Termination w.r.t. Q proof of /tmp/tmproQI3i/ExAppendixB_AEL03

Termination w.r.t. Q of the given QTRS could be proven:

0 QTRS

↳1 DependencyPairsProof (⇔)

↳2 QDP

↳3 QDPOrderProof (⇔)

↳4 QDP

↳5 QDPOrderProof (⇔)

↳6 QDP

↳7 QDPOrderProof (⇔)

↳8 QDP

↳9 QDPOrderProof (⇔)

↳10 QDP

↳11 DependencyGraphProof (⇔)

↳12 QDP

↳13 QDPOrderProof (⇔)

↳14 QDP

↳15 QDPOrderProof (⇔)

↳16 QDP

↳17 DependencyGraphProof (⇔)

↳18 QDP

↳19 QDPOrderProof (⇔)

↳20 QDP

↳21 QDPOrderProof (⇔)

↳22 QDP

↳23 DependencyGraphProof (⇔)

↳24 QDP

↳25 QDPOrderProof (⇔)

↳26 QDP

↳27 QDPOrderProof (⇔)

↳28 QDP

↳29 QDPOrderProof (⇔)

↳30 QDP

↳31 QDPOrderProof (⇔)

↳32 QDP

↳33 QDPOrderProof (⇔)

↳34 QDP

↳35 QDPOrderProof (⇔)

↳36 QDP

↳37 QDPOrderProof (⇔)

↳38 QDP

↳39 QDPOrderProof (⇔)

↳40 QDP

↳41 QDPOrderProof (⇔)

↳42 QDP

↳43 DependencyGraphProof (⇔)

↳44 AND

  ↳45 QDP

    ↳46 QDPOrderProof (⇔)

    ↳47 QDP

    ↳48 QDPOrderProof (⇔)

    ↳49 QDP

    ↳50 DependencyGraphProof (⇔)

    ↳51 AND

      ↳52 QDP

        ↳53 UsableRulesProof (⇔)

        ↳54 QDP

        ↳55 QDPSizeChangeProof (⇔)

        ↳56 TRUE

      ↳57 QDP

        ↳58 QDPOrderProof (⇔)

        ↳59 QDP

        ↳60 PisEmptyProof (⇔)

        ↳61 TRUE

  ↳62 QDP

    ↳63 QDPOrderProof (⇔)

    ↳64 QDP

    ↳65 DependencyGraphProof (⇔)

    ↳66 TRUE

(0) Obligation:

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.

(1) DependencyPairsProof (EQUIVALENT transformation)

Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.

(2) Obligation:

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

A__FROM(X) → MARK(X)
A__2NDSPOS(s(N), cons(X, Z)) → A__2NDSPOS(s(mark(N)), cons2(X, mark(Z)))
A__2NDSPOS(s(N), cons(X, Z)) → MARK(N)
A__2NDSPOS(s(N), cons(X, Z)) → MARK(Z)
A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(Y)
A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → A__2NDSNEG(mark(N), mark(Z))
A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(N)
A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(Z)
A__2NDSNEG(s(N), cons(X, Z)) → A__2NDSNEG(s(mark(N)), cons2(X, mark(Z)))
A__2NDSNEG(s(N), cons(X, Z)) → MARK(N)
A__2NDSNEG(s(N), cons(X, Z)) → MARK(Z)
A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(Y)
A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → A__2NDSPOS(mark(N), mark(Z))
A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(N)
A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(Z)
A__PI(X) → A__2NDSPOS(mark(X), a__from(0))
A__PI(X) → MARK(X)
A__PI(X) → A__FROM(0)
A__PLUS(0, Y) → MARK(Y)
A__PLUS(s(X), Y) → A__PLUS(mark(X), mark(Y))
A__PLUS(s(X), Y) → MARK(X)
A__PLUS(s(X), Y) → MARK(Y)
A__TIMES(s(X), Y) → A__PLUS(mark(Y), a__times(mark(X), mark(Y)))
A__TIMES(s(X), Y) → MARK(Y)
A__TIMES(s(X), Y) → A__TIMES(mark(X), mark(Y))
A__TIMES(s(X), Y) → MARK(X)
A__SQUARE(X) → A__TIMES(mark(X), mark(X))
A__SQUARE(X) → MARK(X)
MARK(from(X)) → A__FROM(mark(X))
MARK(from(X)) → MARK(X)
MARK(2ndspos(X1, X2)) → A__2NDSPOS(mark(X1), mark(X2))
MARK(2ndspos(X1, X2)) → MARK(X1)
MARK(2ndspos(X1, X2)) → MARK(X2)
MARK(2ndsneg(X1, X2)) → A__2NDSNEG(mark(X1), mark(X2))
MARK(2ndsneg(X1, X2)) → MARK(X1)
MARK(2ndsneg(X1, X2)) → MARK(X2)
MARK(pi(X)) → A__PI(mark(X))
MARK(pi(X)) → MARK(X)
MARK(plus(X1, X2)) → A__PLUS(mark(X1), mark(X2))
MARK(plus(X1, X2)) → MARK(X1)
MARK(plus(X1, X2)) → MARK(X2)
MARK(times(X1, X2)) → A__TIMES(mark(X1), mark(X2))
MARK(times(X1, X2)) → MARK(X1)
MARK(times(X1, X2)) → MARK(X2)
MARK(square(X)) → A__SQUARE(mark(X))
MARK(square(X)) → MARK(X)
MARK(s(X)) → MARK(X)
MARK(posrecip(X)) → MARK(X)
MARK(negrecip(X)) → MARK(X)
MARK(cons(X1, X2)) → MARK(X1)
MARK(cons2(X1, X2)) → MARK(X2)
MARK(rcons(X1, X2)) → MARK(X1)
MARK(rcons(X1, X2)) → MARK(X2)

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.

(3) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

A__PI(X) → A__FROM(0)

The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:

POL(A__FROM(x₁)) = -I + 0A · x₁

POL(MARK(x₁)) = -I + 0A · x₁

POL(A__2NDSPOS(x₁, x₂)) = -I + 0A · x₁ + 0A · x₂

POL(s(x₁)) = -I + 0A · x₁

POL(cons(x₁, x₂)) = -I + 0A · x₁ + 0A · x₂

POL(mark(x₁)) = -I + 0A · x₁

POL(cons2(x₁, x₂)) = -I + -I · x₁ + 0A · x₂

POL(A__2NDSNEG(x₁, x₂)) = -I + 0A · x₁ + 0A · x₂

POL(A__PI(x₁)) = 1A + 0A · x₁

POL(a__from(x₁)) = 0A + 0A · x₁

POL(0) = 0A

POL(A__PLUS(x₁, x₂)) = 0A + 0A · x₁ + 0A · x₂

POL(A__TIMES(x₁, x₂)) = 0A + 0A · x₁ + 0A · x₂

POL(a__times(x₁, x₂)) = 0A + 0A · x₁ + 0A · x₂

POL(A__SQUARE(x₁)) = 1A + 0A · x₁

POL(from(x₁)) = 0A + 0A · x₁

POL(2ndspos(x₁, x₂)) = -I + 0A · x₁ + 0A · x₂

POL(2ndsneg(x₁, x₂)) = -I + 0A · x₁ + 0A · x₂

POL(pi(x₁)) = 1A + 0A · x₁

POL(plus(x₁, x₂)) = 0A + 0A · x₁ + 0A · x₂

POL(times(x₁, x₂)) = 0A + 0A · x₁ + 0A · x₂

POL(square(x₁)) = 1A + 0A · x₁

POL(posrecip(x₁)) = -I + 0A · x₁

POL(negrecip(x₁)) = -I + 0A · x₁

POL(rcons(x₁, x₂)) = -I + 0A · x₁ + 0A · x₂

POL(nil) = 3A

POL(rnil) = 0A

POL(a__plus(x₁, x₂)) = 0A + 0A · x₁ + 0A · x₂

POL(a__pi(x₁)) = 1A + 0A · x₁

POL(a__2ndsneg(x₁, x₂)) = -I + 0A · x₁ + 0A · x₂

POL(a__2ndspos(x₁, x₂)) = -I + 0A · x₁ + 0A · x₂

POL(a__square(x₁)) = 1A + 0A · x₁

The following usable rules [FROCOS05] were oriented:

mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(nil) → nil
mark(negrecip(X)) → negrecip(mark(X))
mark(posrecip(X)) → posrecip(mark(X))
a__from(X) → from(X)
mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))
mark(rnil) → rnil
mark(cons2(X1, X2)) → cons2(X1, mark(X2))
a__plus(X1, X2) → plus(X1, X2)
a__pi(X) → pi(X)
a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)
a__2ndspos(X1, X2) → 2ndspos(X1, X2)
a__square(X) → square(X)
a__times(X1, X2) → times(X1, X2)
a__2ndspos(s(N), cons2(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))
a__2ndspos(s(N), cons(X, Z)) → a__2ndspos(s(mark(N)), cons2(X, mark(Z)))
a__2ndsneg(s(N), cons(X, Z)) → a__2ndsneg(s(mark(N)), cons2(X, mark(Z)))
a__2ndsneg(0, Z) → rnil
a__2ndspos(0, Z) → rnil
a__from(X) → cons(mark(X), from(s(X)))
a__times(0, Y) → 0
mark(from(X)) → a__from(mark(X))
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(s(X), Y) → s(a__plus(mark(X), mark(Y)))
mark(0) → 0
mark(s(X)) → s(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))
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))

(4) Obligation:

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

A__FROM(X) → MARK(X)
A__2NDSPOS(s(N), cons(X, Z)) → A__2NDSPOS(s(mark(N)), cons2(X, mark(Z)))
A__2NDSPOS(s(N), cons(X, Z)) → MARK(N)
A__2NDSPOS(s(N), cons(X, Z)) → MARK(Z)
A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(Y)
A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → A__2NDSNEG(mark(N), mark(Z))
A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(N)
A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(Z)
A__2NDSNEG(s(N), cons(X, Z)) → A__2NDSNEG(s(mark(N)), cons2(X, mark(Z)))
A__2NDSNEG(s(N), cons(X, Z)) → MARK(N)
A__2NDSNEG(s(N), cons(X, Z)) → MARK(Z)
A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(Y)
A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → A__2NDSPOS(mark(N), mark(Z))
A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(N)
A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(Z)
A__PI(X) → A__2NDSPOS(mark(X), a__from(0))
A__PI(X) → MARK(X)
A__PLUS(0, Y) → MARK(Y)
A__PLUS(s(X), Y) → A__PLUS(mark(X), mark(Y))
A__PLUS(s(X), Y) → MARK(X)
A__PLUS(s(X), Y) → MARK(Y)
A__TIMES(s(X), Y) → A__PLUS(mark(Y), a__times(mark(X), mark(Y)))
A__TIMES(s(X), Y) → MARK(Y)
A__TIMES(s(X), Y) → A__TIMES(mark(X), mark(Y))
A__TIMES(s(X), Y) → MARK(X)
A__SQUARE(X) → A__TIMES(mark(X), mark(X))
A__SQUARE(X) → MARK(X)
MARK(from(X)) → A__FROM(mark(X))
MARK(from(X)) → MARK(X)
MARK(2ndspos(X1, X2)) → A__2NDSPOS(mark(X1), mark(X2))
MARK(2ndspos(X1, X2)) → MARK(X1)
MARK(2ndspos(X1, X2)) → MARK(X2)
MARK(2ndsneg(X1, X2)) → A__2NDSNEG(mark(X1), mark(X2))
MARK(2ndsneg(X1, X2)) → MARK(X1)
MARK(2ndsneg(X1, X2)) → MARK(X2)
MARK(pi(X)) → A__PI(mark(X))
MARK(pi(X)) → MARK(X)
MARK(plus(X1, X2)) → A__PLUS(mark(X1), mark(X2))
MARK(plus(X1, X2)) → MARK(X1)
MARK(plus(X1, X2)) → MARK(X2)
MARK(times(X1, X2)) → A__TIMES(mark(X1), mark(X2))
MARK(times(X1, X2)) → MARK(X1)
MARK(times(X1, X2)) → MARK(X2)
MARK(square(X)) → A__SQUARE(mark(X))
MARK(square(X)) → MARK(X)
MARK(s(X)) → MARK(X)
MARK(posrecip(X)) → MARK(X)
MARK(negrecip(X)) → MARK(X)
MARK(cons(X1, X2)) → MARK(X1)
MARK(cons2(X1, X2)) → MARK(X2)
MARK(rcons(X1, X2)) → MARK(X1)
MARK(rcons(X1, X2)) → MARK(X2)

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.

(5) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

MARK(pi(X)) → MARK(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:

POL(A__FROM(x₁)) = 0A + 0A · x₁

POL(MARK(x₁)) = 0A + 0A · x₁

POL(A__2NDSPOS(x₁, x₂)) = -I + 0A · x₁ + 0A · x₂

POL(s(x₁)) = 0A + 0A · x₁

POL(cons(x₁, x₂)) = -I + 0A · x₁ + 0A · x₂

POL(mark(x₁)) = 0A + 0A · x₁

POL(cons2(x₁, x₂)) = -I + 0A · x₁ + 0A · x₂

POL(A__2NDSNEG(x₁, x₂)) = -I + 0A · x₁ + 0A · x₂

POL(A__PI(x₁)) = 0A + 1A · x₁

POL(a__from(x₁)) = 0A + 0A · x₁

POL(0) = 0A

POL(A__PLUS(x₁, x₂)) = -I + 0A · x₁ + 0A · x₂

POL(A__TIMES(x₁, x₂)) = -I + 0A · x₁ + 0A · x₂

POL(a__times(x₁, x₂)) = 0A + 0A · x₁ + 0A · x₂

POL(A__SQUARE(x₁)) = 0A + 0A · x₁

POL(from(x₁)) = 0A + 0A · x₁

POL(2ndspos(x₁, x₂)) = -I + 0A · x₁ + 0A · x₂

POL(2ndsneg(x₁, x₂)) = 0A + 0A · x₁ + 0A · x₂

POL(pi(x₁)) = 1A + 1A · x₁

POL(plus(x₁, x₂)) = -I + 0A · x₁ + 0A · x₂

POL(times(x₁, x₂)) = 0A + 0A · x₁ + 0A · x₂

POL(square(x₁)) = -I + 0A · x₁

POL(posrecip(x₁)) = -I + 0A · x₁

POL(negrecip(x₁)) = -I + 0A · x₁

POL(rcons(x₁, x₂)) = -I + 0A · x₁ + 0A · x₂

POL(nil) = 2A

POL(rnil) = 0A

POL(a__plus(x₁, x₂)) = 0A + 0A · x₁ + 0A · x₂

POL(a__pi(x₁)) = 1A + 1A · x₁

POL(a__2ndsneg(x₁, x₂)) = 0A + 0A · x₁ + 0A · x₂

POL(a__2ndspos(x₁, x₂)) = -I + 0A · x₁ + 0A · x₂

POL(a__square(x₁)) = 0A + 0A · x₁

The following usable rules [FROCOS05] were oriented:

mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(nil) → nil
mark(negrecip(X)) → negrecip(mark(X))
mark(posrecip(X)) → posrecip(mark(X))
a__from(X) → from(X)
mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))
mark(rnil) → rnil
mark(cons2(X1, X2)) → cons2(X1, mark(X2))
a__plus(X1, X2) → plus(X1, X2)
a__pi(X) → pi(X)
a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)
a__2ndspos(X1, X2) → 2ndspos(X1, X2)
a__square(X) → square(X)
a__times(X1, X2) → times(X1, X2)
a__2ndspos(s(N), cons2(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))
a__2ndspos(s(N), cons(X, Z)) → a__2ndspos(s(mark(N)), cons2(X, mark(Z)))
a__2ndsneg(s(N), cons(X, Z)) → a__2ndsneg(s(mark(N)), cons2(X, mark(Z)))
a__2ndsneg(0, Z) → rnil
a__2ndspos(0, Z) → rnil
a__from(X) → cons(mark(X), from(s(X)))
a__times(0, Y) → 0
mark(from(X)) → a__from(mark(X))
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(s(X), Y) → s(a__plus(mark(X), mark(Y)))
mark(0) → 0
mark(s(X)) → s(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))
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))

(6) Obligation:

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

A__FROM(X) → MARK(X)
A__2NDSPOS(s(N), cons(X, Z)) → A__2NDSPOS(s(mark(N)), cons2(X, mark(Z)))
A__2NDSPOS(s(N), cons(X, Z)) → MARK(N)
A__2NDSPOS(s(N), cons(X, Z)) → MARK(Z)
A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(Y)
A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → A__2NDSNEG(mark(N), mark(Z))
A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(N)
A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(Z)
A__2NDSNEG(s(N), cons(X, Z)) → A__2NDSNEG(s(mark(N)), cons2(X, mark(Z)))
A__2NDSNEG(s(N), cons(X, Z)) → MARK(N)
A__2NDSNEG(s(N), cons(X, Z)) → MARK(Z)
A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(Y)
A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → A__2NDSPOS(mark(N), mark(Z))
A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(N)
A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(Z)
A__PI(X) → A__2NDSPOS(mark(X), a__from(0))
A__PI(X) → MARK(X)
A__PLUS(0, Y) → MARK(Y)
A__PLUS(s(X), Y) → A__PLUS(mark(X), mark(Y))
A__PLUS(s(X), Y) → MARK(X)
A__PLUS(s(X), Y) → MARK(Y)
A__TIMES(s(X), Y) → A__PLUS(mark(Y), a__times(mark(X), mark(Y)))
A__TIMES(s(X), Y) → MARK(Y)
A__TIMES(s(X), Y) → A__TIMES(mark(X), mark(Y))
A__TIMES(s(X), Y) → MARK(X)
A__SQUARE(X) → A__TIMES(mark(X), mark(X))
A__SQUARE(X) → MARK(X)
MARK(from(X)) → A__FROM(mark(X))
MARK(from(X)) → MARK(X)
MARK(2ndspos(X1, X2)) → A__2NDSPOS(mark(X1), mark(X2))
MARK(2ndspos(X1, X2)) → MARK(X1)
MARK(2ndspos(X1, X2)) → MARK(X2)
MARK(2ndsneg(X1, X2)) → A__2NDSNEG(mark(X1), mark(X2))
MARK(2ndsneg(X1, X2)) → MARK(X1)
MARK(2ndsneg(X1, X2)) → MARK(X2)
MARK(pi(X)) → A__PI(mark(X))
MARK(plus(X1, X2)) → A__PLUS(mark(X1), mark(X2))
MARK(plus(X1, X2)) → MARK(X1)
MARK(plus(X1, X2)) → MARK(X2)
MARK(times(X1, X2)) → A__TIMES(mark(X1), mark(X2))
MARK(times(X1, X2)) → MARK(X1)
MARK(times(X1, X2)) → MARK(X2)
MARK(square(X)) → A__SQUARE(mark(X))
MARK(square(X)) → MARK(X)
MARK(s(X)) → MARK(X)
MARK(posrecip(X)) → MARK(X)
MARK(negrecip(X)) → MARK(X)
MARK(cons(X1, X2)) → MARK(X1)
MARK(cons2(X1, X2)) → MARK(X2)
MARK(rcons(X1, X2)) → MARK(X1)
MARK(rcons(X1, X2)) → MARK(X2)

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.

(7) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

A__TIMES(s(X), Y) → MARK(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:

POL(A__FROM(x₁)) = 0A + 0A · x₁

POL(MARK(x₁)) = 0A + 0A · x₁

POL(A__2NDSPOS(x₁, x₂)) = -I + 0A · x₁ + 0A · x₂

POL(s(x₁)) = 0A + 0A · x₁

POL(cons(x₁, x₂)) = -I + 0A · x₁ + 0A · x₂

POL(mark(x₁)) = -I + 0A · x₁

POL(cons2(x₁, x₂)) = -I + 0A · x₁ + 0A · x₂

POL(A__2NDSNEG(x₁, x₂)) = 0A + 0A · x₁ + 0A · x₂

POL(A__PI(x₁)) = 0A + 0A · x₁

POL(a__from(x₁)) = 0A + 0A · x₁

POL(0) = 0A

POL(A__PLUS(x₁, x₂)) = 0A + 0A · x₁ + 0A · x₂

POL(A__TIMES(x₁, x₂)) = 0A + 1A · x₁ + 0A · x₂

POL(a__times(x₁, x₂)) = -I + 1A · x₁ + 0A · x₂

POL(A__SQUARE(x₁)) = 0A + 1A · x₁

POL(from(x₁)) = 0A + 0A · x₁

POL(2ndspos(x₁, x₂)) = -I + 0A · x₁ + 0A · x₂

POL(2ndsneg(x₁, x₂)) = 0A + 0A · x₁ + 0A · x₂

POL(pi(x₁)) = 0A + 0A · x₁

POL(plus(x₁, x₂)) = -I + 0A · x₁ + 0A · x₂

POL(times(x₁, x₂)) = -I + 1A · x₁ + 0A · x₂

POL(square(x₁)) = -I + 1A · x₁

POL(posrecip(x₁)) = 0A + 0A · x₁

POL(negrecip(x₁)) = 0A + 0A · x₁

POL(rcons(x₁, x₂)) = -I + 0A · x₁ + 0A · x₂

POL(nil) = 0A

POL(rnil) = 0A

POL(a__plus(x₁, x₂)) = -I + 0A · x₁ + 0A · x₂

POL(a__pi(x₁)) = 0A + 0A · x₁

POL(a__2ndsneg(x₁, x₂)) = 0A + 0A · x₁ + 0A · x₂

POL(a__2ndspos(x₁, x₂)) = -I + 0A · x₁ + 0A · x₂

POL(a__square(x₁)) = -I + 1A · x₁

The following usable rules [FROCOS05] were oriented:

mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(nil) → nil
mark(negrecip(X)) → negrecip(mark(X))
mark(posrecip(X)) → posrecip(mark(X))
a__from(X) → from(X)
mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))
mark(rnil) → rnil
mark(cons2(X1, X2)) → cons2(X1, mark(X2))
a__plus(X1, X2) → plus(X1, X2)
a__pi(X) → pi(X)
a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)
a__2ndspos(X1, X2) → 2ndspos(X1, X2)
a__square(X) → square(X)
a__times(X1, X2) → times(X1, X2)
a__2ndspos(s(N), cons2(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))
a__2ndspos(s(N), cons(X, Z)) → a__2ndspos(s(mark(N)), cons2(X, mark(Z)))
a__2ndsneg(s(N), cons(X, Z)) → a__2ndsneg(s(mark(N)), cons2(X, mark(Z)))
a__2ndsneg(0, Z) → rnil
a__2ndspos(0, Z) → rnil
a__from(X) → cons(mark(X), from(s(X)))
a__times(0, Y) → 0
mark(from(X)) → a__from(mark(X))
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(s(X), Y) → s(a__plus(mark(X), mark(Y)))
mark(0) → 0
mark(s(X)) → s(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))
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))

(8) Obligation:

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

A__FROM(X) → MARK(X)
A__2NDSPOS(s(N), cons(X, Z)) → A__2NDSPOS(s(mark(N)), cons2(X, mark(Z)))
A__2NDSPOS(s(N), cons(X, Z)) → MARK(N)
A__2NDSPOS(s(N), cons(X, Z)) → MARK(Z)
A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(Y)
A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → A__2NDSNEG(mark(N), mark(Z))
A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(N)
A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(Z)
A__2NDSNEG(s(N), cons(X, Z)) → A__2NDSNEG(s(mark(N)), cons2(X, mark(Z)))
A__2NDSNEG(s(N), cons(X, Z)) → MARK(N)
A__2NDSNEG(s(N), cons(X, Z)) → MARK(Z)
A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(Y)
A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → A__2NDSPOS(mark(N), mark(Z))
A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(N)
A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(Z)
A__PI(X) → A__2NDSPOS(mark(X), a__from(0))
A__PI(X) → MARK(X)
A__PLUS(0, Y) → MARK(Y)
A__PLUS(s(X), Y) → A__PLUS(mark(X), mark(Y))
A__PLUS(s(X), Y) → MARK(X)
A__PLUS(s(X), Y) → MARK(Y)
A__TIMES(s(X), Y) → A__PLUS(mark(Y), a__times(mark(X), mark(Y)))
A__TIMES(s(X), Y) → MARK(Y)
A__TIMES(s(X), Y) → A__TIMES(mark(X), mark(Y))
A__SQUARE(X) → A__TIMES(mark(X), mark(X))
A__SQUARE(X) → MARK(X)
MARK(from(X)) → A__FROM(mark(X))
MARK(from(X)) → MARK(X)
MARK(2ndspos(X1, X2)) → A__2NDSPOS(mark(X1), mark(X2))
MARK(2ndspos(X1, X2)) → MARK(X1)
MARK(2ndspos(X1, X2)) → MARK(X2)
MARK(2ndsneg(X1, X2)) → A__2NDSNEG(mark(X1), mark(X2))
MARK(2ndsneg(X1, X2)) → MARK(X1)
MARK(2ndsneg(X1, X2)) → MARK(X2)
MARK(pi(X)) → A__PI(mark(X))
MARK(plus(X1, X2)) → A__PLUS(mark(X1), mark(X2))
MARK(plus(X1, X2)) → MARK(X1)
MARK(plus(X1, X2)) → MARK(X2)
MARK(times(X1, X2)) → A__TIMES(mark(X1), mark(X2))
MARK(times(X1, X2)) → MARK(X1)
MARK(times(X1, X2)) → MARK(X2)
MARK(square(X)) → A__SQUARE(mark(X))
MARK(square(X)) → MARK(X)
MARK(s(X)) → MARK(X)
MARK(posrecip(X)) → MARK(X)
MARK(negrecip(X)) → MARK(X)
MARK(cons(X1, X2)) → MARK(X1)
MARK(cons2(X1, X2)) → MARK(X2)
MARK(rcons(X1, X2)) → MARK(X1)
MARK(rcons(X1, X2)) → MARK(X2)

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.

(9) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

MARK(pi(X)) → A__PI(mark(X))

The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:

POL(A__FROM(x₁)) = 0A + 0A · x₁

POL(MARK(x₁)) = 0A + 0A · x₁

POL(A__2NDSPOS(x₁, x₂)) = 0A + 0A · x₁ + 0A · x₂

POL(s(x₁)) = -I + 0A · x₁

POL(cons(x₁, x₂)) = -I + 0A · x₁ + 0A · x₂

POL(mark(x₁)) = 0A + 0A · x₁

POL(cons2(x₁, x₂)) = 0A + 0A · x₁ + 0A · x₂

POL(A__2NDSNEG(x₁, x₂)) = 0A + 0A · x₁ + 0A · x₂

POL(A__PI(x₁)) = 0A + 0A · x₁

POL(a__from(x₁)) = 0A + 0A · x₁

POL(0) = 0A

POL(A__PLUS(x₁, x₂)) = 0A + 0A · x₁ + 0A · x₂

POL(A__TIMES(x₁, x₂)) = 0A + 0A · x₁ + 0A · x₂

POL(a__times(x₁, x₂)) = 0A + 0A · x₁ + 0A · x₂

POL(A__SQUARE(x₁)) = 5A + 0A · x₁

POL(from(x₁)) = -I + 0A · x₁

POL(2ndspos(x₁, x₂)) = -I + 0A · x₁ + 0A · x₂

POL(2ndsneg(x₁, x₂)) = -I + 0A · x₁ + 0A · x₂

POL(pi(x₁)) = 1A + 1A · x₁

POL(plus(x₁, x₂)) = -I + 0A · x₁ + 0A · x₂

POL(times(x₁, x₂)) = -I + 0A · x₁ + 0A · x₂

POL(square(x₁)) = 5A + 0A · x₁

POL(posrecip(x₁)) = 0A + 0A · x₁

POL(negrecip(x₁)) = 0A + 0A · x₁

POL(rcons(x₁, x₂)) = -I + 0A · x₁ + 0A · x₂

POL(nil) = 0A

POL(rnil) = 0A

POL(a__plus(x₁, x₂)) = 0A + 0A · x₁ + 0A · x₂

POL(a__pi(x₁)) = 1A + 1A · x₁

POL(a__2ndsneg(x₁, x₂)) = 0A + 0A · x₁ + 0A · x₂

POL(a__2ndspos(x₁, x₂)) = 0A + 0A · x₁ + 0A · x₂

POL(a__square(x₁)) = 5A + 0A · x₁

The following usable rules [FROCOS05] were oriented:

mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(nil) → nil
mark(negrecip(X)) → negrecip(mark(X))
mark(posrecip(X)) → posrecip(mark(X))
a__from(X) → from(X)
mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))
mark(rnil) → rnil
mark(cons2(X1, X2)) → cons2(X1, mark(X2))
a__plus(X1, X2) → plus(X1, X2)
a__pi(X) → pi(X)
a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)
a__2ndspos(X1, X2) → 2ndspos(X1, X2)
a__square(X) → square(X)
a__times(X1, X2) → times(X1, X2)
a__2ndspos(s(N), cons2(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))
a__2ndspos(s(N), cons(X, Z)) → a__2ndspos(s(mark(N)), cons2(X, mark(Z)))
a__2ndsneg(s(N), cons(X, Z)) → a__2ndsneg(s(mark(N)), cons2(X, mark(Z)))
a__2ndsneg(0, Z) → rnil
a__2ndspos(0, Z) → rnil
a__from(X) → cons(mark(X), from(s(X)))
a__times(0, Y) → 0
mark(from(X)) → a__from(mark(X))
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(s(X), Y) → s(a__plus(mark(X), mark(Y)))
mark(0) → 0
mark(s(X)) → s(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))
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))

(10) Obligation:

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

A__FROM(X) → MARK(X)
A__2NDSPOS(s(N), cons(X, Z)) → A__2NDSPOS(s(mark(N)), cons2(X, mark(Z)))
A__2NDSPOS(s(N), cons(X, Z)) → MARK(N)
A__2NDSPOS(s(N), cons(X, Z)) → MARK(Z)
A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(Y)
A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → A__2NDSNEG(mark(N), mark(Z))
A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(N)
A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(Z)
A__2NDSNEG(s(N), cons(X, Z)) → A__2NDSNEG(s(mark(N)), cons2(X, mark(Z)))
A__2NDSNEG(s(N), cons(X, Z)) → MARK(N)
A__2NDSNEG(s(N), cons(X, Z)) → MARK(Z)
A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(Y)
A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → A__2NDSPOS(mark(N), mark(Z))
A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(N)
A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(Z)
A__PI(X) → A__2NDSPOS(mark(X), a__from(0))
A__PI(X) → MARK(X)
A__PLUS(0, Y) → MARK(Y)
A__PLUS(s(X), Y) → A__PLUS(mark(X), mark(Y))
A__PLUS(s(X), Y) → MARK(X)
A__PLUS(s(X), Y) → MARK(Y)
A__TIMES(s(X), Y) → A__PLUS(mark(Y), a__times(mark(X), mark(Y)))
A__TIMES(s(X), Y) → MARK(Y)
A__TIMES(s(X), Y) → A__TIMES(mark(X), mark(Y))
A__SQUARE(X) → A__TIMES(mark(X), mark(X))
A__SQUARE(X) → MARK(X)
MARK(from(X)) → A__FROM(mark(X))
MARK(from(X)) → MARK(X)
MARK(2ndspos(X1, X2)) → A__2NDSPOS(mark(X1), mark(X2))
MARK(2ndspos(X1, X2)) → MARK(X1)
MARK(2ndspos(X1, X2)) → MARK(X2)
MARK(2ndsneg(X1, X2)) → A__2NDSNEG(mark(X1), mark(X2))
MARK(2ndsneg(X1, X2)) → MARK(X1)
MARK(2ndsneg(X1, X2)) → MARK(X2)
MARK(plus(X1, X2)) → A__PLUS(mark(X1), mark(X2))
MARK(plus(X1, X2)) → MARK(X1)
MARK(plus(X1, X2)) → MARK(X2)
MARK(times(X1, X2)) → A__TIMES(mark(X1), mark(X2))
MARK(times(X1, X2)) → MARK(X1)
MARK(times(X1, X2)) → MARK(X2)
MARK(square(X)) → A__SQUARE(mark(X))
MARK(square(X)) → MARK(X)
MARK(s(X)) → MARK(X)
MARK(posrecip(X)) → MARK(X)
MARK(negrecip(X)) → MARK(X)
MARK(cons(X1, X2)) → MARK(X1)
MARK(cons2(X1, X2)) → MARK(X2)
MARK(rcons(X1, X2)) → MARK(X1)
MARK(rcons(X1, X2)) → MARK(X2)

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.

(11) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes.

(12) Obligation:

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

MARK(from(X)) → A__FROM(mark(X))
A__FROM(X) → MARK(X)
MARK(from(X)) → MARK(X)
MARK(2ndspos(X1, X2)) → A__2NDSPOS(mark(X1), mark(X2))
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))) → MARK(Y)
MARK(2ndspos(X1, X2)) → MARK(X1)
MARK(2ndspos(X1, X2)) → MARK(X2)
MARK(2ndsneg(X1, X2)) → A__2NDSNEG(mark(X1), mark(X2))
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(Y)
MARK(2ndsneg(X1, X2)) → MARK(X1)
MARK(2ndsneg(X1, X2)) → MARK(X2)
MARK(plus(X1, X2)) → A__PLUS(mark(X1), mark(X2))
A__PLUS(0, Y) → MARK(Y)
MARK(plus(X1, X2)) → MARK(X1)
MARK(plus(X1, X2)) → MARK(X2)
MARK(times(X1, X2)) → A__TIMES(mark(X1), mark(X2))
A__TIMES(s(X), Y) → A__PLUS(mark(Y), a__times(mark(X), mark(Y)))
A__PLUS(s(X), Y) → A__PLUS(mark(X), mark(Y))
A__PLUS(s(X), Y) → MARK(X)
MARK(times(X1, X2)) → MARK(X1)
MARK(times(X1, X2)) → MARK(X2)
MARK(square(X)) → A__SQUARE(mark(X))
A__SQUARE(X) → A__TIMES(mark(X), mark(X))
A__TIMES(s(X), Y) → MARK(Y)
MARK(square(X)) → MARK(X)
MARK(s(X)) → MARK(X)
MARK(posrecip(X)) → MARK(X)
MARK(negrecip(X)) → MARK(X)
MARK(cons(X1, X2)) → MARK(X1)
MARK(cons2(X1, X2)) → MARK(X2)
MARK(rcons(X1, X2)) → MARK(X1)
MARK(rcons(X1, X2)) → MARK(X2)
A__TIMES(s(X), Y) → A__TIMES(mark(X), mark(Y))
A__SQUARE(X) → MARK(X)
A__PLUS(s(X), Y) → MARK(Y)
A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → A__2NDSPOS(mark(N), mark(Z))
A__2NDSPOS(s(N), cons(X, Z)) → MARK(N)
A__2NDSPOS(s(N), cons(X, 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)
A__2NDSNEG(s(N), cons(X, Z)) → MARK(Z)
A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(N)
A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(Z)
A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(N)
A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(Z)

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.

(13) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

MARK(posrecip(X)) → MARK(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:

POL(MARK(x₁)) = 0A + 0A · x₁

POL(from(x₁)) = 1A + 0A · x₁

POL(A__FROM(x₁)) = 1A + 0A · x₁

POL(mark(x₁)) = -I + 0A · x₁

POL(2ndspos(x₁, x₂)) = 0A + 0A · x₁ + 4A · x₂

POL(A__2NDSPOS(x₁, x₂)) = 0A + 0A · x₁ + 0A · x₂

POL(s(x₁)) = -I + 0A · x₁

POL(cons(x₁, x₂)) = 0A + 0A · x₁ + 0A · x₂

POL(cons2(x₁, x₂)) = -I + 0A · x₁ + 0A · x₂

POL(2ndsneg(x₁, x₂)) = -I + 0A · x₁ + 4A · x₂

POL(A__2NDSNEG(x₁, x₂)) = -I + 0A · x₁ + 0A · x₂

POL(plus(x₁, x₂)) = -I + 0A · x₁ + 0A · x₂

POL(A__PLUS(x₁, x₂)) = 0A + 0A · x₁ + 0A · x₂

POL(0) = 0A

POL(times(x₁, x₂)) = 0A + 0A · x₁ + 0A · x₂

POL(A__TIMES(x₁, x₂)) = 0A + 0A · x₁ + 0A · x₂

POL(a__times(x₁, x₂)) = 0A + 0A · x₁ + 0A · x₂

POL(square(x₁)) = 1A + 0A · x₁

POL(A__SQUARE(x₁)) = 1A + 0A · x₁

POL(posrecip(x₁)) = 1A + 3A · x₁

POL(negrecip(x₁)) = -I + 0A · x₁

POL(rcons(x₁, x₂)) = -I + 1A · x₁ + 0A · x₂

POL(nil) = 0A

POL(a__from(x₁)) = 1A + 0A · x₁

POL(rnil) = 0A

POL(a__plus(x₁, x₂)) = -I + 0A · x₁ + 0A · x₂

POL(a__pi(x₁)) = 5A + 0A · x₁

POL(pi(x₁)) = 5A + 0A · x₁

POL(a__2ndsneg(x₁, x₂)) = -I + 0A · x₁ + 4A · x₂

POL(a__2ndspos(x₁, x₂)) = 0A + 0A · x₁ + 4A · x₂

POL(a__square(x₁)) = 1A + 0A · x₁

The following usable rules [FROCOS05] were oriented:

mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(nil) → nil
mark(negrecip(X)) → negrecip(mark(X))
mark(posrecip(X)) → posrecip(mark(X))
a__from(X) → from(X)
mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))
mark(rnil) → rnil
mark(cons2(X1, X2)) → cons2(X1, mark(X2))
a__plus(X1, X2) → plus(X1, X2)
a__pi(X) → pi(X)
a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)
a__2ndspos(X1, X2) → 2ndspos(X1, X2)
a__square(X) → square(X)
a__times(X1, X2) → times(X1, X2)
a__2ndspos(s(N), cons2(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))
a__2ndspos(s(N), cons(X, Z)) → a__2ndspos(s(mark(N)), cons2(X, mark(Z)))
a__2ndsneg(s(N), cons(X, Z)) → a__2ndsneg(s(mark(N)), cons2(X, mark(Z)))
a__2ndsneg(0, Z) → rnil
a__2ndspos(0, Z) → rnil
a__from(X) → cons(mark(X), from(s(X)))
a__times(0, Y) → 0
mark(from(X)) → a__from(mark(X))
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(s(X), Y) → s(a__plus(mark(X), mark(Y)))
mark(0) → 0
mark(s(X)) → s(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))
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))

(14) Obligation:

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

MARK(from(X)) → A__FROM(mark(X))
A__FROM(X) → MARK(X)
MARK(from(X)) → MARK(X)
MARK(2ndspos(X1, X2)) → A__2NDSPOS(mark(X1), mark(X2))
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))) → MARK(Y)
MARK(2ndspos(X1, X2)) → MARK(X1)
MARK(2ndspos(X1, X2)) → MARK(X2)
MARK(2ndsneg(X1, X2)) → A__2NDSNEG(mark(X1), mark(X2))
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(Y)
MARK(2ndsneg(X1, X2)) → MARK(X1)
MARK(2ndsneg(X1, X2)) → MARK(X2)
MARK(plus(X1, X2)) → A__PLUS(mark(X1), mark(X2))
A__PLUS(0, Y) → MARK(Y)
MARK(plus(X1, X2)) → MARK(X1)
MARK(plus(X1, X2)) → MARK(X2)
MARK(times(X1, X2)) → A__TIMES(mark(X1), mark(X2))
A__TIMES(s(X), Y) → A__PLUS(mark(Y), a__times(mark(X), mark(Y)))
A__PLUS(s(X), Y) → A__PLUS(mark(X), mark(Y))
A__PLUS(s(X), Y) → MARK(X)
MARK(times(X1, X2)) → MARK(X1)
MARK(times(X1, X2)) → MARK(X2)
MARK(square(X)) → A__SQUARE(mark(X))
A__SQUARE(X) → A__TIMES(mark(X), mark(X))
A__TIMES(s(X), Y) → MARK(Y)
MARK(square(X)) → MARK(X)
MARK(s(X)) → MARK(X)
MARK(negrecip(X)) → MARK(X)
MARK(cons(X1, X2)) → MARK(X1)
MARK(cons2(X1, X2)) → MARK(X2)
MARK(rcons(X1, X2)) → MARK(X1)
MARK(rcons(X1, X2)) → MARK(X2)
A__TIMES(s(X), Y) → A__TIMES(mark(X), mark(Y))
A__SQUARE(X) → MARK(X)
A__PLUS(s(X), Y) → MARK(Y)
A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → A__2NDSPOS(mark(N), mark(Z))
A__2NDSPOS(s(N), cons(X, Z)) → MARK(N)
A__2NDSPOS(s(N), cons(X, 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)
A__2NDSNEG(s(N), cons(X, Z)) → MARK(Z)
A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(N)
A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(Z)
A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(N)
A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(Z)

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.

(15) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

MARK(from(X)) → A__FROM(mark(X))
MARK(from(X)) → MARK(X)
A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(Y)
A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(Y)
MARK(cons(X1, X2)) → MARK(X1)

The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:

POL(MARK(x₁)) = 3A + 3A · x₁

POL(from(x₁)) = 5A + 5A · x₁

POL(A__FROM(x₁)) = 3A + 3A · x₁

POL(mark(x₁)) = 0A + 0A · x₁

POL(2ndspos(x₁, x₂)) = -I + 0A · x₁ + 0A · x₂

POL(A__2NDSPOS(x₁, x₂)) = -I + 3A · x₁ + 3A · x₂

POL(s(x₁)) = -I + 0A · x₁

POL(cons(x₁, x₂)) = 5A + 5A · x₁ + 0A · x₂

POL(cons2(x₁, x₂)) = 5A + -I · x₁ + 0A · x₂

POL(2ndsneg(x₁, x₂)) = -I + 0A · x₁ + 0A · x₂

POL(A__2NDSNEG(x₁, x₂)) = 0A + 3A · x₁ + 3A · x₂

POL(plus(x₁, x₂)) = -I + 0A · x₁ + 0A · x₂

POL(A__PLUS(x₁, x₂)) = 3A + 3A · x₁ + 3A · x₂

POL(0) = 0A

POL(times(x₁, x₂)) = 0A + 0A · x₁ + 0A · x₂

POL(A__TIMES(x₁, x₂)) = 3A + 3A · x₁ + 3A · x₂

POL(a__times(x₁, x₂)) = 0A + 0A · x₁ + 0A · x₂

POL(square(x₁)) = -I + 0A · x₁

POL(A__SQUARE(x₁)) = 3A + 3A · x₁

POL(negrecip(x₁)) = -I + 0A · x₁

POL(rcons(x₁, x₂)) = 5A + 0A · x₁ + 0A · x₂

POL(nil) = 0A

POL(posrecip(x₁)) = 0A + -I · x₁

POL(a__from(x₁)) = 5A + 5A · x₁

POL(rnil) = 0A

POL(a__plus(x₁, x₂)) = 0A + 0A · x₁ + 0A · x₂

POL(a__pi(x₁)) = 5A + 0A · x₁

POL(pi(x₁)) = 5A + 0A · x₁

POL(a__2ndsneg(x₁, x₂)) = -I + 0A · x₁ + 0A · x₂

POL(a__2ndspos(x₁, x₂)) = 0A + 0A · x₁ + 0A · x₂

POL(a__square(x₁)) = 0A + 0A · x₁

The following usable rules [FROCOS05] were oriented:

mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(nil) → nil
mark(negrecip(X)) → negrecip(mark(X))
mark(posrecip(X)) → posrecip(mark(X))
a__from(X) → from(X)
mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))
mark(rnil) → rnil
mark(cons2(X1, X2)) → cons2(X1, mark(X2))
a__plus(X1, X2) → plus(X1, X2)
a__pi(X) → pi(X)
a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)
a__2ndspos(X1, X2) → 2ndspos(X1, X2)
a__square(X) → square(X)
a__times(X1, X2) → times(X1, X2)
a__2ndspos(s(N), cons2(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))
a__2ndspos(s(N), cons(X, Z)) → a__2ndspos(s(mark(N)), cons2(X, mark(Z)))
a__2ndsneg(s(N), cons(X, Z)) → a__2ndsneg(s(mark(N)), cons2(X, mark(Z)))
a__2ndsneg(0, Z) → rnil
a__2ndspos(0, Z) → rnil
a__from(X) → cons(mark(X), from(s(X)))
a__times(0, Y) → 0
mark(from(X)) → a__from(mark(X))
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(s(X), Y) → s(a__plus(mark(X), mark(Y)))
mark(0) → 0
mark(s(X)) → s(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))
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))

(16) Obligation:

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

A__FROM(X) → MARK(X)
MARK(2ndspos(X1, X2)) → A__2NDSPOS(mark(X1), mark(X2))
A__2NDSPOS(s(N), cons(X, Z)) → A__2NDSPOS(s(mark(N)), cons2(X, mark(Z)))
MARK(2ndspos(X1, X2)) → MARK(X1)
MARK(2ndspos(X1, X2)) → MARK(X2)
MARK(2ndsneg(X1, X2)) → A__2NDSNEG(mark(X1), mark(X2))
A__2NDSNEG(s(N), cons(X, Z)) → A__2NDSNEG(s(mark(N)), cons2(X, mark(Z)))
MARK(2ndsneg(X1, X2)) → MARK(X1)
MARK(2ndsneg(X1, X2)) → MARK(X2)
MARK(plus(X1, X2)) → A__PLUS(mark(X1), mark(X2))
A__PLUS(0, Y) → MARK(Y)
MARK(plus(X1, X2)) → MARK(X1)
MARK(plus(X1, X2)) → MARK(X2)
MARK(times(X1, X2)) → A__TIMES(mark(X1), mark(X2))
A__TIMES(s(X), Y) → A__PLUS(mark(Y), a__times(mark(X), mark(Y)))
A__PLUS(s(X), Y) → A__PLUS(mark(X), mark(Y))
A__PLUS(s(X), Y) → MARK(X)
MARK(times(X1, X2)) → MARK(X1)
MARK(times(X1, X2)) → MARK(X2)
MARK(square(X)) → A__SQUARE(mark(X))
A__SQUARE(X) → A__TIMES(mark(X), mark(X))
A__TIMES(s(X), Y) → MARK(Y)
MARK(square(X)) → MARK(X)
MARK(s(X)) → MARK(X)
MARK(negrecip(X)) → MARK(X)
MARK(cons2(X1, X2)) → MARK(X2)
MARK(rcons(X1, X2)) → MARK(X1)
MARK(rcons(X1, X2)) → MARK(X2)
A__TIMES(s(X), Y) → A__TIMES(mark(X), mark(Y))
A__SQUARE(X) → MARK(X)
A__PLUS(s(X), Y) → MARK(Y)
A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → A__2NDSPOS(mark(N), mark(Z))
A__2NDSPOS(s(N), cons(X, Z)) → MARK(N)
A__2NDSPOS(s(N), cons(X, 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)
A__2NDSNEG(s(N), cons(X, Z)) → MARK(Z)
A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(N)
A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(Z)
A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(N)
A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(Z)

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.

(17) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.

(18) Obligation:

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

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))) → A__2NDSNEG(mark(N), mark(Z))
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))) → A__2NDSPOS(mark(N), mark(Z))
A__2NDSPOS(s(N), cons(X, Z)) → MARK(N)
MARK(2ndspos(X1, X2)) → A__2NDSPOS(mark(X1), mark(X2))
A__2NDSPOS(s(N), cons(X, Z)) → MARK(Z)
MARK(2ndspos(X1, X2)) → MARK(X1)
MARK(2ndspos(X1, X2)) → MARK(X2)
MARK(2ndsneg(X1, X2)) → A__2NDSNEG(mark(X1), mark(X2))
A__2NDSNEG(s(N), cons(X, Z)) → MARK(N)
MARK(2ndsneg(X1, X2)) → MARK(X1)
MARK(2ndsneg(X1, X2)) → MARK(X2)
MARK(plus(X1, X2)) → A__PLUS(mark(X1), mark(X2))
A__PLUS(0, Y) → MARK(Y)
MARK(plus(X1, X2)) → MARK(X1)
MARK(plus(X1, X2)) → MARK(X2)
MARK(times(X1, X2)) → A__TIMES(mark(X1), mark(X2))
A__TIMES(s(X), Y) → A__PLUS(mark(Y), a__times(mark(X), mark(Y)))
A__PLUS(s(X), Y) → A__PLUS(mark(X), mark(Y))
A__PLUS(s(X), Y) → MARK(X)
MARK(times(X1, X2)) → MARK(X1)
MARK(times(X1, X2)) → MARK(X2)
MARK(square(X)) → A__SQUARE(mark(X))
A__SQUARE(X) → A__TIMES(mark(X), mark(X))
A__TIMES(s(X), Y) → MARK(Y)
MARK(square(X)) → MARK(X)
MARK(s(X)) → MARK(X)
MARK(negrecip(X)) → MARK(X)
MARK(cons2(X1, X2)) → MARK(X2)
MARK(rcons(X1, X2)) → MARK(X1)
MARK(rcons(X1, X2)) → MARK(X2)
A__TIMES(s(X), Y) → A__TIMES(mark(X), mark(Y))
A__SQUARE(X) → MARK(X)
A__PLUS(s(X), Y) → MARK(Y)
A__2NDSNEG(s(N), cons(X, Z)) → MARK(Z)
A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(N)
A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(Z)
A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(N)
A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(Z)

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.

(19) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

MARK(2ndspos(X1, X2)) → MARK(X1)
MARK(2ndsneg(X1, X2)) → MARK(X1)

The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:

POL(A__2NDSPOS(x₁, x₂)) = 1A + 0A · x₁ + 0A · x₂

POL(s(x₁)) = -I + 0A · x₁

POL(cons(x₁, x₂)) = -I + 0A · x₁ + 0A · x₂

POL(mark(x₁)) = 0A + 0A · x₁

POL(cons2(x₁, x₂)) = -I + 0A · x₁ + 0A · x₂

POL(A__2NDSNEG(x₁, x₂)) = 1A + 0A · x₁ + 0A · x₂

POL(MARK(x₁)) = 0A + 0A · x₁

POL(2ndspos(x₁, x₂)) = 1A + 1A · x₁ + 0A · x₂

POL(2ndsneg(x₁, x₂)) = 1A + 1A · x₁ + 0A · x₂

POL(plus(x₁, x₂)) = 0A + 0A · x₁ + 0A · x₂

POL(A__PLUS(x₁, x₂)) = 0A + 0A · x₁ + 0A · x₂

POL(0) = 0A

POL(times(x₁, x₂)) = -I + 0A · x₁ + 0A · x₂

POL(A__TIMES(x₁, x₂)) = 0A + 0A · x₁ + 0A · x₂

POL(a__times(x₁, x₂)) = 0A + 0A · x₁ + 0A · x₂

POL(square(x₁)) = -I + 0A · x₁

POL(A__SQUARE(x₁)) = 0A + 0A · x₁

POL(negrecip(x₁)) = -I + 0A · x₁

POL(rcons(x₁, x₂)) = -I + 0A · x₁ + 0A · x₂

POL(nil) = 0A

POL(posrecip(x₁)) = 1A + -I · x₁

POL(a__from(x₁)) = 0A + 0A · x₁

POL(from(x₁)) = -I + 0A · x₁

POL(rnil) = 0A

POL(a__plus(x₁, x₂)) = 0A + 0A · x₁ + 0A · x₂

POL(a__pi(x₁)) = 1A + 1A · x₁

POL(pi(x₁)) = 1A + 1A · x₁

POL(a__2ndsneg(x₁, x₂)) = 1A + 1A · x₁ + 0A · x₂

POL(a__2ndspos(x₁, x₂)) = 1A + 1A · x₁ + 0A · x₂

POL(a__square(x₁)) = 0A + 0A · x₁

The following usable rules [FROCOS05] were oriented:

mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(nil) → nil
mark(negrecip(X)) → negrecip(mark(X))
mark(posrecip(X)) → posrecip(mark(X))
a__from(X) → from(X)
mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))
mark(rnil) → rnil
mark(cons2(X1, X2)) → cons2(X1, mark(X2))
a__plus(X1, X2) → plus(X1, X2)
a__pi(X) → pi(X)
a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)
a__2ndspos(X1, X2) → 2ndspos(X1, X2)
a__square(X) → square(X)
a__times(X1, X2) → times(X1, X2)
a__2ndspos(s(N), cons2(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))
a__2ndspos(s(N), cons(X, Z)) → a__2ndspos(s(mark(N)), cons2(X, mark(Z)))
a__2ndsneg(s(N), cons(X, Z)) → a__2ndsneg(s(mark(N)), cons2(X, mark(Z)))
a__2ndsneg(0, Z) → rnil
a__2ndspos(0, Z) → rnil
a__from(X) → cons(mark(X), from(s(X)))
a__times(0, Y) → 0
mark(from(X)) → a__from(mark(X))
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(s(X), Y) → s(a__plus(mark(X), mark(Y)))
mark(0) → 0
mark(s(X)) → s(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))
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))

(20) Obligation:

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

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))) → A__2NDSNEG(mark(N), mark(Z))
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))) → A__2NDSPOS(mark(N), mark(Z))
A__2NDSPOS(s(N), cons(X, Z)) → MARK(N)
MARK(2ndspos(X1, X2)) → A__2NDSPOS(mark(X1), mark(X2))
A__2NDSPOS(s(N), cons(X, Z)) → MARK(Z)
MARK(2ndspos(X1, X2)) → MARK(X2)
MARK(2ndsneg(X1, X2)) → A__2NDSNEG(mark(X1), mark(X2))
A__2NDSNEG(s(N), cons(X, Z)) → MARK(N)
MARK(2ndsneg(X1, X2)) → MARK(X2)
MARK(plus(X1, X2)) → A__PLUS(mark(X1), mark(X2))
A__PLUS(0, Y) → MARK(Y)
MARK(plus(X1, X2)) → MARK(X1)
MARK(plus(X1, X2)) → MARK(X2)
MARK(times(X1, X2)) → A__TIMES(mark(X1), mark(X2))
A__TIMES(s(X), Y) → A__PLUS(mark(Y), a__times(mark(X), mark(Y)))
A__PLUS(s(X), Y) → A__PLUS(mark(X), mark(Y))
A__PLUS(s(X), Y) → MARK(X)
MARK(times(X1, X2)) → MARK(X1)
MARK(times(X1, X2)) → MARK(X2)
MARK(square(X)) → A__SQUARE(mark(X))
A__SQUARE(X) → A__TIMES(mark(X), mark(X))
A__TIMES(s(X), Y) → MARK(Y)
MARK(square(X)) → MARK(X)
MARK(s(X)) → MARK(X)
MARK(negrecip(X)) → MARK(X)
MARK(cons2(X1, X2)) → MARK(X2)
MARK(rcons(X1, X2)) → MARK(X1)
MARK(rcons(X1, X2)) → MARK(X2)
A__TIMES(s(X), Y) → A__TIMES(mark(X), mark(Y))
A__SQUARE(X) → MARK(X)
A__PLUS(s(X), Y) → MARK(Y)
A__2NDSNEG(s(N), cons(X, Z)) → MARK(Z)
A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(N)
A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(Z)
A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(N)
A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(Z)

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.

(21) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

MARK(square(X)) → A__SQUARE(mark(X))
MARK(square(X)) → MARK(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:

POL(A__2NDSPOS(x₁, x₂)) = -I + 0A · x₁ + 0A · x₂

POL(s(x₁)) = 0A + 0A · x₁

POL(cons(x₁, x₂)) = -I + 0A · x₁ + 0A · x₂

POL(mark(x₁)) = 0A + 0A · x₁

POL(cons2(x₁, x₂)) = -I + -I · x₁ + 0A · x₂

POL(A__2NDSNEG(x₁, x₂)) = 0A + 0A · x₁ + 0A · x₂

POL(MARK(x₁)) = 0A + 0A · x₁

POL(2ndspos(x₁, x₂)) = -I + 0A · x₁ + 0A · x₂

POL(2ndsneg(x₁, x₂)) = 0A + 0A · x₁ + 0A · x₂

POL(plus(x₁, x₂)) = 0A + 0A · x₁ + 0A · x₂

POL(A__PLUS(x₁, x₂)) = -I + 0A · x₁ + 0A · x₂

POL(0) = 0A

POL(times(x₁, x₂)) = -I + 0A · x₁ + 0A · x₂

POL(A__TIMES(x₁, x₂)) = -I + 0A · x₁ + 0A · x₂

POL(a__times(x₁, x₂)) = 0A + 0A · x₁ + 0A · x₂

POL(square(x₁)) = 1A + 1A · x₁

POL(A__SQUARE(x₁)) = 0A + 0A · x₁

POL(negrecip(x₁)) = -I + 0A · x₁

POL(rcons(x₁, x₂)) = -I + 0A · x₁ + 0A · x₂

POL(nil) = 0A

POL(posrecip(x₁)) = 0A + 0A · x₁

POL(a__from(x₁)) = 0A + 0A · x₁

POL(from(x₁)) = -I + 0A · x₁

POL(rnil) = 0A

POL(a__plus(x₁, x₂)) = 0A + 0A · x₁ + 0A · x₂

POL(a__pi(x₁)) = 5A + 0A · x₁

POL(pi(x₁)) = 5A + 0A · x₁

POL(a__2ndsneg(x₁, x₂)) = 0A + 0A · x₁ + 0A · x₂

POL(a__2ndspos(x₁, x₂)) = 0A + 0A · x₁ + 0A · x₂

POL(a__square(x₁)) = 1A + 1A · x₁

The following usable rules [FROCOS05] were oriented:

mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(nil) → nil
mark(negrecip(X)) → negrecip(mark(X))
mark(posrecip(X)) → posrecip(mark(X))
a__from(X) → from(X)
mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))
mark(rnil) → rnil
mark(cons2(X1, X2)) → cons2(X1, mark(X2))
a__plus(X1, X2) → plus(X1, X2)
a__pi(X) → pi(X)
a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)
a__2ndspos(X1, X2) → 2ndspos(X1, X2)
a__square(X) → square(X)
a__times(X1, X2) → times(X1, X2)
a__2ndspos(s(N), cons2(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))
a__2ndspos(s(N), cons(X, Z)) → a__2ndspos(s(mark(N)), cons2(X, mark(Z)))
a__2ndsneg(s(N), cons(X, Z)) → a__2ndsneg(s(mark(N)), cons2(X, mark(Z)))
a__2ndsneg(0, Z) → rnil
a__2ndspos(0, Z) → rnil
a__from(X) → cons(mark(X), from(s(X)))
a__times(0, Y) → 0
mark(from(X)) → a__from(mark(X))
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(s(X), Y) → s(a__plus(mark(X), mark(Y)))
mark(0) → 0
mark(s(X)) → s(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))
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))

(22) Obligation:

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

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))) → A__2NDSNEG(mark(N), mark(Z))
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))) → A__2NDSPOS(mark(N), mark(Z))
A__2NDSPOS(s(N), cons(X, Z)) → MARK(N)
MARK(2ndspos(X1, X2)) → A__2NDSPOS(mark(X1), mark(X2))
A__2NDSPOS(s(N), cons(X, Z)) → MARK(Z)
MARK(2ndspos(X1, X2)) → MARK(X2)
MARK(2ndsneg(X1, X2)) → A__2NDSNEG(mark(X1), mark(X2))
A__2NDSNEG(s(N), cons(X, Z)) → MARK(N)
MARK(2ndsneg(X1, X2)) → MARK(X2)
MARK(plus(X1, X2)) → A__PLUS(mark(X1), mark(X2))
A__PLUS(0, Y) → MARK(Y)
MARK(plus(X1, X2)) → MARK(X1)
MARK(plus(X1, X2)) → MARK(X2)
MARK(times(X1, X2)) → A__TIMES(mark(X1), mark(X2))
A__TIMES(s(X), Y) → A__PLUS(mark(Y), a__times(mark(X), mark(Y)))
A__PLUS(s(X), Y) → A__PLUS(mark(X), mark(Y))
A__PLUS(s(X), Y) → MARK(X)
MARK(times(X1, X2)) → MARK(X1)
MARK(times(X1, X2)) → MARK(X2)
A__SQUARE(X) → A__TIMES(mark(X), mark(X))
A__TIMES(s(X), Y) → MARK(Y)
MARK(s(X)) → MARK(X)
MARK(negrecip(X)) → MARK(X)
MARK(cons2(X1, X2)) → MARK(X2)
MARK(rcons(X1, X2)) → MARK(X1)
MARK(rcons(X1, X2)) → MARK(X2)
A__TIMES(s(X), Y) → A__TIMES(mark(X), mark(Y))
A__SQUARE(X) → MARK(X)
A__PLUS(s(X), Y) → MARK(Y)
A__2NDSNEG(s(N), cons(X, Z)) → MARK(Z)
A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(N)
A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(Z)
A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(N)
A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(Z)

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.

(23) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes.

(24) Obligation:

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

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → A__2NDSNEG(mark(N), mark(Z))
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))) → A__2NDSPOS(mark(N), mark(Z))
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))) → MARK(N)
MARK(2ndspos(X1, X2)) → A__2NDSPOS(mark(X1), mark(X2))
A__2NDSPOS(s(N), cons(X, Z)) → MARK(N)
MARK(2ndspos(X1, X2)) → MARK(X2)
MARK(2ndsneg(X1, X2)) → A__2NDSNEG(mark(X1), mark(X2))
A__2NDSNEG(s(N), cons(X, Z)) → MARK(N)
MARK(2ndsneg(X1, X2)) → MARK(X2)
MARK(plus(X1, X2)) → A__PLUS(mark(X1), mark(X2))
A__PLUS(0, Y) → MARK(Y)
MARK(plus(X1, X2)) → MARK(X1)
MARK(plus(X1, X2)) → MARK(X2)
MARK(times(X1, X2)) → A__TIMES(mark(X1), mark(X2))
A__TIMES(s(X), Y) → A__PLUS(mark(Y), a__times(mark(X), mark(Y)))
A__PLUS(s(X), Y) → A__PLUS(mark(X), mark(Y))
A__PLUS(s(X), Y) → MARK(X)
MARK(times(X1, X2)) → MARK(X1)
MARK(times(X1, X2)) → MARK(X2)
MARK(s(X)) → MARK(X)
MARK(negrecip(X)) → MARK(X)
MARK(cons2(X1, X2)) → MARK(X2)
MARK(rcons(X1, X2)) → MARK(X1)
MARK(rcons(X1, X2)) → MARK(X2)
A__PLUS(s(X), Y) → MARK(Y)
A__TIMES(s(X), Y) → MARK(Y)
A__TIMES(s(X), Y) → A__TIMES(mark(X), mark(Y))
A__2NDSNEG(s(N), cons(X, Z)) → MARK(Z)
A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(N)
A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(Z)
A__2NDSPOS(s(N), cons(X, Z)) → MARK(Z)
A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(Z)

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.

(25) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

MARK(times(X1, X2)) → MARK(X2)
A__TIMES(s(X), Y) → MARK(Y)

The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:

POL(A__2NDSPOS(x₁, x₂)) = 0A + 0A · x₁ + 0A · x₂

POL(s(x₁)) = -I + 0A · x₁

POL(cons2(x₁, x₂)) = 0A + -I · x₁ + 0A · x₂

POL(cons(x₁, x₂)) = -I + 0A · x₁ + 0A · x₂

POL(A__2NDSNEG(x₁, x₂)) = 0A + 0A · x₁ + 0A · x₂

POL(mark(x₁)) = -I + 0A · x₁

POL(MARK(x₁)) = 0A + 0A · x₁

POL(2ndspos(x₁, x₂)) = 0A + 0A · x₁ + 0A · x₂

POL(2ndsneg(x₁, x₂)) = 0A + 0A · x₁ + 0A · x₂

POL(plus(x₁, x₂)) = 0A + 1A · x₁ + 0A · x₂

POL(A__PLUS(x₁, x₂)) = 0A + 0A · x₁ + 0A · x₂

POL(0) = 0A

POL(times(x₁, x₂)) = 1A + 0A · x₁ + 5A · x₂

POL(A__TIMES(x₁, x₂)) = 1A + 0A · x₁ + 5A · x₂

POL(a__times(x₁, x₂)) = 1A + 0A · x₁ + 5A · x₂

POL(negrecip(x₁)) = -I + 0A · x₁

POL(rcons(x₁, x₂)) = 0A + 0A · x₁ + 0A · x₂

POL(nil) = 4A

POL(posrecip(x₁)) = -I + 0A · x₁

POL(a__from(x₁)) = -I + 0A · x₁

POL(from(x₁)) = -I + 0A · x₁

POL(rnil) = 0A

POL(a__plus(x₁, x₂)) = 0A + 1A · x₁ + 0A · x₂

POL(a__pi(x₁)) = 5A + 0A · x₁

POL(pi(x₁)) = 5A + 0A · x₁

POL(a__2ndsneg(x₁, x₂)) = 0A + 0A · x₁ + 0A · x₂

POL(a__2ndspos(x₁, x₂)) = 0A + 0A · x₁ + 0A · x₂

POL(a__square(x₁)) = 1A + 5A · x₁

POL(square(x₁)) = 1A + 5A · x₁

The following usable rules [FROCOS05] were oriented:

mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(nil) → nil
mark(negrecip(X)) → negrecip(mark(X))
mark(posrecip(X)) → posrecip(mark(X))
a__from(X) → from(X)
mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))
mark(rnil) → rnil
mark(cons2(X1, X2)) → cons2(X1, mark(X2))
a__plus(X1, X2) → plus(X1, X2)
a__pi(X) → pi(X)
a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)
a__2ndspos(X1, X2) → 2ndspos(X1, X2)
a__square(X) → square(X)
a__times(X1, X2) → times(X1, X2)
a__2ndspos(s(N), cons2(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))
a__2ndspos(s(N), cons(X, Z)) → a__2ndspos(s(mark(N)), cons2(X, mark(Z)))
a__2ndsneg(s(N), cons(X, Z)) → a__2ndsneg(s(mark(N)), cons2(X, mark(Z)))
a__2ndsneg(0, Z) → rnil
a__2ndspos(0, Z) → rnil
a__from(X) → cons(mark(X), from(s(X)))
a__times(0, Y) → 0
mark(from(X)) → a__from(mark(X))
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(s(X), Y) → s(a__plus(mark(X), mark(Y)))
mark(0) → 0
mark(s(X)) → s(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))
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))

(26) Obligation:

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

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → A__2NDSNEG(mark(N), mark(Z))
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))) → A__2NDSPOS(mark(N), mark(Z))
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))) → MARK(N)
MARK(2ndspos(X1, X2)) → A__2NDSPOS(mark(X1), mark(X2))
A__2NDSPOS(s(N), cons(X, Z)) → MARK(N)
MARK(2ndspos(X1, X2)) → MARK(X2)
MARK(2ndsneg(X1, X2)) → A__2NDSNEG(mark(X1), mark(X2))
A__2NDSNEG(s(N), cons(X, Z)) → MARK(N)
MARK(2ndsneg(X1, X2)) → MARK(X2)
MARK(plus(X1, X2)) → A__PLUS(mark(X1), mark(X2))
A__PLUS(0, Y) → MARK(Y)
MARK(plus(X1, X2)) → MARK(X1)
MARK(plus(X1, X2)) → MARK(X2)
MARK(times(X1, X2)) → A__TIMES(mark(X1), mark(X2))
A__TIMES(s(X), Y) → A__PLUS(mark(Y), a__times(mark(X), mark(Y)))
A__PLUS(s(X), Y) → A__PLUS(mark(X), mark(Y))
A__PLUS(s(X), Y) → MARK(X)
MARK(times(X1, X2)) → MARK(X1)
MARK(s(X)) → MARK(X)
MARK(negrecip(X)) → MARK(X)
MARK(cons2(X1, X2)) → MARK(X2)
MARK(rcons(X1, X2)) → MARK(X1)
MARK(rcons(X1, X2)) → MARK(X2)
A__PLUS(s(X), Y) → MARK(Y)
A__TIMES(s(X), Y) → A__TIMES(mark(X), mark(Y))
A__2NDSNEG(s(N), cons(X, Z)) → MARK(Z)
A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(N)
A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(Z)
A__2NDSPOS(s(N), cons(X, Z)) → MARK(Z)
A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(Z)

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.

(27) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

MARK(2ndspos(X1, X2)) → MARK(X2)
MARK(2ndsneg(X1, X2)) → MARK(X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:

POL(A__2NDSPOS(x₁, x₂)) = 0A + 0A · x₁ + 0A · x₂

POL(s(x₁)) = -I + 0A · x₁

POL(cons2(x₁, x₂)) = 0A + 0A · x₁ + 0A · x₂

POL(cons(x₁, x₂)) = -I + 0A · x₁ + 0A · x₂

POL(A__2NDSNEG(x₁, x₂)) = 0A + 0A · x₁ + 0A · x₂

POL(mark(x₁)) = 0A + 0A · x₁

POL(MARK(x₁)) = 0A + 0A · x₁

POL(2ndspos(x₁, x₂)) = 1A + 0A · x₁ + 1A · x₂

POL(2ndsneg(x₁, x₂)) = 1A + 0A · x₁ + 1A · x₂

POL(plus(x₁, x₂)) = -I + 0A · x₁ + 0A · x₂

POL(A__PLUS(x₁, x₂)) = 0A + 0A · x₁ + 0A · x₂

POL(0) = 0A

POL(times(x₁, x₂)) = -I + 0A · x₁ + 0A · x₂

POL(A__TIMES(x₁, x₂)) = 0A + 0A · x₁ + 0A · x₂

POL(a__times(x₁, x₂)) = 0A + 0A · x₁ + 0A · x₂

POL(negrecip(x₁)) = -I + 0A · x₁

POL(rcons(x₁, x₂)) = -I + 0A · x₁ + 0A · x₂

POL(nil) = 0A

POL(posrecip(x₁)) = 1A + -I · x₁

POL(a__from(x₁)) = 0A + 0A · x₁

POL(from(x₁)) = -I + 0A · x₁

POL(rnil) = 1A

POL(a__plus(x₁, x₂)) = 0A + 0A · x₁ + 0A · x₂

POL(a__pi(x₁)) = 2A + 0A · x₁

POL(pi(x₁)) = 2A + 0A · x₁

POL(a__2ndsneg(x₁, x₂)) = 1A + 0A · x₁ + 1A · x₂

POL(a__2ndspos(x₁, x₂)) = 1A + 0A · x₁ + 1A · x₂

POL(a__square(x₁)) = 0A + 0A · x₁

POL(square(x₁)) = -I + 0A · x₁

The following usable rules [FROCOS05] were oriented:

mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(nil) → nil
mark(negrecip(X)) → negrecip(mark(X))
mark(posrecip(X)) → posrecip(mark(X))
a__from(X) → from(X)
mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))
mark(rnil) → rnil
mark(cons2(X1, X2)) → cons2(X1, mark(X2))
a__plus(X1, X2) → plus(X1, X2)
a__pi(X) → pi(X)
a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)
a__2ndspos(X1, X2) → 2ndspos(X1, X2)
a__square(X) → square(X)
a__times(X1, X2) → times(X1, X2)
a__2ndspos(s(N), cons2(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))
a__2ndspos(s(N), cons(X, Z)) → a__2ndspos(s(mark(N)), cons2(X, mark(Z)))
a__2ndsneg(s(N), cons(X, Z)) → a__2ndsneg(s(mark(N)), cons2(X, mark(Z)))
a__2ndsneg(0, Z) → rnil
a__2ndspos(0, Z) → rnil
a__from(X) → cons(mark(X), from(s(X)))
a__times(0, Y) → 0
mark(from(X)) → a__from(mark(X))
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(s(X), Y) → s(a__plus(mark(X), mark(Y)))
mark(0) → 0
mark(s(X)) → s(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))
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))

(28) Obligation:

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

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → A__2NDSNEG(mark(N), mark(Z))
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))) → A__2NDSPOS(mark(N), mark(Z))
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))) → MARK(N)
MARK(2ndspos(X1, X2)) → A__2NDSPOS(mark(X1), mark(X2))
A__2NDSPOS(s(N), cons(X, Z)) → MARK(N)
MARK(2ndsneg(X1, X2)) → A__2NDSNEG(mark(X1), mark(X2))
A__2NDSNEG(s(N), cons(X, Z)) → MARK(N)
MARK(plus(X1, X2)) → A__PLUS(mark(X1), mark(X2))
A__PLUS(0, Y) → MARK(Y)
MARK(plus(X1, X2)) → MARK(X1)
MARK(plus(X1, X2)) → MARK(X2)
MARK(times(X1, X2)) → A__TIMES(mark(X1), mark(X2))
A__TIMES(s(X), Y) → A__PLUS(mark(Y), a__times(mark(X), mark(Y)))
A__PLUS(s(X), Y) → A__PLUS(mark(X), mark(Y))
A__PLUS(s(X), Y) → MARK(X)
MARK(times(X1, X2)) → MARK(X1)
MARK(s(X)) → MARK(X)
MARK(negrecip(X)) → MARK(X)
MARK(cons2(X1, X2)) → MARK(X2)
MARK(rcons(X1, X2)) → MARK(X1)
MARK(rcons(X1, X2)) → MARK(X2)
A__PLUS(s(X), Y) → MARK(Y)
A__TIMES(s(X), Y) → A__TIMES(mark(X), mark(Y))
A__2NDSNEG(s(N), cons(X, Z)) → MARK(Z)
A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(N)
A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(Z)
A__2NDSPOS(s(N), cons(X, Z)) → MARK(Z)
A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(Z)

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.

(29) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

MARK(plus(X1, X2)) → MARK(X1)

The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:

POL(A__2NDSPOS(x₁, x₂)) = -I + 0A · x₁ + 0A · x₂

POL(s(x₁)) = -I + 0A · x₁

POL(cons2(x₁, x₂)) = -I + 0A · x₁ + 0A · x₂

POL(cons(x₁, x₂)) = 0A + 0A · x₁ + 0A · x₂

POL(A__2NDSNEG(x₁, x₂)) = -I + 0A · x₁ + 0A · x₂

POL(mark(x₁)) = 0A + 0A · x₁

POL(MARK(x₁)) = -I + 0A · x₁

POL(2ndspos(x₁, x₂)) = 0A + 0A · x₁ + 0A · x₂

POL(2ndsneg(x₁, x₂)) = 0A + 0A · x₁ + 0A · x₂

POL(plus(x₁, x₂)) = 1A + 1A · x₁ + 0A · x₂

POL(A__PLUS(x₁, x₂)) = 1A + 0A · x₁ + 0A · x₂

POL(0) = 4A

POL(times(x₁, x₂)) = 1A + 0A · x₁ + 1A · x₂

POL(A__TIMES(x₁, x₂)) = 1A + 0A · x₁ + 1A · x₂

POL(a__times(x₁, x₂)) = 1A + 0A · x₁ + 1A · x₂

POL(negrecip(x₁)) = -I + 0A · x₁

POL(rcons(x₁, x₂)) = 0A + 0A · x₁ + 0A · x₂

POL(nil) = 5A

POL(posrecip(x₁)) = 0A + -I · x₁

POL(a__from(x₁)) = 0A + 0A · x₁

POL(from(x₁)) = -I + 0A · x₁

POL(rnil) = 0A

POL(a__plus(x₁, x₂)) = 1A + 1A · x₁ + 0A · x₂

POL(a__pi(x₁)) = 5A + 0A · x₁

POL(pi(x₁)) = 5A + 0A · x₁

POL(a__2ndsneg(x₁, x₂)) = 0A + 0A · x₁ + 0A · x₂

POL(a__2ndspos(x₁, x₂)) = 0A + 0A · x₁ + 0A · x₂

POL(a__square(x₁)) = 1A + 1A · x₁

POL(square(x₁)) = 1A + 1A · x₁

The following usable rules [FROCOS05] were oriented:

mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(nil) → nil
mark(negrecip(X)) → negrecip(mark(X))
mark(posrecip(X)) → posrecip(mark(X))
a__from(X) → from(X)
mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))
mark(rnil) → rnil
mark(cons2(X1, X2)) → cons2(X1, mark(X2))
a__plus(X1, X2) → plus(X1, X2)
a__pi(X) → pi(X)
a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)
a__2ndspos(X1, X2) → 2ndspos(X1, X2)
a__square(X) → square(X)
a__times(X1, X2) → times(X1, X2)
a__2ndspos(s(N), cons2(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))
a__2ndspos(s(N), cons(X, Z)) → a__2ndspos(s(mark(N)), cons2(X, mark(Z)))
a__2ndsneg(s(N), cons(X, Z)) → a__2ndsneg(s(mark(N)), cons2(X, mark(Z)))
a__2ndsneg(0, Z) → rnil
a__2ndspos(0, Z) → rnil
a__from(X) → cons(mark(X), from(s(X)))
a__times(0, Y) → 0
mark(from(X)) → a__from(mark(X))
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(s(X), Y) → s(a__plus(mark(X), mark(Y)))
mark(0) → 0
mark(s(X)) → s(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))
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))

(30) Obligation:

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

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → A__2NDSNEG(mark(N), mark(Z))
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))) → A__2NDSPOS(mark(N), mark(Z))
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))) → MARK(N)
MARK(2ndspos(X1, X2)) → A__2NDSPOS(mark(X1), mark(X2))
A__2NDSPOS(s(N), cons(X, Z)) → MARK(N)
MARK(2ndsneg(X1, X2)) → A__2NDSNEG(mark(X1), mark(X2))
A__2NDSNEG(s(N), cons(X, Z)) → MARK(N)
MARK(plus(X1, X2)) → A__PLUS(mark(X1), mark(X2))
A__PLUS(0, Y) → MARK(Y)
MARK(plus(X1, X2)) → MARK(X2)
MARK(times(X1, X2)) → A__TIMES(mark(X1), mark(X2))
A__TIMES(s(X), Y) → A__PLUS(mark(Y), a__times(mark(X), mark(Y)))
A__PLUS(s(X), Y) → A__PLUS(mark(X), mark(Y))
A__PLUS(s(X), Y) → MARK(X)
MARK(times(X1, X2)) → MARK(X1)
MARK(s(X)) → MARK(X)
MARK(negrecip(X)) → MARK(X)
MARK(cons2(X1, X2)) → MARK(X2)
MARK(rcons(X1, X2)) → MARK(X1)
MARK(rcons(X1, X2)) → MARK(X2)
A__PLUS(s(X), Y) → MARK(Y)
A__TIMES(s(X), Y) → A__TIMES(mark(X), mark(Y))
A__2NDSNEG(s(N), cons(X, Z)) → MARK(Z)
A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(N)
A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(Z)
A__2NDSPOS(s(N), cons(X, Z)) → MARK(Z)
A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → MARK(Z)

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.

(31) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

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

The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:

POL(A__2NDSPOS(x₁, x₂)) = 0A + 0A · x₁ + 1A · x₂

POL(s(x₁)) = -I + 0A · x₁

POL(cons2(x₁, x₂)) = -I + -I · x₁ + 0A · x₂

POL(cons(x₁, x₂)) = 0A + 0A · x₁ + 0A · x₂

POL(A__2NDSNEG(x₁, x₂)) = -I + 0A · x₁ + 1A · x₂

POL(mark(x₁)) = 0A + 0A · x₁

POL(MARK(x₁)) = 0A + 0A · x₁

POL(2ndspos(x₁, x₂)) = 1A + 0A · x₁ + 1A · x₂

POL(2ndsneg(x₁, x₂)) = 1A + 0A · x₁ + 1A · x₂

POL(plus(x₁, x₂)) = 0A + 0A · x₁ + 0A · x₂

POL(A__PLUS(x₁, x₂)) = 0A + 0A · x₁ + 0A · x₂

POL(0) = 0A

POL(times(x₁, x₂)) = 0A + 0A · x₁ + 0A · x₂

POL(A__TIMES(x₁, x₂)) = 0A + 0A · x₁ + 0A · x₂

POL(a__times(x₁, x₂)) = 0A + 0A · x₁ + 0A · x₂

POL(negrecip(x₁)) = 0A + 0A · x₁

POL(rcons(x₁, x₂)) = 0A + 0A · x₁ + 0A · x₂

POL(nil) = 0A

POL(posrecip(x₁)) = 0A + -I · x₁

POL(a__from(x₁)) = 0A + 0A · x₁

POL(from(x₁)) = -I + 0A · x₁

POL(rnil) = 1A

POL(a__plus(x₁, x₂)) = 0A + 0A · x₁ + 0A · x₂

POL(a__pi(x₁)) = 1A + 0A · x₁

POL(pi(x₁)) = 1A + 0A · x₁

POL(a__2ndsneg(x₁, x₂)) = 1A + 0A · x₁ + 1A · x₂

POL(a__2ndspos(x₁, x₂)) = 1A + 0A · x₁ + 1A · x₂

POL(a__square(x₁)) = 0A + 0A · x₁

POL(square(x₁)) = -I + 0A · x₁

The following usable rules [FROCOS05] were oriented:

mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(nil) → nil
mark(negrecip(X)) → negrecip(mark(X))
mark(posrecip(X)) → posrecip(mark(X))
a__from(X) → from(X)
mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))
mark(rnil) → rnil
mark(cons2(X1, X2)) → cons2(X1, mark(X2))
a__plus(X1, X2) → plus(X1, X2)
a__pi(X) → pi(X)
a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)
a__2ndspos(X1, X2) → 2ndspos(X1, X2)
a__square(X) → square(X)
a__times(X1, X2) → times(X1, X2)
a__2ndspos(s(N), cons2(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))
a__2ndspos(s(N), cons(X, Z)) → a__2ndspos(s(mark(N)), cons2(X, mark(Z)))
a__2ndsneg(s(N), cons(X, Z)) → a__2ndsneg(s(mark(N)), cons2(X, mark(Z)))
a__2ndsneg(0, Z) → rnil
a__2ndspos(0, Z) → rnil
a__from(X) → cons(mark(X), from(s(X)))
a__times(0, Y) → 0
mark(from(X)) → a__from(mark(X))
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(s(X), Y) → s(a__plus(mark(X), mark(Y)))
mark(0) → 0
mark(s(X)) → s(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))
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))

(32) Obligation:

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

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → A__2NDSNEG(mark(N), mark(Z))
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))) → A__2NDSPOS(mark(N), mark(Z))
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))) → MARK(N)
MARK(2ndspos(X1, X2)) → A__2NDSPOS(mark(X1), mark(X2))
A__2NDSPOS(s(N), cons(X, Z)) → MARK(N)
MARK(2ndsneg(X1, X2)) → A__2NDSNEG(mark(X1), mark(X2))
A__2NDSNEG(s(N), cons(X, Z)) → MARK(N)
MARK(plus(X1, X2)) → A__PLUS(mark(X1), mark(X2))
A__PLUS(0, Y) → MARK(Y)
MARK(plus(X1, X2)) → MARK(X2)
MARK(times(X1, X2)) → A__TIMES(mark(X1), mark(X2))
A__TIMES(s(X), Y) → A__PLUS(mark(Y), a__times(mark(X), mark(Y)))
A__PLUS(s(X), Y) → A__PLUS(mark(X), mark(Y))
A__PLUS(s(X), Y) → MARK(X)
MARK(times(X1, X2)) → MARK(X1)
MARK(s(X)) → MARK(X)
MARK(negrecip(X)) → MARK(X)
MARK(cons2(X1, X2)) → MARK(X2)
MARK(rcons(X1, X2)) → MARK(X1)
MARK(rcons(X1, X2)) → MARK(X2)
A__PLUS(s(X), Y) → MARK(Y)
A__TIMES(s(X), Y) → A__TIMES(mark(X), mark(Y))
A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(N)

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.

(33) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

A__PLUS(s(X), Y) → MARK(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:

POL(A__2NDSPOS(x₁, x₂)) = -I + 0A · x₁ + -I · x₂

POL(s(x₁)) = 1A + 0A · x₁

POL(cons2(x₁, x₂)) = -I + -I · x₁ + 0A · x₂

POL(cons(x₁, x₂)) = -I + 0A · x₁ + 0A · x₂

POL(A__2NDSNEG(x₁, x₂)) = 1A + 0A · x₁ + -I · x₂

POL(mark(x₁)) = 0A + 0A · x₁

POL(MARK(x₁)) = 1A + 0A · x₁

POL(2ndspos(x₁, x₂)) = -I + 0A · x₁ + 0A · x₂

POL(2ndsneg(x₁, x₂)) = -I + 0A · x₁ + 0A · x₂

POL(plus(x₁, x₂)) = 1A + 1A · x₁ + 0A · x₂

POL(A__PLUS(x₁, x₂)) = -I + 1A · x₁ + 0A · x₂

POL(0) = 0A

POL(times(x₁, x₂)) = 1A + 1A · x₁ + 1A · x₂

POL(A__TIMES(x₁, x₂)) = -I + 1A · x₁ + 1A · x₂

POL(a__times(x₁, x₂)) = 1A + 1A · x₁ + 1A · x₂

POL(negrecip(x₁)) = -I + 0A · x₁

POL(rcons(x₁, x₂)) = -I + 0A · x₁ + 0A · x₂

POL(nil) = 1A

POL(posrecip(x₁)) = -I + 0A · x₁

POL(a__from(x₁)) = 1A + 0A · x₁

POL(from(x₁)) = 1A + 0A · x₁

POL(rnil) = 0A

POL(a__plus(x₁, x₂)) = 1A + 1A · x₁ + 0A · x₂

POL(a__pi(x₁)) = 1A + 0A · x₁

POL(pi(x₁)) = 1A + 0A · x₁

POL(a__2ndsneg(x₁, x₂)) = 0A + 0A · x₁ + 0A · x₂

POL(a__2ndspos(x₁, x₂)) = -I + 0A · x₁ + 0A · x₂

POL(a__square(x₁)) = 1A + 1A · x₁

POL(square(x₁)) = 1A + 1A · x₁

The following usable rules [FROCOS05] were oriented:

mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(nil) → nil
mark(negrecip(X)) → negrecip(mark(X))
mark(posrecip(X)) → posrecip(mark(X))
a__from(X) → from(X)
mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))
mark(rnil) → rnil
mark(cons2(X1, X2)) → cons2(X1, mark(X2))
a__plus(X1, X2) → plus(X1, X2)
a__pi(X) → pi(X)
a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)
a__2ndspos(X1, X2) → 2ndspos(X1, X2)
a__square(X) → square(X)
a__times(X1, X2) → times(X1, X2)
a__2ndspos(s(N), cons2(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))
a__2ndspos(s(N), cons(X, Z)) → a__2ndspos(s(mark(N)), cons2(X, mark(Z)))
a__2ndsneg(s(N), cons(X, Z)) → a__2ndsneg(s(mark(N)), cons2(X, mark(Z)))
a__2ndsneg(0, Z) → rnil
a__2ndspos(0, Z) → rnil
a__from(X) → cons(mark(X), from(s(X)))
a__times(0, Y) → 0
mark(from(X)) → a__from(mark(X))
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(s(X), Y) → s(a__plus(mark(X), mark(Y)))
mark(0) → 0
mark(s(X)) → s(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))
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))

(34) Obligation:

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

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → A__2NDSNEG(mark(N), mark(Z))
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))) → A__2NDSPOS(mark(N), mark(Z))
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))) → MARK(N)
MARK(2ndspos(X1, X2)) → A__2NDSPOS(mark(X1), mark(X2))
A__2NDSPOS(s(N), cons(X, Z)) → MARK(N)
MARK(2ndsneg(X1, X2)) → A__2NDSNEG(mark(X1), mark(X2))
A__2NDSNEG(s(N), cons(X, Z)) → MARK(N)
MARK(plus(X1, X2)) → A__PLUS(mark(X1), mark(X2))
A__PLUS(0, Y) → MARK(Y)
MARK(plus(X1, X2)) → MARK(X2)
MARK(times(X1, X2)) → A__TIMES(mark(X1), mark(X2))
A__TIMES(s(X), Y) → A__PLUS(mark(Y), a__times(mark(X), mark(Y)))
A__PLUS(s(X), Y) → A__PLUS(mark(X), mark(Y))
MARK(times(X1, X2)) → MARK(X1)
MARK(s(X)) → MARK(X)
MARK(negrecip(X)) → MARK(X)
MARK(cons2(X1, X2)) → MARK(X2)
MARK(rcons(X1, X2)) → MARK(X1)
MARK(rcons(X1, X2)) → MARK(X2)
A__PLUS(s(X), Y) → MARK(Y)
A__TIMES(s(X), Y) → A__TIMES(mark(X), mark(Y))
A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(N)

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.

(35) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

MARK(negrecip(X)) → MARK(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:

POL(A__2NDSPOS(x₁, x₂)) = -I + 0A · x₁ + -I · x₂

POL(s(x₁)) = -I + 0A · x₁

POL(cons2(x₁, x₂)) = -I + -I · x₁ + 0A · x₂

POL(cons(x₁, x₂)) = -I + 4A · x₁ + 0A · x₂

POL(A__2NDSNEG(x₁, x₂)) = -I + 0A · x₁ + -I · x₂

POL(mark(x₁)) = -I + 0A · x₁

POL(MARK(x₁)) = -I + 0A · x₁

POL(2ndspos(x₁, x₂)) = -I + 0A · x₁ + 0A · x₂

POL(2ndsneg(x₁, x₂)) = -I + 0A · x₁ + 0A · x₂

POL(plus(x₁, x₂)) = 0A + -I · x₁ + 0A · x₂

POL(A__PLUS(x₁, x₂)) = -I + -I · x₁ + 0A · x₂

POL(0) = 0A

POL(times(x₁, x₂)) = 0A + 0A · x₁ + -I · x₂

POL(A__TIMES(x₁, x₂)) = 0A + 0A · x₁ + -I · x₂

POL(a__times(x₁, x₂)) = 0A + 0A · x₁ + -I · x₂

POL(negrecip(x₁)) = -I + 4A · x₁

POL(rcons(x₁, x₂)) = -I + 0A · x₁ + 0A · x₂

POL(nil) = 0A

POL(posrecip(x₁)) = -I + 0A · x₁

POL(a__from(x₁)) = -I + 4A · x₁

POL(from(x₁)) = -I + 4A · x₁

POL(rnil) = 0A

POL(a__plus(x₁, x₂)) = 0A + -I · x₁ + 0A · x₂

POL(a__pi(x₁)) = 4A + 0A · x₁

POL(pi(x₁)) = 4A + 0A · x₁

POL(a__2ndsneg(x₁, x₂)) = -I + 0A · x₁ + 0A · x₂

POL(a__2ndspos(x₁, x₂)) = -I + 0A · x₁ + 0A · x₂

POL(a__square(x₁)) = 0A + 1A · x₁

POL(square(x₁)) = 0A + 1A · x₁

The following usable rules [FROCOS05] were oriented:

mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(nil) → nil
mark(negrecip(X)) → negrecip(mark(X))
mark(posrecip(X)) → posrecip(mark(X))
a__from(X) → from(X)
mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))
mark(rnil) → rnil
mark(cons2(X1, X2)) → cons2(X1, mark(X2))
a__plus(X1, X2) → plus(X1, X2)
a__pi(X) → pi(X)
a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)
a__2ndspos(X1, X2) → 2ndspos(X1, X2)
a__square(X) → square(X)
a__times(X1, X2) → times(X1, X2)
a__2ndspos(s(N), cons2(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))
a__2ndspos(s(N), cons(X, Z)) → a__2ndspos(s(mark(N)), cons2(X, mark(Z)))
a__2ndsneg(s(N), cons(X, Z)) → a__2ndsneg(s(mark(N)), cons2(X, mark(Z)))
a__2ndsneg(0, Z) → rnil
a__2ndspos(0, Z) → rnil
a__from(X) → cons(mark(X), from(s(X)))
a__times(0, Y) → 0
mark(from(X)) → a__from(mark(X))
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(s(X), Y) → s(a__plus(mark(X), mark(Y)))
mark(0) → 0
mark(s(X)) → s(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))
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))

(36) Obligation:

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

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → A__2NDSNEG(mark(N), mark(Z))
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))) → A__2NDSPOS(mark(N), mark(Z))
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))) → MARK(N)
MARK(2ndspos(X1, X2)) → A__2NDSPOS(mark(X1), mark(X2))
A__2NDSPOS(s(N), cons(X, Z)) → MARK(N)
MARK(2ndsneg(X1, X2)) → A__2NDSNEG(mark(X1), mark(X2))
A__2NDSNEG(s(N), cons(X, Z)) → MARK(N)
MARK(plus(X1, X2)) → A__PLUS(mark(X1), mark(X2))
A__PLUS(0, Y) → MARK(Y)
MARK(plus(X1, X2)) → MARK(X2)
MARK(times(X1, X2)) → A__TIMES(mark(X1), mark(X2))
A__TIMES(s(X), Y) → A__PLUS(mark(Y), a__times(mark(X), mark(Y)))
A__PLUS(s(X), Y) → A__PLUS(mark(X), mark(Y))
MARK(times(X1, X2)) → MARK(X1)
MARK(s(X)) → MARK(X)
MARK(cons2(X1, X2)) → MARK(X2)
MARK(rcons(X1, X2)) → MARK(X1)
MARK(rcons(X1, X2)) → MARK(X2)
A__PLUS(s(X), Y) → MARK(Y)
A__TIMES(s(X), Y) → A__TIMES(mark(X), mark(Y))
A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(N)

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.

(37) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

MARK(cons2(X1, X2)) → MARK(X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:

POL(0) = 0
POL(2ndsneg(x₁, x₂)) = x₁
POL(2ndspos(x₁, x₂)) = x₁
POL(A__2NDSNEG(x₁, x₂)) = x₁
POL(A__2NDSPOS(x₁, x₂)) = x₁
POL(A__PLUS(x₁, x₂)) = x₂
POL(A__TIMES(x₁, x₂)) = x₁
POL(MARK(x₁)) = x₁
POL(a__2ndsneg(x₁, x₂)) = x₁
POL(a__2ndspos(x₁, x₂)) = x₁
POL(a__from(x₁)) = 0
POL(a__pi(x₁)) = x₁
POL(a__plus(x₁, x₂)) = x₂
POL(a__square(x₁)) = x₁
POL(a__times(x₁, x₂)) = x₁
POL(cons(x₁, x₂)) = 0
POL(cons2(x₁, x₂)) = 1 + x₁ + x₂
POL(from(x₁)) = 0
POL(mark(x₁)) = x₁
POL(negrecip(x₁)) = 0
POL(nil) = 0
POL(pi(x₁)) = x₁
POL(plus(x₁, x₂)) = x₂
POL(posrecip(x₁)) = 0
POL(rcons(x₁, x₂)) = x₁ + x₂
POL(rnil) = 0
POL(s(x₁)) = x₁
POL(square(x₁)) = x₁
POL(times(x₁, x₂)) = x₁

The following usable rules [FROCOS05] were oriented:

mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(nil) → nil
mark(negrecip(X)) → negrecip(mark(X))
mark(posrecip(X)) → posrecip(mark(X))
a__from(X) → from(X)
mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))
mark(rnil) → rnil
mark(cons2(X1, X2)) → cons2(X1, mark(X2))
a__plus(X1, X2) → plus(X1, X2)
a__pi(X) → pi(X)
a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)
a__2ndspos(X1, X2) → 2ndspos(X1, X2)
a__square(X) → square(X)
a__times(X1, X2) → times(X1, X2)
a__2ndspos(s(N), cons2(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))
a__2ndspos(s(N), cons(X, Z)) → a__2ndspos(s(mark(N)), cons2(X, mark(Z)))
a__2ndsneg(s(N), cons(X, Z)) → a__2ndsneg(s(mark(N)), cons2(X, mark(Z)))
a__2ndsneg(0, Z) → rnil
a__2ndspos(0, Z) → rnil
a__from(X) → cons(mark(X), from(s(X)))
a__times(0, Y) → 0
mark(from(X)) → a__from(mark(X))
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(s(X), Y) → s(a__plus(mark(X), mark(Y)))
mark(0) → 0
mark(s(X)) → s(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))
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))

(38) Obligation:

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

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → A__2NDSNEG(mark(N), mark(Z))
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))) → A__2NDSPOS(mark(N), mark(Z))
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))) → MARK(N)
MARK(2ndspos(X1, X2)) → A__2NDSPOS(mark(X1), mark(X2))
A__2NDSPOS(s(N), cons(X, Z)) → MARK(N)
MARK(2ndsneg(X1, X2)) → A__2NDSNEG(mark(X1), mark(X2))
A__2NDSNEG(s(N), cons(X, Z)) → MARK(N)
MARK(plus(X1, X2)) → A__PLUS(mark(X1), mark(X2))
A__PLUS(0, Y) → MARK(Y)
MARK(plus(X1, X2)) → MARK(X2)
MARK(times(X1, X2)) → A__TIMES(mark(X1), mark(X2))
A__TIMES(s(X), Y) → A__PLUS(mark(Y), a__times(mark(X), mark(Y)))
A__PLUS(s(X), Y) → A__PLUS(mark(X), mark(Y))
MARK(times(X1, X2)) → MARK(X1)
MARK(s(X)) → MARK(X)
MARK(rcons(X1, X2)) → MARK(X1)
MARK(rcons(X1, X2)) → MARK(X2)
A__PLUS(s(X), Y) → MARK(Y)
A__TIMES(s(X), Y) → A__TIMES(mark(X), mark(Y))
A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(N)

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.

(39) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

MARK(times(X1, X2)) → MARK(X1)

The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:

POL(0) = 0
POL(2ndsneg(x₁, x₂)) = x₁
POL(2ndspos(x₁, x₂)) = x₁
POL(A__2NDSNEG(x₁, x₂)) = x₁
POL(A__2NDSPOS(x₁, x₂)) = x₁
POL(A__PLUS(x₁, x₂)) = x₂
POL(A__TIMES(x₁, x₂)) = 1 + x₁
POL(MARK(x₁)) = x₁
POL(a__2ndsneg(x₁, x₂)) = x₁
POL(a__2ndspos(x₁, x₂)) = x₁
POL(a__from(x₁)) = x₁
POL(a__pi(x₁)) = x₁
POL(a__plus(x₁, x₂)) = x₂
POL(a__square(x₁)) = 1 + x₁
POL(a__times(x₁, x₂)) = 1 + x₁
POL(cons(x₁, x₂)) = 0
POL(cons2(x₁, x₂)) = x₁
POL(from(x₁)) = x₁
POL(mark(x₁)) = x₁
POL(negrecip(x₁)) = 0
POL(nil) = 0
POL(pi(x₁)) = x₁
POL(plus(x₁, x₂)) = x₂
POL(posrecip(x₁)) = 0
POL(rcons(x₁, x₂)) = x₁ + x₂
POL(rnil) = 0
POL(s(x₁)) = x₁
POL(square(x₁)) = 1 + x₁
POL(times(x₁, x₂)) = 1 + x₁

The following usable rules [FROCOS05] were oriented:

mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(nil) → nil
mark(negrecip(X)) → negrecip(mark(X))
mark(posrecip(X)) → posrecip(mark(X))
a__from(X) → from(X)
mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))
mark(rnil) → rnil
mark(cons2(X1, X2)) → cons2(X1, mark(X2))
a__plus(X1, X2) → plus(X1, X2)
a__pi(X) → pi(X)
a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)
a__2ndspos(X1, X2) → 2ndspos(X1, X2)
a__square(X) → square(X)
a__times(X1, X2) → times(X1, X2)
a__2ndspos(s(N), cons2(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))
a__2ndspos(s(N), cons(X, Z)) → a__2ndspos(s(mark(N)), cons2(X, mark(Z)))
a__2ndsneg(s(N), cons(X, Z)) → a__2ndsneg(s(mark(N)), cons2(X, mark(Z)))
a__2ndsneg(0, Z) → rnil
a__2ndspos(0, Z) → rnil
a__from(X) → cons(mark(X), from(s(X)))
a__times(0, Y) → 0
mark(from(X)) → a__from(mark(X))
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(s(X), Y) → s(a__plus(mark(X), mark(Y)))
mark(0) → 0
mark(s(X)) → s(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))
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))

(40) Obligation:

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

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → A__2NDSNEG(mark(N), mark(Z))
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))) → A__2NDSPOS(mark(N), mark(Z))
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))) → MARK(N)
MARK(2ndspos(X1, X2)) → A__2NDSPOS(mark(X1), mark(X2))
A__2NDSPOS(s(N), cons(X, Z)) → MARK(N)
MARK(2ndsneg(X1, X2)) → A__2NDSNEG(mark(X1), mark(X2))
A__2NDSNEG(s(N), cons(X, Z)) → MARK(N)
MARK(plus(X1, X2)) → A__PLUS(mark(X1), mark(X2))
A__PLUS(0, Y) → MARK(Y)
MARK(plus(X1, X2)) → MARK(X2)
MARK(times(X1, X2)) → A__TIMES(mark(X1), mark(X2))
A__TIMES(s(X), Y) → A__PLUS(mark(Y), a__times(mark(X), mark(Y)))
A__PLUS(s(X), Y) → A__PLUS(mark(X), mark(Y))
MARK(s(X)) → MARK(X)
MARK(rcons(X1, X2)) → MARK(X1)
MARK(rcons(X1, X2)) → MARK(X2)
A__PLUS(s(X), Y) → MARK(Y)
A__TIMES(s(X), Y) → A__TIMES(mark(X), mark(Y))
A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(N)

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.

(41) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

MARK(2ndspos(X1, X2)) → A__2NDSPOS(mark(X1), mark(X2))
MARK(2ndsneg(X1, X2)) → A__2NDSNEG(mark(X1), mark(X2))

The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:

POL(0) = 0
POL(2ndsneg(x₁, x₂)) = 1 + x₁
POL(2ndspos(x₁, x₂)) = 1 + x₁
POL(A__2NDSNEG(x₁, x₂)) = x₁
POL(A__2NDSPOS(x₁, x₂)) = x₁
POL(A__PLUS(x₁, x₂)) = x₂
POL(A__TIMES(x₁, x₂)) = x₁
POL(MARK(x₁)) = x₁
POL(a__2ndsneg(x₁, x₂)) = 1 + x₁
POL(a__2ndspos(x₁, x₂)) = 1 + x₁
POL(a__from(x₁)) = 0
POL(a__pi(x₁)) = 1 + x₁
POL(a__plus(x₁, x₂)) = x₂
POL(a__square(x₁)) = x₁
POL(a__times(x₁, x₂)) = x₁
POL(cons(x₁, x₂)) = 0
POL(cons2(x₁, x₂)) = 0
POL(from(x₁)) = 0
POL(mark(x₁)) = x₁
POL(negrecip(x₁)) = 0
POL(nil) = 0
POL(pi(x₁)) = 1 + x₁
POL(plus(x₁, x₂)) = x₂
POL(posrecip(x₁)) = 0
POL(rcons(x₁, x₂)) = x₁ + x₂
POL(rnil) = 1
POL(s(x₁)) = x₁
POL(square(x₁)) = x₁
POL(times(x₁, x₂)) = x₁

The following usable rules [FROCOS05] were oriented:

mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(nil) → nil
mark(negrecip(X)) → negrecip(mark(X))
mark(posrecip(X)) → posrecip(mark(X))
a__from(X) → from(X)
mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))
mark(rnil) → rnil
mark(cons2(X1, X2)) → cons2(X1, mark(X2))
a__plus(X1, X2) → plus(X1, X2)
a__pi(X) → pi(X)
a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)
a__2ndspos(X1, X2) → 2ndspos(X1, X2)
a__square(X) → square(X)
a__times(X1, X2) → times(X1, X2)
a__2ndspos(s(N), cons2(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))
a__2ndspos(s(N), cons(X, Z)) → a__2ndspos(s(mark(N)), cons2(X, mark(Z)))
a__2ndsneg(s(N), cons(X, Z)) → a__2ndsneg(s(mark(N)), cons2(X, mark(Z)))
a__2ndsneg(0, Z) → rnil
a__2ndspos(0, Z) → rnil
a__from(X) → cons(mark(X), from(s(X)))
a__times(0, Y) → 0
mark(from(X)) → a__from(mark(X))
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(s(X), Y) → s(a__plus(mark(X), mark(Y)))
mark(0) → 0
mark(s(X)) → s(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))
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))

(42) Obligation:

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

A__2NDSPOS(s(N), cons2(X, cons(Y, Z))) → A__2NDSNEG(mark(N), mark(Z))
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))) → A__2NDSPOS(mark(N), mark(Z))
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))) → MARK(N)
A__2NDSPOS(s(N), cons(X, Z)) → MARK(N)
A__2NDSNEG(s(N), cons(X, Z)) → MARK(N)
MARK(plus(X1, X2)) → A__PLUS(mark(X1), mark(X2))
A__PLUS(0, Y) → MARK(Y)
MARK(plus(X1, X2)) → MARK(X2)
MARK(times(X1, X2)) → A__TIMES(mark(X1), mark(X2))
A__TIMES(s(X), Y) → A__PLUS(mark(Y), a__times(mark(X), mark(Y)))
A__PLUS(s(X), Y) → A__PLUS(mark(X), mark(Y))
MARK(s(X)) → MARK(X)
MARK(rcons(X1, X2)) → MARK(X1)
MARK(rcons(X1, X2)) → MARK(X2)
A__PLUS(s(X), Y) → MARK(Y)
A__TIMES(s(X), Y) → A__TIMES(mark(X), mark(Y))
A__2NDSNEG(s(N), cons2(X, cons(Y, Z))) → MARK(N)

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.

(43) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 4 less nodes.

(44) Complex Obligation (AND)

(45) Obligation:

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

A__PLUS(0, Y) → MARK(Y)
MARK(plus(X1, X2)) → A__PLUS(mark(X1), mark(X2))
A__PLUS(s(X), Y) → A__PLUS(mark(X), mark(Y))
A__PLUS(s(X), Y) → MARK(Y)
MARK(plus(X1, X2)) → MARK(X2)
MARK(times(X1, X2)) → A__TIMES(mark(X1), mark(X2))
A__TIMES(s(X), Y) → A__PLUS(mark(Y), a__times(mark(X), mark(Y)))
A__TIMES(s(X), Y) → A__TIMES(mark(X), mark(Y))
MARK(s(X)) → MARK(X)
MARK(rcons(X1, X2)) → MARK(X1)
MARK(rcons(X1, X2)) → MARK(X2)

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.

(46) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

MARK(rcons(X1, X2)) → MARK(X1)

The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:

POL(A__PLUS(x₁, x₂)) = 3A + -I · x₁ + 0A · x₂

POL(0) = 0A

POL(MARK(x₁)) = 3A + 0A · x₁

POL(plus(x₁, x₂)) = -I + -I · x₁ + 0A · x₂

POL(mark(x₁)) = -I + 0A · x₁

POL(s(x₁)) = -I + 0A · x₁

POL(times(x₁, x₂)) = -I + 0A · x₁ + 0A · x₂

POL(A__TIMES(x₁, x₂)) = 3A + 0A · x₁ + 0A · x₂

POL(a__times(x₁, x₂)) = -I + 0A · x₁ + 0A · x₂

POL(rcons(x₁, x₂)) = 5A + 1A · x₁ + 0A · x₂

POL(cons(x₁, x₂)) = 0A + 0A · x₁ + -I · x₂

POL(nil) = 0A

POL(negrecip(x₁)) = 0A + -I · x₁

POL(posrecip(x₁)) = 0A + -I · x₁

POL(a__from(x₁)) = 0A + 0A · x₁

POL(from(x₁)) = 0A + 0A · x₁

POL(rnil) = 5A

POL(cons2(x₁, x₂)) = 5A + -I · x₁ + -I · x₂

POL(a__plus(x₁, x₂)) = -I + -I · x₁ + 0A · x₂

POL(a__pi(x₁)) = 5A + 0A · x₁

POL(pi(x₁)) = 5A + 0A · x₁

POL(a__2ndsneg(x₁, x₂)) = 5A + 0A · x₁ + -I · x₂

POL(2ndsneg(x₁, x₂)) = 5A + 0A · x₁ + -I · x₂

POL(a__2ndspos(x₁, x₂)) = 5A + 0A · x₁ + -I · x₂

POL(2ndspos(x₁, x₂)) = 5A + 0A · x₁ + -I · x₂

POL(a__square(x₁)) = -I + 0A · x₁

POL(square(x₁)) = -I + 0A · x₁

The following usable rules [FROCOS05] were oriented:

mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(nil) → nil
mark(negrecip(X)) → negrecip(mark(X))
mark(posrecip(X)) → posrecip(mark(X))
a__from(X) → from(X)
mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))
mark(rnil) → rnil
mark(cons2(X1, X2)) → cons2(X1, mark(X2))
a__plus(X1, X2) → plus(X1, X2)
a__pi(X) → pi(X)
a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)
a__2ndspos(X1, X2) → 2ndspos(X1, X2)
a__square(X) → square(X)
a__times(X1, X2) → times(X1, X2)
a__2ndspos(s(N), cons2(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))
a__2ndspos(s(N), cons(X, Z)) → a__2ndspos(s(mark(N)), cons2(X, mark(Z)))
a__2ndsneg(s(N), cons(X, Z)) → a__2ndsneg(s(mark(N)), cons2(X, mark(Z)))
a__2ndsneg(0, Z) → rnil
a__2ndspos(0, Z) → rnil
a__from(X) → cons(mark(X), from(s(X)))
a__times(0, Y) → 0
mark(from(X)) → a__from(mark(X))
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(s(X), Y) → s(a__plus(mark(X), mark(Y)))
mark(0) → 0
mark(s(X)) → s(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))
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))

(47) Obligation:

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

A__PLUS(0, Y) → MARK(Y)
MARK(plus(X1, X2)) → A__PLUS(mark(X1), mark(X2))
A__PLUS(s(X), Y) → A__PLUS(mark(X), mark(Y))
A__PLUS(s(X), Y) → MARK(Y)
MARK(plus(X1, X2)) → MARK(X2)
MARK(times(X1, X2)) → A__TIMES(mark(X1), mark(X2))
A__TIMES(s(X), Y) → A__PLUS(mark(Y), a__times(mark(X), mark(Y)))
A__TIMES(s(X), Y) → A__TIMES(mark(X), mark(Y))
MARK(s(X)) → MARK(X)
MARK(rcons(X1, X2)) → MARK(X2)

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.

(48) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

MARK(plus(X1, X2)) → A__PLUS(mark(X1), mark(X2))
MARK(plus(X1, X2)) → MARK(X2)
MARK(times(X1, X2)) → A__TIMES(mark(X1), mark(X2))
A__TIMES(s(X), Y) → A__PLUS(mark(Y), a__times(mark(X), mark(Y)))
A__TIMES(s(X), Y) → A__TIMES(mark(X), mark(Y))
MARK(s(X)) → MARK(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
A__PLUS(x1, x2) = A__PLUS(x2)
0 = 0
MARK(x1) = MARK(x1)
plus(x1, x2) = plus(x1, x2)
mark(x1) = x1
s(x1) = s(x1)
times(x1, x2) = times(x1, x2)
A__TIMES(x1, x2) = A__TIMES(x1, x2)
a__times(x1, x2) = a__times(x1, x2)
rcons(x1, x2) = x2
cons(x1, x2) = x1
nil = nil
negrecip(x1) = negrecip
posrecip(x1) = posrecip(x1)
a__from(x1) = x1
from(x1) = x1
rnil = rnil
cons2(x1, x2) = x1
a__plus(x1, x2) = a__plus(x1, x2)
a__pi(x1) = a__pi
pi(x1) = pi
a__2ndsneg(x1, x2) = a__2ndsneg
2ndsneg(x1, x2) = 2ndsneg
a__2ndspos(x1, x2) = a__2ndspos
2ndspos(x1, x2) = 2ndspos
a__square(x1) = a__square(x1)
square(x1) = square(x1)

Recursive path order with status [RPO].
Quasi-Precedence:

[posrecip₁, a_{_pi}, pi, a_{_2ndsneg}, 2ndsneg, a_{_2ndspos}, 2ndspos] > 0
[posrecip₁, a_{_pi}, pi, a_{_2ndsneg}, 2ndsneg, a_{_2ndspos}, 2ndspos] > s₁
[posrecip₁, a_{_pi}, pi, a_{_2ndsneg}, 2ndsneg, a_{_2ndspos}, 2ndspos] > negrecip
[posrecip₁, a_{_pi}, pi, a_{_2ndsneg}, 2ndsneg, a_{_2ndspos}, 2ndspos] > rnil
[a_{_square₁}, square₁] > [times₂, A_{_TIMES₂}, a_{_times₂}] > [A_{_PLUS₁}, MARK₁, plus₂, a_{_plus₂}] > s₁
[a_{_square₁}, square₁] > [times₂, A_{_TIMES₂}, a_{_times₂}] > 0

Status:

a_{_pi}: []
plus₂: multiset
rnil: multiset
A_{_TIMES₂}: multiset
posrecip₁: multiset
square₁: multiset
2ndsneg: []
0: multiset
2ndspos: []
negrecip: []
a_{_plus₂}: multiset
MARK₁: multiset
A_{_PLUS₁}: multiset
a_{_2ndspos}: []
times₂: multiset
pi: []
s₁: multiset
a_{_times₂}: multiset
nil: multiset
a_{_2ndsneg}: []
a_{_square₁}: multiset

The following usable rules [FROCOS05] were oriented:

mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(nil) → nil
mark(negrecip(X)) → negrecip(mark(X))
mark(posrecip(X)) → posrecip(mark(X))
a__from(X) → from(X)
mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))
mark(rnil) → rnil
mark(cons2(X1, X2)) → cons2(X1, mark(X2))
a__plus(X1, X2) → plus(X1, X2)
a__pi(X) → pi(X)
a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)
a__2ndspos(X1, X2) → 2ndspos(X1, X2)
a__square(X) → square(X)
a__times(X1, X2) → times(X1, X2)
a__2ndspos(s(N), cons2(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))
a__2ndspos(s(N), cons(X, Z)) → a__2ndspos(s(mark(N)), cons2(X, mark(Z)))
a__2ndsneg(s(N), cons(X, Z)) → a__2ndsneg(s(mark(N)), cons2(X, mark(Z)))
a__2ndsneg(0, Z) → rnil
a__2ndspos(0, Z) → rnil
a__from(X) → cons(mark(X), from(s(X)))
a__times(0, Y) → 0
mark(from(X)) → a__from(mark(X))
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(s(X), Y) → s(a__plus(mark(X), mark(Y)))
mark(0) → 0
mark(s(X)) → s(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))
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))

(49) Obligation:

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

A__PLUS(0, Y) → MARK(Y)
A__PLUS(s(X), Y) → A__PLUS(mark(X), mark(Y))
A__PLUS(s(X), Y) → MARK(Y)
MARK(rcons(X1, X2)) → MARK(X2)

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.

(50) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 2 less nodes.

(51) Complex Obligation (AND)

(52) Obligation:

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

MARK(rcons(X1, X2)) → MARK(X2)

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.

(53) UsableRulesProof (EQUIVALENT transformation)

We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.

(54) Obligation:

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

MARK(rcons(X1, X2)) → MARK(X2)

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

(55) QDPSizeChangeProof (EQUIVALENT transformation)

By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs:

MARK(rcons(X1, X2)) → MARK(X2)
The graph contains the following edges 1 > 1

(56) TRUE

(57) Obligation:

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

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

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.

(58) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

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

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
A__PLUS(x1, x2) = A__PLUS(x1)
s(x1) = s(x1)
mark(x1) = x1
cons(x1, x2) = x2
nil = nil
negrecip(x1) = x1
posrecip(x1) = x1
a__from(x1) = a__from
from(x1) = from
rcons(x1, x2) = rcons
rnil = rnil
cons2(x1, x2) = cons2
a__plus(x1, x2) = a__plus(x1, x2)
plus(x1, x2) = plus(x1, x2)
a__pi(x1) = a__pi(x1)
pi(x1) = pi(x1)
a__2ndsneg(x1, x2) = a__2ndsneg(x1)
2ndsneg(x1, x2) = 2ndsneg(x1)
a__2ndspos(x1, x2) = a__2ndspos(x1, x2)
2ndspos(x1, x2) = 2ndspos(x1, x2)
a__square(x1) = a__square(x1)
square(x1) = square(x1)
a__times(x1, x2) = a__times(x1, x2)
times(x1, x2) = times(x1, x2)
0 = 0

Recursive path order with status [RPO].
Quasi-Precedence:

nil > cons2
[a_{_pi₁}, pi₁] > [a_{_from}, from] > cons2
[a_{_pi₁}, pi₁] > [rcons, a_{_2ndspos₂}, 2ndspos₂] > [A_{_PLUS₁}, s₁] > cons2
[a_{_2ndsneg₁}, 2ndsneg₁] > [rcons, a_{_2ndspos₂}, 2ndspos₂] > [A_{_PLUS₁}, s₁] > cons2
[a_{_2ndsneg₁}, 2ndsneg₁] > rnil > cons2
[a_{_square₁}, square₁] > [a_{_times₂}, times₂] > [a_{_plus₂}, plus₂] > [A_{_PLUS₁}, s₁] > cons2
0 > rnil > cons2

Status:

plus₂: multiset
rnil: multiset
cons2: []
pi₁: multiset
square₁: multiset
0: multiset
a_{_2ndsneg₁}: multiset
from: []
a_{_plus₂}: multiset
a_{_pi₁}: multiset
a_{_2ndspos₂}: multiset
A_{_PLUS₁}: multiset
2ndspos₂: multiset
rcons: multiset
times₂: multiset
a_{_from}: []
s₁: multiset
a_{_times₂}: multiset
2ndsneg₁: multiset
nil: multiset
a_{_square₁}: multiset

The following usable rules [FROCOS05] were oriented:

mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(nil) → nil
mark(negrecip(X)) → negrecip(mark(X))
mark(posrecip(X)) → posrecip(mark(X))
a__from(X) → from(X)
mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))
mark(rnil) → rnil
mark(cons2(X1, X2)) → cons2(X1, mark(X2))
a__plus(X1, X2) → plus(X1, X2)
a__pi(X) → pi(X)
a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)
a__2ndspos(X1, X2) → 2ndspos(X1, X2)
a__square(X) → square(X)
a__times(X1, X2) → times(X1, X2)
a__2ndspos(s(N), cons2(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))
a__2ndspos(s(N), cons(X, Z)) → a__2ndspos(s(mark(N)), cons2(X, mark(Z)))
a__2ndsneg(s(N), cons(X, Z)) → a__2ndsneg(s(mark(N)), cons2(X, mark(Z)))
a__2ndsneg(0, Z) → rnil
a__2ndspos(0, Z) → rnil
a__from(X) → cons(mark(X), from(s(X)))
a__times(0, Y) → 0
mark(from(X)) → a__from(mark(X))
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(s(X), Y) → s(a__plus(mark(X), mark(Y)))
mark(0) → 0
mark(s(X)) → s(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))
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))

(59) Obligation:

Q DP problem:
P is empty.
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.

(60) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(61) TRUE

(62) Obligation:

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

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))) → A__2NDSPOS(mark(N), mark(Z))
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))) → A__2NDSNEG(mark(N), mark(Z))

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.

(63) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

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

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
A__2NDSNEG(x1, x2) = A__2NDSNEG(x1)
s(x1) = s(x1)
cons(x1, x2) = cons(x1)
mark(x1) = x1
cons2(x1, x2) = cons2
A__2NDSPOS(x1, x2) = A__2NDSPOS(x1)
nil = nil
negrecip(x1) = negrecip
posrecip(x1) = posrecip
a__from(x1) = a__from(x1)
from(x1) = from(x1)
rcons(x1, x2) = x1
rnil = rnil
a__plus(x1, x2) = a__plus(x1, x2)
plus(x1, x2) = plus(x1, x2)
a__pi(x1) = a__pi(x1)
pi(x1) = pi(x1)
a__2ndsneg(x1, x2) = a__2ndsneg(x1, x2)
2ndsneg(x1, x2) = 2ndsneg(x1, x2)
a__2ndspos(x1, x2) = x2
2ndspos(x1, x2) = x2
a__square(x1) = a__square(x1)
square(x1) = square(x1)
a__times(x1, x2) = a__times(x1, x2)
times(x1, x2) = times(x1, x2)
0 = 0

Recursive path order with status [RPO].
Quasi-Precedence:

[A_{_2NDSNEG₁}, A_{_2NDSPOS₁}] > [cons2, posrecip] > [s₁, rnil, 0]
[cons₁, a_{_from₁}, from₁, a_{_pi₁}, pi₁] > [cons2, posrecip] > [s₁, rnil, 0]
nil > [s₁, rnil, 0]
[a_{_2ndsneg₂}, 2ndsneg₂] > [cons2, posrecip] > [s₁, rnil, 0]
[a_{_2ndsneg₂}, 2ndsneg₂] > negrecip > [s₁, rnil, 0]
[a_{_square₁}, square₁] > [a_{_times₂}, times₂] > [a_{_plus₂}, plus₂] > [s₁, rnil, 0]

Status:

from₁: [1]
plus₂: multiset
cons₁: [1]
rnil: multiset
cons2: []
a_{_from₁}: [1]
pi₁: [1]
square₁: multiset
0: multiset
2ndsneg₂: multiset
negrecip: multiset
a_{_plus₂}: multiset
a_{_pi₁}: [1]
A_{_2NDSPOS₁}: multiset
times₂: multiset
s₁: multiset
a_{_times₂}: multiset
posrecip: []
a_{_2ndsneg₂}: multiset
A_{_2NDSNEG₁}: multiset
nil: multiset
a_{_square₁}: multiset

The following usable rules [FROCOS05] were oriented:

mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(nil) → nil
mark(negrecip(X)) → negrecip(mark(X))
mark(posrecip(X)) → posrecip(mark(X))
a__from(X) → from(X)
mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))
mark(rnil) → rnil
mark(cons2(X1, X2)) → cons2(X1, mark(X2))
a__plus(X1, X2) → plus(X1, X2)
a__pi(X) → pi(X)
a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)
a__2ndspos(X1, X2) → 2ndspos(X1, X2)
a__square(X) → square(X)
a__times(X1, X2) → times(X1, X2)
a__2ndspos(s(N), cons2(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))
a__2ndspos(s(N), cons(X, Z)) → a__2ndspos(s(mark(N)), cons2(X, mark(Z)))
a__2ndsneg(s(N), cons(X, Z)) → a__2ndsneg(s(mark(N)), cons2(X, mark(Z)))
a__2ndsneg(0, Z) → rnil
a__2ndspos(0, Z) → rnil
a__from(X) → cons(mark(X), from(s(X)))
a__times(0, Y) → 0
mark(from(X)) → a__from(mark(X))
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(s(X), Y) → s(a__plus(mark(X), mark(Y)))
mark(0) → 0
mark(s(X)) → s(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))
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))

(64) Obligation:

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

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

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.

(65) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 2 less nodes.

(0) Obligation:

(1) DependencyPairsProof (EQUIVALENT transformation)

(2) Obligation:

(3) QDPOrderProof (EQUIVALENT transformation)

(4) Obligation:

(5) QDPOrderProof (EQUIVALENT transformation)

(6) Obligation:

(7) QDPOrderProof (EQUIVALENT transformation)

(8) Obligation:

(9) QDPOrderProof (EQUIVALENT transformation)

(10) Obligation:

(11) DependencyGraphProof (EQUIVALENT transformation)

(12) Obligation:

(13) QDPOrderProof (EQUIVALENT transformation)

(14) Obligation:

(15) QDPOrderProof (EQUIVALENT transformation)

(16) Obligation:

(17) DependencyGraphProof (EQUIVALENT transformation)

(18) Obligation:

(19) QDPOrderProof (EQUIVALENT transformation)

(20) Obligation:

(21) QDPOrderProof (EQUIVALENT transformation)

(22) Obligation:

(23) DependencyGraphProof (EQUIVALENT transformation)

(24) Obligation:

(25) QDPOrderProof (EQUIVALENT transformation)

(26) Obligation:

(27) QDPOrderProof (EQUIVALENT transformation)

(28) Obligation:

(29) QDPOrderProof (EQUIVALENT transformation)

(30) Obligation:

(31) QDPOrderProof (EQUIVALENT transformation)

(32) Obligation:

(33) QDPOrderProof (EQUIVALENT transformation)

(34) Obligation:

(35) QDPOrderProof (EQUIVALENT transformation)

(36) Obligation:

(37) QDPOrderProof (EQUIVALENT transformation)

(38) Obligation:

(39) QDPOrderProof (EQUIVALENT transformation)

(40) Obligation:

(41) QDPOrderProof (EQUIVALENT transformation)

(42) Obligation:

(43) DependencyGraphProof (EQUIVALENT transformation)

(44) Complex Obligation (AND)

(45) Obligation:

(46) QDPOrderProof (EQUIVALENT transformation)

(47) Obligation:

(48) QDPOrderProof (EQUIVALENT transformation)

(49) Obligation:

(50) DependencyGraphProof (EQUIVALENT transformation)

(51) Complex Obligation (AND)

(52) Obligation:

(53) UsableRulesProof (EQUIVALENT transformation)

(54) Obligation:

(55) QDPSizeChangeProof (EQUIVALENT transformation)

(56) TRUE

(57) Obligation:

(58) QDPOrderProof (EQUIVALENT transformation)

(59) Obligation:

(60) PisEmptyProof (EQUIVALENT transformation)

(61) TRUE

(62) Obligation:

(63) QDPOrderProof (EQUIVALENT transformation)

(64) Obligation:

(65) DependencyGraphProof (EQUIVALENT transformation)

(66) TRUE