ndsneg(plus(y0, y1), from(x0))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(times(X1, X2)) → MARK(X2)

A__PI(square(x0)) → A__2NDSPOS(a__square(mark(x0)), a__from(0))

MARK(2ndsneg(square(y0), cons(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), cons(mark(x0), x1))

MARK(2ndspos(s(y0), times(x0, x1))) → A__2NDSPOS(s(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndsneg(X1, X2)) → MARK(X1)

MARK(2ndspos(s(y0), 2ndsneg(x0, x1))) → A__2NDSPOS(s(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndsneg(square(y0), square(x0))) → A__2NDSNEG(a__square(mark(y0)), a__square(mark(x0)))

MARK(2ndspos(plus(y0, y1), 2ndsneg(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndspos(2ndsneg(y0, y1), plus(x0, x1))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

A__PI(2ndsneg(x0, x1)) → A__2NDSPOS(a__2ndsneg(mark(x0), mark(x1)), a__from(0))

MARK(plus(pi(x0), y1)) → A__PLUS(a__pi(mark(x0)), mark(y1))

MARK(times(pi(x0), y1)) → A__TIMES(a__pi(mark(x0)), mark(y1))

MARK(2ndspos(2ndspos(y0, y1), 2ndspos(x0, x1))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndsneg(square(y0), pi(x0))) → A__2NDSNEG(a__square(mark(y0)), a__pi(mark(x0)))

MARK(2ndspos(2ndspos(y0, y1), 2ndsneg(x0, x1))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndspos(2ndsneg(y0, y1), 2ndspos(x0, x1))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__2ndspos(mark(x0), mark(x1)))

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(Y)

MARK(2ndsneg(s(y0), 2ndsneg(x0, x1))) → A__2NDSNEG(s(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndspos(2ndspos(y0, y1), cons(x0, x1))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), cons(mark(x0), x1))

A__FROM(X) → MARK(X)

MARK(2ndsneg(s(y0), plus(x0, x1))) → A__2NDSNEG(s(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(pi(y0), square(x0))) → A__2NDSPOS(a__pi(mark(y0)), a__square(mark(x0)))

MARK(2ndspos(square(y0), pi(x0))) → A__2NDSPOS(a__square(mark(y0)), a__pi(mark(x0)))

MARK(rcons(X1, X2)) → MARK(X2)

A__TIMES(s(X), Y) → A__TIMES(mark(X), mark(Y))

A__2NDSPOS(s(y0), cons(y1, cons(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, cons(mark(x0), x1)))

MARK(plus(times(x0, x1), y1)) → A__PLUS(a__times(mark(x0), mark(x1)), mark(y1))

MARK(2ndsneg(from(y0), from(x0))) → A__2NDSNEG(a__from(mark(y0)), a__from(mark(x0)))

A__2NDSPOS(s(y0), cons(y1, 2ndsneg(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__2ndsneg(mark(x0), mark(x1))))

A__2NDSPOS(s(N), cons(X, Z)) → MARK(N)

MARK(2ndsneg(2ndsneg(y0, y1), plus(x0, x1))) → A__2NDSNEG(a__2ndsneg(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(2ndsneg(plus(y0, y1), 2ndsneg(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__2ndsneg(mark(x0), mark(x1)))

A__PLUS(0, Y) → MARK(Y)

MARK(2ndsneg(s(y0), posrecip(x0))) → A__2NDSNEG(s(mark(y0)), posrecip(mark(x0)))

MARK(posrecip(X)) → MARK(X)

MARK(2ndsneg(pi(x0), y1)) → A__2NDSNEG(a__pi(mark(x0)), mark(y1))

MARK(2ndspos(square(y0), from(x0))) → A__2NDSPOS(a__square(mark(y0)), a__from(mark(x0)))

MARK(2ndspos(from(y0), square(x0))) → A__2NDSPOS(a__from(mark(y0)), a__square(mark(x0)))

A__2NDSPOS(s(y0), cons(y1, square(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__square(mark(x0))))

MARK(plus(0, y1)) → A__PLUS(0, mark(y1))

MARK(2ndsneg(s(y0), times(x0, x1))) → A__2NDSNEG(s(mark(y0)), a__times(mark(x0), mark(x1)))

A__PLUS(s(X), Y) → A__PLUS(mark(X), mark(Y))

MARK(2ndspos(square(y0), square(x0))) → A__2NDSPOS(a__square(mark(y0)), a__square(mark(x0)))

A__PI(pi(x0)) → A__2NDSPOS(a__pi(mark(x0)), a__from(0))

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → A__2NDSPOS(mark(N), mark(Z))

MARK(plus(from(x0), y1)) → A__PLUS(a__from(mark(x0)), mark(y1))

MARK(2ndsneg(times(y0, y1), pi(x0))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__pi(mark(x0)))

MARK(square(X)) → A__SQUARE(mark(X))

MARK(2ndsneg(times(y0, y1), plus(x0, x1))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(2ndsneg(plus(y0, y1), times(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(plus(y0, y1), plus(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(2ndsneg(2ndsneg(y0, y1), cons2(x0, x1))) → A__2NDSNEG(a__2ndsneg(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(2ndspos(2ndsneg(y0, y1), cons(x0, x1))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), cons(mark(x0), x1))

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))

a__2ndspos(0, Z) → rnil

a__2ndspos(s(N), cons(X, Z)) → a__2ndspos(s(mark(N)), cons2(X, mark(Z)))

a__2ndspos(s(N), cons2(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))

a__2ndsneg(0, Z) → rnil

a__2ndsneg(s(N), cons(X, Z)) → a__2ndsneg(s(mark(N)), cons2(X, mark(Z)))

a__2ndsneg(s(N), cons2(X, cons(Y, Z))) → rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z)))

a__pi(X) → a__2ndspos(mark(X), a__from(0))

a__plus(0, Y) → mark(Y)

a__plus(s(X), Y) → s(a__plus(mark(X), mark(Y)))

a__times(0, Y) → 0

a__times(s(X), Y) → a__plus(mark(Y), a__times(mark(X), mark(Y)))

a__square(X) → a__times(mark(X), mark(X))

mark(from(X)) → a__from(mark(X))

mark(2ndspos(X1, X2)) → a__2ndspos(mark(X1), mark(X2))

mark(2ndsneg(X1, X2)) → a__2ndsneg(mark(X1), mark(X2))

mark(pi(X)) → a__pi(mark(X))

mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))

mark(times(X1, X2)) → a__times(mark(X1), mark(X2))

mark(square(X)) → a__square(mark(X))

mark(0) → 0

mark(s(X)) → s(mark(X))

mark(posrecip(X)) → posrecip(mark(X))

mark(negrecip(X)) → negrecip(mark(X))

mark(nil) → nil

mark(cons(X1, X2)) → cons(mark(X1), X2)

mark(cons2(X1, X2)) → cons2(X1, mark(X2))

mark(rnil) → rnil

mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))

a__from(X) → from(X)

a__2ndspos(X1, X2) → 2ndspos(X1, X2)

a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)

a__pi(X) → pi(X)

a__plus(X1, X2) → plus(X1, X2)

a__times(X1, X2) → times(X1, X2)

a__square(X) → square(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 7 less nodes.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ Narrowing
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ QDP
              ↳ Narrowing
                ↳ QDP
                  ↳ DependencyGraphProof
                    ↳ QDP
                      ↳ Narrowing
                        ↳ QDP
                          ↳ DependencyGraphProof
                            ↳ QDP
                              ↳ Narrowing
                                ↳ QDP
                                  ↳ DependencyGraphProof
                                    ↳ QDP
                                      ↳ Narrowing
                                        ↳ QDP
                                          ↳ DependencyGraphProof
                                            ↳ QDP
                                              ↳ Narrowing
                                                ↳ QDP
                                                  ↳ DependencyGraphProof
                                                    ↳ QDP
                                                      ↳ Narrowing
                                                        ↳ QDP
                                                          ↳ DependencyGraphProof
                                                            ↳ QDP
                                                              ↳ Narrowing
                                                                ↳ QDP
                                                                  ↳ DependencyGraphProof
                                                                    ↳ QDP
                                                                      ↳ Narrowing
                                                                        ↳ QDP
                                                                          ↳ DependencyGraphProof
                                                                            ↳ QDP
                                                                              ↳ Narrowing
                                                                                ↳ QDP
                                                                                  ↳ DependencyGraphProof
                                                                                    ↳ QDP
                                                                                      ↳ Narrowing
                                                                                        ↳ QDP
                                                                                          ↳ DependencyGraphProof
                                                                                            ↳ QDP
                                                                                              ↳ Narrowing
                                                                                                ↳ QDP
                                                                                                  ↳ DependencyGraphProof
                                                                                                    ↳ QDP
                                                                                                      ↳ Narrowing
                                                                                                        ↳ QDP
                                                                                                          ↳ DependencyGraphProof
                                                                                                            ↳ QDP
                                                                                                              ↳ Narrowing
                                                                                                                ↳ QDP
                                                                                                                  ↳ DependencyGraphProof
                                                                                                                    ↳ QDP
                                                                                                                      ↳ Narrowing
                                                                                                                        ↳ QDP
                                                                                                                          ↳ DependencyGraphProof
                                                                                                                            ↳ QDP
                                                                                                                              ↳ Narrowing
                                                                                                                                ↳ QDP
                                                                                                                                  ↳ DependencyGraphProof
                                                                                                                                    ↳ QDP
                                                                                                                                      ↳ Narrowing
                                                                                                                                        ↳ QDP
                                                                                                                                          ↳ DependencyGraphProof
                                                                                                                                            ↳ QDP
                                                                                                                                              ↳ Narrowing
                                                                                                                                                ↳ QDP
                                                                                                                                                  ↳ DependencyGraphProof
                                                                                                                                                    ↳ QDP
                                                                                                                                                      ↳ Narrowing
                                                                                                                                                        ↳ QDP
                                                                                                                                                          ↳ DependencyGraphProof
                                                                                                                                                            ↳ QDP
                                                                                                                                                              ↳ Narrowing
                                                                                                                                                                ↳ QDP
                                                                                                                                                                  ↳ DependencyGraphProof
                                                                                                                                                                    ↳ QDP
                                                                                                                                                                      ↳ Narrowing
                                                                                                                                                                        ↳ QDP
                                                                                                                                                                          ↳ DependencyGraphProof
                                                                                                                                                                            ↳ QDP
                                                                                                                                                                              ↳ Narrowing

Q DP problem:
The TRS P consists of the following rules:

MARK(times(times(x0, x1), y1)) → A__TIMES(a__times(mark(x0), mark(x1)), mark(y1))

MARK(2ndsneg(2ndspos(y0, y1), 2ndsneg(x0, x1))) → A__2NDSNEG(a__2ndspos(mark(y0), mark(y1)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndsneg(2ndsneg(y0, y1), 2ndspos(x0, x1))) → A__2NDSNEG(a__2ndsneg(mark(y0), mark(y1)), a__2ndspos(mark(x0), mark(x1)))

MARK(plus(plus(x0, x1), y1)) → A__PLUS(a__plus(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(2ndsneg(y0, y1), from(x0))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndspos(from(y0), 2ndsneg(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

A__2NDSNEG(s(N), cons(X, Z)) → MARK(Z)

MARK(2ndsneg(square(y0), 2ndspos(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndsneg(2ndspos(y0, y1), square(x0))) → A__2NDSNEG(a__2ndspos(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndsneg(s(y0), pi(x0))) → A__2NDSNEG(s(mark(y0)), a__pi(mark(x0)))

MARK(2ndsneg(2ndspos(y0, y1), pi(x0))) → A__2NDSNEG(a__2ndspos(mark(y0), mark(y1)), a__pi(mark(x0)))

A__TIMES(s(X), Y) → MARK(Y)

MARK(2ndspos(square(y0), 2ndsneg(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndspos(2ndsneg(y0, y1), square(x0))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__square(mark(x0)))

A__2NDSNEG(s(N), cons(X, Z)) → MARK(N)

MARK(times(2ndsneg(x0, x1), y1)) → A__TIMES(a__2ndsneg(mark(x0), mark(x1)), mark(y1))

MARK(2ndsneg(times(y0, y1), times(x0, x1))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(pi(y0), pi(x0))) → A__2NDSPOS(a__pi(mark(y0)), a__pi(mark(x0)))

MARK(2ndsneg(square(y0), 2ndsneg(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndsneg(2ndsneg(y0, y1), square(x0))) → A__2NDSNEG(a__2ndsneg(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndsneg(from(y0), cons(x0, x1))) → A__2NDSNEG(a__from(mark(y0)), cons(mark(x0), x1))

MARK(2ndspos(s(y0), pi(x0))) → A__2NDSPOS(s(mark(y0)), a__pi(mark(x0)))

MARK(plus(square(x0), y1)) → A__PLUS(a__square(mark(x0)), mark(y1))

MARK(2ndsneg(times(y0, y1), cons2(x0, x1))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(cons2(X1, X2)) → MARK(X2)

MARK(2ndsneg(square(y0), cons2(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), cons2(x0, mark(x1)))

MARK(2ndsneg(2ndspos(y0, y1), times(x0, x1))) → A__2NDSNEG(a__2ndspos(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndsneg(times(y0, y1), 2ndspos(x0, x1))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndspos(X1, X2)) → MARK(X1)

MARK(pi(X)) → MARK(X)

MARK(2ndsneg(s(y0), from(x0))) → A__2NDSNEG(s(mark(y0)), a__from(mark(x0)))

MARK(pi(X)) → A__PI(mark(X))

MARK(2ndspos(plus(y0, y1), cons(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), cons(mark(x0), x1))

MARK(2ndsneg(square(y0), times(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndsneg(times(y0, y1), square(x0))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(plus(2ndsneg(x0, x1), y1)) → A__PLUS(a__2ndsneg(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(2ndspos(y0, y1), cons2(x0, x1))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(2ndsneg(X1, X2)) → MARK(X2)

MARK(2ndspos(square(y0), times(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(times(y0, y1), square(x0))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(times(square(x0), y1)) → A__TIMES(a__square(mark(x0)), mark(y1))

MARK(2ndspos(pi(y0), cons(x0, x1))) → A__2NDSPOS(a__pi(mark(y0)), cons(mark(x0), x1))

A__PI(X) → MARK(X)

MARK(times(s(x0), y1)) → A__TIMES(s(mark(x0)), mark(y1))

MARK(2ndspos(2ndspos(y0, y1), plus(x0, x1))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(plus(y0, y1), 2ndspos(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__2ndspos(mark(x0), mark(x1)))

A__PLUS(s(X), Y) → MARK(Y)

MARK(2ndsneg(plus(y0, y1), pi(x0))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__pi(mark(x0)))

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(N)

MARK(2ndspos(square(y0), 2ndspos(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndspos(2ndspos(y0, y1), square(x0))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndspos(s(y0), 2ndspos(x0, x1))) → A__2NDSPOS(s(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndspos(times(y0, y1), cons(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), cons(mark(x0), x1))

MARK(2ndspos(2ndsneg(y0, y1), 2ndsneg(x0, x1))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndspos(times(y0, y1), plus(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(plus(y0, y1), times(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndsneg(2ndspos(y0, y1), from(x0))) → A__2NDSNEG(a__2ndspos(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndsneg(from(y0), 2ndspos(x0, x1))) → A__2NDSNEG(a__from(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndsneg(2ndspos(y0, y1), cons2(x0, x1))) → A__2NDSNEG(a__2ndspos(mark(y0), mark(y1)), cons2(x0, mark(x1)))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(Y)

MARK(rcons(X1, X2)) → MARK(X1)

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(Z)

MARK(square(X)) → MARK(X)

MARK(2ndspos(pi(y0), times(x0, x1))) → A__2NDSPOS(a__pi(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(times(y0, y1), pi(x0))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__pi(mark(x0)))

A__PI(s(x0)) → A__2NDSPOS(s(mark(x0)), a__from(0))

MARK(2ndsneg(2ndsneg(y0, y1), pi(x0))) → A__2NDSNEG(a__2ndsneg(mark(y0), mark(y1)), a__pi(mark(x0)))

MARK(times(plus(x0, x1), y1)) → A__TIMES(a__plus(mark(x0), mark(x1)), mark(y1))

A__2NDSPOS(s(y0), cons(y1, from(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__from(mark(x0))))

MARK(2ndspos(times(y0, y1), 2ndspos(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndspos(2ndspos(y0, y1), times(x0, x1))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndsneg(from(y0), cons2(x0, x1))) → A__2NDSNEG(a__from(mark(y0)), cons2(x0, mark(x1)))

MARK(2ndspos(s(y0), square(x0))) → A__2NDSPOS(s(mark(y0)), a__square(mark(x0)))

MARK(times(2ndspos(x0, x1), y1)) → A__TIMES(a__2ndspos(mark(x0), mark(x1)), mark(y1))

MARK(2ndsneg(2ndsneg(y0, y1), from(x0))) → A__2NDSNEG(a__2ndsneg(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndsneg(from(y0), 2ndsneg(x0, x1))) → A__2NDSNEG(a__from(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndspos(s(y0), cons2(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(x0, mark(x1)))

MARK(2ndsneg(plus(y0, y1), cons2(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), cons2(x0, mark(x1)))

A__PLUS(s(X), Y) → MARK(X)

MARK(times(X1, X2)) → MARK(X1)

A__SQUARE(X) → MARK(X)

MARK(negrecip(X)) → MARK(X)

MARK(2ndspos(from(y0), times(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(times(y0, y1), from(x0))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndspos(2ndsneg(y0, y1), times(x0, x1))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(times(y0, y1), 2ndsneg(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__2ndsneg(mark(x0), mark(x1)))

MARK(from(X)) → MARK(X)

A__SQUARE(X) → A__TIMES(mark(X), mark(X))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(Z)

MARK(2ndsneg(plus(y0, y1), plus(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(from(y0), plus(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(plus(y0, y1), from(x0))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndsneg(2ndsneg(y0, y1), times(x0, x1))) → A__2NDSNEG(a__2ndsneg(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndsneg(times(y0, y1), 2ndsneg(x0, x1))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndsneg(times(y0, y1), cons(x0, x1))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), cons(mark(x0), x1))

MARK(2ndsneg(2ndsneg(y0, y1), cons(x0, x1))) → A__2NDSNEG(a__2ndsneg(mark(y0), mark(y1)), cons(mark(x0), x1))

MARK(2ndspos(from(y0), 2ndspos(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndspos(2ndspos(y0, y1), from(x0))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndspos(square(y0), plus(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(plus(y0, y1), square(x0))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__square(mark(x0)))

A__PI(times(x0, x1)) → A__2NDSPOS(a__times(mark(x0), mark(x1)), a__from(0))

A__2NDSPOS(s(y0), cons(y1, 2ndspos(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__2ndspos(mark(x0), mark(x1))))

A__2NDSNEG(s(N), cons(X, Z)) → A__2NDSNEG(s(mark(N)), cons2(X, mark(Z)))

MARK(2ndsneg(s(y0), square(x0))) → A__2NDSNEG(s(mark(y0)), a__square(mark(x0)))

MARK(2ndspos(X1, X2)) → MARK(X2)

MARK(2ndspos(pi(y0), cons2(x0, x1))) → A__2NDSPOS(a__pi(mark(y0)), cons2(x0, mark(x1)))

MARK(2ndspos(from(y0), cons2(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), cons2(x0, mark(x1)))

MARK(2ndsneg(from(y0), pi(x0))) → A__2NDSNEG(a__from(mark(y0)), a__pi(mark(x0)))

MARK(2ndsneg(2ndspos(y0, y1), 2ndspos(x0, x1))) → A__2NDSNEG(a__2ndspos(mark(y0), mark(y1)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndspos(s(y0), from(x0))) → A__2NDSPOS(s(mark(y0)), a__from(mark(x0)))

MARK(2ndspos(2ndsneg(y0, y1), pi(x0))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__pi(mark(x0)))

MARK(2ndspos(pi(y0), 2ndsneg(x0, x1))) → A__2NDSPOS(a__pi(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

A__2NDSPOS(s(y0), cons(y1, plus(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__plus(mark(x0), mark(x1))))

MARK(2ndspos(from(y0), pi(x0))) → A__2NDSPOS(a__from(mark(y0)), a__pi(mark(x0)))

MARK(2ndspos(pi(y0), from(x0))) → A__2NDSPOS(a__pi(mark(y0)), a__from(mark(x0)))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → A__2NDSNEG(mark(N), mark(Z))

MARK(2ndspos(from(y0), from(x0))) → A__2NDSPOS(a__from(mark(y0)), a__from(mark(x0)))

MARK(2ndspos(times(y0, y1), times(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

A__TIMES(s(X), Y) → A__PLUS(mark(Y), a__times(mark(X), mark(Y)))

A__PI(2ndspos(x0, x1)) → A__2NDSPOS(a__2ndspos(mark(x0), mark(x1)), a__from(0))

MARK(2ndspos(square(y0), cons2(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), cons2(x0, mark(x1)))

MARK(plus(s(x0), y1)) → A__PLUS(s(mark(x0)), mark(y1))

A__PI(plus(x0, x1)) → A__2NDSPOS(a__plus(mark(x0), mark(x1)), a__from(0))

MARK(2ndspos(square(y0), cons(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), cons(mark(x0), x1))

MARK(s(X)) → MARK(X)

MARK(2ndsneg(square(y0), plus(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(2ndsneg(plus(y0, y1), square(x0))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndspos(plus(y0, y1), cons2(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(plus(X1, X2)) → MARK(X2)

A__2NDSPOS(s(N), cons(X, Z)) → MARK(Z)

MARK(2ndsneg(from(y0), times(x0, x1))) → A__2NDSNEG(a__from(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndsneg(times(y0, y1), from(x0))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndspos(times(y0, y1), cons2(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(2ndsneg(square(y0), from(x0))) → A__2NDSNEG(a__square(mark(y0)), a__from(mark(x0)))

MARK(2ndsneg(from(y0), square(x0))) → A__2NDSNEG(a__from(mark(y0)), a__square(mark(x0)))

MARK(2ndspos(2ndsneg(y0, y1), cons2(x0, x1))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(2ndsneg(s(y0), cons(x0, x1))) → A__2NDSNEG(s(mark(y0)), cons(mark(x0), x1))

A__PI(X) → A__FROM(0)

MARK(2ndspos(s(y0), plus(x0, x1))) → A__2NDSPOS(s(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(from(X)) → A__FROM(mark(X))

MARK(plus(X1, X2)) → MARK(X1)

MARK(times(from(x0), y1)) → A__TIMES(a__from(mark(x0)), mark(y1))

A__2NDSPOS(s(y0), cons(y1, times(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__times(mark(x0), mark(x1))))

MARK(2ndsneg(2ndsneg(y0, y1), 2ndsneg(x0, x1))) → A__2NDSNEG(a__2ndsneg(mark(y0), mark(y1)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndspos(s(y0), cons(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons(mark(x0), x1))

MARK(2ndsneg(2ndspos(y0, y1), plus(x0, x1))) → A__2NDSNEG(a__2ndspos(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(2ndsneg(plus(y0, y1), 2ndspos(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndspos(pi(y0), 2ndspos(x0, x1))) → A__2NDSPOS(a__pi(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndspos(2ndspos(y0, y1), pi(x0))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), a__pi(mark(x0)))

MARK(2ndspos(from(y0), cons(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), cons(mark(x0), x1))

MARK(2ndsneg(2ndspos(y0, y1), cons(x0, x1))) → A__2NDSNEG(a__2ndspos(mark(y0), mark(y1)), cons(mark(x0), x1))

MARK(2ndsneg(plus(y0, y1), cons(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), cons(mark(x0), x1))

A__2NDSPOS(s(y0), cons(y1, pi(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__pi(mark(x0))))

A__TIMES(s(X), Y) → MARK(X)

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(N)

MARK(2ndspos(pi(y0), plus(x0, x1))) → A__2NDSPOS(a__pi(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(plus(y0, y1), pi(x0))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__pi(mark(x0)))

MARK(2ndsneg(s(y0), cons2(x0, x1))) → A__2NDSNEG(s(mark(y0)), cons2(x0, mark(x1)))

MARK(plus(2ndspos(x0, x1), y1)) → A__PLUS(a__2ndspos(mark(x0), mark(x1)), mark(y1))

A__PI(from(x0)) → A__2NDSPOS(a__from(mark(x0)), a__from(0))

MARK(cons(X1, X2)) → MARK(X1)

MARK(2ndsneg(s(y0), 2ndspos(x0, x1))) → A__2NDSNEG(s(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndsneg(plus(y0, y1), from(x0))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndsneg(from(y0), plus(x0, x1))) → A__2NDSNEG(a__from(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(times(X1, X2)) → MARK(X2)

A__PI(square(x0)) → A__2NDSPOS(a__square(mark(x0)), a__from(0))

MARK(2ndsneg(square(y0), cons(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), cons(mark(x0), x1))

MARK(2ndspos(s(y0), times(x0, x1))) → A__2NDSPOS(s(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndsneg(X1, X2)) → MARK(X1)

MARK(2ndspos(s(y0), 2ndsneg(x0, x1))) → A__2NDSPOS(s(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndsneg(square(y0), square(x0))) → A__2NDSNEG(a__square(mark(y0)), a__square(mark(x0)))

MARK(2ndspos(2ndsneg(y0, y1), plus(x0, x1))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(plus(y0, y1), 2ndsneg(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__2ndsneg(mark(x0), mark(x1)))

A__PI(2ndsneg(x0, x1)) → A__2NDSPOS(a__2ndsneg(mark(x0), mark(x1)), a__from(0))

MARK(times(pi(x0), y1)) → A__TIMES(a__pi(mark(x0)), mark(y1))

MARK(plus(pi(x0), y1)) → A__PLUS(a__pi(mark(x0)), mark(y1))

MARK(2ndspos(2ndspos(y0, y1), 2ndspos(x0, x1))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndsneg(square(y0), pi(x0))) → A__2NDSNEG(a__square(mark(y0)), a__pi(mark(x0)))

MARK(2ndspos(2ndsneg(y0, y1), 2ndspos(x0, x1))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndspos(2ndspos(y0, y1), 2ndsneg(x0, x1))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), a__2ndsneg(mark(x0), mark(x1)))

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(Y)

MARK(2ndsneg(s(y0), 2ndsneg(x0, x1))) → A__2NDSNEG(s(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

A__FROM(X) → MARK(X)

MARK(2ndspos(2ndspos(y0, y1), cons(x0, x1))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), cons(mark(x0), x1))

MARK(2ndsneg(s(y0), plus(x0, x1))) → A__2NDSNEG(s(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(square(y0), pi(x0))) → A__2NDSPOS(a__square(mark(y0)), a__pi(mark(x0)))

MARK(2ndspos(pi(y0), square(x0))) → A__2NDSPOS(a__pi(mark(y0)), a__square(mark(x0)))

MARK(rcons(X1, X2)) → MARK(X2)

A__TIMES(s(X), Y) → A__TIMES(mark(X), mark(Y))

A__2NDSPOS(s(y0), cons(y1, cons(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, cons(mark(x0), x1)))

MARK(plus(times(x0, x1), y1)) → A__PLUS(a__times(mark(x0), mark(x1)), mark(y1))

MARK(2ndsneg(from(y0), from(x0))) → A__2NDSNEG(a__from(mark(y0)), a__from(mark(x0)))

A__2NDSPOS(s(y0), cons(y1, 2ndsneg(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__2ndsneg(mark(x0), mark(x1))))

A__2NDSPOS(s(N), cons(X, Z)) → MARK(N)

MARK(2ndsneg(plus(y0, y1), 2ndsneg(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndsneg(2ndsneg(y0, y1), plus(x0, x1))) → A__2NDSNEG(a__2ndsneg(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

A__PLUS(0, Y) → MARK(Y)

MARK(posrecip(X)) → MARK(X)

MARK(2ndspos(from(y0), square(x0))) → A__2NDSPOS(a__from(mark(y0)), a__square(mark(x0)))

MARK(2ndspos(square(y0), from(x0))) → A__2NDSPOS(a__square(mark(y0)), a__from(mark(x0)))

MARK(2ndsneg(pi(x0), y1)) → A__2NDSNEG(a__pi(mark(x0)), mark(y1))

A__2NDSPOS(s(y0), cons(y1, square(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__square(mark(x0))))

MARK(plus(0, y1)) → A__PLUS(0, mark(y1))

MARK(2ndsneg(s(y0), times(x0, x1))) → A__2NDSNEG(s(mark(y0)), a__times(mark(x0), mark(x1)))

A__PLUS(s(X), Y) → A__PLUS(mark(X), mark(Y))

MARK(2ndspos(square(y0), square(x0))) → A__2NDSPOS(a__square(mark(y0)), a__square(mark(x0)))

A__PI(pi(x0)) → A__2NDSPOS(a__pi(mark(x0)), a__from(0))

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → A__2NDSPOS(mark(N), mark(Z))

MARK(plus(from(x0), y1)) → A__PLUS(a__from(mark(x0)), mark(y1))

MARK(square(X)) → A__SQUARE(mark(X))

MARK(2ndsneg(times(y0, y1), pi(x0))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__pi(mark(x0)))

MARK(2ndsneg(plus(y0, y1), times(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndsneg(times(y0, y1), plus(x0, x1))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(plus(y0, y1), plus(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(2ndsneg(2ndsneg(y0, y1), cons2(x0, x1))) → A__2NDSNEG(a__2ndsneg(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(2ndspos(2ndsneg(y0, y1), cons(x0, x1))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), cons(mark(x0), x1))

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))

a__2ndspos(0, Z) → rnil

a__2ndspos(s(N), cons(X, Z)) → a__2ndspos(s(mark(N)), cons2(X, mark(Z)))

a__2ndspos(s(N), cons2(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))

a__2ndsneg(0, Z) → rnil

a__2ndsneg(s(N), cons(X, Z)) → a__2ndsneg(s(mark(N)), cons2(X, mark(Z)))

a__2ndsneg(s(N), cons2(X, cons(Y, Z))) → rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z)))

a__pi(X) → a__2ndspos(mark(X), a__from(0))

a__plus(0, Y) → mark(Y)

a__plus(s(X), Y) → s(a__plus(mark(X), mark(Y)))

a__times(0, Y) → 0

a__times(s(X), Y) → a__plus(mark(Y), a__times(mark(X), mark(Y)))

a__square(X) → a__times(mark(X), mark(X))

mark(from(X)) → a__from(mark(X))

mark(2ndspos(X1, X2)) → a__2ndspos(mark(X1), mark(X2))

mark(2ndsneg(X1, X2)) → a__2ndsneg(mark(X1), mark(X2))

mark(pi(X)) → a__pi(mark(X))

mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))

mark(times(X1, X2)) → a__times(mark(X1), mark(X2))

mark(square(X)) → a__square(mark(X))

mark(0) → 0

mark(s(X)) → s(mark(X))

mark(posrecip(X)) → posrecip(mark(X))

mark(negrecip(X)) → negrecip(mark(X))

mark(nil) → nil

mark(cons(X1, X2)) → cons(mark(X1), X2)

mark(cons2(X1, X2)) → cons2(X1, mark(X2))

mark(rnil) → rnil

mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))

a__from(X) → from(X)

a__2ndspos(X1, X2) → 2ndspos(X1, X2)

a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)

a__pi(X) → pi(X)

a__plus(X1, X2) → plus(X1, X2)

a__times(X1, X2) → times(X1, X2)

a__square(X) → square(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule MARK(2ndsneg(pi(x0), y1)) → A__2NDSNEG(a__pi(mark(x0)), mark(y1)) at position [1] we obtained the following new rules:

MARK(2ndsneg(pi(y0), 2ndspos(x0, x1))) → A__2NDSNEG(a__pi(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndsneg(pi(y0), cons(x0, x1))) → A__2NDSNEG(a__pi(mark(y0)), cons(mark(x0), x1))

MARK(2ndsneg(pi(y0), square(x0))) → A__2NDSNEG(a__pi(mark(y0)), a__square(mark(x0)))

MARK(2ndsneg(pi(y0), 2ndsneg(x0, x1))) → A__2NDSNEG(a__pi(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndsneg(pi(y0), from(x0))) → A__2NDSNEG(a__pi(mark(y0)), a__from(mark(x0)))

MARK(2ndsneg(pi(y0), times(x0, x1))) → A__2NDSNEG(a__pi(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndsneg(pi(y0), pi(x0))) → A__2NDSNEG(a__pi(mark(y0)), a__pi(mark(x0)))

MARK(2ndsneg(pi(y0), rcons(x0, x1))) → A__2NDSNEG(a__pi(mark(y0)), rcons(mark(x0), mark(x1)))

MARK(2ndsneg(pi(y0), cons2(x0, x1))) → A__2NDSNEG(a__pi(mark(y0)), cons2(x0, mark(x1)))

MARK(2ndsneg(pi(y0), rnil)) → A__2NDSNEG(a__pi(mark(y0)), rnil)

MARK(2ndsneg(pi(y0), negrecip(x0))) → A__2NDSNEG(a__pi(mark(y0)), negrecip(mark(x0)))

MARK(2ndsneg(pi(y0), posrecip(x0))) → A__2NDSNEG(a__pi(mark(y0)), posrecip(mark(x0)))

MARK(2ndsneg(pi(y0), nil)) → A__2NDSNEG(a__pi(mark(y0)), nil)

MARK(2ndsneg(pi(y0), s(x0))) → A__2NDSNEG(a__pi(mark(y0)), s(mark(x0)))

MARK(2ndsneg(pi(y0), 0)) → A__2NDSNEG(a__pi(mark(y0)), 0)

MARK(2ndsneg(pi(y0), plus(x0, x1))) → A__2NDSNEG(a__pi(mark(y0)), a__plus(mark(x0), mark(x1)))



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ Narrowing
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ QDP
              ↳ Narrowing
                ↳ QDP
                  ↳ DependencyGraphProof
                    ↳ QDP
                      ↳ Narrowing
                        ↳ QDP
                          ↳ DependencyGraphProof
                            ↳ QDP
                              ↳ Narrowing
                                ↳ QDP
                                  ↳ DependencyGraphProof
                                    ↳ QDP
                                      ↳ Narrowing
                                        ↳ QDP
                                          ↳ DependencyGraphProof
                                            ↳ QDP
                                              ↳ Narrowing
                                                ↳ QDP
                                                  ↳ DependencyGraphProof
                                                    ↳ QDP
                                                      ↳ Narrowing
                                                        ↳ QDP
                                                          ↳ DependencyGraphProof
                                                            ↳ QDP
                                                              ↳ Narrowing
                                                                ↳ QDP
                                                                  ↳ DependencyGraphProof
                                                                    ↳ QDP
                                                                      ↳ Narrowing
                                                                        ↳ QDP
                                                                          ↳ DependencyGraphProof
                                                                            ↳ QDP
                                                                              ↳ Narrowing
                                                                                ↳ QDP
                                                                                  ↳ DependencyGraphProof
                                                                                    ↳ QDP
                                                                                      ↳ Narrowing
                                                                                        ↳ QDP
                                                                                          ↳ DependencyGraphProof
                                                                                            ↳ QDP
                                                                                              ↳ Narrowing
                                                                                                ↳ QDP
                                                                                                  ↳ DependencyGraphProof
                                                                                                    ↳ QDP
                                                                                                      ↳ Narrowing
                                                                                                        ↳ QDP
                                                                                                          ↳ DependencyGraphProof
                                                                                                            ↳ QDP
                                                                                                              ↳ Narrowing
                                                                                                                ↳ QDP
                                                                                                                  ↳ DependencyGraphProof
                                                                                                                    ↳ QDP
                                                                                                                      ↳ Narrowing
                                                                                                                        ↳ QDP
                                                                                                                          ↳ DependencyGraphProof
                                                                                                                            ↳ QDP
                                                                                                                              ↳ Narrowing
                                                                                                                                ↳ QDP
                                                                                                                                  ↳ DependencyGraphProof
                                                                                                                                    ↳ QDP
                                                                                                                                      ↳ Narrowing
                                                                                                                                        ↳ QDP
                                                                                                                                          ↳ DependencyGraphProof
                                                                                                                                            ↳ QDP
                                                                                                                                              ↳ Narrowing
                                                                                                                                                ↳ QDP
                                                                                                                                                  ↳ DependencyGraphProof
                                                                                                                                                    ↳ QDP
                                                                                                                                                      ↳ Narrowing
                                                                                                                                                        ↳ QDP
                                                                                                                                                          ↳ DependencyGraphProof
                                                                                                                                                            ↳ QDP
                                                                                                                                                              ↳ Narrowing
                                                                                                                                                                ↳ QDP
                                                                                                                                                                  ↳ DependencyGraphProof
                                                                                                                                                                    ↳ QDP
                                                                                                                                                                      ↳ Narrowing
                                                                                                                                                                        ↳ QDP
                                                                                                                                                                          ↳ DependencyGraphProof
                                                                                                                                                                            ↳ QDP
                                                                                                                                                                              ↳ Narrowing
                                                                                                                                                                                ↳ QDP
                                                                                                                                                                                  ↳ DependencyGraphProof

Q DP problem:
The TRS P consists of the following rules:

MARK(times(times(x0, x1), y1)) → A__TIMES(a__times(mark(x0), mark(x1)), mark(y1))

MARK(2ndsneg(2ndsneg(y0, y1), 2ndspos(x0, x1))) → A__2NDSNEG(a__2ndsneg(mark(y0), mark(y1)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndsneg(2ndspos(y0, y1), 2ndsneg(x0, x1))) → A__2NDSNEG(a__2ndspos(mark(y0), mark(y1)), a__2ndsneg(mark(x0), mark(x1)))

MARK(plus(plus(x0, x1), y1)) → A__PLUS(a__plus(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(from(y0), 2ndsneg(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndspos(2ndsneg(y0, y1), from(x0))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndsneg(2ndspos(y0, y1), square(x0))) → A__2NDSNEG(a__2ndspos(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndsneg(square(y0), 2ndspos(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

A__2NDSNEG(s(N), cons(X, Z)) → MARK(Z)

MARK(2ndsneg(pi(y0), s(x0))) → A__2NDSNEG(a__pi(mark(y0)), s(mark(x0)))

MARK(2ndsneg(s(y0), pi(x0))) → A__2NDSNEG(s(mark(y0)), a__pi(mark(x0)))

MARK(2ndsneg(pi(y0), 2ndspos(x0, x1))) → A__2NDSNEG(a__pi(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndsneg(2ndspos(y0, y1), pi(x0))) → A__2NDSNEG(a__2ndspos(mark(y0), mark(y1)), a__pi(mark(x0)))

A__TIMES(s(X), Y) → MARK(Y)

MARK(2ndspos(2ndsneg(y0, y1), square(x0))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndspos(square(y0), 2ndsneg(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

A__2NDSNEG(s(N), cons(X, Z)) → MARK(N)

MARK(times(2ndsneg(x0, x1), y1)) → A__TIMES(a__2ndsneg(mark(x0), mark(x1)), mark(y1))

MARK(2ndsneg(times(y0, y1), times(x0, x1))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(pi(y0), pi(x0))) → A__2NDSPOS(a__pi(mark(y0)), a__pi(mark(x0)))

MARK(2ndsneg(2ndsneg(y0, y1), square(x0))) → A__2NDSNEG(a__2ndsneg(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndsneg(square(y0), 2ndsneg(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndsneg(from(y0), cons(x0, x1))) → A__2NDSNEG(a__from(mark(y0)), cons(mark(x0), x1))

MARK(2ndspos(s(y0), pi(x0))) → A__2NDSPOS(s(mark(y0)), a__pi(mark(x0)))

MARK(plus(square(x0), y1)) → A__PLUS(a__square(mark(x0)), mark(y1))

MARK(2ndsneg(pi(y0), posrecip(x0))) → A__2NDSNEG(a__pi(mark(y0)), posrecip(mark(x0)))

MARK(2ndsneg(times(y0, y1), cons2(x0, x1))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(cons2(X1, X2)) → MARK(X2)

MARK(2ndsneg(times(y0, y1), 2ndspos(x0, x1))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndsneg(2ndspos(y0, y1), times(x0, x1))) → A__2NDSNEG(a__2ndspos(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndsneg(square(y0), cons2(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), cons2(x0, mark(x1)))

MARK(2ndspos(X1, X2)) → MARK(X1)

MARK(pi(X)) → MARK(X)

MARK(2ndsneg(s(y0), from(x0))) → A__2NDSNEG(s(mark(y0)), a__from(mark(x0)))

MARK(pi(X)) → A__PI(mark(X))

MARK(2ndsneg(times(y0, y1), square(x0))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndsneg(square(y0), times(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(plus(y0, y1), cons(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), cons(mark(x0), x1))

MARK(plus(2ndsneg(x0, x1), y1)) → A__PLUS(a__2ndsneg(mark(x0), mark(x1)), mark(y1))

MARK(2ndsneg(pi(y0), rnil)) → A__2NDSNEG(a__pi(mark(y0)), rnil)

MARK(2ndspos(2ndspos(y0, y1), cons2(x0, x1))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(2ndsneg(X1, X2)) → MARK(X2)

MARK(2ndspos(times(y0, y1), square(x0))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndspos(square(y0), times(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(times(square(x0), y1)) → A__TIMES(a__square(mark(x0)), mark(y1))

MARK(2ndspos(pi(y0), cons(x0, x1))) → A__2NDSPOS(a__pi(mark(y0)), cons(mark(x0), x1))

A__PI(X) → MARK(X)

MARK(times(s(x0), y1)) → A__TIMES(s(mark(x0)), mark(y1))

MARK(2ndspos(plus(y0, y1), 2ndspos(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndspos(2ndspos(y0, y1), plus(x0, x1))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

A__PLUS(s(X), Y) → MARK(Y)

MARK(2ndsneg(pi(y0), plus(x0, x1))) → A__2NDSNEG(a__pi(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(2ndsneg(plus(y0, y1), pi(x0))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__pi(mark(x0)))

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(N)

MARK(2ndspos(2ndspos(y0, y1), square(x0))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndspos(square(y0), 2ndspos(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndspos(times(y0, y1), cons(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), cons(mark(x0), x1))

MARK(2ndspos(s(y0), 2ndspos(x0, x1))) → A__2NDSPOS(s(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndspos(2ndsneg(y0, y1), 2ndsneg(x0, x1))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndsneg(from(y0), 2ndspos(x0, x1))) → A__2NDSNEG(a__from(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndsneg(2ndspos(y0, y1), from(x0))) → A__2NDSNEG(a__2ndspos(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndspos(plus(y0, y1), times(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(times(y0, y1), plus(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(2ndsneg(2ndspos(y0, y1), cons2(x0, x1))) → A__2NDSNEG(a__2ndspos(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(2ndsneg(pi(y0), negrecip(x0))) → A__2NDSNEG(a__pi(mark(y0)), negrecip(mark(x0)))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(Y)

MARK(rcons(X1, X2)) → MARK(X1)

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(Z)

MARK(square(X)) → MARK(X)

MARK(2ndspos(times(y0, y1), pi(x0))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__pi(mark(x0)))

MARK(2ndspos(pi(y0), times(x0, x1))) → A__2NDSPOS(a__pi(mark(y0)), a__times(mark(x0), mark(x1)))

A__PI(s(x0)) → A__2NDSPOS(s(mark(x0)), a__from(0))

MARK(2ndsneg(pi(y0), 2ndsneg(x0, x1))) → A__2NDSNEG(a__pi(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndsneg(2ndsneg(y0, y1), pi(x0))) → A__2NDSNEG(a__2ndsneg(mark(y0), mark(y1)), a__pi(mark(x0)))

MARK(times(plus(x0, x1), y1)) → A__TIMES(a__plus(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(2ndspos(y0, y1), times(x0, x1))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(times(y0, y1), 2ndspos(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__2ndspos(mark(x0), mark(x1)))

A__2NDSPOS(s(y0), cons(y1, from(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__from(mark(x0))))

MARK(2ndsneg(from(y0), cons2(x0, x1))) → A__2NDSNEG(a__from(mark(y0)), cons2(x0, mark(x1)))

MARK(2ndspos(s(y0), square(x0))) → A__2NDSPOS(s(mark(y0)), a__square(mark(x0)))

MARK(times(2ndspos(x0, x1), y1)) → A__TIMES(a__2ndspos(mark(x0), mark(x1)), mark(y1))

MARK(2ndsneg(from(y0), 2ndsneg(x0, x1))) → A__2NDSNEG(a__from(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndsneg(2ndsneg(y0, y1), from(x0))) → A__2NDSNEG(a__2ndsneg(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndsneg(pi(y0), 0)) → A__2NDSNEG(a__pi(mark(y0)), 0)

MARK(2ndspos(s(y0), cons2(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(x0, mark(x1)))

MARK(2ndsneg(plus(y0, y1), cons2(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), cons2(x0, mark(x1)))

A__PLUS(s(X), Y) → MARK(X)

MARK(times(X1, X2)) → MARK(X1)

A__SQUARE(X) → MARK(X)

MARK(2ndsneg(pi(y0), pi(x0))) → A__2NDSNEG(a__pi(mark(y0)), a__pi(mark(x0)))

MARK(negrecip(X)) → MARK(X)

MARK(2ndsneg(pi(y0), rcons(x0, x1))) → A__2NDSNEG(a__pi(mark(y0)), rcons(mark(x0), mark(x1)))

MARK(2ndspos(times(y0, y1), from(x0))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndspos(from(y0), times(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(times(y0, y1), 2ndsneg(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndspos(2ndsneg(y0, y1), times(x0, x1))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(from(X)) → MARK(X)

A__SQUARE(X) → A__TIMES(mark(X), mark(X))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(Z)

MARK(2ndsneg(plus(y0, y1), plus(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(plus(y0, y1), from(x0))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndspos(from(y0), plus(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(2ndsneg(times(y0, y1), 2ndsneg(x0, x1))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndsneg(2ndsneg(y0, y1), times(x0, x1))) → A__2NDSNEG(a__2ndsneg(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndsneg(times(y0, y1), cons(x0, x1))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), cons(mark(x0), x1))

MARK(2ndsneg(2ndsneg(y0, y1), cons(x0, x1))) → A__2NDSNEG(a__2ndsneg(mark(y0), mark(y1)), cons(mark(x0), x1))

MARK(2ndspos(2ndspos(y0, y1), from(x0))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndspos(from(y0), 2ndspos(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndspos(plus(y0, y1), square(x0))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndspos(square(y0), plus(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__plus(mark(x0), mark(x1)))

A__2NDSPOS(s(y0), cons(y1, 2ndspos(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__2ndspos(mark(x0), mark(x1))))

A__PI(times(x0, x1)) → A__2NDSPOS(a__times(mark(x0), mark(x1)), a__from(0))

A__2NDSNEG(s(N), cons(X, Z)) → A__2NDSNEG(s(mark(N)), cons2(X, mark(Z)))

MARK(2ndsneg(s(y0), square(x0))) → A__2NDSNEG(s(mark(y0)), a__square(mark(x0)))

MARK(2ndspos(pi(y0), cons2(x0, x1))) → A__2NDSPOS(a__pi(mark(y0)), cons2(x0, mark(x1)))

MARK(2ndspos(X1, X2)) → MARK(X2)

MARK(2ndspos(from(y0), cons2(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), cons2(x0, mark(x1)))

MARK(2ndsneg(pi(y0), from(x0))) → A__2NDSNEG(a__pi(mark(y0)), a__from(mark(x0)))

MARK(2ndsneg(from(y0), pi(x0))) → A__2NDSNEG(a__from(mark(y0)), a__pi(mark(x0)))

MARK(2ndsneg(2ndspos(y0, y1), 2ndspos(x0, x1))) → A__2NDSNEG(a__2ndspos(mark(y0), mark(y1)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndspos(s(y0), from(x0))) → A__2NDSPOS(s(mark(y0)), a__from(mark(x0)))

MARK(2ndspos(pi(y0), 2ndsneg(x0, x1))) → A__2NDSPOS(a__pi(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndspos(2ndsneg(y0, y1), pi(x0))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__pi(mark(x0)))

A__2NDSPOS(s(y0), cons(y1, plus(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__plus(mark(x0), mark(x1))))

MARK(2ndspos(pi(y0), from(x0))) → A__2NDSPOS(a__pi(mark(y0)), a__from(mark(x0)))

MARK(2ndspos(from(y0), pi(x0))) → A__2NDSPOS(a__from(mark(y0)), a__pi(mark(x0)))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → A__2NDSNEG(mark(N), mark(Z))

MARK(2ndspos(from(y0), from(x0))) → A__2NDSPOS(a__from(mark(y0)), a__from(mark(x0)))

MARK(2ndspos(times(y0, y1), times(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

A__TIMES(s(X), Y) → A__PLUS(mark(Y), a__times(mark(X), mark(Y)))

MARK(2ndsneg(pi(y0), cons2(x0, x1))) → A__2NDSNEG(a__pi(mark(y0)), cons2(x0, mark(x1)))

MARK(2ndspos(square(y0), cons2(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), cons2(x0, mark(x1)))

A__PI(2ndspos(x0, x1)) → A__2NDSPOS(a__2ndspos(mark(x0), mark(x1)), a__from(0))

MARK(plus(s(x0), y1)) → A__PLUS(s(mark(x0)), mark(y1))

MARK(2ndspos(square(y0), cons(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), cons(mark(x0), x1))

A__PI(plus(x0, x1)) → A__2NDSPOS(a__plus(mark(x0), mark(x1)), a__from(0))

MARK(2ndsneg(pi(y0), nil)) → A__2NDSNEG(a__pi(mark(y0)), nil)

MARK(2ndsneg(plus(y0, y1), square(x0))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndsneg(square(y0), plus(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(s(X)) → MARK(X)

MARK(2ndspos(plus(y0, y1), cons2(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(plus(X1, X2)) → MARK(X2)

A__2NDSPOS(s(N), cons(X, Z)) → MARK(Z)

MARK(2ndsneg(times(y0, y1), from(x0))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndsneg(from(y0), times(x0, x1))) → A__2NDSNEG(a__from(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(times(y0, y1), cons2(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(2ndsneg(from(y0), square(x0))) → A__2NDSNEG(a__from(mark(y0)), a__square(mark(x0)))

MARK(2ndsneg(square(y0), from(x0))) → A__2NDSNEG(a__square(mark(y0)), a__from(mark(x0)))

MARK(2ndspos(2ndsneg(y0, y1), cons2(x0, x1))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(2ndsneg(s(y0), cons(x0, x1))) → A__2NDSNEG(s(mark(y0)), cons(mark(x0), x1))

MARK(from(X)) → A__FROM(mark(X))

MARK(2ndspos(s(y0), plus(x0, x1))) → A__2NDSPOS(s(mark(y0)), a__plus(mark(x0), mark(x1)))

A__PI(X) → A__FROM(0)

MARK(times(from(x0), y1)) → A__TIMES(a__from(mark(x0)), mark(y1))

MARK(plus(X1, X2)) → MARK(X1)

A__2NDSPOS(s(y0), cons(y1, times(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__times(mark(x0), mark(x1))))

MARK(2ndsneg(2ndsneg(y0, y1), 2ndsneg(x0, x1))) → A__2NDSNEG(a__2ndsneg(mark(y0), mark(y1)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndspos(s(y0), cons(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons(mark(x0), x1))

MARK(2ndsneg(plus(y0, y1), 2ndspos(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndsneg(2ndspos(y0, y1), plus(x0, x1))) → A__2NDSNEG(a__2ndspos(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(2ndspos(y0, y1), pi(x0))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), a__pi(mark(x0)))

MARK(2ndspos(pi(y0), 2ndspos(x0, x1))) → A__2NDSPOS(a__pi(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndspos(from(y0), cons(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), cons(mark(x0), x1))

MARK(2ndsneg(plus(y0, y1), cons(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), cons(mark(x0), x1))

MARK(2ndsneg(2ndspos(y0, y1), cons(x0, x1))) → A__2NDSNEG(a__2ndspos(mark(y0), mark(y1)), cons(mark(x0), x1))

A__TIMES(s(X), Y) → MARK(X)

A__2NDSPOS(s(y0), cons(y1, pi(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__pi(mark(x0))))

MARK(2ndspos(plus(y0, y1), pi(x0))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__pi(mark(x0)))

MARK(2ndspos(pi(y0), plus(x0, x1))) → A__2NDSPOS(a__pi(mark(y0)), a__plus(mark(x0), mark(x1)))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(N)

MARK(2ndsneg(s(y0), cons2(x0, x1))) → A__2NDSNEG(s(mark(y0)), cons2(x0, mark(x1)))

MARK(plus(2ndspos(x0, x1), y1)) → A__PLUS(a__2ndspos(mark(x0), mark(x1)), mark(y1))

MARK(2ndsneg(pi(y0), cons(x0, x1))) → A__2NDSNEG(a__pi(mark(y0)), cons(mark(x0), x1))

MARK(cons(X1, X2)) → MARK(X1)

A__PI(from(x0)) → A__2NDSPOS(a__from(mark(x0)), a__from(0))

MARK(2ndsneg(from(y0), plus(x0, x1))) → A__2NDSNEG(a__from(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(2ndsneg(plus(y0, y1), from(x0))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndsneg(s(y0), 2ndspos(x0, x1))) → A__2NDSNEG(s(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

MARK(times(X1, X2)) → MARK(X2)

A__PI(square(x0)) → A__2NDSPOS(a__square(mark(x0)), a__from(0))

MARK(2ndsneg(square(y0), cons(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), cons(mark(x0), x1))

MARK(2ndspos(s(y0), times(x0, x1))) → A__2NDSPOS(s(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndsneg(X1, X2)) → MARK(X1)

MARK(2ndspos(s(y0), 2ndsneg(x0, x1))) → A__2NDSPOS(s(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndsneg(square(y0), square(x0))) → A__2NDSNEG(a__square(mark(y0)), a__square(mark(x0)))

MARK(2ndspos(plus(y0, y1), 2ndsneg(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndspos(2ndsneg(y0, y1), plus(x0, x1))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

A__PI(2ndsneg(x0, x1)) → A__2NDSPOS(a__2ndsneg(mark(x0), mark(x1)), a__from(0))

MARK(plus(pi(x0), y1)) → A__PLUS(a__pi(mark(x0)), mark(y1))

MARK(times(pi(x0), y1)) → A__TIMES(a__pi(mark(x0)), mark(y1))

MARK(2ndspos(2ndspos(y0, y1), 2ndspos(x0, x1))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndsneg(pi(y0), square(x0))) → A__2NDSNEG(a__pi(mark(y0)), a__square(mark(x0)))

MARK(2ndsneg(square(y0), pi(x0))) → A__2NDSNEG(a__square(mark(y0)), a__pi(mark(x0)))

MARK(2ndspos(2ndspos(y0, y1), 2ndsneg(x0, x1))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndspos(2ndsneg(y0, y1), 2ndspos(x0, x1))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__2ndspos(mark(x0), mark(x1)))

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(Y)

MARK(2ndspos(2ndspos(y0, y1), cons(x0, x1))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), cons(mark(x0), x1))

A__FROM(X) → MARK(X)

MARK(2ndsneg(s(y0), 2ndsneg(x0, x1))) → A__2NDSNEG(s(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndsneg(s(y0), plus(x0, x1))) → A__2NDSNEG(s(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(pi(y0), square(x0))) → A__2NDSPOS(a__pi(mark(y0)), a__square(mark(x0)))

MARK(2ndspos(square(y0), pi(x0))) → A__2NDSPOS(a__square(mark(y0)), a__pi(mark(x0)))

MARK(rcons(X1, X2)) → MARK(X2)

A__TIMES(s(X), Y) → A__TIMES(mark(X), mark(Y))

A__2NDSPOS(s(y0), cons(y1, cons(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, cons(mark(x0), x1)))

MARK(plus(times(x0, x1), y1)) → A__PLUS(a__times(mark(x0), mark(x1)), mark(y1))

MARK(2ndsneg(from(y0), from(x0))) → A__2NDSNEG(a__from(mark(y0)), a__from(mark(x0)))

A__2NDSPOS(s(y0), cons(y1, 2ndsneg(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__2ndsneg(mark(x0), mark(x1))))

A__2NDSPOS(s(N), cons(X, Z)) → MARK(N)

MARK(2ndsneg(2ndsneg(y0, y1), plus(x0, x1))) → A__2NDSNEG(a__2ndsneg(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(2ndsneg(plus(y0, y1), 2ndsneg(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__2ndsneg(mark(x0), mark(x1)))

A__PLUS(0, Y) → MARK(Y)

MARK(posrecip(X)) → MARK(X)

MARK(2ndspos(square(y0), from(x0))) → A__2NDSPOS(a__square(mark(y0)), a__from(mark(x0)))

MARK(2ndspos(from(y0), square(x0))) → A__2NDSPOS(a__from(mark(y0)), a__square(mark(x0)))

A__2NDSPOS(s(y0), cons(y1, square(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__square(mark(x0))))

MARK(plus(0, y1)) → A__PLUS(0, mark(y1))

A__PLUS(s(X), Y) → A__PLUS(mark(X), mark(Y))

MARK(2ndsneg(s(y0), times(x0, x1))) → A__2NDSNEG(s(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(square(y0), square(x0))) → A__2NDSPOS(a__square(mark(y0)), a__square(mark(x0)))

A__PI(pi(x0)) → A__2NDSPOS(a__pi(mark(x0)), a__from(0))

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → A__2NDSPOS(mark(N), mark(Z))

MARK(plus(from(x0), y1)) → A__PLUS(a__from(mark(x0)), mark(y1))

MARK(2ndsneg(pi(y0), times(x0, x1))) → A__2NDSNEG(a__pi(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndsneg(times(y0, y1), pi(x0))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__pi(mark(x0)))

MARK(square(X)) → A__SQUARE(mark(X))

MARK(2ndsneg(times(y0, y1), plus(x0, x1))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(2ndsneg(plus(y0, y1), times(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(plus(y0, y1), plus(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(2ndsneg(2ndsneg(y0, y1), cons2(x0, x1))) → A__2NDSNEG(a__2ndsneg(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(2ndspos(2ndsneg(y0, y1), cons(x0, x1))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), cons(mark(x0), x1))

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))

a__2ndspos(0, Z) → rnil

a__2ndspos(s(N), cons(X, Z)) → a__2ndspos(s(mark(N)), cons2(X, mark(Z)))

a__2ndspos(s(N), cons2(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))

a__2ndsneg(0, Z) → rnil

a__2ndsneg(s(N), cons(X, Z)) → a__2ndsneg(s(mark(N)), cons2(X, mark(Z)))

a__2ndsneg(s(N), cons2(X, cons(Y, Z))) → rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z)))

a__pi(X) → a__2ndspos(mark(X), a__from(0))

a__plus(0, Y) → mark(Y)

a__plus(s(X), Y) → s(a__plus(mark(X), mark(Y)))

a__times(0, Y) → 0

a__times(s(X), Y) → a__plus(mark(Y), a__times(mark(X), mark(Y)))

a__square(X) → a__times(mark(X), mark(X))

mark(from(X)) → a__from(mark(X))

mark(2ndspos(X1, X2)) → a__2ndspos(mark(X1), mark(X2))

mark(2ndsneg(X1, X2)) → a__2ndsneg(mark(X1), mark(X2))

mark(pi(X)) → a__pi(mark(X))

mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))

mark(times(X1, X2)) → a__times(mark(X1), mark(X2))

mark(square(X)) → a__square(mark(X))

mark(0) → 0

mark(s(X)) → s(mark(X))

mark(posrecip(X)) → posrecip(mark(X))

mark(negrecip(X)) → negrecip(mark(X))

mark(nil) → nil

mark(cons(X1, X2)) → cons(mark(X1), X2)

mark(cons2(X1, X2)) → cons2(X1, mark(X2))

mark(rnil) → rnil

mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))

a__from(X) → from(X)

a__2ndspos(X1, X2) → 2ndspos(X1, X2)

a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)

a__pi(X) → pi(X)

a__plus(X1, X2) → plus(X1, X2)

a__times(X1, X2) → times(X1, X2)

a__square(X) → square(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 7 less nodes.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ Narrowing
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ QDP
              ↳ Narrowing
                ↳ QDP
                  ↳ DependencyGraphProof
                    ↳ QDP
                      ↳ Narrowing
                        ↳ QDP
                          ↳ DependencyGraphProof
                            ↳ QDP
                              ↳ Narrowing
                                ↳ QDP
                                  ↳ DependencyGraphProof
                                    ↳ QDP
                                      ↳ Narrowing
                                        ↳ QDP
                                          ↳ DependencyGraphProof
                                            ↳ QDP
                                              ↳ Narrowing
                                                ↳ QDP
                                                  ↳ DependencyGraphProof
                                                    ↳ QDP
                                                      ↳ Narrowing
                                                        ↳ QDP
                                                          ↳ DependencyGraphProof
                                                            ↳ QDP
                                                              ↳ Narrowing
                                                                ↳ QDP
                                                                  ↳ DependencyGraphProof
                                                                    ↳ QDP
                                                                      ↳ Narrowing
                                                                        ↳ QDP
                                                                          ↳ DependencyGraphProof
                                                                            ↳ QDP
                                                                              ↳ Narrowing
                                                                                ↳ QDP
                                                                                  ↳ DependencyGraphProof
                                                                                    ↳ QDP
                                                                                      ↳ Narrowing
                                                                                        ↳ QDP
                                                                                          ↳ DependencyGraphProof
                                                                                            ↳ QDP
                                                                                              ↳ Narrowing
                                                                                                ↳ QDP
                                                                                                  ↳ DependencyGraphProof
                                                                                                    ↳ QDP
                                                                                                      ↳ Narrowing
                                                                                                        ↳ QDP
                                                                                                          ↳ DependencyGraphProof
                                                                                                            ↳ QDP
                                                                                                              ↳ Narrowing
                                                                                                                ↳ QDP
                                                                                                                  ↳ DependencyGraphProof
                                                                                                                    ↳ QDP
                                                                                                                      ↳ Narrowing
                                                                                                                        ↳ QDP
                                                                                                                          ↳ DependencyGraphProof
                                                                                                                            ↳ QDP
                                                                                                                              ↳ Narrowing
                                                                                                                                ↳ QDP
                                                                                                                                  ↳ DependencyGraphProof
                                                                                                                                    ↳ QDP
                                                                                                                                      ↳ Narrowing
                                                                                                                                        ↳ QDP
                                                                                                                                          ↳ DependencyGraphProof
                                                                                                                                            ↳ QDP
                                                                                                                                              ↳ Narrowing
                                                                                                                                                ↳ QDP
                                                                                                                                                  ↳ DependencyGraphProof
                                                                                                                                                    ↳ QDP
                                                                                                                                                      ↳ Narrowing
                                                                                                                                                        ↳ QDP
                                                                                                                                                          ↳ DependencyGraphProof
                                                                                                                                                            ↳ QDP
                                                                                                                                                              ↳ Narrowing
                                                                                                                                                                ↳ QDP
                                                                                                                                                                  ↳ DependencyGraphProof
                                                                                                                                                                    ↳ QDP
                                                                                                                                                                      ↳ Narrowing
                                                                                                                                                                        ↳ QDP
                                                                                                                                                                          ↳ DependencyGraphProof
                                                                                                                                                                            ↳ QDP
                                                                                                                                                                              ↳ Narrowing
                                                                                                                                                                                ↳ QDP
                                                                                                                                                                                  ↳ DependencyGraphProof
                                                                                                                                                                                    ↳ QDP
                                                                                                                                                                                      ↳ Narrowing

Q DP problem:
The TRS P consists of the following rules:

MARK(times(times(x0, x1), y1)) → A__TIMES(a__times(mark(x0), mark(x1)), mark(y1))

MARK(2ndsneg(2ndspos(y0, y1), 2ndsneg(x0, x1))) → A__2NDSNEG(a__2ndspos(mark(y0), mark(y1)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndsneg(2ndsneg(y0, y1), 2ndspos(x0, x1))) → A__2NDSNEG(a__2ndsneg(mark(y0), mark(y1)), a__2ndspos(mark(x0), mark(x1)))

MARK(plus(plus(x0, x1), y1)) → A__PLUS(a__plus(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(2ndsneg(y0, y1), from(x0))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndspos(from(y0), 2ndsneg(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

A__2NDSNEG(s(N), cons(X, Z)) → MARK(Z)

MARK(2ndsneg(square(y0), 2ndspos(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndsneg(2ndspos(y0, y1), square(x0))) → A__2NDSNEG(a__2ndspos(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndsneg(s(y0), pi(x0))) → A__2NDSNEG(s(mark(y0)), a__pi(mark(x0)))

MARK(2ndsneg(pi(y0), 2ndspos(x0, x1))) → A__2NDSNEG(a__pi(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndsneg(2ndspos(y0, y1), pi(x0))) → A__2NDSNEG(a__2ndspos(mark(y0), mark(y1)), a__pi(mark(x0)))

A__TIMES(s(X), Y) → MARK(Y)

MARK(2ndspos(square(y0), 2ndsneg(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndspos(2ndsneg(y0, y1), square(x0))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__square(mark(x0)))

A__2NDSNEG(s(N), cons(X, Z)) → MARK(N)

MARK(times(2ndsneg(x0, x1), y1)) → A__TIMES(a__2ndsneg(mark(x0), mark(x1)), mark(y1))

MARK(2ndsneg(times(y0, y1), times(x0, x1))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(pi(y0), pi(x0))) → A__2NDSPOS(a__pi(mark(y0)), a__pi(mark(x0)))

MARK(2ndsneg(square(y0), 2ndsneg(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndsneg(2ndsneg(y0, y1), square(x0))) → A__2NDSNEG(a__2ndsneg(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndsneg(from(y0), cons(x0, x1))) → A__2NDSNEG(a__from(mark(y0)), cons(mark(x0), x1))

MARK(2ndspos(s(y0), pi(x0))) → A__2NDSPOS(s(mark(y0)), a__pi(mark(x0)))

MARK(plus(square(x0), y1)) → A__PLUS(a__square(mark(x0)), mark(y1))

MARK(2ndsneg(times(y0, y1), cons2(x0, x1))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(cons2(X1, X2)) → MARK(X2)

MARK(2ndsneg(square(y0), cons2(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), cons2(x0, mark(x1)))

MARK(2ndsneg(2ndspos(y0, y1), times(x0, x1))) → A__2NDSNEG(a__2ndspos(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndsneg(times(y0, y1), 2ndspos(x0, x1))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndspos(X1, X2)) → MARK(X1)

MARK(pi(X)) → MARK(X)

MARK(2ndsneg(s(y0), from(x0))) → A__2NDSNEG(s(mark(y0)), a__from(mark(x0)))

MARK(pi(X)) → A__PI(mark(X))

MARK(2ndspos(plus(y0, y1), cons(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), cons(mark(x0), x1))

MARK(2ndsneg(square(y0), times(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndsneg(times(y0, y1), square(x0))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(plus(2ndsneg(x0, x1), y1)) → A__PLUS(a__2ndsneg(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(2ndspos(y0, y1), cons2(x0, x1))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(2ndsneg(X1, X2)) → MARK(X2)

MARK(2ndspos(square(y0), times(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(times(y0, y1), square(x0))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(times(square(x0), y1)) → A__TIMES(a__square(mark(x0)), mark(y1))

MARK(2ndspos(pi(y0), cons(x0, x1))) → A__2NDSPOS(a__pi(mark(y0)), cons(mark(x0), x1))

A__PI(X) → MARK(X)

MARK(times(s(x0), y1)) → A__TIMES(s(mark(x0)), mark(y1))

MARK(2ndspos(2ndspos(y0, y1), plus(x0, x1))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(plus(y0, y1), 2ndspos(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__2ndspos(mark(x0), mark(x1)))

A__PLUS(s(X), Y) → MARK(Y)

MARK(2ndsneg(pi(y0), plus(x0, x1))) → A__2NDSNEG(a__pi(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(2ndsneg(plus(y0, y1), pi(x0))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__pi(mark(x0)))

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(N)

MARK(2ndspos(square(y0), 2ndspos(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndspos(2ndspos(y0, y1), square(x0))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndspos(s(y0), 2ndspos(x0, x1))) → A__2NDSPOS(s(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndspos(times(y0, y1), cons(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), cons(mark(x0), x1))

MARK(2ndspos(2ndsneg(y0, y1), 2ndsneg(x0, x1))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndspos(times(y0, y1), plus(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(plus(y0, y1), times(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndsneg(2ndspos(y0, y1), from(x0))) → A__2NDSNEG(a__2ndspos(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndsneg(from(y0), 2ndspos(x0, x1))) → A__2NDSNEG(a__from(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndsneg(2ndspos(y0, y1), cons2(x0, x1))) → A__2NDSNEG(a__2ndspos(mark(y0), mark(y1)), cons2(x0, mark(x1)))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(Y)

MARK(rcons(X1, X2)) → MARK(X1)

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(Z)

MARK(square(X)) → MARK(X)

MARK(2ndspos(pi(y0), times(x0, x1))) → A__2NDSPOS(a__pi(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(times(y0, y1), pi(x0))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__pi(mark(x0)))

A__PI(s(x0)) → A__2NDSPOS(s(mark(x0)), a__from(0))

MARK(2ndsneg(pi(y0), 2ndsneg(x0, x1))) → A__2NDSNEG(a__pi(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndsneg(2ndsneg(y0, y1), pi(x0))) → A__2NDSNEG(a__2ndsneg(mark(y0), mark(y1)), a__pi(mark(x0)))

MARK(times(plus(x0, x1), y1)) → A__TIMES(a__plus(mark(x0), mark(x1)), mark(y1))

A__2NDSPOS(s(y0), cons(y1, from(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__from(mark(x0))))

MARK(2ndspos(times(y0, y1), 2ndspos(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndspos(2ndspos(y0, y1), times(x0, x1))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndsneg(from(y0), cons2(x0, x1))) → A__2NDSNEG(a__from(mark(y0)), cons2(x0, mark(x1)))

MARK(2ndspos(s(y0), square(x0))) → A__2NDSPOS(s(mark(y0)), a__square(mark(x0)))

MARK(times(2ndspos(x0, x1), y1)) → A__TIMES(a__2ndspos(mark(x0), mark(x1)), mark(y1))

MARK(2ndsneg(2ndsneg(y0, y1), from(x0))) → A__2NDSNEG(a__2ndsneg(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndsneg(from(y0), 2ndsneg(x0, x1))) → A__2NDSNEG(a__from(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndspos(s(y0), cons2(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(x0, mark(x1)))

MARK(2ndsneg(plus(y0, y1), cons2(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), cons2(x0, mark(x1)))

A__PLUS(s(X), Y) → MARK(X)

MARK(times(X1, X2)) → MARK(X1)

A__SQUARE(X) → MARK(X)

MARK(2ndsneg(pi(y0), pi(x0))) → A__2NDSNEG(a__pi(mark(y0)), a__pi(mark(x0)))

MARK(negrecip(X)) → MARK(X)

MARK(2ndspos(from(y0), times(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(times(y0, y1), from(x0))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndspos(2ndsneg(y0, y1), times(x0, x1))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(times(y0, y1), 2ndsneg(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__2ndsneg(mark(x0), mark(x1)))

MARK(from(X)) → MARK(X)

A__SQUARE(X) → A__TIMES(mark(X), mark(X))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(Z)

MARK(2ndsneg(plus(y0, y1), plus(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(from(y0), plus(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(plus(y0, y1), from(x0))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndsneg(2ndsneg(y0, y1), times(x0, x1))) → A__2NDSNEG(a__2ndsneg(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndsneg(times(y0, y1), 2ndsneg(x0, x1))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndsneg(times(y0, y1), cons(x0, x1))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), cons(mark(x0), x1))

MARK(2ndsneg(2ndsneg(y0, y1), cons(x0, x1))) → A__2NDSNEG(a__2ndsneg(mark(y0), mark(y1)), cons(mark(x0), x1))

MARK(2ndspos(from(y0), 2ndspos(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndspos(2ndspos(y0, y1), from(x0))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndspos(square(y0), plus(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(plus(y0, y1), square(x0))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__square(mark(x0)))

A__PI(times(x0, x1)) → A__2NDSPOS(a__times(mark(x0), mark(x1)), a__from(0))

A__2NDSPOS(s(y0), cons(y1, 2ndspos(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__2ndspos(mark(x0), mark(x1))))

A__2NDSNEG(s(N), cons(X, Z)) → A__2NDSNEG(s(mark(N)), cons2(X, mark(Z)))

MARK(2ndsneg(s(y0), square(x0))) → A__2NDSNEG(s(mark(y0)), a__square(mark(x0)))

MARK(2ndspos(X1, X2)) → MARK(X2)

MARK(2ndspos(pi(y0), cons2(x0, x1))) → A__2NDSPOS(a__pi(mark(y0)), cons2(x0, mark(x1)))

MARK(2ndspos(from(y0), cons2(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), cons2(x0, mark(x1)))

MARK(2ndsneg(pi(y0), from(x0))) → A__2NDSNEG(a__pi(mark(y0)), a__from(mark(x0)))

MARK(2ndsneg(from(y0), pi(x0))) → A__2NDSNEG(a__from(mark(y0)), a__pi(mark(x0)))

MARK(2ndsneg(2ndspos(y0, y1), 2ndspos(x0, x1))) → A__2NDSNEG(a__2ndspos(mark(y0), mark(y1)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndspos(s(y0), from(x0))) → A__2NDSPOS(s(mark(y0)), a__from(mark(x0)))

MARK(2ndspos(2ndsneg(y0, y1), pi(x0))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__pi(mark(x0)))

MARK(2ndspos(pi(y0), 2ndsneg(x0, x1))) → A__2NDSPOS(a__pi(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

A__2NDSPOS(s(y0), cons(y1, plus(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__plus(mark(x0), mark(x1))))

MARK(2ndspos(from(y0), pi(x0))) → A__2NDSPOS(a__from(mark(y0)), a__pi(mark(x0)))

MARK(2ndspos(pi(y0), from(x0))) → A__2NDSPOS(a__pi(mark(y0)), a__from(mark(x0)))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → A__2NDSNEG(mark(N), mark(Z))

MARK(2ndspos(from(y0), from(x0))) → A__2NDSPOS(a__from(mark(y0)), a__from(mark(x0)))

MARK(2ndspos(times(y0, y1), times(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

A__TIMES(s(X), Y) → A__PLUS(mark(Y), a__times(mark(X), mark(Y)))

MARK(2ndsneg(pi(y0), cons2(x0, x1))) → A__2NDSNEG(a__pi(mark(y0)), cons2(x0, mark(x1)))

A__PI(2ndspos(x0, x1)) → A__2NDSPOS(a__2ndspos(mark(x0), mark(x1)), a__from(0))

MARK(2ndspos(square(y0), cons2(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), cons2(x0, mark(x1)))

MARK(plus(s(x0), y1)) → A__PLUS(s(mark(x0)), mark(y1))

A__PI(plus(x0, x1)) → A__2NDSPOS(a__plus(mark(x0), mark(x1)), a__from(0))

MARK(2ndspos(square(y0), cons(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), cons(mark(x0), x1))

MARK(s(X)) → MARK(X)

MARK(2ndsneg(square(y0), plus(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(2ndsneg(plus(y0, y1), square(x0))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndspos(plus(y0, y1), cons2(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(plus(X1, X2)) → MARK(X2)

A__2NDSPOS(s(N), cons(X, Z)) → MARK(Z)

MARK(2ndsneg(from(y0), times(x0, x1))) → A__2NDSNEG(a__from(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndsneg(times(y0, y1), from(x0))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndspos(times(y0, y1), cons2(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(2ndsneg(square(y0), from(x0))) → A__2NDSNEG(a__square(mark(y0)), a__from(mark(x0)))

MARK(2ndsneg(from(y0), square(x0))) → A__2NDSNEG(a__from(mark(y0)), a__square(mark(x0)))

MARK(2ndspos(2ndsneg(y0, y1), cons2(x0, x1))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(2ndsneg(s(y0), cons(x0, x1))) → A__2NDSNEG(s(mark(y0)), cons(mark(x0), x1))

A__PI(X) → A__FROM(0)

MARK(2ndspos(s(y0), plus(x0, x1))) → A__2NDSPOS(s(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(from(X)) → A__FROM(mark(X))

MARK(plus(X1, X2)) → MARK(X1)

MARK(times(from(x0), y1)) → A__TIMES(a__from(mark(x0)), mark(y1))

A__2NDSPOS(s(y0), cons(y1, times(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__times(mark(x0), mark(x1))))

MARK(2ndsneg(2ndsneg(y0, y1), 2ndsneg(x0, x1))) → A__2NDSNEG(a__2ndsneg(mark(y0), mark(y1)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndspos(s(y0), cons(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons(mark(x0), x1))

MARK(2ndsneg(2ndspos(y0, y1), plus(x0, x1))) → A__2NDSNEG(a__2ndspos(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(2ndsneg(plus(y0, y1), 2ndspos(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndspos(pi(y0), 2ndspos(x0, x1))) → A__2NDSPOS(a__pi(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndspos(2ndspos(y0, y1), pi(x0))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), a__pi(mark(x0)))

MARK(2ndspos(from(y0), cons(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), cons(mark(x0), x1))

MARK(2ndsneg(2ndspos(y0, y1), cons(x0, x1))) → A__2NDSNEG(a__2ndspos(mark(y0), mark(y1)), cons(mark(x0), x1))

MARK(2ndsneg(plus(y0, y1), cons(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), cons(mark(x0), x1))

A__2NDSPOS(s(y0), cons(y1, pi(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__pi(mark(x0))))

A__TIMES(s(X), Y) → MARK(X)

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(N)

MARK(2ndspos(pi(y0), plus(x0, x1))) → A__2NDSPOS(a__pi(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(plus(y0, y1), pi(x0))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__pi(mark(x0)))

MARK(2ndsneg(s(y0), cons2(x0, x1))) → A__2NDSNEG(s(mark(y0)), cons2(x0, mark(x1)))

MARK(plus(2ndspos(x0, x1), y1)) → A__PLUS(a__2ndspos(mark(x0), mark(x1)), mark(y1))

MARK(2ndsneg(pi(y0), cons(x0, x1))) → A__2NDSNEG(a__pi(mark(y0)), cons(mark(x0), x1))

A__PI(from(x0)) → A__2NDSPOS(a__from(mark(x0)), a__from(0))

MARK(cons(X1, X2)) → MARK(X1)

MARK(2ndsneg(s(y0), 2ndspos(x0, x1))) → A__2NDSNEG(s(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndsneg(plus(y0, y1), from(x0))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndsneg(from(y0), plus(x0, x1))) → A__2NDSNEG(a__from(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(times(X1, X2)) → MARK(X2)

A__PI(square(x0)) → A__2NDSPOS(a__square(mark(x0)), a__from(0))

MARK(2ndsneg(square(y0), cons(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), cons(mark(x0), x1))

MARK(2ndspos(s(y0), times(x0, x1))) → A__2NDSPOS(s(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndsneg(X1, X2)) → MARK(X1)

MARK(2ndspos(s(y0), 2ndsneg(x0, x1))) → A__2NDSPOS(s(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndsneg(square(y0), square(x0))) → A__2NDSNEG(a__square(mark(y0)), a__square(mark(x0)))

MARK(2ndspos(2ndsneg(y0, y1), plus(x0, x1))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(plus(y0, y1), 2ndsneg(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__2ndsneg(mark(x0), mark(x1)))

A__PI(2ndsneg(x0, x1)) → A__2NDSPOS(a__2ndsneg(mark(x0), mark(x1)), a__from(0))

MARK(plus(pi(x0), y1)) → A__PLUS(a__pi(mark(x0)), mark(y1))

MARK(times(pi(x0), y1)) → A__TIMES(a__pi(mark(x0)), mark(y1))

MARK(2ndspos(2ndspos(y0, y1), 2ndspos(x0, x1))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndsneg(pi(y0), square(x0))) → A__2NDSNEG(a__pi(mark(y0)), a__square(mark(x0)))

MARK(2ndsneg(square(y0), pi(x0))) → A__2NDSNEG(a__square(mark(y0)), a__pi(mark(x0)))

MARK(2ndspos(2ndsneg(y0, y1), 2ndspos(x0, x1))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndspos(2ndspos(y0, y1), 2ndsneg(x0, x1))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), a__2ndsneg(mark(x0), mark(x1)))

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(Y)

MARK(2ndsneg(s(y0), 2ndsneg(x0, x1))) → A__2NDSNEG(s(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

A__FROM(X) → MARK(X)

MARK(2ndspos(2ndspos(y0, y1), cons(x0, x1))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), cons(mark(x0), x1))

MARK(2ndsneg(s(y0), plus(x0, x1))) → A__2NDSNEG(s(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(square(y0), pi(x0))) → A__2NDSPOS(a__square(mark(y0)), a__pi(mark(x0)))

MARK(2ndspos(pi(y0), square(x0))) → A__2NDSPOS(a__pi(mark(y0)), a__square(mark(x0)))

MARK(rcons(X1, X2)) → MARK(X2)

A__TIMES(s(X), Y) → A__TIMES(mark(X), mark(Y))

A__2NDSPOS(s(y0), cons(y1, cons(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, cons(mark(x0), x1)))

MARK(plus(times(x0, x1), y1)) → A__PLUS(a__times(mark(x0), mark(x1)), mark(y1))

MARK(2ndsneg(from(y0), from(x0))) → A__2NDSNEG(a__from(mark(y0)), a__from(mark(x0)))

A__2NDSPOS(s(y0), cons(y1, 2ndsneg(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__2ndsneg(mark(x0), mark(x1))))

A__2NDSPOS(s(N), cons(X, Z)) → MARK(N)

MARK(2ndsneg(plus(y0, y1), 2ndsneg(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndsneg(2ndsneg(y0, y1), plus(x0, x1))) → A__2NDSNEG(a__2ndsneg(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

A__PLUS(0, Y) → MARK(Y)

MARK(posrecip(X)) → MARK(X)

MARK(2ndspos(from(y0), square(x0))) → A__2NDSPOS(a__from(mark(y0)), a__square(mark(x0)))

MARK(2ndspos(square(y0), from(x0))) → A__2NDSPOS(a__square(mark(y0)), a__from(mark(x0)))

A__2NDSPOS(s(y0), cons(y1, square(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__square(mark(x0))))

MARK(plus(0, y1)) → A__PLUS(0, mark(y1))

MARK(2ndsneg(s(y0), times(x0, x1))) → A__2NDSNEG(s(mark(y0)), a__times(mark(x0), mark(x1)))

A__PLUS(s(X), Y) → A__PLUS(mark(X), mark(Y))

MARK(2ndspos(square(y0), square(x0))) → A__2NDSPOS(a__square(mark(y0)), a__square(mark(x0)))

A__PI(pi(x0)) → A__2NDSPOS(a__pi(mark(x0)), a__from(0))

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → A__2NDSPOS(mark(N), mark(Z))

MARK(plus(from(x0), y1)) → A__PLUS(a__from(mark(x0)), mark(y1))

MARK(2ndsneg(pi(y0), times(x0, x1))) → A__2NDSNEG(a__pi(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(square(X)) → A__SQUARE(mark(X))

MARK(2ndsneg(times(y0, y1), pi(x0))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__pi(mark(x0)))

MARK(2ndsneg(plus(y0, y1), times(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndsneg(times(y0, y1), plus(x0, x1))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(plus(y0, y1), plus(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(2ndsneg(2ndsneg(y0, y1), cons2(x0, x1))) → A__2NDSNEG(a__2ndsneg(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(2ndspos(2ndsneg(y0, y1), cons(x0, x1))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), cons(mark(x0), x1))

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))

a__2ndspos(0, Z) → rnil

a__2ndspos(s(N), cons(X, Z)) → a__2ndspos(s(mark(N)), cons2(X, mark(Z)))

a__2ndspos(s(N), cons2(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))

a__2ndsneg(0, Z) → rnil

a__2ndsneg(s(N), cons(X, Z)) → a__2ndsneg(s(mark(N)), cons2(X, mark(Z)))

a__2ndsneg(s(N), cons2(X, cons(Y, Z))) → rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z)))

a__pi(X) → a__2ndspos(mark(X), a__from(0))

a__plus(0, Y) → mark(Y)

a__plus(s(X), Y) → s(a__plus(mark(X), mark(Y)))

a__times(0, Y) → 0

a__times(s(X), Y) → a__plus(mark(Y), a__times(mark(X), mark(Y)))

a__square(X) → a__times(mark(X), mark(X))

mark(from(X)) → a__from(mark(X))

mark(2ndspos(X1, X2)) → a__2ndspos(mark(X1), mark(X2))

mark(2ndsneg(X1, X2)) → a__2ndsneg(mark(X1), mark(X2))

mark(pi(X)) → a__pi(mark(X))

mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))

mark(times(X1, X2)) → a__times(mark(X1), mark(X2))

mark(square(X)) → a__square(mark(X))

mark(0) → 0

mark(s(X)) → s(mark(X))

mark(posrecip(X)) → posrecip(mark(X))

mark(negrecip(X)) → negrecip(mark(X))

mark(nil) → nil

mark(cons(X1, X2)) → cons(mark(X1), X2)

mark(cons2(X1, X2)) → cons2(X1, mark(X2))

mark(rnil) → rnil

mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))

a__from(X) → from(X)

a__2ndspos(X1, X2) → 2ndspos(X1, X2)

a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)

a__pi(X) → pi(X)

a__plus(X1, X2) → plus(X1, X2)

a__times(X1, X2) → times(X1, X2)

a__square(X) → square(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule A__PLUS(s(X), Y) → A__PLUS(mark(X), mark(Y)) at position [0] we obtained the following new rules:

A__PLUS(s(square(x0)), y1) → A__PLUS(a__square(mark(x0)), mark(y1))

A__PLUS(s(2ndsneg(x0, x1)), y1) → A__PLUS(a__2ndsneg(mark(x0), mark(x1)), mark(y1))

A__PLUS(s(plus(x0, x1)), y1) → A__PLUS(a__plus(mark(x0), mark(x1)), mark(y1))

A__PLUS(s(rnil), y1) → A__PLUS(rnil, mark(y1))

A__PLUS(s(cons(x0, x1)), y1) → A__PLUS(cons(mark(x0), x1), mark(y1))

A__PLUS(s(posrecip(x0)), y1) → A__PLUS(posrecip(mark(x0)), mark(y1))

A__PLUS(s(negrecip(x0)), y1) → A__PLUS(negrecip(mark(x0)), mark(y1))

A__PLUS(s(from(x0)), y1) → A__PLUS(a__from(mark(x0)), mark(y1))

A__PLUS(s(pi(x0)), y1) → A__PLUS(a__pi(mark(x0)), mark(y1))

A__PLUS(s(nil), y1) → A__PLUS(nil, mark(y1))

A__PLUS(s(s(x0)), y1) → A__PLUS(s(mark(x0)), mark(y1))

A__PLUS(s(2ndspos(x0, x1)), y1) → A__PLUS(a__2ndspos(mark(x0), mark(x1)), mark(y1))

A__PLUS(s(rcons(x0, x1)), y1) → A__PLUS(rcons(mark(x0), mark(x1)), mark(y1))

A__PLUS(s(cons2(x0, x1)), y1) → A__PLUS(cons2(x0, mark(x1)), mark(y1))

A__PLUS(s(times(x0, x1)), y1) → A__PLUS(a__times(mark(x0), mark(x1)), mark(y1))

A__PLUS(s(0), y1) → A__PLUS(0, mark(y1))



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ Narrowing
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ QDP
              ↳ Narrowing
                ↳ QDP
                  ↳ DependencyGraphProof
                    ↳ QDP
                      ↳ Narrowing
                        ↳ QDP
                          ↳ DependencyGraphProof
                            ↳ QDP
                              ↳ Narrowing
                                ↳ QDP
                                  ↳ DependencyGraphProof
                                    ↳ QDP
                                      ↳ Narrowing
                                        ↳ QDP
                                          ↳ DependencyGraphProof
                                            ↳ QDP
                                              ↳ Narrowing
                                                ↳ QDP
                                                  ↳ DependencyGraphProof
                                                    ↳ QDP
                                                      ↳ Narrowing
                                                        ↳ QDP
                                                          ↳ DependencyGraphProof
                                                            ↳ QDP
                                                              ↳ Narrowing
                                                                ↳ QDP
                                                                  ↳ DependencyGraphProof
                                                                    ↳ QDP
                                                                      ↳ Narrowing
                                                                        ↳ QDP
                                                                          ↳ DependencyGraphProof
                                                                            ↳ QDP
                                                                              ↳ Narrowing
                                                                                ↳ QDP
                                                                                  ↳ DependencyGraphProof
                                                                                    ↳ QDP
                                                                                      ↳ Narrowing
                                                                                        ↳ QDP
                                                                                          ↳ DependencyGraphProof
                                                                                            ↳ QDP
                                                                                              ↳ Narrowing
                                                                                                ↳ QDP
                                                                                                  ↳ DependencyGraphProof
                                                                                                    ↳ QDP
                                                                                                      ↳ Narrowing
                                                                                                        ↳ QDP
                                                                                                          ↳ DependencyGraphProof
                                                                                                            ↳ QDP
                                                                                                              ↳ Narrowing
                                                                                                                ↳ QDP
                                                                                                                  ↳ DependencyGraphProof
                                                                                                                    ↳ QDP
                                                                                                                      ↳ Narrowing
                                                                                                                        ↳ QDP
                                                                                                                          ↳ DependencyGraphProof
                                                                                                                            ↳ QDP
                                                                                                                              ↳ Narrowing
                                                                                                                                ↳ QDP
                                                                                                                                  ↳ DependencyGraphProof
                                                                                                                                    ↳ QDP
                                                                                                                                      ↳ Narrowing
                                                                                                                                        ↳ QDP
                                                                                                                                          ↳ DependencyGraphProof
                                                                                                                                            ↳ QDP
                                                                                                                                              ↳ Narrowing
                                                                                                                                                ↳ QDP
                                                                                                                                                  ↳ DependencyGraphProof
                                                                                                                                                    ↳ QDP
                                                                                                                                                      ↳ Narrowing
                                                                                                                                                        ↳ QDP
                                                                                                                                                          ↳ DependencyGraphProof
                                                                                                                                                            ↳ QDP
                                                                                                                                                              ↳ Narrowing
                                                                                                                                                                ↳ QDP
                                                                                                                                                                  ↳ DependencyGraphProof
                                                                                                                                                                    ↳ QDP
                                                                                                                                                                      ↳ Narrowing
                                                                                                                                                                        ↳ QDP
                                                                                                                                                                          ↳ DependencyGraphProof
                                                                                                                                                                            ↳ QDP
                                                                                                                                                                              ↳ Narrowing
                                                                                                                                                                                ↳ QDP
                                                                                                                                                                                  ↳ DependencyGraphProof
                                                                                                                                                                                    ↳ QDP
                                                                                                                                                                                      ↳ Narrowing
                                                                                                                                                                                        ↳ QDP
                                                                                                                                                                                          ↳ DependencyGraphProof

Q DP problem:
The TRS P consists of the following rules:

MARK(times(times(x0, x1), y1)) → A__TIMES(a__times(mark(x0), mark(x1)), mark(y1))

MARK(2ndsneg(2ndsneg(y0, y1), 2ndspos(x0, x1))) → A__2NDSNEG(a__2ndsneg(mark(y0), mark(y1)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndsneg(2ndspos(y0, y1), 2ndsneg(x0, x1))) → A__2NDSNEG(a__2ndspos(mark(y0), mark(y1)), a__2ndsneg(mark(x0), mark(x1)))

MARK(plus(plus(x0, x1), y1)) → A__PLUS(a__plus(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(from(y0), 2ndsneg(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndspos(2ndsneg(y0, y1), from(x0))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndsneg(2ndspos(y0, y1), square(x0))) → A__2NDSNEG(a__2ndspos(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndsneg(square(y0), 2ndspos(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

A__2NDSNEG(s(N), cons(X, Z)) → MARK(Z)

A__PLUS(s(pi(x0)), y1) → A__PLUS(a__pi(mark(x0)), mark(y1))

MARK(2ndsneg(s(y0), pi(x0))) → A__2NDSNEG(s(mark(y0)), a__pi(mark(x0)))

MARK(2ndsneg(2ndspos(y0, y1), pi(x0))) → A__2NDSNEG(a__2ndspos(mark(y0), mark(y1)), a__pi(mark(x0)))

MARK(2ndsneg(pi(y0), 2ndspos(x0, x1))) → A__2NDSNEG(a__pi(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

A__TIMES(s(X), Y) → MARK(Y)

MARK(2ndspos(2ndsneg(y0, y1), square(x0))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndspos(square(y0), 2ndsneg(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

A__2NDSNEG(s(N), cons(X, Z)) → MARK(N)

MARK(times(2ndsneg(x0, x1), y1)) → A__TIMES(a__2ndsneg(mark(x0), mark(x1)), mark(y1))

MARK(2ndsneg(times(y0, y1), times(x0, x1))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(pi(y0), pi(x0))) → A__2NDSPOS(a__pi(mark(y0)), a__pi(mark(x0)))

MARK(2ndsneg(2ndsneg(y0, y1), square(x0))) → A__2NDSNEG(a__2ndsneg(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndsneg(square(y0), 2ndsneg(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndsneg(from(y0), cons(x0, x1))) → A__2NDSNEG(a__from(mark(y0)), cons(mark(x0), x1))

MARK(2ndspos(s(y0), pi(x0))) → A__2NDSPOS(s(mark(y0)), a__pi(mark(x0)))

MARK(plus(square(x0), y1)) → A__PLUS(a__square(mark(x0)), mark(y1))

MARK(2ndsneg(times(y0, y1), cons2(x0, x1))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(cons2(X1, X2)) → MARK(X2)

MARK(2ndsneg(times(y0, y1), 2ndspos(x0, x1))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndsneg(2ndspos(y0, y1), times(x0, x1))) → A__2NDSNEG(a__2ndspos(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndsneg(square(y0), cons2(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), cons2(x0, mark(x1)))

A__PLUS(s(square(x0)), y1) → A__PLUS(a__square(mark(x0)), mark(y1))

MARK(2ndspos(X1, X2)) → MARK(X1)

A__PLUS(s(rnil), y1) → A__PLUS(rnil, mark(y1))

MARK(pi(X)) → MARK(X)

MARK(2ndsneg(s(y0), from(x0))) → A__2NDSNEG(s(mark(y0)), a__from(mark(x0)))

MARK(pi(X)) → A__PI(mark(X))

MARK(2ndsneg(times(y0, y1), square(x0))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndsneg(square(y0), times(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(plus(y0, y1), cons(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), cons(mark(x0), x1))

MARK(plus(2ndsneg(x0, x1), y1)) → A__PLUS(a__2ndsneg(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(2ndspos(y0, y1), cons2(x0, x1))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(2ndsneg(X1, X2)) → MARK(X2)

MARK(2ndspos(times(y0, y1), square(x0))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndspos(square(y0), times(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(times(square(x0), y1)) → A__TIMES(a__square(mark(x0)), mark(y1))

MARK(2ndspos(pi(y0), cons(x0, x1))) → A__2NDSPOS(a__pi(mark(y0)), cons(mark(x0), x1))

A__PI(X) → MARK(X)

MARK(times(s(x0), y1)) → A__TIMES(s(mark(x0)), mark(y1))

A__PLUS(s(nil), y1) → A__PLUS(nil, mark(y1))

MARK(2ndspos(plus(y0, y1), 2ndspos(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndspos(2ndspos(y0, y1), plus(x0, x1))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

A__PLUS(s(X), Y) → MARK(Y)

A__PLUS(s(2ndspos(x0, x1)), y1) → A__PLUS(a__2ndspos(mark(x0), mark(x1)), mark(y1))

MARK(2ndsneg(plus(y0, y1), pi(x0))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__pi(mark(x0)))

MARK(2ndsneg(pi(y0), plus(x0, x1))) → A__2NDSNEG(a__pi(mark(y0)), a__plus(mark(x0), mark(x1)))

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(N)

MARK(2ndspos(2ndspos(y0, y1), square(x0))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndspos(square(y0), 2ndspos(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndspos(times(y0, y1), cons(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), cons(mark(x0), x1))

MARK(2ndspos(s(y0), 2ndspos(x0, x1))) → A__2NDSPOS(s(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndspos(2ndsneg(y0, y1), 2ndsneg(x0, x1))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndsneg(from(y0), 2ndspos(x0, x1))) → A__2NDSNEG(a__from(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndsneg(2ndspos(y0, y1), from(x0))) → A__2NDSNEG(a__2ndspos(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndspos(plus(y0, y1), times(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(times(y0, y1), plus(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(2ndsneg(2ndspos(y0, y1), cons2(x0, x1))) → A__2NDSNEG(a__2ndspos(mark(y0), mark(y1)), cons2(x0, mark(x1)))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(Y)

MARK(rcons(X1, X2)) → MARK(X1)

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(Z)

MARK(square(X)) → MARK(X)

MARK(2ndspos(times(y0, y1), pi(x0))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__pi(mark(x0)))

MARK(2ndspos(pi(y0), times(x0, x1))) → A__2NDSPOS(a__pi(mark(y0)), a__times(mark(x0), mark(x1)))

A__PI(s(x0)) → A__2NDSPOS(s(mark(x0)), a__from(0))

MARK(2ndsneg(2ndsneg(y0, y1), pi(x0))) → A__2NDSNEG(a__2ndsneg(mark(y0), mark(y1)), a__pi(mark(x0)))

MARK(2ndsneg(pi(y0), 2ndsneg(x0, x1))) → A__2NDSNEG(a__pi(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

MARK(times(plus(x0, x1), y1)) → A__TIMES(a__plus(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(2ndspos(y0, y1), times(x0, x1))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(times(y0, y1), 2ndspos(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__2ndspos(mark(x0), mark(x1)))

A__2NDSPOS(s(y0), cons(y1, from(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__from(mark(x0))))

MARK(2ndsneg(from(y0), cons2(x0, x1))) → A__2NDSNEG(a__from(mark(y0)), cons2(x0, mark(x1)))

MARK(2ndspos(s(y0), square(x0))) → A__2NDSPOS(s(mark(y0)), a__square(mark(x0)))

MARK(times(2ndspos(x0, x1), y1)) → A__TIMES(a__2ndspos(mark(x0), mark(x1)), mark(y1))

MARK(2ndsneg(from(y0), 2ndsneg(x0, x1))) → A__2NDSNEG(a__from(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndsneg(2ndsneg(y0, y1), from(x0))) → A__2NDSNEG(a__2ndsneg(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndspos(s(y0), cons2(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(x0, mark(x1)))

MARK(2ndsneg(plus(y0, y1), cons2(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), cons2(x0, mark(x1)))

A__PLUS(s(X), Y) → MARK(X)

MARK(times(X1, X2)) → MARK(X1)

A__SQUARE(X) → MARK(X)

MARK(2ndsneg(pi(y0), pi(x0))) → A__2NDSNEG(a__pi(mark(y0)), a__pi(mark(x0)))

MARK(negrecip(X)) → MARK(X)

A__PLUS(s(from(x0)), y1) → A__PLUS(a__from(mark(x0)), mark(y1))

MARK(2ndspos(times(y0, y1), from(x0))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndspos(from(y0), times(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), a__times(mark(x0), mark(x1)))

A__PLUS(s(times(x0, x1)), y1) → A__PLUS(a__times(mark(x0), mark(x1)), mark(y1))

A__PLUS(s(0), y1) → A__PLUS(0, mark(y1))

MARK(2ndspos(times(y0, y1), 2ndsneg(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndspos(2ndsneg(y0, y1), times(x0, x1))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(from(X)) → MARK(X)

A__SQUARE(X) → A__TIMES(mark(X), mark(X))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(Z)

MARK(2ndsneg(plus(y0, y1), plus(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

A__PLUS(s(posrecip(x0)), y1) → A__PLUS(posrecip(mark(x0)), mark(y1))

MARK(2ndspos(plus(y0, y1), from(x0))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndspos(from(y0), plus(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(2ndsneg(times(y0, y1), 2ndsneg(x0, x1))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndsneg(2ndsneg(y0, y1), times(x0, x1))) → A__2NDSNEG(a__2ndsneg(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

A__PLUS(s(s(x0)), y1) → A__PLUS(s(mark(x0)), mark(y1))

MARK(2ndsneg(times(y0, y1), cons(x0, x1))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), cons(mark(x0), x1))

MARK(2ndsneg(2ndsneg(y0, y1), cons(x0, x1))) → A__2NDSNEG(a__2ndsneg(mark(y0), mark(y1)), cons(mark(x0), x1))

MARK(2ndspos(2ndspos(y0, y1), from(x0))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndspos(from(y0), 2ndspos(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndspos(plus(y0, y1), square(x0))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndspos(square(y0), plus(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__plus(mark(x0), mark(x1)))

A__2NDSPOS(s(y0), cons(y1, 2ndspos(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__2ndspos(mark(x0), mark(x1))))

A__PI(times(x0, x1)) → A__2NDSPOS(a__times(mark(x0), mark(x1)), a__from(0))

A__2NDSNEG(s(N), cons(X, Z)) → A__2NDSNEG(s(mark(N)), cons2(X, mark(Z)))

MARK(2ndsneg(s(y0), square(x0))) → A__2NDSNEG(s(mark(y0)), a__square(mark(x0)))

MARK(2ndspos(pi(y0), cons2(x0, x1))) → A__2NDSPOS(a__pi(mark(y0)), cons2(x0, mark(x1)))

MARK(2ndspos(X1, X2)) → MARK(X2)

MARK(2ndspos(from(y0), cons2(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), cons2(x0, mark(x1)))

MARK(2ndsneg(from(y0), pi(x0))) → A__2NDSNEG(a__from(mark(y0)), a__pi(mark(x0)))

MARK(2ndsneg(pi(y0), from(x0))) → A__2NDSNEG(a__pi(mark(y0)), a__from(mark(x0)))

MARK(2ndsneg(2ndspos(y0, y1), 2ndspos(x0, x1))) → A__2NDSNEG(a__2ndspos(mark(y0), mark(y1)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndspos(s(y0), from(x0))) → A__2NDSPOS(s(mark(y0)), a__from(mark(x0)))

MARK(2ndspos(pi(y0), 2ndsneg(x0, x1))) → A__2NDSPOS(a__pi(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndspos(2ndsneg(y0, y1), pi(x0))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__pi(mark(x0)))

A__2NDSPOS(s(y0), cons(y1, plus(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__plus(mark(x0), mark(x1))))

MARK(2ndspos(pi(y0), from(x0))) → A__2NDSPOS(a__pi(mark(y0)), a__from(mark(x0)))

MARK(2ndspos(from(y0), pi(x0))) → A__2NDSPOS(a__from(mark(y0)), a__pi(mark(x0)))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → A__2NDSNEG(mark(N), mark(Z))

MARK(2ndspos(from(y0), from(x0))) → A__2NDSPOS(a__from(mark(y0)), a__from(mark(x0)))

MARK(2ndspos(times(y0, y1), times(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

A__TIMES(s(X), Y) → A__PLUS(mark(Y), a__times(mark(X), mark(Y)))

MARK(2ndsneg(pi(y0), cons2(x0, x1))) → A__2NDSNEG(a__pi(mark(y0)), cons2(x0, mark(x1)))

MARK(2ndspos(square(y0), cons2(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), cons2(x0, mark(x1)))

A__PI(2ndspos(x0, x1)) → A__2NDSPOS(a__2ndspos(mark(x0), mark(x1)), a__from(0))

MARK(plus(s(x0), y1)) → A__PLUS(s(mark(x0)), mark(y1))

A__PLUS(s(rcons(x0, x1)), y1) → A__PLUS(rcons(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(square(y0), cons(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), cons(mark(x0), x1))

A__PI(plus(x0, x1)) → A__2NDSPOS(a__plus(mark(x0), mark(x1)), a__from(0))

MARK(2ndsneg(plus(y0, y1), square(x0))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndsneg(square(y0), plus(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(s(X)) → MARK(X)

MARK(2ndspos(plus(y0, y1), cons2(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(plus(X1, X2)) → MARK(X2)

A__2NDSPOS(s(N), cons(X, Z)) → MARK(Z)

MARK(2ndsneg(times(y0, y1), from(x0))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndsneg(from(y0), times(x0, x1))) → A__2NDSNEG(a__from(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(times(y0, y1), cons2(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(2ndsneg(from(y0), square(x0))) → A__2NDSNEG(a__from(mark(y0)), a__square(mark(x0)))

MARK(2ndsneg(square(y0), from(x0))) → A__2NDSNEG(a__square(mark(y0)), a__from(mark(x0)))

MARK(2ndspos(2ndsneg(y0, y1), cons2(x0, x1))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(2ndsneg(s(y0), cons(x0, x1))) → A__2NDSNEG(s(mark(y0)), cons(mark(x0), x1))

MARK(from(X)) → A__FROM(mark(X))

MARK(2ndspos(s(y0), plus(x0, x1))) → A__2NDSPOS(s(mark(y0)), a__plus(mark(x0), mark(x1)))

A__PI(X) → A__FROM(0)

A__PLUS(s(2ndsneg(x0, x1)), y1) → A__PLUS(a__2ndsneg(mark(x0), mark(x1)), mark(y1))

MARK(times(from(x0), y1)) → A__TIMES(a__from(mark(x0)), mark(y1))

MARK(plus(X1, X2)) → MARK(X1)

A__2NDSPOS(s(y0), cons(y1, times(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__times(mark(x0), mark(x1))))

MARK(2ndsneg(2ndsneg(y0, y1), 2ndsneg(x0, x1))) → A__2NDSNEG(a__2ndsneg(mark(y0), mark(y1)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndspos(s(y0), cons(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons(mark(x0), x1))

MARK(2ndsneg(plus(y0, y1), 2ndspos(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndsneg(2ndspos(y0, y1), plus(x0, x1))) → A__2NDSNEG(a__2ndspos(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

A__PLUS(s(negrecip(x0)), y1) → A__PLUS(negrecip(mark(x0)), mark(y1))

MARK(2ndspos(2ndspos(y0, y1), pi(x0))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), a__pi(mark(x0)))

MARK(2ndspos(pi(y0), 2ndspos(x0, x1))) → A__2NDSPOS(a__pi(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndspos(from(y0), cons(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), cons(mark(x0), x1))

MARK(2ndsneg(plus(y0, y1), cons(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), cons(mark(x0), x1))

MARK(2ndsneg(2ndspos(y0, y1), cons(x0, x1))) → A__2NDSNEG(a__2ndspos(mark(y0), mark(y1)), cons(mark(x0), x1))

A__TIMES(s(X), Y) → MARK(X)

A__2NDSPOS(s(y0), cons(y1, pi(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__pi(mark(x0))))

MARK(2ndspos(plus(y0, y1), pi(x0))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__pi(mark(x0)))

MARK(2ndspos(pi(y0), plus(x0, x1))) → A__2NDSPOS(a__pi(mark(y0)), a__plus(mark(x0), mark(x1)))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(N)

MARK(2ndsneg(s(y0), cons2(x0, x1))) → A__2NDSNEG(s(mark(y0)), cons2(x0, mark(x1)))

MARK(plus(2ndspos(x0, x1), y1)) → A__PLUS(a__2ndspos(mark(x0), mark(x1)), mark(y1))

MARK(2ndsneg(pi(y0), cons(x0, x1))) → A__2NDSNEG(a__pi(mark(y0)), cons(mark(x0), x1))

MARK(cons(X1, X2)) → MARK(X1)

A__PI(from(x0)) → A__2NDSPOS(a__from(mark(x0)), a__from(0))

MARK(2ndsneg(from(y0), plus(x0, x1))) → A__2NDSNEG(a__from(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(2ndsneg(plus(y0, y1), from(x0))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndsneg(s(y0), 2ndspos(x0, x1))) → A__2NDSNEG(s(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

MARK(times(X1, X2)) → MARK(X2)

A__PI(square(x0)) → A__2NDSPOS(a__square(mark(x0)), a__from(0))

MARK(2ndsneg(square(y0), cons(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), cons(mark(x0), x1))

MARK(2ndspos(s(y0), times(x0, x1))) → A__2NDSPOS(s(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndsneg(X1, X2)) → MARK(X1)

MARK(2ndspos(s(y0), 2ndsneg(x0, x1))) → A__2NDSPOS(s(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndsneg(square(y0), square(x0))) → A__2NDSNEG(a__square(mark(y0)), a__square(mark(x0)))

MARK(2ndspos(plus(y0, y1), 2ndsneg(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndspos(2ndsneg(y0, y1), plus(x0, x1))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

A__PLUS(s(cons2(x0, x1)), y1) → A__PLUS(cons2(x0, mark(x1)), mark(y1))

A__PI(2ndsneg(x0, x1)) → A__2NDSPOS(a__2ndsneg(mark(x0), mark(x1)), a__from(0))

MARK(times(pi(x0), y1)) → A__TIMES(a__pi(mark(x0)), mark(y1))

MARK(plus(pi(x0), y1)) → A__PLUS(a__pi(mark(x0)), mark(y1))

MARK(2ndspos(2ndspos(y0, y1), 2ndspos(x0, x1))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndsneg(square(y0), pi(x0))) → A__2NDSNEG(a__square(mark(y0)), a__pi(mark(x0)))

MARK(2ndsneg(pi(y0), square(x0))) → A__2NDSNEG(a__pi(mark(y0)), a__square(mark(x0)))

MARK(2ndspos(2ndspos(y0, y1), 2ndsneg(x0, x1))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndspos(2ndsneg(y0, y1), 2ndspos(x0, x1))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__2ndspos(mark(x0), mark(x1)))

A__PLUS(s(cons(x0, x1)), y1) → A__PLUS(cons(mark(x0), x1), mark(y1))

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(Y)

MARK(2ndspos(2ndspos(y0, y1), cons(x0, x1))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), cons(mark(x0), x1))

A__FROM(X) → MARK(X)

MARK(2ndsneg(s(y0), 2ndsneg(x0, x1))) → A__2NDSNEG(s(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndsneg(s(y0), plus(x0, x1))) → A__2NDSNEG(s(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(pi(y0), square(x0))) → A__2NDSPOS(a__pi(mark(y0)), a__square(mark(x0)))

MARK(2ndspos(square(y0), pi(x0))) → A__2NDSPOS(a__square(mark(y0)), a__pi(mark(x0)))

MARK(rcons(X1, X2)) → MARK(X2)

A__TIMES(s(X), Y) → A__TIMES(mark(X), mark(Y))

A__2NDSPOS(s(y0), cons(y1, cons(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, cons(mark(x0), x1)))

MARK(plus(times(x0, x1), y1)) → A__PLUS(a__times(mark(x0), mark(x1)), mark(y1))

MARK(2ndsneg(from(y0), from(x0))) → A__2NDSNEG(a__from(mark(y0)), a__from(mark(x0)))

A__2NDSPOS(s(y0), cons(y1, 2ndsneg(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__2ndsneg(mark(x0), mark(x1))))

A__2NDSPOS(s(N), cons(X, Z)) → MARK(N)

MARK(2ndsneg(2ndsneg(y0, y1), plus(x0, x1))) → A__2NDSNEG(a__2ndsneg(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(2ndsneg(plus(y0, y1), 2ndsneg(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__2ndsneg(mark(x0), mark(x1)))

A__PLUS(0, Y) → MARK(Y)

MARK(posrecip(X)) → MARK(X)

MARK(2ndspos(square(y0), from(x0))) → A__2NDSPOS(a__square(mark(y0)), a__from(mark(x0)))

MARK(2ndspos(from(y0), square(x0))) → A__2NDSPOS(a__from(mark(y0)), a__square(mark(x0)))

A__2NDSPOS(s(y0), cons(y1, square(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__square(mark(x0))))

MARK(plus(0, y1)) → A__PLUS(0, mark(y1))

MARK(2ndsneg(s(y0), times(x0, x1))) → A__2NDSNEG(s(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(square(y0), square(x0))) → A__2NDSPOS(a__square(mark(y0)), a__square(mark(x0)))

A__PLUS(s(plus(x0, x1)), y1) → A__PLUS(a__plus(mark(x0), mark(x1)), mark(y1))

A__PI(pi(x0)) → A__2NDSPOS(a__pi(mark(x0)), a__from(0))

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → A__2NDSPOS(mark(N), mark(Z))

MARK(plus(from(x0), y1)) → A__PLUS(a__from(mark(x0)), mark(y1))

MARK(2ndsneg(times(y0, y1), pi(x0))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__pi(mark(x0)))

MARK(square(X)) → A__SQUARE(mark(X))

MARK(2ndsneg(pi(y0), times(x0, x1))) → A__2NDSNEG(a__pi(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndsneg(times(y0, y1), plus(x0, x1))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(2ndsneg(plus(y0, y1), times(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(plus(y0, y1), plus(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(2ndsneg(2ndsneg(y0, y1), cons2(x0, x1))) → A__2NDSNEG(a__2ndsneg(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(2ndspos(2ndsneg(y0, y1), cons(x0, x1))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), cons(mark(x0), x1))

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))

a__2ndspos(0, Z) → rnil

a__2ndspos(s(N), cons(X, Z)) → a__2ndspos(s(mark(N)), cons2(X, mark(Z)))

a__2ndspos(s(N), cons2(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))

a__2ndsneg(0, Z) → rnil

a__2ndsneg(s(N), cons(X, Z)) → a__2ndsneg(s(mark(N)), cons2(X, mark(Z)))

a__2ndsneg(s(N), cons2(X, cons(Y, Z))) → rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z)))

a__pi(X) → a__2ndspos(mark(X), a__from(0))

a__plus(0, Y) → mark(Y)

a__plus(s(X), Y) → s(a__plus(mark(X), mark(Y)))

a__times(0, Y) → 0

a__times(s(X), Y) → a__plus(mark(Y), a__times(mark(X), mark(Y)))

a__square(X) → a__times(mark(X), mark(X))

mark(from(X)) → a__from(mark(X))

mark(2ndspos(X1, X2)) → a__2ndspos(mark(X1), mark(X2))

mark(2ndsneg(X1, X2)) → a__2ndsneg(mark(X1), mark(X2))

mark(pi(X)) → a__pi(mark(X))

mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))

mark(times(X1, X2)) → a__times(mark(X1), mark(X2))

mark(square(X)) → a__square(mark(X))

mark(0) → 0

mark(s(X)) → s(mark(X))

mark(posrecip(X)) → posrecip(mark(X))

mark(negrecip(X)) → negrecip(mark(X))

mark(nil) → nil

mark(cons(X1, X2)) → cons(mark(X1), X2)

mark(cons2(X1, X2)) → cons2(X1, mark(X2))

mark(rnil) → rnil

mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))

a__from(X) → from(X)

a__2ndspos(X1, X2) → 2ndspos(X1, X2)

a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)

a__pi(X) → pi(X)

a__plus(X1, X2) → plus(X1, X2)

a__times(X1, X2) → times(X1, X2)

a__square(X) → square(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 7 less nodes.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ Narrowing
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ QDP
              ↳ Narrowing
                ↳ QDP
                  ↳ DependencyGraphProof
                    ↳ QDP
                      ↳ Narrowing
                        ↳ QDP
                          ↳ DependencyGraphProof
                            ↳ QDP
                              ↳ Narrowing
                                ↳ QDP
                                  ↳ DependencyGraphProof
                                    ↳ QDP
                                      ↳ Narrowing
                                        ↳ QDP
                                          ↳ DependencyGraphProof
                                            ↳ QDP
                                              ↳ Narrowing
                                                ↳ QDP
                                                  ↳ DependencyGraphProof
                                                    ↳ QDP
                                                      ↳ Narrowing
                                                        ↳ QDP
                                                          ↳ DependencyGraphProof
                                                            ↳ QDP
                                                              ↳ Narrowing
                                                                ↳ QDP
                                                                  ↳ DependencyGraphProof
                                                                    ↳ QDP
                                                                      ↳ Narrowing
                                                                        ↳ QDP
                                                                          ↳ DependencyGraphProof
                                                                            ↳ QDP
                                                                              ↳ Narrowing
                                                                                ↳ QDP
                                                                                  ↳ DependencyGraphProof
                                                                                    ↳ QDP
                                                                                      ↳ Narrowing
                                                                                        ↳ QDP
                                                                                          ↳ DependencyGraphProof
                                                                                            ↳ QDP
                                                                                              ↳ Narrowing
                                                                                                ↳ QDP
                                                                                                  ↳ DependencyGraphProof
                                                                                                    ↳ QDP
                                                                                                      ↳ Narrowing
                                                                                                        ↳ QDP
                                                                                                          ↳ DependencyGraphProof
                                                                                                            ↳ QDP
                                                                                                              ↳ Narrowing
                                                                                                                ↳ QDP
                                                                                                                  ↳ DependencyGraphProof
                                                                                                                    ↳ QDP
                                                                                                                      ↳ Narrowing
                                                                                                                        ↳ QDP
                                                                                                                          ↳ DependencyGraphProof
                                                                                                                            ↳ QDP
                                                                                                                              ↳ Narrowing
                                                                                                                                ↳ QDP
                                                                                                                                  ↳ DependencyGraphProof
                                                                                                                                    ↳ QDP
                                                                                                                                      ↳ Narrowing
                                                                                                                                        ↳ QDP
                                                                                                                                          ↳ DependencyGraphProof
                                                                                                                                            ↳ QDP
                                                                                                                                              ↳ Narrowing
                                                                                                                                                ↳ QDP
                                                                                                                                                  ↳ DependencyGraphProof
                                                                                                                                                    ↳ QDP
                                                                                                                                                      ↳ Narrowing
                                                                                                                                                        ↳ QDP
                                                                                                                                                          ↳ DependencyGraphProof
                                                                                                                                                            ↳ QDP
                                                                                                                                                              ↳ Narrowing
                                                                                                                                                                ↳ QDP
                                                                                                                                                                  ↳ DependencyGraphProof
                                                                                                                                                                    ↳ QDP
                                                                                                                                                                      ↳ Narrowing
                                                                                                                                                                        ↳ QDP
                                                                                                                                                                          ↳ DependencyGraphProof
                                                                                                                                                                            ↳ QDP
                                                                                                                                                                              ↳ Narrowing
                                                                                                                                                                                ↳ QDP
                                                                                                                                                                                  ↳ DependencyGraphProof
                                                                                                                                                                                    ↳ QDP
                                                                                                                                                                                      ↳ Narrowing
                                                                                                                                                                                        ↳ QDP
                                                                                                                                                                                          ↳ DependencyGraphProof
                                                                                                                                                                                            ↳ QDP
                                                                                                                                                                                              ↳ QDPOrderProof

Q DP problem:
The TRS P consists of the following rules:

MARK(times(times(x0, x1), y1)) → A__TIMES(a__times(mark(x0), mark(x1)), mark(y1))

MARK(2ndsneg(2ndsneg(y0, y1), 2ndspos(x0, x1))) → A__2NDSNEG(a__2ndsneg(mark(y0), mark(y1)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndsneg(2ndspos(y0, y1), 2ndsneg(x0, x1))) → A__2NDSNEG(a__2ndspos(mark(y0), mark(y1)), a__2ndsneg(mark(x0), mark(x1)))

MARK(plus(plus(x0, x1), y1)) → A__PLUS(a__plus(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(from(y0), 2ndsneg(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndspos(2ndsneg(y0, y1), from(x0))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__from(mark(x0)))

A__2NDSNEG(s(N), cons(X, Z)) → MARK(Z)

MARK(2ndsneg(square(y0), 2ndspos(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndsneg(2ndspos(y0, y1), square(x0))) → A__2NDSNEG(a__2ndspos(mark(y0), mark(y1)), a__square(mark(x0)))

A__PLUS(s(pi(x0)), y1) → A__PLUS(a__pi(mark(x0)), mark(y1))

MARK(2ndsneg(s(y0), pi(x0))) → A__2NDSNEG(s(mark(y0)), a__pi(mark(x0)))

MARK(2ndsneg(pi(y0), 2ndspos(x0, x1))) → A__2NDSNEG(a__pi(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndsneg(2ndspos(y0, y1), pi(x0))) → A__2NDSNEG(a__2ndspos(mark(y0), mark(y1)), a__pi(mark(x0)))

A__TIMES(s(X), Y) → MARK(Y)

MARK(2ndspos(square(y0), 2ndsneg(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndspos(2ndsneg(y0, y1), square(x0))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__square(mark(x0)))

A__2NDSNEG(s(N), cons(X, Z)) → MARK(N)

MARK(times(2ndsneg(x0, x1), y1)) → A__TIMES(a__2ndsneg(mark(x0), mark(x1)), mark(y1))

MARK(2ndsneg(times(y0, y1), times(x0, x1))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(pi(y0), pi(x0))) → A__2NDSPOS(a__pi(mark(y0)), a__pi(mark(x0)))

MARK(2ndsneg(2ndsneg(y0, y1), square(x0))) → A__2NDSNEG(a__2ndsneg(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndsneg(square(y0), 2ndsneg(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndsneg(from(y0), cons(x0, x1))) → A__2NDSNEG(a__from(mark(y0)), cons(mark(x0), x1))

MARK(2ndspos(s(y0), pi(x0))) → A__2NDSPOS(s(mark(y0)), a__pi(mark(x0)))

MARK(plus(square(x0), y1)) → A__PLUS(a__square(mark(x0)), mark(y1))

MARK(2ndsneg(times(y0, y1), cons2(x0, x1))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(cons2(X1, X2)) → MARK(X2)

MARK(2ndsneg(square(y0), cons2(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), cons2(x0, mark(x1)))

MARK(2ndsneg(2ndspos(y0, y1), times(x0, x1))) → A__2NDSNEG(a__2ndspos(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndsneg(times(y0, y1), 2ndspos(x0, x1))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndspos(X1, X2)) → MARK(X1)

A__PLUS(s(square(x0)), y1) → A__PLUS(a__square(mark(x0)), mark(y1))

MARK(pi(X)) → MARK(X)

MARK(2ndsneg(s(y0), from(x0))) → A__2NDSNEG(s(mark(y0)), a__from(mark(x0)))

MARK(pi(X)) → A__PI(mark(X))

MARK(2ndspos(plus(y0, y1), cons(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), cons(mark(x0), x1))

MARK(2ndsneg(square(y0), times(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndsneg(times(y0, y1), square(x0))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(plus(2ndsneg(x0, x1), y1)) → A__PLUS(a__2ndsneg(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(2ndspos(y0, y1), cons2(x0, x1))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(2ndsneg(X1, X2)) → MARK(X2)

MARK(2ndspos(square(y0), times(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(times(y0, y1), square(x0))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(times(square(x0), y1)) → A__TIMES(a__square(mark(x0)), mark(y1))

MARK(2ndspos(pi(y0), cons(x0, x1))) → A__2NDSPOS(a__pi(mark(y0)), cons(mark(x0), x1))

A__PI(X) → MARK(X)

MARK(times(s(x0), y1)) → A__TIMES(s(mark(x0)), mark(y1))

MARK(2ndspos(2ndspos(y0, y1), plus(x0, x1))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(plus(y0, y1), 2ndspos(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__2ndspos(mark(x0), mark(x1)))

A__PLUS(s(X), Y) → MARK(Y)

A__PLUS(s(2ndspos(x0, x1)), y1) → A__PLUS(a__2ndspos(mark(x0), mark(x1)), mark(y1))

MARK(2ndsneg(pi(y0), plus(x0, x1))) → A__2NDSNEG(a__pi(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(2ndsneg(plus(y0, y1), pi(x0))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__pi(mark(x0)))

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(N)

MARK(2ndspos(square(y0), 2ndspos(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndspos(2ndspos(y0, y1), square(x0))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndspos(s(y0), 2ndspos(x0, x1))) → A__2NDSPOS(s(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndspos(times(y0, y1), cons(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), cons(mark(x0), x1))

MARK(2ndspos(2ndsneg(y0, y1), 2ndsneg(x0, x1))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndspos(times(y0, y1), plus(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(plus(y0, y1), times(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndsneg(2ndspos(y0, y1), from(x0))) → A__2NDSNEG(a__2ndspos(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndsneg(from(y0), 2ndspos(x0, x1))) → A__2NDSNEG(a__from(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndsneg(2ndspos(y0, y1), cons2(x0, x1))) → A__2NDSNEG(a__2ndspos(mark(y0), mark(y1)), cons2(x0, mark(x1)))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(Y)

MARK(rcons(X1, X2)) → MARK(X1)

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(Z)

MARK(square(X)) → MARK(X)

MARK(2ndspos(pi(y0), times(x0, x1))) → A__2NDSPOS(a__pi(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(times(y0, y1), pi(x0))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__pi(mark(x0)))

A__PI(s(x0)) → A__2NDSPOS(s(mark(x0)), a__from(0))

MARK(2ndsneg(pi(y0), 2ndsneg(x0, x1))) → A__2NDSNEG(a__pi(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndsneg(2ndsneg(y0, y1), pi(x0))) → A__2NDSNEG(a__2ndsneg(mark(y0), mark(y1)), a__pi(mark(x0)))

MARK(times(plus(x0, x1), y1)) → A__TIMES(a__plus(mark(x0), mark(x1)), mark(y1))

A__2NDSPOS(s(y0), cons(y1, from(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__from(mark(x0))))

MARK(2ndspos(times(y0, y1), 2ndspos(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndspos(2ndspos(y0, y1), times(x0, x1))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndsneg(from(y0), cons2(x0, x1))) → A__2NDSNEG(a__from(mark(y0)), cons2(x0, mark(x1)))

MARK(2ndspos(s(y0), square(x0))) → A__2NDSPOS(s(mark(y0)), a__square(mark(x0)))

MARK(times(2ndspos(x0, x1), y1)) → A__TIMES(a__2ndspos(mark(x0), mark(x1)), mark(y1))

MARK(2ndsneg(2ndsneg(y0, y1), from(x0))) → A__2NDSNEG(a__2ndsneg(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndsneg(from(y0), 2ndsneg(x0, x1))) → A__2NDSNEG(a__from(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndspos(s(y0), cons2(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(x0, mark(x1)))

MARK(2ndsneg(plus(y0, y1), cons2(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), cons2(x0, mark(x1)))

A__PLUS(s(X), Y) → MARK(X)

MARK(times(X1, X2)) → MARK(X1)

A__SQUARE(X) → MARK(X)

MARK(2ndsneg(pi(y0), pi(x0))) → A__2NDSNEG(a__pi(mark(y0)), a__pi(mark(x0)))

MARK(negrecip(X)) → MARK(X)

A__PLUS(s(from(x0)), y1) → A__PLUS(a__from(mark(x0)), mark(y1))

MARK(2ndspos(from(y0), times(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(times(y0, y1), from(x0))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__from(mark(x0)))

A__PLUS(s(times(x0, x1)), y1) → A__PLUS(a__times(mark(x0), mark(x1)), mark(y1))

A__PLUS(s(0), y1) → A__PLUS(0, mark(y1))

MARK(2ndspos(2ndsneg(y0, y1), times(x0, x1))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(times(y0, y1), 2ndsneg(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__2ndsneg(mark(x0), mark(x1)))

MARK(from(X)) → MARK(X)

A__SQUARE(X) → A__TIMES(mark(X), mark(X))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(Z)

MARK(2ndsneg(plus(y0, y1), plus(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(from(y0), plus(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(plus(y0, y1), from(x0))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndsneg(2ndsneg(y0, y1), times(x0, x1))) → A__2NDSNEG(a__2ndsneg(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndsneg(times(y0, y1), 2ndsneg(x0, x1))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndsneg(times(y0, y1), cons(x0, x1))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), cons(mark(x0), x1))

A__PLUS(s(s(x0)), y1) → A__PLUS(s(mark(x0)), mark(y1))

MARK(2ndsneg(2ndsneg(y0, y1), cons(x0, x1))) → A__2NDSNEG(a__2ndsneg(mark(y0), mark(y1)), cons(mark(x0), x1))

MARK(2ndspos(from(y0), 2ndspos(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndspos(2ndspos(y0, y1), from(x0))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndspos(square(y0), plus(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(plus(y0, y1), square(x0))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__square(mark(x0)))

A__PI(times(x0, x1)) → A__2NDSPOS(a__times(mark(x0), mark(x1)), a__from(0))

A__2NDSPOS(s(y0), cons(y1, 2ndspos(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__2ndspos(mark(x0), mark(x1))))

A__2NDSNEG(s(N), cons(X, Z)) → A__2NDSNEG(s(mark(N)), cons2(X, mark(Z)))

MARK(2ndsneg(s(y0), square(x0))) → A__2NDSNEG(s(mark(y0)), a__square(mark(x0)))

MARK(2ndspos(X1, X2)) → MARK(X2)

MARK(2ndspos(pi(y0), cons2(x0, x1))) → A__2NDSPOS(a__pi(mark(y0)), cons2(x0, mark(x1)))

MARK(2ndspos(from(y0), cons2(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), cons2(x0, mark(x1)))

MARK(2ndsneg(pi(y0), from(x0))) → A__2NDSNEG(a__pi(mark(y0)), a__from(mark(x0)))

MARK(2ndsneg(from(y0), pi(x0))) → A__2NDSNEG(a__from(mark(y0)), a__pi(mark(x0)))

MARK(2ndsneg(2ndspos(y0, y1), 2ndspos(x0, x1))) → A__2NDSNEG(a__2ndspos(mark(y0), mark(y1)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndspos(s(y0), from(x0))) → A__2NDSPOS(s(mark(y0)), a__from(mark(x0)))

MARK(2ndspos(2ndsneg(y0, y1), pi(x0))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__pi(mark(x0)))

MARK(2ndspos(pi(y0), 2ndsneg(x0, x1))) → A__2NDSPOS(a__pi(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

A__2NDSPOS(s(y0), cons(y1, plus(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__plus(mark(x0), mark(x1))))

MARK(2ndspos(from(y0), pi(x0))) → A__2NDSPOS(a__from(mark(y0)), a__pi(mark(x0)))

MARK(2ndspos(pi(y0), from(x0))) → A__2NDSPOS(a__pi(mark(y0)), a__from(mark(x0)))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → A__2NDSNEG(mark(N), mark(Z))

MARK(2ndspos(from(y0), from(x0))) → A__2NDSPOS(a__from(mark(y0)), a__from(mark(x0)))

MARK(2ndspos(times(y0, y1), times(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

A__TIMES(s(X), Y) → A__PLUS(mark(Y), a__times(mark(X), mark(Y)))

MARK(2ndsneg(pi(y0), cons2(x0, x1))) → A__2NDSNEG(a__pi(mark(y0)), cons2(x0, mark(x1)))

A__PI(2ndspos(x0, x1)) → A__2NDSPOS(a__2ndspos(mark(x0), mark(x1)), a__from(0))

MARK(2ndspos(square(y0), cons2(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), cons2(x0, mark(x1)))

MARK(plus(s(x0), y1)) → A__PLUS(s(mark(x0)), mark(y1))

A__PI(plus(x0, x1)) → A__2NDSPOS(a__plus(mark(x0), mark(x1)), a__from(0))

MARK(2ndspos(square(y0), cons(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), cons(mark(x0), x1))

MARK(s(X)) → MARK(X)

MARK(2ndsneg(square(y0), plus(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(2ndsneg(plus(y0, y1), square(x0))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndspos(plus(y0, y1), cons2(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(plus(X1, X2)) → MARK(X2)

A__2NDSPOS(s(N), cons(X, Z)) → MARK(Z)

MARK(2ndsneg(from(y0), times(x0, x1))) → A__2NDSNEG(a__from(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndsneg(times(y0, y1), from(x0))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndspos(times(y0, y1), cons2(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(2ndsneg(square(y0), from(x0))) → A__2NDSNEG(a__square(mark(y0)), a__from(mark(x0)))

MARK(2ndsneg(from(y0), square(x0))) → A__2NDSNEG(a__from(mark(y0)), a__square(mark(x0)))

MARK(2ndspos(2ndsneg(y0, y1), cons2(x0, x1))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(2ndsneg(s(y0), cons(x0, x1))) → A__2NDSNEG(s(mark(y0)), cons(mark(x0), x1))

A__PI(X) → A__FROM(0)

MARK(2ndspos(s(y0), plus(x0, x1))) → A__2NDSPOS(s(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(from(X)) → A__FROM(mark(X))

A__PLUS(s(2ndsneg(x0, x1)), y1) → A__PLUS(a__2ndsneg(mark(x0), mark(x1)), mark(y1))

MARK(plus(X1, X2)) → MARK(X1)

MARK(times(from(x0), y1)) → A__TIMES(a__from(mark(x0)), mark(y1))

A__2NDSPOS(s(y0), cons(y1, times(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__times(mark(x0), mark(x1))))

MARK(2ndsneg(2ndsneg(y0, y1), 2ndsneg(x0, x1))) → A__2NDSNEG(a__2ndsneg(mark(y0), mark(y1)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndspos(s(y0), cons(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons(mark(x0), x1))

MARK(2ndsneg(2ndspos(y0, y1), plus(x0, x1))) → A__2NDSNEG(a__2ndspos(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(2ndsneg(plus(y0, y1), 2ndspos(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndspos(pi(y0), 2ndspos(x0, x1))) → A__2NDSPOS(a__pi(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndspos(2ndspos(y0, y1), pi(x0))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), a__pi(mark(x0)))

MARK(2ndspos(from(y0), cons(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), cons(mark(x0), x1))

MARK(2ndsneg(2ndspos(y0, y1), cons(x0, x1))) → A__2NDSNEG(a__2ndspos(mark(y0), mark(y1)), cons(mark(x0), x1))

MARK(2ndsneg(plus(y0, y1), cons(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), cons(mark(x0), x1))

A__2NDSPOS(s(y0), cons(y1, pi(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__pi(mark(x0))))

A__TIMES(s(X), Y) → MARK(X)

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(N)

MARK(2ndspos(pi(y0), plus(x0, x1))) → A__2NDSPOS(a__pi(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(plus(y0, y1), pi(x0))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__pi(mark(x0)))

MARK(2ndsneg(s(y0), cons2(x0, x1))) → A__2NDSNEG(s(mark(y0)), cons2(x0, mark(x1)))

MARK(plus(2ndspos(x0, x1), y1)) → A__PLUS(a__2ndspos(mark(x0), mark(x1)), mark(y1))

MARK(2ndsneg(pi(y0), cons(x0, x1))) → A__2NDSNEG(a__pi(mark(y0)), cons(mark(x0), x1))

A__PI(from(x0)) → A__2NDSPOS(a__from(mark(x0)), a__from(0))

MARK(cons(X1, X2)) → MARK(X1)

MARK(2ndsneg(s(y0), 2ndspos(x0, x1))) → A__2NDSNEG(s(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndsneg(plus(y0, y1), from(x0))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndsneg(from(y0), plus(x0, x1))) → A__2NDSNEG(a__from(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(times(X1, X2)) → MARK(X2)

A__PI(square(x0)) → A__2NDSPOS(a__square(mark(x0)), a__from(0))

MARK(2ndsneg(square(y0), cons(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), cons(mark(x0), x1))

MARK(2ndspos(s(y0), times(x0, x1))) → A__2NDSPOS(s(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndsneg(X1, X2)) → MARK(X1)

MARK(2ndspos(s(y0), 2ndsneg(x0, x1))) → A__2NDSPOS(s(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndsneg(square(y0), square(x0))) → A__2NDSNEG(a__square(mark(y0)), a__square(mark(x0)))

MARK(2ndspos(2ndsneg(y0, y1), plus(x0, x1))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(plus(y0, y1), 2ndsneg(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__2ndsneg(mark(x0), mark(x1)))

A__PI(2ndsneg(x0, x1)) → A__2NDSPOS(a__2ndsneg(mark(x0), mark(x1)), a__from(0))

MARK(times(pi(x0), y1)) → A__TIMES(a__pi(mark(x0)), mark(y1))

MARK(plus(pi(x0), y1)) → A__PLUS(a__pi(mark(x0)), mark(y1))

MARK(2ndspos(2ndspos(y0, y1), 2ndspos(x0, x1))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndsneg(pi(y0), square(x0))) → A__2NDSNEG(a__pi(mark(y0)), a__square(mark(x0)))

MARK(2ndsneg(square(y0), pi(x0))) → A__2NDSNEG(a__square(mark(y0)), a__pi(mark(x0)))

MARK(2ndspos(2ndsneg(y0, y1), 2ndspos(x0, x1))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndspos(2ndspos(y0, y1), 2ndsneg(x0, x1))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), a__2ndsneg(mark(x0), mark(x1)))

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(Y)

MARK(2ndsneg(s(y0), 2ndsneg(x0, x1))) → A__2NDSNEG(s(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

A__FROM(X) → MARK(X)

MARK(2ndspos(2ndspos(y0, y1), cons(x0, x1))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), cons(mark(x0), x1))

MARK(2ndsneg(s(y0), plus(x0, x1))) → A__2NDSNEG(s(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(square(y0), pi(x0))) → A__2NDSPOS(a__square(mark(y0)), a__pi(mark(x0)))

MARK(2ndspos(pi(y0), square(x0))) → A__2NDSPOS(a__pi(mark(y0)), a__square(mark(x0)))

MARK(rcons(X1, X2)) → MARK(X2)

A__TIMES(s(X), Y) → A__TIMES(mark(X), mark(Y))

A__2NDSPOS(s(y0), cons(y1, cons(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, cons(mark(x0), x1)))

MARK(plus(times(x0, x1), y1)) → A__PLUS(a__times(mark(x0), mark(x1)), mark(y1))

MARK(2ndsneg(from(y0), from(x0))) → A__2NDSNEG(a__from(mark(y0)), a__from(mark(x0)))

A__2NDSPOS(s(y0), cons(y1, 2ndsneg(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__2ndsneg(mark(x0), mark(x1))))

A__2NDSPOS(s(N), cons(X, Z)) → MARK(N)

MARK(2ndsneg(2ndsneg(y0, y1), plus(x0, x1))) → A__2NDSNEG(a__2ndsneg(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(2ndsneg(plus(y0, y1), 2ndsneg(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__2ndsneg(mark(x0), mark(x1)))

A__PLUS(0, Y) → MARK(Y)

MARK(posrecip(X)) → MARK(X)

MARK(2ndspos(from(y0), square(x0))) → A__2NDSPOS(a__from(mark(y0)), a__square(mark(x0)))

MARK(2ndspos(square(y0), from(x0))) → A__2NDSPOS(a__square(mark(y0)), a__from(mark(x0)))

A__2NDSPOS(s(y0), cons(y1, square(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__square(mark(x0))))

MARK(plus(0, y1)) → A__PLUS(0, mark(y1))

MARK(2ndsneg(s(y0), times(x0, x1))) → A__2NDSNEG(s(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(square(y0), square(x0))) → A__2NDSPOS(a__square(mark(y0)), a__square(mark(x0)))

A__PI(pi(x0)) → A__2NDSPOS(a__pi(mark(x0)), a__from(0))

A__PLUS(s(plus(x0, x1)), y1) → A__PLUS(a__plus(mark(x0), mark(x1)), mark(y1))

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → A__2NDSPOS(mark(N), mark(Z))

MARK(plus(from(x0), y1)) → A__PLUS(a__from(mark(x0)), mark(y1))

MARK(2ndsneg(pi(y0), times(x0, x1))) → A__2NDSNEG(a__pi(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(square(X)) → A__SQUARE(mark(X))

MARK(2ndsneg(times(y0, y1), pi(x0))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__pi(mark(x0)))

MARK(2ndsneg(plus(y0, y1), times(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndsneg(times(y0, y1), plus(x0, x1))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(plus(y0, y1), plus(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(2ndsneg(2ndsneg(y0, y1), cons2(x0, x1))) → A__2NDSNEG(a__2ndsneg(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(2ndspos(2ndsneg(y0, y1), cons(x0, x1))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), cons(mark(x0), x1))

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))

a__2ndspos(0, Z) → rnil

a__2ndspos(s(N), cons(X, Z)) → a__2ndspos(s(mark(N)), cons2(X, mark(Z)))

a__2ndspos(s(N), cons2(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))

a__2ndsneg(0, Z) → rnil

a__2ndsneg(s(N), cons(X, Z)) → a__2ndsneg(s(mark(N)), cons2(X, mark(Z)))

a__2ndsneg(s(N), cons2(X, cons(Y, Z))) → rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z)))

a__pi(X) → a__2ndspos(mark(X), a__from(0))

a__plus(0, Y) → mark(Y)

a__plus(s(X), Y) → s(a__plus(mark(X), mark(Y)))

a__times(0, Y) → 0

a__times(s(X), Y) → a__plus(mark(Y), a__times(mark(X), mark(Y)))

a__square(X) → a__times(mark(X), mark(X))

mark(from(X)) → a__from(mark(X))

mark(2ndspos(X1, X2)) → a__2ndspos(mark(X1), mark(X2))

mark(2ndsneg(X1, X2)) → a__2ndsneg(mark(X1), mark(X2))

mark(pi(X)) → a__pi(mark(X))

mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))

mark(times(X1, X2)) → a__times(mark(X1), mark(X2))

mark(square(X)) → a__square(mark(X))

mark(0) → 0

mark(s(X)) → s(mark(X))

mark(posrecip(X)) → posrecip(mark(X))

mark(negrecip(X)) → negrecip(mark(X))

mark(nil) → nil

mark(cons(X1, X2)) → cons(mark(X1), X2)

mark(cons2(X1, X2)) → cons2(X1, mark(X2))

mark(rnil) → rnil

mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))

a__from(X) → from(X)

a__2ndspos(X1, X2) → 2ndspos(X1, X2)

a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)

a__pi(X) → pi(X)

a__plus(X1, X2) → plus(X1, X2)

a__times(X1, X2) → times(X1, X2)

a__square(X) → square(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

MARK(2ndsneg(2ndspos(y0, y1), 2ndsneg(x0, x1))) → A__2NDSNEG(a__2ndspos(mark(y0), mark(y1)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndspos(from(y0), 2ndsneg(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndsneg(2ndspos(y0, y1), square(x0))) → A__2NDSNEG(a__2ndspos(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndsneg(2ndspos(y0, y1), pi(x0))) → A__2NDSNEG(a__2ndspos(mark(y0), mark(y1)), a__pi(mark(x0)))

MARK(2ndsneg(from(y0), cons(x0, x1))) → A__2NDSNEG(a__from(mark(y0)), cons(mark(x0), x1))

MARK(2ndsneg(2ndspos(y0, y1), times(x0, x1))) → A__2NDSNEG(a__2ndspos(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(2ndspos(y0, y1), cons2(x0, x1))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(2ndspos(2ndspos(y0, y1), plus(x0, x1))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(2ndspos(y0, y1), square(x0))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndsneg(2ndspos(y0, y1), from(x0))) → A__2NDSNEG(a__2ndspos(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndsneg(from(y0), 2ndspos(x0, x1))) → A__2NDSNEG(a__from(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndsneg(2ndspos(y0, y1), cons2(x0, x1))) → A__2NDSNEG(a__2ndspos(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(2ndspos(2ndspos(y0, y1), times(x0, x1))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndsneg(from(y0), cons2(x0, x1))) → A__2NDSNEG(a__from(mark(y0)), cons2(x0, mark(x1)))

MARK(2ndsneg(from(y0), 2ndsneg(x0, x1))) → A__2NDSNEG(a__from(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndspos(from(y0), times(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(from(y0), plus(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(from(y0), 2ndspos(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndspos(2ndspos(y0, y1), from(x0))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndspos(from(y0), cons2(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), cons2(x0, mark(x1)))

MARK(2ndsneg(from(y0), pi(x0))) → A__2NDSNEG(a__from(mark(y0)), a__pi(mark(x0)))

MARK(2ndsneg(2ndspos(y0, y1), 2ndspos(x0, x1))) → A__2NDSNEG(a__2ndspos(mark(y0), mark(y1)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndspos(from(y0), pi(x0))) → A__2NDSPOS(a__from(mark(y0)), a__pi(mark(x0)))

MARK(2ndspos(from(y0), from(x0))) → A__2NDSPOS(a__from(mark(y0)), a__from(mark(x0)))

A__PI(2ndspos(x0, x1)) → A__2NDSPOS(a__2ndspos(mark(x0), mark(x1)), a__from(0))

MARK(2ndsneg(from(y0), times(x0, x1))) → A__2NDSNEG(a__from(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndsneg(from(y0), square(x0))) → A__2NDSNEG(a__from(mark(y0)), a__square(mark(x0)))

MARK(2ndsneg(2ndspos(y0, y1), plus(x0, x1))) → A__2NDSNEG(a__2ndspos(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(2ndspos(y0, y1), pi(x0))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), a__pi(mark(x0)))

MARK(2ndspos(from(y0), cons(x0, x1))) → A__2NDSPOS(a__from(mark(y0)), cons(mark(x0), x1))

MARK(2ndsneg(2ndspos(y0, y1), cons(x0, x1))) → A__2NDSNEG(a__2ndspos(mark(y0), mark(y1)), cons(mark(x0), x1))

A__PI(from(x0)) → A__2NDSPOS(a__from(mark(x0)), a__from(0))

MARK(2ndsneg(from(y0), plus(x0, x1))) → A__2NDSNEG(a__from(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(2ndspos(y0, y1), 2ndspos(x0, x1))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndspos(2ndspos(y0, y1), 2ndsneg(x0, x1))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndspos(2ndspos(y0, y1), cons(x0, x1))) → A__2NDSPOS(a__2ndspos(mark(y0), mark(y1)), cons(mark(x0), x1))

MARK(2ndsneg(from(y0), from(x0))) → A__2NDSNEG(a__from(mark(y0)), a__from(mark(x0)))

MARK(2ndspos(from(y0), square(x0))) → A__2NDSPOS(a__from(mark(y0)), a__square(mark(x0)))
The remaining pairs can at least be oriented weakly.

MARK(times(times(x0, x1), y1)) → A__TIMES(a__times(mark(x0), mark(x1)), mark(y1))

MARK(2ndsneg(2ndsneg(y0, y1), 2ndspos(x0, x1))) → A__2NDSNEG(a__2ndsneg(mark(y0), mark(y1)), a__2ndspos(mark(x0), mark(x1)))

MARK(plus(plus(x0, x1), y1)) → A__PLUS(a__plus(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(2ndsneg(y0, y1), from(x0))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__from(mark(x0)))

A__2NDSNEG(s(N), cons(X, Z)) → MARK(Z)

MARK(2ndsneg(square(y0), 2ndspos(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

A__PLUS(s(pi(x0)), y1) → A__PLUS(a__pi(mark(x0)), mark(y1))

MARK(2ndsneg(s(y0), pi(x0))) → A__2NDSNEG(s(mark(y0)), a__pi(mark(x0)))

MARK(2ndsneg(pi(y0), 2ndspos(x0, x1))) → A__2NDSNEG(a__pi(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

A__TIMES(s(X), Y) → MARK(Y)

MARK(2ndspos(square(y0), 2ndsneg(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndspos(2ndsneg(y0, y1), square(x0))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__square(mark(x0)))

A__2NDSNEG(s(N), cons(X, Z)) → MARK(N)

MARK(times(2ndsneg(x0, x1), y1)) → A__TIMES(a__2ndsneg(mark(x0), mark(x1)), mark(y1))

MARK(2ndsneg(times(y0, y1), times(x0, x1))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(pi(y0), pi(x0))) → A__2NDSPOS(a__pi(mark(y0)), a__pi(mark(x0)))

MARK(2ndsneg(2ndsneg(y0, y1), square(x0))) → A__2NDSNEG(a__2ndsneg(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndsneg(square(y0), 2ndsneg(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndspos(s(y0), pi(x0))) → A__2NDSPOS(s(mark(y0)), a__pi(mark(x0)))

MARK(plus(square(x0), y1)) → A__PLUS(a__square(mark(x0)), mark(y1))

MARK(2ndsneg(times(y0, y1), cons2(x0, x1))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(cons2(X1, X2)) → MARK(X2)

MARK(2ndsneg(square(y0), cons2(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), cons2(x0, mark(x1)))

MARK(2ndsneg(times(y0, y1), 2ndspos(x0, x1))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndspos(X1, X2)) → MARK(X1)

A__PLUS(s(square(x0)), y1) → A__PLUS(a__square(mark(x0)), mark(y1))

MARK(pi(X)) → MARK(X)

MARK(2ndsneg(s(y0), from(x0))) → A__2NDSNEG(s(mark(y0)), a__from(mark(x0)))

MARK(pi(X)) → A__PI(mark(X))

MARK(2ndspos(plus(y0, y1), cons(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), cons(mark(x0), x1))

MARK(2ndsneg(square(y0), times(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndsneg(times(y0, y1), square(x0))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(plus(2ndsneg(x0, x1), y1)) → A__PLUS(a__2ndsneg(mark(x0), mark(x1)), mark(y1))

MARK(2ndsneg(X1, X2)) → MARK(X2)

MARK(2ndspos(square(y0), times(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(times(y0, y1), square(x0))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(times(square(x0), y1)) → A__TIMES(a__square(mark(x0)), mark(y1))

MARK(2ndspos(pi(y0), cons(x0, x1))) → A__2NDSPOS(a__pi(mark(y0)), cons(mark(x0), x1))

A__PI(X) → MARK(X)

MARK(times(s(x0), y1)) → A__TIMES(s(mark(x0)), mark(y1))

MARK(2ndspos(plus(y0, y1), 2ndspos(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__2ndspos(mark(x0), mark(x1)))

A__PLUS(s(X), Y) → MARK(Y)

A__PLUS(s(2ndspos(x0, x1)), y1) → A__PLUS(a__2ndspos(mark(x0), mark(x1)), mark(y1))

MARK(2ndsneg(pi(y0), plus(x0, x1))) → A__2NDSNEG(a__pi(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(2ndsneg(plus(y0, y1), pi(x0))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__pi(mark(x0)))

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(N)

MARK(2ndspos(square(y0), 2ndspos(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndspos(s(y0), 2ndspos(x0, x1))) → A__2NDSPOS(s(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndspos(times(y0, y1), cons(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), cons(mark(x0), x1))

MARK(2ndspos(2ndsneg(y0, y1), 2ndsneg(x0, x1))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndspos(times(y0, y1), plus(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(plus(y0, y1), times(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(Y)

MARK(rcons(X1, X2)) → MARK(X1)

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(Z)

MARK(square(X)) → MARK(X)

MARK(2ndspos(pi(y0), times(x0, x1))) → A__2NDSPOS(a__pi(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(times(y0, y1), pi(x0))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__pi(mark(x0)))

A__PI(s(x0)) → A__2NDSPOS(s(mark(x0)), a__from(0))

MARK(2ndsneg(pi(y0), 2ndsneg(x0, x1))) → A__2NDSNEG(a__pi(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndsneg(2ndsneg(y0, y1), pi(x0))) → A__2NDSNEG(a__2ndsneg(mark(y0), mark(y1)), a__pi(mark(x0)))

MARK(times(plus(x0, x1), y1)) → A__TIMES(a__plus(mark(x0), mark(x1)), mark(y1))

A__2NDSPOS(s(y0), cons(y1, from(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__from(mark(x0))))

MARK(2ndspos(times(y0, y1), 2ndspos(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndspos(s(y0), square(x0))) → A__2NDSPOS(s(mark(y0)), a__square(mark(x0)))

MARK(times(2ndspos(x0, x1), y1)) → A__TIMES(a__2ndspos(mark(x0), mark(x1)), mark(y1))

MARK(2ndsneg(2ndsneg(y0, y1), from(x0))) → A__2NDSNEG(a__2ndsneg(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndspos(s(y0), cons2(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(x0, mark(x1)))

MARK(2ndsneg(plus(y0, y1), cons2(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), cons2(x0, mark(x1)))

A__PLUS(s(X), Y) → MARK(X)

MARK(times(X1, X2)) → MARK(X1)

A__SQUARE(X) → MARK(X)

MARK(2ndsneg(pi(y0), pi(x0))) → A__2NDSNEG(a__pi(mark(y0)), a__pi(mark(x0)))

MARK(negrecip(X)) → MARK(X)

A__PLUS(s(from(x0)), y1) → A__PLUS(a__from(mark(x0)), mark(y1))

MARK(2ndspos(times(y0, y1), from(x0))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__from(mark(x0)))

A__PLUS(s(times(x0, x1)), y1) → A__PLUS(a__times(mark(x0), mark(x1)), mark(y1))

A__PLUS(s(0), y1) → A__PLUS(0, mark(y1))

MARK(2ndspos(2ndsneg(y0, y1), times(x0, x1))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(times(y0, y1), 2ndsneg(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__2ndsneg(mark(x0), mark(x1)))

MARK(from(X)) → MARK(X)

A__SQUARE(X) → A__TIMES(mark(X), mark(X))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(Z)

MARK(2ndsneg(plus(y0, y1), plus(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(plus(y0, y1), from(x0))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndsneg(2ndsneg(y0, y1), times(x0, x1))) → A__2NDSNEG(a__2ndsneg(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndsneg(times(y0, y1), 2ndsneg(x0, x1))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndsneg(times(y0, y1), cons(x0, x1))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), cons(mark(x0), x1))

A__PLUS(s(s(x0)), y1) → A__PLUS(s(mark(x0)), mark(y1))

MARK(2ndsneg(2ndsneg(y0, y1), cons(x0, x1))) → A__2NDSNEG(a__2ndsneg(mark(y0), mark(y1)), cons(mark(x0), x1))

MARK(2ndspos(square(y0), plus(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(plus(y0, y1), square(x0))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__square(mark(x0)))

A__PI(times(x0, x1)) → A__2NDSPOS(a__times(mark(x0), mark(x1)), a__from(0))

A__2NDSPOS(s(y0), cons(y1, 2ndspos(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__2ndspos(mark(x0), mark(x1))))

A__2NDSNEG(s(N), cons(X, Z)) → A__2NDSNEG(s(mark(N)), cons2(X, mark(Z)))

MARK(2ndsneg(s(y0), square(x0))) → A__2NDSNEG(s(mark(y0)), a__square(mark(x0)))

MARK(2ndspos(X1, X2)) → MARK(X2)

MARK(2ndspos(pi(y0), cons2(x0, x1))) → A__2NDSPOS(a__pi(mark(y0)), cons2(x0, mark(x1)))

MARK(2ndsneg(pi(y0), from(x0))) → A__2NDSNEG(a__pi(mark(y0)), a__from(mark(x0)))

MARK(2ndspos(s(y0), from(x0))) → A__2NDSPOS(s(mark(y0)), a__from(mark(x0)))

MARK(2ndspos(2ndsneg(y0, y1), pi(x0))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__pi(mark(x0)))

MARK(2ndspos(pi(y0), 2ndsneg(x0, x1))) → A__2NDSPOS(a__pi(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

A__2NDSPOS(s(y0), cons(y1, plus(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__plus(mark(x0), mark(x1))))

MARK(2ndspos(pi(y0), from(x0))) → A__2NDSPOS(a__pi(mark(y0)), a__from(mark(x0)))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → A__2NDSNEG(mark(N), mark(Z))

MARK(2ndspos(times(y0, y1), times(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

A__TIMES(s(X), Y) → A__PLUS(mark(Y), a__times(mark(X), mark(Y)))

MARK(2ndsneg(pi(y0), cons2(x0, x1))) → A__2NDSNEG(a__pi(mark(y0)), cons2(x0, mark(x1)))

MARK(2ndspos(square(y0), cons2(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), cons2(x0, mark(x1)))

MARK(plus(s(x0), y1)) → A__PLUS(s(mark(x0)), mark(y1))

A__PI(plus(x0, x1)) → A__2NDSPOS(a__plus(mark(x0), mark(x1)), a__from(0))

MARK(2ndspos(square(y0), cons(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), cons(mark(x0), x1))

MARK(s(X)) → MARK(X)

MARK(2ndsneg(square(y0), plus(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(2ndsneg(plus(y0, y1), square(x0))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndspos(plus(y0, y1), cons2(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(plus(X1, X2)) → MARK(X2)

A__2NDSPOS(s(N), cons(X, Z)) → MARK(Z)

MARK(2ndsneg(times(y0, y1), from(x0))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndspos(times(y0, y1), cons2(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(2ndsneg(square(y0), from(x0))) → A__2NDSNEG(a__square(mark(y0)), a__from(mark(x0)))

MARK(2ndspos(2ndsneg(y0, y1), cons2(x0, x1))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(2ndsneg(s(y0), cons(x0, x1))) → A__2NDSNEG(s(mark(y0)), cons(mark(x0), x1))

A__PI(X) → A__FROM(0)

MARK(2ndspos(s(y0), plus(x0, x1))) → A__2NDSPOS(s(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(from(X)) → A__FROM(mark(X))

A__PLUS(s(2ndsneg(x0, x1)), y1) → A__PLUS(a__2ndsneg(mark(x0), mark(x1)), mark(y1))

MARK(plus(X1, X2)) → MARK(X1)

MARK(times(from(x0), y1)) → A__TIMES(a__from(mark(x0)), mark(y1))

A__2NDSPOS(s(y0), cons(y1, times(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__times(mark(x0), mark(x1))))

MARK(2ndsneg(2ndsneg(y0, y1), 2ndsneg(x0, x1))) → A__2NDSNEG(a__2ndsneg(mark(y0), mark(y1)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndspos(s(y0), cons(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons(mark(x0), x1))

MARK(2ndsneg(plus(y0, y1), 2ndspos(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndspos(pi(y0), 2ndspos(x0, x1))) → A__2NDSPOS(a__pi(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndsneg(plus(y0, y1), cons(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), cons(mark(x0), x1))

A__2NDSPOS(s(y0), cons(y1, pi(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__pi(mark(x0))))

A__TIMES(s(X), Y) → MARK(X)

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(N)

MARK(2ndspos(pi(y0), plus(x0, x1))) → A__2NDSPOS(a__pi(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(plus(y0, y1), pi(x0))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__pi(mark(x0)))

MARK(2ndsneg(s(y0), cons2(x0, x1))) → A__2NDSNEG(s(mark(y0)), cons2(x0, mark(x1)))

MARK(plus(2ndspos(x0, x1), y1)) → A__PLUS(a__2ndspos(mark(x0), mark(x1)), mark(y1))

MARK(2ndsneg(pi(y0), cons(x0, x1))) → A__2NDSNEG(a__pi(mark(y0)), cons(mark(x0), x1))

MARK(cons(X1, X2)) → MARK(X1)

MARK(2ndsneg(s(y0), 2ndspos(x0, x1))) → A__2NDSNEG(s(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndsneg(plus(y0, y1), from(x0))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(times(X1, X2)) → MARK(X2)

A__PI(square(x0)) → A__2NDSPOS(a__square(mark(x0)), a__from(0))

MARK(2ndsneg(square(y0), cons(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), cons(mark(x0), x1))

MARK(2ndspos(s(y0), times(x0, x1))) → A__2NDSPOS(s(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndsneg(X1, X2)) → MARK(X1)

MARK(2ndspos(s(y0), 2ndsneg(x0, x1))) → A__2NDSPOS(s(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndsneg(square(y0), square(x0))) → A__2NDSNEG(a__square(mark(y0)), a__square(mark(x0)))

MARK(2ndspos(2ndsneg(y0, y1), plus(x0, x1))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(plus(y0, y1), 2ndsneg(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__2ndsneg(mark(x0), mark(x1)))

A__PI(2ndsneg(x0, x1)) → A__2NDSPOS(a__2ndsneg(mark(x0), mark(x1)), a__from(0))

MARK(times(pi(x0), y1)) → A__TIMES(a__pi(mark(x0)), mark(y1))

MARK(plus(pi(x0), y1)) → A__PLUS(a__pi(mark(x0)), mark(y1))

MARK(2ndsneg(pi(y0), square(x0))) → A__2NDSNEG(a__pi(mark(y0)), a__square(mark(x0)))

MARK(2ndsneg(square(y0), pi(x0))) → A__2NDSNEG(a__square(mark(y0)), a__pi(mark(x0)))

MARK(2ndspos(2ndsneg(y0, y1), 2ndspos(x0, x1))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__2ndspos(mark(x0), mark(x1)))

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(Y)

MARK(2ndsneg(s(y0), 2ndsneg(x0, x1))) → A__2NDSNEG(s(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

A__FROM(X) → MARK(X)

MARK(2ndsneg(s(y0), plus(x0, x1))) → A__2NDSNEG(s(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(square(y0), pi(x0))) → A__2NDSPOS(a__square(mark(y0)), a__pi(mark(x0)))

MARK(2ndspos(pi(y0), square(x0))) → A__2NDSPOS(a__pi(mark(y0)), a__square(mark(x0)))

MARK(rcons(X1, X2)) → MARK(X2)

A__TIMES(s(X), Y) → A__TIMES(mark(X), mark(Y))

A__2NDSPOS(s(y0), cons(y1, cons(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, cons(mark(x0), x1)))

MARK(plus(times(x0, x1), y1)) → A__PLUS(a__times(mark(x0), mark(x1)), mark(y1))

A__2NDSPOS(s(y0), cons(y1, 2ndsneg(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__2ndsneg(mark(x0), mark(x1))))

A__2NDSPOS(s(N), cons(X, Z)) → MARK(N)

MARK(2ndsneg(2ndsneg(y0, y1), plus(x0, x1))) → A__2NDSNEG(a__2ndsneg(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(2ndsneg(plus(y0, y1), 2ndsneg(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__2ndsneg(mark(x0), mark(x1)))

A__PLUS(0, Y) → MARK(Y)

MARK(posrecip(X)) → MARK(X)

MARK(2ndspos(square(y0), from(x0))) → A__2NDSPOS(a__square(mark(y0)), a__from(mark(x0)))

A__2NDSPOS(s(y0), cons(y1, square(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__square(mark(x0))))

MARK(plus(0, y1)) → A__PLUS(0, mark(y1))

MARK(2ndsneg(s(y0), times(x0, x1))) → A__2NDSNEG(s(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(square(y0), square(x0))) → A__2NDSPOS(a__square(mark(y0)), a__square(mark(x0)))

A__PI(pi(x0)) → A__2NDSPOS(a__pi(mark(x0)), a__from(0))

A__PLUS(s(plus(x0, x1)), y1) → A__PLUS(a__plus(mark(x0), mark(x1)), mark(y1))

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → A__2NDSPOS(mark(N), mark(Z))

MARK(plus(from(x0), y1)) → A__PLUS(a__from(mark(x0)), mark(y1))

MARK(2ndsneg(pi(y0), times(x0, x1))) → A__2NDSNEG(a__pi(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(square(X)) → A__SQUARE(mark(X))

MARK(2ndsneg(times(y0, y1), pi(x0))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__pi(mark(x0)))

MARK(2ndsneg(plus(y0, y1), times(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndsneg(times(y0, y1), plus(x0, x1))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(plus(y0, y1), plus(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(2ndsneg(2ndsneg(y0, y1), cons2(x0, x1))) → A__2NDSNEG(a__2ndsneg(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(2ndspos(2ndsneg(y0, y1), cons(x0, x1))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), cons(mark(x0), x1))
Used ordering:  Polynomial interpretation [25]:

POL(0) = 1   

POL(2ndsneg(x1, x2)) = 1   

POL(2ndspos(x1, x2)) = 0   

POL(A__2NDSNEG(x1, x2)) = x1   

POL(A__2NDSPOS(x1, x2)) = x1   

POL(A__FROM(x1)) = 1   

POL(A__PI(x1)) = 1   

POL(A__PLUS(x1, x2)) = 1   

POL(A__SQUARE(x1)) = 1   

POL(A__TIMES(x1, x2)) = 1   

POL(MARK(x1)) = 1   

POL(a__2ndsneg(x1, x2)) = 1   

POL(a__2ndspos(x1, x2)) = 0   

POL(a__from(x1)) = 0   

POL(a__pi(x1)) = 1   

POL(a__plus(x1, x2)) = x1   

POL(a__square(x1)) = 1   

POL(a__times(x1, x2)) = 1   

POL(cons(x1, x2)) = 0   

POL(cons2(x1, x2)) = 0   

POL(from(x1)) = 0   

POL(mark(x1)) = 1   

POL(negrecip(x1)) = 0   

POL(nil) = 0   

POL(pi(x1)) = 0   

POL(plus(x1, x2)) = 0   

POL(posrecip(x1)) = 0   

POL(rcons(x1, x2)) = 0   

POL(rnil) = 0   

POL(s(x1)) = 1   

POL(square(x1)) = 0   

POL(times(x1, x2)) = 0   

The following usable rules [17] were oriented:

a__from(X) → cons(mark(X), from(s(X)))

a__pi(X) → a__2ndspos(mark(X), a__from(0))

a__2ndsneg(s(N), cons(X, Z)) → a__2ndsneg(s(mark(N)), cons2(X, mark(Z)))

a__2ndsneg(s(N), cons2(X, cons(Y, Z))) → rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z)))

a__2ndspos(s(N), cons2(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))

a__2ndsneg(0, Z) → rnil

a__2ndspos(0, Z) → rnil

a__2ndspos(s(N), cons(X, Z)) → a__2ndspos(s(mark(N)), cons2(X, mark(Z)))

mark(pi(X)) → a__pi(mark(X))

mark(2ndsneg(X1, X2)) → a__2ndsneg(mark(X1), mark(X2))

mark(2ndspos(X1, X2)) → a__2ndspos(mark(X1), mark(X2))

mark(from(X)) → a__from(mark(X))

a__times(0, Y) → 0

a__plus(s(X), Y) → s(a__plus(mark(X), mark(Y)))

mark(nil) → nil

mark(negrecip(X)) → negrecip(mark(X))

mark(posrecip(X)) → posrecip(mark(X))

mark(s(X)) → s(mark(X))

mark(0) → 0

mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))

a__times(s(X), Y) → a__plus(mark(Y), a__times(mark(X), mark(Y)))

a__plus(0, Y) → mark(Y)

mark(square(X)) → a__square(mark(X))

mark(times(X1, X2)) → a__times(mark(X1), mark(X2))

a__square(X) → a__times(mark(X), mark(X))

a__times(X1, X2) → times(X1, X2)

a__plus(X1, X2) → plus(X1, X2)

a__square(X) → square(X)

mark(cons2(X1, X2)) → cons2(X1, mark(X2))

mark(cons(X1, X2)) → cons(mark(X1), X2)

mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))

mark(rnil) → rnil

a__2ndspos(X1, X2) → 2ndspos(X1, X2)

a__from(X) → from(X)

a__pi(X) → pi(X)

a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ Narrowing
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ QDP
              ↳ Narrowing
                ↳ QDP
                  ↳ DependencyGraphProof
                    ↳ QDP
                      ↳ Narrowing
                        ↳ QDP
                          ↳ DependencyGraphProof
                            ↳ QDP
                              ↳ Narrowing
                                ↳ QDP
                                  ↳ DependencyGraphProof
                                    ↳ QDP
                                      ↳ Narrowing
                                        ↳ QDP
                                          ↳ DependencyGraphProof
                                            ↳ QDP
                                              ↳ Narrowing
                                                ↳ QDP
                                                  ↳ DependencyGraphProof
                                                    ↳ QDP
                                                      ↳ Narrowing
                                                        ↳ QDP
                                                          ↳ DependencyGraphProof
                                                            ↳ QDP
                                                              ↳ Narrowing
                                                                ↳ QDP
                                                                  ↳ DependencyGraphProof
                                                                    ↳ QDP
                                                                      ↳ Narrowing
                                                                        ↳ QDP
                                                                          ↳ DependencyGraphProof
                                                                            ↳ QDP
                                                                              ↳ Narrowing
                                                                                ↳ QDP
                                                                                  ↳ DependencyGraphProof
                                                                                    ↳ QDP
                                                                                      ↳ Narrowing
                                                                                        ↳ QDP
                                                                                          ↳ DependencyGraphProof
                                                                                            ↳ QDP
                                                                                              ↳ Narrowing
                                                                                                ↳ QDP
                                                                                                  ↳ DependencyGraphProof
                                                                                                    ↳ QDP
                                                                                                      ↳ Narrowing
                                                                                                        ↳ QDP
                                                                                                          ↳ DependencyGraphProof
                                                                                                            ↳ QDP
                                                                                                              ↳ Narrowing
                                                                                                                ↳ QDP
                                                                                                                  ↳ DependencyGraphProof
                                                                                                                    ↳ QDP
                                                                                                                      ↳ Narrowing
                                                                                                                        ↳ QDP
                                                                                                                          ↳ DependencyGraphProof
                                                                                                                            ↳ QDP
                                                                                                                              ↳ Narrowing
                                                                                                                                ↳ QDP
                                                                                                                                  ↳ DependencyGraphProof
                                                                                                                                    ↳ QDP
                                                                                                                                      ↳ Narrowing
                                                                                                                                        ↳ QDP
                                                                                                                                          ↳ DependencyGraphProof
                                                                                                                                            ↳ QDP
                                                                                                                                              ↳ Narrowing
                                                                                                                                                ↳ QDP
                                                                                                                                                  ↳ DependencyGraphProof
                                                                                                                                                    ↳ QDP
                                                                                                                                                      ↳ Narrowing
                                                                                                                                                        ↳ QDP
                                                                                                                                                          ↳ DependencyGraphProof
                                                                                                                                                            ↳ QDP
                                                                                                                                                              ↳ Narrowing
                                                                                                                                                                ↳ QDP
                                                                                                                                                                  ↳ DependencyGraphProof
                                                                                                                                                                    ↳ QDP
                                                                                                                                                                      ↳ Narrowing
                                                                                                                                                                        ↳ QDP
                                                                                                                                                                          ↳ DependencyGraphProof
                                                                                                                                                                            ↳ QDP
                                                                                                                                                                              ↳ Narrowing
                                                                                                                                                                                ↳ QDP
                                                                                                                                                                                  ↳ DependencyGraphProof
                                                                                                                                                                                    ↳ QDP
                                                                                                                                                                                      ↳ Narrowing
                                                                                                                                                                                        ↳ QDP
                                                                                                                                                                                          ↳ DependencyGraphProof
                                                                                                                                                                                            ↳ QDP
                                                                                                                                                                                              ↳ QDPOrderProof
                                                                                                                                                                                                ↳ QDP
                                                                                                                                                                                                  ↳ QDPOrderProof

Q DP problem:
The TRS P consists of the following rules:

MARK(times(times(x0, x1), y1)) → A__TIMES(a__times(mark(x0), mark(x1)), mark(y1))

MARK(2ndsneg(2ndsneg(y0, y1), 2ndspos(x0, x1))) → A__2NDSNEG(a__2ndsneg(mark(y0), mark(y1)), a__2ndspos(mark(x0), mark(x1)))

MARK(plus(plus(x0, x1), y1)) → A__PLUS(a__plus(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(2ndsneg(y0, y1), from(x0))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__from(mark(x0)))

A__2NDSNEG(s(N), cons(X, Z)) → MARK(Z)

MARK(2ndsneg(square(y0), 2ndspos(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

A__PLUS(s(pi(x0)), y1) → A__PLUS(a__pi(mark(x0)), mark(y1))

MARK(2ndsneg(s(y0), pi(x0))) → A__2NDSNEG(s(mark(y0)), a__pi(mark(x0)))

MARK(2ndsneg(pi(y0), 2ndspos(x0, x1))) → A__2NDSNEG(a__pi(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

A__TIMES(s(X), Y) → MARK(Y)

MARK(2ndspos(square(y0), 2ndsneg(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndspos(2ndsneg(y0, y1), square(x0))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__square(mark(x0)))

A__2NDSNEG(s(N), cons(X, Z)) → MARK(N)

MARK(times(2ndsneg(x0, x1), y1)) → A__TIMES(a__2ndsneg(mark(x0), mark(x1)), mark(y1))

MARK(2ndsneg(times(y0, y1), times(x0, x1))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(pi(y0), pi(x0))) → A__2NDSPOS(a__pi(mark(y0)), a__pi(mark(x0)))

MARK(2ndsneg(square(y0), 2ndsneg(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndsneg(2ndsneg(y0, y1), square(x0))) → A__2NDSNEG(a__2ndsneg(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndspos(s(y0), pi(x0))) → A__2NDSPOS(s(mark(y0)), a__pi(mark(x0)))

MARK(plus(square(x0), y1)) → A__PLUS(a__square(mark(x0)), mark(y1))

MARK(2ndsneg(times(y0, y1), cons2(x0, x1))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(cons2(X1, X2)) → MARK(X2)

MARK(2ndsneg(times(y0, y1), 2ndspos(x0, x1))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndsneg(square(y0), cons2(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), cons2(x0, mark(x1)))

MARK(2ndspos(X1, X2)) → MARK(X1)

A__PLUS(s(square(x0)), y1) → A__PLUS(a__square(mark(x0)), mark(y1))

MARK(pi(X)) → MARK(X)

MARK(2ndsneg(s(y0), from(x0))) → A__2NDSNEG(s(mark(y0)), a__from(mark(x0)))

MARK(pi(X)) → A__PI(mark(X))

MARK(2ndspos(plus(y0, y1), cons(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), cons(mark(x0), x1))

MARK(2ndsneg(square(y0), times(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndsneg(times(y0, y1), square(x0))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(plus(2ndsneg(x0, x1), y1)) → A__PLUS(a__2ndsneg(mark(x0), mark(x1)), mark(y1))

MARK(2ndsneg(X1, X2)) → MARK(X2)

MARK(2ndspos(square(y0), times(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(times(y0, y1), square(x0))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(times(square(x0), y1)) → A__TIMES(a__square(mark(x0)), mark(y1))

MARK(2ndspos(pi(y0), cons(x0, x1))) → A__2NDSPOS(a__pi(mark(y0)), cons(mark(x0), x1))

A__PI(X) → MARK(X)

MARK(times(s(x0), y1)) → A__TIMES(s(mark(x0)), mark(y1))

MARK(2ndspos(plus(y0, y1), 2ndspos(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__2ndspos(mark(x0), mark(x1)))

A__PLUS(s(X), Y) → MARK(Y)

A__PLUS(s(2ndspos(x0, x1)), y1) → A__PLUS(a__2ndspos(mark(x0), mark(x1)), mark(y1))

MARK(2ndsneg(pi(y0), plus(x0, x1))) → A__2NDSNEG(a__pi(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(2ndsneg(plus(y0, y1), pi(x0))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__pi(mark(x0)))

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(N)

MARK(2ndspos(square(y0), 2ndspos(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndspos(s(y0), 2ndspos(x0, x1))) → A__2NDSPOS(s(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndspos(times(y0, y1), cons(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), cons(mark(x0), x1))

MARK(2ndspos(2ndsneg(y0, y1), 2ndsneg(x0, x1))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndspos(plus(y0, y1), times(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(times(y0, y1), plus(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(Y)

MARK(rcons(X1, X2)) → MARK(X1)

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(Z)

MARK(square(X)) → MARK(X)

MARK(2ndspos(times(y0, y1), pi(x0))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__pi(mark(x0)))

MARK(2ndspos(pi(y0), times(x0, x1))) → A__2NDSPOS(a__pi(mark(y0)), a__times(mark(x0), mark(x1)))

A__PI(s(x0)) → A__2NDSPOS(s(mark(x0)), a__from(0))

MARK(2ndsneg(2ndsneg(y0, y1), pi(x0))) → A__2NDSNEG(a__2ndsneg(mark(y0), mark(y1)), a__pi(mark(x0)))

MARK(2ndsneg(pi(y0), 2ndsneg(x0, x1))) → A__2NDSNEG(a__pi(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

MARK(times(plus(x0, x1), y1)) → A__TIMES(a__plus(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(times(y0, y1), 2ndspos(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__2ndspos(mark(x0), mark(x1)))

A__2NDSPOS(s(y0), cons(y1, from(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__from(mark(x0))))

MARK(2ndspos(s(y0), square(x0))) → A__2NDSPOS(s(mark(y0)), a__square(mark(x0)))

MARK(times(2ndspos(x0, x1), y1)) → A__TIMES(a__2ndspos(mark(x0), mark(x1)), mark(y1))

MARK(2ndsneg(2ndsneg(y0, y1), from(x0))) → A__2NDSNEG(a__2ndsneg(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndspos(s(y0), cons2(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(x0, mark(x1)))

MARK(2ndsneg(plus(y0, y1), cons2(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), cons2(x0, mark(x1)))

A__PLUS(s(X), Y) → MARK(X)

MARK(times(X1, X2)) → MARK(X1)

A__SQUARE(X) → MARK(X)

MARK(2ndsneg(pi(y0), pi(x0))) → A__2NDSNEG(a__pi(mark(y0)), a__pi(mark(x0)))

MARK(negrecip(X)) → MARK(X)

A__PLUS(s(from(x0)), y1) → A__PLUS(a__from(mark(x0)), mark(y1))

MARK(2ndspos(times(y0, y1), from(x0))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__from(mark(x0)))

A__PLUS(s(times(x0, x1)), y1) → A__PLUS(a__times(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(times(y0, y1), 2ndsneg(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndspos(2ndsneg(y0, y1), times(x0, x1))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

A__PLUS(s(0), y1) → A__PLUS(0, mark(y1))

MARK(from(X)) → MARK(X)

A__SQUARE(X) → A__TIMES(mark(X), mark(X))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(Z)

MARK(2ndsneg(plus(y0, y1), plus(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(plus(y0, y1), from(x0))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndsneg(times(y0, y1), 2ndsneg(x0, x1))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndsneg(2ndsneg(y0, y1), times(x0, x1))) → A__2NDSNEG(a__2ndsneg(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

A__PLUS(s(s(x0)), y1) → A__PLUS(s(mark(x0)), mark(y1))

MARK(2ndsneg(times(y0, y1), cons(x0, x1))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), cons(mark(x0), x1))

MARK(2ndsneg(2ndsneg(y0, y1), cons(x0, x1))) → A__2NDSNEG(a__2ndsneg(mark(y0), mark(y1)), cons(mark(x0), x1))

MARK(2ndspos(plus(y0, y1), square(x0))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndspos(square(y0), plus(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__plus(mark(x0), mark(x1)))

A__2NDSPOS(s(y0), cons(y1, 2ndspos(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__2ndspos(mark(x0), mark(x1))))

A__PI(times(x0, x1)) → A__2NDSPOS(a__times(mark(x0), mark(x1)), a__from(0))

A__2NDSNEG(s(N), cons(X, Z)) → A__2NDSNEG(s(mark(N)), cons2(X, mark(Z)))

MARK(2ndsneg(s(y0), square(x0))) → A__2NDSNEG(s(mark(y0)), a__square(mark(x0)))

MARK(2ndspos(X1, X2)) → MARK(X2)

MARK(2ndspos(pi(y0), cons2(x0, x1))) → A__2NDSPOS(a__pi(mark(y0)), cons2(x0, mark(x1)))

MARK(2ndsneg(pi(y0), from(x0))) → A__2NDSNEG(a__pi(mark(y0)), a__from(mark(x0)))

MARK(2ndspos(s(y0), from(x0))) → A__2NDSPOS(s(mark(y0)), a__from(mark(x0)))

MARK(2ndspos(2ndsneg(y0, y1), pi(x0))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__pi(mark(x0)))

MARK(2ndspos(pi(y0), 2ndsneg(x0, x1))) → A__2NDSPOS(a__pi(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

A__2NDSPOS(s(y0), cons(y1, plus(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__plus(mark(x0), mark(x1))))

MARK(2ndspos(pi(y0), from(x0))) → A__2NDSPOS(a__pi(mark(y0)), a__from(mark(x0)))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → A__2NDSNEG(mark(N), mark(Z))

MARK(2ndspos(times(y0, y1), times(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

A__TIMES(s(X), Y) → A__PLUS(mark(Y), a__times(mark(X), mark(Y)))

MARK(2ndsneg(pi(y0), cons2(x0, x1))) → A__2NDSNEG(a__pi(mark(y0)), cons2(x0, mark(x1)))

MARK(2ndspos(square(y0), cons2(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), cons2(x0, mark(x1)))

MARK(plus(s(x0), y1)) → A__PLUS(s(mark(x0)), mark(y1))

A__PI(plus(x0, x1)) → A__2NDSPOS(a__plus(mark(x0), mark(x1)), a__from(0))

MARK(2ndspos(square(y0), cons(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), cons(mark(x0), x1))

MARK(s(X)) → MARK(X)

MARK(2ndsneg(square(y0), plus(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(2ndsneg(plus(y0, y1), square(x0))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndspos(plus(y0, y1), cons2(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(plus(X1, X2)) → MARK(X2)

A__2NDSPOS(s(N), cons(X, Z)) → MARK(Z)

MARK(2ndsneg(times(y0, y1), from(x0))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndspos(times(y0, y1), cons2(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(2ndsneg(square(y0), from(x0))) → A__2NDSNEG(a__square(mark(y0)), a__from(mark(x0)))

MARK(2ndspos(2ndsneg(y0, y1), cons2(x0, x1))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(2ndsneg(s(y0), cons(x0, x1))) → A__2NDSNEG(s(mark(y0)), cons(mark(x0), x1))

A__PI(X) → A__FROM(0)

MARK(2ndspos(s(y0), plus(x0, x1))) → A__2NDSPOS(s(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(from(X)) → A__FROM(mark(X))

A__PLUS(s(2ndsneg(x0, x1)), y1) → A__PLUS(a__2ndsneg(mark(x0), mark(x1)), mark(y1))

MARK(plus(X1, X2)) → MARK(X1)

MARK(times(from(x0), y1)) → A__TIMES(a__from(mark(x0)), mark(y1))

A__2NDSPOS(s(y0), cons(y1, times(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__times(mark(x0), mark(x1))))

MARK(2ndsneg(2ndsneg(y0, y1), 2ndsneg(x0, x1))) → A__2NDSNEG(a__2ndsneg(mark(y0), mark(y1)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndspos(s(y0), cons(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons(mark(x0), x1))

MARK(2ndsneg(plus(y0, y1), 2ndspos(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndspos(pi(y0), 2ndspos(x0, x1))) → A__2NDSPOS(a__pi(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndsneg(plus(y0, y1), cons(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), cons(mark(x0), x1))

A__2NDSPOS(s(y0), cons(y1, pi(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__pi(mark(x0))))

A__TIMES(s(X), Y) → MARK(X)

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(N)

MARK(2ndspos(pi(y0), plus(x0, x1))) → A__2NDSPOS(a__pi(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(plus(y0, y1), pi(x0))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__pi(mark(x0)))

MARK(2ndsneg(s(y0), cons2(x0, x1))) → A__2NDSNEG(s(mark(y0)), cons2(x0, mark(x1)))

MARK(plus(2ndspos(x0, x1), y1)) → A__PLUS(a__2ndspos(mark(x0), mark(x1)), mark(y1))

MARK(2ndsneg(pi(y0), cons(x0, x1))) → A__2NDSNEG(a__pi(mark(y0)), cons(mark(x0), x1))

MARK(cons(X1, X2)) → MARK(X1)

MARK(2ndsneg(s(y0), 2ndspos(x0, x1))) → A__2NDSNEG(s(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndsneg(plus(y0, y1), from(x0))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(times(X1, X2)) → MARK(X2)

A__PI(square(x0)) → A__2NDSPOS(a__square(mark(x0)), a__from(0))

MARK(2ndsneg(square(y0), cons(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), cons(mark(x0), x1))

MARK(2ndspos(s(y0), times(x0, x1))) → A__2NDSPOS(s(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndsneg(X1, X2)) → MARK(X1)

MARK(2ndspos(s(y0), 2ndsneg(x0, x1))) → A__2NDSPOS(s(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndsneg(square(y0), square(x0))) → A__2NDSNEG(a__square(mark(y0)), a__square(mark(x0)))

MARK(2ndspos(2ndsneg(y0, y1), plus(x0, x1))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(plus(y0, y1), 2ndsneg(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__2ndsneg(mark(x0), mark(x1)))

A__PI(2ndsneg(x0, x1)) → A__2NDSPOS(a__2ndsneg(mark(x0), mark(x1)), a__from(0))

MARK(times(pi(x0), y1)) → A__TIMES(a__pi(mark(x0)), mark(y1))

MARK(plus(pi(x0), y1)) → A__PLUS(a__pi(mark(x0)), mark(y1))

MARK(2ndsneg(pi(y0), square(x0))) → A__2NDSNEG(a__pi(mark(y0)), a__square(mark(x0)))

MARK(2ndsneg(square(y0), pi(x0))) → A__2NDSNEG(a__square(mark(y0)), a__pi(mark(x0)))

MARK(2ndspos(2ndsneg(y0, y1), 2ndspos(x0, x1))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__2ndspos(mark(x0), mark(x1)))

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(Y)

MARK(2ndsneg(s(y0), 2ndsneg(x0, x1))) → A__2NDSNEG(s(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

A__FROM(X) → MARK(X)

MARK(2ndsneg(s(y0), plus(x0, x1))) → A__2NDSNEG(s(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(square(y0), pi(x0))) → A__2NDSPOS(a__square(mark(y0)), a__pi(mark(x0)))

MARK(2ndspos(pi(y0), square(x0))) → A__2NDSPOS(a__pi(mark(y0)), a__square(mark(x0)))

MARK(rcons(X1, X2)) → MARK(X2)

A__TIMES(s(X), Y) → A__TIMES(mark(X), mark(Y))

A__2NDSPOS(s(y0), cons(y1, cons(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, cons(mark(x0), x1)))

MARK(plus(times(x0, x1), y1)) → A__PLUS(a__times(mark(x0), mark(x1)), mark(y1))

A__2NDSPOS(s(y0), cons(y1, 2ndsneg(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__2ndsneg(mark(x0), mark(x1))))

A__2NDSPOS(s(N), cons(X, Z)) → MARK(N)

MARK(2ndsneg(2ndsneg(y0, y1), plus(x0, x1))) → A__2NDSNEG(a__2ndsneg(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(2ndsneg(plus(y0, y1), 2ndsneg(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__2ndsneg(mark(x0), mark(x1)))

A__PLUS(0, Y) → MARK(Y)

MARK(posrecip(X)) → MARK(X)

MARK(2ndspos(square(y0), from(x0))) → A__2NDSPOS(a__square(mark(y0)), a__from(mark(x0)))

A__2NDSPOS(s(y0), cons(y1, square(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__square(mark(x0))))

MARK(plus(0, y1)) → A__PLUS(0, mark(y1))

MARK(2ndsneg(s(y0), times(x0, x1))) → A__2NDSNEG(s(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(square(y0), square(x0))) → A__2NDSPOS(a__square(mark(y0)), a__square(mark(x0)))

A__PI(pi(x0)) → A__2NDSPOS(a__pi(mark(x0)), a__from(0))

A__PLUS(s(plus(x0, x1)), y1) → A__PLUS(a__plus(mark(x0), mark(x1)), mark(y1))

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → A__2NDSPOS(mark(N), mark(Z))

MARK(plus(from(x0), y1)) → A__PLUS(a__from(mark(x0)), mark(y1))

MARK(2ndsneg(pi(y0), times(x0, x1))) → A__2NDSNEG(a__pi(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(square(X)) → A__SQUARE(mark(X))

MARK(2ndsneg(times(y0, y1), pi(x0))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__pi(mark(x0)))

MARK(2ndsneg(plus(y0, y1), times(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndsneg(times(y0, y1), plus(x0, x1))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(plus(y0, y1), plus(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(2ndsneg(2ndsneg(y0, y1), cons2(x0, x1))) → A__2NDSNEG(a__2ndsneg(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(2ndspos(2ndsneg(y0, y1), cons(x0, x1))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), cons(mark(x0), x1))

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))

a__2ndspos(0, Z) → rnil

a__2ndspos(s(N), cons(X, Z)) → a__2ndspos(s(mark(N)), cons2(X, mark(Z)))

a__2ndspos(s(N), cons2(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))

a__2ndsneg(0, Z) → rnil

a__2ndsneg(s(N), cons(X, Z)) → a__2ndsneg(s(mark(N)), cons2(X, mark(Z)))

a__2ndsneg(s(N), cons2(X, cons(Y, Z))) → rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z)))

a__pi(X) → a__2ndspos(mark(X), a__from(0))

a__plus(0, Y) → mark(Y)

a__plus(s(X), Y) → s(a__plus(mark(X), mark(Y)))

a__times(0, Y) → 0

a__times(s(X), Y) → a__plus(mark(Y), a__times(mark(X), mark(Y)))

a__square(X) → a__times(mark(X), mark(X))

mark(from(X)) → a__from(mark(X))

mark(2ndspos(X1, X2)) → a__2ndspos(mark(X1), mark(X2))

mark(2ndsneg(X1, X2)) → a__2ndsneg(mark(X1), mark(X2))

mark(pi(X)) → a__pi(mark(X))

mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))

mark(times(X1, X2)) → a__times(mark(X1), mark(X2))

mark(square(X)) → a__square(mark(X))

mark(0) → 0

mark(s(X)) → s(mark(X))

mark(posrecip(X)) → posrecip(mark(X))

mark(negrecip(X)) → negrecip(mark(X))

mark(nil) → nil

mark(cons(X1, X2)) → cons(mark(X1), X2)

mark(cons2(X1, X2)) → cons2(X1, mark(X2))

mark(rnil) → rnil

mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))

a__from(X) → from(X)

a__2ndspos(X1, X2) → 2ndspos(X1, X2)

a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)

a__pi(X) → pi(X)

a__plus(X1, X2) → plus(X1, X2)

a__times(X1, X2) → times(X1, X2)

a__square(X) → square(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

MARK(2ndspos(2ndsneg(y0, y1), from(x0))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndspos(2ndsneg(y0, y1), square(x0))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndspos(pi(y0), pi(x0))) → A__2NDSPOS(a__pi(mark(y0)), a__pi(mark(x0)))

MARK(2ndspos(pi(y0), cons(x0, x1))) → A__2NDSPOS(a__pi(mark(y0)), cons(mark(x0), x1))

MARK(2ndspos(2ndsneg(y0, y1), 2ndsneg(x0, x1))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndspos(pi(y0), times(x0, x1))) → A__2NDSPOS(a__pi(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(2ndsneg(y0, y1), times(x0, x1))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(pi(y0), cons2(x0, x1))) → A__2NDSPOS(a__pi(mark(y0)), cons2(x0, mark(x1)))

MARK(2ndspos(2ndsneg(y0, y1), pi(x0))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__pi(mark(x0)))

MARK(2ndspos(pi(y0), 2ndsneg(x0, x1))) → A__2NDSPOS(a__pi(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndspos(pi(y0), from(x0))) → A__2NDSPOS(a__pi(mark(y0)), a__from(mark(x0)))

MARK(2ndspos(2ndsneg(y0, y1), cons2(x0, x1))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(2ndspos(pi(y0), 2ndspos(x0, x1))) → A__2NDSPOS(a__pi(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndspos(pi(y0), plus(x0, x1))) → A__2NDSPOS(a__pi(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(2ndsneg(y0, y1), plus(x0, x1))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

A__PI(2ndsneg(x0, x1)) → A__2NDSPOS(a__2ndsneg(mark(x0), mark(x1)), a__from(0))

MARK(2ndspos(2ndsneg(y0, y1), 2ndspos(x0, x1))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndspos(pi(y0), square(x0))) → A__2NDSPOS(a__pi(mark(y0)), a__square(mark(x0)))

A__PI(pi(x0)) → A__2NDSPOS(a__pi(mark(x0)), a__from(0))

MARK(2ndspos(2ndsneg(y0, y1), cons(x0, x1))) → A__2NDSPOS(a__2ndsneg(mark(y0), mark(y1)), cons(mark(x0), x1))
The remaining pairs can at least be oriented weakly.

MARK(times(times(x0, x1), y1)) → A__TIMES(a__times(mark(x0), mark(x1)), mark(y1))

MARK(2ndsneg(2ndsneg(y0, y1), 2ndspos(x0, x1))) → A__2NDSNEG(a__2ndsneg(mark(y0), mark(y1)), a__2ndspos(mark(x0), mark(x1)))

MARK(plus(plus(x0, x1), y1)) → A__PLUS(a__plus(mark(x0), mark(x1)), mark(y1))

A__2NDSNEG(s(N), cons(X, Z)) → MARK(Z)

MARK(2ndsneg(square(y0), 2ndspos(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

A__PLUS(s(pi(x0)), y1) → A__PLUS(a__pi(mark(x0)), mark(y1))

MARK(2ndsneg(s(y0), pi(x0))) → A__2NDSNEG(s(mark(y0)), a__pi(mark(x0)))

MARK(2ndsneg(pi(y0), 2ndspos(x0, x1))) → A__2NDSNEG(a__pi(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

A__TIMES(s(X), Y) → MARK(Y)

MARK(2ndspos(square(y0), 2ndsneg(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

A__2NDSNEG(s(N), cons(X, Z)) → MARK(N)

MARK(times(2ndsneg(x0, x1), y1)) → A__TIMES(a__2ndsneg(mark(x0), mark(x1)), mark(y1))

MARK(2ndsneg(times(y0, y1), times(x0, x1))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndsneg(square(y0), 2ndsneg(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndsneg(2ndsneg(y0, y1), square(x0))) → A__2NDSNEG(a__2ndsneg(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndspos(s(y0), pi(x0))) → A__2NDSPOS(s(mark(y0)), a__pi(mark(x0)))

MARK(plus(square(x0), y1)) → A__PLUS(a__square(mark(x0)), mark(y1))

MARK(2ndsneg(times(y0, y1), cons2(x0, x1))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(cons2(X1, X2)) → MARK(X2)

MARK(2ndsneg(times(y0, y1), 2ndspos(x0, x1))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndsneg(square(y0), cons2(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), cons2(x0, mark(x1)))

MARK(2ndspos(X1, X2)) → MARK(X1)

A__PLUS(s(square(x0)), y1) → A__PLUS(a__square(mark(x0)), mark(y1))

MARK(pi(X)) → MARK(X)

MARK(2ndsneg(s(y0), from(x0))) → A__2NDSNEG(s(mark(y0)), a__from(mark(x0)))

MARK(pi(X)) → A__PI(mark(X))

MARK(2ndspos(plus(y0, y1), cons(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), cons(mark(x0), x1))

MARK(2ndsneg(square(y0), times(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndsneg(times(y0, y1), square(x0))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(plus(2ndsneg(x0, x1), y1)) → A__PLUS(a__2ndsneg(mark(x0), mark(x1)), mark(y1))

MARK(2ndsneg(X1, X2)) → MARK(X2)

MARK(2ndspos(square(y0), times(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(times(y0, y1), square(x0))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(times(square(x0), y1)) → A__TIMES(a__square(mark(x0)), mark(y1))

A__PI(X) → MARK(X)

MARK(times(s(x0), y1)) → A__TIMES(s(mark(x0)), mark(y1))

MARK(2ndspos(plus(y0, y1), 2ndspos(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__2ndspos(mark(x0), mark(x1)))

A__PLUS(s(X), Y) → MARK(Y)

A__PLUS(s(2ndspos(x0, x1)), y1) → A__PLUS(a__2ndspos(mark(x0), mark(x1)), mark(y1))

MARK(2ndsneg(pi(y0), plus(x0, x1))) → A__2NDSNEG(a__pi(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(2ndsneg(plus(y0, y1), pi(x0))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__pi(mark(x0)))

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(N)

MARK(2ndspos(square(y0), 2ndspos(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndspos(s(y0), 2ndspos(x0, x1))) → A__2NDSPOS(s(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndspos(times(y0, y1), cons(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), cons(mark(x0), x1))

MARK(2ndspos(plus(y0, y1), times(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(times(y0, y1), plus(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(Y)

MARK(rcons(X1, X2)) → MARK(X1)

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(Z)

MARK(square(X)) → MARK(X)

MARK(2ndspos(times(y0, y1), pi(x0))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__pi(mark(x0)))

A__PI(s(x0)) → A__2NDSPOS(s(mark(x0)), a__from(0))

MARK(2ndsneg(2ndsneg(y0, y1), pi(x0))) → A__2NDSNEG(a__2ndsneg(mark(y0), mark(y1)), a__pi(mark(x0)))

MARK(2ndsneg(pi(y0), 2ndsneg(x0, x1))) → A__2NDSNEG(a__pi(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

MARK(times(plus(x0, x1), y1)) → A__TIMES(a__plus(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(times(y0, y1), 2ndspos(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__2ndspos(mark(x0), mark(x1)))

A__2NDSPOS(s(y0), cons(y1, from(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__from(mark(x0))))

MARK(2ndspos(s(y0), square(x0))) → A__2NDSPOS(s(mark(y0)), a__square(mark(x0)))

MARK(times(2ndspos(x0, x1), y1)) → A__TIMES(a__2ndspos(mark(x0), mark(x1)), mark(y1))

MARK(2ndsneg(2ndsneg(y0, y1), from(x0))) → A__2NDSNEG(a__2ndsneg(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndspos(s(y0), cons2(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(x0, mark(x1)))

MARK(2ndsneg(plus(y0, y1), cons2(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), cons2(x0, mark(x1)))

A__PLUS(s(X), Y) → MARK(X)

MARK(times(X1, X2)) → MARK(X1)

A__SQUARE(X) → MARK(X)

MARK(2ndsneg(pi(y0), pi(x0))) → A__2NDSNEG(a__pi(mark(y0)), a__pi(mark(x0)))

MARK(negrecip(X)) → MARK(X)

A__PLUS(s(from(x0)), y1) → A__PLUS(a__from(mark(x0)), mark(y1))

MARK(2ndspos(times(y0, y1), from(x0))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__from(mark(x0)))

A__PLUS(s(times(x0, x1)), y1) → A__PLUS(a__times(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(times(y0, y1), 2ndsneg(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__2ndsneg(mark(x0), mark(x1)))

A__PLUS(s(0), y1) → A__PLUS(0, mark(y1))

MARK(from(X)) → MARK(X)

A__SQUARE(X) → A__TIMES(mark(X), mark(X))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(Z)

MARK(2ndsneg(plus(y0, y1), plus(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(plus(y0, y1), from(x0))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndsneg(times(y0, y1), 2ndsneg(x0, x1))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndsneg(2ndsneg(y0, y1), times(x0, x1))) → A__2NDSNEG(a__2ndsneg(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

A__PLUS(s(s(x0)), y1) → A__PLUS(s(mark(x0)), mark(y1))

MARK(2ndsneg(times(y0, y1), cons(x0, x1))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), cons(mark(x0), x1))

MARK(2ndsneg(2ndsneg(y0, y1), cons(x0, x1))) → A__2NDSNEG(a__2ndsneg(mark(y0), mark(y1)), cons(mark(x0), x1))

MARK(2ndspos(plus(y0, y1), square(x0))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndspos(square(y0), plus(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__plus(mark(x0), mark(x1)))

A__2NDSPOS(s(y0), cons(y1, 2ndspos(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__2ndspos(mark(x0), mark(x1))))

A__PI(times(x0, x1)) → A__2NDSPOS(a__times(mark(x0), mark(x1)), a__from(0))

A__2NDSNEG(s(N), cons(X, Z)) → A__2NDSNEG(s(mark(N)), cons2(X, mark(Z)))

MARK(2ndsneg(s(y0), square(x0))) → A__2NDSNEG(s(mark(y0)), a__square(mark(x0)))

MARK(2ndspos(X1, X2)) → MARK(X2)

MARK(2ndsneg(pi(y0), from(x0))) → A__2NDSNEG(a__pi(mark(y0)), a__from(mark(x0)))

MARK(2ndspos(s(y0), from(x0))) → A__2NDSPOS(s(mark(y0)), a__from(mark(x0)))

A__2NDSPOS(s(y0), cons(y1, plus(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__plus(mark(x0), mark(x1))))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → A__2NDSNEG(mark(N), mark(Z))

MARK(2ndspos(times(y0, y1), times(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

A__TIMES(s(X), Y) → A__PLUS(mark(Y), a__times(mark(X), mark(Y)))

MARK(2ndsneg(pi(y0), cons2(x0, x1))) → A__2NDSNEG(a__pi(mark(y0)), cons2(x0, mark(x1)))

MARK(2ndspos(square(y0), cons2(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), cons2(x0, mark(x1)))

MARK(plus(s(x0), y1)) → A__PLUS(s(mark(x0)), mark(y1))

A__PI(plus(x0, x1)) → A__2NDSPOS(a__plus(mark(x0), mark(x1)), a__from(0))

MARK(2ndspos(square(y0), cons(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), cons(mark(x0), x1))

MARK(s(X)) → MARK(X)

MARK(2ndsneg(square(y0), plus(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(2ndsneg(plus(y0, y1), square(x0))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndspos(plus(y0, y1), cons2(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(plus(X1, X2)) → MARK(X2)

A__2NDSPOS(s(N), cons(X, Z)) → MARK(Z)

MARK(2ndsneg(times(y0, y1), from(x0))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndspos(times(y0, y1), cons2(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(2ndsneg(square(y0), from(x0))) → A__2NDSNEG(a__square(mark(y0)), a__from(mark(x0)))

MARK(2ndsneg(s(y0), cons(x0, x1))) → A__2NDSNEG(s(mark(y0)), cons(mark(x0), x1))

A__PI(X) → A__FROM(0)

MARK(2ndspos(s(y0), plus(x0, x1))) → A__2NDSPOS(s(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(from(X)) → A__FROM(mark(X))

A__PLUS(s(2ndsneg(x0, x1)), y1) → A__PLUS(a__2ndsneg(mark(x0), mark(x1)), mark(y1))

MARK(plus(X1, X2)) → MARK(X1)

MARK(times(from(x0), y1)) → A__TIMES(a__from(mark(x0)), mark(y1))

A__2NDSPOS(s(y0), cons(y1, times(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__times(mark(x0), mark(x1))))

MARK(2ndsneg(2ndsneg(y0, y1), 2ndsneg(x0, x1))) → A__2NDSNEG(a__2ndsneg(mark(y0), mark(y1)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndspos(s(y0), cons(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons(mark(x0), x1))

MARK(2ndsneg(plus(y0, y1), 2ndspos(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndsneg(plus(y0, y1), cons(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), cons(mark(x0), x1))

A__2NDSPOS(s(y0), cons(y1, pi(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__pi(mark(x0))))

A__TIMES(s(X), Y) → MARK(X)

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(N)

MARK(2ndspos(plus(y0, y1), pi(x0))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__pi(mark(x0)))

MARK(2ndsneg(s(y0), cons2(x0, x1))) → A__2NDSNEG(s(mark(y0)), cons2(x0, mark(x1)))

MARK(plus(2ndspos(x0, x1), y1)) → A__PLUS(a__2ndspos(mark(x0), mark(x1)), mark(y1))

MARK(2ndsneg(pi(y0), cons(x0, x1))) → A__2NDSNEG(a__pi(mark(y0)), cons(mark(x0), x1))

MARK(cons(X1, X2)) → MARK(X1)

MARK(2ndsneg(s(y0), 2ndspos(x0, x1))) → A__2NDSNEG(s(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndsneg(plus(y0, y1), from(x0))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(times(X1, X2)) → MARK(X2)

A__PI(square(x0)) → A__2NDSPOS(a__square(mark(x0)), a__from(0))

MARK(2ndsneg(square(y0), cons(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), cons(mark(x0), x1))

MARK(2ndspos(s(y0), times(x0, x1))) → A__2NDSPOS(s(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndsneg(X1, X2)) → MARK(X1)

MARK(2ndspos(s(y0), 2ndsneg(x0, x1))) → A__2NDSPOS(s(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndsneg(square(y0), square(x0))) → A__2NDSNEG(a__square(mark(y0)), a__square(mark(x0)))

MARK(2ndspos(plus(y0, y1), 2ndsneg(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__2ndsneg(mark(x0), mark(x1)))

MARK(times(pi(x0), y1)) → A__TIMES(a__pi(mark(x0)), mark(y1))

MARK(plus(pi(x0), y1)) → A__PLUS(a__pi(mark(x0)), mark(y1))

MARK(2ndsneg(pi(y0), square(x0))) → A__2NDSNEG(a__pi(mark(y0)), a__square(mark(x0)))

MARK(2ndsneg(square(y0), pi(x0))) → A__2NDSNEG(a__square(mark(y0)), a__pi(mark(x0)))

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(Y)

MARK(2ndsneg(s(y0), 2ndsneg(x0, x1))) → A__2NDSNEG(s(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

A__FROM(X) → MARK(X)

MARK(2ndsneg(s(y0), plus(x0, x1))) → A__2NDSNEG(s(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(square(y0), pi(x0))) → A__2NDSPOS(a__square(mark(y0)), a__pi(mark(x0)))

MARK(rcons(X1, X2)) → MARK(X2)

A__TIMES(s(X), Y) → A__TIMES(mark(X), mark(Y))

A__2NDSPOS(s(y0), cons(y1, cons(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, cons(mark(x0), x1)))

MARK(plus(times(x0, x1), y1)) → A__PLUS(a__times(mark(x0), mark(x1)), mark(y1))

A__2NDSPOS(s(y0), cons(y1, 2ndsneg(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__2ndsneg(mark(x0), mark(x1))))

A__2NDSPOS(s(N), cons(X, Z)) → MARK(N)

MARK(2ndsneg(2ndsneg(y0, y1), plus(x0, x1))) → A__2NDSNEG(a__2ndsneg(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(2ndsneg(plus(y0, y1), 2ndsneg(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__2ndsneg(mark(x0), mark(x1)))

A__PLUS(0, Y) → MARK(Y)

MARK(posrecip(X)) → MARK(X)

MARK(2ndspos(square(y0), from(x0))) → A__2NDSPOS(a__square(mark(y0)), a__from(mark(x0)))

A__2NDSPOS(s(y0), cons(y1, square(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__square(mark(x0))))

MARK(plus(0, y1)) → A__PLUS(0, mark(y1))

MARK(2ndsneg(s(y0), times(x0, x1))) → A__2NDSNEG(s(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(square(y0), square(x0))) → A__2NDSPOS(a__square(mark(y0)), a__square(mark(x0)))

A__PLUS(s(plus(x0, x1)), y1) → A__PLUS(a__plus(mark(x0), mark(x1)), mark(y1))

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → A__2NDSPOS(mark(N), mark(Z))

MARK(plus(from(x0), y1)) → A__PLUS(a__from(mark(x0)), mark(y1))

MARK(2ndsneg(pi(y0), times(x0, x1))) → A__2NDSNEG(a__pi(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(square(X)) → A__SQUARE(mark(X))

MARK(2ndsneg(times(y0, y1), pi(x0))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__pi(mark(x0)))

MARK(2ndsneg(plus(y0, y1), times(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndsneg(times(y0, y1), plus(x0, x1))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(plus(y0, y1), plus(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(2ndsneg(2ndsneg(y0, y1), cons2(x0, x1))) → A__2NDSNEG(a__2ndsneg(mark(y0), mark(y1)), cons2(x0, mark(x1)))
Used ordering:  Polynomial interpretation [25]:

POL(0) = 1   

POL(2ndsneg(x1, x2)) = 0   

POL(2ndspos(x1, x2)) = 0   

POL(A__2NDSNEG(x1, x2)) = 1   

POL(A__2NDSPOS(x1, x2)) = x1   

POL(A__FROM(x1)) = 1   

POL(A__PI(x1)) = 1   

POL(A__PLUS(x1, x2)) = 1   

POL(A__SQUARE(x1)) = 1   

POL(A__TIMES(x1, x2)) = 1   

POL(MARK(x1)) = 1   

POL(a__2ndsneg(x1, x2)) = 0   

POL(a__2ndspos(x1, x2)) = 0   

POL(a__from(x1)) = 0   

POL(a__pi(x1)) = 0   

POL(a__plus(x1, x2)) = x1   

POL(a__square(x1)) = 1   

POL(a__times(x1, x2)) = 1   

POL(cons(x1, x2)) = 0   

POL(cons2(x1, x2)) = 0   

POL(from(x1)) = 0   

POL(mark(x1)) = 1   

POL(negrecip(x1)) = 0   

POL(nil) = 1   

POL(pi(x1)) = 0   

POL(plus(x1, x2)) = 0   

POL(posrecip(x1)) = 0   

POL(rcons(x1, x2)) = 0   

POL(rnil) = 0   

POL(s(x1)) = 1   

POL(square(x1)) = 0   

POL(times(x1, x2)) = 0   

The following usable rules [17] were oriented:

a__from(X) → cons(mark(X), from(s(X)))

a__pi(X) → a__2ndspos(mark(X), a__from(0))

a__2ndsneg(s(N), cons(X, Z)) → a__2ndsneg(s(mark(N)), cons2(X, mark(Z)))

a__2ndsneg(s(N), cons2(X, cons(Y, Z))) → rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z)))

a__2ndspos(s(N), cons2(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))

a__2ndsneg(0, Z) → rnil

a__2ndspos(0, Z) → rnil

a__2ndspos(s(N), cons(X, Z)) → a__2ndspos(s(mark(N)), cons2(X, mark(Z)))

mark(pi(X)) → a__pi(mark(X))

mark(2ndsneg(X1, X2)) → a__2ndsneg(mark(X1), mark(X2))

mark(2ndspos(X1, X2)) → a__2ndspos(mark(X1), mark(X2))

mark(from(X)) → a__from(mark(X))

a__times(0, Y) → 0

a__plus(s(X), Y) → s(a__plus(mark(X), mark(Y)))

mark(nil) → nil

mark(negrecip(X)) → negrecip(mark(X))

mark(posrecip(X)) → posrecip(mark(X))

mark(s(X)) → s(mark(X))

mark(0) → 0

mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))

a__times(s(X), Y) → a__plus(mark(Y), a__times(mark(X), mark(Y)))

a__plus(0, Y) → mark(Y)

mark(square(X)) → a__square(mark(X))

mark(times(X1, X2)) → a__times(mark(X1), mark(X2))

a__square(X) → a__times(mark(X), mark(X))

a__times(X1, X2) → times(X1, X2)

a__plus(X1, X2) → plus(X1, X2)

a__square(X) → square(X)

mark(cons2(X1, X2)) → cons2(X1, mark(X2))

mark(cons(X1, X2)) → cons(mark(X1), X2)

mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))

mark(rnil) → rnil

a__2ndspos(X1, X2) → 2ndspos(X1, X2)

a__from(X) → from(X)

a__pi(X) → pi(X)

a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ Narrowing
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ QDP
              ↳ Narrowing
                ↳ QDP
                  ↳ DependencyGraphProof
                    ↳ QDP
                      ↳ Narrowing
                        ↳ QDP
                          ↳ DependencyGraphProof
                            ↳ QDP
                              ↳ Narrowing
                                ↳ QDP
                                  ↳ DependencyGraphProof
                                    ↳ QDP
                                      ↳ Narrowing
                                        ↳ QDP
                                          ↳ DependencyGraphProof
                                            ↳ QDP
                                              ↳ Narrowing
                                                ↳ QDP
                                                  ↳ DependencyGraphProof
                                                    ↳ QDP
                                                      ↳ Narrowing
                                                        ↳ QDP
                                                          ↳ DependencyGraphProof
                                                            ↳ QDP
                                                              ↳ Narrowing
                                                                ↳ QDP
                                                                  ↳ DependencyGraphProof
                                                                    ↳ QDP
                                                                      ↳ Narrowing
                                                                        ↳ QDP
                                                                          ↳ DependencyGraphProof
                                                                            ↳ QDP
                                                                              ↳ Narrowing
                                                                                ↳ QDP
                                                                                  ↳ DependencyGraphProof
                                                                                    ↳ QDP
                                                                                      ↳ Narrowing
                                                                                        ↳ QDP
                                                                                          ↳ DependencyGraphProof
                                                                                            ↳ QDP
                                                                                              ↳ Narrowing
                                                                                                ↳ QDP
                                                                                                  ↳ DependencyGraphProof
                                                                                                    ↳ QDP
                                                                                                      ↳ Narrowing
                                                                                                        ↳ QDP
                                                                                                          ↳ DependencyGraphProof
                                                                                                            ↳ QDP
                                                                                                              ↳ Narrowing
                                                                                                                ↳ QDP
                                                                                                                  ↳ DependencyGraphProof
                                                                                                                    ↳ QDP
                                                                                                                      ↳ Narrowing
                                                                                                                        ↳ QDP
                                                                                                                          ↳ DependencyGraphProof
                                                                                                                            ↳ QDP
                                                                                                                              ↳ Narrowing
                                                                                                                                ↳ QDP
                                                                                                                                  ↳ DependencyGraphProof
                                                                                                                                    ↳ QDP
                                                                                                                                      ↳ Narrowing
                                                                                                                                        ↳ QDP
                                                                                                                                          ↳ DependencyGraphProof
                                                                                                                                            ↳ QDP
                                                                                                                                              ↳ Narrowing
                                                                                                                                                ↳ QDP
                                                                                                                                                  ↳ DependencyGraphProof
                                                                                                                                                    ↳ QDP
                                                                                                                                                      ↳ Narrowing
                                                                                                                                                        ↳ QDP
                                                                                                                                                          ↳ DependencyGraphProof
                                                                                                                                                            ↳ QDP
                                                                                                                                                              ↳ Narrowing
                                                                                                                                                                ↳ QDP
                                                                                                                                                                  ↳ DependencyGraphProof
                                                                                                                                                                    ↳ QDP
                                                                                                                                                                      ↳ Narrowing
                                                                                                                                                                        ↳ QDP
                                                                                                                                                                          ↳ DependencyGraphProof
                                                                                                                                                                            ↳ QDP
                                                                                                                                                                              ↳ Narrowing
                                                                                                                                                                                ↳ QDP
                                                                                                                                                                                  ↳ DependencyGraphProof
                                                                                                                                                                                    ↳ QDP
                                                                                                                                                                                      ↳ Narrowing
                                                                                                                                                                                        ↳ QDP
                                                                                                                                                                                          ↳ DependencyGraphProof
                                                                                                                                                                                            ↳ QDP
                                                                                                                                                                                              ↳ QDPOrderProof
                                                                                                                                                                                                ↳ QDP
                                                                                                                                                                                                  ↳ QDPOrderProof
                                                                                                                                                                                                    ↳ QDP
                                                                                                                                                                                                      ↳ QDPOrderProof

Q DP problem:
The TRS P consists of the following rules:

MARK(2ndspos(plus(y0, y1), square(x0))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndspos(square(y0), plus(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__plus(mark(x0), mark(x1)))

A__2NDSPOS(s(y0), cons(y1, 2ndspos(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__2ndspos(mark(x0), mark(x1))))

A__PI(times(x0, x1)) → A__2NDSPOS(a__times(mark(x0), mark(x1)), a__from(0))

A__2NDSNEG(s(N), cons(X, Z)) → A__2NDSNEG(s(mark(N)), cons2(X, mark(Z)))

MARK(2ndsneg(s(y0), square(x0))) → A__2NDSNEG(s(mark(y0)), a__square(mark(x0)))

MARK(times(times(x0, x1), y1)) → A__TIMES(a__times(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(X1, X2)) → MARK(X2)

MARK(2ndsneg(2ndsneg(y0, y1), 2ndspos(x0, x1))) → A__2NDSNEG(a__2ndsneg(mark(y0), mark(y1)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndsneg(pi(y0), from(x0))) → A__2NDSNEG(a__pi(mark(y0)), a__from(mark(x0)))

MARK(plus(plus(x0, x1), y1)) → A__PLUS(a__plus(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(s(y0), from(x0))) → A__2NDSPOS(s(mark(y0)), a__from(mark(x0)))

MARK(2ndsneg(square(y0), 2ndspos(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

A__2NDSNEG(s(N), cons(X, Z)) → MARK(Z)

A__PLUS(s(pi(x0)), y1) → A__PLUS(a__pi(mark(x0)), mark(y1))

MARK(2ndsneg(s(y0), pi(x0))) → A__2NDSNEG(s(mark(y0)), a__pi(mark(x0)))

MARK(2ndsneg(pi(y0), 2ndspos(x0, x1))) → A__2NDSNEG(a__pi(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

A__TIMES(s(X), Y) → MARK(Y)

A__2NDSPOS(s(y0), cons(y1, plus(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__plus(mark(x0), mark(x1))))

MARK(2ndspos(square(y0), 2ndsneg(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

A__2NDSNEG(s(N), cons(X, Z)) → MARK(N)

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → A__2NDSNEG(mark(N), mark(Z))

MARK(times(2ndsneg(x0, x1), y1)) → A__TIMES(a__2ndsneg(mark(x0), mark(x1)), mark(y1))

MARK(2ndsneg(times(y0, y1), times(x0, x1))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndsneg(square(y0), 2ndsneg(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndsneg(2ndsneg(y0, y1), square(x0))) → A__2NDSNEG(a__2ndsneg(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndspos(times(y0, y1), times(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(s(y0), pi(x0))) → A__2NDSPOS(s(mark(y0)), a__pi(mark(x0)))

A__TIMES(s(X), Y) → A__PLUS(mark(Y), a__times(mark(X), mark(Y)))

MARK(2ndsneg(pi(y0), cons2(x0, x1))) → A__2NDSNEG(a__pi(mark(y0)), cons2(x0, mark(x1)))

MARK(2ndspos(square(y0), cons2(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), cons2(x0, mark(x1)))

MARK(plus(square(x0), y1)) → A__PLUS(a__square(mark(x0)), mark(y1))

MARK(plus(s(x0), y1)) → A__PLUS(s(mark(x0)), mark(y1))

MARK(2ndspos(square(y0), cons(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), cons(mark(x0), x1))

A__PI(plus(x0, x1)) → A__2NDSPOS(a__plus(mark(x0), mark(x1)), a__from(0))

MARK(2ndsneg(times(y0, y1), cons2(x0, x1))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(cons2(X1, X2)) → MARK(X2)

MARK(2ndsneg(times(y0, y1), 2ndspos(x0, x1))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndsneg(square(y0), cons2(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), cons2(x0, mark(x1)))

MARK(2ndsneg(plus(y0, y1), square(x0))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndsneg(square(y0), plus(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(s(X)) → MARK(X)

MARK(2ndspos(X1, X2)) → MARK(X1)

A__PLUS(s(square(x0)), y1) → A__PLUS(a__square(mark(x0)), mark(y1))

MARK(2ndspos(plus(y0, y1), cons2(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(pi(X)) → MARK(X)

MARK(2ndsneg(s(y0), from(x0))) → A__2NDSNEG(s(mark(y0)), a__from(mark(x0)))

MARK(plus(X1, X2)) → MARK(X2)

A__2NDSPOS(s(N), cons(X, Z)) → MARK(Z)

MARK(pi(X)) → A__PI(mark(X))

MARK(2ndsneg(times(y0, y1), from(x0))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndsneg(times(y0, y1), square(x0))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndsneg(square(y0), times(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(plus(y0, y1), cons(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), cons(mark(x0), x1))

MARK(plus(2ndsneg(x0, x1), y1)) → A__PLUS(a__2ndsneg(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(times(y0, y1), cons2(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(2ndsneg(X1, X2)) → MARK(X2)

MARK(2ndsneg(square(y0), from(x0))) → A__2NDSNEG(a__square(mark(y0)), a__from(mark(x0)))

MARK(2ndspos(times(y0, y1), square(x0))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndspos(square(y0), times(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndsneg(s(y0), cons(x0, x1))) → A__2NDSNEG(s(mark(y0)), cons(mark(x0), x1))

MARK(times(square(x0), y1)) → A__TIMES(a__square(mark(x0)), mark(y1))

MARK(from(X)) → A__FROM(mark(X))

MARK(2ndspos(s(y0), plus(x0, x1))) → A__2NDSPOS(s(mark(y0)), a__plus(mark(x0), mark(x1)))

A__PI(X) → A__FROM(0)

A__PLUS(s(2ndsneg(x0, x1)), y1) → A__PLUS(a__2ndsneg(mark(x0), mark(x1)), mark(y1))

MARK(times(from(x0), y1)) → A__TIMES(a__from(mark(x0)), mark(y1))

MARK(plus(X1, X2)) → MARK(X1)

A__2NDSPOS(s(y0), cons(y1, times(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__times(mark(x0), mark(x1))))

MARK(2ndsneg(2ndsneg(y0, y1), 2ndsneg(x0, x1))) → A__2NDSNEG(a__2ndsneg(mark(y0), mark(y1)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndspos(s(y0), cons(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons(mark(x0), x1))

A__PI(X) → MARK(X)

MARK(2ndsneg(plus(y0, y1), 2ndspos(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__2ndspos(mark(x0), mark(x1)))

MARK(times(s(x0), y1)) → A__TIMES(s(mark(x0)), mark(y1))

MARK(2ndsneg(plus(y0, y1), cons(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), cons(mark(x0), x1))

MARK(2ndspos(plus(y0, y1), 2ndspos(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__2ndspos(mark(x0), mark(x1)))

A__PLUS(s(X), Y) → MARK(Y)

A__PLUS(s(2ndspos(x0, x1)), y1) → A__PLUS(a__2ndspos(mark(x0), mark(x1)), mark(y1))

A__TIMES(s(X), Y) → MARK(X)

A__2NDSPOS(s(y0), cons(y1, pi(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__pi(mark(x0))))

MARK(2ndsneg(plus(y0, y1), pi(x0))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__pi(mark(x0)))

MARK(2ndsneg(pi(y0), plus(x0, x1))) → A__2NDSNEG(a__pi(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(plus(y0, y1), pi(x0))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__pi(mark(x0)))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(N)

MARK(2ndsneg(s(y0), cons2(x0, x1))) → A__2NDSNEG(s(mark(y0)), cons2(x0, mark(x1)))

MARK(plus(2ndspos(x0, x1), y1)) → A__PLUS(a__2ndspos(mark(x0), mark(x1)), mark(y1))

MARK(2ndsneg(pi(y0), cons(x0, x1))) → A__2NDSNEG(a__pi(mark(y0)), cons(mark(x0), x1))

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(N)

MARK(cons(X1, X2)) → MARK(X1)

MARK(2ndsneg(plus(y0, y1), from(x0))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndsneg(s(y0), 2ndspos(x0, x1))) → A__2NDSNEG(s(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndspos(square(y0), 2ndspos(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

MARK(times(X1, X2)) → MARK(X2)

MARK(2ndspos(times(y0, y1), cons(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), cons(mark(x0), x1))

MARK(2ndspos(s(y0), 2ndspos(x0, x1))) → A__2NDSPOS(s(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

A__PI(square(x0)) → A__2NDSPOS(a__square(mark(x0)), a__from(0))

MARK(2ndsneg(square(y0), cons(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), cons(mark(x0), x1))

MARK(2ndspos(times(y0, y1), plus(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(plus(y0, y1), times(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(s(y0), times(x0, x1))) → A__2NDSPOS(s(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndsneg(X1, X2)) → MARK(X1)

MARK(2ndspos(s(y0), 2ndsneg(x0, x1))) → A__2NDSPOS(s(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndsneg(square(y0), square(x0))) → A__2NDSNEG(a__square(mark(y0)), a__square(mark(x0)))

MARK(2ndspos(plus(y0, y1), 2ndsneg(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__2ndsneg(mark(x0), mark(x1)))

MARK(plus(pi(x0), y1)) → A__PLUS(a__pi(mark(x0)), mark(y1))

MARK(times(pi(x0), y1)) → A__TIMES(a__pi(mark(x0)), mark(y1))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(Y)

MARK(rcons(X1, X2)) → MARK(X1)

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(Z)

MARK(2ndsneg(square(y0), pi(x0))) → A__2NDSNEG(a__square(mark(y0)), a__pi(mark(x0)))

MARK(2ndsneg(pi(y0), square(x0))) → A__2NDSNEG(a__pi(mark(y0)), a__square(mark(x0)))

MARK(square(X)) → MARK(X)

MARK(2ndspos(times(y0, y1), pi(x0))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__pi(mark(x0)))

A__PI(s(x0)) → A__2NDSPOS(s(mark(x0)), a__from(0))

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(Y)

A__FROM(X) → MARK(X)

MARK(2ndsneg(s(y0), 2ndsneg(x0, x1))) → A__2NDSNEG(s(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndsneg(2ndsneg(y0, y1), pi(x0))) → A__2NDSNEG(a__2ndsneg(mark(y0), mark(y1)), a__pi(mark(x0)))

MARK(2ndsneg(pi(y0), 2ndsneg(x0, x1))) → A__2NDSNEG(a__pi(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndsneg(s(y0), plus(x0, x1))) → A__2NDSNEG(s(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(times(plus(x0, x1), y1)) → A__TIMES(a__plus(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(times(y0, y1), 2ndspos(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__2ndspos(mark(x0), mark(x1)))

A__2NDSPOS(s(y0), cons(y1, from(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__from(mark(x0))))

MARK(2ndspos(square(y0), pi(x0))) → A__2NDSPOS(a__square(mark(y0)), a__pi(mark(x0)))

MARK(rcons(X1, X2)) → MARK(X2)

MARK(2ndspos(s(y0), square(x0))) → A__2NDSPOS(s(mark(y0)), a__square(mark(x0)))

MARK(times(2ndspos(x0, x1), y1)) → A__TIMES(a__2ndspos(mark(x0), mark(x1)), mark(y1))

MARK(2ndsneg(2ndsneg(y0, y1), from(x0))) → A__2NDSNEG(a__2ndsneg(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndspos(s(y0), cons2(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(x0, mark(x1)))

A__TIMES(s(X), Y) → A__TIMES(mark(X), mark(Y))

MARK(2ndsneg(plus(y0, y1), cons2(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), cons2(x0, mark(x1)))

A__2NDSPOS(s(y0), cons(y1, cons(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, cons(mark(x0), x1)))

A__PLUS(s(X), Y) → MARK(X)

MARK(times(X1, X2)) → MARK(X1)

MARK(plus(times(x0, x1), y1)) → A__PLUS(a__times(mark(x0), mark(x1)), mark(y1))

A__SQUARE(X) → MARK(X)

MARK(2ndsneg(pi(y0), pi(x0))) → A__2NDSNEG(a__pi(mark(y0)), a__pi(mark(x0)))

A__2NDSPOS(s(y0), cons(y1, 2ndsneg(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__2ndsneg(mark(x0), mark(x1))))

A__2NDSPOS(s(N), cons(X, Z)) → MARK(N)

MARK(negrecip(X)) → MARK(X)

MARK(2ndsneg(plus(y0, y1), 2ndsneg(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndsneg(2ndsneg(y0, y1), plus(x0, x1))) → A__2NDSNEG(a__2ndsneg(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

A__PLUS(s(from(x0)), y1) → A__PLUS(a__from(mark(x0)), mark(y1))

A__PLUS(0, Y) → MARK(Y)

MARK(posrecip(X)) → MARK(X)

MARK(2ndspos(square(y0), from(x0))) → A__2NDSPOS(a__square(mark(y0)), a__from(mark(x0)))

MARK(2ndspos(times(y0, y1), from(x0))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__from(mark(x0)))

A__PLUS(s(times(x0, x1)), y1) → A__PLUS(a__times(mark(x0), mark(x1)), mark(y1))

A__2NDSPOS(s(y0), cons(y1, square(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__square(mark(x0))))

MARK(2ndspos(times(y0, y1), 2ndsneg(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__2ndsneg(mark(x0), mark(x1)))

A__PLUS(s(0), y1) → A__PLUS(0, mark(y1))

MARK(plus(0, y1)) → A__PLUS(0, mark(y1))

MARK(from(X)) → MARK(X)

MARK(2ndsneg(s(y0), times(x0, x1))) → A__2NDSNEG(s(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(square(y0), square(x0))) → A__2NDSPOS(a__square(mark(y0)), a__square(mark(x0)))

A__SQUARE(X) → A__TIMES(mark(X), mark(X))

A__PLUS(s(plus(x0, x1)), y1) → A__PLUS(a__plus(mark(x0), mark(x1)), mark(y1))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(Z)

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → A__2NDSPOS(mark(N), mark(Z))

MARK(plus(from(x0), y1)) → A__PLUS(a__from(mark(x0)), mark(y1))

MARK(2ndsneg(plus(y0, y1), plus(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(2ndsneg(times(y0, y1), pi(x0))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__pi(mark(x0)))

MARK(square(X)) → A__SQUARE(mark(X))

MARK(2ndsneg(pi(y0), times(x0, x1))) → A__2NDSNEG(a__pi(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(plus(y0, y1), from(x0))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndsneg(times(y0, y1), plus(x0, x1))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(2ndsneg(plus(y0, y1), times(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(plus(y0, y1), plus(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(2ndsneg(2ndsneg(y0, y1), cons2(x0, x1))) → A__2NDSNEG(a__2ndsneg(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(2ndsneg(times(y0, y1), 2ndsneg(x0, x1))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndsneg(2ndsneg(y0, y1), times(x0, x1))) → A__2NDSNEG(a__2ndsneg(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

A__PLUS(s(s(x0)), y1) → A__PLUS(s(mark(x0)), mark(y1))

MARK(2ndsneg(times(y0, y1), cons(x0, x1))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), cons(mark(x0), x1))

MARK(2ndsneg(2ndsneg(y0, y1), cons(x0, x1))) → A__2NDSNEG(a__2ndsneg(mark(y0), mark(y1)), cons(mark(x0), x1))

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))

a__2ndspos(0, Z) → rnil

a__2ndspos(s(N), cons(X, Z)) → a__2ndspos(s(mark(N)), cons2(X, mark(Z)))

a__2ndspos(s(N), cons2(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))

a__2ndsneg(0, Z) → rnil

a__2ndsneg(s(N), cons(X, Z)) → a__2ndsneg(s(mark(N)), cons2(X, mark(Z)))

a__2ndsneg(s(N), cons2(X, cons(Y, Z))) → rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z)))

a__pi(X) → a__2ndspos(mark(X), a__from(0))

a__plus(0, Y) → mark(Y)

a__plus(s(X), Y) → s(a__plus(mark(X), mark(Y)))

a__times(0, Y) → 0

a__times(s(X), Y) → a__plus(mark(Y), a__times(mark(X), mark(Y)))

a__square(X) → a__times(mark(X), mark(X))

mark(from(X)) → a__from(mark(X))

mark(2ndspos(X1, X2)) → a__2ndspos(mark(X1), mark(X2))

mark(2ndsneg(X1, X2)) → a__2ndsneg(mark(X1), mark(X2))

mark(pi(X)) → a__pi(mark(X))

mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))

mark(times(X1, X2)) → a__times(mark(X1), mark(X2))

mark(square(X)) → a__square(mark(X))

mark(0) → 0

mark(s(X)) → s(mark(X))

mark(posrecip(X)) → posrecip(mark(X))

mark(negrecip(X)) → negrecip(mark(X))

mark(nil) → nil

mark(cons(X1, X2)) → cons(mark(X1), X2)

mark(cons2(X1, X2)) → cons2(X1, mark(X2))

mark(rnil) → rnil

mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))

a__from(X) → from(X)

a__2ndspos(X1, X2) → 2ndspos(X1, X2)

a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)

a__pi(X) → pi(X)

a__plus(X1, X2) → plus(X1, X2)

a__times(X1, X2) → times(X1, X2)

a__square(X) → square(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

MARK(2ndsneg(pi(y0), from(x0))) → A__2NDSNEG(a__pi(mark(y0)), a__from(mark(x0)))

MARK(2ndsneg(pi(y0), 2ndspos(x0, x1))) → A__2NDSNEG(a__pi(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndsneg(pi(y0), cons2(x0, x1))) → A__2NDSNEG(a__pi(mark(y0)), cons2(x0, mark(x1)))

MARK(times(from(x0), y1)) → A__TIMES(a__from(mark(x0)), mark(y1))

MARK(2ndsneg(pi(y0), plus(x0, x1))) → A__2NDSNEG(a__pi(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(2ndsneg(pi(y0), cons(x0, x1))) → A__2NDSNEG(a__pi(mark(y0)), cons(mark(x0), x1))

MARK(times(pi(x0), y1)) → A__TIMES(a__pi(mark(x0)), mark(y1))

MARK(2ndsneg(pi(y0), square(x0))) → A__2NDSNEG(a__pi(mark(y0)), a__square(mark(x0)))

MARK(2ndsneg(pi(y0), 2ndsneg(x0, x1))) → A__2NDSNEG(a__pi(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

MARK(times(2ndspos(x0, x1), y1)) → A__TIMES(a__2ndspos(mark(x0), mark(x1)), mark(y1))

MARK(2ndsneg(pi(y0), pi(x0))) → A__2NDSNEG(a__pi(mark(y0)), a__pi(mark(x0)))

MARK(2ndsneg(pi(y0), times(x0, x1))) → A__2NDSNEG(a__pi(mark(y0)), a__times(mark(x0), mark(x1)))
The remaining pairs can at least be oriented weakly.

MARK(2ndspos(plus(y0, y1), square(x0))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndspos(square(y0), plus(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__plus(mark(x0), mark(x1)))

A__2NDSPOS(s(y0), cons(y1, 2ndspos(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__2ndspos(mark(x0), mark(x1))))

A__PI(times(x0, x1)) → A__2NDSPOS(a__times(mark(x0), mark(x1)), a__from(0))

A__2NDSNEG(s(N), cons(X, Z)) → A__2NDSNEG(s(mark(N)), cons2(X, mark(Z)))

MARK(2ndsneg(s(y0), square(x0))) → A__2NDSNEG(s(mark(y0)), a__square(mark(x0)))

MARK(times(times(x0, x1), y1)) → A__TIMES(a__times(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(X1, X2)) → MARK(X2)

MARK(2ndsneg(2ndsneg(y0, y1), 2ndspos(x0, x1))) → A__2NDSNEG(a__2ndsneg(mark(y0), mark(y1)), a__2ndspos(mark(x0), mark(x1)))

MARK(plus(plus(x0, x1), y1)) → A__PLUS(a__plus(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(s(y0), from(x0))) → A__2NDSPOS(s(mark(y0)), a__from(mark(x0)))

MARK(2ndsneg(square(y0), 2ndspos(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

A__2NDSNEG(s(N), cons(X, Z)) → MARK(Z)

A__PLUS(s(pi(x0)), y1) → A__PLUS(a__pi(mark(x0)), mark(y1))

MARK(2ndsneg(s(y0), pi(x0))) → A__2NDSNEG(s(mark(y0)), a__pi(mark(x0)))

A__TIMES(s(X), Y) → MARK(Y)

A__2NDSPOS(s(y0), cons(y1, plus(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__plus(mark(x0), mark(x1))))

MARK(2ndspos(square(y0), 2ndsneg(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

A__2NDSNEG(s(N), cons(X, Z)) → MARK(N)

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → A__2NDSNEG(mark(N), mark(Z))

MARK(times(2ndsneg(x0, x1), y1)) → A__TIMES(a__2ndsneg(mark(x0), mark(x1)), mark(y1))

MARK(2ndsneg(times(y0, y1), times(x0, x1))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndsneg(square(y0), 2ndsneg(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndsneg(2ndsneg(y0, y1), square(x0))) → A__2NDSNEG(a__2ndsneg(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndspos(times(y0, y1), times(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(s(y0), pi(x0))) → A__2NDSPOS(s(mark(y0)), a__pi(mark(x0)))

A__TIMES(s(X), Y) → A__PLUS(mark(Y), a__times(mark(X), mark(Y)))

MARK(2ndspos(square(y0), cons2(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), cons2(x0, mark(x1)))

MARK(plus(square(x0), y1)) → A__PLUS(a__square(mark(x0)), mark(y1))

MARK(plus(s(x0), y1)) → A__PLUS(s(mark(x0)), mark(y1))

MARK(2ndspos(square(y0), cons(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), cons(mark(x0), x1))

A__PI(plus(x0, x1)) → A__2NDSPOS(a__plus(mark(x0), mark(x1)), a__from(0))

MARK(2ndsneg(times(y0, y1), cons2(x0, x1))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(cons2(X1, X2)) → MARK(X2)

MARK(2ndsneg(times(y0, y1), 2ndspos(x0, x1))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndsneg(square(y0), cons2(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), cons2(x0, mark(x1)))

MARK(2ndsneg(plus(y0, y1), square(x0))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndsneg(square(y0), plus(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(s(X)) → MARK(X)

MARK(2ndspos(X1, X2)) → MARK(X1)

A__PLUS(s(square(x0)), y1) → A__PLUS(a__square(mark(x0)), mark(y1))

MARK(2ndspos(plus(y0, y1), cons2(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(pi(X)) → MARK(X)

MARK(2ndsneg(s(y0), from(x0))) → A__2NDSNEG(s(mark(y0)), a__from(mark(x0)))

MARK(plus(X1, X2)) → MARK(X2)

A__2NDSPOS(s(N), cons(X, Z)) → MARK(Z)

MARK(pi(X)) → A__PI(mark(X))

MARK(2ndsneg(times(y0, y1), from(x0))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndsneg(times(y0, y1), square(x0))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndsneg(square(y0), times(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(plus(y0, y1), cons(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), cons(mark(x0), x1))

MARK(plus(2ndsneg(x0, x1), y1)) → A__PLUS(a__2ndsneg(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(times(y0, y1), cons2(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(2ndsneg(X1, X2)) → MARK(X2)

MARK(2ndsneg(square(y0), from(x0))) → A__2NDSNEG(a__square(mark(y0)), a__from(mark(x0)))

MARK(2ndspos(times(y0, y1), square(x0))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndspos(square(y0), times(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndsneg(s(y0), cons(x0, x1))) → A__2NDSNEG(s(mark(y0)), cons(mark(x0), x1))

MARK(times(square(x0), y1)) → A__TIMES(a__square(mark(x0)), mark(y1))

MARK(from(X)) → A__FROM(mark(X))

MARK(2ndspos(s(y0), plus(x0, x1))) → A__2NDSPOS(s(mark(y0)), a__plus(mark(x0), mark(x1)))

A__PI(X) → A__FROM(0)

A__PLUS(s(2ndsneg(x0, x1)), y1) → A__PLUS(a__2ndsneg(mark(x0), mark(x1)), mark(y1))

MARK(plus(X1, X2)) → MARK(X1)

A__2NDSPOS(s(y0), cons(y1, times(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__times(mark(x0), mark(x1))))

MARK(2ndsneg(2ndsneg(y0, y1), 2ndsneg(x0, x1))) → A__2NDSNEG(a__2ndsneg(mark(y0), mark(y1)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndspos(s(y0), cons(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons(mark(x0), x1))

A__PI(X) → MARK(X)

MARK(2ndsneg(plus(y0, y1), 2ndspos(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__2ndspos(mark(x0), mark(x1)))

MARK(times(s(x0), y1)) → A__TIMES(s(mark(x0)), mark(y1))

MARK(2ndsneg(plus(y0, y1), cons(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), cons(mark(x0), x1))

MARK(2ndspos(plus(y0, y1), 2ndspos(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__2ndspos(mark(x0), mark(x1)))

A__PLUS(s(X), Y) → MARK(Y)

A__PLUS(s(2ndspos(x0, x1)), y1) → A__PLUS(a__2ndspos(mark(x0), mark(x1)), mark(y1))

A__TIMES(s(X), Y) → MARK(X)

A__2NDSPOS(s(y0), cons(y1, pi(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__pi(mark(x0))))

MARK(2ndsneg(plus(y0, y1), pi(x0))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__pi(mark(x0)))

MARK(2ndspos(plus(y0, y1), pi(x0))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__pi(mark(x0)))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(N)

MARK(2ndsneg(s(y0), cons2(x0, x1))) → A__2NDSNEG(s(mark(y0)), cons2(x0, mark(x1)))

MARK(plus(2ndspos(x0, x1), y1)) → A__PLUS(a__2ndspos(mark(x0), mark(x1)), mark(y1))

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(N)

MARK(cons(X1, X2)) → MARK(X1)

MARK(2ndsneg(plus(y0, y1), from(x0))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndsneg(s(y0), 2ndspos(x0, x1))) → A__2NDSNEG(s(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndspos(square(y0), 2ndspos(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

MARK(times(X1, X2)) → MARK(X2)

MARK(2ndspos(times(y0, y1), cons(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), cons(mark(x0), x1))

MARK(2ndspos(s(y0), 2ndspos(x0, x1))) → A__2NDSPOS(s(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

A__PI(square(x0)) → A__2NDSPOS(a__square(mark(x0)), a__from(0))

MARK(2ndsneg(square(y0), cons(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), cons(mark(x0), x1))

MARK(2ndspos(times(y0, y1), plus(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(plus(y0, y1), times(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(s(y0), times(x0, x1))) → A__2NDSPOS(s(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndsneg(X1, X2)) → MARK(X1)

MARK(2ndspos(s(y0), 2ndsneg(x0, x1))) → A__2NDSPOS(s(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndsneg(square(y0), square(x0))) → A__2NDSNEG(a__square(mark(y0)), a__square(mark(x0)))

MARK(2ndspos(plus(y0, y1), 2ndsneg(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__2ndsneg(mark(x0), mark(x1)))

MARK(plus(pi(x0), y1)) → A__PLUS(a__pi(mark(x0)), mark(y1))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(Y)

MARK(rcons(X1, X2)) → MARK(X1)

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(Z)

MARK(2ndsneg(square(y0), pi(x0))) → A__2NDSNEG(a__square(mark(y0)), a__pi(mark(x0)))

MARK(square(X)) → MARK(X)

MARK(2ndspos(times(y0, y1), pi(x0))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__pi(mark(x0)))

A__PI(s(x0)) → A__2NDSPOS(s(mark(x0)), a__from(0))

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(Y)

A__FROM(X) → MARK(X)

MARK(2ndsneg(s(y0), 2ndsneg(x0, x1))) → A__2NDSNEG(s(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndsneg(2ndsneg(y0, y1), pi(x0))) → A__2NDSNEG(a__2ndsneg(mark(y0), mark(y1)), a__pi(mark(x0)))

MARK(2ndsneg(s(y0), plus(x0, x1))) → A__2NDSNEG(s(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(times(plus(x0, x1), y1)) → A__TIMES(a__plus(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(times(y0, y1), 2ndspos(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__2ndspos(mark(x0), mark(x1)))

A__2NDSPOS(s(y0), cons(y1, from(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__from(mark(x0))))

MARK(2ndspos(square(y0), pi(x0))) → A__2NDSPOS(a__square(mark(y0)), a__pi(mark(x0)))

MARK(rcons(X1, X2)) → MARK(X2)

MARK(2ndspos(s(y0), square(x0))) → A__2NDSPOS(s(mark(y0)), a__square(mark(x0)))

MARK(2ndsneg(2ndsneg(y0, y1), from(x0))) → A__2NDSNEG(a__2ndsneg(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndspos(s(y0), cons2(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(x0, mark(x1)))

A__TIMES(s(X), Y) → A__TIMES(mark(X), mark(Y))

MARK(2ndsneg(plus(y0, y1), cons2(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), cons2(x0, mark(x1)))

A__2NDSPOS(s(y0), cons(y1, cons(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, cons(mark(x0), x1)))

A__PLUS(s(X), Y) → MARK(X)

MARK(times(X1, X2)) → MARK(X1)

MARK(plus(times(x0, x1), y1)) → A__PLUS(a__times(mark(x0), mark(x1)), mark(y1))

A__SQUARE(X) → MARK(X)

A__2NDSPOS(s(y0), cons(y1, 2ndsneg(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__2ndsneg(mark(x0), mark(x1))))

A__2NDSPOS(s(N), cons(X, Z)) → MARK(N)

MARK(negrecip(X)) → MARK(X)

MARK(2ndsneg(plus(y0, y1), 2ndsneg(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndsneg(2ndsneg(y0, y1), plus(x0, x1))) → A__2NDSNEG(a__2ndsneg(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

A__PLUS(s(from(x0)), y1) → A__PLUS(a__from(mark(x0)), mark(y1))

A__PLUS(0, Y) → MARK(Y)

MARK(posrecip(X)) → MARK(X)

MARK(2ndspos(square(y0), from(x0))) → A__2NDSPOS(a__square(mark(y0)), a__from(mark(x0)))

MARK(2ndspos(times(y0, y1), from(x0))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__from(mark(x0)))

A__PLUS(s(times(x0, x1)), y1) → A__PLUS(a__times(mark(x0), mark(x1)), mark(y1))

A__2NDSPOS(s(y0), cons(y1, square(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__square(mark(x0))))

MARK(2ndspos(times(y0, y1), 2ndsneg(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__2ndsneg(mark(x0), mark(x1)))

A__PLUS(s(0), y1) → A__PLUS(0, mark(y1))

MARK(plus(0, y1)) → A__PLUS(0, mark(y1))

MARK(from(X)) → MARK(X)

MARK(2ndsneg(s(y0), times(x0, x1))) → A__2NDSNEG(s(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(square(y0), square(x0))) → A__2NDSPOS(a__square(mark(y0)), a__square(mark(x0)))

A__SQUARE(X) → A__TIMES(mark(X), mark(X))

A__PLUS(s(plus(x0, x1)), y1) → A__PLUS(a__plus(mark(x0), mark(x1)), mark(y1))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(Z)

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → A__2NDSPOS(mark(N), mark(Z))

MARK(plus(from(x0), y1)) → A__PLUS(a__from(mark(x0)), mark(y1))

MARK(2ndsneg(plus(y0, y1), plus(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(2ndsneg(times(y0, y1), pi(x0))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__pi(mark(x0)))

MARK(square(X)) → A__SQUARE(mark(X))

MARK(2ndspos(plus(y0, y1), from(x0))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndsneg(times(y0, y1), plus(x0, x1))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(2ndsneg(plus(y0, y1), times(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(plus(y0, y1), plus(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(2ndsneg(2ndsneg(y0, y1), cons2(x0, x1))) → A__2NDSNEG(a__2ndsneg(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(2ndsneg(times(y0, y1), 2ndsneg(x0, x1))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndsneg(2ndsneg(y0, y1), times(x0, x1))) → A__2NDSNEG(a__2ndsneg(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

A__PLUS(s(s(x0)), y1) → A__PLUS(s(mark(x0)), mark(y1))

MARK(2ndsneg(times(y0, y1), cons(x0, x1))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), cons(mark(x0), x1))

MARK(2ndsneg(2ndsneg(y0, y1), cons(x0, x1))) → A__2NDSNEG(a__2ndsneg(mark(y0), mark(y1)), cons(mark(x0), x1))
Used ordering:  Polynomial interpretation [25]:

POL(0) = 0   

POL(2ndsneg(x1, x2)) = 0   

POL(2ndspos(x1, x2)) = 0   

POL(A__2NDSNEG(x1, x2)) = x1   

POL(A__2NDSPOS(x1, x2)) = x1   

POL(A__FROM(x1)) = 1   

POL(A__PI(x1)) = 1   

POL(A__PLUS(x1, x2)) = 1   

POL(A__SQUARE(x1)) = 1   

POL(A__TIMES(x1, x2)) = x1   

POL(MARK(x1)) = 1   

POL(a__2ndsneg(x1, x2)) = 1   

POL(a__2ndspos(x1, x2)) = 0   

POL(a__from(x1)) = 0   

POL(a__pi(x1)) = 0   

POL(a__plus(x1, x2)) = 1   

POL(a__square(x1)) = 1   

POL(a__times(x1, x2)) = 1   

POL(cons(x1, x2)) = 0   

POL(cons2(x1, x2)) = x2   

POL(from(x1)) = 0   

POL(mark(x1)) = 1   

POL(negrecip(x1)) = 1   

POL(nil) = 1   

POL(pi(x1)) = 0   

POL(plus(x1, x2)) = 0   

POL(posrecip(x1)) = 1   

POL(rcons(x1, x2)) = 0   

POL(rnil) = 0   

POL(s(x1)) = 1   

POL(square(x1)) = 0   

POL(times(x1, x2)) = 0   

The following usable rules [17] were oriented:

a__from(X) → cons(mark(X), from(s(X)))

a__pi(X) → a__2ndspos(mark(X), a__from(0))

a__2ndsneg(s(N), cons(X, Z)) → a__2ndsneg(s(mark(N)), cons2(X, mark(Z)))

a__2ndsneg(s(N), cons2(X, cons(Y, Z))) → rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z)))

a__2ndspos(s(N), cons2(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))

a__2ndsneg(0, Z) → rnil

a__2ndspos(0, Z) → rnil

a__2ndspos(s(N), cons(X, Z)) → a__2ndspos(s(mark(N)), cons2(X, mark(Z)))

mark(pi(X)) → a__pi(mark(X))

mark(2ndsneg(X1, X2)) → a__2ndsneg(mark(X1), mark(X2))

mark(2ndspos(X1, X2)) → a__2ndspos(mark(X1), mark(X2))

mark(from(X)) → a__from(mark(X))

a__times(0, Y) → 0

a__plus(s(X), Y) → s(a__plus(mark(X), mark(Y)))

mark(nil) → nil

mark(negrecip(X)) → negrecip(mark(X))

mark(posrecip(X)) → posrecip(mark(X))

mark(s(X)) → s(mark(X))

mark(0) → 0

mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))

a__times(s(X), Y) → a__plus(mark(Y), a__times(mark(X), mark(Y)))

a__plus(0, Y) → mark(Y)

mark(square(X)) → a__square(mark(X))

mark(times(X1, X2)) → a__times(mark(X1), mark(X2))

a__square(X) → a__times(mark(X), mark(X))

a__times(X1, X2) → times(X1, X2)

a__plus(X1, X2) → plus(X1, X2)

a__square(X) → square(X)

mark(cons2(X1, X2)) → cons2(X1, mark(X2))

mark(cons(X1, X2)) → cons(mark(X1), X2)

mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))

mark(rnil) → rnil

a__2ndspos(X1, X2) → 2ndspos(X1, X2)

a__from(X) → from(X)

a__pi(X) → pi(X)

a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ Narrowing
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ QDP
              ↳ Narrowing
                ↳ QDP
                  ↳ DependencyGraphProof
                    ↳ QDP
                      ↳ Narrowing
                        ↳ QDP
                          ↳ DependencyGraphProof
                            ↳ QDP
                              ↳ Narrowing
                                ↳ QDP
                                  ↳ DependencyGraphProof
                                    ↳ QDP
                                      ↳ Narrowing
                                        ↳ QDP
                                          ↳ DependencyGraphProof
                                            ↳ QDP
                                              ↳ Narrowing
                                                ↳ QDP
                                                  ↳ DependencyGraphProof
                                                    ↳ QDP
                                                      ↳ Narrowing
                                                        ↳ QDP
                                                          ↳ DependencyGraphProof
                                                            ↳ QDP
                                                              ↳ Narrowing
                                                                ↳ QDP
                                                                  ↳ DependencyGraphProof
                                                                    ↳ QDP
                                                                      ↳ Narrowing
                                                                        ↳ QDP
                                                                          ↳ DependencyGraphProof
                                                                            ↳ QDP
                                                                              ↳ Narrowing
                                                                                ↳ QDP
                                                                                  ↳ DependencyGraphProof
                                                                                    ↳ QDP
                                                                                      ↳ Narrowing
                                                                                        ↳ QDP
                                                                                          ↳ DependencyGraphProof
                                                                                            ↳ QDP
                                                                                              ↳ Narrowing
                                                                                                ↳ QDP
                                                                                                  ↳ DependencyGraphProof
                                                                                                    ↳ QDP
                                                                                                      ↳ Narrowing
                                                                                                        ↳ QDP
                                                                                                          ↳ DependencyGraphProof
                                                                                                            ↳ QDP
                                                                                                              ↳ Narrowing
                                                                                                                ↳ QDP
                                                                                                                  ↳ DependencyGraphProof
                                                                                                                    ↳ QDP
                                                                                                                      ↳ Narrowing
                                                                                                                        ↳ QDP
                                                                                                                          ↳ DependencyGraphProof
                                                                                                                            ↳ QDP
                                                                                                                              ↳ Narrowing
                                                                                                                                ↳ QDP
                                                                                                                                  ↳ DependencyGraphProof
                                                                                                                                    ↳ QDP
                                                                                                                                      ↳ Narrowing
                                                                                                                                        ↳ QDP
                                                                                                                                          ↳ DependencyGraphProof
                                                                                                                                            ↳ QDP
                                                                                                                                              ↳ Narrowing
                                                                                                                                                ↳ QDP
                                                                                                                                                  ↳ DependencyGraphProof
                                                                                                                                                    ↳ QDP
                                                                                                                                                      ↳ Narrowing
                                                                                                                                                        ↳ QDP
                                                                                                                                                          ↳ DependencyGraphProof
                                                                                                                                                            ↳ QDP
                                                                                                                                                              ↳ Narrowing
                                                                                                                                                                ↳ QDP
                                                                                                                                                                  ↳ DependencyGraphProof
                                                                                                                                                                    ↳ QDP
                                                                                                                                                                      ↳ Narrowing
                                                                                                                                                                        ↳ QDP
                                                                                                                                                                          ↳ DependencyGraphProof
                                                                                                                                                                            ↳ QDP
                                                                                                                                                                              ↳ Narrowing
                                                                                                                                                                                ↳ QDP
                                                                                                                                                                                  ↳ DependencyGraphProof
                                                                                                                                                                                    ↳ QDP
                                                                                                                                                                                      ↳ Narrowing
                                                                                                                                                                                        ↳ QDP
                                                                                                                                                                                          ↳ DependencyGraphProof
                                                                                                                                                                                            ↳ QDP
                                                                                                                                                                                              ↳ QDPOrderProof
                                                                                                                                                                                                ↳ QDP
                                                                                                                                                                                                  ↳ QDPOrderProof
                                                                                                                                                                                                    ↳ QDP
                                                                                                                                                                                                      ↳ QDPOrderProof
                                                                                                                                                                                                        ↳ QDP
                                                                                                                                                                                                          ↳ QDPOrderProof

Q DP problem:
The TRS P consists of the following rules:

MARK(2ndspos(square(y0), plus(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(plus(y0, y1), square(x0))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__square(mark(x0)))

A__PI(times(x0, x1)) → A__2NDSPOS(a__times(mark(x0), mark(x1)), a__from(0))

A__2NDSPOS(s(y0), cons(y1, 2ndspos(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__2ndspos(mark(x0), mark(x1))))

A__2NDSNEG(s(N), cons(X, Z)) → A__2NDSNEG(s(mark(N)), cons2(X, mark(Z)))

MARK(2ndsneg(s(y0), square(x0))) → A__2NDSNEG(s(mark(y0)), a__square(mark(x0)))

MARK(2ndspos(X1, X2)) → MARK(X2)

MARK(times(times(x0, x1), y1)) → A__TIMES(a__times(mark(x0), mark(x1)), mark(y1))

MARK(2ndsneg(2ndsneg(y0, y1), 2ndspos(x0, x1))) → A__2NDSNEG(a__2ndsneg(mark(y0), mark(y1)), a__2ndspos(mark(x0), mark(x1)))

MARK(plus(plus(x0, x1), y1)) → A__PLUS(a__plus(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(s(y0), from(x0))) → A__2NDSPOS(s(mark(y0)), a__from(mark(x0)))

A__2NDSNEG(s(N), cons(X, Z)) → MARK(Z)

MARK(2ndsneg(square(y0), 2ndspos(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

A__PLUS(s(pi(x0)), y1) → A__PLUS(a__pi(mark(x0)), mark(y1))

MARK(2ndsneg(s(y0), pi(x0))) → A__2NDSNEG(s(mark(y0)), a__pi(mark(x0)))

A__TIMES(s(X), Y) → MARK(Y)

A__2NDSPOS(s(y0), cons(y1, plus(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__plus(mark(x0), mark(x1))))

MARK(2ndspos(square(y0), 2ndsneg(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

A__2NDSNEG(s(N), cons(X, Z)) → MARK(N)

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → A__2NDSNEG(mark(N), mark(Z))

MARK(times(2ndsneg(x0, x1), y1)) → A__TIMES(a__2ndsneg(mark(x0), mark(x1)), mark(y1))

MARK(2ndsneg(times(y0, y1), times(x0, x1))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndsneg(square(y0), 2ndsneg(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndsneg(2ndsneg(y0, y1), square(x0))) → A__2NDSNEG(a__2ndsneg(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndspos(times(y0, y1), times(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

A__TIMES(s(X), Y) → A__PLUS(mark(Y), a__times(mark(X), mark(Y)))

MARK(2ndspos(s(y0), pi(x0))) → A__2NDSPOS(s(mark(y0)), a__pi(mark(x0)))

MARK(2ndspos(square(y0), cons2(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), cons2(x0, mark(x1)))

MARK(plus(square(x0), y1)) → A__PLUS(a__square(mark(x0)), mark(y1))

MARK(plus(s(x0), y1)) → A__PLUS(s(mark(x0)), mark(y1))

A__PI(plus(x0, x1)) → A__2NDSPOS(a__plus(mark(x0), mark(x1)), a__from(0))

MARK(2ndspos(square(y0), cons(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), cons(mark(x0), x1))

MARK(2ndsneg(times(y0, y1), cons2(x0, x1))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(cons2(X1, X2)) → MARK(X2)

MARK(2ndsneg(square(y0), cons2(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), cons2(x0, mark(x1)))

MARK(2ndsneg(times(y0, y1), 2ndspos(x0, x1))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__2ndspos(mark(x0), mark(x1)))

A__PLUS(s(square(x0)), y1) → A__PLUS(a__square(mark(x0)), mark(y1))

MARK(2ndspos(X1, X2)) → MARK(X1)

MARK(s(X)) → MARK(X)

MARK(2ndsneg(square(y0), plus(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(2ndsneg(plus(y0, y1), square(x0))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndspos(plus(y0, y1), cons2(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(pi(X)) → MARK(X)

MARK(2ndsneg(s(y0), from(x0))) → A__2NDSNEG(s(mark(y0)), a__from(mark(x0)))

MARK(plus(X1, X2)) → MARK(X2)

A__2NDSPOS(s(N), cons(X, Z)) → MARK(Z)

MARK(pi(X)) → A__PI(mark(X))

MARK(2ndsneg(times(y0, y1), square(x0))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndsneg(times(y0, y1), from(x0))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndsneg(square(y0), times(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(plus(y0, y1), cons(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), cons(mark(x0), x1))

MARK(plus(2ndsneg(x0, x1), y1)) → A__PLUS(a__2ndsneg(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(times(y0, y1), cons2(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(2ndsneg(X1, X2)) → MARK(X2)

MARK(2ndsneg(square(y0), from(x0))) → A__2NDSNEG(a__square(mark(y0)), a__from(mark(x0)))

MARK(2ndspos(times(y0, y1), square(x0))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndspos(square(y0), times(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndsneg(s(y0), cons(x0, x1))) → A__2NDSNEG(s(mark(y0)), cons(mark(x0), x1))

MARK(times(square(x0), y1)) → A__TIMES(a__square(mark(x0)), mark(y1))

MARK(from(X)) → A__FROM(mark(X))

MARK(2ndspos(s(y0), plus(x0, x1))) → A__2NDSPOS(s(mark(y0)), a__plus(mark(x0), mark(x1)))

A__PI(X) → A__FROM(0)

A__PLUS(s(2ndsneg(x0, x1)), y1) → A__PLUS(a__2ndsneg(mark(x0), mark(x1)), mark(y1))

MARK(plus(X1, X2)) → MARK(X1)

A__2NDSPOS(s(y0), cons(y1, times(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__times(mark(x0), mark(x1))))

MARK(2ndsneg(2ndsneg(y0, y1), 2ndsneg(x0, x1))) → A__2NDSNEG(a__2ndsneg(mark(y0), mark(y1)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndspos(s(y0), cons(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons(mark(x0), x1))

A__PI(X) → MARK(X)

MARK(2ndsneg(plus(y0, y1), 2ndspos(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__2ndspos(mark(x0), mark(x1)))

MARK(times(s(x0), y1)) → A__TIMES(s(mark(x0)), mark(y1))

MARK(2ndsneg(plus(y0, y1), cons(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), cons(mark(x0), x1))

MARK(2ndspos(plus(y0, y1), 2ndspos(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__2ndspos(mark(x0), mark(x1)))

A__PLUS(s(X), Y) → MARK(Y)

A__PLUS(s(2ndspos(x0, x1)), y1) → A__PLUS(a__2ndspos(mark(x0), mark(x1)), mark(y1))

A__TIMES(s(X), Y) → MARK(X)

A__2NDSPOS(s(y0), cons(y1, pi(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__pi(mark(x0))))

MARK(2ndsneg(plus(y0, y1), pi(x0))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__pi(mark(x0)))

MARK(2ndspos(plus(y0, y1), pi(x0))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__pi(mark(x0)))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(N)

MARK(2ndsneg(s(y0), cons2(x0, x1))) → A__2NDSNEG(s(mark(y0)), cons2(x0, mark(x1)))

MARK(plus(2ndspos(x0, x1), y1)) → A__PLUS(a__2ndspos(mark(x0), mark(x1)), mark(y1))

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(N)

MARK(cons(X1, X2)) → MARK(X1)

MARK(2ndsneg(plus(y0, y1), from(x0))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndsneg(s(y0), 2ndspos(x0, x1))) → A__2NDSNEG(s(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndspos(square(y0), 2ndspos(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

MARK(times(X1, X2)) → MARK(X2)

MARK(2ndspos(times(y0, y1), cons(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), cons(mark(x0), x1))

MARK(2ndspos(s(y0), 2ndspos(x0, x1))) → A__2NDSPOS(s(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

A__PI(square(x0)) → A__2NDSPOS(a__square(mark(x0)), a__from(0))

MARK(2ndsneg(square(y0), cons(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), cons(mark(x0), x1))

MARK(2ndspos(times(y0, y1), plus(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(plus(y0, y1), times(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(s(y0), times(x0, x1))) → A__2NDSPOS(s(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndsneg(X1, X2)) → MARK(X1)

MARK(2ndspos(s(y0), 2ndsneg(x0, x1))) → A__2NDSPOS(s(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndsneg(square(y0), square(x0))) → A__2NDSNEG(a__square(mark(y0)), a__square(mark(x0)))

MARK(2ndspos(plus(y0, y1), 2ndsneg(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__2ndsneg(mark(x0), mark(x1)))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(Y)

MARK(plus(pi(x0), y1)) → A__PLUS(a__pi(mark(x0)), mark(y1))

MARK(rcons(X1, X2)) → MARK(X1)

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(Z)

MARK(square(X)) → MARK(X)

MARK(2ndsneg(square(y0), pi(x0))) → A__2NDSNEG(a__square(mark(y0)), a__pi(mark(x0)))

MARK(2ndspos(times(y0, y1), pi(x0))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__pi(mark(x0)))

A__PI(s(x0)) → A__2NDSPOS(s(mark(x0)), a__from(0))

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(Y)

MARK(2ndsneg(2ndsneg(y0, y1), pi(x0))) → A__2NDSNEG(a__2ndsneg(mark(y0), mark(y1)), a__pi(mark(x0)))

MARK(2ndsneg(s(y0), 2ndsneg(x0, x1))) → A__2NDSNEG(s(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

A__FROM(X) → MARK(X)

MARK(times(plus(x0, x1), y1)) → A__TIMES(a__plus(mark(x0), mark(x1)), mark(y1))

MARK(2ndsneg(s(y0), plus(x0, x1))) → A__2NDSNEG(s(mark(y0)), a__plus(mark(x0), mark(x1)))

A__2NDSPOS(s(y0), cons(y1, from(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__from(mark(x0))))

MARK(2ndspos(times(y0, y1), 2ndspos(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndspos(square(y0), pi(x0))) → A__2NDSPOS(a__square(mark(y0)), a__pi(mark(x0)))

MARK(2ndspos(s(y0), square(x0))) → A__2NDSPOS(s(mark(y0)), a__square(mark(x0)))

MARK(rcons(X1, X2)) → MARK(X2)

MARK(2ndsneg(2ndsneg(y0, y1), from(x0))) → A__2NDSNEG(a__2ndsneg(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndspos(s(y0), cons2(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(x0, mark(x1)))

MARK(2ndsneg(plus(y0, y1), cons2(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), cons2(x0, mark(x1)))

A__TIMES(s(X), Y) → A__TIMES(mark(X), mark(Y))

A__PLUS(s(X), Y) → MARK(X)

A__2NDSPOS(s(y0), cons(y1, cons(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, cons(mark(x0), x1)))

MARK(times(X1, X2)) → MARK(X1)

MARK(plus(times(x0, x1), y1)) → A__PLUS(a__times(mark(x0), mark(x1)), mark(y1))

A__SQUARE(X) → MARK(X)

A__2NDSPOS(s(y0), cons(y1, 2ndsneg(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__2ndsneg(mark(x0), mark(x1))))

MARK(negrecip(X)) → MARK(X)

A__2NDSPOS(s(N), cons(X, Z)) → MARK(N)

MARK(2ndsneg(2ndsneg(y0, y1), plus(x0, x1))) → A__2NDSNEG(a__2ndsneg(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(2ndsneg(plus(y0, y1), 2ndsneg(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__2ndsneg(mark(x0), mark(x1)))

A__PLUS(s(from(x0)), y1) → A__PLUS(a__from(mark(x0)), mark(y1))

A__PLUS(0, Y) → MARK(Y)

MARK(posrecip(X)) → MARK(X)

MARK(2ndspos(square(y0), from(x0))) → A__2NDSPOS(a__square(mark(y0)), a__from(mark(x0)))

MARK(2ndspos(times(y0, y1), from(x0))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__from(mark(x0)))

A__PLUS(s(times(x0, x1)), y1) → A__PLUS(a__times(mark(x0), mark(x1)), mark(y1))

A__2NDSPOS(s(y0), cons(y1, square(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__square(mark(x0))))

A__PLUS(s(0), y1) → A__PLUS(0, mark(y1))

MARK(2ndspos(times(y0, y1), 2ndsneg(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__2ndsneg(mark(x0), mark(x1)))

MARK(plus(0, y1)) → A__PLUS(0, mark(y1))

MARK(from(X)) → MARK(X)

MARK(2ndsneg(s(y0), times(x0, x1))) → A__2NDSNEG(s(mark(y0)), a__times(mark(x0), mark(x1)))

A__SQUARE(X) → A__TIMES(mark(X), mark(X))

MARK(2ndspos(square(y0), square(x0))) → A__2NDSPOS(a__square(mark(y0)), a__square(mark(x0)))

A__PLUS(s(plus(x0, x1)), y1) → A__PLUS(a__plus(mark(x0), mark(x1)), mark(y1))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(Z)

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → A__2NDSPOS(mark(N), mark(Z))

MARK(2ndsneg(plus(y0, y1), plus(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(plus(from(x0), y1)) → A__PLUS(a__from(mark(x0)), mark(y1))

MARK(square(X)) → A__SQUARE(mark(X))

MARK(2ndsneg(times(y0, y1), pi(x0))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__pi(mark(x0)))

MARK(2ndspos(plus(y0, y1), from(x0))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndsneg(plus(y0, y1), times(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndsneg(times(y0, y1), plus(x0, x1))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(plus(y0, y1), plus(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(2ndsneg(2ndsneg(y0, y1), times(x0, x1))) → A__2NDSNEG(a__2ndsneg(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndsneg(times(y0, y1), 2ndsneg(x0, x1))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndsneg(2ndsneg(y0, y1), cons2(x0, x1))) → A__2NDSNEG(a__2ndsneg(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(2ndsneg(times(y0, y1), cons(x0, x1))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), cons(mark(x0), x1))

A__PLUS(s(s(x0)), y1) → A__PLUS(s(mark(x0)), mark(y1))

MARK(2ndsneg(2ndsneg(y0, y1), cons(x0, x1))) → A__2NDSNEG(a__2ndsneg(mark(y0), mark(y1)), cons(mark(x0), x1))

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))

a__2ndspos(0, Z) → rnil

a__2ndspos(s(N), cons(X, Z)) → a__2ndspos(s(mark(N)), cons2(X, mark(Z)))

a__2ndspos(s(N), cons2(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))

a__2ndsneg(0, Z) → rnil

a__2ndsneg(s(N), cons(X, Z)) → a__2ndsneg(s(mark(N)), cons2(X, mark(Z)))

a__2ndsneg(s(N), cons2(X, cons(Y, Z))) → rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z)))

a__pi(X) → a__2ndspos(mark(X), a__from(0))

a__plus(0, Y) → mark(Y)

a__plus(s(X), Y) → s(a__plus(mark(X), mark(Y)))

a__times(0, Y) → 0

a__times(s(X), Y) → a__plus(mark(Y), a__times(mark(X), mark(Y)))

a__square(X) → a__times(mark(X), mark(X))

mark(from(X)) → a__from(mark(X))

mark(2ndspos(X1, X2)) → a__2ndspos(mark(X1), mark(X2))

mark(2ndsneg(X1, X2)) → a__2ndsneg(mark(X1), mark(X2))

mark(pi(X)) → a__pi(mark(X))

mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))

mark(times(X1, X2)) → a__times(mark(X1), mark(X2))

mark(square(X)) → a__square(mark(X))

mark(0) → 0

mark(s(X)) → s(mark(X))

mark(posrecip(X)) → posrecip(mark(X))

mark(negrecip(X)) → negrecip(mark(X))

mark(nil) → nil

mark(cons(X1, X2)) → cons(mark(X1), X2)

mark(cons2(X1, X2)) → cons2(X1, mark(X2))

mark(rnil) → rnil

mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))

a__from(X) → from(X)

a__2ndspos(X1, X2) → 2ndspos(X1, X2)

a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)

a__pi(X) → pi(X)

a__plus(X1, X2) → plus(X1, X2)

a__times(X1, X2) → times(X1, X2)

a__square(X) → square(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

MARK(2ndsneg(2ndsneg(y0, y1), 2ndspos(x0, x1))) → A__2NDSNEG(a__2ndsneg(mark(y0), mark(y1)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndsneg(square(y0), 2ndspos(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndsneg(s(y0), pi(x0))) → A__2NDSNEG(s(mark(y0)), a__pi(mark(x0)))

MARK(2ndsneg(square(y0), 2ndsneg(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndsneg(times(y0, y1), 2ndspos(x0, x1))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndsneg(2ndsneg(y0, y1), 2ndsneg(x0, x1))) → A__2NDSNEG(a__2ndsneg(mark(y0), mark(y1)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndsneg(plus(y0, y1), 2ndspos(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndsneg(plus(y0, y1), pi(x0))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__pi(mark(x0)))

MARK(2ndsneg(s(y0), 2ndspos(x0, x1))) → A__2NDSNEG(s(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndsneg(square(y0), pi(x0))) → A__2NDSNEG(a__square(mark(y0)), a__pi(mark(x0)))

MARK(2ndsneg(2ndsneg(y0, y1), pi(x0))) → A__2NDSNEG(a__2ndsneg(mark(y0), mark(y1)), a__pi(mark(x0)))

MARK(2ndsneg(s(y0), 2ndsneg(x0, x1))) → A__2NDSNEG(s(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndsneg(plus(y0, y1), 2ndsneg(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndsneg(times(y0, y1), pi(x0))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__pi(mark(x0)))

MARK(2ndsneg(times(y0, y1), 2ndsneg(x0, x1))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__2ndsneg(mark(x0), mark(x1)))
The remaining pairs can at least be oriented weakly.

MARK(2ndspos(square(y0), plus(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(plus(y0, y1), square(x0))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__square(mark(x0)))

A__PI(times(x0, x1)) → A__2NDSPOS(a__times(mark(x0), mark(x1)), a__from(0))

A__2NDSPOS(s(y0), cons(y1, 2ndspos(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__2ndspos(mark(x0), mark(x1))))

A__2NDSNEG(s(N), cons(X, Z)) → A__2NDSNEG(s(mark(N)), cons2(X, mark(Z)))

MARK(2ndsneg(s(y0), square(x0))) → A__2NDSNEG(s(mark(y0)), a__square(mark(x0)))

MARK(2ndspos(X1, X2)) → MARK(X2)

MARK(times(times(x0, x1), y1)) → A__TIMES(a__times(mark(x0), mark(x1)), mark(y1))

MARK(plus(plus(x0, x1), y1)) → A__PLUS(a__plus(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(s(y0), from(x0))) → A__2NDSPOS(s(mark(y0)), a__from(mark(x0)))

A__2NDSNEG(s(N), cons(X, Z)) → MARK(Z)

A__PLUS(s(pi(x0)), y1) → A__PLUS(a__pi(mark(x0)), mark(y1))

A__TIMES(s(X), Y) → MARK(Y)

A__2NDSPOS(s(y0), cons(y1, plus(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__plus(mark(x0), mark(x1))))

MARK(2ndspos(square(y0), 2ndsneg(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

A__2NDSNEG(s(N), cons(X, Z)) → MARK(N)

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → A__2NDSNEG(mark(N), mark(Z))

MARK(times(2ndsneg(x0, x1), y1)) → A__TIMES(a__2ndsneg(mark(x0), mark(x1)), mark(y1))

MARK(2ndsneg(times(y0, y1), times(x0, x1))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndsneg(2ndsneg(y0, y1), square(x0))) → A__2NDSNEG(a__2ndsneg(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndspos(times(y0, y1), times(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

A__TIMES(s(X), Y) → A__PLUS(mark(Y), a__times(mark(X), mark(Y)))

MARK(2ndspos(s(y0), pi(x0))) → A__2NDSPOS(s(mark(y0)), a__pi(mark(x0)))

MARK(2ndspos(square(y0), cons2(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), cons2(x0, mark(x1)))

MARK(plus(square(x0), y1)) → A__PLUS(a__square(mark(x0)), mark(y1))

MARK(plus(s(x0), y1)) → A__PLUS(s(mark(x0)), mark(y1))

A__PI(plus(x0, x1)) → A__2NDSPOS(a__plus(mark(x0), mark(x1)), a__from(0))

MARK(2ndspos(square(y0), cons(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), cons(mark(x0), x1))

MARK(2ndsneg(times(y0, y1), cons2(x0, x1))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(cons2(X1, X2)) → MARK(X2)

MARK(2ndsneg(square(y0), cons2(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), cons2(x0, mark(x1)))

A__PLUS(s(square(x0)), y1) → A__PLUS(a__square(mark(x0)), mark(y1))

MARK(2ndspos(X1, X2)) → MARK(X1)

MARK(s(X)) → MARK(X)

MARK(2ndsneg(square(y0), plus(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(2ndsneg(plus(y0, y1), square(x0))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndspos(plus(y0, y1), cons2(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(pi(X)) → MARK(X)

MARK(2ndsneg(s(y0), from(x0))) → A__2NDSNEG(s(mark(y0)), a__from(mark(x0)))

MARK(plus(X1, X2)) → MARK(X2)

A__2NDSPOS(s(N), cons(X, Z)) → MARK(Z)

MARK(pi(X)) → A__PI(mark(X))

MARK(2ndsneg(times(y0, y1), square(x0))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndsneg(times(y0, y1), from(x0))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndsneg(square(y0), times(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(plus(y0, y1), cons(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), cons(mark(x0), x1))

MARK(plus(2ndsneg(x0, x1), y1)) → A__PLUS(a__2ndsneg(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(times(y0, y1), cons2(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(2ndsneg(X1, X2)) → MARK(X2)

MARK(2ndsneg(square(y0), from(x0))) → A__2NDSNEG(a__square(mark(y0)), a__from(mark(x0)))

MARK(2ndspos(times(y0, y1), square(x0))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndspos(square(y0), times(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndsneg(s(y0), cons(x0, x1))) → A__2NDSNEG(s(mark(y0)), cons(mark(x0), x1))

MARK(times(square(x0), y1)) → A__TIMES(a__square(mark(x0)), mark(y1))

MARK(from(X)) → A__FROM(mark(X))

MARK(2ndspos(s(y0), plus(x0, x1))) → A__2NDSPOS(s(mark(y0)), a__plus(mark(x0), mark(x1)))

A__PI(X) → A__FROM(0)

A__PLUS(s(2ndsneg(x0, x1)), y1) → A__PLUS(a__2ndsneg(mark(x0), mark(x1)), mark(y1))

MARK(plus(X1, X2)) → MARK(X1)

A__2NDSPOS(s(y0), cons(y1, times(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__times(mark(x0), mark(x1))))

MARK(2ndspos(s(y0), cons(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons(mark(x0), x1))

A__PI(X) → MARK(X)

MARK(times(s(x0), y1)) → A__TIMES(s(mark(x0)), mark(y1))

MARK(2ndsneg(plus(y0, y1), cons(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), cons(mark(x0), x1))

MARK(2ndspos(plus(y0, y1), 2ndspos(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__2ndspos(mark(x0), mark(x1)))

A__PLUS(s(X), Y) → MARK(Y)

A__PLUS(s(2ndspos(x0, x1)), y1) → A__PLUS(a__2ndspos(mark(x0), mark(x1)), mark(y1))

A__TIMES(s(X), Y) → MARK(X)

A__2NDSPOS(s(y0), cons(y1, pi(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__pi(mark(x0))))

MARK(2ndspos(plus(y0, y1), pi(x0))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__pi(mark(x0)))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(N)

MARK(2ndsneg(s(y0), cons2(x0, x1))) → A__2NDSNEG(s(mark(y0)), cons2(x0, mark(x1)))

MARK(plus(2ndspos(x0, x1), y1)) → A__PLUS(a__2ndspos(mark(x0), mark(x1)), mark(y1))

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(N)

MARK(cons(X1, X2)) → MARK(X1)

MARK(2ndsneg(plus(y0, y1), from(x0))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndspos(square(y0), 2ndspos(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

MARK(times(X1, X2)) → MARK(X2)

MARK(2ndspos(times(y0, y1), cons(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), cons(mark(x0), x1))

MARK(2ndspos(s(y0), 2ndspos(x0, x1))) → A__2NDSPOS(s(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

A__PI(square(x0)) → A__2NDSPOS(a__square(mark(x0)), a__from(0))

MARK(2ndsneg(square(y0), cons(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), cons(mark(x0), x1))

MARK(2ndspos(times(y0, y1), plus(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(plus(y0, y1), times(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(s(y0), times(x0, x1))) → A__2NDSPOS(s(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndsneg(X1, X2)) → MARK(X1)

MARK(2ndspos(s(y0), 2ndsneg(x0, x1))) → A__2NDSPOS(s(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndsneg(square(y0), square(x0))) → A__2NDSNEG(a__square(mark(y0)), a__square(mark(x0)))

MARK(2ndspos(plus(y0, y1), 2ndsneg(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__2ndsneg(mark(x0), mark(x1)))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(Y)

MARK(plus(pi(x0), y1)) → A__PLUS(a__pi(mark(x0)), mark(y1))

MARK(rcons(X1, X2)) → MARK(X1)

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(Z)

MARK(square(X)) → MARK(X)

MARK(2ndspos(times(y0, y1), pi(x0))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__pi(mark(x0)))

A__PI(s(x0)) → A__2NDSPOS(s(mark(x0)), a__from(0))

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(Y)

A__FROM(X) → MARK(X)

MARK(times(plus(x0, x1), y1)) → A__TIMES(a__plus(mark(x0), mark(x1)), mark(y1))

MARK(2ndsneg(s(y0), plus(x0, x1))) → A__2NDSNEG(s(mark(y0)), a__plus(mark(x0), mark(x1)))

A__2NDSPOS(s(y0), cons(y1, from(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__from(mark(x0))))

MARK(2ndspos(times(y0, y1), 2ndspos(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndspos(square(y0), pi(x0))) → A__2NDSPOS(a__square(mark(y0)), a__pi(mark(x0)))

MARK(2ndspos(s(y0), square(x0))) → A__2NDSPOS(s(mark(y0)), a__square(mark(x0)))

MARK(rcons(X1, X2)) → MARK(X2)

MARK(2ndsneg(2ndsneg(y0, y1), from(x0))) → A__2NDSNEG(a__2ndsneg(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndspos(s(y0), cons2(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(x0, mark(x1)))

MARK(2ndsneg(plus(y0, y1), cons2(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), cons2(x0, mark(x1)))

A__TIMES(s(X), Y) → A__TIMES(mark(X), mark(Y))

A__PLUS(s(X), Y) → MARK(X)

A__2NDSPOS(s(y0), cons(y1, cons(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, cons(mark(x0), x1)))

MARK(times(X1, X2)) → MARK(X1)

MARK(plus(times(x0, x1), y1)) → A__PLUS(a__times(mark(x0), mark(x1)), mark(y1))

A__SQUARE(X) → MARK(X)

A__2NDSPOS(s(y0), cons(y1, 2ndsneg(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__2ndsneg(mark(x0), mark(x1))))

MARK(negrecip(X)) → MARK(X)

A__2NDSPOS(s(N), cons(X, Z)) → MARK(N)

MARK(2ndsneg(2ndsneg(y0, y1), plus(x0, x1))) → A__2NDSNEG(a__2ndsneg(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

A__PLUS(s(from(x0)), y1) → A__PLUS(a__from(mark(x0)), mark(y1))

A__PLUS(0, Y) → MARK(Y)

MARK(posrecip(X)) → MARK(X)

MARK(2ndspos(square(y0), from(x0))) → A__2NDSPOS(a__square(mark(y0)), a__from(mark(x0)))

MARK(2ndspos(times(y0, y1), from(x0))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__from(mark(x0)))

A__PLUS(s(times(x0, x1)), y1) → A__PLUS(a__times(mark(x0), mark(x1)), mark(y1))

A__2NDSPOS(s(y0), cons(y1, square(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__square(mark(x0))))

A__PLUS(s(0), y1) → A__PLUS(0, mark(y1))

MARK(2ndspos(times(y0, y1), 2ndsneg(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__2ndsneg(mark(x0), mark(x1)))

MARK(plus(0, y1)) → A__PLUS(0, mark(y1))

MARK(from(X)) → MARK(X)

MARK(2ndsneg(s(y0), times(x0, x1))) → A__2NDSNEG(s(mark(y0)), a__times(mark(x0), mark(x1)))

A__SQUARE(X) → A__TIMES(mark(X), mark(X))

MARK(2ndspos(square(y0), square(x0))) → A__2NDSPOS(a__square(mark(y0)), a__square(mark(x0)))

A__PLUS(s(plus(x0, x1)), y1) → A__PLUS(a__plus(mark(x0), mark(x1)), mark(y1))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(Z)

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → A__2NDSPOS(mark(N), mark(Z))

MARK(2ndsneg(plus(y0, y1), plus(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(plus(from(x0), y1)) → A__PLUS(a__from(mark(x0)), mark(y1))

MARK(square(X)) → A__SQUARE(mark(X))

MARK(2ndspos(plus(y0, y1), from(x0))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndsneg(plus(y0, y1), times(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndsneg(times(y0, y1), plus(x0, x1))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(plus(y0, y1), plus(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(2ndsneg(2ndsneg(y0, y1), times(x0, x1))) → A__2NDSNEG(a__2ndsneg(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndsneg(2ndsneg(y0, y1), cons2(x0, x1))) → A__2NDSNEG(a__2ndsneg(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(2ndsneg(times(y0, y1), cons(x0, x1))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), cons(mark(x0), x1))

A__PLUS(s(s(x0)), y1) → A__PLUS(s(mark(x0)), mark(y1))

MARK(2ndsneg(2ndsneg(y0, y1), cons(x0, x1))) → A__2NDSNEG(a__2ndsneg(mark(y0), mark(y1)), cons(mark(x0), x1))
Used ordering:  Polynomial interpretation [25]:

POL(0) = 1   

POL(2ndsneg(x1, x2)) = 0   

POL(2ndspos(x1, x2)) = 0   

POL(A__2NDSNEG(x1, x2)) = x2   

POL(A__2NDSPOS(x1, x2)) = 1   

POL(A__FROM(x1)) = 1   

POL(A__PI(x1)) = 1   

POL(A__PLUS(x1, x2)) = 1   

POL(A__SQUARE(x1)) = 1   

POL(A__TIMES(x1, x2)) = 1   

POL(MARK(x1)) = 1   

POL(a__2ndsneg(x1, x2)) = 0   

POL(a__2ndspos(x1, x2)) = 0   

POL(a__from(x1)) = 1   

POL(a__pi(x1)) = 0   

POL(a__plus(x1, x2)) = x1   

POL(a__square(x1)) = 1   

POL(a__times(x1, x2)) = 1   

POL(cons(x1, x2)) = 1   

POL(cons2(x1, x2)) = 1   

POL(from(x1)) = 0   

POL(mark(x1)) = 1   

POL(negrecip(x1)) = 0   

POL(nil) = 0   

POL(pi(x1)) = 0   

POL(plus(x1, x2)) = x1   

POL(posrecip(x1)) = 0   

POL(rcons(x1, x2)) = x1   

POL(rnil) = 0   

POL(s(x1)) = 0   

POL(square(x1)) = 0   

POL(times(x1, x2)) = 0   

The following usable rules [17] were oriented:

a__from(X) → cons(mark(X), from(s(X)))

a__pi(X) → a__2ndspos(mark(X), a__from(0))

a__2ndsneg(s(N), cons(X, Z)) → a__2ndsneg(s(mark(N)), cons2(X, mark(Z)))

a__2ndsneg(s(N), cons2(X, cons(Y, Z))) → rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z)))

a__2ndspos(s(N), cons2(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))

a__2ndsneg(0, Z) → rnil

a__2ndspos(0, Z) → rnil

a__2ndspos(s(N), cons(X, Z)) → a__2ndspos(s(mark(N)), cons2(X, mark(Z)))

mark(pi(X)) → a__pi(mark(X))

mark(2ndsneg(X1, X2)) → a__2ndsneg(mark(X1), mark(X2))

mark(2ndspos(X1, X2)) → a__2ndspos(mark(X1), mark(X2))

mark(from(X)) → a__from(mark(X))

a__times(0, Y) → 0

a__plus(s(X), Y) → s(a__plus(mark(X), mark(Y)))

mark(nil) → nil

mark(negrecip(X)) → negrecip(mark(X))

mark(posrecip(X)) → posrecip(mark(X))

mark(s(X)) → s(mark(X))

mark(0) → 0

mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))

a__times(s(X), Y) → a__plus(mark(Y), a__times(mark(X), mark(Y)))

a__plus(0, Y) → mark(Y)

mark(square(X)) → a__square(mark(X))

mark(times(X1, X2)) → a__times(mark(X1), mark(X2))

a__square(X) → a__times(mark(X), mark(X))

a__times(X1, X2) → times(X1, X2)

a__plus(X1, X2) → plus(X1, X2)

a__square(X) → square(X)

mark(cons2(X1, X2)) → cons2(X1, mark(X2))

mark(cons(X1, X2)) → cons(mark(X1), X2)

mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))

mark(rnil) → rnil

a__2ndspos(X1, X2) → 2ndspos(X1, X2)

a__from(X) → from(X)

a__pi(X) → pi(X)

a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ Narrowing
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ QDP
              ↳ Narrowing
                ↳ QDP
                  ↳ DependencyGraphProof
                    ↳ QDP
                      ↳ Narrowing
                        ↳ QDP
                          ↳ DependencyGraphProof
                            ↳ QDP
                              ↳ Narrowing
                                ↳ QDP
                                  ↳ DependencyGraphProof
                                    ↳ QDP
                                      ↳ Narrowing
                                        ↳ QDP
                                          ↳ DependencyGraphProof
                                            ↳ QDP
                                              ↳ Narrowing
                                                ↳ QDP
                                                  ↳ DependencyGraphProof
                                                    ↳ QDP
                                                      ↳ Narrowing
                                                        ↳ QDP
                                                          ↳ DependencyGraphProof
                                                            ↳ QDP
                                                              ↳ Narrowing
                                                                ↳ QDP
                                                                  ↳ DependencyGraphProof
                                                                    ↳ QDP
                                                                      ↳ Narrowing
                                                                        ↳ QDP
                                                                          ↳ DependencyGraphProof
                                                                            ↳ QDP
                                                                              ↳ Narrowing
                                                                                ↳ QDP
                                                                                  ↳ DependencyGraphProof
                                                                                    ↳ QDP
                                                                                      ↳ Narrowing
                                                                                        ↳ QDP
                                                                                          ↳ DependencyGraphProof
                                                                                            ↳ QDP
                                                                                              ↳ Narrowing
                                                                                                ↳ QDP
                                                                                                  ↳ DependencyGraphProof
                                                                                                    ↳ QDP
                                                                                                      ↳ Narrowing
                                                                                                        ↳ QDP
                                                                                                          ↳ DependencyGraphProof
                                                                                                            ↳ QDP
                                                                                                              ↳ Narrowing
                                                                                                                ↳ QDP
                                                                                                                  ↳ DependencyGraphProof
                                                                                                                    ↳ QDP
                                                                                                                      ↳ Narrowing
                                                                                                                        ↳ QDP
                                                                                                                          ↳ DependencyGraphProof
                                                                                                                            ↳ QDP
                                                                                                                              ↳ Narrowing
                                                                                                                                ↳ QDP
                                                                                                                                  ↳ DependencyGraphProof
                                                                                                                                    ↳ QDP
                                                                                                                                      ↳ Narrowing
                                                                                                                                        ↳ QDP
                                                                                                                                          ↳ DependencyGraphProof
                                                                                                                                            ↳ QDP
                                                                                                                                              ↳ Narrowing
                                                                                                                                                ↳ QDP
                                                                                                                                                  ↳ DependencyGraphProof
                                                                                                                                                    ↳ QDP
                                                                                                                                                      ↳ Narrowing
                                                                                                                                                        ↳ QDP
                                                                                                                                                          ↳ DependencyGraphProof
                                                                                                                                                            ↳ QDP
                                                                                                                                                              ↳ Narrowing
                                                                                                                                                                ↳ QDP
                                                                                                                                                                  ↳ DependencyGraphProof
                                                                                                                                                                    ↳ QDP
                                                                                                                                                                      ↳ Narrowing
                                                                                                                                                                        ↳ QDP
                                                                                                                                                                          ↳ DependencyGraphProof
                                                                                                                                                                            ↳ QDP
                                                                                                                                                                              ↳ Narrowing
                                                                                                                                                                                ↳ QDP
                                                                                                                                                                                  ↳ DependencyGraphProof
                                                                                                                                                                                    ↳ QDP
                                                                                                                                                                                      ↳ Narrowing
                                                                                                                                                                                        ↳ QDP
                                                                                                                                                                                          ↳ DependencyGraphProof
                                                                                                                                                                                            ↳ QDP
                                                                                                                                                                                              ↳ QDPOrderProof
                                                                                                                                                                                                ↳ QDP
                                                                                                                                                                                                  ↳ QDPOrderProof
                                                                                                                                                                                                    ↳ QDP
                                                                                                                                                                                                      ↳ QDPOrderProof
                                                                                                                                                                                                        ↳ QDP
                                                                                                                                                                                                          ↳ QDPOrderProof
                                                                                                                                                                                                            ↳ QDP
                                                                                                                                                                                                              ↳ QDPOrderProof

Q DP problem:
The TRS P consists of the following rules:

MARK(2ndspos(plus(y0, y1), square(x0))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndspos(square(y0), plus(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__plus(mark(x0), mark(x1)))

A__2NDSPOS(s(y0), cons(y1, 2ndspos(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__2ndspos(mark(x0), mark(x1))))

A__PI(times(x0, x1)) → A__2NDSPOS(a__times(mark(x0), mark(x1)), a__from(0))

A__2NDSNEG(s(N), cons(X, Z)) → A__2NDSNEG(s(mark(N)), cons2(X, mark(Z)))

MARK(2ndsneg(s(y0), square(x0))) → A__2NDSNEG(s(mark(y0)), a__square(mark(x0)))

MARK(times(times(x0, x1), y1)) → A__TIMES(a__times(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(X1, X2)) → MARK(X2)

MARK(plus(plus(x0, x1), y1)) → A__PLUS(a__plus(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(s(y0), from(x0))) → A__2NDSPOS(s(mark(y0)), a__from(mark(x0)))

A__2NDSNEG(s(N), cons(X, Z)) → MARK(Z)

A__PLUS(s(pi(x0)), y1) → A__PLUS(a__pi(mark(x0)), mark(y1))

A__TIMES(s(X), Y) → MARK(Y)

A__2NDSPOS(s(y0), cons(y1, plus(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__plus(mark(x0), mark(x1))))

MARK(2ndspos(square(y0), 2ndsneg(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

A__2NDSNEG(s(N), cons(X, Z)) → MARK(N)

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → A__2NDSNEG(mark(N), mark(Z))

MARK(times(2ndsneg(x0, x1), y1)) → A__TIMES(a__2ndsneg(mark(x0), mark(x1)), mark(y1))

MARK(2ndsneg(times(y0, y1), times(x0, x1))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndsneg(2ndsneg(y0, y1), square(x0))) → A__2NDSNEG(a__2ndsneg(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndspos(times(y0, y1), times(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

A__TIMES(s(X), Y) → A__PLUS(mark(Y), a__times(mark(X), mark(Y)))

MARK(2ndspos(s(y0), pi(x0))) → A__2NDSPOS(s(mark(y0)), a__pi(mark(x0)))

MARK(2ndspos(square(y0), cons2(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), cons2(x0, mark(x1)))

MARK(plus(square(x0), y1)) → A__PLUS(a__square(mark(x0)), mark(y1))

MARK(plus(s(x0), y1)) → A__PLUS(s(mark(x0)), mark(y1))

MARK(2ndspos(square(y0), cons(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), cons(mark(x0), x1))

A__PI(plus(x0, x1)) → A__2NDSPOS(a__plus(mark(x0), mark(x1)), a__from(0))

MARK(2ndsneg(times(y0, y1), cons2(x0, x1))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(cons2(X1, X2)) → MARK(X2)

MARK(2ndsneg(square(y0), cons2(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), cons2(x0, mark(x1)))

MARK(2ndsneg(plus(y0, y1), square(x0))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndsneg(square(y0), plus(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(s(X)) → MARK(X)

MARK(2ndspos(X1, X2)) → MARK(X1)

A__PLUS(s(square(x0)), y1) → A__PLUS(a__square(mark(x0)), mark(y1))

MARK(2ndspos(plus(y0, y1), cons2(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(pi(X)) → MARK(X)

MARK(2ndsneg(s(y0), from(x0))) → A__2NDSNEG(s(mark(y0)), a__from(mark(x0)))

MARK(plus(X1, X2)) → MARK(X2)

A__2NDSPOS(s(N), cons(X, Z)) → MARK(Z)

MARK(pi(X)) → A__PI(mark(X))

MARK(2ndspos(plus(y0, y1), cons(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), cons(mark(x0), x1))

MARK(2ndsneg(square(y0), times(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndsneg(times(y0, y1), from(x0))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndsneg(times(y0, y1), square(x0))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(plus(2ndsneg(x0, x1), y1)) → A__PLUS(a__2ndsneg(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(times(y0, y1), cons2(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(2ndsneg(X1, X2)) → MARK(X2)

MARK(2ndsneg(square(y0), from(x0))) → A__2NDSNEG(a__square(mark(y0)), a__from(mark(x0)))

MARK(2ndspos(times(y0, y1), square(x0))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndspos(square(y0), times(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndsneg(s(y0), cons(x0, x1))) → A__2NDSNEG(s(mark(y0)), cons(mark(x0), x1))

MARK(times(square(x0), y1)) → A__TIMES(a__square(mark(x0)), mark(y1))

MARK(from(X)) → A__FROM(mark(X))

MARK(2ndspos(s(y0), plus(x0, x1))) → A__2NDSPOS(s(mark(y0)), a__plus(mark(x0), mark(x1)))

A__PI(X) → A__FROM(0)

A__PLUS(s(2ndsneg(x0, x1)), y1) → A__PLUS(a__2ndsneg(mark(x0), mark(x1)), mark(y1))

MARK(plus(X1, X2)) → MARK(X1)

A__2NDSPOS(s(y0), cons(y1, times(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__times(mark(x0), mark(x1))))

MARK(2ndspos(s(y0), cons(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons(mark(x0), x1))

A__PI(X) → MARK(X)

MARK(times(s(x0), y1)) → A__TIMES(s(mark(x0)), mark(y1))

MARK(2ndsneg(plus(y0, y1), cons(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), cons(mark(x0), x1))

MARK(2ndspos(plus(y0, y1), 2ndspos(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__2ndspos(mark(x0), mark(x1)))

A__PLUS(s(X), Y) → MARK(Y)

A__PLUS(s(2ndspos(x0, x1)), y1) → A__PLUS(a__2ndspos(mark(x0), mark(x1)), mark(y1))

A__TIMES(s(X), Y) → MARK(X)

A__2NDSPOS(s(y0), cons(y1, pi(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__pi(mark(x0))))

MARK(2ndspos(plus(y0, y1), pi(x0))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__pi(mark(x0)))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(N)

MARK(2ndsneg(s(y0), cons2(x0, x1))) → A__2NDSNEG(s(mark(y0)), cons2(x0, mark(x1)))

MARK(plus(2ndspos(x0, x1), y1)) → A__PLUS(a__2ndspos(mark(x0), mark(x1)), mark(y1))

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(N)

MARK(cons(X1, X2)) → MARK(X1)

MARK(2ndsneg(plus(y0, y1), from(x0))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndspos(square(y0), 2ndspos(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

MARK(times(X1, X2)) → MARK(X2)

MARK(2ndspos(times(y0, y1), cons(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), cons(mark(x0), x1))

MARK(2ndspos(s(y0), 2ndspos(x0, x1))) → A__2NDSPOS(s(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

A__PI(square(x0)) → A__2NDSPOS(a__square(mark(x0)), a__from(0))

MARK(2ndsneg(square(y0), cons(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), cons(mark(x0), x1))

MARK(2ndspos(times(y0, y1), plus(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(plus(y0, y1), times(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(s(y0), times(x0, x1))) → A__2NDSPOS(s(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndsneg(X1, X2)) → MARK(X1)

MARK(2ndspos(s(y0), 2ndsneg(x0, x1))) → A__2NDSPOS(s(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndsneg(square(y0), square(x0))) → A__2NDSNEG(a__square(mark(y0)), a__square(mark(x0)))

MARK(2ndspos(plus(y0, y1), 2ndsneg(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__2ndsneg(mark(x0), mark(x1)))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(Y)

MARK(plus(pi(x0), y1)) → A__PLUS(a__pi(mark(x0)), mark(y1))

MARK(rcons(X1, X2)) → MARK(X1)

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(Z)

MARK(square(X)) → MARK(X)

MARK(2ndspos(times(y0, y1), pi(x0))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__pi(mark(x0)))

A__PI(s(x0)) → A__2NDSPOS(s(mark(x0)), a__from(0))

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(Y)

A__FROM(X) → MARK(X)

MARK(2ndsneg(s(y0), plus(x0, x1))) → A__2NDSNEG(s(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(times(plus(x0, x1), y1)) → A__TIMES(a__plus(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(times(y0, y1), 2ndspos(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__2ndspos(mark(x0), mark(x1)))

A__2NDSPOS(s(y0), cons(y1, from(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__from(mark(x0))))

MARK(2ndspos(square(y0), pi(x0))) → A__2NDSPOS(a__square(mark(y0)), a__pi(mark(x0)))

MARK(rcons(X1, X2)) → MARK(X2)

MARK(2ndspos(s(y0), square(x0))) → A__2NDSPOS(s(mark(y0)), a__square(mark(x0)))

MARK(2ndsneg(2ndsneg(y0, y1), from(x0))) → A__2NDSNEG(a__2ndsneg(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndspos(s(y0), cons2(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(x0, mark(x1)))

A__TIMES(s(X), Y) → A__TIMES(mark(X), mark(Y))

MARK(2ndsneg(plus(y0, y1), cons2(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), cons2(x0, mark(x1)))

A__2NDSPOS(s(y0), cons(y1, cons(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, cons(mark(x0), x1)))

A__PLUS(s(X), Y) → MARK(X)

MARK(times(X1, X2)) → MARK(X1)

MARK(plus(times(x0, x1), y1)) → A__PLUS(a__times(mark(x0), mark(x1)), mark(y1))

A__SQUARE(X) → MARK(X)

A__2NDSPOS(s(y0), cons(y1, 2ndsneg(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__2ndsneg(mark(x0), mark(x1))))

A__2NDSPOS(s(N), cons(X, Z)) → MARK(N)

MARK(negrecip(X)) → MARK(X)

MARK(2ndsneg(2ndsneg(y0, y1), plus(x0, x1))) → A__2NDSNEG(a__2ndsneg(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

A__PLUS(s(from(x0)), y1) → A__PLUS(a__from(mark(x0)), mark(y1))

A__PLUS(0, Y) → MARK(Y)

MARK(posrecip(X)) → MARK(X)

MARK(2ndspos(square(y0), from(x0))) → A__2NDSPOS(a__square(mark(y0)), a__from(mark(x0)))

MARK(2ndspos(times(y0, y1), from(x0))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__from(mark(x0)))

A__PLUS(s(times(x0, x1)), y1) → A__PLUS(a__times(mark(x0), mark(x1)), mark(y1))

A__2NDSPOS(s(y0), cons(y1, square(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__square(mark(x0))))

MARK(2ndspos(times(y0, y1), 2ndsneg(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__2ndsneg(mark(x0), mark(x1)))

A__PLUS(s(0), y1) → A__PLUS(0, mark(y1))

MARK(plus(0, y1)) → A__PLUS(0, mark(y1))

MARK(from(X)) → MARK(X)

MARK(2ndsneg(s(y0), times(x0, x1))) → A__2NDSNEG(s(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(square(y0), square(x0))) → A__2NDSPOS(a__square(mark(y0)), a__square(mark(x0)))

A__SQUARE(X) → A__TIMES(mark(X), mark(X))

A__PLUS(s(plus(x0, x1)), y1) → A__PLUS(a__plus(mark(x0), mark(x1)), mark(y1))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(Z)

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → A__2NDSPOS(mark(N), mark(Z))

MARK(plus(from(x0), y1)) → A__PLUS(a__from(mark(x0)), mark(y1))

MARK(2ndsneg(plus(y0, y1), plus(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(square(X)) → A__SQUARE(mark(X))

MARK(2ndspos(plus(y0, y1), from(x0))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndsneg(times(y0, y1), plus(x0, x1))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(2ndsneg(plus(y0, y1), times(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(plus(y0, y1), plus(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(2ndsneg(2ndsneg(y0, y1), cons2(x0, x1))) → A__2NDSNEG(a__2ndsneg(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(2ndsneg(2ndsneg(y0, y1), times(x0, x1))) → A__2NDSNEG(a__2ndsneg(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

A__PLUS(s(s(x0)), y1) → A__PLUS(s(mark(x0)), mark(y1))

MARK(2ndsneg(times(y0, y1), cons(x0, x1))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), cons(mark(x0), x1))

MARK(2ndsneg(2ndsneg(y0, y1), cons(x0, x1))) → A__2NDSNEG(a__2ndsneg(mark(y0), mark(y1)), cons(mark(x0), x1))

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))

a__2ndspos(0, Z) → rnil

a__2ndspos(s(N), cons(X, Z)) → a__2ndspos(s(mark(N)), cons2(X, mark(Z)))

a__2ndspos(s(N), cons2(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))

a__2ndsneg(0, Z) → rnil

a__2ndsneg(s(N), cons(X, Z)) → a__2ndsneg(s(mark(N)), cons2(X, mark(Z)))

a__2ndsneg(s(N), cons2(X, cons(Y, Z))) → rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z)))

a__pi(X) → a__2ndspos(mark(X), a__from(0))

a__plus(0, Y) → mark(Y)

a__plus(s(X), Y) → s(a__plus(mark(X), mark(Y)))

a__times(0, Y) → 0

a__times(s(X), Y) → a__plus(mark(Y), a__times(mark(X), mark(Y)))

a__square(X) → a__times(mark(X), mark(X))

mark(from(X)) → a__from(mark(X))

mark(2ndspos(X1, X2)) → a__2ndspos(mark(X1), mark(X2))

mark(2ndsneg(X1, X2)) → a__2ndsneg(mark(X1), mark(X2))

mark(pi(X)) → a__pi(mark(X))

mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))

mark(times(X1, X2)) → a__times(mark(X1), mark(X2))

mark(square(X)) → a__square(mark(X))

mark(0) → 0

mark(s(X)) → s(mark(X))

mark(posrecip(X)) → posrecip(mark(X))

mark(negrecip(X)) → negrecip(mark(X))

mark(nil) → nil

mark(cons(X1, X2)) → cons(mark(X1), X2)

mark(cons2(X1, X2)) → cons2(X1, mark(X2))

mark(rnil) → rnil

mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))

a__from(X) → from(X)

a__2ndspos(X1, X2) → 2ndspos(X1, X2)

a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)

a__pi(X) → pi(X)

a__plus(X1, X2) → plus(X1, X2)

a__times(X1, X2) → times(X1, X2)

a__square(X) → square(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

A__2NDSPOS(s(y0), cons(y1, 2ndspos(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__2ndspos(mark(x0), mark(x1))))

MARK(2ndspos(square(y0), 2ndsneg(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndspos(s(y0), pi(x0))) → A__2NDSPOS(s(mark(y0)), a__pi(mark(x0)))

MARK(2ndspos(plus(y0, y1), 2ndspos(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__2ndspos(mark(x0), mark(x1)))

A__2NDSPOS(s(y0), cons(y1, pi(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__pi(mark(x0))))

MARK(2ndspos(plus(y0, y1), pi(x0))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__pi(mark(x0)))

MARK(2ndspos(square(y0), 2ndspos(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndspos(s(y0), 2ndspos(x0, x1))) → A__2NDSPOS(s(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndspos(s(y0), 2ndsneg(x0, x1))) → A__2NDSPOS(s(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndspos(plus(y0, y1), 2ndsneg(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndspos(times(y0, y1), pi(x0))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__pi(mark(x0)))

MARK(2ndspos(times(y0, y1), 2ndspos(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__2ndspos(mark(x0), mark(x1)))

MARK(2ndspos(square(y0), pi(x0))) → A__2NDSPOS(a__square(mark(y0)), a__pi(mark(x0)))

A__2NDSPOS(s(y0), cons(y1, 2ndsneg(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__2ndsneg(mark(x0), mark(x1))))

MARK(2ndspos(times(y0, y1), 2ndsneg(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__2ndsneg(mark(x0), mark(x1)))
The remaining pairs can at least be oriented weakly.

MARK(2ndspos(plus(y0, y1), square(x0))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndspos(square(y0), plus(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__plus(mark(x0), mark(x1)))

A__PI(times(x0, x1)) → A__2NDSPOS(a__times(mark(x0), mark(x1)), a__from(0))

A__2NDSNEG(s(N), cons(X, Z)) → A__2NDSNEG(s(mark(N)), cons2(X, mark(Z)))

MARK(2ndsneg(s(y0), square(x0))) → A__2NDSNEG(s(mark(y0)), a__square(mark(x0)))

MARK(times(times(x0, x1), y1)) → A__TIMES(a__times(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(X1, X2)) → MARK(X2)

MARK(plus(plus(x0, x1), y1)) → A__PLUS(a__plus(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(s(y0), from(x0))) → A__2NDSPOS(s(mark(y0)), a__from(mark(x0)))

A__2NDSNEG(s(N), cons(X, Z)) → MARK(Z)

A__PLUS(s(pi(x0)), y1) → A__PLUS(a__pi(mark(x0)), mark(y1))

A__TIMES(s(X), Y) → MARK(Y)

A__2NDSPOS(s(y0), cons(y1, plus(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__plus(mark(x0), mark(x1))))

A__2NDSNEG(s(N), cons(X, Z)) → MARK(N)

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → A__2NDSNEG(mark(N), mark(Z))

MARK(times(2ndsneg(x0, x1), y1)) → A__TIMES(a__2ndsneg(mark(x0), mark(x1)), mark(y1))

MARK(2ndsneg(times(y0, y1), times(x0, x1))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndsneg(2ndsneg(y0, y1), square(x0))) → A__2NDSNEG(a__2ndsneg(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndspos(times(y0, y1), times(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

A__TIMES(s(X), Y) → A__PLUS(mark(Y), a__times(mark(X), mark(Y)))

MARK(2ndspos(square(y0), cons2(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), cons2(x0, mark(x1)))

MARK(plus(square(x0), y1)) → A__PLUS(a__square(mark(x0)), mark(y1))

MARK(plus(s(x0), y1)) → A__PLUS(s(mark(x0)), mark(y1))

MARK(2ndspos(square(y0), cons(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), cons(mark(x0), x1))

A__PI(plus(x0, x1)) → A__2NDSPOS(a__plus(mark(x0), mark(x1)), a__from(0))

MARK(2ndsneg(times(y0, y1), cons2(x0, x1))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(cons2(X1, X2)) → MARK(X2)

MARK(2ndsneg(square(y0), cons2(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), cons2(x0, mark(x1)))

MARK(2ndsneg(plus(y0, y1), square(x0))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndsneg(square(y0), plus(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(s(X)) → MARK(X)

MARK(2ndspos(X1, X2)) → MARK(X1)

A__PLUS(s(square(x0)), y1) → A__PLUS(a__square(mark(x0)), mark(y1))

MARK(2ndspos(plus(y0, y1), cons2(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(pi(X)) → MARK(X)

MARK(2ndsneg(s(y0), from(x0))) → A__2NDSNEG(s(mark(y0)), a__from(mark(x0)))

MARK(plus(X1, X2)) → MARK(X2)

A__2NDSPOS(s(N), cons(X, Z)) → MARK(Z)

MARK(pi(X)) → A__PI(mark(X))

MARK(2ndspos(plus(y0, y1), cons(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), cons(mark(x0), x1))

MARK(2ndsneg(square(y0), times(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndsneg(times(y0, y1), from(x0))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndsneg(times(y0, y1), square(x0))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(plus(2ndsneg(x0, x1), y1)) → A__PLUS(a__2ndsneg(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(times(y0, y1), cons2(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(2ndsneg(X1, X2)) → MARK(X2)

MARK(2ndsneg(square(y0), from(x0))) → A__2NDSNEG(a__square(mark(y0)), a__from(mark(x0)))

MARK(2ndspos(times(y0, y1), square(x0))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndspos(square(y0), times(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndsneg(s(y0), cons(x0, x1))) → A__2NDSNEG(s(mark(y0)), cons(mark(x0), x1))

MARK(times(square(x0), y1)) → A__TIMES(a__square(mark(x0)), mark(y1))

MARK(from(X)) → A__FROM(mark(X))

MARK(2ndspos(s(y0), plus(x0, x1))) → A__2NDSPOS(s(mark(y0)), a__plus(mark(x0), mark(x1)))

A__PI(X) → A__FROM(0)

A__PLUS(s(2ndsneg(x0, x1)), y1) → A__PLUS(a__2ndsneg(mark(x0), mark(x1)), mark(y1))

MARK(plus(X1, X2)) → MARK(X1)

A__2NDSPOS(s(y0), cons(y1, times(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__times(mark(x0), mark(x1))))

MARK(2ndspos(s(y0), cons(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons(mark(x0), x1))

A__PI(X) → MARK(X)

MARK(times(s(x0), y1)) → A__TIMES(s(mark(x0)), mark(y1))

MARK(2ndsneg(plus(y0, y1), cons(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), cons(mark(x0), x1))

A__PLUS(s(X), Y) → MARK(Y)

A__PLUS(s(2ndspos(x0, x1)), y1) → A__PLUS(a__2ndspos(mark(x0), mark(x1)), mark(y1))

A__TIMES(s(X), Y) → MARK(X)

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(N)

MARK(2ndsneg(s(y0), cons2(x0, x1))) → A__2NDSNEG(s(mark(y0)), cons2(x0, mark(x1)))

MARK(plus(2ndspos(x0, x1), y1)) → A__PLUS(a__2ndspos(mark(x0), mark(x1)), mark(y1))

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(N)

MARK(cons(X1, X2)) → MARK(X1)

MARK(2ndsneg(plus(y0, y1), from(x0))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(times(X1, X2)) → MARK(X2)

MARK(2ndspos(times(y0, y1), cons(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), cons(mark(x0), x1))

A__PI(square(x0)) → A__2NDSPOS(a__square(mark(x0)), a__from(0))

MARK(2ndsneg(square(y0), cons(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), cons(mark(x0), x1))

MARK(2ndspos(times(y0, y1), plus(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(plus(y0, y1), times(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(s(y0), times(x0, x1))) → A__2NDSPOS(s(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndsneg(X1, X2)) → MARK(X1)

MARK(2ndsneg(square(y0), square(x0))) → A__2NDSNEG(a__square(mark(y0)), a__square(mark(x0)))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(Y)

MARK(plus(pi(x0), y1)) → A__PLUS(a__pi(mark(x0)), mark(y1))

MARK(rcons(X1, X2)) → MARK(X1)

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(Z)

MARK(square(X)) → MARK(X)

A__PI(s(x0)) → A__2NDSPOS(s(mark(x0)), a__from(0))

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(Y)

A__FROM(X) → MARK(X)

MARK(2ndsneg(s(y0), plus(x0, x1))) → A__2NDSNEG(s(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(times(plus(x0, x1), y1)) → A__TIMES(a__plus(mark(x0), mark(x1)), mark(y1))

A__2NDSPOS(s(y0), cons(y1, from(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__from(mark(x0))))

MARK(rcons(X1, X2)) → MARK(X2)

MARK(2ndspos(s(y0), square(x0))) → A__2NDSPOS(s(mark(y0)), a__square(mark(x0)))

MARK(2ndsneg(2ndsneg(y0, y1), from(x0))) → A__2NDSNEG(a__2ndsneg(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndspos(s(y0), cons2(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(x0, mark(x1)))

A__TIMES(s(X), Y) → A__TIMES(mark(X), mark(Y))

MARK(2ndsneg(plus(y0, y1), cons2(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), cons2(x0, mark(x1)))

A__2NDSPOS(s(y0), cons(y1, cons(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, cons(mark(x0), x1)))

A__PLUS(s(X), Y) → MARK(X)

MARK(times(X1, X2)) → MARK(X1)

MARK(plus(times(x0, x1), y1)) → A__PLUS(a__times(mark(x0), mark(x1)), mark(y1))

A__SQUARE(X) → MARK(X)

A__2NDSPOS(s(N), cons(X, Z)) → MARK(N)

MARK(negrecip(X)) → MARK(X)

MARK(2ndsneg(2ndsneg(y0, y1), plus(x0, x1))) → A__2NDSNEG(a__2ndsneg(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

A__PLUS(s(from(x0)), y1) → A__PLUS(a__from(mark(x0)), mark(y1))

A__PLUS(0, Y) → MARK(Y)

MARK(posrecip(X)) → MARK(X)

MARK(2ndspos(square(y0), from(x0))) → A__2NDSPOS(a__square(mark(y0)), a__from(mark(x0)))

MARK(2ndspos(times(y0, y1), from(x0))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__from(mark(x0)))

A__PLUS(s(times(x0, x1)), y1) → A__PLUS(a__times(mark(x0), mark(x1)), mark(y1))

A__2NDSPOS(s(y0), cons(y1, square(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__square(mark(x0))))

A__PLUS(s(0), y1) → A__PLUS(0, mark(y1))

MARK(plus(0, y1)) → A__PLUS(0, mark(y1))

MARK(from(X)) → MARK(X)

MARK(2ndsneg(s(y0), times(x0, x1))) → A__2NDSNEG(s(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(square(y0), square(x0))) → A__2NDSPOS(a__square(mark(y0)), a__square(mark(x0)))

A__SQUARE(X) → A__TIMES(mark(X), mark(X))

A__PLUS(s(plus(x0, x1)), y1) → A__PLUS(a__plus(mark(x0), mark(x1)), mark(y1))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(Z)

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → A__2NDSPOS(mark(N), mark(Z))

MARK(plus(from(x0), y1)) → A__PLUS(a__from(mark(x0)), mark(y1))

MARK(2ndsneg(plus(y0, y1), plus(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(square(X)) → A__SQUARE(mark(X))

MARK(2ndspos(plus(y0, y1), from(x0))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndsneg(times(y0, y1), plus(x0, x1))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(2ndsneg(plus(y0, y1), times(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(plus(y0, y1), plus(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(2ndsneg(2ndsneg(y0, y1), cons2(x0, x1))) → A__2NDSNEG(a__2ndsneg(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(2ndsneg(2ndsneg(y0, y1), times(x0, x1))) → A__2NDSNEG(a__2ndsneg(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

A__PLUS(s(s(x0)), y1) → A__PLUS(s(mark(x0)), mark(y1))

MARK(2ndsneg(times(y0, y1), cons(x0, x1))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), cons(mark(x0), x1))

MARK(2ndsneg(2ndsneg(y0, y1), cons(x0, x1))) → A__2NDSNEG(a__2ndsneg(mark(y0), mark(y1)), cons(mark(x0), x1))
Used ordering:  Polynomial interpretation [25]:

POL(0) = 0   

POL(2ndsneg(x1, x2)) = 0   

POL(2ndspos(x1, x2)) = 0   

POL(A__2NDSNEG(x1, x2)) = 1   

POL(A__2NDSPOS(x1, x2)) = x2   

POL(A__FROM(x1)) = 1   

POL(A__PI(x1)) = 1   

POL(A__PLUS(x1, x2)) = 1   

POL(A__SQUARE(x1)) = 1   

POL(A__TIMES(x1, x2)) = 1   

POL(MARK(x1)) = 1   

POL(a__2ndsneg(x1, x2)) = 0   

POL(a__2ndspos(x1, x2)) = 0   

POL(a__from(x1)) = 1   

POL(a__pi(x1)) = 0   

POL(a__plus(x1, x2)) = 1   

POL(a__square(x1)) = 1   

POL(a__times(x1, x2)) = 1   

POL(cons(x1, x2)) = 1   

POL(cons2(x1, x2)) = x2   

POL(from(x1)) = 0   

POL(mark(x1)) = 1   

POL(negrecip(x1)) = 0   

POL(nil) = 0   

POL(pi(x1)) = 0   

POL(plus(x1, x2)) = 0   

POL(posrecip(x1)) = x1   

POL(rcons(x1, x2)) = 0   

POL(rnil) = 0   

POL(s(x1)) = 0   

POL(square(x1)) = 0   

POL(times(x1, x2)) = 0   

The following usable rules [17] were oriented:

a__from(X) → cons(mark(X), from(s(X)))

a__pi(X) → a__2ndspos(mark(X), a__from(0))

a__2ndsneg(s(N), cons(X, Z)) → a__2ndsneg(s(mark(N)), cons2(X, mark(Z)))

a__2ndsneg(s(N), cons2(X, cons(Y, Z))) → rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z)))

a__2ndspos(s(N), cons2(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))

a__2ndsneg(0, Z) → rnil

a__2ndspos(0, Z) → rnil

a__2ndspos(s(N), cons(X, Z)) → a__2ndspos(s(mark(N)), cons2(X, mark(Z)))

mark(pi(X)) → a__pi(mark(X))

mark(2ndsneg(X1, X2)) → a__2ndsneg(mark(X1), mark(X2))

mark(2ndspos(X1, X2)) → a__2ndspos(mark(X1), mark(X2))

mark(from(X)) → a__from(mark(X))

a__times(0, Y) → 0

a__plus(s(X), Y) → s(a__plus(mark(X), mark(Y)))

mark(nil) → nil

mark(negrecip(X)) → negrecip(mark(X))

mark(posrecip(X)) → posrecip(mark(X))

mark(s(X)) → s(mark(X))

mark(0) → 0

mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))

a__times(s(X), Y) → a__plus(mark(Y), a__times(mark(X), mark(Y)))

a__plus(0, Y) → mark(Y)

mark(square(X)) → a__square(mark(X))

mark(times(X1, X2)) → a__times(mark(X1), mark(X2))

a__square(X) → a__times(mark(X), mark(X))

a__times(X1, X2) → times(X1, X2)

a__plus(X1, X2) → plus(X1, X2)

a__square(X) → square(X)

mark(cons2(X1, X2)) → cons2(X1, mark(X2))

mark(cons(X1, X2)) → cons(mark(X1), X2)

mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))

mark(rnil) → rnil

a__2ndspos(X1, X2) → 2ndspos(X1, X2)

a__from(X) → from(X)

a__pi(X) → pi(X)

a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ Narrowing
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ QDP
              ↳ Narrowing
                ↳ QDP
                  ↳ DependencyGraphProof
                    ↳ QDP
                      ↳ Narrowing
                        ↳ QDP
                          ↳ DependencyGraphProof
                            ↳ QDP
                              ↳ Narrowing
                                ↳ QDP
                                  ↳ DependencyGraphProof
                                    ↳ QDP
                                      ↳ Narrowing
                                        ↳ QDP
                                          ↳ DependencyGraphProof
                                            ↳ QDP
                                              ↳ Narrowing
                                                ↳ QDP
                                                  ↳ DependencyGraphProof
                                                    ↳ QDP
                                                      ↳ Narrowing
                                                        ↳ QDP
                                                          ↳ DependencyGraphProof
                                                            ↳ QDP
                                                              ↳ Narrowing
                                                                ↳ QDP
                                                                  ↳ DependencyGraphProof
                                                                    ↳ QDP
                                                                      ↳ Narrowing
                                                                        ↳ QDP
                                                                          ↳ DependencyGraphProof
                                                                            ↳ QDP
                                                                              ↳ Narrowing
                                                                                ↳ QDP
                                                                                  ↳ DependencyGraphProof
                                                                                    ↳ QDP
                                                                                      ↳ Narrowing
                                                                                        ↳ QDP
                                                                                          ↳ DependencyGraphProof
                                                                                            ↳ QDP
                                                                                              ↳ Narrowing
                                                                                                ↳ QDP
                                                                                                  ↳ DependencyGraphProof
                                                                                                    ↳ QDP
                                                                                                      ↳ Narrowing
                                                                                                        ↳ QDP
                                                                                                          ↳ DependencyGraphProof
                                                                                                            ↳ QDP
                                                                                                              ↳ Narrowing
                                                                                                                ↳ QDP
                                                                                                                  ↳ DependencyGraphProof
                                                                                                                    ↳ QDP
                                                                                                                      ↳ Narrowing
                                                                                                                        ↳ QDP
                                                                                                                          ↳ DependencyGraphProof
                                                                                                                            ↳ QDP
                                                                                                                              ↳ Narrowing
                                                                                                                                ↳ QDP
                                                                                                                                  ↳ DependencyGraphProof
                                                                                                                                    ↳ QDP
                                                                                                                                      ↳ Narrowing
                                                                                                                                        ↳ QDP
                                                                                                                                          ↳ DependencyGraphProof
                                                                                                                                            ↳ QDP
                                                                                                                                              ↳ Narrowing
                                                                                                                                                ↳ QDP
                                                                                                                                                  ↳ DependencyGraphProof
                                                                                                                                                    ↳ QDP
                                                                                                                                                      ↳ Narrowing
                                                                                                                                                        ↳ QDP
                                                                                                                                                          ↳ DependencyGraphProof
                                                                                                                                                            ↳ QDP
                                                                                                                                                              ↳ Narrowing
                                                                                                                                                                ↳ QDP
                                                                                                                                                                  ↳ DependencyGraphProof
                                                                                                                                                                    ↳ QDP
                                                                                                                                                                      ↳ Narrowing
                                                                                                                                                                        ↳ QDP
                                                                                                                                                                          ↳ DependencyGraphProof
                                                                                                                                                                            ↳ QDP
                                                                                                                                                                              ↳ Narrowing
                                                                                                                                                                                ↳ QDP
                                                                                                                                                                                  ↳ DependencyGraphProof
                                                                                                                                                                                    ↳ QDP
                                                                                                                                                                                      ↳ Narrowing
                                                                                                                                                                                        ↳ QDP
                                                                                                                                                                                          ↳ DependencyGraphProof
                                                                                                                                                                                            ↳ QDP
                                                                                                                                                                                              ↳ QDPOrderProof
                                                                                                                                                                                                ↳ QDP
                                                                                                                                                                                                  ↳ QDPOrderProof
                                                                                                                                                                                                    ↳ QDP
                                                                                                                                                                                                      ↳ QDPOrderProof
                                                                                                                                                                                                        ↳ QDP
                                                                                                                                                                                                          ↳ QDPOrderProof
                                                                                                                                                                                                            ↳ QDP
                                                                                                                                                                                                              ↳ QDPOrderProof
                                                                                                                                                                                                                ↳ QDP
                                                                                                                                                                                                                  ↳ QDPOrderProof

Q DP problem:
The TRS P consists of the following rules:

MARK(2ndspos(square(y0), plus(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(plus(y0, y1), square(x0))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__square(mark(x0)))

A__PI(times(x0, x1)) → A__2NDSPOS(a__times(mark(x0), mark(x1)), a__from(0))

A__2NDSNEG(s(N), cons(X, Z)) → A__2NDSNEG(s(mark(N)), cons2(X, mark(Z)))

MARK(2ndsneg(s(y0), square(x0))) → A__2NDSNEG(s(mark(y0)), a__square(mark(x0)))

MARK(2ndspos(X1, X2)) → MARK(X2)

MARK(times(times(x0, x1), y1)) → A__TIMES(a__times(mark(x0), mark(x1)), mark(y1))

MARK(plus(plus(x0, x1), y1)) → A__PLUS(a__plus(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(s(y0), from(x0))) → A__2NDSPOS(s(mark(y0)), a__from(mark(x0)))

A__2NDSNEG(s(N), cons(X, Z)) → MARK(Z)

A__PLUS(s(pi(x0)), y1) → A__PLUS(a__pi(mark(x0)), mark(y1))

A__TIMES(s(X), Y) → MARK(Y)

A__2NDSPOS(s(y0), cons(y1, plus(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__plus(mark(x0), mark(x1))))

A__2NDSNEG(s(N), cons(X, Z)) → MARK(N)

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → A__2NDSNEG(mark(N), mark(Z))

MARK(times(2ndsneg(x0, x1), y1)) → A__TIMES(a__2ndsneg(mark(x0), mark(x1)), mark(y1))

MARK(2ndsneg(times(y0, y1), times(x0, x1))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndsneg(2ndsneg(y0, y1), square(x0))) → A__2NDSNEG(a__2ndsneg(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndspos(times(y0, y1), times(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

A__TIMES(s(X), Y) → A__PLUS(mark(Y), a__times(mark(X), mark(Y)))

MARK(2ndspos(square(y0), cons2(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), cons2(x0, mark(x1)))

MARK(plus(square(x0), y1)) → A__PLUS(a__square(mark(x0)), mark(y1))

MARK(plus(s(x0), y1)) → A__PLUS(s(mark(x0)), mark(y1))

MARK(2ndspos(square(y0), cons(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), cons(mark(x0), x1))

A__PI(plus(x0, x1)) → A__2NDSPOS(a__plus(mark(x0), mark(x1)), a__from(0))

MARK(2ndsneg(times(y0, y1), cons2(x0, x1))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(cons2(X1, X2)) → MARK(X2)

MARK(2ndsneg(square(y0), cons2(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), cons2(x0, mark(x1)))

A__PLUS(s(square(x0)), y1) → A__PLUS(a__square(mark(x0)), mark(y1))

MARK(2ndspos(X1, X2)) → MARK(X1)

MARK(s(X)) → MARK(X)

MARK(2ndsneg(square(y0), plus(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(2ndsneg(plus(y0, y1), square(x0))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndspos(plus(y0, y1), cons2(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(pi(X)) → MARK(X)

MARK(2ndsneg(s(y0), from(x0))) → A__2NDSNEG(s(mark(y0)), a__from(mark(x0)))

MARK(plus(X1, X2)) → MARK(X2)

A__2NDSPOS(s(N), cons(X, Z)) → MARK(Z)

MARK(pi(X)) → A__PI(mark(X))

MARK(2ndsneg(times(y0, y1), square(x0))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndsneg(times(y0, y1), from(x0))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndsneg(square(y0), times(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(plus(y0, y1), cons(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), cons(mark(x0), x1))

MARK(plus(2ndsneg(x0, x1), y1)) → A__PLUS(a__2ndsneg(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(times(y0, y1), cons2(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(2ndsneg(X1, X2)) → MARK(X2)

MARK(2ndsneg(square(y0), from(x0))) → A__2NDSNEG(a__square(mark(y0)), a__from(mark(x0)))

MARK(2ndspos(square(y0), times(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(times(y0, y1), square(x0))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndsneg(s(y0), cons(x0, x1))) → A__2NDSNEG(s(mark(y0)), cons(mark(x0), x1))

MARK(times(square(x0), y1)) → A__TIMES(a__square(mark(x0)), mark(y1))

MARK(from(X)) → A__FROM(mark(X))

MARK(2ndspos(s(y0), plus(x0, x1))) → A__2NDSPOS(s(mark(y0)), a__plus(mark(x0), mark(x1)))

A__PI(X) → A__FROM(0)

A__PLUS(s(2ndsneg(x0, x1)), y1) → A__PLUS(a__2ndsneg(mark(x0), mark(x1)), mark(y1))

MARK(plus(X1, X2)) → MARK(X1)

A__2NDSPOS(s(y0), cons(y1, times(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__times(mark(x0), mark(x1))))

MARK(2ndspos(s(y0), cons(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons(mark(x0), x1))

A__PI(X) → MARK(X)

MARK(times(s(x0), y1)) → A__TIMES(s(mark(x0)), mark(y1))

MARK(2ndsneg(plus(y0, y1), cons(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), cons(mark(x0), x1))

A__PLUS(s(X), Y) → MARK(Y)

A__PLUS(s(2ndspos(x0, x1)), y1) → A__PLUS(a__2ndspos(mark(x0), mark(x1)), mark(y1))

A__TIMES(s(X), Y) → MARK(X)

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(N)

MARK(2ndsneg(s(y0), cons2(x0, x1))) → A__2NDSNEG(s(mark(y0)), cons2(x0, mark(x1)))

MARK(plus(2ndspos(x0, x1), y1)) → A__PLUS(a__2ndspos(mark(x0), mark(x1)), mark(y1))

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(N)

MARK(cons(X1, X2)) → MARK(X1)

MARK(2ndsneg(plus(y0, y1), from(x0))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(times(X1, X2)) → MARK(X2)

MARK(2ndspos(times(y0, y1), cons(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), cons(mark(x0), x1))

A__PI(square(x0)) → A__2NDSPOS(a__square(mark(x0)), a__from(0))

MARK(2ndsneg(square(y0), cons(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), cons(mark(x0), x1))

MARK(2ndspos(times(y0, y1), plus(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(plus(y0, y1), times(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(s(y0), times(x0, x1))) → A__2NDSPOS(s(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndsneg(X1, X2)) → MARK(X1)

MARK(2ndsneg(square(y0), square(x0))) → A__2NDSNEG(a__square(mark(y0)), a__square(mark(x0)))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(Y)

MARK(plus(pi(x0), y1)) → A__PLUS(a__pi(mark(x0)), mark(y1))

MARK(rcons(X1, X2)) → MARK(X1)

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(Z)

MARK(square(X)) → MARK(X)

A__PI(s(x0)) → A__2NDSPOS(s(mark(x0)), a__from(0))

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(Y)

A__FROM(X) → MARK(X)

MARK(2ndsneg(s(y0), plus(x0, x1))) → A__2NDSNEG(s(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(times(plus(x0, x1), y1)) → A__TIMES(a__plus(mark(x0), mark(x1)), mark(y1))

A__2NDSPOS(s(y0), cons(y1, from(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__from(mark(x0))))

MARK(rcons(X1, X2)) → MARK(X2)

MARK(2ndspos(s(y0), square(x0))) → A__2NDSPOS(s(mark(y0)), a__square(mark(x0)))

MARK(2ndsneg(2ndsneg(y0, y1), from(x0))) → A__2NDSNEG(a__2ndsneg(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndspos(s(y0), cons2(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(x0, mark(x1)))

A__TIMES(s(X), Y) → A__TIMES(mark(X), mark(Y))

MARK(2ndsneg(plus(y0, y1), cons2(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), cons2(x0, mark(x1)))

A__PLUS(s(X), Y) → MARK(X)

A__2NDSPOS(s(y0), cons(y1, cons(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, cons(mark(x0), x1)))

MARK(times(X1, X2)) → MARK(X1)

MARK(plus(times(x0, x1), y1)) → A__PLUS(a__times(mark(x0), mark(x1)), mark(y1))

A__SQUARE(X) → MARK(X)

MARK(negrecip(X)) → MARK(X)

A__2NDSPOS(s(N), cons(X, Z)) → MARK(N)

MARK(2ndsneg(2ndsneg(y0, y1), plus(x0, x1))) → A__2NDSNEG(a__2ndsneg(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

A__PLUS(s(from(x0)), y1) → A__PLUS(a__from(mark(x0)), mark(y1))

A__PLUS(0, Y) → MARK(Y)

MARK(posrecip(X)) → MARK(X)

MARK(2ndspos(square(y0), from(x0))) → A__2NDSPOS(a__square(mark(y0)), a__from(mark(x0)))

MARK(2ndspos(times(y0, y1), from(x0))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__from(mark(x0)))

A__PLUS(s(times(x0, x1)), y1) → A__PLUS(a__times(mark(x0), mark(x1)), mark(y1))

A__2NDSPOS(s(y0), cons(y1, square(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__square(mark(x0))))

A__PLUS(s(0), y1) → A__PLUS(0, mark(y1))

MARK(plus(0, y1)) → A__PLUS(0, mark(y1))

MARK(from(X)) → MARK(X)

MARK(2ndsneg(s(y0), times(x0, x1))) → A__2NDSNEG(s(mark(y0)), a__times(mark(x0), mark(x1)))

A__SQUARE(X) → A__TIMES(mark(X), mark(X))

MARK(2ndspos(square(y0), square(x0))) → A__2NDSPOS(a__square(mark(y0)), a__square(mark(x0)))

A__PLUS(s(plus(x0, x1)), y1) → A__PLUS(a__plus(mark(x0), mark(x1)), mark(y1))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(Z)

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → A__2NDSPOS(mark(N), mark(Z))

MARK(2ndsneg(plus(y0, y1), plus(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(plus(from(x0), y1)) → A__PLUS(a__from(mark(x0)), mark(y1))

MARK(square(X)) → A__SQUARE(mark(X))

MARK(2ndspos(plus(y0, y1), from(x0))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndsneg(plus(y0, y1), times(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndsneg(times(y0, y1), plus(x0, x1))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(plus(y0, y1), plus(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(2ndsneg(2ndsneg(y0, y1), times(x0, x1))) → A__2NDSNEG(a__2ndsneg(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndsneg(2ndsneg(y0, y1), cons2(x0, x1))) → A__2NDSNEG(a__2ndsneg(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(2ndsneg(times(y0, y1), cons(x0, x1))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), cons(mark(x0), x1))

A__PLUS(s(s(x0)), y1) → A__PLUS(s(mark(x0)), mark(y1))

MARK(2ndsneg(2ndsneg(y0, y1), cons(x0, x1))) → A__2NDSNEG(a__2ndsneg(mark(y0), mark(y1)), cons(mark(x0), x1))

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))

a__2ndspos(0, Z) → rnil

a__2ndspos(s(N), cons(X, Z)) → a__2ndspos(s(mark(N)), cons2(X, mark(Z)))

a__2ndspos(s(N), cons2(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))

a__2ndsneg(0, Z) → rnil

a__2ndsneg(s(N), cons(X, Z)) → a__2ndsneg(s(mark(N)), cons2(X, mark(Z)))

a__2ndsneg(s(N), cons2(X, cons(Y, Z))) → rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z)))

a__pi(X) → a__2ndspos(mark(X), a__from(0))

a__plus(0, Y) → mark(Y)

a__plus(s(X), Y) → s(a__plus(mark(X), mark(Y)))

a__times(0, Y) → 0

a__times(s(X), Y) → a__plus(mark(Y), a__times(mark(X), mark(Y)))

a__square(X) → a__times(mark(X), mark(X))

mark(from(X)) → a__from(mark(X))

mark(2ndspos(X1, X2)) → a__2ndspos(mark(X1), mark(X2))

mark(2ndsneg(X1, X2)) → a__2ndsneg(mark(X1), mark(X2))

mark(pi(X)) → a__pi(mark(X))

mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))

mark(times(X1, X2)) → a__times(mark(X1), mark(X2))

mark(square(X)) → a__square(mark(X))

mark(0) → 0

mark(s(X)) → s(mark(X))

mark(posrecip(X)) → posrecip(mark(X))

mark(negrecip(X)) → negrecip(mark(X))

mark(nil) → nil

mark(cons(X1, X2)) → cons(mark(X1), X2)

mark(cons2(X1, X2)) → cons2(X1, mark(X2))

mark(rnil) → rnil

mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))

a__from(X) → from(X)

a__2ndspos(X1, X2) → 2ndspos(X1, X2)

a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)

a__pi(X) → pi(X)

a__plus(X1, X2) → plus(X1, X2)

a__times(X1, X2) → times(X1, X2)

a__square(X) → square(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

MARK(times(2ndsneg(x0, x1), y1)) → A__TIMES(a__2ndsneg(mark(x0), mark(x1)), mark(y1))
The remaining pairs can at least be oriented weakly.

MARK(2ndspos(square(y0), plus(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(plus(y0, y1), square(x0))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__square(mark(x0)))

A__PI(times(x0, x1)) → A__2NDSPOS(a__times(mark(x0), mark(x1)), a__from(0))

A__2NDSNEG(s(N), cons(X, Z)) → A__2NDSNEG(s(mark(N)), cons2(X, mark(Z)))

MARK(2ndsneg(s(y0), square(x0))) → A__2NDSNEG(s(mark(y0)), a__square(mark(x0)))

MARK(2ndspos(X1, X2)) → MARK(X2)

MARK(times(times(x0, x1), y1)) → A__TIMES(a__times(mark(x0), mark(x1)), mark(y1))

MARK(plus(plus(x0, x1), y1)) → A__PLUS(a__plus(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(s(y0), from(x0))) → A__2NDSPOS(s(mark(y0)), a__from(mark(x0)))

A__2NDSNEG(s(N), cons(X, Z)) → MARK(Z)

A__PLUS(s(pi(x0)), y1) → A__PLUS(a__pi(mark(x0)), mark(y1))

A__TIMES(s(X), Y) → MARK(Y)

A__2NDSPOS(s(y0), cons(y1, plus(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__plus(mark(x0), mark(x1))))

A__2NDSNEG(s(N), cons(X, Z)) → MARK(N)

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → A__2NDSNEG(mark(N), mark(Z))

MARK(2ndsneg(times(y0, y1), times(x0, x1))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndsneg(2ndsneg(y0, y1), square(x0))) → A__2NDSNEG(a__2ndsneg(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndspos(times(y0, y1), times(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

A__TIMES(s(X), Y) → A__PLUS(mark(Y), a__times(mark(X), mark(Y)))

MARK(2ndspos(square(y0), cons2(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), cons2(x0, mark(x1)))

MARK(plus(square(x0), y1)) → A__PLUS(a__square(mark(x0)), mark(y1))

MARK(plus(s(x0), y1)) → A__PLUS(s(mark(x0)), mark(y1))

MARK(2ndspos(square(y0), cons(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), cons(mark(x0), x1))

A__PI(plus(x0, x1)) → A__2NDSPOS(a__plus(mark(x0), mark(x1)), a__from(0))

MARK(2ndsneg(times(y0, y1), cons2(x0, x1))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(cons2(X1, X2)) → MARK(X2)

MARK(2ndsneg(square(y0), cons2(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), cons2(x0, mark(x1)))

A__PLUS(s(square(x0)), y1) → A__PLUS(a__square(mark(x0)), mark(y1))

MARK(2ndspos(X1, X2)) → MARK(X1)

MARK(s(X)) → MARK(X)

MARK(2ndsneg(square(y0), plus(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(2ndsneg(plus(y0, y1), square(x0))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndspos(plus(y0, y1), cons2(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(pi(X)) → MARK(X)

MARK(2ndsneg(s(y0), from(x0))) → A__2NDSNEG(s(mark(y0)), a__from(mark(x0)))

MARK(plus(X1, X2)) → MARK(X2)

A__2NDSPOS(s(N), cons(X, Z)) → MARK(Z)

MARK(pi(X)) → A__PI(mark(X))

MARK(2ndsneg(times(y0, y1), square(x0))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndsneg(times(y0, y1), from(x0))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndsneg(square(y0), times(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(plus(y0, y1), cons(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), cons(mark(x0), x1))

MARK(plus(2ndsneg(x0, x1), y1)) → A__PLUS(a__2ndsneg(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(times(y0, y1), cons2(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(2ndsneg(X1, X2)) → MARK(X2)

MARK(2ndsneg(square(y0), from(x0))) → A__2NDSNEG(a__square(mark(y0)), a__from(mark(x0)))

MARK(2ndspos(square(y0), times(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(times(y0, y1), square(x0))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndsneg(s(y0), cons(x0, x1))) → A__2NDSNEG(s(mark(y0)), cons(mark(x0), x1))

MARK(times(square(x0), y1)) → A__TIMES(a__square(mark(x0)), mark(y1))

MARK(from(X)) → A__FROM(mark(X))

MARK(2ndspos(s(y0), plus(x0, x1))) → A__2NDSPOS(s(mark(y0)), a__plus(mark(x0), mark(x1)))

A__PI(X) → A__FROM(0)

A__PLUS(s(2ndsneg(x0, x1)), y1) → A__PLUS(a__2ndsneg(mark(x0), mark(x1)), mark(y1))

MARK(plus(X1, X2)) → MARK(X1)

A__2NDSPOS(s(y0), cons(y1, times(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__times(mark(x0), mark(x1))))

MARK(2ndspos(s(y0), cons(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons(mark(x0), x1))

A__PI(X) → MARK(X)

MARK(times(s(x0), y1)) → A__TIMES(s(mark(x0)), mark(y1))

MARK(2ndsneg(plus(y0, y1), cons(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), cons(mark(x0), x1))

A__PLUS(s(X), Y) → MARK(Y)

A__PLUS(s(2ndspos(x0, x1)), y1) → A__PLUS(a__2ndspos(mark(x0), mark(x1)), mark(y1))

A__TIMES(s(X), Y) → MARK(X)

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(N)

MARK(2ndsneg(s(y0), cons2(x0, x1))) → A__2NDSNEG(s(mark(y0)), cons2(x0, mark(x1)))

MARK(plus(2ndspos(x0, x1), y1)) → A__PLUS(a__2ndspos(mark(x0), mark(x1)), mark(y1))

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(N)

MARK(cons(X1, X2)) → MARK(X1)

MARK(2ndsneg(plus(y0, y1), from(x0))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(times(X1, X2)) → MARK(X2)

MARK(2ndspos(times(y0, y1), cons(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), cons(mark(x0), x1))

A__PI(square(x0)) → A__2NDSPOS(a__square(mark(x0)), a__from(0))

MARK(2ndsneg(square(y0), cons(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), cons(mark(x0), x1))

MARK(2ndspos(times(y0, y1), plus(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(plus(y0, y1), times(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(s(y0), times(x0, x1))) → A__2NDSPOS(s(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndsneg(X1, X2)) → MARK(X1)

MARK(2ndsneg(square(y0), square(x0))) → A__2NDSNEG(a__square(mark(y0)), a__square(mark(x0)))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(Y)

MARK(plus(pi(x0), y1)) → A__PLUS(a__pi(mark(x0)), mark(y1))

MARK(rcons(X1, X2)) → MARK(X1)

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(Z)

MARK(square(X)) → MARK(X)

A__PI(s(x0)) → A__2NDSPOS(s(mark(x0)), a__from(0))

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(Y)

A__FROM(X) → MARK(X)

MARK(2ndsneg(s(y0), plus(x0, x1))) → A__2NDSNEG(s(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(times(plus(x0, x1), y1)) → A__TIMES(a__plus(mark(x0), mark(x1)), mark(y1))

A__2NDSPOS(s(y0), cons(y1, from(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__from(mark(x0))))

MARK(rcons(X1, X2)) → MARK(X2)

MARK(2ndspos(s(y0), square(x0))) → A__2NDSPOS(s(mark(y0)), a__square(mark(x0)))

MARK(2ndsneg(2ndsneg(y0, y1), from(x0))) → A__2NDSNEG(a__2ndsneg(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndspos(s(y0), cons2(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(x0, mark(x1)))

A__TIMES(s(X), Y) → A__TIMES(mark(X), mark(Y))

MARK(2ndsneg(plus(y0, y1), cons2(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), cons2(x0, mark(x1)))

A__PLUS(s(X), Y) → MARK(X)

A__2NDSPOS(s(y0), cons(y1, cons(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, cons(mark(x0), x1)))

MARK(times(X1, X2)) → MARK(X1)

MARK(plus(times(x0, x1), y1)) → A__PLUS(a__times(mark(x0), mark(x1)), mark(y1))

A__SQUARE(X) → MARK(X)

MARK(negrecip(X)) → MARK(X)

A__2NDSPOS(s(N), cons(X, Z)) → MARK(N)

MARK(2ndsneg(2ndsneg(y0, y1), plus(x0, x1))) → A__2NDSNEG(a__2ndsneg(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

A__PLUS(s(from(x0)), y1) → A__PLUS(a__from(mark(x0)), mark(y1))

A__PLUS(0, Y) → MARK(Y)

MARK(posrecip(X)) → MARK(X)

MARK(2ndspos(square(y0), from(x0))) → A__2NDSPOS(a__square(mark(y0)), a__from(mark(x0)))

MARK(2ndspos(times(y0, y1), from(x0))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__from(mark(x0)))

A__PLUS(s(times(x0, x1)), y1) → A__PLUS(a__times(mark(x0), mark(x1)), mark(y1))

A__2NDSPOS(s(y0), cons(y1, square(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__square(mark(x0))))

A__PLUS(s(0), y1) → A__PLUS(0, mark(y1))

MARK(plus(0, y1)) → A__PLUS(0, mark(y1))

MARK(from(X)) → MARK(X)

MARK(2ndsneg(s(y0), times(x0, x1))) → A__2NDSNEG(s(mark(y0)), a__times(mark(x0), mark(x1)))

A__SQUARE(X) → A__TIMES(mark(X), mark(X))

MARK(2ndspos(square(y0), square(x0))) → A__2NDSPOS(a__square(mark(y0)), a__square(mark(x0)))

A__PLUS(s(plus(x0, x1)), y1) → A__PLUS(a__plus(mark(x0), mark(x1)), mark(y1))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(Z)

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → A__2NDSPOS(mark(N), mark(Z))

MARK(2ndsneg(plus(y0, y1), plus(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(plus(from(x0), y1)) → A__PLUS(a__from(mark(x0)), mark(y1))

MARK(square(X)) → A__SQUARE(mark(X))

MARK(2ndspos(plus(y0, y1), from(x0))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndsneg(plus(y0, y1), times(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndsneg(times(y0, y1), plus(x0, x1))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(plus(y0, y1), plus(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(2ndsneg(2ndsneg(y0, y1), times(x0, x1))) → A__2NDSNEG(a__2ndsneg(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndsneg(2ndsneg(y0, y1), cons2(x0, x1))) → A__2NDSNEG(a__2ndsneg(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(2ndsneg(times(y0, y1), cons(x0, x1))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), cons(mark(x0), x1))

A__PLUS(s(s(x0)), y1) → A__PLUS(s(mark(x0)), mark(y1))

MARK(2ndsneg(2ndsneg(y0, y1), cons(x0, x1))) → A__2NDSNEG(a__2ndsneg(mark(y0), mark(y1)), cons(mark(x0), x1))
Used ordering:  Polynomial interpretation [25]:

POL(0) = 0   

POL(2ndsneg(x1, x2)) = 0   

POL(2ndspos(x1, x2)) = 0   

POL(A__2NDSNEG(x1, x2)) = 1   

POL(A__2NDSPOS(x1, x2)) = x2   

POL(A__FROM(x1)) = 1   

POL(A__PI(x1)) = 1   

POL(A__PLUS(x1, x2)) = 1   

POL(A__SQUARE(x1)) = 1   

POL(A__TIMES(x1, x2)) = x1   

POL(MARK(x1)) = 1   

POL(a__2ndsneg(x1, x2)) = 0   

POL(a__2ndspos(x1, x2)) = x1   

POL(a__from(x1)) = 1   

POL(a__pi(x1)) = 1   

POL(a__plus(x1, x2)) = 1   

POL(a__square(x1)) = 1   

POL(a__times(x1, x2)) = 1   

POL(cons(x1, x2)) = 1   

POL(cons2(x1, x2)) = 1   

POL(from(x1)) = 0   

POL(mark(x1)) = 1   

POL(negrecip(x1)) = 0   

POL(nil) = 0   

POL(pi(x1)) = 0   

POL(plus(x1, x2)) = 1   

POL(posrecip(x1)) = 0   

POL(rcons(x1, x2)) = 0   

POL(rnil) = 0   

POL(s(x1)) = 1   

POL(square(x1)) = 0   

POL(times(x1, x2)) = 1   

The following usable rules [17] were oriented:

a__from(X) → cons(mark(X), from(s(X)))

a__pi(X) → a__2ndspos(mark(X), a__from(0))

a__2ndsneg(s(N), cons(X, Z)) → a__2ndsneg(s(mark(N)), cons2(X, mark(Z)))

a__2ndsneg(s(N), cons2(X, cons(Y, Z))) → rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z)))

a__2ndspos(s(N), cons2(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))

a__2ndsneg(0, Z) → rnil

a__2ndspos(0, Z) → rnil

a__2ndspos(s(N), cons(X, Z)) → a__2ndspos(s(mark(N)), cons2(X, mark(Z)))

mark(pi(X)) → a__pi(mark(X))

mark(2ndsneg(X1, X2)) → a__2ndsneg(mark(X1), mark(X2))

mark(2ndspos(X1, X2)) → a__2ndspos(mark(X1), mark(X2))

mark(from(X)) → a__from(mark(X))

a__times(0, Y) → 0

a__plus(s(X), Y) → s(a__plus(mark(X), mark(Y)))

mark(nil) → nil

mark(negrecip(X)) → negrecip(mark(X))

mark(posrecip(X)) → posrecip(mark(X))

mark(s(X)) → s(mark(X))

mark(0) → 0

mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))

a__times(s(X), Y) → a__plus(mark(Y), a__times(mark(X), mark(Y)))

a__plus(0, Y) → mark(Y)

mark(square(X)) → a__square(mark(X))

mark(times(X1, X2)) → a__times(mark(X1), mark(X2))

a__square(X) → a__times(mark(X), mark(X))

a__times(X1, X2) → times(X1, X2)

a__plus(X1, X2) → plus(X1, X2)

a__square(X) → square(X)

mark(cons2(X1, X2)) → cons2(X1, mark(X2))

mark(cons(X1, X2)) → cons(mark(X1), X2)

mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))

mark(rnil) → rnil

a__2ndspos(X1, X2) → 2ndspos(X1, X2)

a__from(X) → from(X)

a__pi(X) → pi(X)

a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ Narrowing
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ QDP
              ↳ Narrowing
                ↳ QDP
                  ↳ DependencyGraphProof
                    ↳ QDP
                      ↳ Narrowing
                        ↳ QDP
                          ↳ DependencyGraphProof
                            ↳ QDP
                              ↳ Narrowing
                                ↳ QDP
                                  ↳ DependencyGraphProof
                                    ↳ QDP
                                      ↳ Narrowing
                                        ↳ QDP
                                          ↳ DependencyGraphProof
                                            ↳ QDP
                                              ↳ Narrowing
                                                ↳ QDP
                                                  ↳ DependencyGraphProof
                                                    ↳ QDP
                                                      ↳ Narrowing
                                                        ↳ QDP
                                                          ↳ DependencyGraphProof
                                                            ↳ QDP
                                                              ↳ Narrowing
                                                                ↳ QDP
                                                                  ↳ DependencyGraphProof
                                                                    ↳ QDP
                                                                      ↳ Narrowing
                                                                        ↳ QDP
                                                                          ↳ DependencyGraphProof
                                                                            ↳ QDP
                                                                              ↳ Narrowing
                                                                                ↳ QDP
                                                                                  ↳ DependencyGraphProof
                                                                                    ↳ QDP
                                                                                      ↳ Narrowing
                                                                                        ↳ QDP
                                                                                          ↳ DependencyGraphProof
                                                                                            ↳ QDP
                                                                                              ↳ Narrowing
                                                                                                ↳ QDP
                                                                                                  ↳ DependencyGraphProof
                                                                                                    ↳ QDP
                                                                                                      ↳ Narrowing
                                                                                                        ↳ QDP
                                                                                                          ↳ DependencyGraphProof
                                                                                                            ↳ QDP
                                                                                                              ↳ Narrowing
                                                                                                                ↳ QDP
                                                                                                                  ↳ DependencyGraphProof
                                                                                                                    ↳ QDP
                                                                                                                      ↳ Narrowing
                                                                                                                        ↳ QDP
                                                                                                                          ↳ DependencyGraphProof
                                                                                                                            ↳ QDP
                                                                                                                              ↳ Narrowing
                                                                                                                                ↳ QDP
                                                                                                                                  ↳ DependencyGraphProof
                                                                                                                                    ↳ QDP
                                                                                                                                      ↳ Narrowing
                                                                                                                                        ↳ QDP
                                                                                                                                          ↳ DependencyGraphProof
                                                                                                                                            ↳ QDP
                                                                                                                                              ↳ Narrowing
                                                                                                                                                ↳ QDP
                                                                                                                                                  ↳ DependencyGraphProof
                                                                                                                                                    ↳ QDP
                                                                                                                                                      ↳ Narrowing
                                                                                                                                                        ↳ QDP
                                                                                                                                                          ↳ DependencyGraphProof
                                                                                                                                                            ↳ QDP
                                                                                                                                                              ↳ Narrowing
                                                                                                                                                                ↳ QDP
                                                                                                                                                                  ↳ DependencyGraphProof
                                                                                                                                                                    ↳ QDP
                                                                                                                                                                      ↳ Narrowing
                                                                                                                                                                        ↳ QDP
                                                                                                                                                                          ↳ DependencyGraphProof
                                                                                                                                                                            ↳ QDP
                                                                                                                                                                              ↳ Narrowing
                                                                                                                                                                                ↳ QDP
                                                                                                                                                                                  ↳ DependencyGraphProof
                                                                                                                                                                                    ↳ QDP
                                                                                                                                                                                      ↳ Narrowing
                                                                                                                                                                                        ↳ QDP
                                                                                                                                                                                          ↳ DependencyGraphProof
                                                                                                                                                                                            ↳ QDP
                                                                                                                                                                                              ↳ QDPOrderProof
                                                                                                                                                                                                ↳ QDP
                                                                                                                                                                                                  ↳ QDPOrderProof
                                                                                                                                                                                                    ↳ QDP
                                                                                                                                                                                                      ↳ QDPOrderProof
                                                                                                                                                                                                        ↳ QDP
                                                                                                                                                                                                          ↳ QDPOrderProof
                                                                                                                                                                                                            ↳ QDP
                                                                                                                                                                                                              ↳ QDPOrderProof
                                                                                                                                                                                                                ↳ QDP
                                                                                                                                                                                                                  ↳ QDPOrderProof
                                                                                                                                                                                                                    ↳ QDP
                                                                                                                                                                                                                      ↳ QDPOrderProof

Q DP problem:
The TRS P consists of the following rules:

MARK(2ndspos(plus(y0, y1), square(x0))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndspos(square(y0), plus(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__plus(mark(x0), mark(x1)))

A__PI(times(x0, x1)) → A__2NDSPOS(a__times(mark(x0), mark(x1)), a__from(0))

A__2NDSNEG(s(N), cons(X, Z)) → A__2NDSNEG(s(mark(N)), cons2(X, mark(Z)))

MARK(2ndsneg(s(y0), square(x0))) → A__2NDSNEG(s(mark(y0)), a__square(mark(x0)))

MARK(times(times(x0, x1), y1)) → A__TIMES(a__times(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(X1, X2)) → MARK(X2)

MARK(plus(plus(x0, x1), y1)) → A__PLUS(a__plus(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(s(y0), from(x0))) → A__2NDSPOS(s(mark(y0)), a__from(mark(x0)))

A__2NDSNEG(s(N), cons(X, Z)) → MARK(Z)

A__PLUS(s(pi(x0)), y1) → A__PLUS(a__pi(mark(x0)), mark(y1))

A__TIMES(s(X), Y) → MARK(Y)

A__2NDSPOS(s(y0), cons(y1, plus(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__plus(mark(x0), mark(x1))))

A__2NDSNEG(s(N), cons(X, Z)) → MARK(N)

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → A__2NDSNEG(mark(N), mark(Z))

MARK(2ndsneg(times(y0, y1), times(x0, x1))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndsneg(2ndsneg(y0, y1), square(x0))) → A__2NDSNEG(a__2ndsneg(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndspos(times(y0, y1), times(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

A__TIMES(s(X), Y) → A__PLUS(mark(Y), a__times(mark(X), mark(Y)))

MARK(2ndspos(square(y0), cons2(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), cons2(x0, mark(x1)))

MARK(plus(square(x0), y1)) → A__PLUS(a__square(mark(x0)), mark(y1))

MARK(plus(s(x0), y1)) → A__PLUS(s(mark(x0)), mark(y1))

MARK(2ndspos(square(y0), cons(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), cons(mark(x0), x1))

A__PI(plus(x0, x1)) → A__2NDSPOS(a__plus(mark(x0), mark(x1)), a__from(0))

MARK(2ndsneg(times(y0, y1), cons2(x0, x1))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(cons2(X1, X2)) → MARK(X2)

MARK(2ndsneg(square(y0), cons2(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), cons2(x0, mark(x1)))

MARK(2ndsneg(plus(y0, y1), square(x0))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndsneg(square(y0), plus(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(s(X)) → MARK(X)

MARK(2ndspos(X1, X2)) → MARK(X1)

A__PLUS(s(square(x0)), y1) → A__PLUS(a__square(mark(x0)), mark(y1))

MARK(2ndspos(plus(y0, y1), cons2(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(pi(X)) → MARK(X)

MARK(2ndsneg(s(y0), from(x0))) → A__2NDSNEG(s(mark(y0)), a__from(mark(x0)))

MARK(plus(X1, X2)) → MARK(X2)

A__2NDSPOS(s(N), cons(X, Z)) → MARK(Z)

MARK(pi(X)) → A__PI(mark(X))

MARK(2ndspos(plus(y0, y1), cons(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), cons(mark(x0), x1))

MARK(2ndsneg(square(y0), times(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndsneg(times(y0, y1), from(x0))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndsneg(times(y0, y1), square(x0))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(plus(2ndsneg(x0, x1), y1)) → A__PLUS(a__2ndsneg(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(times(y0, y1), cons2(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(2ndsneg(X1, X2)) → MARK(X2)

MARK(2ndsneg(square(y0), from(x0))) → A__2NDSNEG(a__square(mark(y0)), a__from(mark(x0)))

MARK(2ndspos(times(y0, y1), square(x0))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndspos(square(y0), times(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndsneg(s(y0), cons(x0, x1))) → A__2NDSNEG(s(mark(y0)), cons(mark(x0), x1))

MARK(times(square(x0), y1)) → A__TIMES(a__square(mark(x0)), mark(y1))

MARK(from(X)) → A__FROM(mark(X))

MARK(2ndspos(s(y0), plus(x0, x1))) → A__2NDSPOS(s(mark(y0)), a__plus(mark(x0), mark(x1)))

A__PI(X) → A__FROM(0)

A__PLUS(s(2ndsneg(x0, x1)), y1) → A__PLUS(a__2ndsneg(mark(x0), mark(x1)), mark(y1))

MARK(plus(X1, X2)) → MARK(X1)

A__2NDSPOS(s(y0), cons(y1, times(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__times(mark(x0), mark(x1))))

MARK(2ndspos(s(y0), cons(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons(mark(x0), x1))

A__PI(X) → MARK(X)

MARK(times(s(x0), y1)) → A__TIMES(s(mark(x0)), mark(y1))

MARK(2ndsneg(plus(y0, y1), cons(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), cons(mark(x0), x1))

A__PLUS(s(X), Y) → MARK(Y)

A__PLUS(s(2ndspos(x0, x1)), y1) → A__PLUS(a__2ndspos(mark(x0), mark(x1)), mark(y1))

A__TIMES(s(X), Y) → MARK(X)

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(N)

MARK(2ndsneg(s(y0), cons2(x0, x1))) → A__2NDSNEG(s(mark(y0)), cons2(x0, mark(x1)))

MARK(plus(2ndspos(x0, x1), y1)) → A__PLUS(a__2ndspos(mark(x0), mark(x1)), mark(y1))

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(N)

MARK(cons(X1, X2)) → MARK(X1)

MARK(2ndsneg(plus(y0, y1), from(x0))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(times(X1, X2)) → MARK(X2)

MARK(2ndspos(times(y0, y1), cons(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), cons(mark(x0), x1))

A__PI(square(x0)) → A__2NDSPOS(a__square(mark(x0)), a__from(0))

MARK(2ndsneg(square(y0), cons(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), cons(mark(x0), x1))

MARK(2ndspos(times(y0, y1), plus(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(plus(y0, y1), times(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(s(y0), times(x0, x1))) → A__2NDSPOS(s(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndsneg(X1, X2)) → MARK(X1)

MARK(2ndsneg(square(y0), square(x0))) → A__2NDSNEG(a__square(mark(y0)), a__square(mark(x0)))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(Y)

MARK(plus(pi(x0), y1)) → A__PLUS(a__pi(mark(x0)), mark(y1))

MARK(rcons(X1, X2)) → MARK(X1)

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(Z)

MARK(square(X)) → MARK(X)

A__PI(s(x0)) → A__2NDSPOS(s(mark(x0)), a__from(0))

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(Y)

A__FROM(X) → MARK(X)

MARK(2ndsneg(s(y0), plus(x0, x1))) → A__2NDSNEG(s(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(times(plus(x0, x1), y1)) → A__TIMES(a__plus(mark(x0), mark(x1)), mark(y1))

A__2NDSPOS(s(y0), cons(y1, from(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__from(mark(x0))))

MARK(rcons(X1, X2)) → MARK(X2)

MARK(2ndspos(s(y0), square(x0))) → A__2NDSPOS(s(mark(y0)), a__square(mark(x0)))

MARK(2ndsneg(2ndsneg(y0, y1), from(x0))) → A__2NDSNEG(a__2ndsneg(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndspos(s(y0), cons2(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(x0, mark(x1)))

A__TIMES(s(X), Y) → A__TIMES(mark(X), mark(Y))

MARK(2ndsneg(plus(y0, y1), cons2(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), cons2(x0, mark(x1)))

A__PLUS(s(X), Y) → MARK(X)

A__2NDSPOS(s(y0), cons(y1, cons(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, cons(mark(x0), x1)))

MARK(times(X1, X2)) → MARK(X1)

MARK(plus(times(x0, x1), y1)) → A__PLUS(a__times(mark(x0), mark(x1)), mark(y1))

A__SQUARE(X) → MARK(X)

A__2NDSPOS(s(N), cons(X, Z)) → MARK(N)

MARK(negrecip(X)) → MARK(X)

MARK(2ndsneg(2ndsneg(y0, y1), plus(x0, x1))) → A__2NDSNEG(a__2ndsneg(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

A__PLUS(s(from(x0)), y1) → A__PLUS(a__from(mark(x0)), mark(y1))

A__PLUS(0, Y) → MARK(Y)

MARK(posrecip(X)) → MARK(X)

MARK(2ndspos(square(y0), from(x0))) → A__2NDSPOS(a__square(mark(y0)), a__from(mark(x0)))

MARK(2ndspos(times(y0, y1), from(x0))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__from(mark(x0)))

A__PLUS(s(times(x0, x1)), y1) → A__PLUS(a__times(mark(x0), mark(x1)), mark(y1))

A__2NDSPOS(s(y0), cons(y1, square(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__square(mark(x0))))

A__PLUS(s(0), y1) → A__PLUS(0, mark(y1))

MARK(plus(0, y1)) → A__PLUS(0, mark(y1))

MARK(from(X)) → MARK(X)

MARK(2ndsneg(s(y0), times(x0, x1))) → A__2NDSNEG(s(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(square(y0), square(x0))) → A__2NDSPOS(a__square(mark(y0)), a__square(mark(x0)))

A__SQUARE(X) → A__TIMES(mark(X), mark(X))

A__PLUS(s(plus(x0, x1)), y1) → A__PLUS(a__plus(mark(x0), mark(x1)), mark(y1))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(Z)

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → A__2NDSPOS(mark(N), mark(Z))

MARK(plus(from(x0), y1)) → A__PLUS(a__from(mark(x0)), mark(y1))

MARK(2ndsneg(plus(y0, y1), plus(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(square(X)) → A__SQUARE(mark(X))

MARK(2ndspos(plus(y0, y1), from(x0))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndsneg(times(y0, y1), plus(x0, x1))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(2ndsneg(plus(y0, y1), times(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(plus(y0, y1), plus(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(2ndsneg(2ndsneg(y0, y1), cons2(x0, x1))) → A__2NDSNEG(a__2ndsneg(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(2ndsneg(2ndsneg(y0, y1), times(x0, x1))) → A__2NDSNEG(a__2ndsneg(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

A__PLUS(s(s(x0)), y1) → A__PLUS(s(mark(x0)), mark(y1))

MARK(2ndsneg(times(y0, y1), cons(x0, x1))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), cons(mark(x0), x1))

MARK(2ndsneg(2ndsneg(y0, y1), cons(x0, x1))) → A__2NDSNEG(a__2ndsneg(mark(y0), mark(y1)), cons(mark(x0), x1))

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))

a__2ndspos(0, Z) → rnil

a__2ndspos(s(N), cons(X, Z)) → a__2ndspos(s(mark(N)), cons2(X, mark(Z)))

a__2ndspos(s(N), cons2(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))

a__2ndsneg(0, Z) → rnil

a__2ndsneg(s(N), cons(X, Z)) → a__2ndsneg(s(mark(N)), cons2(X, mark(Z)))

a__2ndsneg(s(N), cons2(X, cons(Y, Z))) → rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z)))

a__pi(X) → a__2ndspos(mark(X), a__from(0))

a__plus(0, Y) → mark(Y)

a__plus(s(X), Y) → s(a__plus(mark(X), mark(Y)))

a__times(0, Y) → 0

a__times(s(X), Y) → a__plus(mark(Y), a__times(mark(X), mark(Y)))

a__square(X) → a__times(mark(X), mark(X))

mark(from(X)) → a__from(mark(X))

mark(2ndspos(X1, X2)) → a__2ndspos(mark(X1), mark(X2))

mark(2ndsneg(X1, X2)) → a__2ndsneg(mark(X1), mark(X2))

mark(pi(X)) → a__pi(mark(X))

mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))

mark(times(X1, X2)) → a__times(mark(X1), mark(X2))

mark(square(X)) → a__square(mark(X))

mark(0) → 0

mark(s(X)) → s(mark(X))

mark(posrecip(X)) → posrecip(mark(X))

mark(negrecip(X)) → negrecip(mark(X))

mark(nil) → nil

mark(cons(X1, X2)) → cons(mark(X1), X2)

mark(cons2(X1, X2)) → cons2(X1, mark(X2))

mark(rnil) → rnil

mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))

a__from(X) → from(X)

a__2ndspos(X1, X2) → 2ndspos(X1, X2)

a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)

a__pi(X) → pi(X)

a__plus(X1, X2) → plus(X1, X2)

a__times(X1, X2) → times(X1, X2)

a__square(X) → square(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

MARK(2ndsneg(2ndsneg(y0, y1), square(x0))) → A__2NDSNEG(a__2ndsneg(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndsneg(2ndsneg(y0, y1), from(x0))) → A__2NDSNEG(a__2ndsneg(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndsneg(2ndsneg(y0, y1), plus(x0, x1))) → A__2NDSNEG(a__2ndsneg(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(2ndsneg(2ndsneg(y0, y1), cons2(x0, x1))) → A__2NDSNEG(a__2ndsneg(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(2ndsneg(2ndsneg(y0, y1), times(x0, x1))) → A__2NDSNEG(a__2ndsneg(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndsneg(2ndsneg(y0, y1), cons(x0, x1))) → A__2NDSNEG(a__2ndsneg(mark(y0), mark(y1)), cons(mark(x0), x1))
The remaining pairs can at least be oriented weakly.

MARK(2ndspos(plus(y0, y1), square(x0))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndspos(square(y0), plus(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__plus(mark(x0), mark(x1)))

A__PI(times(x0, x1)) → A__2NDSPOS(a__times(mark(x0), mark(x1)), a__from(0))

A__2NDSNEG(s(N), cons(X, Z)) → A__2NDSNEG(s(mark(N)), cons2(X, mark(Z)))

MARK(2ndsneg(s(y0), square(x0))) → A__2NDSNEG(s(mark(y0)), a__square(mark(x0)))

MARK(times(times(x0, x1), y1)) → A__TIMES(a__times(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(X1, X2)) → MARK(X2)

MARK(plus(plus(x0, x1), y1)) → A__PLUS(a__plus(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(s(y0), from(x0))) → A__2NDSPOS(s(mark(y0)), a__from(mark(x0)))

A__2NDSNEG(s(N), cons(X, Z)) → MARK(Z)

A__PLUS(s(pi(x0)), y1) → A__PLUS(a__pi(mark(x0)), mark(y1))

A__TIMES(s(X), Y) → MARK(Y)

A__2NDSPOS(s(y0), cons(y1, plus(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__plus(mark(x0), mark(x1))))

A__2NDSNEG(s(N), cons(X, Z)) → MARK(N)

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → A__2NDSNEG(mark(N), mark(Z))

MARK(2ndsneg(times(y0, y1), times(x0, x1))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(times(y0, y1), times(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

A__TIMES(s(X), Y) → A__PLUS(mark(Y), a__times(mark(X), mark(Y)))

MARK(2ndspos(square(y0), cons2(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), cons2(x0, mark(x1)))

MARK(plus(square(x0), y1)) → A__PLUS(a__square(mark(x0)), mark(y1))

MARK(plus(s(x0), y1)) → A__PLUS(s(mark(x0)), mark(y1))

MARK(2ndspos(square(y0), cons(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), cons(mark(x0), x1))

A__PI(plus(x0, x1)) → A__2NDSPOS(a__plus(mark(x0), mark(x1)), a__from(0))

MARK(2ndsneg(times(y0, y1), cons2(x0, x1))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(cons2(X1, X2)) → MARK(X2)

MARK(2ndsneg(square(y0), cons2(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), cons2(x0, mark(x1)))

MARK(2ndsneg(plus(y0, y1), square(x0))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndsneg(square(y0), plus(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(s(X)) → MARK(X)

MARK(2ndspos(X1, X2)) → MARK(X1)

A__PLUS(s(square(x0)), y1) → A__PLUS(a__square(mark(x0)), mark(y1))

MARK(2ndspos(plus(y0, y1), cons2(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(pi(X)) → MARK(X)

MARK(2ndsneg(s(y0), from(x0))) → A__2NDSNEG(s(mark(y0)), a__from(mark(x0)))

MARK(plus(X1, X2)) → MARK(X2)

A__2NDSPOS(s(N), cons(X, Z)) → MARK(Z)

MARK(pi(X)) → A__PI(mark(X))

MARK(2ndspos(plus(y0, y1), cons(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), cons(mark(x0), x1))

MARK(2ndsneg(square(y0), times(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndsneg(times(y0, y1), from(x0))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndsneg(times(y0, y1), square(x0))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(plus(2ndsneg(x0, x1), y1)) → A__PLUS(a__2ndsneg(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(times(y0, y1), cons2(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(2ndsneg(X1, X2)) → MARK(X2)

MARK(2ndsneg(square(y0), from(x0))) → A__2NDSNEG(a__square(mark(y0)), a__from(mark(x0)))

MARK(2ndspos(times(y0, y1), square(x0))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndspos(square(y0), times(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndsneg(s(y0), cons(x0, x1))) → A__2NDSNEG(s(mark(y0)), cons(mark(x0), x1))

MARK(times(square(x0), y1)) → A__TIMES(a__square(mark(x0)), mark(y1))

MARK(from(X)) → A__FROM(mark(X))

MARK(2ndspos(s(y0), plus(x0, x1))) → A__2NDSPOS(s(mark(y0)), a__plus(mark(x0), mark(x1)))

A__PI(X) → A__FROM(0)

A__PLUS(s(2ndsneg(x0, x1)), y1) → A__PLUS(a__2ndsneg(mark(x0), mark(x1)), mark(y1))

MARK(plus(X1, X2)) → MARK(X1)

A__2NDSPOS(s(y0), cons(y1, times(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__times(mark(x0), mark(x1))))

MARK(2ndspos(s(y0), cons(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons(mark(x0), x1))

A__PI(X) → MARK(X)

MARK(times(s(x0), y1)) → A__TIMES(s(mark(x0)), mark(y1))

MARK(2ndsneg(plus(y0, y1), cons(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), cons(mark(x0), x1))

A__PLUS(s(X), Y) → MARK(Y)

A__PLUS(s(2ndspos(x0, x1)), y1) → A__PLUS(a__2ndspos(mark(x0), mark(x1)), mark(y1))

A__TIMES(s(X), Y) → MARK(X)

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(N)

MARK(2ndsneg(s(y0), cons2(x0, x1))) → A__2NDSNEG(s(mark(y0)), cons2(x0, mark(x1)))

MARK(plus(2ndspos(x0, x1), y1)) → A__PLUS(a__2ndspos(mark(x0), mark(x1)), mark(y1))

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(N)

MARK(cons(X1, X2)) → MARK(X1)

MARK(2ndsneg(plus(y0, y1), from(x0))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(times(X1, X2)) → MARK(X2)

MARK(2ndspos(times(y0, y1), cons(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), cons(mark(x0), x1))

A__PI(square(x0)) → A__2NDSPOS(a__square(mark(x0)), a__from(0))

MARK(2ndsneg(square(y0), cons(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), cons(mark(x0), x1))

MARK(2ndspos(times(y0, y1), plus(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(plus(y0, y1), times(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(s(y0), times(x0, x1))) → A__2NDSPOS(s(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndsneg(X1, X2)) → MARK(X1)

MARK(2ndsneg(square(y0), square(x0))) → A__2NDSNEG(a__square(mark(y0)), a__square(mark(x0)))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(Y)

MARK(plus(pi(x0), y1)) → A__PLUS(a__pi(mark(x0)), mark(y1))

MARK(rcons(X1, X2)) → MARK(X1)

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(Z)

MARK(square(X)) → MARK(X)

A__PI(s(x0)) → A__2NDSPOS(s(mark(x0)), a__from(0))

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(Y)

A__FROM(X) → MARK(X)

MARK(2ndsneg(s(y0), plus(x0, x1))) → A__2NDSNEG(s(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(times(plus(x0, x1), y1)) → A__TIMES(a__plus(mark(x0), mark(x1)), mark(y1))

A__2NDSPOS(s(y0), cons(y1, from(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__from(mark(x0))))

MARK(rcons(X1, X2)) → MARK(X2)

MARK(2ndspos(s(y0), square(x0))) → A__2NDSPOS(s(mark(y0)), a__square(mark(x0)))

MARK(2ndspos(s(y0), cons2(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(x0, mark(x1)))

A__TIMES(s(X), Y) → A__TIMES(mark(X), mark(Y))

MARK(2ndsneg(plus(y0, y1), cons2(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), cons2(x0, mark(x1)))

A__PLUS(s(X), Y) → MARK(X)

A__2NDSPOS(s(y0), cons(y1, cons(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, cons(mark(x0), x1)))

MARK(times(X1, X2)) → MARK(X1)

MARK(plus(times(x0, x1), y1)) → A__PLUS(a__times(mark(x0), mark(x1)), mark(y1))

A__SQUARE(X) → MARK(X)

A__2NDSPOS(s(N), cons(X, Z)) → MARK(N)

MARK(negrecip(X)) → MARK(X)

A__PLUS(s(from(x0)), y1) → A__PLUS(a__from(mark(x0)), mark(y1))

A__PLUS(0, Y) → MARK(Y)

MARK(posrecip(X)) → MARK(X)

MARK(2ndspos(square(y0), from(x0))) → A__2NDSPOS(a__square(mark(y0)), a__from(mark(x0)))

MARK(2ndspos(times(y0, y1), from(x0))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__from(mark(x0)))

A__PLUS(s(times(x0, x1)), y1) → A__PLUS(a__times(mark(x0), mark(x1)), mark(y1))

A__2NDSPOS(s(y0), cons(y1, square(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__square(mark(x0))))

A__PLUS(s(0), y1) → A__PLUS(0, mark(y1))

MARK(plus(0, y1)) → A__PLUS(0, mark(y1))

MARK(from(X)) → MARK(X)

MARK(2ndsneg(s(y0), times(x0, x1))) → A__2NDSNEG(s(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(square(y0), square(x0))) → A__2NDSPOS(a__square(mark(y0)), a__square(mark(x0)))

A__SQUARE(X) → A__TIMES(mark(X), mark(X))

A__PLUS(s(plus(x0, x1)), y1) → A__PLUS(a__plus(mark(x0), mark(x1)), mark(y1))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(Z)

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → A__2NDSPOS(mark(N), mark(Z))

MARK(plus(from(x0), y1)) → A__PLUS(a__from(mark(x0)), mark(y1))

MARK(2ndsneg(plus(y0, y1), plus(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(square(X)) → A__SQUARE(mark(X))

MARK(2ndspos(plus(y0, y1), from(x0))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndsneg(times(y0, y1), plus(x0, x1))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(2ndsneg(plus(y0, y1), times(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(plus(y0, y1), plus(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

A__PLUS(s(s(x0)), y1) → A__PLUS(s(mark(x0)), mark(y1))

MARK(2ndsneg(times(y0, y1), cons(x0, x1))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), cons(mark(x0), x1))
Used ordering:  Polynomial interpretation [25]:

POL(0) = 0   

POL(2ndsneg(x1, x2)) = 0   

POL(2ndspos(x1, x2)) = 0   

POL(A__2NDSNEG(x1, x2)) = x1   

POL(A__2NDSPOS(x1, x2)) = 1   

POL(A__FROM(x1)) = 1   

POL(A__PI(x1)) = 1   

POL(A__PLUS(x1, x2)) = 1   

POL(A__SQUARE(x1)) = 1   

POL(A__TIMES(x1, x2)) = 1   

POL(MARK(x1)) = 1   

POL(a__2ndsneg(x1, x2)) = 0   

POL(a__2ndspos(x1, x2)) = 0   

POL(a__from(x1)) = 0   

POL(a__pi(x1)) = 0   

POL(a__plus(x1, x2)) = 1   

POL(a__square(x1)) = 1   

POL(a__times(x1, x2)) = x1   

POL(cons(x1, x2)) = 0   

POL(cons2(x1, x2)) = 0   

POL(from(x1)) = 0   

POL(mark(x1)) = 1   

POL(negrecip(x1)) = 1   

POL(nil) = 0   

POL(pi(x1)) = 0   

POL(plus(x1, x2)) = 1   

POL(posrecip(x1)) = 0   

POL(rcons(x1, x2)) = 0   

POL(rnil) = 0   

POL(s(x1)) = 1   

POL(square(x1)) = 1   

POL(times(x1, x2)) = 0   

The following usable rules [17] were oriented:

a__from(X) → cons(mark(X), from(s(X)))

a__pi(X) → a__2ndspos(mark(X), a__from(0))

a__2ndsneg(s(N), cons(X, Z)) → a__2ndsneg(s(mark(N)), cons2(X, mark(Z)))

a__2ndsneg(s(N), cons2(X, cons(Y, Z))) → rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z)))

a__2ndspos(s(N), cons2(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))

a__2ndsneg(0, Z) → rnil

a__2ndspos(0, Z) → rnil

a__2ndspos(s(N), cons(X, Z)) → a__2ndspos(s(mark(N)), cons2(X, mark(Z)))

mark(pi(X)) → a__pi(mark(X))

mark(2ndsneg(X1, X2)) → a__2ndsneg(mark(X1), mark(X2))

mark(2ndspos(X1, X2)) → a__2ndspos(mark(X1), mark(X2))

mark(from(X)) → a__from(mark(X))

a__times(0, Y) → 0

a__plus(s(X), Y) → s(a__plus(mark(X), mark(Y)))

mark(nil) → nil

mark(negrecip(X)) → negrecip(mark(X))

mark(posrecip(X)) → posrecip(mark(X))

mark(s(X)) → s(mark(X))

mark(0) → 0

mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))

a__times(s(X), Y) → a__plus(mark(Y), a__times(mark(X), mark(Y)))

a__plus(0, Y) → mark(Y)

mark(square(X)) → a__square(mark(X))

mark(times(X1, X2)) → a__times(mark(X1), mark(X2))

a__square(X) → a__times(mark(X), mark(X))

a__times(X1, X2) → times(X1, X2)

a__plus(X1, X2) → plus(X1, X2)

a__square(X) → square(X)

mark(cons2(X1, X2)) → cons2(X1, mark(X2))

mark(cons(X1, X2)) → cons(mark(X1), X2)

mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))

mark(rnil) → rnil

a__2ndspos(X1, X2) → 2ndspos(X1, X2)

a__from(X) → from(X)

a__pi(X) → pi(X)

a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ Narrowing
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ QDP
              ↳ Narrowing
                ↳ QDP
                  ↳ DependencyGraphProof
                    ↳ QDP
                      ↳ Narrowing
                        ↳ QDP
                          ↳ DependencyGraphProof
                            ↳ QDP
                              ↳ Narrowing
                                ↳ QDP
                                  ↳ DependencyGraphProof
                                    ↳ QDP
                                      ↳ Narrowing
                                        ↳ QDP
                                          ↳ DependencyGraphProof
                                            ↳ QDP
                                              ↳ Narrowing
                                                ↳ QDP
                                                  ↳ DependencyGraphProof
                                                    ↳ QDP
                                                      ↳ Narrowing
                                                        ↳ QDP
                                                          ↳ DependencyGraphProof
                                                            ↳ QDP
                                                              ↳ Narrowing
                                                                ↳ QDP
                                                                  ↳ DependencyGraphProof
                                                                    ↳ QDP
                                                                      ↳ Narrowing
                                                                        ↳ QDP
                                                                          ↳ DependencyGraphProof
                                                                            ↳ QDP
                                                                              ↳ Narrowing
                                                                                ↳ QDP
                                                                                  ↳ DependencyGraphProof
                                                                                    ↳ QDP
                                                                                      ↳ Narrowing
                                                                                        ↳ QDP
                                                                                          ↳ DependencyGraphProof
                                                                                            ↳ QDP
                                                                                              ↳ Narrowing
                                                                                                ↳ QDP
                                                                                                  ↳ DependencyGraphProof
                                                                                                    ↳ QDP
                                                                                                      ↳ Narrowing
                                                                                                        ↳ QDP
                                                                                                          ↳ DependencyGraphProof
                                                                                                            ↳ QDP
                                                                                                              ↳ Narrowing
                                                                                                                ↳ QDP
                                                                                                                  ↳ DependencyGraphProof
                                                                                                                    ↳ QDP
                                                                                                                      ↳ Narrowing
                                                                                                                        ↳ QDP
                                                                                                                          ↳ DependencyGraphProof
                                                                                                                            ↳ QDP
                                                                                                                              ↳ Narrowing
                                                                                                                                ↳ QDP
                                                                                                                                  ↳ DependencyGraphProof
                                                                                                                                    ↳ QDP
                                                                                                                                      ↳ Narrowing
                                                                                                                                        ↳ QDP
                                                                                                                                          ↳ DependencyGraphProof
                                                                                                                                            ↳ QDP
                                                                                                                                              ↳ Narrowing
                                                                                                                                                ↳ QDP
                                                                                                                                                  ↳ DependencyGraphProof
                                                                                                                                                    ↳ QDP
                                                                                                                                                      ↳ Narrowing
                                                                                                                                                        ↳ QDP
                                                                                                                                                          ↳ DependencyGraphProof
                                                                                                                                                            ↳ QDP
                                                                                                                                                              ↳ Narrowing
                                                                                                                                                                ↳ QDP
                                                                                                                                                                  ↳ DependencyGraphProof
                                                                                                                                                                    ↳ QDP
                                                                                                                                                                      ↳ Narrowing
                                                                                                                                                                        ↳ QDP
                                                                                                                                                                          ↳ DependencyGraphProof
                                                                                                                                                                            ↳ QDP
                                                                                                                                                                              ↳ Narrowing
                                                                                                                                                                                ↳ QDP
                                                                                                                                                                                  ↳ DependencyGraphProof
                                                                                                                                                                                    ↳ QDP
                                                                                                                                                                                      ↳ Narrowing
                                                                                                                                                                                        ↳ QDP
                                                                                                                                                                                          ↳ DependencyGraphProof
                                                                                                                                                                                            ↳ QDP
                                                                                                                                                                                              ↳ QDPOrderProof
                                                                                                                                                                                                ↳ QDP
                                                                                                                                                                                                  ↳ QDPOrderProof
                                                                                                                                                                                                    ↳ QDP
                                                                                                                                                                                                      ↳ QDPOrderProof
                                                                                                                                                                                                        ↳ QDP
                                                                                                                                                                                                          ↳ QDPOrderProof
                                                                                                                                                                                                            ↳ QDP
                                                                                                                                                                                                              ↳ QDPOrderProof
                                                                                                                                                                                                                ↳ QDP
                                                                                                                                                                                                                  ↳ QDPOrderProof
                                                                                                                                                                                                                    ↳ QDP
                                                                                                                                                                                                                      ↳ QDPOrderProof
                                                                                                                                                                                                                        ↳ QDP
                                                                                                                                                                                                                          ↳ QDPOrderProof

Q DP problem:
The TRS P consists of the following rules:

MARK(2ndspos(square(y0), plus(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(plus(y0, y1), square(x0))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__square(mark(x0)))

A__PI(times(x0, x1)) → A__2NDSPOS(a__times(mark(x0), mark(x1)), a__from(0))

A__2NDSNEG(s(N), cons(X, Z)) → A__2NDSNEG(s(mark(N)), cons2(X, mark(Z)))

MARK(2ndsneg(s(y0), square(x0))) → A__2NDSNEG(s(mark(y0)), a__square(mark(x0)))

MARK(2ndspos(X1, X2)) → MARK(X2)

MARK(times(times(x0, x1), y1)) → A__TIMES(a__times(mark(x0), mark(x1)), mark(y1))

MARK(plus(plus(x0, x1), y1)) → A__PLUS(a__plus(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(s(y0), from(x0))) → A__2NDSPOS(s(mark(y0)), a__from(mark(x0)))

A__2NDSNEG(s(N), cons(X, Z)) → MARK(Z)

A__PLUS(s(pi(x0)), y1) → A__PLUS(a__pi(mark(x0)), mark(y1))

A__TIMES(s(X), Y) → MARK(Y)

A__2NDSPOS(s(y0), cons(y1, plus(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__plus(mark(x0), mark(x1))))

A__2NDSNEG(s(N), cons(X, Z)) → MARK(N)

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → A__2NDSNEG(mark(N), mark(Z))

MARK(2ndsneg(times(y0, y1), times(x0, x1))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(times(y0, y1), times(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

A__TIMES(s(X), Y) → A__PLUS(mark(Y), a__times(mark(X), mark(Y)))

MARK(2ndspos(square(y0), cons2(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), cons2(x0, mark(x1)))

MARK(plus(square(x0), y1)) → A__PLUS(a__square(mark(x0)), mark(y1))

MARK(plus(s(x0), y1)) → A__PLUS(s(mark(x0)), mark(y1))

MARK(2ndspos(square(y0), cons(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), cons(mark(x0), x1))

A__PI(plus(x0, x1)) → A__2NDSPOS(a__plus(mark(x0), mark(x1)), a__from(0))

MARK(2ndsneg(times(y0, y1), cons2(x0, x1))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(cons2(X1, X2)) → MARK(X2)

MARK(2ndsneg(square(y0), cons2(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), cons2(x0, mark(x1)))

A__PLUS(s(square(x0)), y1) → A__PLUS(a__square(mark(x0)), mark(y1))

MARK(2ndspos(X1, X2)) → MARK(X1)

MARK(s(X)) → MARK(X)

MARK(2ndsneg(square(y0), plus(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(2ndsneg(plus(y0, y1), square(x0))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndspos(plus(y0, y1), cons2(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(pi(X)) → MARK(X)

MARK(2ndsneg(s(y0), from(x0))) → A__2NDSNEG(s(mark(y0)), a__from(mark(x0)))

MARK(plus(X1, X2)) → MARK(X2)

A__2NDSPOS(s(N), cons(X, Z)) → MARK(Z)

MARK(pi(X)) → A__PI(mark(X))

MARK(2ndsneg(times(y0, y1), square(x0))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndsneg(times(y0, y1), from(x0))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndsneg(square(y0), times(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(plus(y0, y1), cons(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), cons(mark(x0), x1))

MARK(plus(2ndsneg(x0, x1), y1)) → A__PLUS(a__2ndsneg(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(times(y0, y1), cons2(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(2ndsneg(X1, X2)) → MARK(X2)

MARK(2ndsneg(square(y0), from(x0))) → A__2NDSNEG(a__square(mark(y0)), a__from(mark(x0)))

MARK(2ndspos(square(y0), times(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(times(y0, y1), square(x0))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(times(square(x0), y1)) → A__TIMES(a__square(mark(x0)), mark(y1))

MARK(2ndsneg(s(y0), cons(x0, x1))) → A__2NDSNEG(s(mark(y0)), cons(mark(x0), x1))

MARK(from(X)) → A__FROM(mark(X))

MARK(2ndspos(s(y0), plus(x0, x1))) → A__2NDSPOS(s(mark(y0)), a__plus(mark(x0), mark(x1)))

A__PI(X) → A__FROM(0)

A__PLUS(s(2ndsneg(x0, x1)), y1) → A__PLUS(a__2ndsneg(mark(x0), mark(x1)), mark(y1))

MARK(plus(X1, X2)) → MARK(X1)

A__2NDSPOS(s(y0), cons(y1, times(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__times(mark(x0), mark(x1))))

MARK(2ndspos(s(y0), cons(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons(mark(x0), x1))

A__PI(X) → MARK(X)

MARK(times(s(x0), y1)) → A__TIMES(s(mark(x0)), mark(y1))

MARK(2ndsneg(plus(y0, y1), cons(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), cons(mark(x0), x1))

A__PLUS(s(X), Y) → MARK(Y)

A__PLUS(s(2ndspos(x0, x1)), y1) → A__PLUS(a__2ndspos(mark(x0), mark(x1)), mark(y1))

A__TIMES(s(X), Y) → MARK(X)

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(N)

MARK(2ndsneg(s(y0), cons2(x0, x1))) → A__2NDSNEG(s(mark(y0)), cons2(x0, mark(x1)))

MARK(plus(2ndspos(x0, x1), y1)) → A__PLUS(a__2ndspos(mark(x0), mark(x1)), mark(y1))

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(N)

MARK(cons(X1, X2)) → MARK(X1)

MARK(2ndsneg(plus(y0, y1), from(x0))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(times(X1, X2)) → MARK(X2)

MARK(2ndspos(times(y0, y1), cons(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), cons(mark(x0), x1))

A__PI(square(x0)) → A__2NDSPOS(a__square(mark(x0)), a__from(0))

MARK(2ndsneg(square(y0), cons(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), cons(mark(x0), x1))

MARK(2ndspos(times(y0, y1), plus(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(plus(y0, y1), times(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(s(y0), times(x0, x1))) → A__2NDSPOS(s(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndsneg(X1, X2)) → MARK(X1)

MARK(2ndsneg(square(y0), square(x0))) → A__2NDSNEG(a__square(mark(y0)), a__square(mark(x0)))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(Y)

MARK(plus(pi(x0), y1)) → A__PLUS(a__pi(mark(x0)), mark(y1))

MARK(rcons(X1, X2)) → MARK(X1)

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(Z)

MARK(square(X)) → MARK(X)

A__PI(s(x0)) → A__2NDSPOS(s(mark(x0)), a__from(0))

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(Y)

A__FROM(X) → MARK(X)

MARK(2ndsneg(s(y0), plus(x0, x1))) → A__2NDSNEG(s(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(times(plus(x0, x1), y1)) → A__TIMES(a__plus(mark(x0), mark(x1)), mark(y1))

A__2NDSPOS(s(y0), cons(y1, from(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__from(mark(x0))))

MARK(rcons(X1, X2)) → MARK(X2)

MARK(2ndspos(s(y0), square(x0))) → A__2NDSPOS(s(mark(y0)), a__square(mark(x0)))

MARK(2ndspos(s(y0), cons2(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(x0, mark(x1)))

A__TIMES(s(X), Y) → A__TIMES(mark(X), mark(Y))

MARK(2ndsneg(plus(y0, y1), cons2(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), cons2(x0, mark(x1)))

A__PLUS(s(X), Y) → MARK(X)

A__2NDSPOS(s(y0), cons(y1, cons(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, cons(mark(x0), x1)))

MARK(times(X1, X2)) → MARK(X1)

MARK(plus(times(x0, x1), y1)) → A__PLUS(a__times(mark(x0), mark(x1)), mark(y1))

A__SQUARE(X) → MARK(X)

MARK(negrecip(X)) → MARK(X)

A__2NDSPOS(s(N), cons(X, Z)) → MARK(N)

A__PLUS(s(from(x0)), y1) → A__PLUS(a__from(mark(x0)), mark(y1))

A__PLUS(0, Y) → MARK(Y)

MARK(posrecip(X)) → MARK(X)

MARK(2ndspos(square(y0), from(x0))) → A__2NDSPOS(a__square(mark(y0)), a__from(mark(x0)))

MARK(2ndspos(times(y0, y1), from(x0))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__from(mark(x0)))

A__PLUS(s(times(x0, x1)), y1) → A__PLUS(a__times(mark(x0), mark(x1)), mark(y1))

A__2NDSPOS(s(y0), cons(y1, square(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__square(mark(x0))))

A__PLUS(s(0), y1) → A__PLUS(0, mark(y1))

MARK(plus(0, y1)) → A__PLUS(0, mark(y1))

MARK(from(X)) → MARK(X)

MARK(2ndsneg(s(y0), times(x0, x1))) → A__2NDSNEG(s(mark(y0)), a__times(mark(x0), mark(x1)))

A__SQUARE(X) → A__TIMES(mark(X), mark(X))

MARK(2ndspos(square(y0), square(x0))) → A__2NDSPOS(a__square(mark(y0)), a__square(mark(x0)))

A__PLUS(s(plus(x0, x1)), y1) → A__PLUS(a__plus(mark(x0), mark(x1)), mark(y1))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(Z)

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → A__2NDSPOS(mark(N), mark(Z))

MARK(2ndsneg(plus(y0, y1), plus(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(plus(from(x0), y1)) → A__PLUS(a__from(mark(x0)), mark(y1))

MARK(square(X)) → A__SQUARE(mark(X))

MARK(2ndspos(plus(y0, y1), from(x0))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndsneg(plus(y0, y1), times(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndsneg(times(y0, y1), plus(x0, x1))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(plus(y0, y1), plus(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(2ndsneg(times(y0, y1), cons(x0, x1))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), cons(mark(x0), x1))

A__PLUS(s(s(x0)), y1) → A__PLUS(s(mark(x0)), mark(y1))

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))

a__2ndspos(0, Z) → rnil

a__2ndspos(s(N), cons(X, Z)) → a__2ndspos(s(mark(N)), cons2(X, mark(Z)))

a__2ndspos(s(N), cons2(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))

a__2ndsneg(0, Z) → rnil

a__2ndsneg(s(N), cons(X, Z)) → a__2ndsneg(s(mark(N)), cons2(X, mark(Z)))

a__2ndsneg(s(N), cons2(X, cons(Y, Z))) → rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z)))

a__pi(X) → a__2ndspos(mark(X), a__from(0))

a__plus(0, Y) → mark(Y)

a__plus(s(X), Y) → s(a__plus(mark(X), mark(Y)))

a__times(0, Y) → 0

a__times(s(X), Y) → a__plus(mark(Y), a__times(mark(X), mark(Y)))

a__square(X) → a__times(mark(X), mark(X))

mark(from(X)) → a__from(mark(X))

mark(2ndspos(X1, X2)) → a__2ndspos(mark(X1), mark(X2))

mark(2ndsneg(X1, X2)) → a__2ndsneg(mark(X1), mark(X2))

mark(pi(X)) → a__pi(mark(X))

mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))

mark(times(X1, X2)) → a__times(mark(X1), mark(X2))

mark(square(X)) → a__square(mark(X))

mark(0) → 0

mark(s(X)) → s(mark(X))

mark(posrecip(X)) → posrecip(mark(X))

mark(negrecip(X)) → negrecip(mark(X))

mark(nil) → nil

mark(cons(X1, X2)) → cons(mark(X1), X2)

mark(cons2(X1, X2)) → cons2(X1, mark(X2))

mark(rnil) → rnil

mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))

a__from(X) → from(X)

a__2ndspos(X1, X2) → 2ndspos(X1, X2)

a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)

a__pi(X) → pi(X)

a__plus(X1, X2) → plus(X1, X2)

a__times(X1, X2) → times(X1, X2)

a__square(X) → square(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

A__PLUS(s(pi(x0)), y1) → A__PLUS(a__pi(mark(x0)), mark(y1))

MARK(plus(2ndsneg(x0, x1), y1)) → A__PLUS(a__2ndsneg(mark(x0), mark(x1)), mark(y1))

A__PLUS(s(2ndsneg(x0, x1)), y1) → A__PLUS(a__2ndsneg(mark(x0), mark(x1)), mark(y1))

A__PLUS(s(2ndspos(x0, x1)), y1) → A__PLUS(a__2ndspos(mark(x0), mark(x1)), mark(y1))

MARK(plus(2ndspos(x0, x1), y1)) → A__PLUS(a__2ndspos(mark(x0), mark(x1)), mark(y1))

MARK(plus(pi(x0), y1)) → A__PLUS(a__pi(mark(x0)), mark(y1))

A__PLUS(s(from(x0)), y1) → A__PLUS(a__from(mark(x0)), mark(y1))

MARK(plus(from(x0), y1)) → A__PLUS(a__from(mark(x0)), mark(y1))
The remaining pairs can at least be oriented weakly.

MARK(2ndspos(square(y0), plus(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(plus(y0, y1), square(x0))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__square(mark(x0)))

A__PI(times(x0, x1)) → A__2NDSPOS(a__times(mark(x0), mark(x1)), a__from(0))

A__2NDSNEG(s(N), cons(X, Z)) → A__2NDSNEG(s(mark(N)), cons2(X, mark(Z)))

MARK(2ndsneg(s(y0), square(x0))) → A__2NDSNEG(s(mark(y0)), a__square(mark(x0)))

MARK(2ndspos(X1, X2)) → MARK(X2)

MARK(times(times(x0, x1), y1)) → A__TIMES(a__times(mark(x0), mark(x1)), mark(y1))

MARK(plus(plus(x0, x1), y1)) → A__PLUS(a__plus(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(s(y0), from(x0))) → A__2NDSPOS(s(mark(y0)), a__from(mark(x0)))

A__2NDSNEG(s(N), cons(X, Z)) → MARK(Z)

A__TIMES(s(X), Y) → MARK(Y)

A__2NDSPOS(s(y0), cons(y1, plus(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__plus(mark(x0), mark(x1))))

A__2NDSNEG(s(N), cons(X, Z)) → MARK(N)

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → A__2NDSNEG(mark(N), mark(Z))

MARK(2ndsneg(times(y0, y1), times(x0, x1))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(times(y0, y1), times(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

A__TIMES(s(X), Y) → A__PLUS(mark(Y), a__times(mark(X), mark(Y)))

MARK(2ndspos(square(y0), cons2(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), cons2(x0, mark(x1)))

MARK(plus(square(x0), y1)) → A__PLUS(a__square(mark(x0)), mark(y1))

MARK(plus(s(x0), y1)) → A__PLUS(s(mark(x0)), mark(y1))

MARK(2ndspos(square(y0), cons(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), cons(mark(x0), x1))

A__PI(plus(x0, x1)) → A__2NDSPOS(a__plus(mark(x0), mark(x1)), a__from(0))

MARK(2ndsneg(times(y0, y1), cons2(x0, x1))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(cons2(X1, X2)) → MARK(X2)

MARK(2ndsneg(square(y0), cons2(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), cons2(x0, mark(x1)))

A__PLUS(s(square(x0)), y1) → A__PLUS(a__square(mark(x0)), mark(y1))

MARK(2ndspos(X1, X2)) → MARK(X1)

MARK(s(X)) → MARK(X)

MARK(2ndsneg(square(y0), plus(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(2ndsneg(plus(y0, y1), square(x0))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndspos(plus(y0, y1), cons2(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(pi(X)) → MARK(X)

MARK(2ndsneg(s(y0), from(x0))) → A__2NDSNEG(s(mark(y0)), a__from(mark(x0)))

MARK(plus(X1, X2)) → MARK(X2)

A__2NDSPOS(s(N), cons(X, Z)) → MARK(Z)

MARK(pi(X)) → A__PI(mark(X))

MARK(2ndsneg(times(y0, y1), square(x0))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndsneg(times(y0, y1), from(x0))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndsneg(square(y0), times(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(plus(y0, y1), cons(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), cons(mark(x0), x1))

MARK(2ndspos(times(y0, y1), cons2(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(2ndsneg(X1, X2)) → MARK(X2)

MARK(2ndsneg(square(y0), from(x0))) → A__2NDSNEG(a__square(mark(y0)), a__from(mark(x0)))

MARK(2ndspos(square(y0), times(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(times(y0, y1), square(x0))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(times(square(x0), y1)) → A__TIMES(a__square(mark(x0)), mark(y1))

MARK(2ndsneg(s(y0), cons(x0, x1))) → A__2NDSNEG(s(mark(y0)), cons(mark(x0), x1))

MARK(from(X)) → A__FROM(mark(X))

MARK(2ndspos(s(y0), plus(x0, x1))) → A__2NDSPOS(s(mark(y0)), a__plus(mark(x0), mark(x1)))

A__PI(X) → A__FROM(0)

MARK(plus(X1, X2)) → MARK(X1)

A__2NDSPOS(s(y0), cons(y1, times(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__times(mark(x0), mark(x1))))

MARK(2ndspos(s(y0), cons(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons(mark(x0), x1))

A__PI(X) → MARK(X)

MARK(times(s(x0), y1)) → A__TIMES(s(mark(x0)), mark(y1))

MARK(2ndsneg(plus(y0, y1), cons(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), cons(mark(x0), x1))

A__PLUS(s(X), Y) → MARK(Y)

A__TIMES(s(X), Y) → MARK(X)

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(N)

MARK(2ndsneg(s(y0), cons2(x0, x1))) → A__2NDSNEG(s(mark(y0)), cons2(x0, mark(x1)))

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(N)

MARK(cons(X1, X2)) → MARK(X1)

MARK(2ndsneg(plus(y0, y1), from(x0))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(times(X1, X2)) → MARK(X2)

MARK(2ndspos(times(y0, y1), cons(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), cons(mark(x0), x1))

A__PI(square(x0)) → A__2NDSPOS(a__square(mark(x0)), a__from(0))

MARK(2ndsneg(square(y0), cons(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), cons(mark(x0), x1))

MARK(2ndspos(times(y0, y1), plus(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(plus(y0, y1), times(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(s(y0), times(x0, x1))) → A__2NDSPOS(s(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndsneg(X1, X2)) → MARK(X1)

MARK(2ndsneg(square(y0), square(x0))) → A__2NDSNEG(a__square(mark(y0)), a__square(mark(x0)))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(Y)

MARK(rcons(X1, X2)) → MARK(X1)

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(Z)

MARK(square(X)) → MARK(X)

A__PI(s(x0)) → A__2NDSPOS(s(mark(x0)), a__from(0))

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(Y)

A__FROM(X) → MARK(X)

MARK(2ndsneg(s(y0), plus(x0, x1))) → A__2NDSNEG(s(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(times(plus(x0, x1), y1)) → A__TIMES(a__plus(mark(x0), mark(x1)), mark(y1))

A__2NDSPOS(s(y0), cons(y1, from(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__from(mark(x0))))

MARK(rcons(X1, X2)) → MARK(X2)

MARK(2ndspos(s(y0), square(x0))) → A__2NDSPOS(s(mark(y0)), a__square(mark(x0)))

MARK(2ndspos(s(y0), cons2(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(x0, mark(x1)))

A__TIMES(s(X), Y) → A__TIMES(mark(X), mark(Y))

MARK(2ndsneg(plus(y0, y1), cons2(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), cons2(x0, mark(x1)))

A__PLUS(s(X), Y) → MARK(X)

A__2NDSPOS(s(y0), cons(y1, cons(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, cons(mark(x0), x1)))

MARK(times(X1, X2)) → MARK(X1)

MARK(plus(times(x0, x1), y1)) → A__PLUS(a__times(mark(x0), mark(x1)), mark(y1))

A__SQUARE(X) → MARK(X)

MARK(negrecip(X)) → MARK(X)

A__2NDSPOS(s(N), cons(X, Z)) → MARK(N)

A__PLUS(0, Y) → MARK(Y)

MARK(posrecip(X)) → MARK(X)

MARK(2ndspos(square(y0), from(x0))) → A__2NDSPOS(a__square(mark(y0)), a__from(mark(x0)))

MARK(2ndspos(times(y0, y1), from(x0))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__from(mark(x0)))

A__PLUS(s(times(x0, x1)), y1) → A__PLUS(a__times(mark(x0), mark(x1)), mark(y1))

A__2NDSPOS(s(y0), cons(y1, square(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__square(mark(x0))))

A__PLUS(s(0), y1) → A__PLUS(0, mark(y1))

MARK(plus(0, y1)) → A__PLUS(0, mark(y1))

MARK(from(X)) → MARK(X)

MARK(2ndsneg(s(y0), times(x0, x1))) → A__2NDSNEG(s(mark(y0)), a__times(mark(x0), mark(x1)))

A__SQUARE(X) → A__TIMES(mark(X), mark(X))

MARK(2ndspos(square(y0), square(x0))) → A__2NDSPOS(a__square(mark(y0)), a__square(mark(x0)))

A__PLUS(s(plus(x0, x1)), y1) → A__PLUS(a__plus(mark(x0), mark(x1)), mark(y1))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(Z)

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → A__2NDSPOS(mark(N), mark(Z))

MARK(2ndsneg(plus(y0, y1), plus(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(square(X)) → A__SQUARE(mark(X))

MARK(2ndspos(plus(y0, y1), from(x0))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndsneg(plus(y0, y1), times(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndsneg(times(y0, y1), plus(x0, x1))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(plus(y0, y1), plus(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(2ndsneg(times(y0, y1), cons(x0, x1))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), cons(mark(x0), x1))

A__PLUS(s(s(x0)), y1) → A__PLUS(s(mark(x0)), mark(y1))
Used ordering:  Polynomial interpretation [25]:

POL(0) = 1   

POL(2ndsneg(x1, x2)) = 0   

POL(2ndspos(x1, x2)) = 0   

POL(A__2NDSNEG(x1, x2)) = x1   

POL(A__2NDSPOS(x1, x2)) = x1   

POL(A__FROM(x1)) = 1   

POL(A__PI(x1)) = 1   

POL(A__PLUS(x1, x2)) = x1   

POL(A__SQUARE(x1)) = 1   

POL(A__TIMES(x1, x2)) = 1   

POL(MARK(x1)) = 1   

POL(a__2ndsneg(x1, x2)) = 0   

POL(a__2ndspos(x1, x2)) = 0   

POL(a__from(x1)) = 0   

POL(a__pi(x1)) = 0   

POL(a__plus(x1, x2)) = 1   

POL(a__square(x1)) = 1   

POL(a__times(x1, x2)) = 1   

POL(cons(x1, x2)) = 0   

POL(cons2(x1, x2)) = 0   

POL(from(x1)) = 0   

POL(mark(x1)) = 1   

POL(negrecip(x1)) = 0   

POL(nil) = 0   

POL(pi(x1)) = 0   

POL(plus(x1, x2)) = 1   

POL(posrecip(x1)) = 0   

POL(rcons(x1, x2)) = x1   

POL(rnil) = 0   

POL(s(x1)) = 1   

POL(square(x1)) = 0   

POL(times(x1, x2)) = 0   

The following usable rules [17] were oriented:

a__from(X) → cons(mark(X), from(s(X)))

a__pi(X) → a__2ndspos(mark(X), a__from(0))

a__2ndsneg(s(N), cons(X, Z)) → a__2ndsneg(s(mark(N)), cons2(X, mark(Z)))

a__2ndsneg(s(N), cons2(X, cons(Y, Z))) → rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z)))

a__2ndspos(s(N), cons2(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))

a__2ndsneg(0, Z) → rnil

a__2ndspos(0, Z) → rnil

a__2ndspos(s(N), cons(X, Z)) → a__2ndspos(s(mark(N)), cons2(X, mark(Z)))

mark(pi(X)) → a__pi(mark(X))

mark(2ndsneg(X1, X2)) → a__2ndsneg(mark(X1), mark(X2))

mark(2ndspos(X1, X2)) → a__2ndspos(mark(X1), mark(X2))

mark(from(X)) → a__from(mark(X))

a__times(0, Y) → 0

a__plus(s(X), Y) → s(a__plus(mark(X), mark(Y)))

mark(nil) → nil

mark(negrecip(X)) → negrecip(mark(X))

mark(posrecip(X)) → posrecip(mark(X))

mark(s(X)) → s(mark(X))

mark(0) → 0

mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))

a__times(s(X), Y) → a__plus(mark(Y), a__times(mark(X), mark(Y)))

a__plus(0, Y) → mark(Y)

mark(square(X)) → a__square(mark(X))

mark(times(X1, X2)) → a__times(mark(X1), mark(X2))

a__square(X) → a__times(mark(X), mark(X))

a__times(X1, X2) → times(X1, X2)

a__plus(X1, X2) → plus(X1, X2)

a__square(X) → square(X)

mark(cons2(X1, X2)) → cons2(X1, mark(X2))

mark(cons(X1, X2)) → cons(mark(X1), X2)

mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))

mark(rnil) → rnil

a__2ndspos(X1, X2) → 2ndspos(X1, X2)

a__from(X) → from(X)

a__pi(X) → pi(X)

a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ Narrowing
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ QDP
              ↳ Narrowing
                ↳ QDP
                  ↳ DependencyGraphProof
                    ↳ QDP
                      ↳ Narrowing
                        ↳ QDP
                          ↳ DependencyGraphProof
                            ↳ QDP
                              ↳ Narrowing
                                ↳ QDP
                                  ↳ DependencyGraphProof
                                    ↳ QDP
                                      ↳ Narrowing
                                        ↳ QDP
                                          ↳ DependencyGraphProof
                                            ↳ QDP
                                              ↳ Narrowing
                                                ↳ QDP
                                                  ↳ DependencyGraphProof
                                                    ↳ QDP
                                                      ↳ Narrowing
                                                        ↳ QDP
                                                          ↳ DependencyGraphProof
                                                            ↳ QDP
                                                              ↳ Narrowing
                                                                ↳ QDP
                                                                  ↳ DependencyGraphProof
                                                                    ↳ QDP
                                                                      ↳ Narrowing
                                                                        ↳ QDP
                                                                          ↳ DependencyGraphProof
                                                                            ↳ QDP
                                                                              ↳ Narrowing
                                                                                ↳ QDP
                                                                                  ↳ DependencyGraphProof
                                                                                    ↳ QDP
                                                                                      ↳ Narrowing
                                                                                        ↳ QDP
                                                                                          ↳ DependencyGraphProof
                                                                                            ↳ QDP
                                                                                              ↳ Narrowing
                                                                                                ↳ QDP
                                                                                                  ↳ DependencyGraphProof
                                                                                                    ↳ QDP
                                                                                                      ↳ Narrowing
                                                                                                        ↳ QDP
                                                                                                          ↳ DependencyGraphProof
                                                                                                            ↳ QDP
                                                                                                              ↳ Narrowing
                                                                                                                ↳ QDP
                                                                                                                  ↳ DependencyGraphProof
                                                                                                                    ↳ QDP
                                                                                                                      ↳ Narrowing
                                                                                                                        ↳ QDP
                                                                                                                          ↳ DependencyGraphProof
                                                                                                                            ↳ QDP
                                                                                                                              ↳ Narrowing
                                                                                                                                ↳ QDP
                                                                                                                                  ↳ DependencyGraphProof
                                                                                                                                    ↳ QDP
                                                                                                                                      ↳ Narrowing
                                                                                                                                        ↳ QDP
                                                                                                                                          ↳ DependencyGraphProof
                                                                                                                                            ↳ QDP
                                                                                                                                              ↳ Narrowing
                                                                                                                                                ↳ QDP
                                                                                                                                                  ↳ DependencyGraphProof
                                                                                                                                                    ↳ QDP
                                                                                                                                                      ↳ Narrowing
                                                                                                                                                        ↳ QDP
                                                                                                                                                          ↳ DependencyGraphProof
                                                                                                                                                            ↳ QDP
                                                                                                                                                              ↳ Narrowing
                                                                                                                                                                ↳ QDP
                                                                                                                                                                  ↳ DependencyGraphProof
                                                                                                                                                                    ↳ QDP
                                                                                                                                                                      ↳ Narrowing
                                                                                                                                                                        ↳ QDP
                                                                                                                                                                          ↳ DependencyGraphProof
                                                                                                                                                                            ↳ QDP
                                                                                                                                                                              ↳ Narrowing
                                                                                                                                                                                ↳ QDP
                                                                                                                                                                                  ↳ DependencyGraphProof
                                                                                                                                                                                    ↳ QDP
                                                                                                                                                                                      ↳ Narrowing
                                                                                                                                                                                        ↳ QDP
                                                                                                                                                                                          ↳ DependencyGraphProof
                                                                                                                                                                                            ↳ QDP
                                                                                                                                                                                              ↳ QDPOrderProof
                                                                                                                                                                                                ↳ QDP
                                                                                                                                                                                                  ↳ QDPOrderProof
                                                                                                                                                                                                    ↳ QDP
                                                                                                                                                                                                      ↳ QDPOrderProof
                                                                                                                                                                                                        ↳ QDP
                                                                                                                                                                                                          ↳ QDPOrderProof
                                                                                                                                                                                                            ↳ QDP
                                                                                                                                                                                                              ↳ QDPOrderProof
                                                                                                                                                                                                                ↳ QDP
                                                                                                                                                                                                                  ↳ QDPOrderProof
                                                                                                                                                                                                                    ↳ QDP
                                                                                                                                                                                                                      ↳ QDPOrderProof
                                                                                                                                                                                                                        ↳ QDP
                                                                                                                                                                                                                          ↳ QDPOrderProof
                                                                                                                                                                                                                            ↳ QDP
                                                                                                                                                                                                                              ↳ Narrowing

Q DP problem:
The TRS P consists of the following rules:

MARK(2ndspos(plus(y0, y1), square(x0))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndspos(square(y0), plus(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__plus(mark(x0), mark(x1)))

A__PI(times(x0, x1)) → A__2NDSPOS(a__times(mark(x0), mark(x1)), a__from(0))

A__2NDSNEG(s(N), cons(X, Z)) → A__2NDSNEG(s(mark(N)), cons2(X, mark(Z)))

MARK(2ndsneg(s(y0), square(x0))) → A__2NDSNEG(s(mark(y0)), a__square(mark(x0)))

MARK(times(times(x0, x1), y1)) → A__TIMES(a__times(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(X1, X2)) → MARK(X2)

MARK(plus(plus(x0, x1), y1)) → A__PLUS(a__plus(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(s(y0), from(x0))) → A__2NDSPOS(s(mark(y0)), a__from(mark(x0)))

A__2NDSNEG(s(N), cons(X, Z)) → MARK(Z)

A__TIMES(s(X), Y) → MARK(Y)

A__2NDSPOS(s(y0), cons(y1, plus(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__plus(mark(x0), mark(x1))))

A__2NDSNEG(s(N), cons(X, Z)) → MARK(N)

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → A__2NDSNEG(mark(N), mark(Z))

MARK(2ndsneg(times(y0, y1), times(x0, x1))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(times(y0, y1), times(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

A__TIMES(s(X), Y) → A__PLUS(mark(Y), a__times(mark(X), mark(Y)))

MARK(2ndspos(square(y0), cons2(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), cons2(x0, mark(x1)))

MARK(plus(square(x0), y1)) → A__PLUS(a__square(mark(x0)), mark(y1))

MARK(plus(s(x0), y1)) → A__PLUS(s(mark(x0)), mark(y1))

MARK(2ndspos(square(y0), cons(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), cons(mark(x0), x1))

A__PI(plus(x0, x1)) → A__2NDSPOS(a__plus(mark(x0), mark(x1)), a__from(0))

MARK(2ndsneg(times(y0, y1), cons2(x0, x1))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(cons2(X1, X2)) → MARK(X2)

MARK(2ndsneg(square(y0), cons2(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), cons2(x0, mark(x1)))

MARK(2ndsneg(plus(y0, y1), square(x0))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndsneg(square(y0), plus(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(s(X)) → MARK(X)

MARK(2ndspos(X1, X2)) → MARK(X1)

A__PLUS(s(square(x0)), y1) → A__PLUS(a__square(mark(x0)), mark(y1))

MARK(2ndspos(plus(y0, y1), cons2(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(pi(X)) → MARK(X)

MARK(2ndsneg(s(y0), from(x0))) → A__2NDSNEG(s(mark(y0)), a__from(mark(x0)))

MARK(plus(X1, X2)) → MARK(X2)

A__2NDSPOS(s(N), cons(X, Z)) → MARK(Z)

MARK(pi(X)) → A__PI(mark(X))

MARK(2ndspos(plus(y0, y1), cons(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), cons(mark(x0), x1))

MARK(2ndsneg(square(y0), times(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndsneg(times(y0, y1), from(x0))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndsneg(times(y0, y1), square(x0))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndspos(times(y0, y1), cons2(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(2ndsneg(X1, X2)) → MARK(X2)

MARK(2ndsneg(square(y0), from(x0))) → A__2NDSNEG(a__square(mark(y0)), a__from(mark(x0)))

MARK(2ndspos(times(y0, y1), square(x0))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndspos(square(y0), times(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndsneg(s(y0), cons(x0, x1))) → A__2NDSNEG(s(mark(y0)), cons(mark(x0), x1))

MARK(times(square(x0), y1)) → A__TIMES(a__square(mark(x0)), mark(y1))

MARK(2ndspos(s(y0), plus(x0, x1))) → A__2NDSPOS(s(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(from(X)) → A__FROM(mark(X))

A__PI(X) → A__FROM(0)

MARK(plus(X1, X2)) → MARK(X1)

A__2NDSPOS(s(y0), cons(y1, times(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__times(mark(x0), mark(x1))))

MARK(2ndspos(s(y0), cons(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons(mark(x0), x1))

A__PI(X) → MARK(X)

MARK(times(s(x0), y1)) → A__TIMES(s(mark(x0)), mark(y1))

MARK(2ndsneg(plus(y0, y1), cons(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), cons(mark(x0), x1))

A__PLUS(s(X), Y) → MARK(Y)

A__TIMES(s(X), Y) → MARK(X)

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(N)

MARK(2ndsneg(s(y0), cons2(x0, x1))) → A__2NDSNEG(s(mark(y0)), cons2(x0, mark(x1)))

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(N)

MARK(cons(X1, X2)) → MARK(X1)

MARK(2ndsneg(plus(y0, y1), from(x0))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(times(X1, X2)) → MARK(X2)

MARK(2ndspos(times(y0, y1), cons(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), cons(mark(x0), x1))

A__PI(square(x0)) → A__2NDSPOS(a__square(mark(x0)), a__from(0))

MARK(2ndsneg(square(y0), cons(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), cons(mark(x0), x1))

MARK(2ndspos(times(y0, y1), plus(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(plus(y0, y1), times(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(s(y0), times(x0, x1))) → A__2NDSPOS(s(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndsneg(X1, X2)) → MARK(X1)

MARK(2ndsneg(square(y0), square(x0))) → A__2NDSNEG(a__square(mark(y0)), a__square(mark(x0)))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(Y)

MARK(rcons(X1, X2)) → MARK(X1)

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(Z)

MARK(square(X)) → MARK(X)

A__PI(s(x0)) → A__2NDSPOS(s(mark(x0)), a__from(0))

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(Y)

A__FROM(X) → MARK(X)

MARK(2ndsneg(s(y0), plus(x0, x1))) → A__2NDSNEG(s(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(times(plus(x0, x1), y1)) → A__TIMES(a__plus(mark(x0), mark(x1)), mark(y1))

A__2NDSPOS(s(y0), cons(y1, from(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__from(mark(x0))))

MARK(rcons(X1, X2)) → MARK(X2)

MARK(2ndspos(s(y0), square(x0))) → A__2NDSPOS(s(mark(y0)), a__square(mark(x0)))

MARK(2ndspos(s(y0), cons2(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(x0, mark(x1)))

A__TIMES(s(X), Y) → A__TIMES(mark(X), mark(Y))

MARK(2ndsneg(plus(y0, y1), cons2(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), cons2(x0, mark(x1)))

A__PLUS(s(X), Y) → MARK(X)

A__2NDSPOS(s(y0), cons(y1, cons(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, cons(mark(x0), x1)))

MARK(times(X1, X2)) → MARK(X1)

MARK(plus(times(x0, x1), y1)) → A__PLUS(a__times(mark(x0), mark(x1)), mark(y1))

A__SQUARE(X) → MARK(X)

MARK(negrecip(X)) → MARK(X)

A__2NDSPOS(s(N), cons(X, Z)) → MARK(N)

A__PLUS(0, Y) → MARK(Y)

MARK(posrecip(X)) → MARK(X)

MARK(2ndspos(square(y0), from(x0))) → A__2NDSPOS(a__square(mark(y0)), a__from(mark(x0)))

MARK(2ndspos(times(y0, y1), from(x0))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__from(mark(x0)))

A__PLUS(s(times(x0, x1)), y1) → A__PLUS(a__times(mark(x0), mark(x1)), mark(y1))

A__2NDSPOS(s(y0), cons(y1, square(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__square(mark(x0))))

A__PLUS(s(0), y1) → A__PLUS(0, mark(y1))

MARK(plus(0, y1)) → A__PLUS(0, mark(y1))

MARK(from(X)) → MARK(X)

MARK(2ndsneg(s(y0), times(x0, x1))) → A__2NDSNEG(s(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(square(y0), square(x0))) → A__2NDSPOS(a__square(mark(y0)), a__square(mark(x0)))

A__SQUARE(X) → A__TIMES(mark(X), mark(X))

A__PLUS(s(plus(x0, x1)), y1) → A__PLUS(a__plus(mark(x0), mark(x1)), mark(y1))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(Z)

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → A__2NDSPOS(mark(N), mark(Z))

MARK(2ndsneg(plus(y0, y1), plus(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(square(X)) → A__SQUARE(mark(X))

MARK(2ndspos(plus(y0, y1), from(x0))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndsneg(times(y0, y1), plus(x0, x1))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(2ndsneg(plus(y0, y1), times(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(plus(y0, y1), plus(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

A__PLUS(s(s(x0)), y1) → A__PLUS(s(mark(x0)), mark(y1))

MARK(2ndsneg(times(y0, y1), cons(x0, x1))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), cons(mark(x0), x1))

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))

a__2ndspos(0, Z) → rnil

a__2ndspos(s(N), cons(X, Z)) → a__2ndspos(s(mark(N)), cons2(X, mark(Z)))

a__2ndspos(s(N), cons2(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))

a__2ndsneg(0, Z) → rnil

a__2ndsneg(s(N), cons(X, Z)) → a__2ndsneg(s(mark(N)), cons2(X, mark(Z)))

a__2ndsneg(s(N), cons2(X, cons(Y, Z))) → rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z)))

a__pi(X) → a__2ndspos(mark(X), a__from(0))

a__plus(0, Y) → mark(Y)

a__plus(s(X), Y) → s(a__plus(mark(X), mark(Y)))

a__times(0, Y) → 0

a__times(s(X), Y) → a__plus(mark(Y), a__times(mark(X), mark(Y)))

a__square(X) → a__times(mark(X), mark(X))

mark(from(X)) → a__from(mark(X))

mark(2ndspos(X1, X2)) → a__2ndspos(mark(X1), mark(X2))

mark(2ndsneg(X1, X2)) → a__2ndsneg(mark(X1), mark(X2))

mark(pi(X)) → a__pi(mark(X))

mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))

mark(times(X1, X2)) → a__times(mark(X1), mark(X2))

mark(square(X)) → a__square(mark(X))

mark(0) → 0

mark(s(X)) → s(mark(X))

mark(posrecip(X)) → posrecip(mark(X))

mark(negrecip(X)) → negrecip(mark(X))

mark(nil) → nil

mark(cons(X1, X2)) → cons(mark(X1), X2)

mark(cons2(X1, X2)) → cons2(X1, mark(X2))

mark(rnil) → rnil

mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))

a__from(X) → from(X)

a__2ndspos(X1, X2) → 2ndspos(X1, X2)

a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)

a__pi(X) → pi(X)

a__plus(X1, X2) → plus(X1, X2)

a__times(X1, X2) → times(X1, X2)

a__square(X) → square(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule A__PI(times(x0, x1)) → A__2NDSPOS(a__times(mark(x0), mark(x1)), a__from(0)) at position [1] we obtained the following new rules:

A__PI(times(y0, y1)) → A__2NDSPOS(a__times(mark(y0), mark(y1)), from(0))

A__PI(times(y0, y1)) → A__2NDSPOS(a__times(mark(y0), mark(y1)), cons(mark(0), from(s(0))))



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ Narrowing
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ QDP
              ↳ Narrowing
                ↳ QDP
                  ↳ DependencyGraphProof
                    ↳ QDP
                      ↳ Narrowing
                        ↳ QDP
                          ↳ DependencyGraphProof
                            ↳ QDP
                              ↳ Narrowing
                                ↳ QDP
                                  ↳ DependencyGraphProof
                                    ↳ QDP
                                      ↳ Narrowing
                                        ↳ QDP
                                          ↳ DependencyGraphProof
                                            ↳ QDP
                                              ↳ Narrowing
                                                ↳ QDP
                                                  ↳ DependencyGraphProof
                                                    ↳ QDP
                                                      ↳ Narrowing
                                                        ↳ QDP
                                                          ↳ DependencyGraphProof
                                                            ↳ QDP
                                                              ↳ Narrowing
                                                                ↳ QDP
                                                                  ↳ DependencyGraphProof
                                                                    ↳ QDP
                                                                      ↳ Narrowing
                                                                        ↳ QDP
                                                                          ↳ DependencyGraphProof
                                                                            ↳ QDP
                                                                              ↳ Narrowing
                                                                                ↳ QDP
                                                                                  ↳ DependencyGraphProof
                                                                                    ↳ QDP
                                                                                      ↳ Narrowing
                                                                                        ↳ QDP
                                                                                          ↳ DependencyGraphProof
                                                                                            ↳ QDP
                                                                                              ↳ Narrowing
                                                                                                ↳ QDP
                                                                                                  ↳ DependencyGraphProof
                                                                                                    ↳ QDP
                                                                                                      ↳ Narrowing
                                                                                                        ↳ QDP
                                                                                                          ↳ DependencyGraphProof
                                                                                                            ↳ QDP
                                                                                                              ↳ Narrowing
                                                                                                                ↳ QDP
                                                                                                                  ↳ DependencyGraphProof
                                                                                                                    ↳ QDP
                                                                                                                      ↳ Narrowing
                                                                                                                        ↳ QDP
                                                                                                                          ↳ DependencyGraphProof
                                                                                                                            ↳ QDP
                                                                                                                              ↳ Narrowing
                                                                                                                                ↳ QDP
                                                                                                                                  ↳ DependencyGraphProof
                                                                                                                                    ↳ QDP
                                                                                                                                      ↳ Narrowing
                                                                                                                                        ↳ QDP
                                                                                                                                          ↳ DependencyGraphProof
                                                                                                                                            ↳ QDP
                                                                                                                                              ↳ Narrowing
                                                                                                                                                ↳ QDP
                                                                                                                                                  ↳ DependencyGraphProof
                                                                                                                                                    ↳ QDP
                                                                                                                                                      ↳ Narrowing
                                                                                                                                                        ↳ QDP
                                                                                                                                                          ↳ DependencyGraphProof
                                                                                                                                                            ↳ QDP
                                                                                                                                                              ↳ Narrowing
                                                                                                                                                                ↳ QDP
                                                                                                                                                                  ↳ DependencyGraphProof
                                                                                                                                                                    ↳ QDP
                                                                                                                                                                      ↳ Narrowing
                                                                                                                                                                        ↳ QDP
                                                                                                                                                                          ↳ DependencyGraphProof
                                                                                                                                                                            ↳ QDP
                                                                                                                                                                              ↳ Narrowing
                                                                                                                                                                                ↳ QDP
                                                                                                                                                                                  ↳ DependencyGraphProof
                                                                                                                                                                                    ↳ QDP
                                                                                                                                                                                      ↳ Narrowing
                                                                                                                                                                                        ↳ QDP
                                                                                                                                                                                          ↳ DependencyGraphProof
                                                                                                                                                                                            ↳ QDP
                                                                                                                                                                                              ↳ QDPOrderProof
                                                                                                                                                                                                ↳ QDP
                                                                                                                                                                                                  ↳ QDPOrderProof
                                                                                                                                                                                                    ↳ QDP
                                                                                                                                                                                                      ↳ QDPOrderProof
                                                                                                                                                                                                        ↳ QDP
                                                                                                                                                                                                          ↳ QDPOrderProof
                                                                                                                                                                                                            ↳ QDP
                                                                                                                                                                                                              ↳ QDPOrderProof
                                                                                                                                                                                                                ↳ QDP
                                                                                                                                                                                                                  ↳ QDPOrderProof
                                                                                                                                                                                                                    ↳ QDP
                                                                                                                                                                                                                      ↳ QDPOrderProof
                                                                                                                                                                                                                        ↳ QDP
                                                                                                                                                                                                                          ↳ QDPOrderProof
                                                                                                                                                                                                                            ↳ QDP
                                                                                                                                                                                                                              ↳ Narrowing
                                                                                                                                                                                                                                ↳ QDP
                                                                                                                                                                                                                                  ↳ DependencyGraphProof

Q DP problem:
The TRS P consists of the following rules:

MARK(2ndspos(square(y0), plus(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(plus(y0, y1), square(x0))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__square(mark(x0)))

A__2NDSNEG(s(N), cons(X, Z)) → A__2NDSNEG(s(mark(N)), cons2(X, mark(Z)))

MARK(2ndsneg(s(y0), square(x0))) → A__2NDSNEG(s(mark(y0)), a__square(mark(x0)))

MARK(2ndspos(X1, X2)) → MARK(X2)

MARK(times(times(x0, x1), y1)) → A__TIMES(a__times(mark(x0), mark(x1)), mark(y1))

MARK(plus(plus(x0, x1), y1)) → A__PLUS(a__plus(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(s(y0), from(x0))) → A__2NDSPOS(s(mark(y0)), a__from(mark(x0)))

A__2NDSNEG(s(N), cons(X, Z)) → MARK(Z)

A__TIMES(s(X), Y) → MARK(Y)

A__2NDSPOS(s(y0), cons(y1, plus(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__plus(mark(x0), mark(x1))))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → A__2NDSNEG(mark(N), mark(Z))

A__2NDSNEG(s(N), cons(X, Z)) → MARK(N)

MARK(2ndsneg(times(y0, y1), times(x0, x1))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(times(y0, y1), times(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

A__TIMES(s(X), Y) → A__PLUS(mark(Y), a__times(mark(X), mark(Y)))

MARK(2ndspos(square(y0), cons2(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), cons2(x0, mark(x1)))

MARK(plus(square(x0), y1)) → A__PLUS(a__square(mark(x0)), mark(y1))

MARK(plus(s(x0), y1)) → A__PLUS(s(mark(x0)), mark(y1))

A__PI(plus(x0, x1)) → A__2NDSPOS(a__plus(mark(x0), mark(x1)), a__from(0))

MARK(2ndspos(square(y0), cons(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), cons(mark(x0), x1))

MARK(2ndsneg(times(y0, y1), cons2(x0, x1))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), cons2(x0, mark(x1)))

A__PI(times(y0, y1)) → A__2NDSPOS(a__times(mark(y0), mark(y1)), cons(mark(0), from(s(0))))

MARK(cons2(X1, X2)) → MARK(X2)

MARK(2ndsneg(square(y0), cons2(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), cons2(x0, mark(x1)))

A__PLUS(s(square(x0)), y1) → A__PLUS(a__square(mark(x0)), mark(y1))

MARK(2ndspos(X1, X2)) → MARK(X1)

MARK(s(X)) → MARK(X)

MARK(2ndsneg(square(y0), plus(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(2ndsneg(plus(y0, y1), square(x0))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndspos(plus(y0, y1), cons2(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(pi(X)) → MARK(X)

MARK(2ndsneg(s(y0), from(x0))) → A__2NDSNEG(s(mark(y0)), a__from(mark(x0)))

MARK(plus(X1, X2)) → MARK(X2)

A__2NDSPOS(s(N), cons(X, Z)) → MARK(Z)

MARK(pi(X)) → A__PI(mark(X))

MARK(2ndsneg(times(y0, y1), square(x0))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndsneg(times(y0, y1), from(x0))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndsneg(square(y0), times(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(plus(y0, y1), cons(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), cons(mark(x0), x1))

MARK(2ndspos(times(y0, y1), cons2(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(2ndsneg(X1, X2)) → MARK(X2)

MARK(2ndsneg(square(y0), from(x0))) → A__2NDSNEG(a__square(mark(y0)), a__from(mark(x0)))

MARK(2ndspos(square(y0), times(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(times(y0, y1), square(x0))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(times(square(x0), y1)) → A__TIMES(a__square(mark(x0)), mark(y1))

MARK(2ndsneg(s(y0), cons(x0, x1))) → A__2NDSNEG(s(mark(y0)), cons(mark(x0), x1))

A__PI(X) → A__FROM(0)

MARK(from(X)) → A__FROM(mark(X))

MARK(2ndspos(s(y0), plus(x0, x1))) → A__2NDSPOS(s(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(plus(X1, X2)) → MARK(X1)

A__2NDSPOS(s(y0), cons(y1, times(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__times(mark(x0), mark(x1))))

A__PI(X) → MARK(X)

MARK(2ndspos(s(y0), cons(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons(mark(x0), x1))

MARK(times(s(x0), y1)) → A__TIMES(s(mark(x0)), mark(y1))

MARK(2ndsneg(plus(y0, y1), cons(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), cons(mark(x0), x1))

A__PLUS(s(X), Y) → MARK(Y)

A__TIMES(s(X), Y) → MARK(X)

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(N)

MARK(2ndsneg(s(y0), cons2(x0, x1))) → A__2NDSNEG(s(mark(y0)), cons2(x0, mark(x1)))

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(N)

A__PI(times(y0, y1)) → A__2NDSPOS(a__times(mark(y0), mark(y1)), from(0))

MARK(cons(X1, X2)) → MARK(X1)

MARK(2ndsneg(plus(y0, y1), from(x0))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndspos(times(y0, y1), cons(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), cons(mark(x0), x1))

MARK(times(X1, X2)) → MARK(X2)

A__PI(square(x0)) → A__2NDSPOS(a__square(mark(x0)), a__from(0))

MARK(2ndspos(plus(y0, y1), times(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(times(y0, y1), plus(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(2ndsneg(square(y0), cons(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), cons(mark(x0), x1))

MARK(2ndspos(s(y0), times(x0, x1))) → A__2NDSPOS(s(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndsneg(X1, X2)) → MARK(X1)

MARK(2ndsneg(square(y0), square(x0))) → A__2NDSNEG(a__square(mark(y0)), a__square(mark(x0)))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(Y)

MARK(rcons(X1, X2)) → MARK(X1)

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(Z)

MARK(square(X)) → MARK(X)

A__PI(s(x0)) → A__2NDSPOS(s(mark(x0)), a__from(0))

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(Y)

A__FROM(X) → MARK(X)

MARK(times(plus(x0, x1), y1)) → A__TIMES(a__plus(mark(x0), mark(x1)), mark(y1))

MARK(2ndsneg(s(y0), plus(x0, x1))) → A__2NDSNEG(s(mark(y0)), a__plus(mark(x0), mark(x1)))

A__2NDSPOS(s(y0), cons(y1, from(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__from(mark(x0))))

MARK(2ndspos(s(y0), square(x0))) → A__2NDSPOS(s(mark(y0)), a__square(mark(x0)))

MARK(rcons(X1, X2)) → MARK(X2)

MARK(2ndspos(s(y0), cons2(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(x0, mark(x1)))

MARK(2ndsneg(plus(y0, y1), cons2(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), cons2(x0, mark(x1)))

A__TIMES(s(X), Y) → A__TIMES(mark(X), mark(Y))

A__2NDSPOS(s(y0), cons(y1, cons(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, cons(mark(x0), x1)))

A__PLUS(s(X), Y) → MARK(X)

MARK(times(X1, X2)) → MARK(X1)

MARK(plus(times(x0, x1), y1)) → A__PLUS(a__times(mark(x0), mark(x1)), mark(y1))

A__SQUARE(X) → MARK(X)

A__2NDSPOS(s(N), cons(X, Z)) → MARK(N)

MARK(negrecip(X)) → MARK(X)

A__PLUS(0, Y) → MARK(Y)

MARK(posrecip(X)) → MARK(X)

MARK(2ndspos(square(y0), from(x0))) → A__2NDSPOS(a__square(mark(y0)), a__from(mark(x0)))

MARK(2ndspos(times(y0, y1), from(x0))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__from(mark(x0)))

A__PLUS(s(times(x0, x1)), y1) → A__PLUS(a__times(mark(x0), mark(x1)), mark(y1))

A__2NDSPOS(s(y0), cons(y1, square(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__square(mark(x0))))

A__PLUS(s(0), y1) → A__PLUS(0, mark(y1))

MARK(plus(0, y1)) → A__PLUS(0, mark(y1))

MARK(from(X)) → MARK(X)

MARK(2ndsneg(s(y0), times(x0, x1))) → A__2NDSNEG(s(mark(y0)), a__times(mark(x0), mark(x1)))

A__SQUARE(X) → A__TIMES(mark(X), mark(X))

MARK(2ndspos(square(y0), square(x0))) → A__2NDSPOS(a__square(mark(y0)), a__square(mark(x0)))

A__PLUS(s(plus(x0, x1)), y1) → A__PLUS(a__plus(mark(x0), mark(x1)), mark(y1))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(Z)

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → A__2NDSPOS(mark(N), mark(Z))

MARK(2ndsneg(plus(y0, y1), plus(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(square(X)) → A__SQUARE(mark(X))

MARK(2ndspos(plus(y0, y1), from(x0))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndsneg(plus(y0, y1), times(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndsneg(times(y0, y1), plus(x0, x1))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(plus(y0, y1), plus(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(2ndsneg(times(y0, y1), cons(x0, x1))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), cons(mark(x0), x1))

A__PLUS(s(s(x0)), y1) → A__PLUS(s(mark(x0)), mark(y1))

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))

a__2ndspos(0, Z) → rnil

a__2ndspos(s(N), cons(X, Z)) → a__2ndspos(s(mark(N)), cons2(X, mark(Z)))

a__2ndspos(s(N), cons2(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))

a__2ndsneg(0, Z) → rnil

a__2ndsneg(s(N), cons(X, Z)) → a__2ndsneg(s(mark(N)), cons2(X, mark(Z)))

a__2ndsneg(s(N), cons2(X, cons(Y, Z))) → rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z)))

a__pi(X) → a__2ndspos(mark(X), a__from(0))

a__plus(0, Y) → mark(Y)

a__plus(s(X), Y) → s(a__plus(mark(X), mark(Y)))

a__times(0, Y) → 0

a__times(s(X), Y) → a__plus(mark(Y), a__times(mark(X), mark(Y)))

a__square(X) → a__times(mark(X), mark(X))

mark(from(X)) → a__from(mark(X))

mark(2ndspos(X1, X2)) → a__2ndspos(mark(X1), mark(X2))

mark(2ndsneg(X1, X2)) → a__2ndsneg(mark(X1), mark(X2))

mark(pi(X)) → a__pi(mark(X))

mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))

mark(times(X1, X2)) → a__times(mark(X1), mark(X2))

mark(square(X)) → a__square(mark(X))

mark(0) → 0

mark(s(X)) → s(mark(X))

mark(posrecip(X)) → posrecip(mark(X))

mark(negrecip(X)) → negrecip(mark(X))

mark(nil) → nil

mark(cons(X1, X2)) → cons(mark(X1), X2)

mark(cons2(X1, X2)) → cons2(X1, mark(X2))

mark(rnil) → rnil

mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))

a__from(X) → from(X)

a__2ndspos(X1, X2) → 2ndspos(X1, X2)

a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)

a__pi(X) → pi(X)

a__plus(X1, X2) → plus(X1, X2)

a__times(X1, X2) → times(X1, X2)

a__square(X) → square(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 1 less node.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ Narrowing
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ QDP
              ↳ Narrowing
                ↳ QDP
                  ↳ DependencyGraphProof
                    ↳ QDP
                      ↳ Narrowing
                        ↳ QDP
                          ↳ DependencyGraphProof
                            ↳ QDP
                              ↳ Narrowing
                                ↳ QDP
                                  ↳ DependencyGraphProof
                                    ↳ QDP
                                      ↳ Narrowing
                                        ↳ QDP
                                          ↳ DependencyGraphProof
                                            ↳ QDP
                                              ↳ Narrowing
                                                ↳ QDP
                                                  ↳ DependencyGraphProof
                                                    ↳ QDP
                                                      ↳ Narrowing
                                                        ↳ QDP
                                                          ↳ DependencyGraphProof
                                                            ↳ QDP
                                                              ↳ Narrowing
                                                                ↳ QDP
                                                                  ↳ DependencyGraphProof
                                                                    ↳ QDP
                                                                      ↳ Narrowing
                                                                        ↳ QDP
                                                                          ↳ DependencyGraphProof
                                                                            ↳ QDP
                                                                              ↳ Narrowing
                                                                                ↳ QDP
                                                                                  ↳ DependencyGraphProof
                                                                                    ↳ QDP
                                                                                      ↳ Narrowing
                                                                                        ↳ QDP
                                                                                          ↳ DependencyGraphProof
                                                                                            ↳ QDP
                                                                                              ↳ Narrowing
                                                                                                ↳ QDP
                                                                                                  ↳ DependencyGraphProof
                                                                                                    ↳ QDP
                                                                                                      ↳ Narrowing
                                                                                                        ↳ QDP
                                                                                                          ↳ DependencyGraphProof
                                                                                                            ↳ QDP
                                                                                                              ↳ Narrowing
                                                                                                                ↳ QDP
                                                                                                                  ↳ DependencyGraphProof
                                                                                                                    ↳ QDP
                                                                                                                      ↳ Narrowing
                                                                                                                        ↳ QDP
                                                                                                                          ↳ DependencyGraphProof
                                                                                                                            ↳ QDP
                                                                                                                              ↳ Narrowing
                                                                                                                                ↳ QDP
                                                                                                                                  ↳ DependencyGraphProof
                                                                                                                                    ↳ QDP
                                                                                                                                      ↳ Narrowing
                                                                                                                                        ↳ QDP
                                                                                                                                          ↳ DependencyGraphProof
                                                                                                                                            ↳ QDP
                                                                                                                                              ↳ Narrowing
                                                                                                                                                ↳ QDP
                                                                                                                                                  ↳ DependencyGraphProof
                                                                                                                                                    ↳ QDP
                                                                                                                                                      ↳ Narrowing
                                                                                                                                                        ↳ QDP
                                                                                                                                                          ↳ DependencyGraphProof
                                                                                                                                                            ↳ QDP
                                                                                                                                                              ↳ Narrowing
                                                                                                                                                                ↳ QDP
                                                                                                                                                                  ↳ DependencyGraphProof
                                                                                                                                                                    ↳ QDP
                                                                                                                                                                      ↳ Narrowing
                                                                                                                                                                        ↳ QDP
                                                                                                                                                                          ↳ DependencyGraphProof
                                                                                                                                                                            ↳ QDP
                                                                                                                                                                              ↳ Narrowing
                                                                                                                                                                                ↳ QDP
                                                                                                                                                                                  ↳ DependencyGraphProof
                                                                                                                                                                                    ↳ QDP
                                                                                                                                                                                      ↳ Narrowing
                                                                                                                                                                                        ↳ QDP
                                                                                                                                                                                          ↳ DependencyGraphProof
                                                                                                                                                                                            ↳ QDP
                                                                                                                                                                                              ↳ QDPOrderProof
                                                                                                                                                                                                ↳ QDP
                                                                                                                                                                                                  ↳ QDPOrderProof
                                                                                                                                                                                                    ↳ QDP
                                                                                                                                                                                                      ↳ QDPOrderProof
                                                                                                                                                                                                        ↳ QDP
                                                                                                                                                                                                          ↳ QDPOrderProof
                                                                                                                                                                                                            ↳ QDP
                                                                                                                                                                                                              ↳ QDPOrderProof
                                                                                                                                                                                                                ↳ QDP
                                                                                                                                                                                                                  ↳ QDPOrderProof
                                                                                                                                                                                                                    ↳ QDP
                                                                                                                                                                                                                      ↳ QDPOrderProof
                                                                                                                                                                                                                        ↳ QDP
                                                                                                                                                                                                                          ↳ QDPOrderProof
                                                                                                                                                                                                                            ↳ QDP
                                                                                                                                                                                                                              ↳ Narrowing
                                                                                                                                                                                                                                ↳ QDP
                                                                                                                                                                                                                                  ↳ DependencyGraphProof
                                                                                                                                                                                                                                    ↳ QDP
                                                                                                                                                                                                                                      ↳ Narrowing

Q DP problem:
The TRS P consists of the following rules:

MARK(2ndspos(square(y0), plus(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(plus(y0, y1), square(x0))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__square(mark(x0)))

A__2NDSNEG(s(N), cons(X, Z)) → A__2NDSNEG(s(mark(N)), cons2(X, mark(Z)))

MARK(2ndsneg(s(y0), square(x0))) → A__2NDSNEG(s(mark(y0)), a__square(mark(x0)))

MARK(times(times(x0, x1), y1)) → A__TIMES(a__times(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(X1, X2)) → MARK(X2)

MARK(plus(plus(x0, x1), y1)) → A__PLUS(a__plus(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(s(y0), from(x0))) → A__2NDSPOS(s(mark(y0)), a__from(mark(x0)))

A__2NDSNEG(s(N), cons(X, Z)) → MARK(Z)

A__TIMES(s(X), Y) → MARK(Y)

A__2NDSPOS(s(y0), cons(y1, plus(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__plus(mark(x0), mark(x1))))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → A__2NDSNEG(mark(N), mark(Z))

A__2NDSNEG(s(N), cons(X, Z)) → MARK(N)

MARK(2ndsneg(times(y0, y1), times(x0, x1))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(times(y0, y1), times(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

A__TIMES(s(X), Y) → A__PLUS(mark(Y), a__times(mark(X), mark(Y)))

MARK(2ndspos(square(y0), cons2(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), cons2(x0, mark(x1)))

MARK(plus(square(x0), y1)) → A__PLUS(a__square(mark(x0)), mark(y1))

MARK(plus(s(x0), y1)) → A__PLUS(s(mark(x0)), mark(y1))

A__PI(plus(x0, x1)) → A__2NDSPOS(a__plus(mark(x0), mark(x1)), a__from(0))

MARK(2ndspos(square(y0), cons(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), cons(mark(x0), x1))

MARK(2ndsneg(times(y0, y1), cons2(x0, x1))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), cons2(x0, mark(x1)))

A__PI(times(y0, y1)) → A__2NDSPOS(a__times(mark(y0), mark(y1)), cons(mark(0), from(s(0))))

MARK(cons2(X1, X2)) → MARK(X2)

MARK(2ndsneg(square(y0), cons2(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), cons2(x0, mark(x1)))

A__PLUS(s(square(x0)), y1) → A__PLUS(a__square(mark(x0)), mark(y1))

MARK(2ndspos(X1, X2)) → MARK(X1)

MARK(s(X)) → MARK(X)

MARK(2ndsneg(square(y0), plus(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(2ndsneg(plus(y0, y1), square(x0))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndspos(plus(y0, y1), cons2(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(pi(X)) → MARK(X)

MARK(2ndsneg(s(y0), from(x0))) → A__2NDSNEG(s(mark(y0)), a__from(mark(x0)))

MARK(plus(X1, X2)) → MARK(X2)

A__2NDSPOS(s(N), cons(X, Z)) → MARK(Z)

MARK(pi(X)) → A__PI(mark(X))

MARK(2ndsneg(times(y0, y1), from(x0))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndsneg(times(y0, y1), square(x0))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndspos(plus(y0, y1), cons(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), cons(mark(x0), x1))

MARK(2ndsneg(square(y0), times(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(times(y0, y1), cons2(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(2ndsneg(X1, X2)) → MARK(X2)

MARK(2ndsneg(square(y0), from(x0))) → A__2NDSNEG(a__square(mark(y0)), a__from(mark(x0)))

MARK(2ndspos(square(y0), times(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(times(y0, y1), square(x0))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndsneg(s(y0), cons(x0, x1))) → A__2NDSNEG(s(mark(y0)), cons(mark(x0), x1))

MARK(times(square(x0), y1)) → A__TIMES(a__square(mark(x0)), mark(y1))

A__PI(X) → A__FROM(0)

MARK(from(X)) → A__FROM(mark(X))

MARK(2ndspos(s(y0), plus(x0, x1))) → A__2NDSPOS(s(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(plus(X1, X2)) → MARK(X1)

A__2NDSPOS(s(y0), cons(y1, times(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__times(mark(x0), mark(x1))))

A__PI(X) → MARK(X)

MARK(2ndspos(s(y0), cons(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons(mark(x0), x1))

MARK(times(s(x0), y1)) → A__TIMES(s(mark(x0)), mark(y1))

MARK(2ndsneg(plus(y0, y1), cons(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), cons(mark(x0), x1))

A__PLUS(s(X), Y) → MARK(Y)

A__TIMES(s(X), Y) → MARK(X)

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(N)

MARK(2ndsneg(s(y0), cons2(x0, x1))) → A__2NDSNEG(s(mark(y0)), cons2(x0, mark(x1)))

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(N)

MARK(cons(X1, X2)) → MARK(X1)

MARK(2ndsneg(plus(y0, y1), from(x0))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(times(X1, X2)) → MARK(X2)

MARK(2ndspos(times(y0, y1), cons(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), cons(mark(x0), x1))

A__PI(square(x0)) → A__2NDSPOS(a__square(mark(x0)), a__from(0))

MARK(2ndspos(times(y0, y1), plus(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(plus(y0, y1), times(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndsneg(square(y0), cons(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), cons(mark(x0), x1))

MARK(2ndspos(s(y0), times(x0, x1))) → A__2NDSPOS(s(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndsneg(X1, X2)) → MARK(X1)

MARK(2ndsneg(square(y0), square(x0))) → A__2NDSNEG(a__square(mark(y0)), a__square(mark(x0)))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(Y)

MARK(rcons(X1, X2)) → MARK(X1)

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(Z)

MARK(square(X)) → MARK(X)

A__PI(s(x0)) → A__2NDSPOS(s(mark(x0)), a__from(0))

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(Y)

A__FROM(X) → MARK(X)

MARK(2ndsneg(s(y0), plus(x0, x1))) → A__2NDSNEG(s(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(times(plus(x0, x1), y1)) → A__TIMES(a__plus(mark(x0), mark(x1)), mark(y1))

A__2NDSPOS(s(y0), cons(y1, from(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__from(mark(x0))))

MARK(rcons(X1, X2)) → MARK(X2)

MARK(2ndspos(s(y0), square(x0))) → A__2NDSPOS(s(mark(y0)), a__square(mark(x0)))

MARK(2ndspos(s(y0), cons2(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(x0, mark(x1)))

A__TIMES(s(X), Y) → A__TIMES(mark(X), mark(Y))

MARK(2ndsneg(plus(y0, y1), cons2(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), cons2(x0, mark(x1)))

A__PLUS(s(X), Y) → MARK(X)

A__2NDSPOS(s(y0), cons(y1, cons(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, cons(mark(x0), x1)))

MARK(times(X1, X2)) → MARK(X1)

MARK(plus(times(x0, x1), y1)) → A__PLUS(a__times(mark(x0), mark(x1)), mark(y1))

A__SQUARE(X) → MARK(X)

A__2NDSPOS(s(N), cons(X, Z)) → MARK(N)

MARK(negrecip(X)) → MARK(X)

A__PLUS(0, Y) → MARK(Y)

MARK(posrecip(X)) → MARK(X)

MARK(2ndspos(square(y0), from(x0))) → A__2NDSPOS(a__square(mark(y0)), a__from(mark(x0)))

MARK(2ndspos(times(y0, y1), from(x0))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__from(mark(x0)))

A__PLUS(s(times(x0, x1)), y1) → A__PLUS(a__times(mark(x0), mark(x1)), mark(y1))

A__2NDSPOS(s(y0), cons(y1, square(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__square(mark(x0))))

A__PLUS(s(0), y1) → A__PLUS(0, mark(y1))

MARK(plus(0, y1)) → A__PLUS(0, mark(y1))

MARK(from(X)) → MARK(X)

MARK(2ndsneg(s(y0), times(x0, x1))) → A__2NDSNEG(s(mark(y0)), a__times(mark(x0), mark(x1)))

A__SQUARE(X) → A__TIMES(mark(X), mark(X))

MARK(2ndspos(square(y0), square(x0))) → A__2NDSPOS(a__square(mark(y0)), a__square(mark(x0)))

A__PLUS(s(plus(x0, x1)), y1) → A__PLUS(a__plus(mark(x0), mark(x1)), mark(y1))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(Z)

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → A__2NDSPOS(mark(N), mark(Z))

MARK(2ndsneg(plus(y0, y1), plus(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(square(X)) → A__SQUARE(mark(X))

MARK(2ndspos(plus(y0, y1), from(x0))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndsneg(times(y0, y1), plus(x0, x1))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(2ndsneg(plus(y0, y1), times(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(plus(y0, y1), plus(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(2ndsneg(times(y0, y1), cons(x0, x1))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), cons(mark(x0), x1))

A__PLUS(s(s(x0)), y1) → A__PLUS(s(mark(x0)), mark(y1))

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))

a__2ndspos(0, Z) → rnil

a__2ndspos(s(N), cons(X, Z)) → a__2ndspos(s(mark(N)), cons2(X, mark(Z)))

a__2ndspos(s(N), cons2(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))

a__2ndsneg(0, Z) → rnil

a__2ndsneg(s(N), cons(X, Z)) → a__2ndsneg(s(mark(N)), cons2(X, mark(Z)))

a__2ndsneg(s(N), cons2(X, cons(Y, Z))) → rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z)))

a__pi(X) → a__2ndspos(mark(X), a__from(0))

a__plus(0, Y) → mark(Y)

a__plus(s(X), Y) → s(a__plus(mark(X), mark(Y)))

a__times(0, Y) → 0

a__times(s(X), Y) → a__plus(mark(Y), a__times(mark(X), mark(Y)))

a__square(X) → a__times(mark(X), mark(X))

mark(from(X)) → a__from(mark(X))

mark(2ndspos(X1, X2)) → a__2ndspos(mark(X1), mark(X2))

mark(2ndsneg(X1, X2)) → a__2ndsneg(mark(X1), mark(X2))

mark(pi(X)) → a__pi(mark(X))

mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))

mark(times(X1, X2)) → a__times(mark(X1), mark(X2))

mark(square(X)) → a__square(mark(X))

mark(0) → 0

mark(s(X)) → s(mark(X))

mark(posrecip(X)) → posrecip(mark(X))

mark(negrecip(X)) → negrecip(mark(X))

mark(nil) → nil

mark(cons(X1, X2)) → cons(mark(X1), X2)

mark(cons2(X1, X2)) → cons2(X1, mark(X2))

mark(rnil) → rnil

mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))

a__from(X) → from(X)

a__2ndspos(X1, X2) → 2ndspos(X1, X2)

a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)

a__pi(X) → pi(X)

a__plus(X1, X2) → plus(X1, X2)

a__times(X1, X2) → times(X1, X2)

a__square(X) → square(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule A__TIMES(s(X), Y) → A__TIMES(mark(X), mark(Y)) at position [0] we obtained the following new rules:

A__TIMES(s(s(x0)), y1) → A__TIMES(s(mark(x0)), mark(y1))

A__TIMES(s(cons2(x0, x1)), y1) → A__TIMES(cons2(x0, mark(x1)), mark(y1))

A__TIMES(s(nil), y1) → A__TIMES(nil, mark(y1))

A__TIMES(s(square(x0)), y1) → A__TIMES(a__square(mark(x0)), mark(y1))

A__TIMES(s(plus(x0, x1)), y1) → A__TIMES(a__plus(mark(x0), mark(x1)), mark(y1))

A__TIMES(s(pi(x0)), y1) → A__TIMES(a__pi(mark(x0)), mark(y1))

A__TIMES(s(2ndsneg(x0, x1)), y1) → A__TIMES(a__2ndsneg(mark(x0), mark(x1)), mark(y1))

A__TIMES(s(0), y1) → A__TIMES(0, mark(y1))

A__TIMES(s(2ndspos(x0, x1)), y1) → A__TIMES(a__2ndspos(mark(x0), mark(x1)), mark(y1))

A__TIMES(s(from(x0)), y1) → A__TIMES(a__from(mark(x0)), mark(y1))

A__TIMES(s(rcons(x0, x1)), y1) → A__TIMES(rcons(mark(x0), mark(x1)), mark(y1))

A__TIMES(s(posrecip(x0)), y1) → A__TIMES(posrecip(mark(x0)), mark(y1))

A__TIMES(s(cons(x0, x1)), y1) → A__TIMES(cons(mark(x0), x1), mark(y1))

A__TIMES(s(negrecip(x0)), y1) → A__TIMES(negrecip(mark(x0)), mark(y1))

A__TIMES(s(rnil), y1) → A__TIMES(rnil, mark(y1))

A__TIMES(s(times(x0, x1)), y1) → A__TIMES(a__times(mark(x0), mark(x1)), mark(y1))



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ Narrowing
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ QDP
              ↳ Narrowing
                ↳ QDP
                  ↳ DependencyGraphProof
                    ↳ QDP
                      ↳ Narrowing
                        ↳ QDP
                          ↳ DependencyGraphProof
                            ↳ QDP
                              ↳ Narrowing
                                ↳ QDP
                                  ↳ DependencyGraphProof
                                    ↳ QDP
                                      ↳ Narrowing
                                        ↳ QDP
                                          ↳ DependencyGraphProof
                                            ↳ QDP
                                              ↳ Narrowing
                                                ↳ QDP
                                                  ↳ DependencyGraphProof
                                                    ↳ QDP
                                                      ↳ Narrowing
                                                        ↳ QDP
                                                          ↳ DependencyGraphProof
                                                            ↳ QDP
                                                              ↳ Narrowing
                                                                ↳ QDP
                                                                  ↳ DependencyGraphProof
                                                                    ↳ QDP
                                                                      ↳ Narrowing
                                                                        ↳ QDP
                                                                          ↳ DependencyGraphProof
                                                                            ↳ QDP
                                                                              ↳ Narrowing
                                                                                ↳ QDP
                                                                                  ↳ DependencyGraphProof
                                                                                    ↳ QDP
                                                                                      ↳ Narrowing
                                                                                        ↳ QDP
                                                                                          ↳ DependencyGraphProof
                                                                                            ↳ QDP
                                                                                              ↳ Narrowing
                                                                                                ↳ QDP
                                                                                                  ↳ DependencyGraphProof
                                                                                                    ↳ QDP
                                                                                                      ↳ Narrowing
                                                                                                        ↳ QDP
                                                                                                          ↳ DependencyGraphProof
                                                                                                            ↳ QDP
                                                                                                              ↳ Narrowing
                                                                                                                ↳ QDP
                                                                                                                  ↳ DependencyGraphProof
                                                                                                                    ↳ QDP
                                                                                                                      ↳ Narrowing
                                                                                                                        ↳ QDP
                                                                                                                          ↳ DependencyGraphProof
                                                                                                                            ↳ QDP
                                                                                                                              ↳ Narrowing
                                                                                                                                ↳ QDP
                                                                                                                                  ↳ DependencyGraphProof
                                                                                                                                    ↳ QDP
                                                                                                                                      ↳ Narrowing
                                                                                                                                        ↳ QDP
                                                                                                                                          ↳ DependencyGraphProof
                                                                                                                                            ↳ QDP
                                                                                                                                              ↳ Narrowing
                                                                                                                                                ↳ QDP
                                                                                                                                                  ↳ DependencyGraphProof
                                                                                                                                                    ↳ QDP
                                                                                                                                                      ↳ Narrowing
                                                                                                                                                        ↳ QDP
                                                                                                                                                          ↳ DependencyGraphProof
                                                                                                                                                            ↳ QDP
                                                                                                                                                              ↳ Narrowing
                                                                                                                                                                ↳ QDP
                                                                                                                                                                  ↳ DependencyGraphProof
                                                                                                                                                                    ↳ QDP
                                                                                                                                                                      ↳ Narrowing
                                                                                                                                                                        ↳ QDP
                                                                                                                                                                          ↳ DependencyGraphProof
                                                                                                                                                                            ↳ QDP
                                                                                                                                                                              ↳ Narrowing
                                                                                                                                                                                ↳ QDP
                                                                                                                                                                                  ↳ DependencyGraphProof
                                                                                                                                                                                    ↳ QDP
                                                                                                                                                                                      ↳ Narrowing
                                                                                                                                                                                        ↳ QDP
                                                                                                                                                                                          ↳ DependencyGraphProof
                                                                                                                                                                                            ↳ QDP
                                                                                                                                                                                              ↳ QDPOrderProof
                                                                                                                                                                                                ↳ QDP
                                                                                                                                                                                                  ↳ QDPOrderProof
                                                                                                                                                                                                    ↳ QDP
                                                                                                                                                                                                      ↳ QDPOrderProof
                                                                                                                                                                                                        ↳ QDP
                                                                                                                                                                                                          ↳ QDPOrderProof
                                                                                                                                                                                                            ↳ QDP
                                                                                                                                                                                                              ↳ QDPOrderProof
                                                                                                                                                                                                                ↳ QDP
                                                                                                                                                                                                                  ↳ QDPOrderProof
                                                                                                                                                                                                                    ↳ QDP
                                                                                                                                                                                                                      ↳ QDPOrderProof
                                                                                                                                                                                                                        ↳ QDP
                                                                                                                                                                                                                          ↳ QDPOrderProof
                                                                                                                                                                                                                            ↳ QDP
                                                                                                                                                                                                                              ↳ Narrowing
                                                                                                                                                                                                                                ↳ QDP
                                                                                                                                                                                                                                  ↳ DependencyGraphProof
                                                                                                                                                                                                                                    ↳ QDP
                                                                                                                                                                                                                                      ↳ Narrowing
                                                                                                                                                                                                                                        ↳ QDP
                                                                                                                                                                                                                                          ↳ DependencyGraphProof

Q DP problem:
The TRS P consists of the following rules:

MARK(2ndspos(plus(y0, y1), square(x0))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndspos(square(y0), plus(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__plus(mark(x0), mark(x1)))

A__2NDSNEG(s(N), cons(X, Z)) → A__2NDSNEG(s(mark(N)), cons2(X, mark(Z)))

MARK(2ndsneg(s(y0), square(x0))) → A__2NDSNEG(s(mark(y0)), a__square(mark(x0)))

MARK(2ndspos(X1, X2)) → MARK(X2)

MARK(times(times(x0, x1), y1)) → A__TIMES(a__times(mark(x0), mark(x1)), mark(y1))

MARK(plus(plus(x0, x1), y1)) → A__PLUS(a__plus(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(s(y0), from(x0))) → A__2NDSPOS(s(mark(y0)), a__from(mark(x0)))

A__2NDSNEG(s(N), cons(X, Z)) → MARK(Z)

A__TIMES(s(rcons(x0, x1)), y1) → A__TIMES(rcons(mark(x0), mark(x1)), mark(y1))

A__TIMES(s(rnil), y1) → A__TIMES(rnil, mark(y1))

A__TIMES(s(X), Y) → MARK(Y)

A__2NDSPOS(s(y0), cons(y1, plus(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__plus(mark(x0), mark(x1))))

A__2NDSNEG(s(N), cons(X, Z)) → MARK(N)

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → A__2NDSNEG(mark(N), mark(Z))

MARK(2ndsneg(times(y0, y1), times(x0, x1))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(times(y0, y1), times(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

A__TIMES(s(X), Y) → A__PLUS(mark(Y), a__times(mark(X), mark(Y)))

A__TIMES(s(from(x0)), y1) → A__TIMES(a__from(mark(x0)), mark(y1))

MARK(2ndspos(square(y0), cons2(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), cons2(x0, mark(x1)))

MARK(plus(square(x0), y1)) → A__PLUS(a__square(mark(x0)), mark(y1))

MARK(plus(s(x0), y1)) → A__PLUS(s(mark(x0)), mark(y1))

MARK(2ndspos(square(y0), cons(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), cons(mark(x0), x1))

A__PI(plus(x0, x1)) → A__2NDSPOS(a__plus(mark(x0), mark(x1)), a__from(0))

MARK(2ndsneg(times(y0, y1), cons2(x0, x1))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), cons2(x0, mark(x1)))

A__TIMES(s(negrecip(x0)), y1) → A__TIMES(negrecip(mark(x0)), mark(y1))

A__PI(times(y0, y1)) → A__2NDSPOS(a__times(mark(y0), mark(y1)), cons(mark(0), from(s(0))))

MARK(cons2(X1, X2)) → MARK(X2)

MARK(2ndsneg(square(y0), cons2(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), cons2(x0, mark(x1)))

A__TIMES(s(square(x0)), y1) → A__TIMES(a__square(mark(x0)), mark(y1))

MARK(2ndsneg(plus(y0, y1), square(x0))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndsneg(square(y0), plus(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(s(X)) → MARK(X)

MARK(2ndspos(X1, X2)) → MARK(X1)

A__PLUS(s(square(x0)), y1) → A__PLUS(a__square(mark(x0)), mark(y1))

MARK(2ndspos(plus(y0, y1), cons2(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(pi(X)) → MARK(X)

MARK(2ndsneg(s(y0), from(x0))) → A__2NDSNEG(s(mark(y0)), a__from(mark(x0)))

MARK(plus(X1, X2)) → MARK(X2)

A__TIMES(s(2ndsneg(x0, x1)), y1) → A__TIMES(a__2ndsneg(mark(x0), mark(x1)), mark(y1))

A__2NDSPOS(s(N), cons(X, Z)) → MARK(Z)

MARK(pi(X)) → A__PI(mark(X))

MARK(2ndsneg(square(y0), times(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(plus(y0, y1), cons(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), cons(mark(x0), x1))

MARK(2ndsneg(times(y0, y1), square(x0))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndsneg(times(y0, y1), from(x0))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndspos(times(y0, y1), cons2(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(2ndsneg(X1, X2)) → MARK(X2)

MARK(2ndsneg(square(y0), from(x0))) → A__2NDSNEG(a__square(mark(y0)), a__from(mark(x0)))

A__TIMES(s(times(x0, x1)), y1) → A__TIMES(a__times(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(times(y0, y1), square(x0))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndspos(square(y0), times(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(times(square(x0), y1)) → A__TIMES(a__square(mark(x0)), mark(y1))

MARK(2ndsneg(s(y0), cons(x0, x1))) → A__2NDSNEG(s(mark(y0)), cons(mark(x0), x1))

MARK(2ndspos(s(y0), plus(x0, x1))) → A__2NDSPOS(s(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(from(X)) → A__FROM(mark(X))

A__PI(X) → A__FROM(0)

MARK(plus(X1, X2)) → MARK(X1)

A__2NDSPOS(s(y0), cons(y1, times(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__times(mark(x0), mark(x1))))

MARK(2ndspos(s(y0), cons(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons(mark(x0), x1))

A__PI(X) → MARK(X)

MARK(times(s(x0), y1)) → A__TIMES(s(mark(x0)), mark(y1))

MARK(2ndsneg(plus(y0, y1), cons(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), cons(mark(x0), x1))

A__PLUS(s(X), Y) → MARK(Y)

A__TIMES(s(X), Y) → MARK(X)

A__TIMES(s(s(x0)), y1) → A__TIMES(s(mark(x0)), mark(y1))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(N)

MARK(2ndsneg(s(y0), cons2(x0, x1))) → A__2NDSNEG(s(mark(y0)), cons2(x0, mark(x1)))

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(N)

MARK(cons(X1, X2)) → MARK(X1)

MARK(2ndsneg(plus(y0, y1), from(x0))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndspos(times(y0, y1), cons(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), cons(mark(x0), x1))

MARK(times(X1, X2)) → MARK(X2)

A__TIMES(s(0), y1) → A__TIMES(0, mark(y1))

A__PI(square(x0)) → A__2NDSPOS(a__square(mark(x0)), a__from(0))

MARK(2ndsneg(square(y0), cons(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), cons(mark(x0), x1))

MARK(2ndspos(plus(y0, y1), times(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(times(y0, y1), plus(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(s(y0), times(x0, x1))) → A__2NDSPOS(s(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndsneg(X1, X2)) → MARK(X1)

MARK(2ndsneg(square(y0), square(x0))) → A__2NDSNEG(a__square(mark(y0)), a__square(mark(x0)))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(Y)

MARK(rcons(X1, X2)) → MARK(X1)

A__TIMES(s(nil), y1) → A__TIMES(nil, mark(y1))

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(Z)

MARK(square(X)) → MARK(X)

A__PI(s(x0)) → A__2NDSPOS(s(mark(x0)), a__from(0))

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(Y)

A__FROM(X) → MARK(X)

MARK(times(plus(x0, x1), y1)) → A__TIMES(a__plus(mark(x0), mark(x1)), mark(y1))

MARK(2ndsneg(s(y0), plus(x0, x1))) → A__2NDSNEG(s(mark(y0)), a__plus(mark(x0), mark(x1)))

A__2NDSPOS(s(y0), cons(y1, from(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__from(mark(x0))))

A__TIMES(s(posrecip(x0)), y1) → A__TIMES(posrecip(mark(x0)), mark(y1))

MARK(2ndspos(s(y0), square(x0))) → A__2NDSPOS(s(mark(y0)), a__square(mark(x0)))

MARK(rcons(X1, X2)) → MARK(X2)

MARK(2ndspos(s(y0), cons2(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(x0, mark(x1)))

MARK(2ndsneg(plus(y0, y1), cons2(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), cons2(x0, mark(x1)))

A__2NDSPOS(s(y0), cons(y1, cons(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, cons(mark(x0), x1)))

A__PLUS(s(X), Y) → MARK(X)

MARK(times(X1, X2)) → MARK(X1)

A__TIMES(s(plus(x0, x1)), y1) → A__TIMES(a__plus(mark(x0), mark(x1)), mark(y1))

MARK(plus(times(x0, x1), y1)) → A__PLUS(a__times(mark(x0), mark(x1)), mark(y1))

A__SQUARE(X) → MARK(X)

A__TIMES(s(2ndspos(x0, x1)), y1) → A__TIMES(a__2ndspos(mark(x0), mark(x1)), mark(y1))

MARK(negrecip(X)) → MARK(X)

A__2NDSPOS(s(N), cons(X, Z)) → MARK(N)

A__PLUS(0, Y) → MARK(Y)

MARK(posrecip(X)) → MARK(X)

MARK(2ndspos(square(y0), from(x0))) → A__2NDSPOS(a__square(mark(y0)), a__from(mark(x0)))

MARK(2ndspos(times(y0, y1), from(x0))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__from(mark(x0)))

A__PLUS(s(times(x0, x1)), y1) → A__PLUS(a__times(mark(x0), mark(x1)), mark(y1))

A__2NDSPOS(s(y0), cons(y1, square(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__square(mark(x0))))

A__PLUS(s(0), y1) → A__PLUS(0, mark(y1))

A__TIMES(s(cons2(x0, x1)), y1) → A__TIMES(cons2(x0, mark(x1)), mark(y1))

MARK(plus(0, y1)) → A__PLUS(0, mark(y1))

MARK(from(X)) → MARK(X)

MARK(2ndsneg(s(y0), times(x0, x1))) → A__2NDSNEG(s(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(square(y0), square(x0))) → A__2NDSPOS(a__square(mark(y0)), a__square(mark(x0)))

A__SQUARE(X) → A__TIMES(mark(X), mark(X))

A__PLUS(s(plus(x0, x1)), y1) → A__PLUS(a__plus(mark(x0), mark(x1)), mark(y1))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(Z)

A__TIMES(s(pi(x0)), y1) → A__TIMES(a__pi(mark(x0)), mark(y1))

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → A__2NDSPOS(mark(N), mark(Z))

MARK(2ndsneg(plus(y0, y1), plus(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(square(X)) → A__SQUARE(mark(X))

MARK(2ndspos(plus(y0, y1), from(x0))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndsneg(plus(y0, y1), times(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndsneg(times(y0, y1), plus(x0, x1))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(plus(y0, y1), plus(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

A__PLUS(s(s(x0)), y1) → A__PLUS(s(mark(x0)), mark(y1))

MARK(2ndsneg(times(y0, y1), cons(x0, x1))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), cons(mark(x0), x1))

A__TIMES(s(cons(x0, x1)), y1) → A__TIMES(cons(mark(x0), x1), mark(y1))

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))

a__2ndspos(0, Z) → rnil

a__2ndspos(s(N), cons(X, Z)) → a__2ndspos(s(mark(N)), cons2(X, mark(Z)))

a__2ndspos(s(N), cons2(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))

a__2ndsneg(0, Z) → rnil

a__2ndsneg(s(N), cons(X, Z)) → a__2ndsneg(s(mark(N)), cons2(X, mark(Z)))

a__2ndsneg(s(N), cons2(X, cons(Y, Z))) → rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z)))

a__pi(X) → a__2ndspos(mark(X), a__from(0))

a__plus(0, Y) → mark(Y)

a__plus(s(X), Y) → s(a__plus(mark(X), mark(Y)))

a__times(0, Y) → 0

a__times(s(X), Y) → a__plus(mark(Y), a__times(mark(X), mark(Y)))

a__square(X) → a__times(mark(X), mark(X))

mark(from(X)) → a__from(mark(X))

mark(2ndspos(X1, X2)) → a__2ndspos(mark(X1), mark(X2))

mark(2ndsneg(X1, X2)) → a__2ndsneg(mark(X1), mark(X2))

mark(pi(X)) → a__pi(mark(X))

mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))

mark(times(X1, X2)) → a__times(mark(X1), mark(X2))

mark(square(X)) → a__square(mark(X))

mark(0) → 0

mark(s(X)) → s(mark(X))

mark(posrecip(X)) → posrecip(mark(X))

mark(negrecip(X)) → negrecip(mark(X))

mark(nil) → nil

mark(cons(X1, X2)) → cons(mark(X1), X2)

mark(cons2(X1, X2)) → cons2(X1, mark(X2))

mark(rnil) → rnil

mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))

a__from(X) → from(X)

a__2ndspos(X1, X2) → 2ndspos(X1, X2)

a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)

a__pi(X) → pi(X)

a__plus(X1, X2) → plus(X1, X2)

a__times(X1, X2) → times(X1, X2)

a__square(X) → square(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 8 less nodes.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ Narrowing
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ QDP
              ↳ Narrowing
                ↳ QDP
                  ↳ DependencyGraphProof
                    ↳ QDP
                      ↳ Narrowing
                        ↳ QDP
                          ↳ DependencyGraphProof
                            ↳ QDP
                              ↳ Narrowing
                                ↳ QDP
                                  ↳ DependencyGraphProof
                                    ↳ QDP
                                      ↳ Narrowing
                                        ↳ QDP
                                          ↳ DependencyGraphProof
                                            ↳ QDP
                                              ↳ Narrowing
                                                ↳ QDP
                                                  ↳ DependencyGraphProof
                                                    ↳ QDP
                                                      ↳ Narrowing
                                                        ↳ QDP
                                                          ↳ DependencyGraphProof
                                                            ↳ QDP
                                                              ↳ Narrowing
                                                                ↳ QDP
                                                                  ↳ DependencyGraphProof
                                                                    ↳ QDP
                                                                      ↳ Narrowing
                                                                        ↳ QDP
                                                                          ↳ DependencyGraphProof
                                                                            ↳ QDP
                                                                              ↳ Narrowing
                                                                                ↳ QDP
                                                                                  ↳ DependencyGraphProof
                                                                                    ↳ QDP
                                                                                      ↳ Narrowing
                                                                                        ↳ QDP
                                                                                          ↳ DependencyGraphProof
                                                                                            ↳ QDP
                                                                                              ↳ Narrowing
                                                                                                ↳ QDP
                                                                                                  ↳ DependencyGraphProof
                                                                                                    ↳ QDP
                                                                                                      ↳ Narrowing
                                                                                                        ↳ QDP
                                                                                                          ↳ DependencyGraphProof
                                                                                                            ↳ QDP
                                                                                                              ↳ Narrowing
                                                                                                                ↳ QDP
                                                                                                                  ↳ DependencyGraphProof
                                                                                                                    ↳ QDP
                                                                                                                      ↳ Narrowing
                                                                                                                        ↳ QDP
                                                                                                                          ↳ DependencyGraphProof
                                                                                                                            ↳ QDP
                                                                                                                              ↳ Narrowing
                                                                                                                                ↳ QDP
                                                                                                                                  ↳ DependencyGraphProof
                                                                                                                                    ↳ QDP
                                                                                                                                      ↳ Narrowing
                                                                                                                                        ↳ QDP
                                                                                                                                          ↳ DependencyGraphProof
                                                                                                                                            ↳ QDP
                                                                                                                                              ↳ Narrowing
                                                                                                                                                ↳ QDP
                                                                                                                                                  ↳ DependencyGraphProof
                                                                                                                                                    ↳ QDP
                                                                                                                                                      ↳ Narrowing
                                                                                                                                                        ↳ QDP
                                                                                                                                                          ↳ DependencyGraphProof
                                                                                                                                                            ↳ QDP
                                                                                                                                                              ↳ Narrowing
                                                                                                                                                                ↳ QDP
                                                                                                                                                                  ↳ DependencyGraphProof
                                                                                                                                                                    ↳ QDP
                                                                                                                                                                      ↳ Narrowing
                                                                                                                                                                        ↳ QDP
                                                                                                                                                                          ↳ DependencyGraphProof
                                                                                                                                                                            ↳ QDP
                                                                                                                                                                              ↳ Narrowing
                                                                                                                                                                                ↳ QDP
                                                                                                                                                                                  ↳ DependencyGraphProof
                                                                                                                                                                                    ↳ QDP
                                                                                                                                                                                      ↳ Narrowing
                                                                                                                                                                                        ↳ QDP
                                                                                                                                                                                          ↳ DependencyGraphProof
                                                                                                                                                                                            ↳ QDP
                                                                                                                                                                                              ↳ QDPOrderProof
                                                                                                                                                                                                ↳ QDP
                                                                                                                                                                                                  ↳ QDPOrderProof
                                                                                                                                                                                                    ↳ QDP
                                                                                                                                                                                                      ↳ QDPOrderProof
                                                                                                                                                                                                        ↳ QDP
                                                                                                                                                                                                          ↳ QDPOrderProof
                                                                                                                                                                                                            ↳ QDP
                                                                                                                                                                                                              ↳ QDPOrderProof
                                                                                                                                                                                                                ↳ QDP
                                                                                                                                                                                                                  ↳ QDPOrderProof
                                                                                                                                                                                                                    ↳ QDP
                                                                                                                                                                                                                      ↳ QDPOrderProof
                                                                                                                                                                                                                        ↳ QDP
                                                                                                                                                                                                                          ↳ QDPOrderProof
                                                                                                                                                                                                                            ↳ QDP
                                                                                                                                                                                                                              ↳ Narrowing
                                                                                                                                                                                                                                ↳ QDP
                                                                                                                                                                                                                                  ↳ DependencyGraphProof
                                                                                                                                                                                                                                    ↳ QDP
                                                                                                                                                                                                                                      ↳ Narrowing
                                                                                                                                                                                                                                        ↳ QDP
                                                                                                                                                                                                                                          ↳ DependencyGraphProof
                                                                                                                                                                                                                                            ↳ QDP
                                                                                                                                                                                                                                              ↳ QDPOrderProof

Q DP problem:
The TRS P consists of the following rules:

MARK(2ndspos(square(y0), plus(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(plus(y0, y1), square(x0))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__square(mark(x0)))

A__2NDSNEG(s(N), cons(X, Z)) → A__2NDSNEG(s(mark(N)), cons2(X, mark(Z)))

MARK(2ndsneg(s(y0), square(x0))) → A__2NDSNEG(s(mark(y0)), a__square(mark(x0)))

MARK(times(times(x0, x1), y1)) → A__TIMES(a__times(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(X1, X2)) → MARK(X2)

MARK(plus(plus(x0, x1), y1)) → A__PLUS(a__plus(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(s(y0), from(x0))) → A__2NDSPOS(s(mark(y0)), a__from(mark(x0)))

A__2NDSNEG(s(N), cons(X, Z)) → MARK(Z)

A__TIMES(s(X), Y) → MARK(Y)

A__2NDSPOS(s(y0), cons(y1, plus(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__plus(mark(x0), mark(x1))))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → A__2NDSNEG(mark(N), mark(Z))

A__2NDSNEG(s(N), cons(X, Z)) → MARK(N)

MARK(2ndsneg(times(y0, y1), times(x0, x1))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(times(y0, y1), times(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

A__TIMES(s(X), Y) → A__PLUS(mark(Y), a__times(mark(X), mark(Y)))

A__TIMES(s(from(x0)), y1) → A__TIMES(a__from(mark(x0)), mark(y1))

MARK(2ndspos(square(y0), cons2(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), cons2(x0, mark(x1)))

MARK(plus(square(x0), y1)) → A__PLUS(a__square(mark(x0)), mark(y1))

MARK(plus(s(x0), y1)) → A__PLUS(s(mark(x0)), mark(y1))

A__PI(plus(x0, x1)) → A__2NDSPOS(a__plus(mark(x0), mark(x1)), a__from(0))

MARK(2ndspos(square(y0), cons(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), cons(mark(x0), x1))

MARK(2ndsneg(times(y0, y1), cons2(x0, x1))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), cons2(x0, mark(x1)))

A__PI(times(y0, y1)) → A__2NDSPOS(a__times(mark(y0), mark(y1)), cons(mark(0), from(s(0))))

MARK(cons2(X1, X2)) → MARK(X2)

MARK(2ndsneg(square(y0), cons2(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), cons2(x0, mark(x1)))

MARK(2ndsneg(plus(y0, y1), square(x0))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__square(mark(x0)))

A__TIMES(s(square(x0)), y1) → A__TIMES(a__square(mark(x0)), mark(y1))

A__PLUS(s(square(x0)), y1) → A__PLUS(a__square(mark(x0)), mark(y1))

MARK(2ndspos(X1, X2)) → MARK(X1)

MARK(s(X)) → MARK(X)

MARK(2ndsneg(square(y0), plus(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(plus(y0, y1), cons2(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(pi(X)) → MARK(X)

MARK(2ndsneg(s(y0), from(x0))) → A__2NDSNEG(s(mark(y0)), a__from(mark(x0)))

MARK(plus(X1, X2)) → MARK(X2)

A__TIMES(s(2ndsneg(x0, x1)), y1) → A__TIMES(a__2ndsneg(mark(x0), mark(x1)), mark(y1))

A__2NDSPOS(s(N), cons(X, Z)) → MARK(Z)

MARK(pi(X)) → A__PI(mark(X))

MARK(2ndsneg(times(y0, y1), from(x0))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndsneg(times(y0, y1), square(x0))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndspos(plus(y0, y1), cons(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), cons(mark(x0), x1))

MARK(2ndsneg(square(y0), times(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(times(y0, y1), cons2(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(2ndsneg(X1, X2)) → MARK(X2)

MARK(2ndsneg(square(y0), from(x0))) → A__2NDSNEG(a__square(mark(y0)), a__from(mark(x0)))

A__TIMES(s(times(x0, x1)), y1) → A__TIMES(a__times(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(square(y0), times(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(times(y0, y1), square(x0))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndsneg(s(y0), cons(x0, x1))) → A__2NDSNEG(s(mark(y0)), cons(mark(x0), x1))

MARK(times(square(x0), y1)) → A__TIMES(a__square(mark(x0)), mark(y1))

A__PI(X) → A__FROM(0)

MARK(from(X)) → A__FROM(mark(X))

MARK(2ndspos(s(y0), plus(x0, x1))) → A__2NDSPOS(s(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(plus(X1, X2)) → MARK(X1)

A__2NDSPOS(s(y0), cons(y1, times(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__times(mark(x0), mark(x1))))

A__PI(X) → MARK(X)

MARK(2ndspos(s(y0), cons(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons(mark(x0), x1))

MARK(times(s(x0), y1)) → A__TIMES(s(mark(x0)), mark(y1))

MARK(2ndsneg(plus(y0, y1), cons(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), cons(mark(x0), x1))

A__PLUS(s(X), Y) → MARK(Y)

A__TIMES(s(X), Y) → MARK(X)

A__TIMES(s(s(x0)), y1) → A__TIMES(s(mark(x0)), mark(y1))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(N)

MARK(2ndsneg(s(y0), cons2(x0, x1))) → A__2NDSNEG(s(mark(y0)), cons2(x0, mark(x1)))

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(N)

MARK(cons(X1, X2)) → MARK(X1)

MARK(2ndsneg(plus(y0, y1), from(x0))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(times(X1, X2)) → MARK(X2)

MARK(2ndspos(times(y0, y1), cons(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), cons(mark(x0), x1))

A__PI(square(x0)) → A__2NDSPOS(a__square(mark(x0)), a__from(0))

MARK(2ndspos(times(y0, y1), plus(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(plus(y0, y1), times(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndsneg(square(y0), cons(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), cons(mark(x0), x1))

MARK(2ndspos(s(y0), times(x0, x1))) → A__2NDSPOS(s(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndsneg(X1, X2)) → MARK(X1)

MARK(2ndsneg(square(y0), square(x0))) → A__2NDSNEG(a__square(mark(y0)), a__square(mark(x0)))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(Y)

MARK(rcons(X1, X2)) → MARK(X1)

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(Z)

MARK(square(X)) → MARK(X)

A__PI(s(x0)) → A__2NDSPOS(s(mark(x0)), a__from(0))

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(Y)

A__FROM(X) → MARK(X)

MARK(2ndsneg(s(y0), plus(x0, x1))) → A__2NDSNEG(s(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(times(plus(x0, x1), y1)) → A__TIMES(a__plus(mark(x0), mark(x1)), mark(y1))

A__2NDSPOS(s(y0), cons(y1, from(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__from(mark(x0))))

MARK(rcons(X1, X2)) → MARK(X2)

MARK(2ndspos(s(y0), square(x0))) → A__2NDSPOS(s(mark(y0)), a__square(mark(x0)))

MARK(2ndspos(s(y0), cons2(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(x0, mark(x1)))

MARK(2ndsneg(plus(y0, y1), cons2(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), cons2(x0, mark(x1)))

A__PLUS(s(X), Y) → MARK(X)

A__2NDSPOS(s(y0), cons(y1, cons(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, cons(mark(x0), x1)))

MARK(times(X1, X2)) → MARK(X1)

A__TIMES(s(plus(x0, x1)), y1) → A__TIMES(a__plus(mark(x0), mark(x1)), mark(y1))

MARK(plus(times(x0, x1), y1)) → A__PLUS(a__times(mark(x0), mark(x1)), mark(y1))

A__SQUARE(X) → MARK(X)

A__TIMES(s(2ndspos(x0, x1)), y1) → A__TIMES(a__2ndspos(mark(x0), mark(x1)), mark(y1))

MARK(negrecip(X)) → MARK(X)

A__2NDSPOS(s(N), cons(X, Z)) → MARK(N)

A__PLUS(0, Y) → MARK(Y)

MARK(posrecip(X)) → MARK(X)

MARK(2ndspos(square(y0), from(x0))) → A__2NDSPOS(a__square(mark(y0)), a__from(mark(x0)))

MARK(2ndspos(times(y0, y1), from(x0))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__from(mark(x0)))

A__PLUS(s(times(x0, x1)), y1) → A__PLUS(a__times(mark(x0), mark(x1)), mark(y1))

A__2NDSPOS(s(y0), cons(y1, square(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__square(mark(x0))))

A__PLUS(s(0), y1) → A__PLUS(0, mark(y1))

MARK(plus(0, y1)) → A__PLUS(0, mark(y1))

MARK(from(X)) → MARK(X)

MARK(2ndsneg(s(y0), times(x0, x1))) → A__2NDSNEG(s(mark(y0)), a__times(mark(x0), mark(x1)))

A__SQUARE(X) → A__TIMES(mark(X), mark(X))

MARK(2ndspos(square(y0), square(x0))) → A__2NDSPOS(a__square(mark(y0)), a__square(mark(x0)))

A__PLUS(s(plus(x0, x1)), y1) → A__PLUS(a__plus(mark(x0), mark(x1)), mark(y1))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(Z)

A__TIMES(s(pi(x0)), y1) → A__TIMES(a__pi(mark(x0)), mark(y1))

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → A__2NDSPOS(mark(N), mark(Z))

MARK(2ndsneg(plus(y0, y1), plus(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(square(X)) → A__SQUARE(mark(X))

MARK(2ndspos(plus(y0, y1), from(x0))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndsneg(times(y0, y1), plus(x0, x1))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(2ndsneg(plus(y0, y1), times(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(plus(y0, y1), plus(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(2ndsneg(times(y0, y1), cons(x0, x1))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), cons(mark(x0), x1))

A__PLUS(s(s(x0)), y1) → A__PLUS(s(mark(x0)), mark(y1))

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))

a__2ndspos(0, Z) → rnil

a__2ndspos(s(N), cons(X, Z)) → a__2ndspos(s(mark(N)), cons2(X, mark(Z)))

a__2ndspos(s(N), cons2(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))

a__2ndsneg(0, Z) → rnil

a__2ndsneg(s(N), cons(X, Z)) → a__2ndsneg(s(mark(N)), cons2(X, mark(Z)))

a__2ndsneg(s(N), cons2(X, cons(Y, Z))) → rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z)))

a__pi(X) → a__2ndspos(mark(X), a__from(0))

a__plus(0, Y) → mark(Y)

a__plus(s(X), Y) → s(a__plus(mark(X), mark(Y)))

a__times(0, Y) → 0

a__times(s(X), Y) → a__plus(mark(Y), a__times(mark(X), mark(Y)))

a__square(X) → a__times(mark(X), mark(X))

mark(from(X)) → a__from(mark(X))

mark(2ndspos(X1, X2)) → a__2ndspos(mark(X1), mark(X2))

mark(2ndsneg(X1, X2)) → a__2ndsneg(mark(X1), mark(X2))

mark(pi(X)) → a__pi(mark(X))

mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))

mark(times(X1, X2)) → a__times(mark(X1), mark(X2))

mark(square(X)) → a__square(mark(X))

mark(0) → 0

mark(s(X)) → s(mark(X))

mark(posrecip(X)) → posrecip(mark(X))

mark(negrecip(X)) → negrecip(mark(X))

mark(nil) → nil

mark(cons(X1, X2)) → cons(mark(X1), X2)

mark(cons2(X1, X2)) → cons2(X1, mark(X2))

mark(rnil) → rnil

mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))

a__from(X) → from(X)

a__2ndspos(X1, X2) → 2ndspos(X1, X2)

a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)

a__pi(X) → pi(X)

a__plus(X1, X2) → plus(X1, X2)

a__times(X1, X2) → times(X1, X2)

a__square(X) → square(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

A__TIMES(s(from(x0)), y1) → A__TIMES(a__from(mark(x0)), mark(y1))

A__TIMES(s(2ndsneg(x0, x1)), y1) → A__TIMES(a__2ndsneg(mark(x0), mark(x1)), mark(y1))

A__TIMES(s(2ndspos(x0, x1)), y1) → A__TIMES(a__2ndspos(mark(x0), mark(x1)), mark(y1))

A__TIMES(s(pi(x0)), y1) → A__TIMES(a__pi(mark(x0)), mark(y1))
The remaining pairs can at least be oriented weakly.

MARK(2ndspos(square(y0), plus(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(plus(y0, y1), square(x0))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__square(mark(x0)))

A__2NDSNEG(s(N), cons(X, Z)) → A__2NDSNEG(s(mark(N)), cons2(X, mark(Z)))

MARK(2ndsneg(s(y0), square(x0))) → A__2NDSNEG(s(mark(y0)), a__square(mark(x0)))

MARK(times(times(x0, x1), y1)) → A__TIMES(a__times(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(X1, X2)) → MARK(X2)

MARK(plus(plus(x0, x1), y1)) → A__PLUS(a__plus(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(s(y0), from(x0))) → A__2NDSPOS(s(mark(y0)), a__from(mark(x0)))

A__2NDSNEG(s(N), cons(X, Z)) → MARK(Z)

A__TIMES(s(X), Y) → MARK(Y)

A__2NDSPOS(s(y0), cons(y1, plus(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__plus(mark(x0), mark(x1))))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → A__2NDSNEG(mark(N), mark(Z))

A__2NDSNEG(s(N), cons(X, Z)) → MARK(N)

MARK(2ndsneg(times(y0, y1), times(x0, x1))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(times(y0, y1), times(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

A__TIMES(s(X), Y) → A__PLUS(mark(Y), a__times(mark(X), mark(Y)))

MARK(2ndspos(square(y0), cons2(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), cons2(x0, mark(x1)))

MARK(plus(square(x0), y1)) → A__PLUS(a__square(mark(x0)), mark(y1))

MARK(plus(s(x0), y1)) → A__PLUS(s(mark(x0)), mark(y1))

A__PI(plus(x0, x1)) → A__2NDSPOS(a__plus(mark(x0), mark(x1)), a__from(0))

MARK(2ndspos(square(y0), cons(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), cons(mark(x0), x1))

MARK(2ndsneg(times(y0, y1), cons2(x0, x1))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), cons2(x0, mark(x1)))

A__PI(times(y0, y1)) → A__2NDSPOS(a__times(mark(y0), mark(y1)), cons(mark(0), from(s(0))))

MARK(cons2(X1, X2)) → MARK(X2)

MARK(2ndsneg(square(y0), cons2(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), cons2(x0, mark(x1)))

MARK(2ndsneg(plus(y0, y1), square(x0))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__square(mark(x0)))

A__TIMES(s(square(x0)), y1) → A__TIMES(a__square(mark(x0)), mark(y1))

A__PLUS(s(square(x0)), y1) → A__PLUS(a__square(mark(x0)), mark(y1))

MARK(2ndspos(X1, X2)) → MARK(X1)

MARK(s(X)) → MARK(X)

MARK(2ndsneg(square(y0), plus(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(plus(y0, y1), cons2(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(pi(X)) → MARK(X)

MARK(2ndsneg(s(y0), from(x0))) → A__2NDSNEG(s(mark(y0)), a__from(mark(x0)))

MARK(plus(X1, X2)) → MARK(X2)

A__2NDSPOS(s(N), cons(X, Z)) → MARK(Z)

MARK(pi(X)) → A__PI(mark(X))

MARK(2ndsneg(times(y0, y1), from(x0))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndsneg(times(y0, y1), square(x0))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndspos(plus(y0, y1), cons(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), cons(mark(x0), x1))

MARK(2ndsneg(square(y0), times(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(times(y0, y1), cons2(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(2ndsneg(X1, X2)) → MARK(X2)

MARK(2ndsneg(square(y0), from(x0))) → A__2NDSNEG(a__square(mark(y0)), a__from(mark(x0)))

A__TIMES(s(times(x0, x1)), y1) → A__TIMES(a__times(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(square(y0), times(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(times(y0, y1), square(x0))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndsneg(s(y0), cons(x0, x1))) → A__2NDSNEG(s(mark(y0)), cons(mark(x0), x1))

MARK(times(square(x0), y1)) → A__TIMES(a__square(mark(x0)), mark(y1))

A__PI(X) → A__FROM(0)

MARK(from(X)) → A__FROM(mark(X))

MARK(2ndspos(s(y0), plus(x0, x1))) → A__2NDSPOS(s(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(plus(X1, X2)) → MARK(X1)

A__2NDSPOS(s(y0), cons(y1, times(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__times(mark(x0), mark(x1))))

A__PI(X) → MARK(X)

MARK(2ndspos(s(y0), cons(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons(mark(x0), x1))

MARK(times(s(x0), y1)) → A__TIMES(s(mark(x0)), mark(y1))

MARK(2ndsneg(plus(y0, y1), cons(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), cons(mark(x0), x1))

A__PLUS(s(X), Y) → MARK(Y)

A__TIMES(s(X), Y) → MARK(X)

A__TIMES(s(s(x0)), y1) → A__TIMES(s(mark(x0)), mark(y1))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(N)

MARK(2ndsneg(s(y0), cons2(x0, x1))) → A__2NDSNEG(s(mark(y0)), cons2(x0, mark(x1)))

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(N)

MARK(cons(X1, X2)) → MARK(X1)

MARK(2ndsneg(plus(y0, y1), from(x0))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(times(X1, X2)) → MARK(X2)

MARK(2ndspos(times(y0, y1), cons(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), cons(mark(x0), x1))

A__PI(square(x0)) → A__2NDSPOS(a__square(mark(x0)), a__from(0))

MARK(2ndspos(times(y0, y1), plus(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(plus(y0, y1), times(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndsneg(square(y0), cons(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), cons(mark(x0), x1))

MARK(2ndspos(s(y0), times(x0, x1))) → A__2NDSPOS(s(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndsneg(X1, X2)) → MARK(X1)

MARK(2ndsneg(square(y0), square(x0))) → A__2NDSNEG(a__square(mark(y0)), a__square(mark(x0)))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(Y)

MARK(rcons(X1, X2)) → MARK(X1)

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(Z)

MARK(square(X)) → MARK(X)

A__PI(s(x0)) → A__2NDSPOS(s(mark(x0)), a__from(0))

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(Y)

A__FROM(X) → MARK(X)

MARK(2ndsneg(s(y0), plus(x0, x1))) → A__2NDSNEG(s(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(times(plus(x0, x1), y1)) → A__TIMES(a__plus(mark(x0), mark(x1)), mark(y1))

A__2NDSPOS(s(y0), cons(y1, from(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__from(mark(x0))))

MARK(rcons(X1, X2)) → MARK(X2)

MARK(2ndspos(s(y0), square(x0))) → A__2NDSPOS(s(mark(y0)), a__square(mark(x0)))

MARK(2ndspos(s(y0), cons2(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(x0, mark(x1)))

MARK(2ndsneg(plus(y0, y1), cons2(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), cons2(x0, mark(x1)))

A__PLUS(s(X), Y) → MARK(X)

A__2NDSPOS(s(y0), cons(y1, cons(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, cons(mark(x0), x1)))

MARK(times(X1, X2)) → MARK(X1)

A__TIMES(s(plus(x0, x1)), y1) → A__TIMES(a__plus(mark(x0), mark(x1)), mark(y1))

MARK(plus(times(x0, x1), y1)) → A__PLUS(a__times(mark(x0), mark(x1)), mark(y1))

A__SQUARE(X) → MARK(X)

MARK(negrecip(X)) → MARK(X)

A__2NDSPOS(s(N), cons(X, Z)) → MARK(N)

A__PLUS(0, Y) → MARK(Y)

MARK(posrecip(X)) → MARK(X)

MARK(2ndspos(square(y0), from(x0))) → A__2NDSPOS(a__square(mark(y0)), a__from(mark(x0)))

MARK(2ndspos(times(y0, y1), from(x0))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__from(mark(x0)))

A__PLUS(s(times(x0, x1)), y1) → A__PLUS(a__times(mark(x0), mark(x1)), mark(y1))

A__2NDSPOS(s(y0), cons(y1, square(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__square(mark(x0))))

A__PLUS(s(0), y1) → A__PLUS(0, mark(y1))

MARK(plus(0, y1)) → A__PLUS(0, mark(y1))

MARK(from(X)) → MARK(X)

MARK(2ndsneg(s(y0), times(x0, x1))) → A__2NDSNEG(s(mark(y0)), a__times(mark(x0), mark(x1)))

A__SQUARE(X) → A__TIMES(mark(X), mark(X))

MARK(2ndspos(square(y0), square(x0))) → A__2NDSPOS(a__square(mark(y0)), a__square(mark(x0)))

A__PLUS(s(plus(x0, x1)), y1) → A__PLUS(a__plus(mark(x0), mark(x1)), mark(y1))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(Z)

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → A__2NDSPOS(mark(N), mark(Z))

MARK(2ndsneg(plus(y0, y1), plus(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(square(X)) → A__SQUARE(mark(X))

MARK(2ndspos(plus(y0, y1), from(x0))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndsneg(times(y0, y1), plus(x0, x1))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(2ndsneg(plus(y0, y1), times(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(plus(y0, y1), plus(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(2ndsneg(times(y0, y1), cons(x0, x1))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), cons(mark(x0), x1))

A__PLUS(s(s(x0)), y1) → A__PLUS(s(mark(x0)), mark(y1))
Used ordering:  Polynomial interpretation [25]:

POL(0) = 1   

POL(2ndsneg(x1, x2)) = 0   

POL(2ndspos(x1, x2)) = 0   

POL(A__2NDSNEG(x1, x2)) = 1   

POL(A__2NDSPOS(x1, x2)) = x1   

POL(A__FROM(x1)) = 1   

POL(A__PI(x1)) = 1   

POL(A__PLUS(x1, x2)) = 1   

POL(A__SQUARE(x1)) = 1   

POL(A__TIMES(x1, x2)) = x1   

POL(MARK(x1)) = 1   

POL(a__2ndsneg(x1, x2)) = 0   

POL(a__2ndspos(x1, x2)) = 0   

POL(a__from(x1)) = 0   

POL(a__pi(x1)) = 0   

POL(a__plus(x1, x2)) = x1   

POL(a__square(x1)) = 1   

POL(a__times(x1, x2)) = x1   

POL(cons(x1, x2)) = 0   

POL(cons2(x1, x2)) = 0   

POL(from(x1)) = 0   

POL(mark(x1)) = 1   

POL(negrecip(x1)) = 0   

POL(nil) = 1   

POL(pi(x1)) = 0   

POL(plus(x1, x2)) = 0   

POL(posrecip(x1)) = 0   

POL(rcons(x1, x2)) = x1   

POL(rnil) = 0   

POL(s(x1)) = 1   

POL(square(x1)) = 0   

POL(times(x1, x2)) = 0   

The following usable rules [17] were oriented:

a__from(X) → cons(mark(X), from(s(X)))

a__pi(X) → a__2ndspos(mark(X), a__from(0))

a__2ndsneg(s(N), cons(X, Z)) → a__2ndsneg(s(mark(N)), cons2(X, mark(Z)))

a__2ndsneg(s(N), cons2(X, cons(Y, Z))) → rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z)))

a__2ndspos(s(N), cons2(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))

a__2ndsneg(0, Z) → rnil

a__2ndspos(0, Z) → rnil

a__2ndspos(s(N), cons(X, Z)) → a__2ndspos(s(mark(N)), cons2(X, mark(Z)))

mark(pi(X)) → a__pi(mark(X))

mark(2ndsneg(X1, X2)) → a__2ndsneg(mark(X1), mark(X2))

mark(2ndspos(X1, X2)) → a__2ndspos(mark(X1), mark(X2))

mark(from(X)) → a__from(mark(X))

a__times(0, Y) → 0

a__plus(s(X), Y) → s(a__plus(mark(X), mark(Y)))

mark(nil) → nil

mark(negrecip(X)) → negrecip(mark(X))

mark(posrecip(X)) → posrecip(mark(X))

mark(s(X)) → s(mark(X))

mark(0) → 0

mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))

a__times(s(X), Y) → a__plus(mark(Y), a__times(mark(X), mark(Y)))

a__plus(0, Y) → mark(Y)

mark(square(X)) → a__square(mark(X))

mark(times(X1, X2)) → a__times(mark(X1), mark(X2))

a__square(X) → a__times(mark(X), mark(X))

a__times(X1, X2) → times(X1, X2)

a__plus(X1, X2) → plus(X1, X2)

a__square(X) → square(X)

mark(cons2(X1, X2)) → cons2(X1, mark(X2))

mark(cons(X1, X2)) → cons(mark(X1), X2)

mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))

mark(rnil) → rnil

a__2ndspos(X1, X2) → 2ndspos(X1, X2)

a__from(X) → from(X)

a__pi(X) → pi(X)

a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ Narrowing
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ QDP
              ↳ Narrowing
                ↳ QDP
                  ↳ DependencyGraphProof
                    ↳ QDP
                      ↳ Narrowing
                        ↳ QDP
                          ↳ DependencyGraphProof
                            ↳ QDP
                              ↳ Narrowing
                                ↳ QDP
                                  ↳ DependencyGraphProof
                                    ↳ QDP
                                      ↳ Narrowing
                                        ↳ QDP
                                          ↳ DependencyGraphProof
                                            ↳ QDP
                                              ↳ Narrowing
                                                ↳ QDP
                                                  ↳ DependencyGraphProof
                                                    ↳ QDP
                                                      ↳ Narrowing
                                                        ↳ QDP
                                                          ↳ DependencyGraphProof
                                                            ↳ QDP
                                                              ↳ Narrowing
                                                                ↳ QDP
                                                                  ↳ DependencyGraphProof
                                                                    ↳ QDP
                                                                      ↳ Narrowing
                                                                        ↳ QDP
                                                                          ↳ DependencyGraphProof
                                                                            ↳ QDP
                                                                              ↳ Narrowing
                                                                                ↳ QDP
                                                                                  ↳ DependencyGraphProof
                                                                                    ↳ QDP
                                                                                      ↳ Narrowing
                                                                                        ↳ QDP
                                                                                          ↳ DependencyGraphProof
                                                                                            ↳ QDP
                                                                                              ↳ Narrowing
                                                                                                ↳ QDP
                                                                                                  ↳ DependencyGraphProof
                                                                                                    ↳ QDP
                                                                                                      ↳ Narrowing
                                                                                                        ↳ QDP
                                                                                                          ↳ DependencyGraphProof
                                                                                                            ↳ QDP
                                                                                                              ↳ Narrowing
                                                                                                                ↳ QDP
                                                                                                                  ↳ DependencyGraphProof
                                                                                                                    ↳ QDP
                                                                                                                      ↳ Narrowing
                                                                                                                        ↳ QDP
                                                                                                                          ↳ DependencyGraphProof
                                                                                                                            ↳ QDP
                                                                                                                              ↳ Narrowing
                                                                                                                                ↳ QDP
                                                                                                                                  ↳ DependencyGraphProof
                                                                                                                                    ↳ QDP
                                                                                                                                      ↳ Narrowing
                                                                                                                                        ↳ QDP
                                                                                                                                          ↳ DependencyGraphProof
                                                                                                                                            ↳ QDP
                                                                                                                                              ↳ Narrowing
                                                                                                                                                ↳ QDP
                                                                                                                                                  ↳ DependencyGraphProof
                                                                                                                                                    ↳ QDP
                                                                                                                                                      ↳ Narrowing
                                                                                                                                                        ↳ QDP
                                                                                                                                                          ↳ DependencyGraphProof
                                                                                                                                                            ↳ QDP
                                                                                                                                                              ↳ Narrowing
                                                                                                                                                                ↳ QDP
                                                                                                                                                                  ↳ DependencyGraphProof
                                                                                                                                                                    ↳ QDP
                                                                                                                                                                      ↳ Narrowing
                                                                                                                                                                        ↳ QDP
                                                                                                                                                                          ↳ DependencyGraphProof
                                                                                                                                                                            ↳ QDP
                                                                                                                                                                              ↳ Narrowing
                                                                                                                                                                                ↳ QDP
                                                                                                                                                                                  ↳ DependencyGraphProof
                                                                                                                                                                                    ↳ QDP
                                                                                                                                                                                      ↳ Narrowing
                                                                                                                                                                                        ↳ QDP
                                                                                                                                                                                          ↳ DependencyGraphProof
                                                                                                                                                                                            ↳ QDP
                                                                                                                                                                                              ↳ QDPOrderProof
                                                                                                                                                                                                ↳ QDP
                                                                                                                                                                                                  ↳ QDPOrderProof
                                                                                                                                                                                                    ↳ QDP
                                                                                                                                                                                                      ↳ QDPOrderProof
                                                                                                                                                                                                        ↳ QDP
                                                                                                                                                                                                          ↳ QDPOrderProof
                                                                                                                                                                                                            ↳ QDP
                                                                                                                                                                                                              ↳ QDPOrderProof
                                                                                                                                                                                                                ↳ QDP
                                                                                                                                                                                                                  ↳ QDPOrderProof
                                                                                                                                                                                                                    ↳ QDP
                                                                                                                                                                                                                      ↳ QDPOrderProof
                                                                                                                                                                                                                        ↳ QDP
                                                                                                                                                                                                                          ↳ QDPOrderProof
                                                                                                                                                                                                                            ↳ QDP
                                                                                                                                                                                                                              ↳ Narrowing
                                                                                                                                                                                                                                ↳ QDP
                                                                                                                                                                                                                                  ↳ DependencyGraphProof
                                                                                                                                                                                                                                    ↳ QDP
                                                                                                                                                                                                                                      ↳ Narrowing
                                                                                                                                                                                                                                        ↳ QDP
                                                                                                                                                                                                                                          ↳ DependencyGraphProof
                                                                                                                                                                                                                                            ↳ QDP
                                                                                                                                                                                                                                              ↳ QDPOrderProof
                                                                                                                                                                                                                                                ↳ QDP
                                                                                                                                                                                                                                                  ↳ Narrowing

Q DP problem:
The TRS P consists of the following rules:

MARK(2ndspos(plus(y0, y1), square(x0))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndspos(square(y0), plus(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__plus(mark(x0), mark(x1)))

A__2NDSNEG(s(N), cons(X, Z)) → A__2NDSNEG(s(mark(N)), cons2(X, mark(Z)))

MARK(2ndsneg(s(y0), square(x0))) → A__2NDSNEG(s(mark(y0)), a__square(mark(x0)))

MARK(2ndspos(X1, X2)) → MARK(X2)

MARK(times(times(x0, x1), y1)) → A__TIMES(a__times(mark(x0), mark(x1)), mark(y1))

MARK(plus(plus(x0, x1), y1)) → A__PLUS(a__plus(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(s(y0), from(x0))) → A__2NDSPOS(s(mark(y0)), a__from(mark(x0)))

A__2NDSNEG(s(N), cons(X, Z)) → MARK(Z)

A__TIMES(s(X), Y) → MARK(Y)

A__2NDSPOS(s(y0), cons(y1, plus(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__plus(mark(x0), mark(x1))))

A__2NDSNEG(s(N), cons(X, Z)) → MARK(N)

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → A__2NDSNEG(mark(N), mark(Z))

MARK(2ndsneg(times(y0, y1), times(x0, x1))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(times(y0, y1), times(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

A__TIMES(s(X), Y) → A__PLUS(mark(Y), a__times(mark(X), mark(Y)))

MARK(2ndspos(square(y0), cons2(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), cons2(x0, mark(x1)))

MARK(plus(square(x0), y1)) → A__PLUS(a__square(mark(x0)), mark(y1))

MARK(plus(s(x0), y1)) → A__PLUS(s(mark(x0)), mark(y1))

A__PI(plus(x0, x1)) → A__2NDSPOS(a__plus(mark(x0), mark(x1)), a__from(0))

MARK(2ndspos(square(y0), cons(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), cons(mark(x0), x1))

MARK(2ndsneg(times(y0, y1), cons2(x0, x1))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), cons2(x0, mark(x1)))

A__PI(times(y0, y1)) → A__2NDSPOS(a__times(mark(y0), mark(y1)), cons(mark(0), from(s(0))))

MARK(cons2(X1, X2)) → MARK(X2)

MARK(2ndsneg(square(y0), cons2(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), cons2(x0, mark(x1)))

MARK(2ndsneg(square(y0), plus(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(s(X)) → MARK(X)

MARK(2ndspos(X1, X2)) → MARK(X1)

A__PLUS(s(square(x0)), y1) → A__PLUS(a__square(mark(x0)), mark(y1))

A__TIMES(s(square(x0)), y1) → A__TIMES(a__square(mark(x0)), mark(y1))

MARK(2ndsneg(plus(y0, y1), square(x0))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndspos(plus(y0, y1), cons2(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(pi(X)) → MARK(X)

MARK(2ndsneg(s(y0), from(x0))) → A__2NDSNEG(s(mark(y0)), a__from(mark(x0)))

MARK(plus(X1, X2)) → MARK(X2)

A__2NDSPOS(s(N), cons(X, Z)) → MARK(Z)

MARK(pi(X)) → A__PI(mark(X))

MARK(2ndsneg(square(y0), times(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(plus(y0, y1), cons(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), cons(mark(x0), x1))

MARK(2ndsneg(times(y0, y1), square(x0))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndsneg(times(y0, y1), from(x0))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndspos(times(y0, y1), cons2(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(2ndsneg(X1, X2)) → MARK(X2)

MARK(2ndsneg(square(y0), from(x0))) → A__2NDSNEG(a__square(mark(y0)), a__from(mark(x0)))

A__TIMES(s(times(x0, x1)), y1) → A__TIMES(a__times(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(times(y0, y1), square(x0))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndspos(square(y0), times(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(times(square(x0), y1)) → A__TIMES(a__square(mark(x0)), mark(y1))

MARK(2ndsneg(s(y0), cons(x0, x1))) → A__2NDSNEG(s(mark(y0)), cons(mark(x0), x1))

A__PI(X) → A__FROM(0)

MARK(from(X)) → A__FROM(mark(X))

MARK(2ndspos(s(y0), plus(x0, x1))) → A__2NDSPOS(s(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(plus(X1, X2)) → MARK(X1)

A__2NDSPOS(s(y0), cons(y1, times(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__times(mark(x0), mark(x1))))

A__PI(X) → MARK(X)

MARK(2ndspos(s(y0), cons(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons(mark(x0), x1))

MARK(times(s(x0), y1)) → A__TIMES(s(mark(x0)), mark(y1))

MARK(2ndsneg(plus(y0, y1), cons(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), cons(mark(x0), x1))

A__PLUS(s(X), Y) → MARK(Y)

A__TIMES(s(X), Y) → MARK(X)

A__TIMES(s(s(x0)), y1) → A__TIMES(s(mark(x0)), mark(y1))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(N)

MARK(2ndsneg(s(y0), cons2(x0, x1))) → A__2NDSNEG(s(mark(y0)), cons2(x0, mark(x1)))

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(N)

MARK(cons(X1, X2)) → MARK(X1)

MARK(2ndsneg(plus(y0, y1), from(x0))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(times(X1, X2)) → MARK(X2)

MARK(2ndspos(times(y0, y1), cons(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), cons(mark(x0), x1))

A__PI(square(x0)) → A__2NDSPOS(a__square(mark(x0)), a__from(0))

MARK(2ndspos(times(y0, y1), plus(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(plus(y0, y1), times(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndsneg(square(y0), cons(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), cons(mark(x0), x1))

MARK(2ndspos(s(y0), times(x0, x1))) → A__2NDSPOS(s(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndsneg(X1, X2)) → MARK(X1)

MARK(2ndsneg(square(y0), square(x0))) → A__2NDSNEG(a__square(mark(y0)), a__square(mark(x0)))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(Y)

MARK(rcons(X1, X2)) → MARK(X1)

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(Z)

MARK(square(X)) → MARK(X)

A__PI(s(x0)) → A__2NDSPOS(s(mark(x0)), a__from(0))

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(Y)

A__FROM(X) → MARK(X)

MARK(2ndsneg(s(y0), plus(x0, x1))) → A__2NDSNEG(s(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(times(plus(x0, x1), y1)) → A__TIMES(a__plus(mark(x0), mark(x1)), mark(y1))

A__2NDSPOS(s(y0), cons(y1, from(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__from(mark(x0))))

MARK(rcons(X1, X2)) → MARK(X2)

MARK(2ndspos(s(y0), square(x0))) → A__2NDSPOS(s(mark(y0)), a__square(mark(x0)))

MARK(2ndspos(s(y0), cons2(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(x0, mark(x1)))

MARK(2ndsneg(plus(y0, y1), cons2(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), cons2(x0, mark(x1)))

A__PLUS(s(X), Y) → MARK(X)

A__2NDSPOS(s(y0), cons(y1, cons(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, cons(mark(x0), x1)))

MARK(times(X1, X2)) → MARK(X1)

A__TIMES(s(plus(x0, x1)), y1) → A__TIMES(a__plus(mark(x0), mark(x1)), mark(y1))

MARK(plus(times(x0, x1), y1)) → A__PLUS(a__times(mark(x0), mark(x1)), mark(y1))

A__SQUARE(X) → MARK(X)

MARK(negrecip(X)) → MARK(X)

A__2NDSPOS(s(N), cons(X, Z)) → MARK(N)

A__PLUS(0, Y) → MARK(Y)

MARK(posrecip(X)) → MARK(X)

MARK(2ndspos(square(y0), from(x0))) → A__2NDSPOS(a__square(mark(y0)), a__from(mark(x0)))

MARK(2ndspos(times(y0, y1), from(x0))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__from(mark(x0)))

A__PLUS(s(times(x0, x1)), y1) → A__PLUS(a__times(mark(x0), mark(x1)), mark(y1))

A__2NDSPOS(s(y0), cons(y1, square(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__square(mark(x0))))

A__PLUS(s(0), y1) → A__PLUS(0, mark(y1))

MARK(plus(0, y1)) → A__PLUS(0, mark(y1))

MARK(from(X)) → MARK(X)

MARK(2ndsneg(s(y0), times(x0, x1))) → A__2NDSNEG(s(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(square(y0), square(x0))) → A__2NDSPOS(a__square(mark(y0)), a__square(mark(x0)))

A__SQUARE(X) → A__TIMES(mark(X), mark(X))

A__PLUS(s(plus(x0, x1)), y1) → A__PLUS(a__plus(mark(x0), mark(x1)), mark(y1))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(Z)

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → A__2NDSPOS(mark(N), mark(Z))

MARK(2ndsneg(plus(y0, y1), plus(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(square(X)) → A__SQUARE(mark(X))

MARK(2ndspos(plus(y0, y1), from(x0))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndsneg(plus(y0, y1), times(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndsneg(times(y0, y1), plus(x0, x1))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(plus(y0, y1), plus(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

A__PLUS(s(s(x0)), y1) → A__PLUS(s(mark(x0)), mark(y1))

MARK(2ndsneg(times(y0, y1), cons(x0, x1))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), cons(mark(x0), x1))

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))

a__2ndspos(0, Z) → rnil

a__2ndspos(s(N), cons(X, Z)) → a__2ndspos(s(mark(N)), cons2(X, mark(Z)))

a__2ndspos(s(N), cons2(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))

a__2ndsneg(0, Z) → rnil

a__2ndsneg(s(N), cons(X, Z)) → a__2ndsneg(s(mark(N)), cons2(X, mark(Z)))

a__2ndsneg(s(N), cons2(X, cons(Y, Z))) → rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z)))

a__pi(X) → a__2ndspos(mark(X), a__from(0))

a__plus(0, Y) → mark(Y)

a__plus(s(X), Y) → s(a__plus(mark(X), mark(Y)))

a__times(0, Y) → 0

a__times(s(X), Y) → a__plus(mark(Y), a__times(mark(X), mark(Y)))

a__square(X) → a__times(mark(X), mark(X))

mark(from(X)) → a__from(mark(X))

mark(2ndspos(X1, X2)) → a__2ndspos(mark(X1), mark(X2))

mark(2ndsneg(X1, X2)) → a__2ndsneg(mark(X1), mark(X2))

mark(pi(X)) → a__pi(mark(X))

mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))

mark(times(X1, X2)) → a__times(mark(X1), mark(X2))

mark(square(X)) → a__square(mark(X))

mark(0) → 0

mark(s(X)) → s(mark(X))

mark(posrecip(X)) → posrecip(mark(X))

mark(negrecip(X)) → negrecip(mark(X))

mark(nil) → nil

mark(cons(X1, X2)) → cons(mark(X1), X2)

mark(cons2(X1, X2)) → cons2(X1, mark(X2))

mark(rnil) → rnil

mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))

a__from(X) → from(X)

a__2ndspos(X1, X2) → 2ndspos(X1, X2)

a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)

a__pi(X) → pi(X)

a__plus(X1, X2) → plus(X1, X2)

a__times(X1, X2) → times(X1, X2)

a__square(X) → square(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule A__2NDSNEG(s(N), cons(X, Z)) → A__2NDSNEG(s(mark(N)), cons2(X, mark(Z))) at position [1,1] we obtained the following new rules:

A__2NDSNEG(s(y0), cons(y1, 2ndspos(x0, x1))) → A__2NDSNEG(s(mark(y0)), cons2(y1, a__2ndspos(mark(x0), mark(x1))))

A__2NDSNEG(s(y0), cons(y1, from(x0))) → A__2NDSNEG(s(mark(y0)), cons2(y1, a__from(mark(x0))))

A__2NDSNEG(s(y0), cons(y1, nil)) → A__2NDSNEG(s(mark(y0)), cons2(y1, nil))

A__2NDSNEG(s(y0), cons(y1, plus(x0, x1))) → A__2NDSNEG(s(mark(y0)), cons2(y1, a__plus(mark(x0), mark(x1))))

A__2NDSNEG(s(y0), cons(y1, s(x0))) → A__2NDSNEG(s(mark(y0)), cons2(y1, s(mark(x0))))

A__2NDSNEG(s(y0), cons(y1, times(x0, x1))) → A__2NDSNEG(s(mark(y0)), cons2(y1, a__times(mark(x0), mark(x1))))

A__2NDSNEG(s(y0), cons(y1, pi(x0))) → A__2NDSNEG(s(mark(y0)), cons2(y1, a__pi(mark(x0))))

A__2NDSNEG(s(y0), cons(y1, 0)) → A__2NDSNEG(s(mark(y0)), cons2(y1, 0))

A__2NDSNEG(s(y0), cons(y1, cons(x0, x1))) → A__2NDSNEG(s(mark(y0)), cons2(y1, cons(mark(x0), x1)))

A__2NDSNEG(s(y0), cons(y1, negrecip(x0))) → A__2NDSNEG(s(mark(y0)), cons2(y1, negrecip(mark(x0))))

A__2NDSNEG(s(y0), cons(y1, 2ndsneg(x0, x1))) → A__2NDSNEG(s(mark(y0)), cons2(y1, a__2ndsneg(mark(x0), mark(x1))))

A__2NDSNEG(s(y0), cons(y1, posrecip(x0))) → A__2NDSNEG(s(mark(y0)), cons2(y1, posrecip(mark(x0))))

A__2NDSNEG(s(y0), cons(y1, cons2(x0, x1))) → A__2NDSNEG(s(mark(y0)), cons2(y1, cons2(x0, mark(x1))))

A__2NDSNEG(s(y0), cons(y1, square(x0))) → A__2NDSNEG(s(mark(y0)), cons2(y1, a__square(mark(x0))))

A__2NDSNEG(s(y0), cons(y1, rcons(x0, x1))) → A__2NDSNEG(s(mark(y0)), cons2(y1, rcons(mark(x0), mark(x1))))

A__2NDSNEG(s(y0), cons(y1, rnil)) → A__2NDSNEG(s(mark(y0)), cons2(y1, rnil))



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ Narrowing
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ QDP
              ↳ Narrowing
                ↳ QDP
                  ↳ DependencyGraphProof
                    ↳ QDP
                      ↳ Narrowing
                        ↳ QDP
                          ↳ DependencyGraphProof
                            ↳ QDP
                              ↳ Narrowing
                                ↳ QDP
                                  ↳ DependencyGraphProof
                                    ↳ QDP
                                      ↳ Narrowing
                                        ↳ QDP
                                          ↳ DependencyGraphProof
                                            ↳ QDP
                                              ↳ Narrowing
                                                ↳ QDP
                                                  ↳ DependencyGraphProof
                                                    ↳ QDP
                                                      ↳ Narrowing
                                                        ↳ QDP
                                                          ↳ DependencyGraphProof
                                                            ↳ QDP
                                                              ↳ Narrowing
                                                                ↳ QDP
                                                                  ↳ DependencyGraphProof
                                                                    ↳ QDP
                                                                      ↳ Narrowing
                                                                        ↳ QDP
                                                                          ↳ DependencyGraphProof
                                                                            ↳ QDP
                                                                              ↳ Narrowing
                                                                                ↳ QDP
                                                                                  ↳ DependencyGraphProof
                                                                                    ↳ QDP
                                                                                      ↳ Narrowing
                                                                                        ↳ QDP
                                                                                          ↳ DependencyGraphProof
                                                                                            ↳ QDP
                                                                                              ↳ Narrowing
                                                                                                ↳ QDP
                                                                                                  ↳ DependencyGraphProof
                                                                                                    ↳ QDP
                                                                                                      ↳ Narrowing
                                                                                                        ↳ QDP
                                                                                                          ↳ DependencyGraphProof
                                                                                                            ↳ QDP
                                                                                                              ↳ Narrowing
                                                                                                                ↳ QDP
                                                                                                                  ↳ DependencyGraphProof
                                                                                                                    ↳ QDP
                                                                                                                      ↳ Narrowing
                                                                                                                        ↳ QDP
                                                                                                                          ↳ DependencyGraphProof
                                                                                                                            ↳ QDP
                                                                                                                              ↳ Narrowing
                                                                                                                                ↳ QDP
                                                                                                                                  ↳ DependencyGraphProof
                                                                                                                                    ↳ QDP
                                                                                                                                      ↳ Narrowing
                                                                                                                                        ↳ QDP
                                                                                                                                          ↳ DependencyGraphProof
                                                                                                                                            ↳ QDP
                                                                                                                                              ↳ Narrowing
                                                                                                                                                ↳ QDP
                                                                                                                                                  ↳ DependencyGraphProof
                                                                                                                                                    ↳ QDP
                                                                                                                                                      ↳ Narrowing
                                                                                                                                                        ↳ QDP
                                                                                                                                                          ↳ DependencyGraphProof
                                                                                                                                                            ↳ QDP
                                                                                                                                                              ↳ Narrowing
                                                                                                                                                                ↳ QDP
                                                                                                                                                                  ↳ DependencyGraphProof
                                                                                                                                                                    ↳ QDP
                                                                                                                                                                      ↳ Narrowing
                                                                                                                                                                        ↳ QDP
                                                                                                                                                                          ↳ DependencyGraphProof
                                                                                                                                                                            ↳ QDP
                                                                                                                                                                              ↳ Narrowing
                                                                                                                                                                                ↳ QDP
                                                                                                                                                                                  ↳ DependencyGraphProof
                                                                                                                                                                                    ↳ QDP
                                                                                                                                                                                      ↳ Narrowing
                                                                                                                                                                                        ↳ QDP
                                                                                                                                                                                          ↳ DependencyGraphProof
                                                                                                                                                                                            ↳ QDP
                                                                                                                                                                                              ↳ QDPOrderProof
                                                                                                                                                                                                ↳ QDP
                                                                                                                                                                                                  ↳ QDPOrderProof
                                                                                                                                                                                                    ↳ QDP
                                                                                                                                                                                                      ↳ QDPOrderProof
                                                                                                                                                                                                        ↳ QDP
                                                                                                                                                                                                          ↳ QDPOrderProof
                                                                                                                                                                                                            ↳ QDP
                                                                                                                                                                                                              ↳ QDPOrderProof
                                                                                                                                                                                                                ↳ QDP
                                                                                                                                                                                                                  ↳ QDPOrderProof
                                                                                                                                                                                                                    ↳ QDP
                                                                                                                                                                                                                      ↳ QDPOrderProof
                                                                                                                                                                                                                        ↳ QDP
                                                                                                                                                                                                                          ↳ QDPOrderProof
                                                                                                                                                                                                                            ↳ QDP
                                                                                                                                                                                                                              ↳ Narrowing
                                                                                                                                                                                                                                ↳ QDP
                                                                                                                                                                                                                                  ↳ DependencyGraphProof
                                                                                                                                                                                                                                    ↳ QDP
                                                                                                                                                                                                                                      ↳ Narrowing
                                                                                                                                                                                                                                        ↳ QDP
                                                                                                                                                                                                                                          ↳ DependencyGraphProof
                                                                                                                                                                                                                                            ↳ QDP
                                                                                                                                                                                                                                              ↳ QDPOrderProof
                                                                                                                                                                                                                                                ↳ QDP
                                                                                                                                                                                                                                                  ↳ Narrowing
                                                                                                                                                                                                                                                    ↳ QDP
                                                                                                                                                                                                                                                      ↳ DependencyGraphProof

Q DP problem:
The TRS P consists of the following rules:

MARK(2ndspos(square(y0), plus(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(plus(y0, y1), square(x0))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndsneg(s(y0), square(x0))) → A__2NDSNEG(s(mark(y0)), a__square(mark(x0)))

MARK(times(times(x0, x1), y1)) → A__TIMES(a__times(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(X1, X2)) → MARK(X2)

MARK(plus(plus(x0, x1), y1)) → A__PLUS(a__plus(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(s(y0), from(x0))) → A__2NDSPOS(s(mark(y0)), a__from(mark(x0)))

A__2NDSNEG(s(y0), cons(y1, 2ndsneg(x0, x1))) → A__2NDSNEG(s(mark(y0)), cons2(y1, a__2ndsneg(mark(x0), mark(x1))))

A__2NDSNEG(s(N), cons(X, Z)) → MARK(Z)

A__2NDSNEG(s(y0), cons(y1, rcons(x0, x1))) → A__2NDSNEG(s(mark(y0)), cons2(y1, rcons(mark(x0), mark(x1))))

A__2NDSNEG(s(y0), cons(y1, 2ndspos(x0, x1))) → A__2NDSNEG(s(mark(y0)), cons2(y1, a__2ndspos(mark(x0), mark(x1))))

A__TIMES(s(X), Y) → MARK(Y)

A__2NDSNEG(s(y0), cons(y1, s(x0))) → A__2NDSNEG(s(mark(y0)), cons2(y1, s(mark(x0))))

A__2NDSPOS(s(y0), cons(y1, plus(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__plus(mark(x0), mark(x1))))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → A__2NDSNEG(mark(N), mark(Z))

A__2NDSNEG(s(N), cons(X, Z)) → MARK(N)

MARK(2ndsneg(times(y0, y1), times(x0, x1))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

A__2NDSNEG(s(y0), cons(y1, negrecip(x0))) → A__2NDSNEG(s(mark(y0)), cons2(y1, negrecip(mark(x0))))

MARK(2ndspos(times(y0, y1), times(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

A__TIMES(s(X), Y) → A__PLUS(mark(Y), a__times(mark(X), mark(Y)))

MARK(2ndspos(square(y0), cons2(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), cons2(x0, mark(x1)))

MARK(plus(square(x0), y1)) → A__PLUS(a__square(mark(x0)), mark(y1))

MARK(plus(s(x0), y1)) → A__PLUS(s(mark(x0)), mark(y1))

MARK(2ndspos(square(y0), cons(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), cons(mark(x0), x1))

A__PI(plus(x0, x1)) → A__2NDSPOS(a__plus(mark(x0), mark(x1)), a__from(0))

MARK(2ndsneg(times(y0, y1), cons2(x0, x1))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), cons2(x0, mark(x1)))

A__PI(times(y0, y1)) → A__2NDSPOS(a__times(mark(y0), mark(y1)), cons(mark(0), from(s(0))))

MARK(cons2(X1, X2)) → MARK(X2)

MARK(2ndsneg(square(y0), cons2(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), cons2(x0, mark(x1)))

A__2NDSNEG(s(y0), cons(y1, nil)) → A__2NDSNEG(s(mark(y0)), cons2(y1, nil))

MARK(2ndsneg(plus(y0, y1), square(x0))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__square(mark(x0)))

A__TIMES(s(square(x0)), y1) → A__TIMES(a__square(mark(x0)), mark(y1))

A__PLUS(s(square(x0)), y1) → A__PLUS(a__square(mark(x0)), mark(y1))

MARK(2ndspos(X1, X2)) → MARK(X1)

MARK(s(X)) → MARK(X)

MARK(2ndsneg(square(y0), plus(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(plus(y0, y1), cons2(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(pi(X)) → MARK(X)

MARK(2ndsneg(s(y0), from(x0))) → A__2NDSNEG(s(mark(y0)), a__from(mark(x0)))

MARK(plus(X1, X2)) → MARK(X2)

A__2NDSPOS(s(N), cons(X, Z)) → MARK(Z)

MARK(pi(X)) → A__PI(mark(X))

MARK(2ndsneg(times(y0, y1), from(x0))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndsneg(times(y0, y1), square(x0))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndspos(plus(y0, y1), cons(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), cons(mark(x0), x1))

MARK(2ndsneg(square(y0), times(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), a__times(mark(x0), mark(x1)))

A__2NDSNEG(s(y0), cons(y1, cons2(x0, x1))) → A__2NDSNEG(s(mark(y0)), cons2(y1, cons2(x0, mark(x1))))

MARK(2ndspos(times(y0, y1), cons2(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(2ndsneg(X1, X2)) → MARK(X2)

MARK(2ndsneg(square(y0), from(x0))) → A__2NDSNEG(a__square(mark(y0)), a__from(mark(x0)))

A__TIMES(s(times(x0, x1)), y1) → A__TIMES(a__times(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(square(y0), times(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(times(y0, y1), square(x0))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__square(mark(x0)))

A__2NDSNEG(s(y0), cons(y1, from(x0))) → A__2NDSNEG(s(mark(y0)), cons2(y1, a__from(mark(x0))))

MARK(2ndsneg(s(y0), cons(x0, x1))) → A__2NDSNEG(s(mark(y0)), cons(mark(x0), x1))

MARK(times(square(x0), y1)) → A__TIMES(a__square(mark(x0)), mark(y1))

MARK(2ndspos(s(y0), plus(x0, x1))) → A__2NDSPOS(s(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(from(X)) → A__FROM(mark(X))

A__PI(X) → A__FROM(0)

A__2NDSNEG(s(y0), cons(y1, pi(x0))) → A__2NDSNEG(s(mark(y0)), cons2(y1, a__pi(mark(x0))))

MARK(plus(X1, X2)) → MARK(X1)

A__2NDSPOS(s(y0), cons(y1, times(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__times(mark(x0), mark(x1))))

MARK(2ndspos(s(y0), cons(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons(mark(x0), x1))

A__PI(X) → MARK(X)

MARK(times(s(x0), y1)) → A__TIMES(s(mark(x0)), mark(y1))

MARK(2ndsneg(plus(y0, y1), cons(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), cons(mark(x0), x1))

A__PLUS(s(X), Y) → MARK(Y)

A__TIMES(s(X), Y) → MARK(X)

A__TIMES(s(s(x0)), y1) → A__TIMES(s(mark(x0)), mark(y1))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(N)

A__2NDSNEG(s(y0), cons(y1, times(x0, x1))) → A__2NDSNEG(s(mark(y0)), cons2(y1, a__times(mark(x0), mark(x1))))

MARK(2ndsneg(s(y0), cons2(x0, x1))) → A__2NDSNEG(s(mark(y0)), cons2(x0, mark(x1)))

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(N)

MARK(cons(X1, X2)) → MARK(X1)

MARK(2ndsneg(plus(y0, y1), from(x0))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndspos(times(y0, y1), cons(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), cons(mark(x0), x1))

MARK(times(X1, X2)) → MARK(X2)

A__PI(square(x0)) → A__2NDSPOS(a__square(mark(x0)), a__from(0))

MARK(2ndsneg(square(y0), cons(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), cons(mark(x0), x1))

MARK(2ndspos(plus(y0, y1), times(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(times(y0, y1), plus(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(s(y0), times(x0, x1))) → A__2NDSPOS(s(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndsneg(X1, X2)) → MARK(X1)

MARK(2ndsneg(square(y0), square(x0))) → A__2NDSNEG(a__square(mark(y0)), a__square(mark(x0)))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(Y)

MARK(rcons(X1, X2)) → MARK(X1)

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(Z)

MARK(square(X)) → MARK(X)

A__PI(s(x0)) → A__2NDSPOS(s(mark(x0)), a__from(0))

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(Y)

A__FROM(X) → MARK(X)

MARK(times(plus(x0, x1), y1)) → A__TIMES(a__plus(mark(x0), mark(x1)), mark(y1))

MARK(2ndsneg(s(y0), plus(x0, x1))) → A__2NDSNEG(s(mark(y0)), a__plus(mark(x0), mark(x1)))

A__2NDSNEG(s(y0), cons(y1, cons(x0, x1))) → A__2NDSNEG(s(mark(y0)), cons2(y1, cons(mark(x0), x1)))

A__2NDSPOS(s(y0), cons(y1, from(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__from(mark(x0))))

MARK(2ndspos(s(y0), square(x0))) → A__2NDSPOS(s(mark(y0)), a__square(mark(x0)))

MARK(rcons(X1, X2)) → MARK(X2)

MARK(2ndspos(s(y0), cons2(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(x0, mark(x1)))

A__2NDSNEG(s(y0), cons(y1, plus(x0, x1))) → A__2NDSNEG(s(mark(y0)), cons2(y1, a__plus(mark(x0), mark(x1))))

MARK(2ndsneg(plus(y0, y1), cons2(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), cons2(x0, mark(x1)))

A__2NDSPOS(s(y0), cons(y1, cons(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, cons(mark(x0), x1)))

A__PLUS(s(X), Y) → MARK(X)

MARK(times(X1, X2)) → MARK(X1)

A__2NDSNEG(s(y0), cons(y1, 0)) → A__2NDSNEG(s(mark(y0)), cons2(y1, 0))

A__TIMES(s(plus(x0, x1)), y1) → A__TIMES(a__plus(mark(x0), mark(x1)), mark(y1))

MARK(plus(times(x0, x1), y1)) → A__PLUS(a__times(mark(x0), mark(x1)), mark(y1))

A__SQUARE(X) → MARK(X)

A__2NDSPOS(s(N), cons(X, Z)) → MARK(N)

MARK(negrecip(X)) → MARK(X)

A__2NDSNEG(s(y0), cons(y1, square(x0))) → A__2NDSNEG(s(mark(y0)), cons2(y1, a__square(mark(x0))))

A__PLUS(0, Y) → MARK(Y)

MARK(posrecip(X)) → MARK(X)

MARK(2ndspos(square(y0), from(x0))) → A__2NDSPOS(a__square(mark(y0)), a__from(mark(x0)))

MARK(2ndspos(times(y0, y1), from(x0))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__from(mark(x0)))

A__PLUS(s(times(x0, x1)), y1) → A__PLUS(a__times(mark(x0), mark(x1)), mark(y1))

A__2NDSPOS(s(y0), cons(y1, square(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__square(mark(x0))))

A__2NDSNEG(s(y0), cons(y1, rnil)) → A__2NDSNEG(s(mark(y0)), cons2(y1, rnil))

A__PLUS(s(0), y1) → A__PLUS(0, mark(y1))

MARK(plus(0, y1)) → A__PLUS(0, mark(y1))

MARK(from(X)) → MARK(X)

MARK(2ndsneg(s(y0), times(x0, x1))) → A__2NDSNEG(s(mark(y0)), a__times(mark(x0), mark(x1)))

A__SQUARE(X) → A__TIMES(mark(X), mark(X))

MARK(2ndspos(square(y0), square(x0))) → A__2NDSPOS(a__square(mark(y0)), a__square(mark(x0)))

A__PLUS(s(plus(x0, x1)), y1) → A__PLUS(a__plus(mark(x0), mark(x1)), mark(y1))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(Z)

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → A__2NDSPOS(mark(N), mark(Z))

MARK(2ndsneg(plus(y0, y1), plus(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(square(X)) → A__SQUARE(mark(X))

MARK(2ndspos(plus(y0, y1), from(x0))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndsneg(times(y0, y1), plus(x0, x1))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(2ndsneg(plus(y0, y1), times(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

A__2NDSNEG(s(y0), cons(y1, posrecip(x0))) → A__2NDSNEG(s(mark(y0)), cons2(y1, posrecip(mark(x0))))

MARK(2ndspos(plus(y0, y1), plus(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(2ndsneg(times(y0, y1), cons(x0, x1))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), cons(mark(x0), x1))

A__PLUS(s(s(x0)), y1) → A__PLUS(s(mark(x0)), mark(y1))

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))

a__2ndspos(0, Z) → rnil

a__2ndspos(s(N), cons(X, Z)) → a__2ndspos(s(mark(N)), cons2(X, mark(Z)))

a__2ndspos(s(N), cons2(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))

a__2ndsneg(0, Z) → rnil

a__2ndsneg(s(N), cons(X, Z)) → a__2ndsneg(s(mark(N)), cons2(X, mark(Z)))

a__2ndsneg(s(N), cons2(X, cons(Y, Z))) → rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z)))

a__pi(X) → a__2ndspos(mark(X), a__from(0))

a__plus(0, Y) → mark(Y)

a__plus(s(X), Y) → s(a__plus(mark(X), mark(Y)))

a__times(0, Y) → 0

a__times(s(X), Y) → a__plus(mark(Y), a__times(mark(X), mark(Y)))

a__square(X) → a__times(mark(X), mark(X))

mark(from(X)) → a__from(mark(X))

mark(2ndspos(X1, X2)) → a__2ndspos(mark(X1), mark(X2))

mark(2ndsneg(X1, X2)) → a__2ndsneg(mark(X1), mark(X2))

mark(pi(X)) → a__pi(mark(X))

mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))

mark(times(X1, X2)) → a__times(mark(X1), mark(X2))

mark(square(X)) → a__square(mark(X))

mark(0) → 0

mark(s(X)) → s(mark(X))

mark(posrecip(X)) → posrecip(mark(X))

mark(negrecip(X)) → negrecip(mark(X))

mark(nil) → nil

mark(cons(X1, X2)) → cons(mark(X1), X2)

mark(cons2(X1, X2)) → cons2(X1, mark(X2))

mark(rnil) → rnil

mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))

a__from(X) → from(X)

a__2ndspos(X1, X2) → 2ndspos(X1, X2)

a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)

a__pi(X) → pi(X)

a__plus(X1, X2) → plus(X1, X2)

a__times(X1, X2) → times(X1, X2)

a__square(X) → square(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 8 less nodes.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ Narrowing
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ QDP
              ↳ Narrowing
                ↳ QDP
                  ↳ DependencyGraphProof
                    ↳ QDP
                      ↳ Narrowing
                        ↳ QDP
                          ↳ DependencyGraphProof
                            ↳ QDP
                              ↳ Narrowing
                                ↳ QDP
                                  ↳ DependencyGraphProof
                                    ↳ QDP
                                      ↳ Narrowing
                                        ↳ QDP
                                          ↳ DependencyGraphProof
                                            ↳ QDP
                                              ↳ Narrowing
                                                ↳ QDP
                                                  ↳ DependencyGraphProof
                                                    ↳ QDP
                                                      ↳ Narrowing
                                                        ↳ QDP
                                                          ↳ DependencyGraphProof
                                                            ↳ QDP
                                                              ↳ Narrowing
                                                                ↳ QDP
                                                                  ↳ DependencyGraphProof
                                                                    ↳ QDP
                                                                      ↳ Narrowing
                                                                        ↳ QDP
                                                                          ↳ DependencyGraphProof
                                                                            ↳ QDP
                                                                              ↳ Narrowing
                                                                                ↳ QDP
                                                                                  ↳ DependencyGraphProof
                                                                                    ↳ QDP
                                                                                      ↳ Narrowing
                                                                                        ↳ QDP
                                                                                          ↳ DependencyGraphProof
                                                                                            ↳ QDP
                                                                                              ↳ Narrowing
                                                                                                ↳ QDP
                                                                                                  ↳ DependencyGraphProof
                                                                                                    ↳ QDP
                                                                                                      ↳ Narrowing
                                                                                                        ↳ QDP
                                                                                                          ↳ DependencyGraphProof
                                                                                                            ↳ QDP
                                                                                                              ↳ Narrowing
                                                                                                                ↳ QDP
                                                                                                                  ↳ DependencyGraphProof
                                                                                                                    ↳ QDP
                                                                                                                      ↳ Narrowing
                                                                                                                        ↳ QDP
                                                                                                                          ↳ DependencyGraphProof
                                                                                                                            ↳ QDP
                                                                                                                              ↳ Narrowing
                                                                                                                                ↳ QDP
                                                                                                                                  ↳ DependencyGraphProof
                                                                                                                                    ↳ QDP
                                                                                                                                      ↳ Narrowing
                                                                                                                                        ↳ QDP
                                                                                                                                          ↳ DependencyGraphProof
                                                                                                                                            ↳ QDP
                                                                                                                                              ↳ Narrowing
                                                                                                                                                ↳ QDP
                                                                                                                                                  ↳ DependencyGraphProof
                                                                                                                                                    ↳ QDP
                                                                                                                                                      ↳ Narrowing
                                                                                                                                                        ↳ QDP
                                                                                                                                                          ↳ DependencyGraphProof
                                                                                                                                                            ↳ QDP
                                                                                                                                                              ↳ Narrowing
                                                                                                                                                                ↳ QDP
                                                                                                                                                                  ↳ DependencyGraphProof
                                                                                                                                                                    ↳ QDP
                                                                                                                                                                      ↳ Narrowing
                                                                                                                                                                        ↳ QDP
                                                                                                                                                                          ↳ DependencyGraphProof
                                                                                                                                                                            ↳ QDP
                                                                                                                                                                              ↳ Narrowing
                                                                                                                                                                                ↳ QDP
                                                                                                                                                                                  ↳ DependencyGraphProof
                                                                                                                                                                                    ↳ QDP
                                                                                                                                                                                      ↳ Narrowing
                                                                                                                                                                                        ↳ QDP
                                                                                                                                                                                          ↳ DependencyGraphProof
                                                                                                                                                                                            ↳ QDP
                                                                                                                                                                                              ↳ QDPOrderProof
                                                                                                                                                                                                ↳ QDP
                                                                                                                                                                                                  ↳ QDPOrderProof
                                                                                                                                                                                                    ↳ QDP
                                                                                                                                                                                                      ↳ QDPOrderProof
                                                                                                                                                                                                        ↳ QDP
                                                                                                                                                                                                          ↳ QDPOrderProof
                                                                                                                                                                                                            ↳ QDP
                                                                                                                                                                                                              ↳ QDPOrderProof
                                                                                                                                                                                                                ↳ QDP
                                                                                                                                                                                                                  ↳ QDPOrderProof
                                                                                                                                                                                                                    ↳ QDP
                                                                                                                                                                                                                      ↳ QDPOrderProof
                                                                                                                                                                                                                        ↳ QDP
                                                                                                                                                                                                                          ↳ QDPOrderProof
                                                                                                                                                                                                                            ↳ QDP
                                                                                                                                                                                                                              ↳ Narrowing
                                                                                                                                                                                                                                ↳ QDP
                                                                                                                                                                                                                                  ↳ DependencyGraphProof
                                                                                                                                                                                                                                    ↳ QDP
                                                                                                                                                                                                                                      ↳ Narrowing
                                                                                                                                                                                                                                        ↳ QDP
                                                                                                                                                                                                                                          ↳ DependencyGraphProof
                                                                                                                                                                                                                                            ↳ QDP
                                                                                                                                                                                                                                              ↳ QDPOrderProof
                                                                                                                                                                                                                                                ↳ QDP
                                                                                                                                                                                                                                                  ↳ Narrowing
                                                                                                                                                                                                                                                    ↳ QDP
                                                                                                                                                                                                                                                      ↳ DependencyGraphProof
                                                                                                                                                                                                                                                        ↳ QDP
                                                                                                                                                                                                                                                          ↳ Narrowing

Q DP problem:
The TRS P consists of the following rules:

MARK(2ndspos(square(y0), plus(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(plus(y0, y1), square(x0))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndsneg(s(y0), square(x0))) → A__2NDSNEG(s(mark(y0)), a__square(mark(x0)))

MARK(times(times(x0, x1), y1)) → A__TIMES(a__times(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(X1, X2)) → MARK(X2)

MARK(plus(plus(x0, x1), y1)) → A__PLUS(a__plus(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(s(y0), from(x0))) → A__2NDSPOS(s(mark(y0)), a__from(mark(x0)))

A__2NDSNEG(s(y0), cons(y1, 2ndsneg(x0, x1))) → A__2NDSNEG(s(mark(y0)), cons2(y1, a__2ndsneg(mark(x0), mark(x1))))

A__2NDSNEG(s(N), cons(X, Z)) → MARK(Z)

A__2NDSNEG(s(y0), cons(y1, 2ndspos(x0, x1))) → A__2NDSNEG(s(mark(y0)), cons2(y1, a__2ndspos(mark(x0), mark(x1))))

A__TIMES(s(X), Y) → MARK(Y)

A__2NDSPOS(s(y0), cons(y1, plus(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__plus(mark(x0), mark(x1))))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → A__2NDSNEG(mark(N), mark(Z))

A__2NDSNEG(s(N), cons(X, Z)) → MARK(N)

MARK(2ndsneg(times(y0, y1), times(x0, x1))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(times(y0, y1), times(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

A__TIMES(s(X), Y) → A__PLUS(mark(Y), a__times(mark(X), mark(Y)))

MARK(2ndspos(square(y0), cons2(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), cons2(x0, mark(x1)))

MARK(plus(square(x0), y1)) → A__PLUS(a__square(mark(x0)), mark(y1))

MARK(plus(s(x0), y1)) → A__PLUS(s(mark(x0)), mark(y1))

A__PI(plus(x0, x1)) → A__2NDSPOS(a__plus(mark(x0), mark(x1)), a__from(0))

MARK(2ndspos(square(y0), cons(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), cons(mark(x0), x1))

MARK(2ndsneg(times(y0, y1), cons2(x0, x1))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), cons2(x0, mark(x1)))

A__PI(times(y0, y1)) → A__2NDSPOS(a__times(mark(y0), mark(y1)), cons(mark(0), from(s(0))))

MARK(cons2(X1, X2)) → MARK(X2)

MARK(2ndsneg(square(y0), cons2(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), cons2(x0, mark(x1)))

MARK(2ndsneg(plus(y0, y1), square(x0))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__square(mark(x0)))

A__TIMES(s(square(x0)), y1) → A__TIMES(a__square(mark(x0)), mark(y1))

A__PLUS(s(square(x0)), y1) → A__PLUS(a__square(mark(x0)), mark(y1))

MARK(2ndspos(X1, X2)) → MARK(X1)

MARK(s(X)) → MARK(X)

MARK(2ndsneg(square(y0), plus(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(plus(y0, y1), cons2(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(pi(X)) → MARK(X)

MARK(2ndsneg(s(y0), from(x0))) → A__2NDSNEG(s(mark(y0)), a__from(mark(x0)))

MARK(plus(X1, X2)) → MARK(X2)

A__2NDSPOS(s(N), cons(X, Z)) → MARK(Z)

MARK(pi(X)) → A__PI(mark(X))

MARK(2ndsneg(times(y0, y1), from(x0))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndsneg(times(y0, y1), square(x0))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndspos(plus(y0, y1), cons(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), cons(mark(x0), x1))

MARK(2ndsneg(square(y0), times(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(times(y0, y1), cons2(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(2ndsneg(X1, X2)) → MARK(X2)

MARK(2ndsneg(square(y0), from(x0))) → A__2NDSNEG(a__square(mark(y0)), a__from(mark(x0)))

A__TIMES(s(times(x0, x1)), y1) → A__TIMES(a__times(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(square(y0), times(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(times(y0, y1), square(x0))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndsneg(s(y0), cons(x0, x1))) → A__2NDSNEG(s(mark(y0)), cons(mark(x0), x1))

MARK(times(square(x0), y1)) → A__TIMES(a__square(mark(x0)), mark(y1))

A__2NDSNEG(s(y0), cons(y1, from(x0))) → A__2NDSNEG(s(mark(y0)), cons2(y1, a__from(mark(x0))))

A__PI(X) → A__FROM(0)

MARK(from(X)) → A__FROM(mark(X))

MARK(2ndspos(s(y0), plus(x0, x1))) → A__2NDSPOS(s(mark(y0)), a__plus(mark(x0), mark(x1)))

A__2NDSNEG(s(y0), cons(y1, pi(x0))) → A__2NDSNEG(s(mark(y0)), cons2(y1, a__pi(mark(x0))))

MARK(plus(X1, X2)) → MARK(X1)

A__2NDSPOS(s(y0), cons(y1, times(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__times(mark(x0), mark(x1))))

A__PI(X) → MARK(X)

MARK(2ndspos(s(y0), cons(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons(mark(x0), x1))

MARK(times(s(x0), y1)) → A__TIMES(s(mark(x0)), mark(y1))

MARK(2ndsneg(plus(y0, y1), cons(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), cons(mark(x0), x1))

A__PLUS(s(X), Y) → MARK(Y)

A__TIMES(s(X), Y) → MARK(X)

A__TIMES(s(s(x0)), y1) → A__TIMES(s(mark(x0)), mark(y1))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(N)

MARK(2ndsneg(s(y0), cons2(x0, x1))) → A__2NDSNEG(s(mark(y0)), cons2(x0, mark(x1)))

A__2NDSNEG(s(y0), cons(y1, times(x0, x1))) → A__2NDSNEG(s(mark(y0)), cons2(y1, a__times(mark(x0), mark(x1))))

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(N)

MARK(cons(X1, X2)) → MARK(X1)

MARK(2ndsneg(plus(y0, y1), from(x0))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(times(X1, X2)) → MARK(X2)

MARK(2ndspos(times(y0, y1), cons(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), cons(mark(x0), x1))

A__PI(square(x0)) → A__2NDSPOS(a__square(mark(x0)), a__from(0))

MARK(2ndspos(times(y0, y1), plus(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(plus(y0, y1), times(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndsneg(square(y0), cons(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), cons(mark(x0), x1))

MARK(2ndspos(s(y0), times(x0, x1))) → A__2NDSPOS(s(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndsneg(X1, X2)) → MARK(X1)

MARK(2ndsneg(square(y0), square(x0))) → A__2NDSNEG(a__square(mark(y0)), a__square(mark(x0)))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(Y)

MARK(rcons(X1, X2)) → MARK(X1)

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(Z)

MARK(square(X)) → MARK(X)

A__PI(s(x0)) → A__2NDSPOS(s(mark(x0)), a__from(0))

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(Y)

A__FROM(X) → MARK(X)

MARK(2ndsneg(s(y0), plus(x0, x1))) → A__2NDSNEG(s(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(times(plus(x0, x1), y1)) → A__TIMES(a__plus(mark(x0), mark(x1)), mark(y1))

A__2NDSNEG(s(y0), cons(y1, cons(x0, x1))) → A__2NDSNEG(s(mark(y0)), cons2(y1, cons(mark(x0), x1)))

A__2NDSPOS(s(y0), cons(y1, from(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__from(mark(x0))))

MARK(rcons(X1, X2)) → MARK(X2)

MARK(2ndspos(s(y0), square(x0))) → A__2NDSPOS(s(mark(y0)), a__square(mark(x0)))

MARK(2ndspos(s(y0), cons2(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(x0, mark(x1)))

MARK(2ndsneg(plus(y0, y1), cons2(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), cons2(x0, mark(x1)))

A__2NDSNEG(s(y0), cons(y1, plus(x0, x1))) → A__2NDSNEG(s(mark(y0)), cons2(y1, a__plus(mark(x0), mark(x1))))

A__PLUS(s(X), Y) → MARK(X)

A__2NDSPOS(s(y0), cons(y1, cons(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, cons(mark(x0), x1)))

MARK(times(X1, X2)) → MARK(X1)

A__TIMES(s(plus(x0, x1)), y1) → A__TIMES(a__plus(mark(x0), mark(x1)), mark(y1))

MARK(plus(times(x0, x1), y1)) → A__PLUS(a__times(mark(x0), mark(x1)), mark(y1))

A__SQUARE(X) → MARK(X)

A__2NDSPOS(s(N), cons(X, Z)) → MARK(N)

MARK(negrecip(X)) → MARK(X)

A__2NDSNEG(s(y0), cons(y1, square(x0))) → A__2NDSNEG(s(mark(y0)), cons2(y1, a__square(mark(x0))))

A__PLUS(0, Y) → MARK(Y)

MARK(posrecip(X)) → MARK(X)

MARK(2ndspos(square(y0), from(x0))) → A__2NDSPOS(a__square(mark(y0)), a__from(mark(x0)))

MARK(2ndspos(times(y0, y1), from(x0))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__from(mark(x0)))

A__PLUS(s(times(x0, x1)), y1) → A__PLUS(a__times(mark(x0), mark(x1)), mark(y1))

A__2NDSPOS(s(y0), cons(y1, square(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__square(mark(x0))))

A__PLUS(s(0), y1) → A__PLUS(0, mark(y1))

MARK(plus(0, y1)) → A__PLUS(0, mark(y1))

MARK(from(X)) → MARK(X)

MARK(2ndsneg(s(y0), times(x0, x1))) → A__2NDSNEG(s(mark(y0)), a__times(mark(x0), mark(x1)))

A__SQUARE(X) → A__TIMES(mark(X), mark(X))

MARK(2ndspos(square(y0), square(x0))) → A__2NDSPOS(a__square(mark(y0)), a__square(mark(x0)))

A__PLUS(s(plus(x0, x1)), y1) → A__PLUS(a__plus(mark(x0), mark(x1)), mark(y1))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(Z)

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → A__2NDSPOS(mark(N), mark(Z))

MARK(2ndsneg(plus(y0, y1), plus(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(square(X)) → A__SQUARE(mark(X))

MARK(2ndspos(plus(y0, y1), from(x0))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndsneg(times(y0, y1), plus(x0, x1))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(2ndsneg(plus(y0, y1), times(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(plus(y0, y1), plus(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(2ndsneg(times(y0, y1), cons(x0, x1))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), cons(mark(x0), x1))

A__PLUS(s(s(x0)), y1) → A__PLUS(s(mark(x0)), mark(y1))

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))

a__2ndspos(0, Z) → rnil

a__2ndspos(s(N), cons(X, Z)) → a__2ndspos(s(mark(N)), cons2(X, mark(Z)))

a__2ndspos(s(N), cons2(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))

a__2ndsneg(0, Z) → rnil

a__2ndsneg(s(N), cons(X, Z)) → a__2ndsneg(s(mark(N)), cons2(X, mark(Z)))

a__2ndsneg(s(N), cons2(X, cons(Y, Z))) → rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z)))

a__pi(X) → a__2ndspos(mark(X), a__from(0))

a__plus(0, Y) → mark(Y)

a__plus(s(X), Y) → s(a__plus(mark(X), mark(Y)))

a__times(0, Y) → 0

a__times(s(X), Y) → a__plus(mark(Y), a__times(mark(X), mark(Y)))

a__square(X) → a__times(mark(X), mark(X))

mark(from(X)) → a__from(mark(X))

mark(2ndspos(X1, X2)) → a__2ndspos(mark(X1), mark(X2))

mark(2ndsneg(X1, X2)) → a__2ndsneg(mark(X1), mark(X2))

mark(pi(X)) → a__pi(mark(X))

mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))

mark(times(X1, X2)) → a__times(mark(X1), mark(X2))

mark(square(X)) → a__square(mark(X))

mark(0) → 0

mark(s(X)) → s(mark(X))

mark(posrecip(X)) → posrecip(mark(X))

mark(negrecip(X)) → negrecip(mark(X))

mark(nil) → nil

mark(cons(X1, X2)) → cons(mark(X1), X2)

mark(cons2(X1, X2)) → cons2(X1, mark(X2))

mark(rnil) → rnil

mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))

a__from(X) → from(X)

a__2ndspos(X1, X2) → 2ndspos(X1, X2)

a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)

a__pi(X) → pi(X)

a__plus(X1, X2) → plus(X1, X2)

a__times(X1, X2) → times(X1, X2)

a__square(X) → square(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → A__2NDSNEG(mark(N), mark(Z)) at position [0] we obtained the following new rules:

A__2NDSPOS(s(s(x0)), cons2(y1, cons(y2, y3))) → A__2NDSNEG(s(mark(x0)), mark(y3))

A__2NDSPOS(s(plus(x0, x1)), cons2(y1, cons(y2, y3))) → A__2NDSNEG(a__plus(mark(x0), mark(x1)), mark(y3))

A__2NDSPOS(s(from(x0)), cons2(y1, cons(y2, y3))) → A__2NDSNEG(a__from(mark(x0)), mark(y3))

A__2NDSPOS(s(nil), cons2(y1, cons(y2, y3))) → A__2NDSNEG(nil, mark(y3))

A__2NDSPOS(s(0), cons2(y1, cons(y2, y3))) → A__2NDSNEG(0, mark(y3))

A__2NDSPOS(s(times(x0, x1)), cons2(y1, cons(y2, y3))) → A__2NDSNEG(a__times(mark(x0), mark(x1)), mark(y3))

A__2NDSPOS(s(2ndspos(x0, x1)), cons2(y1, cons(y2, y3))) → A__2NDSNEG(a__2ndspos(mark(x0), mark(x1)), mark(y3))

A__2NDSPOS(s(cons(x0, x1)), cons2(y1, cons(y2, y3))) → A__2NDSNEG(cons(mark(x0), x1), mark(y3))

A__2NDSPOS(s(posrecip(x0)), cons2(y1, cons(y2, y3))) → A__2NDSNEG(posrecip(mark(x0)), mark(y3))

A__2NDSPOS(s(rnil), cons2(y1, cons(y2, y3))) → A__2NDSNEG(rnil, mark(y3))

A__2NDSPOS(s(2ndsneg(x0, x1)), cons2(y1, cons(y2, y3))) → A__2NDSNEG(a__2ndsneg(mark(x0), mark(x1)), mark(y3))

A__2NDSPOS(s(negrecip(x0)), cons2(y1, cons(y2, y3))) → A__2NDSNEG(negrecip(mark(x0)), mark(y3))

A__2NDSPOS(s(square(x0)), cons2(y1, cons(y2, y3))) → A__2NDSNEG(a__square(mark(x0)), mark(y3))

A__2NDSPOS(s(pi(x0)), cons2(y1, cons(y2, y3))) → A__2NDSNEG(a__pi(mark(x0)), mark(y3))

A__2NDSPOS(s(rcons(x0, x1)), cons2(y1, cons(y2, y3))) → A__2NDSNEG(rcons(mark(x0), mark(x1)), mark(y3))

A__2NDSPOS(s(cons2(x0, x1)), cons2(y1, cons(y2, y3))) → A__2NDSNEG(cons2(x0, mark(x1)), mark(y3))



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ Narrowing
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ QDP
              ↳ Narrowing
                ↳ QDP
                  ↳ DependencyGraphProof
                    ↳ QDP
                      ↳ Narrowing
                        ↳ QDP
                          ↳ DependencyGraphProof
                            ↳ QDP
                              ↳ Narrowing
                                ↳ QDP
                                  ↳ DependencyGraphProof
                                    ↳ QDP
                                      ↳ Narrowing
                                        ↳ QDP
                                          ↳ DependencyGraphProof
                                            ↳ QDP
                                              ↳ Narrowing
                                                ↳ QDP
                                                  ↳ DependencyGraphProof
                                                    ↳ QDP
                                                      ↳ Narrowing
                                                        ↳ QDP
                                                          ↳ DependencyGraphProof
                                                            ↳ QDP
                                                              ↳ Narrowing
                                                                ↳ QDP
                                                                  ↳ DependencyGraphProof
                                                                    ↳ QDP
                                                                      ↳ Narrowing
                                                                        ↳ QDP
                                                                          ↳ DependencyGraphProof
                                                                            ↳ QDP
                                                                              ↳ Narrowing
                                                                                ↳ QDP
                                                                                  ↳ DependencyGraphProof
                                                                                    ↳ QDP
                                                                                      ↳ Narrowing
                                                                                        ↳ QDP
                                                                                          ↳ DependencyGraphProof
                                                                                            ↳ QDP
                                                                                              ↳ Narrowing
                                                                                                ↳ QDP
                                                                                                  ↳ DependencyGraphProof
                                                                                                    ↳ QDP
                                                                                                      ↳ Narrowing
                                                                                                        ↳ QDP
                                                                                                          ↳ DependencyGraphProof
                                                                                                            ↳ QDP
                                                                                                              ↳ Narrowing
                                                                                                                ↳ QDP
                                                                                                                  ↳ DependencyGraphProof
                                                                                                                    ↳ QDP
                                                                                                                      ↳ Narrowing
                                                                                                                        ↳ QDP
                                                                                                                          ↳ DependencyGraphProof
                                                                                                                            ↳ QDP
                                                                                                                              ↳ Narrowing
                                                                                                                                ↳ QDP
                                                                                                                                  ↳ DependencyGraphProof
                                                                                                                                    ↳ QDP
                                                                                                                                      ↳ Narrowing
                                                                                                                                        ↳ QDP
                                                                                                                                          ↳ DependencyGraphProof
                                                                                                                                            ↳ QDP
                                                                                                                                              ↳ Narrowing
                                                                                                                                                ↳ QDP
                                                                                                                                                  ↳ DependencyGraphProof
                                                                                                                                                    ↳ QDP
                                                                                                                                                      ↳ Narrowing
                                                                                                                                                        ↳ QDP
                                                                                                                                                          ↳ DependencyGraphProof
                                                                                                                                                            ↳ QDP
                                                                                                                                                              ↳ Narrowing
                                                                                                                                                                ↳ QDP
                                                                                                                                                                  ↳ DependencyGraphProof
                                                                                                                                                                    ↳ QDP
                                                                                                                                                                      ↳ Narrowing
                                                                                                                                                                        ↳ QDP
                                                                                                                                                                          ↳ DependencyGraphProof
                                                                                                                                                                            ↳ QDP
                                                                                                                                                                              ↳ Narrowing
                                                                                                                                                                                ↳ QDP
                                                                                                                                                                                  ↳ DependencyGraphProof
                                                                                                                                                                                    ↳ QDP
                                                                                                                                                                                      ↳ Narrowing
                                                                                                                                                                                        ↳ QDP
                                                                                                                                                                                          ↳ DependencyGraphProof
                                                                                                                                                                                            ↳ QDP
                                                                                                                                                                                              ↳ QDPOrderProof
                                                                                                                                                                                                ↳ QDP
                                                                                                                                                                                                  ↳ QDPOrderProof
                                                                                                                                                                                                    ↳ QDP
                                                                                                                                                                                                      ↳ QDPOrderProof
                                                                                                                                                                                                        ↳ QDP
                                                                                                                                                                                                          ↳ QDPOrderProof
                                                                                                                                                                                                            ↳ QDP
                                                                                                                                                                                                              ↳ QDPOrderProof
                                                                                                                                                                                                                ↳ QDP
                                                                                                                                                                                                                  ↳ QDPOrderProof
                                                                                                                                                                                                                    ↳ QDP
                                                                                                                                                                                                                      ↳ QDPOrderProof
                                                                                                                                                                                                                        ↳ QDP
                                                                                                                                                                                                                          ↳ QDPOrderProof
                                                                                                                                                                                                                            ↳ QDP
                                                                                                                                                                                                                              ↳ Narrowing
                                                                                                                                                                                                                                ↳ QDP
                                                                                                                                                                                                                                  ↳ DependencyGraphProof
                                                                                                                                                                                                                                    ↳ QDP
                                                                                                                                                                                                                                      ↳ Narrowing
                                                                                                                                                                                                                                        ↳ QDP
                                                                                                                                                                                                                                          ↳ DependencyGraphProof
                                                                                                                                                                                                                                            ↳ QDP
                                                                                                                                                                                                                                              ↳ QDPOrderProof
                                                                                                                                                                                                                                                ↳ QDP
                                                                                                                                                                                                                                                  ↳ Narrowing
                                                                                                                                                                                                                                                    ↳ QDP
                                                                                                                                                                                                                                                      ↳ DependencyGraphProof
                                                                                                                                                                                                                                                        ↳ QDP
                                                                                                                                                                                                                                                          ↳ Narrowing
                                                                                                                                                                                                                                                            ↳ QDP
                                                                                                                                                                                                                                                              ↳ DependencyGraphProof

Q DP problem:
The TRS P consists of the following rules:

MARK(2ndspos(plus(y0, y1), square(x0))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndspos(square(y0), plus(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__plus(mark(x0), mark(x1)))

A__2NDSPOS(s(nil), cons2(y1, cons(y2, y3))) → A__2NDSNEG(nil, mark(y3))

MARK(2ndsneg(s(y0), square(x0))) → A__2NDSNEG(s(mark(y0)), a__square(mark(x0)))

MARK(2ndspos(X1, X2)) → MARK(X2)

MARK(times(times(x0, x1), y1)) → A__TIMES(a__times(mark(x0), mark(x1)), mark(y1))

MARK(plus(plus(x0, x1), y1)) → A__PLUS(a__plus(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(s(y0), from(x0))) → A__2NDSPOS(s(mark(y0)), a__from(mark(x0)))

A__2NDSNEG(s(y0), cons(y1, 2ndsneg(x0, x1))) → A__2NDSNEG(s(mark(y0)), cons2(y1, a__2ndsneg(mark(x0), mark(x1))))

A__2NDSNEG(s(N), cons(X, Z)) → MARK(Z)

A__2NDSPOS(s(s(x0)), cons2(y1, cons(y2, y3))) → A__2NDSNEG(s(mark(x0)), mark(y3))

A__2NDSNEG(s(y0), cons(y1, 2ndspos(x0, x1))) → A__2NDSNEG(s(mark(y0)), cons2(y1, a__2ndspos(mark(x0), mark(x1))))

A__TIMES(s(X), Y) → MARK(Y)

A__2NDSPOS(s(y0), cons(y1, plus(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__plus(mark(x0), mark(x1))))

A__2NDSNEG(s(N), cons(X, Z)) → MARK(N)

MARK(2ndsneg(times(y0, y1), times(x0, x1))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

A__2NDSPOS(s(0), cons2(y1, cons(y2, y3))) → A__2NDSNEG(0, mark(y3))

MARK(2ndspos(times(y0, y1), times(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

A__TIMES(s(X), Y) → A__PLUS(mark(Y), a__times(mark(X), mark(Y)))

MARK(2ndspos(square(y0), cons2(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), cons2(x0, mark(x1)))

MARK(plus(square(x0), y1)) → A__PLUS(a__square(mark(x0)), mark(y1))

MARK(plus(s(x0), y1)) → A__PLUS(s(mark(x0)), mark(y1))

MARK(2ndspos(square(y0), cons(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), cons(mark(x0), x1))

A__PI(plus(x0, x1)) → A__2NDSPOS(a__plus(mark(x0), mark(x1)), a__from(0))

MARK(2ndsneg(times(y0, y1), cons2(x0, x1))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), cons2(x0, mark(x1)))

A__PI(times(y0, y1)) → A__2NDSPOS(a__times(mark(y0), mark(y1)), cons(mark(0), from(s(0))))

MARK(cons2(X1, X2)) → MARK(X2)

MARK(2ndsneg(square(y0), cons2(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), cons2(x0, mark(x1)))

A__2NDSPOS(s(plus(x0, x1)), cons2(y1, cons(y2, y3))) → A__2NDSNEG(a__plus(mark(x0), mark(x1)), mark(y3))

MARK(2ndsneg(square(y0), plus(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(s(X)) → MARK(X)

MARK(2ndspos(X1, X2)) → MARK(X1)

A__PLUS(s(square(x0)), y1) → A__PLUS(a__square(mark(x0)), mark(y1))

A__TIMES(s(square(x0)), y1) → A__TIMES(a__square(mark(x0)), mark(y1))

MARK(2ndsneg(plus(y0, y1), square(x0))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndspos(plus(y0, y1), cons2(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(pi(X)) → MARK(X)

MARK(2ndsneg(s(y0), from(x0))) → A__2NDSNEG(s(mark(y0)), a__from(mark(x0)))

MARK(plus(X1, X2)) → MARK(X2)

A__2NDSPOS(s(N), cons(X, Z)) → MARK(Z)

MARK(pi(X)) → A__PI(mark(X))

MARK(2ndsneg(square(y0), times(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(plus(y0, y1), cons(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), cons(mark(x0), x1))

MARK(2ndsneg(times(y0, y1), square(x0))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndsneg(times(y0, y1), from(x0))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndspos(times(y0, y1), cons2(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(2ndsneg(X1, X2)) → MARK(X2)

MARK(2ndsneg(square(y0), from(x0))) → A__2NDSNEG(a__square(mark(y0)), a__from(mark(x0)))

A__2NDSPOS(s(cons2(x0, x1)), cons2(y1, cons(y2, y3))) → A__2NDSNEG(cons2(x0, mark(x1)), mark(y3))

A__2NDSPOS(s(rcons(x0, x1)), cons2(y1, cons(y2, y3))) → A__2NDSNEG(rcons(mark(x0), mark(x1)), mark(y3))

A__TIMES(s(times(x0, x1)), y1) → A__TIMES(a__times(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(times(y0, y1), square(x0))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndspos(square(y0), times(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__times(mark(x0), mark(x1)))

A__2NDSNEG(s(y0), cons(y1, from(x0))) → A__2NDSNEG(s(mark(y0)), cons2(y1, a__from(mark(x0))))

MARK(times(square(x0), y1)) → A__TIMES(a__square(mark(x0)), mark(y1))

MARK(2ndsneg(s(y0), cons(x0, x1))) → A__2NDSNEG(s(mark(y0)), cons(mark(x0), x1))

MARK(2ndspos(s(y0), plus(x0, x1))) → A__2NDSPOS(s(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(from(X)) → A__FROM(mark(X))

A__PI(X) → A__FROM(0)

A__2NDSNEG(s(y0), cons(y1, pi(x0))) → A__2NDSNEG(s(mark(y0)), cons2(y1, a__pi(mark(x0))))

MARK(plus(X1, X2)) → MARK(X1)

A__2NDSPOS(s(y0), cons(y1, times(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__times(mark(x0), mark(x1))))

MARK(2ndspos(s(y0), cons(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons(mark(x0), x1))

A__PI(X) → MARK(X)

MARK(times(s(x0), y1)) → A__TIMES(s(mark(x0)), mark(y1))

A__2NDSPOS(s(2ndsneg(x0, x1)), cons2(y1, cons(y2, y3))) → A__2NDSNEG(a__2ndsneg(mark(x0), mark(x1)), mark(y3))

A__2NDSPOS(s(negrecip(x0)), cons2(y1, cons(y2, y3))) → A__2NDSNEG(negrecip(mark(x0)), mark(y3))

MARK(2ndsneg(plus(y0, y1), cons(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), cons(mark(x0), x1))

A__PLUS(s(X), Y) → MARK(Y)

A__TIMES(s(X), Y) → MARK(X)

A__TIMES(s(s(x0)), y1) → A__TIMES(s(mark(x0)), mark(y1))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(N)

A__2NDSNEG(s(y0), cons(y1, times(x0, x1))) → A__2NDSNEG(s(mark(y0)), cons2(y1, a__times(mark(x0), mark(x1))))

MARK(2ndsneg(s(y0), cons2(x0, x1))) → A__2NDSNEG(s(mark(y0)), cons2(x0, mark(x1)))

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(N)

MARK(cons(X1, X2)) → MARK(X1)

MARK(2ndsneg(plus(y0, y1), from(x0))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndspos(times(y0, y1), cons(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), cons(mark(x0), x1))

MARK(times(X1, X2)) → MARK(X2)

A__PI(square(x0)) → A__2NDSPOS(a__square(mark(x0)), a__from(0))

MARK(2ndsneg(square(y0), cons(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), cons(mark(x0), x1))

MARK(2ndspos(plus(y0, y1), times(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(times(y0, y1), plus(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(s(y0), times(x0, x1))) → A__2NDSPOS(s(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndsneg(X1, X2)) → MARK(X1)

MARK(2ndsneg(square(y0), square(x0))) → A__2NDSNEG(a__square(mark(y0)), a__square(mark(x0)))

A__2NDSPOS(s(square(x0)), cons2(y1, cons(y2, y3))) → A__2NDSNEG(a__square(mark(x0)), mark(y3))

A__2NDSPOS(s(pi(x0)), cons2(y1, cons(y2, y3))) → A__2NDSNEG(a__pi(mark(x0)), mark(y3))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(Y)

MARK(rcons(X1, X2)) → MARK(X1)

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(Z)

MARK(square(X)) → MARK(X)

A__PI(s(x0)) → A__2NDSPOS(s(mark(x0)), a__from(0))

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(Y)

A__FROM(X) → MARK(X)

MARK(times(plus(x0, x1), y1)) → A__TIMES(a__plus(mark(x0), mark(x1)), mark(y1))

MARK(2ndsneg(s(y0), plus(x0, x1))) → A__2NDSNEG(s(mark(y0)), a__plus(mark(x0), mark(x1)))

A__2NDSPOS(s(times(x0, x1)), cons2(y1, cons(y2, y3))) → A__2NDSNEG(a__times(mark(x0), mark(x1)), mark(y3))

A__2NDSNEG(s(y0), cons(y1, cons(x0, x1))) → A__2NDSNEG(s(mark(y0)), cons2(y1, cons(mark(x0), x1)))

A__2NDSPOS(s(2ndspos(x0, x1)), cons2(y1, cons(y2, y3))) → A__2NDSNEG(a__2ndspos(mark(x0), mark(x1)), mark(y3))

A__2NDSPOS(s(y0), cons(y1, from(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__from(mark(x0))))

A__2NDSPOS(s(cons(x0, x1)), cons2(y1, cons(y2, y3))) → A__2NDSNEG(cons(mark(x0), x1), mark(y3))

A__2NDSPOS(s(posrecip(x0)), cons2(y1, cons(y2, y3))) → A__2NDSNEG(posrecip(mark(x0)), mark(y3))

MARK(2ndspos(s(y0), square(x0))) → A__2NDSPOS(s(mark(y0)), a__square(mark(x0)))

MARK(rcons(X1, X2)) → MARK(X2)

MARK(2ndspos(s(y0), cons2(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(x0, mark(x1)))

A__2NDSNEG(s(y0), cons(y1, plus(x0, x1))) → A__2NDSNEG(s(mark(y0)), cons2(y1, a__plus(mark(x0), mark(x1))))

MARK(2ndsneg(plus(y0, y1), cons2(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), cons2(x0, mark(x1)))

A__2NDSPOS(s(from(x0)), cons2(y1, cons(y2, y3))) → A__2NDSNEG(a__from(mark(x0)), mark(y3))

A__2NDSPOS(s(y0), cons(y1, cons(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, cons(mark(x0), x1)))

A__PLUS(s(X), Y) → MARK(X)

MARK(times(X1, X2)) → MARK(X1)

A__TIMES(s(plus(x0, x1)), y1) → A__TIMES(a__plus(mark(x0), mark(x1)), mark(y1))

MARK(plus(times(x0, x1), y1)) → A__PLUS(a__times(mark(x0), mark(x1)), mark(y1))

A__SQUARE(X) → MARK(X)

MARK(negrecip(X)) → MARK(X)

A__2NDSPOS(s(N), cons(X, Z)) → MARK(N)

A__2NDSPOS(s(rnil), cons2(y1, cons(y2, y3))) → A__2NDSNEG(rnil, mark(y3))

A__PLUS(0, Y) → MARK(Y)

A__2NDSNEG(s(y0), cons(y1, square(x0))) → A__2NDSNEG(s(mark(y0)), cons2(y1, a__square(mark(x0))))

MARK(posrecip(X)) → MARK(X)

MARK(2ndspos(square(y0), from(x0))) → A__2NDSPOS(a__square(mark(y0)), a__from(mark(x0)))

MARK(2ndspos(times(y0, y1), from(x0))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__from(mark(x0)))

A__PLUS(s(times(x0, x1)), y1) → A__PLUS(a__times(mark(x0), mark(x1)), mark(y1))

A__2NDSPOS(s(y0), cons(y1, square(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__square(mark(x0))))

A__PLUS(s(0), y1) → A__PLUS(0, mark(y1))

MARK(plus(0, y1)) → A__PLUS(0, mark(y1))

MARK(from(X)) → MARK(X)

MARK(2ndsneg(s(y0), times(x0, x1))) → A__2NDSNEG(s(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(square(y0), square(x0))) → A__2NDSPOS(a__square(mark(y0)), a__square(mark(x0)))

A__SQUARE(X) → A__TIMES(mark(X), mark(X))

A__PLUS(s(plus(x0, x1)), y1) → A__PLUS(a__plus(mark(x0), mark(x1)), mark(y1))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(Z)

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → A__2NDSPOS(mark(N), mark(Z))

MARK(2ndsneg(plus(y0, y1), plus(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(square(X)) → A__SQUARE(mark(X))

MARK(2ndspos(plus(y0, y1), from(x0))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndsneg(plus(y0, y1), times(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndsneg(times(y0, y1), plus(x0, x1))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(plus(y0, y1), plus(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

A__PLUS(s(s(x0)), y1) → A__PLUS(s(mark(x0)), mark(y1))

MARK(2ndsneg(times(y0, y1), cons(x0, x1))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), cons(mark(x0), x1))

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))

a__2ndspos(0, Z) → rnil

a__2ndspos(s(N), cons(X, Z)) → a__2ndspos(s(mark(N)), cons2(X, mark(Z)))

a__2ndspos(s(N), cons2(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))

a__2ndsneg(0, Z) → rnil

a__2ndsneg(s(N), cons(X, Z)) → a__2ndsneg(s(mark(N)), cons2(X, mark(Z)))

a__2ndsneg(s(N), cons2(X, cons(Y, Z))) → rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z)))

a__pi(X) → a__2ndspos(mark(X), a__from(0))

a__plus(0, Y) → mark(Y)

a__plus(s(X), Y) → s(a__plus(mark(X), mark(Y)))

a__times(0, Y) → 0

a__times(s(X), Y) → a__plus(mark(Y), a__times(mark(X), mark(Y)))

a__square(X) → a__times(mark(X), mark(X))

mark(from(X)) → a__from(mark(X))

mark(2ndspos(X1, X2)) → a__2ndspos(mark(X1), mark(X2))

mark(2ndsneg(X1, X2)) → a__2ndsneg(mark(X1), mark(X2))

mark(pi(X)) → a__pi(mark(X))

mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))

mark(times(X1, X2)) → a__times(mark(X1), mark(X2))

mark(square(X)) → a__square(mark(X))

mark(0) → 0

mark(s(X)) → s(mark(X))

mark(posrecip(X)) → posrecip(mark(X))

mark(negrecip(X)) → negrecip(mark(X))

mark(nil) → nil

mark(cons(X1, X2)) → cons(mark(X1), X2)

mark(cons2(X1, X2)) → cons2(X1, mark(X2))

mark(rnil) → rnil

mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))

a__from(X) → from(X)

a__2ndspos(X1, X2) → 2ndspos(X1, X2)

a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)

a__pi(X) → pi(X)

a__plus(X1, X2) → plus(X1, X2)

a__times(X1, X2) → times(X1, X2)

a__square(X) → square(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 8 less nodes.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ Narrowing
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ QDP
              ↳ Narrowing
                ↳ QDP
                  ↳ DependencyGraphProof
                    ↳ QDP
                      ↳ Narrowing
                        ↳ QDP
                          ↳ DependencyGraphProof
                            ↳ QDP
                              ↳ Narrowing
                                ↳ QDP
                                  ↳ DependencyGraphProof
                                    ↳ QDP
                                      ↳ Narrowing
                                        ↳ QDP
                                          ↳ DependencyGraphProof
                                            ↳ QDP
                                              ↳ Narrowing
                                                ↳ QDP
                                                  ↳ DependencyGraphProof
                                                    ↳ QDP
                                                      ↳ Narrowing
                                                        ↳ QDP
                                                          ↳ DependencyGraphProof
                                                            ↳ QDP
                                                              ↳ Narrowing
                                                                ↳ QDP
                                                                  ↳ DependencyGraphProof
                                                                    ↳ QDP
                                                                      ↳ Narrowing
                                                                        ↳ QDP
                                                                          ↳ DependencyGraphProof
                                                                            ↳ QDP
                                                                              ↳ Narrowing
                                                                                ↳ QDP
                                                                                  ↳ DependencyGraphProof
                                                                                    ↳ QDP
                                                                                      ↳ Narrowing
                                                                                        ↳ QDP
                                                                                          ↳ DependencyGraphProof
                                                                                            ↳ QDP
                                                                                              ↳ Narrowing
                                                                                                ↳ QDP
                                                                                                  ↳ DependencyGraphProof
                                                                                                    ↳ QDP
                                                                                                      ↳ Narrowing
                                                                                                        ↳ QDP
                                                                                                          ↳ DependencyGraphProof
                                                                                                            ↳ QDP
                                                                                                              ↳ Narrowing
                                                                                                                ↳ QDP
                                                                                                                  ↳ DependencyGraphProof
                                                                                                                    ↳ QDP
                                                                                                                      ↳ Narrowing
                                                                                                                        ↳ QDP
                                                                                                                          ↳ DependencyGraphProof
                                                                                                                            ↳ QDP
                                                                                                                              ↳ Narrowing
                                                                                                                                ↳ QDP
                                                                                                                                  ↳ DependencyGraphProof
                                                                                                                                    ↳ QDP
                                                                                                                                      ↳ Narrowing
                                                                                                                                        ↳ QDP
                                                                                                                                          ↳ DependencyGraphProof
                                                                                                                                            ↳ QDP
                                                                                                                                              ↳ Narrowing
                                                                                                                                                ↳ QDP
                                                                                                                                                  ↳ DependencyGraphProof
                                                                                                                                                    ↳ QDP
                                                                                                                                                      ↳ Narrowing
                                                                                                                                                        ↳ QDP
                                                                                                                                                          ↳ DependencyGraphProof
                                                                                                                                                            ↳ QDP
                                                                                                                                                              ↳ Narrowing
                                                                                                                                                                ↳ QDP
                                                                                                                                                                  ↳ DependencyGraphProof
                                                                                                                                                                    ↳ QDP
                                                                                                                                                                      ↳ Narrowing
                                                                                                                                                                        ↳ QDP
                                                                                                                                                                          ↳ DependencyGraphProof
                                                                                                                                                                            ↳ QDP
                                                                                                                                                                              ↳ Narrowing
                                                                                                                                                                                ↳ QDP
                                                                                                                                                                                  ↳ DependencyGraphProof
                                                                                                                                                                                    ↳ QDP
                                                                                                                                                                                      ↳ Narrowing
                                                                                                                                                                                        ↳ QDP
                                                                                                                                                                                          ↳ DependencyGraphProof
                                                                                                                                                                                            ↳ QDP
                                                                                                                                                                                              ↳ QDPOrderProof
                                                                                                                                                                                                ↳ QDP
                                                                                                                                                                                                  ↳ QDPOrderProof
                                                                                                                                                                                                    ↳ QDP
                                                                                                                                                                                                      ↳ QDPOrderProof
                                                                                                                                                                                                        ↳ QDP
                                                                                                                                                                                                          ↳ QDPOrderProof
                                                                                                                                                                                                            ↳ QDP
                                                                                                                                                                                                              ↳ QDPOrderProof
                                                                                                                                                                                                                ↳ QDP
                                                                                                                                                                                                                  ↳ QDPOrderProof
                                                                                                                                                                                                                    ↳ QDP
                                                                                                                                                                                                                      ↳ QDPOrderProof
                                                                                                                                                                                                                        ↳ QDP
                                                                                                                                                                                                                          ↳ QDPOrderProof
                                                                                                                                                                                                                            ↳ QDP
                                                                                                                                                                                                                              ↳ Narrowing
                                                                                                                                                                                                                                ↳ QDP
                                                                                                                                                                                                                                  ↳ DependencyGraphProof
                                                                                                                                                                                                                                    ↳ QDP
                                                                                                                                                                                                                                      ↳ Narrowing
                                                                                                                                                                                                                                        ↳ QDP
                                                                                                                                                                                                                                          ↳ DependencyGraphProof
                                                                                                                                                                                                                                            ↳ QDP
                                                                                                                                                                                                                                              ↳ QDPOrderProof
                                                                                                                                                                                                                                                ↳ QDP
                                                                                                                                                                                                                                                  ↳ Narrowing
                                                                                                                                                                                                                                                    ↳ QDP
                                                                                                                                                                                                                                                      ↳ DependencyGraphProof
                                                                                                                                                                                                                                                        ↳ QDP
                                                                                                                                                                                                                                                          ↳ Narrowing
                                                                                                                                                                                                                                                            ↳ QDP
                                                                                                                                                                                                                                                              ↳ DependencyGraphProof
                                                                                                                                                                                                                                                                ↳ QDP
                                                                                                                                                                                                                                                                  ↳ Narrowing

Q DP problem:
The TRS P consists of the following rules:

MARK(2ndspos(plus(y0, y1), square(x0))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndspos(square(y0), plus(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(2ndsneg(s(y0), square(x0))) → A__2NDSNEG(s(mark(y0)), a__square(mark(x0)))

MARK(times(times(x0, x1), y1)) → A__TIMES(a__times(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(X1, X2)) → MARK(X2)

MARK(plus(plus(x0, x1), y1)) → A__PLUS(a__plus(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(s(y0), from(x0))) → A__2NDSPOS(s(mark(y0)), a__from(mark(x0)))

A__2NDSNEG(s(y0), cons(y1, 2ndsneg(x0, x1))) → A__2NDSNEG(s(mark(y0)), cons2(y1, a__2ndsneg(mark(x0), mark(x1))))

A__2NDSNEG(s(N), cons(X, Z)) → MARK(Z)

A__2NDSPOS(s(s(x0)), cons2(y1, cons(y2, y3))) → A__2NDSNEG(s(mark(x0)), mark(y3))

A__2NDSNEG(s(y0), cons(y1, 2ndspos(x0, x1))) → A__2NDSNEG(s(mark(y0)), cons2(y1, a__2ndspos(mark(x0), mark(x1))))

A__TIMES(s(X), Y) → MARK(Y)

A__2NDSPOS(s(y0), cons(y1, plus(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__plus(mark(x0), mark(x1))))

A__2NDSNEG(s(N), cons(X, Z)) → MARK(N)

MARK(2ndsneg(times(y0, y1), times(x0, x1))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(times(y0, y1), times(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

A__TIMES(s(X), Y) → A__PLUS(mark(Y), a__times(mark(X), mark(Y)))

MARK(2ndspos(square(y0), cons2(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), cons2(x0, mark(x1)))

MARK(plus(square(x0), y1)) → A__PLUS(a__square(mark(x0)), mark(y1))

MARK(plus(s(x0), y1)) → A__PLUS(s(mark(x0)), mark(y1))

A__PI(plus(x0, x1)) → A__2NDSPOS(a__plus(mark(x0), mark(x1)), a__from(0))

MARK(2ndspos(square(y0), cons(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), cons(mark(x0), x1))

MARK(2ndsneg(times(y0, y1), cons2(x0, x1))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), cons2(x0, mark(x1)))

A__PI(times(y0, y1)) → A__2NDSPOS(a__times(mark(y0), mark(y1)), cons(mark(0), from(s(0))))

MARK(cons2(X1, X2)) → MARK(X2)

MARK(2ndsneg(square(y0), cons2(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), cons2(x0, mark(x1)))

A__2NDSPOS(s(plus(x0, x1)), cons2(y1, cons(y2, y3))) → A__2NDSNEG(a__plus(mark(x0), mark(x1)), mark(y3))

MARK(2ndsneg(plus(y0, y1), square(x0))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndsneg(square(y0), plus(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), a__plus(mark(x0), mark(x1)))

A__TIMES(s(square(x0)), y1) → A__TIMES(a__square(mark(x0)), mark(y1))

A__PLUS(s(square(x0)), y1) → A__PLUS(a__square(mark(x0)), mark(y1))

MARK(2ndspos(X1, X2)) → MARK(X1)

MARK(s(X)) → MARK(X)

MARK(2ndspos(plus(y0, y1), cons2(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(pi(X)) → MARK(X)

MARK(2ndsneg(s(y0), from(x0))) → A__2NDSNEG(s(mark(y0)), a__from(mark(x0)))

MARK(plus(X1, X2)) → MARK(X2)

A__2NDSPOS(s(N), cons(X, Z)) → MARK(Z)

MARK(pi(X)) → A__PI(mark(X))

MARK(2ndsneg(times(y0, y1), from(x0))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndsneg(times(y0, y1), square(x0))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndsneg(square(y0), times(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(plus(y0, y1), cons(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), cons(mark(x0), x1))

MARK(2ndspos(times(y0, y1), cons2(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(2ndsneg(X1, X2)) → MARK(X2)

MARK(2ndsneg(square(y0), from(x0))) → A__2NDSNEG(a__square(mark(y0)), a__from(mark(x0)))

A__TIMES(s(times(x0, x1)), y1) → A__TIMES(a__times(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(square(y0), times(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(times(y0, y1), square(x0))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndsneg(s(y0), cons(x0, x1))) → A__2NDSNEG(s(mark(y0)), cons(mark(x0), x1))

MARK(times(square(x0), y1)) → A__TIMES(a__square(mark(x0)), mark(y1))

A__2NDSNEG(s(y0), cons(y1, from(x0))) → A__2NDSNEG(s(mark(y0)), cons2(y1, a__from(mark(x0))))

A__PI(X) → A__FROM(0)

MARK(from(X)) → A__FROM(mark(X))

MARK(2ndspos(s(y0), plus(x0, x1))) → A__2NDSPOS(s(mark(y0)), a__plus(mark(x0), mark(x1)))

A__2NDSNEG(s(y0), cons(y1, pi(x0))) → A__2NDSNEG(s(mark(y0)), cons2(y1, a__pi(mark(x0))))

MARK(plus(X1, X2)) → MARK(X1)

A__2NDSPOS(s(y0), cons(y1, times(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__times(mark(x0), mark(x1))))

A__PI(X) → MARK(X)

MARK(2ndspos(s(y0), cons(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons(mark(x0), x1))

MARK(times(s(x0), y1)) → A__TIMES(s(mark(x0)), mark(y1))

A__2NDSPOS(s(2ndsneg(x0, x1)), cons2(y1, cons(y2, y3))) → A__2NDSNEG(a__2ndsneg(mark(x0), mark(x1)), mark(y3))

MARK(2ndsneg(plus(y0, y1), cons(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), cons(mark(x0), x1))

A__PLUS(s(X), Y) → MARK(Y)

A__TIMES(s(X), Y) → MARK(X)

A__TIMES(s(s(x0)), y1) → A__TIMES(s(mark(x0)), mark(y1))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(N)

MARK(2ndsneg(s(y0), cons2(x0, x1))) → A__2NDSNEG(s(mark(y0)), cons2(x0, mark(x1)))

A__2NDSNEG(s(y0), cons(y1, times(x0, x1))) → A__2NDSNEG(s(mark(y0)), cons2(y1, a__times(mark(x0), mark(x1))))

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(N)

MARK(cons(X1, X2)) → MARK(X1)

MARK(2ndsneg(plus(y0, y1), from(x0))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(times(X1, X2)) → MARK(X2)

MARK(2ndspos(times(y0, y1), cons(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), cons(mark(x0), x1))

A__PI(square(x0)) → A__2NDSPOS(a__square(mark(x0)), a__from(0))

MARK(2ndsneg(square(y0), cons(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), cons(mark(x0), x1))

MARK(2ndspos(plus(y0, y1), times(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(times(y0, y1), plus(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(s(y0), times(x0, x1))) → A__2NDSPOS(s(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndsneg(X1, X2)) → MARK(X1)

MARK(2ndsneg(square(y0), square(x0))) → A__2NDSNEG(a__square(mark(y0)), a__square(mark(x0)))

A__2NDSPOS(s(square(x0)), cons2(y1, cons(y2, y3))) → A__2NDSNEG(a__square(mark(x0)), mark(y3))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(Y)

A__2NDSPOS(s(pi(x0)), cons2(y1, cons(y2, y3))) → A__2NDSNEG(a__pi(mark(x0)), mark(y3))

MARK(rcons(X1, X2)) → MARK(X1)

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(Z)

MARK(square(X)) → MARK(X)

A__PI(s(x0)) → A__2NDSPOS(s(mark(x0)), a__from(0))

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(Y)

A__FROM(X) → MARK(X)

MARK(2ndsneg(s(y0), plus(x0, x1))) → A__2NDSNEG(s(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(times(plus(x0, x1), y1)) → A__TIMES(a__plus(mark(x0), mark(x1)), mark(y1))

A__2NDSPOS(s(times(x0, x1)), cons2(y1, cons(y2, y3))) → A__2NDSNEG(a__times(mark(x0), mark(x1)), mark(y3))

A__2NDSNEG(s(y0), cons(y1, cons(x0, x1))) → A__2NDSNEG(s(mark(y0)), cons2(y1, cons(mark(x0), x1)))

A__2NDSPOS(s(y0), cons(y1, from(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__from(mark(x0))))

A__2NDSPOS(s(2ndspos(x0, x1)), cons2(y1, cons(y2, y3))) → A__2NDSNEG(a__2ndspos(mark(x0), mark(x1)), mark(y3))

MARK(rcons(X1, X2)) → MARK(X2)

MARK(2ndspos(s(y0), square(x0))) → A__2NDSPOS(s(mark(y0)), a__square(mark(x0)))

MARK(2ndspos(s(y0), cons2(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(x0, mark(x1)))

MARK(2ndsneg(plus(y0, y1), cons2(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), cons2(x0, mark(x1)))

A__2NDSNEG(s(y0), cons(y1, plus(x0, x1))) → A__2NDSNEG(s(mark(y0)), cons2(y1, a__plus(mark(x0), mark(x1))))

A__2NDSPOS(s(from(x0)), cons2(y1, cons(y2, y3))) → A__2NDSNEG(a__from(mark(x0)), mark(y3))

A__PLUS(s(X), Y) → MARK(X)

A__2NDSPOS(s(y0), cons(y1, cons(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, cons(mark(x0), x1)))

MARK(times(X1, X2)) → MARK(X1)

A__TIMES(s(plus(x0, x1)), y1) → A__TIMES(a__plus(mark(x0), mark(x1)), mark(y1))

MARK(plus(times(x0, x1), y1)) → A__PLUS(a__times(mark(x0), mark(x1)), mark(y1))

A__SQUARE(X) → MARK(X)

MARK(negrecip(X)) → MARK(X)

A__2NDSPOS(s(N), cons(X, Z)) → MARK(N)

A__PLUS(0, Y) → MARK(Y)

A__2NDSNEG(s(y0), cons(y1, square(x0))) → A__2NDSNEG(s(mark(y0)), cons2(y1, a__square(mark(x0))))

MARK(posrecip(X)) → MARK(X)

MARK(2ndspos(square(y0), from(x0))) → A__2NDSPOS(a__square(mark(y0)), a__from(mark(x0)))

MARK(2ndspos(times(y0, y1), from(x0))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__from(mark(x0)))

A__PLUS(s(times(x0, x1)), y1) → A__PLUS(a__times(mark(x0), mark(x1)), mark(y1))

A__2NDSPOS(s(y0), cons(y1, square(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__square(mark(x0))))

A__PLUS(s(0), y1) → A__PLUS(0, mark(y1))

MARK(plus(0, y1)) → A__PLUS(0, mark(y1))

MARK(from(X)) → MARK(X)

MARK(2ndsneg(s(y0), times(x0, x1))) → A__2NDSNEG(s(mark(y0)), a__times(mark(x0), mark(x1)))

A__SQUARE(X) → A__TIMES(mark(X), mark(X))

MARK(2ndspos(square(y0), square(x0))) → A__2NDSPOS(a__square(mark(y0)), a__square(mark(x0)))

A__PLUS(s(plus(x0, x1)), y1) → A__PLUS(a__plus(mark(x0), mark(x1)), mark(y1))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(Z)

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → A__2NDSPOS(mark(N), mark(Z))

MARK(2ndsneg(plus(y0, y1), plus(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(square(X)) → A__SQUARE(mark(X))

MARK(2ndspos(plus(y0, y1), from(x0))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndsneg(times(y0, y1), plus(x0, x1))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(2ndsneg(plus(y0, y1), times(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(plus(y0, y1), plus(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(2ndsneg(times(y0, y1), cons(x0, x1))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), cons(mark(x0), x1))

A__PLUS(s(s(x0)), y1) → A__PLUS(s(mark(x0)), mark(y1))

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))

a__2ndspos(0, Z) → rnil

a__2ndspos(s(N), cons(X, Z)) → a__2ndspos(s(mark(N)), cons2(X, mark(Z)))

a__2ndspos(s(N), cons2(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))

a__2ndsneg(0, Z) → rnil

a__2ndsneg(s(N), cons(X, Z)) → a__2ndsneg(s(mark(N)), cons2(X, mark(Z)))

a__2ndsneg(s(N), cons2(X, cons(Y, Z))) → rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z)))

a__pi(X) → a__2ndspos(mark(X), a__from(0))

a__plus(0, Y) → mark(Y)

a__plus(s(X), Y) → s(a__plus(mark(X), mark(Y)))

a__times(0, Y) → 0

a__times(s(X), Y) → a__plus(mark(Y), a__times(mark(X), mark(Y)))

a__square(X) → a__times(mark(X), mark(X))

mark(from(X)) → a__from(mark(X))

mark(2ndspos(X1, X2)) → a__2ndspos(mark(X1), mark(X2))

mark(2ndsneg(X1, X2)) → a__2ndsneg(mark(X1), mark(X2))

mark(pi(X)) → a__pi(mark(X))

mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))

mark(times(X1, X2)) → a__times(mark(X1), mark(X2))

mark(square(X)) → a__square(mark(X))

mark(0) → 0

mark(s(X)) → s(mark(X))

mark(posrecip(X)) → posrecip(mark(X))

mark(negrecip(X)) → negrecip(mark(X))

mark(nil) → nil

mark(cons(X1, X2)) → cons(mark(X1), X2)

mark(cons2(X1, X2)) → cons2(X1, mark(X2))

mark(rnil) → rnil

mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))

a__from(X) → from(X)

a__2ndspos(X1, X2) → 2ndspos(X1, X2)

a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)

a__pi(X) → pi(X)

a__plus(X1, X2) → plus(X1, X2)

a__times(X1, X2) → times(X1, X2)

a__square(X) → square(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule A__2NDSPOS(s(s(x0)), cons2(y1, cons(y2, y3))) → A__2NDSNEG(s(mark(x0)), mark(y3)) at position [1] we obtained the following new rules:

A__2NDSPOS(s(s(y0)), cons2(y1, cons(y2, 0))) → A__2NDSNEG(s(mark(y0)), 0)

A__2NDSPOS(s(s(y0)), cons2(y1, cons(y2, s(x0)))) → A__2NDSNEG(s(mark(y0)), s(mark(x0)))

A__2NDSPOS(s(s(y0)), cons2(y1, cons(y2, from(x0)))) → A__2NDSNEG(s(mark(y0)), a__from(mark(x0)))

A__2NDSPOS(s(s(y0)), cons2(y1, cons(y2, pi(x0)))) → A__2NDSNEG(s(mark(y0)), a__pi(mark(x0)))

A__2NDSPOS(s(s(y0)), cons2(y1, cons(y2, cons2(x0, x1)))) → A__2NDSNEG(s(mark(y0)), cons2(x0, mark(x1)))

A__2NDSPOS(s(s(y0)), cons2(y1, cons(y2, square(x0)))) → A__2NDSNEG(s(mark(y0)), a__square(mark(x0)))

A__2NDSPOS(s(s(y0)), cons2(y1, cons(y2, rcons(x0, x1)))) → A__2NDSNEG(s(mark(y0)), rcons(mark(x0), mark(x1)))

A__2NDSPOS(s(s(y0)), cons2(y1, cons(y2, times(x0, x1)))) → A__2NDSNEG(s(mark(y0)), a__times(mark(x0), mark(x1)))

A__2NDSPOS(s(s(y0)), cons2(y1, cons(y2, posrecip(x0)))) → A__2NDSNEG(s(mark(y0)), posrecip(mark(x0)))

A__2NDSPOS(s(s(y0)), cons2(y1, cons(y2, plus(x0, x1)))) → A__2NDSNEG(s(mark(y0)), a__plus(mark(x0), mark(x1)))

A__2NDSPOS(s(s(y0)), cons2(y1, cons(y2, negrecip(x0)))) → A__2NDSNEG(s(mark(y0)), negrecip(mark(x0)))

A__2NDSPOS(s(s(y0)), cons2(y1, cons(y2, cons(x0, x1)))) → A__2NDSNEG(s(mark(y0)), cons(mark(x0), x1))

A__2NDSPOS(s(s(y0)), cons2(y1, cons(y2, 2ndsneg(x0, x1)))) → A__2NDSNEG(s(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

A__2NDSPOS(s(s(y0)), cons2(y1, cons(y2, rnil))) → A__2NDSNEG(s(mark(y0)), rnil)

A__2NDSPOS(s(s(y0)), cons2(y1, cons(y2, 2ndspos(x0, x1)))) → A__2NDSNEG(s(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

A__2NDSPOS(s(s(y0)), cons2(y1, cons(y2, nil))) → A__2NDSNEG(s(mark(y0)), nil)



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ Narrowing
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ QDP
              ↳ Narrowing
                ↳ QDP
                  ↳ DependencyGraphProof
                    ↳ QDP
                      ↳ Narrowing
                        ↳ QDP
                          ↳ DependencyGraphProof
                            ↳ QDP
                              ↳ Narrowing
                                ↳ QDP
                                  ↳ DependencyGraphProof
                                    ↳ QDP
                                      ↳ Narrowing
                                        ↳ QDP
                                          ↳ DependencyGraphProof
                                            ↳ QDP
                                              ↳ Narrowing
                                                ↳ QDP
                                                  ↳ DependencyGraphProof
                                                    ↳ QDP
                                                      ↳ Narrowing
                                                        ↳ QDP
                                                          ↳ DependencyGraphProof
                                                            ↳ QDP
                                                              ↳ Narrowing
                                                                ↳ QDP
                                                                  ↳ DependencyGraphProof
                                                                    ↳ QDP
                                                                      ↳ Narrowing
                                                                        ↳ QDP
                                                                          ↳ DependencyGraphProof
                                                                            ↳ QDP
                                                                              ↳ Narrowing
                                                                                ↳ QDP
                                                                                  ↳ DependencyGraphProof
                                                                                    ↳ QDP
                                                                                      ↳ Narrowing
                                                                                        ↳ QDP
                                                                                          ↳ DependencyGraphProof
                                                                                            ↳ QDP
                                                                                              ↳ Narrowing
                                                                                                ↳ QDP
                                                                                                  ↳ DependencyGraphProof
                                                                                                    ↳ QDP
                                                                                                      ↳ Narrowing
                                                                                                        ↳ QDP
                                                                                                          ↳ DependencyGraphProof
                                                                                                            ↳ QDP
                                                                                                              ↳ Narrowing
                                                                                                                ↳ QDP
                                                                                                                  ↳ DependencyGraphProof
                                                                                                                    ↳ QDP
                                                                                                                      ↳ Narrowing
                                                                                                                        ↳ QDP
                                                                                                                          ↳ DependencyGraphProof
                                                                                                                            ↳ QDP
                                                                                                                              ↳ Narrowing
                                                                                                                                ↳ QDP
                                                                                                                                  ↳ DependencyGraphProof
                                                                                                                                    ↳ QDP
                                                                                                                                      ↳ Narrowing
                                                                                                                                        ↳ QDP
                                                                                                                                          ↳ DependencyGraphProof
                                                                                                                                            ↳ QDP
                                                                                                                                              ↳ Narrowing
                                                                                                                                                ↳ QDP
                                                                                                                                                  ↳ DependencyGraphProof
                                                                                                                                                    ↳ QDP
                                                                                                                                                      ↳ Narrowing
                                                                                                                                                        ↳ QDP
                                                                                                                                                          ↳ DependencyGraphProof
                                                                                                                                                            ↳ QDP
                                                                                                                                                              ↳ Narrowing
                                                                                                                                                                ↳ QDP
                                                                                                                                                                  ↳ DependencyGraphProof
                                                                                                                                                                    ↳ QDP
                                                                                                                                                                      ↳ Narrowing
                                                                                                                                                                        ↳ QDP
                                                                                                                                                                          ↳ DependencyGraphProof
                                                                                                                                                                            ↳ QDP
                                                                                                                                                                              ↳ Narrowing
                                                                                                                                                                                ↳ QDP
                                                                                                                                                                                  ↳ DependencyGraphProof
                                                                                                                                                                                    ↳ QDP
                                                                                                                                                                                      ↳ Narrowing
                                                                                                                                                                                        ↳ QDP
                                                                                                                                                                                          ↳ DependencyGraphProof
                                                                                                                                                                                            ↳ QDP
                                                                                                                                                                                              ↳ QDPOrderProof
                                                                                                                                                                                                ↳ QDP
                                                                                                                                                                                                  ↳ QDPOrderProof
                                                                                                                                                                                                    ↳ QDP
                                                                                                                                                                                                      ↳ QDPOrderProof
                                                                                                                                                                                                        ↳ QDP
                                                                                                                                                                                                          ↳ QDPOrderProof
                                                                                                                                                                                                            ↳ QDP
                                                                                                                                                                                                              ↳ QDPOrderProof
                                                                                                                                                                                                                ↳ QDP
                                                                                                                                                                                                                  ↳ QDPOrderProof
                                                                                                                                                                                                                    ↳ QDP
                                                                                                                                                                                                                      ↳ QDPOrderProof
                                                                                                                                                                                                                        ↳ QDP
                                                                                                                                                                                                                          ↳ QDPOrderProof
                                                                                                                                                                                                                            ↳ QDP
                                                                                                                                                                                                                              ↳ Narrowing
                                                                                                                                                                                                                                ↳ QDP
                                                                                                                                                                                                                                  ↳ DependencyGraphProof
                                                                                                                                                                                                                                    ↳ QDP
                                                                                                                                                                                                                                      ↳ Narrowing
                                                                                                                                                                                                                                        ↳ QDP
                                                                                                                                                                                                                                          ↳ DependencyGraphProof
                                                                                                                                                                                                                                            ↳ QDP
                                                                                                                                                                                                                                              ↳ QDPOrderProof
                                                                                                                                                                                                                                                ↳ QDP
                                                                                                                                                                                                                                                  ↳ Narrowing
                                                                                                                                                                                                                                                    ↳ QDP
                                                                                                                                                                                                                                                      ↳ DependencyGraphProof
                                                                                                                                                                                                                                                        ↳ QDP
                                                                                                                                                                                                                                                          ↳ Narrowing
                                                                                                                                                                                                                                                            ↳ QDP
                                                                                                                                                                                                                                                              ↳ DependencyGraphProof
                                                                                                                                                                                                                                                                ↳ QDP
                                                                                                                                                                                                                                                                  ↳ Narrowing
                                                                                                                                                                                                                                                                    ↳ QDP
                                                                                                                                                                                                                                                                      ↳ DependencyGraphProof

Q DP problem:
The TRS P consists of the following rules:

MARK(2ndspos(square(y0), plus(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(plus(y0, y1), square(x0))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndsneg(s(y0), square(x0))) → A__2NDSNEG(s(mark(y0)), a__square(mark(x0)))

MARK(2ndspos(X1, X2)) → MARK(X2)

MARK(times(times(x0, x1), y1)) → A__TIMES(a__times(mark(x0), mark(x1)), mark(y1))

A__2NDSPOS(s(s(y0)), cons2(y1, cons(y2, cons(x0, x1)))) → A__2NDSNEG(s(mark(y0)), cons(mark(x0), x1))

MARK(plus(plus(x0, x1), y1)) → A__PLUS(a__plus(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(s(y0), from(x0))) → A__2NDSPOS(s(mark(y0)), a__from(mark(x0)))

A__2NDSNEG(s(y0), cons(y1, 2ndsneg(x0, x1))) → A__2NDSNEG(s(mark(y0)), cons2(y1, a__2ndsneg(mark(x0), mark(x1))))

A__2NDSNEG(s(N), cons(X, Z)) → MARK(Z)

A__2NDSNEG(s(y0), cons(y1, 2ndspos(x0, x1))) → A__2NDSNEG(s(mark(y0)), cons2(y1, a__2ndspos(mark(x0), mark(x1))))

A__TIMES(s(X), Y) → MARK(Y)

A__2NDSPOS(s(y0), cons(y1, plus(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__plus(mark(x0), mark(x1))))

A__2NDSNEG(s(N), cons(X, Z)) → MARK(N)

MARK(2ndsneg(times(y0, y1), times(x0, x1))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

A__2NDSPOS(s(s(y0)), cons2(y1, cons(y2, posrecip(x0)))) → A__2NDSNEG(s(mark(y0)), posrecip(mark(x0)))

A__2NDSPOS(s(s(y0)), cons2(y1, cons(y2, plus(x0, x1)))) → A__2NDSNEG(s(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(times(y0, y1), times(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

A__TIMES(s(X), Y) → A__PLUS(mark(Y), a__times(mark(X), mark(Y)))

MARK(2ndspos(square(y0), cons2(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), cons2(x0, mark(x1)))

MARK(plus(square(x0), y1)) → A__PLUS(a__square(mark(x0)), mark(y1))

MARK(plus(s(x0), y1)) → A__PLUS(s(mark(x0)), mark(y1))

MARK(2ndspos(square(y0), cons(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), cons(mark(x0), x1))

A__PI(plus(x0, x1)) → A__2NDSPOS(a__plus(mark(x0), mark(x1)), a__from(0))

MARK(2ndsneg(times(y0, y1), cons2(x0, x1))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), cons2(x0, mark(x1)))

A__PI(times(y0, y1)) → A__2NDSPOS(a__times(mark(y0), mark(y1)), cons(mark(0), from(s(0))))

MARK(cons2(X1, X2)) → MARK(X2)

MARK(2ndsneg(square(y0), cons2(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), cons2(x0, mark(x1)))

A__2NDSPOS(s(s(y0)), cons2(y1, cons(y2, s(x0)))) → A__2NDSNEG(s(mark(y0)), s(mark(x0)))

A__2NDSPOS(s(plus(x0, x1)), cons2(y1, cons(y2, y3))) → A__2NDSNEG(a__plus(mark(x0), mark(x1)), mark(y3))

MARK(s(X)) → MARK(X)

MARK(2ndspos(X1, X2)) → MARK(X1)

A__PLUS(s(square(x0)), y1) → A__PLUS(a__square(mark(x0)), mark(y1))

A__TIMES(s(square(x0)), y1) → A__TIMES(a__square(mark(x0)), mark(y1))

MARK(2ndsneg(square(y0), plus(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(2ndsneg(plus(y0, y1), square(x0))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndspos(plus(y0, y1), cons2(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(pi(X)) → MARK(X)

MARK(2ndsneg(s(y0), from(x0))) → A__2NDSNEG(s(mark(y0)), a__from(mark(x0)))

MARK(plus(X1, X2)) → MARK(X2)

A__2NDSPOS(s(N), cons(X, Z)) → MARK(Z)

MARK(pi(X)) → A__PI(mark(X))

MARK(2ndspos(plus(y0, y1), cons(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), cons(mark(x0), x1))

MARK(2ndsneg(square(y0), times(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndsneg(times(y0, y1), square(x0))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndsneg(times(y0, y1), from(x0))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndspos(times(y0, y1), cons2(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), cons2(x0, mark(x1)))

MARK(2ndsneg(X1, X2)) → MARK(X2)

MARK(2ndsneg(square(y0), from(x0))) → A__2NDSNEG(a__square(mark(y0)), a__from(mark(x0)))

A__TIMES(s(times(x0, x1)), y1) → A__TIMES(a__times(mark(x0), mark(x1)), mark(y1))

MARK(2ndspos(times(y0, y1), square(x0))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__square(mark(x0)))

MARK(2ndspos(square(y0), times(x0, x1))) → A__2NDSPOS(a__square(mark(y0)), a__times(mark(x0), mark(x1)))

A__2NDSPOS(s(s(y0)), cons2(y1, cons(y2, 0))) → A__2NDSNEG(s(mark(y0)), 0)

A__2NDSNEG(s(y0), cons(y1, from(x0))) → A__2NDSNEG(s(mark(y0)), cons2(y1, a__from(mark(x0))))

MARK(times(square(x0), y1)) → A__TIMES(a__square(mark(x0)), mark(y1))

MARK(2ndsneg(s(y0), cons(x0, x1))) → A__2NDSNEG(s(mark(y0)), cons(mark(x0), x1))

MARK(2ndspos(s(y0), plus(x0, x1))) → A__2NDSPOS(s(mark(y0)), a__plus(mark(x0), mark(x1)))

MARK(from(X)) → A__FROM(mark(X))

A__PI(X) → A__FROM(0)

A__2NDSNEG(s(y0), cons(y1, pi(x0))) → A__2NDSNEG(s(mark(y0)), cons2(y1, a__pi(mark(x0))))

A__2NDSPOS(s(s(y0)), cons2(y1, cons(y2, from(x0)))) → A__2NDSNEG(s(mark(y0)), a__from(mark(x0)))

MARK(plus(X1, X2)) → MARK(X1)

A__2NDSPOS(s(s(y0)), cons2(y1, cons(y2, square(x0)))) → A__2NDSNEG(s(mark(y0)), a__square(mark(x0)))

A__2NDSPOS(s(y0), cons(y1, times(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__times(mark(x0), mark(x1))))

A__2NDSPOS(s(s(y0)), cons2(y1, cons(y2, times(x0, x1)))) → A__2NDSNEG(s(mark(y0)), a__times(mark(x0), mark(x1)))

A__2NDSPOS(s(s(y0)), cons2(y1, cons(y2, negrecip(x0)))) → A__2NDSNEG(s(mark(y0)), negrecip(mark(x0)))

MARK(2ndspos(s(y0), cons(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons(mark(x0), x1))

A__PI(X) → MARK(X)

A__2NDSPOS(s(s(y0)), cons2(y1, cons(y2, rnil))) → A__2NDSNEG(s(mark(y0)), rnil)

A__2NDSPOS(s(s(y0)), cons2(y1, cons(y2, 2ndspos(x0, x1)))) → A__2NDSNEG(s(mark(y0)), a__2ndspos(mark(x0), mark(x1)))

MARK(times(s(x0), y1)) → A__TIMES(s(mark(x0)), mark(y1))

A__2NDSPOS(s(2ndsneg(x0, x1)), cons2(y1, cons(y2, y3))) → A__2NDSNEG(a__2ndsneg(mark(x0), mark(x1)), mark(y3))

MARK(2ndsneg(plus(y0, y1), cons(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), cons(mark(x0), x1))

A__2NDSPOS(s(s(y0)), cons2(y1, cons(y2, nil))) → A__2NDSNEG(s(mark(y0)), nil)

A__PLUS(s(X), Y) → MARK(Y)

A__TIMES(s(X), Y) → MARK(X)

A__TIMES(s(s(x0)), y1) → A__TIMES(s(mark(x0)), mark(y1))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(N)

A__2NDSNEG(s(y0), cons(y1, times(x0, x1))) → A__2NDSNEG(s(mark(y0)), cons2(y1, a__times(mark(x0), mark(x1))))

MARK(2ndsneg(s(y0), cons2(x0, x1))) → A__2NDSNEG(s(mark(y0)), cons2(x0, mark(x1)))

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(N)

MARK(cons(X1, X2)) → MARK(X1)

MARK(2ndsneg(plus(y0, y1), from(x0))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndspos(times(y0, y1), cons(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), cons(mark(x0), x1))

MARK(times(X1, X2)) → MARK(X2)

A__PI(square(x0)) → A__2NDSPOS(a__square(mark(x0)), a__from(0))

A__2NDSPOS(s(s(y0)), cons2(y1, cons(y2, 2ndsneg(x0, x1)))) → A__2NDSNEG(s(mark(y0)), a__2ndsneg(mark(x0), mark(x1)))

MARK(2ndspos(times(y0, y1), plus(x0, x1))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(plus(y0, y1), times(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndsneg(square(y0), cons(x0, x1))) → A__2NDSNEG(a__square(mark(y0)), cons(mark(x0), x1))

MARK(2ndspos(s(y0), times(x0, x1))) → A__2NDSPOS(s(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndsneg(X1, X2)) → MARK(X1)

MARK(2ndsneg(square(y0), square(x0))) → A__2NDSNEG(a__square(mark(y0)), a__square(mark(x0)))

A__2NDSPOS(s(square(x0)), cons2(y1, cons(y2, y3))) → A__2NDSNEG(a__square(mark(x0)), mark(y3))

A__2NDSPOS(s(pi(x0)), cons2(y1, cons(y2, y3))) → A__2NDSNEG(a__pi(mark(x0)), mark(y3))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(Y)

MARK(rcons(X1, X2)) → MARK(X1)

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(Z)

A__2NDSPOS(s(s(y0)), cons2(y1, cons(y2, pi(x0)))) → A__2NDSNEG(s(mark(y0)), a__pi(mark(x0)))

MARK(square(X)) → MARK(X)

A__2NDSPOS(s(s(y0)), cons2(y1, cons(y2, cons2(x0, x1)))) → A__2NDSNEG(s(mark(y0)), cons2(x0, mark(x1)))

A__PI(s(x0)) → A__2NDSPOS(s(mark(x0)), a__from(0))

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(Y)

A__FROM(X) → MARK(X)

MARK(times(plus(x0, x1), y1)) → A__TIMES(a__plus(mark(x0), mark(x1)), mark(y1))

MARK(2ndsneg(s(y0), plus(x0, x1))) → A__2NDSNEG(s(mark(y0)), a__plus(mark(x0), mark(x1)))

A__2NDSNEG(s(y0), cons(y1, cons(x0, x1))) → A__2NDSNEG(s(mark(y0)), cons2(y1, cons(mark(x0), x1)))

A__2NDSPOS(s(times(x0, x1)), cons2(y1, cons(y2, y3))) → A__2NDSNEG(a__times(mark(x0), mark(x1)), mark(y3))

A__2NDSPOS(s(2ndspos(x0, x1)), cons2(y1, cons(y2, y3))) → A__2NDSNEG(a__2ndspos(mark(x0), mark(x1)), mark(y3))

A__2NDSPOS(s(y0), cons(y1, from(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__from(mark(x0))))

MARK(2ndspos(s(y0), square(x0))) → A__2NDSPOS(s(mark(y0)), a__square(mark(x0)))

MARK(rcons(X1, X2)) → MARK(X2)

MARK(2ndspos(s(y0), cons2(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(x0, mark(x1)))

A__2NDSNEG(s(y0), cons(y1, plus(x0, x1))) → A__2NDSNEG(s(mark(y0)), cons2(y1, a__plus(mark(x0), mark(x1))))

MARK(2ndsneg(plus(y0, y1), cons2(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), cons2(x0, mark(x1)))

A__2NDSPOS(s(from(x0)), cons2(y1, cons(y2, y3))) → A__2NDSNEG(a__from(mark(x0)), mark(y3))

A__2NDSPOS(s(y0), cons(y1, cons(x0, x1))) → A__2NDSPOS(s(mark(y0)), cons2(y1, cons(mark(x0), x1)))

A__PLUS(s(X), Y) → MARK(X)

MARK(times(X1, X2)) → MARK(X1)

A__2NDSPOS(s(s(y0)), cons2(y1, cons(y2, rcons(x0, x1)))) → A__2NDSNEG(s(mark(y0)), rcons(mark(x0), mark(x1)))

A__TIMES(s(plus(x0, x1)), y1) → A__TIMES(a__plus(mark(x0), mark(x1)), mark(y1))

MARK(plus(times(x0, x1), y1)) → A__PLUS(a__times(mark(x0), mark(x1)), mark(y1))

A__SQUARE(X) → MARK(X)

A__2NDSPOS(s(N), cons(X, Z)) → MARK(N)

MARK(negrecip(X)) → MARK(X)

A__2NDSNEG(s(y0), cons(y1, square(x0))) → A__2NDSNEG(s(mark(y0)), cons2(y1, a__square(mark(x0))))

A__PLUS(0, Y) → MARK(Y)

MARK(posrecip(X)) → MARK(X)

MARK(2ndspos(square(y0), from(x0))) → A__2NDSPOS(a__square(mark(y0)), a__from(mark(x0)))

MARK(2ndspos(times(y0, y1), from(x0))) → A__2NDSPOS(a__times(mark(y0), mark(y1)), a__from(mark(x0)))

A__PLUS(s(times(x0, x1)), y1) → A__PLUS(a__times(mark(x0), mark(x1)), mark(y1))

A__2NDSPOS(s(y0), cons(y1, square(x0))) → A__2NDSPOS(s(mark(y0)), cons2(y1, a__square(mark(x0))))

A__PLUS(s(0), y1) → A__PLUS(0, mark(y1))

MARK(plus(0, y1)) → A__PLUS(0, mark(y1))

MARK(from(X)) → MARK(X)

MARK(2ndsneg(s(y0), times(x0, x1))) → A__2NDSNEG(s(mark(y0)), a__times(mark(x0), mark(x1)))

MARK(2ndspos(square(y0), square(x0))) → A__2NDSPOS(a__square(mark(y0)), a__square(mark(x0)))

A__SQUARE(X) → A__TIMES(mark(X), mark(X))

A__PLUS(s(plus(x0, x1)), y1) → A__PLUS(a__plus(mark(x0), mark(x1)), mark(y1))

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(Z)

A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → A__2NDSPOS(mark(N), mark(Z))

MARK(2ndsneg(plus(y0, y1), plus(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(square(X)) → A__SQUARE(mark(X))

MARK(2ndspos(plus(y0, y1), from(x0))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__from(mark(x0)))

MARK(2ndsneg(plus(y0, y1), times(x0, x1))) → A__2NDSNEG(a__plus(mark(y0), mark(y1)), a__times(mark(x0), mark(x1)))

MARK(2ndsneg(times(y0, y1), plus(x0, x1))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

MARK(2ndspos(plus(y0, y1), plus(x0, x1))) → A__2NDSPOS(a__plus(mark(y0), mark(y1)), a__plus(mark(x0), mark(x1)))

A__PLUS(s(s(x0)), y1) → A__PLUS(s(mark(x0)), mark(y1))

MARK(2ndsneg(times(y0, y1), cons(x0, x1))) → A__2NDSNEG(a__times(mark(y0), mark(y1)), cons(mark(x0), x1))

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))

a__2ndspos(0, Z) → rnil

a__2ndspos(s(N), cons(X, Z)) → a__2ndspos(s(mark(N)), cons2(X, mark(Z)))

a__2ndspos(s(N), cons2(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))

a__2ndsneg(0, Z) → rnil

a__2ndsneg(s(N), cons(X, Z)) → a__2ndsneg(s(mark(N)), cons2(X, mark(Z)))

a__2ndsneg(s(N), cons2(X, cons(Y, Z))) → rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z)))

a__pi(X) → a__2ndspos(mark(X), a__from(0))

a__plus(0, Y) → mark(Y)

a__plus(s(X), Y) → s(a__plus(mark(X), mark(Y)))

a__times(0, Y) → 0

a__times(s(X), Y) → a__plus(mark(Y), a__times(mark(X), mark(Y)))

a__square(X) → a__times(mark(X), mark(X))

mark(from(X)) → a__from(mark(X))

mark(2ndspos(X1, X2)) → a__2ndspos(mark(X1), mark(X2))

mark(2ndsneg(X1, X2)) → a__2ndsneg(mark(X1), mark(X2))

mark(pi(X)) → a__pi(mark(X))

mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))

mark(times(X1, X2)) → a__times(mark(X1), mark(X2))

mark(square(X)) → a__square(mark(X))

mark(0) → 0

mark(s(X)) → s(mark(X))

mark(posrecip(X)) → posrecip(mark(X))

mark(negrecip(X)) → negrecip(mark(X))

mark(nil) → nil

mark(cons(X1, X2)) → cons(mark(X1), X2)

mark(cons2(X1, X2)) → cons2(X1, mark(X2))

mark(rnil) → rnil

mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))

a__from(X) → from(X)

a__2ndspos(X1, X2) → 2ndspos(X1, X2)

a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)

a__pi(X) → pi(X)

a__plus(X1, X2) → plus(X1, X2)

a__times(X1, X2) → times(X1, X2)

a__square(X) → square(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 7 less nodes.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ Narrowing
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ QDP
              ↳ Narrowing
                ↳ QDP
                  ↳ DependencyGraphProof
                    ↳ QDP
                      ↳ Narrowing
                        ↳ QDP
                          ↳ DependencyGraphProof
                            ↳ QDP
                              ↳ Narrowing
                                ↳ QDP
                                  ↳ DependencyGraphProof
                                    ↳ QDP
                                      ↳ Narrowing
                                        ↳ QDP
                                          ↳ DependencyGraphProof
                                            ↳ QDP
                                              ↳ Narrowing
                                                ↳ QDP
                                                  ↳ DependencyGraphProof
                                                    ↳ QDP
                                                      ↳ Narrowing
                                                        ↳ QDP
                                                          ↳ DependencyGraphProof
                                                            ↳ QDP
                                                              ↳ Narrowing
                                                                ↳ QDP
                                                                  ↳ DependencyGraphProof
                                                                    ↳ QDP
                                                                      ↳ Narrowing
                                                                        ↳ QDP
                                                                          ↳ DependencyGraphProof
                                                                            ↳ QDP
                                                                              ↳ Narrowing
                                                                                ↳ QDP
                                                                                  ↳ DependencyGraphProof
                                                                                    ↳ QDP
                                                                                      ↳ Narrowing
                                                                                        ↳ QDP
                                                                                          ↳ DependencyGraphProof
                                                                                            ↳ QDP
                                                                                              ↳ Narrowing
                                                                                                ↳ QDP
                                                                                                  ↳ DependencyGraphProof
                                                                                                    ↳ QDP
                                                                                                      ↳ Narrowing
                                                                                                        ↳ QDP
                                                                                                          ↳ DependencyGraphProof
                                                                                                            ↳ QDP
                                                                                                              ↳ Narrowing
                                                                                                                ↳ QDP
                                                                                                                  ↳ DependencyGraphProof
                                                                                                                    ↳ QDP
                                                                                                                      ↳ Narrowing
                                                                                                                        ↳ QDP
                                                                                                                          ↳ DependencyGraphProof
                                                                                                                            ↳ QDP
                                                                                                                              ↳ Narrowing
                                                                                                                                ↳ QDP
                                                                                                                                  ↳ DependencyGraphProof
                                                                                                                                    ↳ QDP
                                                                                                                                      ↳ Narrowing
                                                                                                                                        ↳ QDP
                                                                                                                                          ↳ DependencyGraphProof
                                                                                                                                            ↳ QDP
                                                                                                                                              ↳ Narrowing
                                                                                                                                                ↳ QDP
                                                                                                                                                  ↳ DependencyGraphProof
                                                                                                                                                    ↳ QDP
                                                                                                                                                      ↳ Narrowing
                                                                                                                                                        ↳ QDP
                                                                                                                                                          ↳ DependencyGraphProof
                                                                                                                                                            ↳ QDP
                                                                                                                                                              ↳ Narrowing
                                                                                                                                                                ↳ QDP
                                                                                                                                                                  ↳ DependencyGraphProof
                                                                                                                                                                    ↳ QDP
                                                                                                                                                                      ↳ Narrowing
                                                                                                                                                                        ↳ QDP
                                                                                                                                                                          ↳ DependencyGraphProof
                                                                                                                                                                            ↳ QDP
                                                                                                                                                                              ↳ Narrowing
                                                                                                                                                                                ↳ QDP
                                                                                                                                                                                  ↳ DependencyGraphProof
                                                                                                                                                                                    ↳ QDP
                                                                                                                                                                                      ↳ Narrowing
                                                                                                                                                                                        ↳ QDP
                                                                                                                                                                                          ↳ DependencyGraphProof
                                                                                                                                                                                            ↳ QDP
                                                                                                                                                                                              ↳ QDPOrderProof
                                                                                                                                                                                                ↳ QDP
                                                                                                                                                                                                  ↳ QDPOrderProof
                                                                                                                                                                                                    ↳ QDP
                                                                                                                                                                                                      ↳ QDPOrderProof
                                                                                                                                                                                                        ↳ QDP
                                                                                                                                                                                                          ↳ QDPOrderProof
                                                                                                                                                                                                            ↳ QDP
                                                                                                                                                                                                              ↳ QDPOrderProof
                                                                                                                                                                                                                ↳ QDP
                                                                                                                                                                                                                  ↳ QDPOrderProof
                                                                                                                                                                                                                    ↳ QDP