Termination w.r.t. Q proof of TRS/TRCSRExSec11_1_Luc02a

Termination w.r.t. Q of the following Term Rewriting System could be proven:

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

a__terms(N) → cons(recip(a__sqr(mark(N))), terms(s(N)))
a__sqr(0) → 0
a__sqr(s(X)) → s(a__add(a__sqr(mark(X)), a__dbl(mark(X))))
a__dbl(0) → 0
a__dbl(s(X)) → s(s(a__dbl(mark(X))))
a__add(0, X) → mark(X)
a__add(s(X), Y) → s(a__add(mark(X), mark(Y)))
a__first(0, X) → nil
a__first(s(X), cons(Y, Z)) → cons(mark(Y), first(X, Z))
a__half(0) → 0
a__half(s(0)) → 0
a__half(s(s(X))) → s(a__half(mark(X)))
a__half(dbl(X)) → mark(X)
mark(terms(X)) → a__terms(mark(X))
mark(sqr(X)) → a__sqr(mark(X))
mark(add(X1, X2)) → a__add(mark(X1), mark(X2))
mark(dbl(X)) → a__dbl(mark(X))
mark(first(X1, X2)) → a__first(mark(X1), mark(X2))
mark(half(X)) → a__half(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(recip(X)) → recip(mark(X))
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(nil) → nil
a__terms(X) → terms(X)
a__sqr(X) → sqr(X)
a__add(X1, X2) → add(X1, X2)
a__dbl(X) → dbl(X)
a__first(X1, X2) → first(X1, X2)
a__half(X) → half(X)

Q is empty.

↳ QTRS

  ↳ DependencyPairsProof

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

a__terms(N) → cons(recip(a__sqr(mark(N))), terms(s(N)))
a__sqr(0) → 0
a__sqr(s(X)) → s(a__add(a__sqr(mark(X)), a__dbl(mark(X))))
a__dbl(0) → 0
a__dbl(s(X)) → s(s(a__dbl(mark(X))))
a__add(0, X) → mark(X)
a__add(s(X), Y) → s(a__add(mark(X), mark(Y)))
a__first(0, X) → nil
a__first(s(X), cons(Y, Z)) → cons(mark(Y), first(X, Z))
a__half(0) → 0
a__half(s(0)) → 0
a__half(s(s(X))) → s(a__half(mark(X)))
a__half(dbl(X)) → mark(X)
mark(terms(X)) → a__terms(mark(X))
mark(sqr(X)) → a__sqr(mark(X))
mark(add(X1, X2)) → a__add(mark(X1), mark(X2))
mark(dbl(X)) → a__dbl(mark(X))
mark(first(X1, X2)) → a__first(mark(X1), mark(X2))
mark(half(X)) → a__half(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(recip(X)) → recip(mark(X))
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(nil) → nil
a__terms(X) → terms(X)
a__sqr(X) → sqr(X)
a__add(X1, X2) → add(X1, X2)
a__dbl(X) → dbl(X)
a__first(X1, X2) → first(X1, X2)
a__half(X) → half(X)

Q is empty.

Using Dependency Pairs [1,13] we result in the following initial DP problem:
Q DP problem:
The TRS P consists of the following rules:

MARK(half(X)) → A__HALF(mark(X))
MARK(first(X1, X2)) → A__FIRST(mark(X1), mark(X2))
A__SQR(s(X)) → MARK(X)
MARK(sqr(X)) → A__SQR(mark(X))
MARK(dbl(X)) → A__DBL(mark(X))
MARK(add(X1, X2)) → MARK(X2)
MARK(first(X1, X2)) → MARK(X2)
MARK(recip(X)) → MARK(X)
A__FIRST(s(X), cons(Y, Z)) → MARK(Y)
A__SQR(s(X)) → A__SQR(mark(X))
MARK(terms(X)) → MARK(X)
MARK(first(X1, X2)) → MARK(X1)
A__ADD(0, X) → MARK(X)
MARK(terms(X)) → A__TERMS(mark(X))
A__ADD(s(X), Y) → A__ADD(mark(X), mark(Y))
A__DBL(s(X)) → MARK(X)
MARK(add(X1, X2)) → A__ADD(mark(X1), mark(X2))
MARK(cons(X1, X2)) → MARK(X1)
MARK(dbl(X)) → MARK(X)
A__HALF(s(s(X))) → A__HALF(mark(X))
MARK(sqr(X)) → MARK(X)
A__SQR(s(X)) → A__ADD(a__sqr(mark(X)), a__dbl(mark(X)))
MARK(half(X)) → MARK(X)
A__ADD(s(X), Y) → MARK(X)
A__HALF(dbl(X)) → MARK(X)
A__TERMS(N) → MARK(N)
MARK(s(X)) → MARK(X)
MARK(add(X1, X2)) → MARK(X1)
A__HALF(s(s(X))) → MARK(X)
A__ADD(s(X), Y) → MARK(Y)
A__DBL(s(X)) → A__DBL(mark(X))
A__TERMS(N) → A__SQR(mark(N))
A__SQR(s(X)) → A__DBL(mark(X))

The TRS R consists of the following rules:

a__terms(N) → cons(recip(a__sqr(mark(N))), terms(s(N)))
a__sqr(0) → 0
a__sqr(s(X)) → s(a__add(a__sqr(mark(X)), a__dbl(mark(X))))
a__dbl(0) → 0
a__dbl(s(X)) → s(s(a__dbl(mark(X))))
a__add(0, X) → mark(X)
a__add(s(X), Y) → s(a__add(mark(X), mark(Y)))
a__first(0, X) → nil
a__first(s(X), cons(Y, Z)) → cons(mark(Y), first(X, Z))
a__half(0) → 0
a__half(s(0)) → 0
a__half(s(s(X))) → s(a__half(mark(X)))
a__half(dbl(X)) → mark(X)
mark(terms(X)) → a__terms(mark(X))
mark(sqr(X)) → a__sqr(mark(X))
mark(add(X1, X2)) → a__add(mark(X1), mark(X2))
mark(dbl(X)) → a__dbl(mark(X))
mark(first(X1, X2)) → a__first(mark(X1), mark(X2))
mark(half(X)) → a__half(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(recip(X)) → recip(mark(X))
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(nil) → nil
a__terms(X) → terms(X)
a__sqr(X) → sqr(X)
a__add(X1, X2) → add(X1, X2)
a__dbl(X) → dbl(X)
a__first(X1, X2) → first(X1, X2)
a__half(X) → half(X)

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ EdgeDeletionProof

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

MARK(half(X)) → A__HALF(mark(X))
MARK(first(X1, X2)) → A__FIRST(mark(X1), mark(X2))
A__SQR(s(X)) → MARK(X)
MARK(sqr(X)) → A__SQR(mark(X))
MARK(dbl(X)) → A__DBL(mark(X))
MARK(add(X1, X2)) → MARK(X2)
MARK(first(X1, X2)) → MARK(X2)
MARK(recip(X)) → MARK(X)
A__FIRST(s(X), cons(Y, Z)) → MARK(Y)
A__SQR(s(X)) → A__SQR(mark(X))
MARK(terms(X)) → MARK(X)
MARK(first(X1, X2)) → MARK(X1)
A__ADD(0, X) → MARK(X)
MARK(terms(X)) → A__TERMS(mark(X))
A__ADD(s(X), Y) → A__ADD(mark(X), mark(Y))
A__DBL(s(X)) → MARK(X)
MARK(add(X1, X2)) → A__ADD(mark(X1), mark(X2))
MARK(cons(X1, X2)) → MARK(X1)
MARK(dbl(X)) → MARK(X)
A__HALF(s(s(X))) → A__HALF(mark(X))
MARK(sqr(X)) → MARK(X)
A__SQR(s(X)) → A__ADD(a__sqr(mark(X)), a__dbl(mark(X)))
MARK(half(X)) → MARK(X)
A__ADD(s(X), Y) → MARK(X)
A__HALF(dbl(X)) → MARK(X)
A__TERMS(N) → MARK(N)
MARK(s(X)) → MARK(X)
MARK(add(X1, X2)) → MARK(X1)
A__HALF(s(s(X))) → MARK(X)
A__ADD(s(X), Y) → MARK(Y)
A__DBL(s(X)) → A__DBL(mark(X))
A__TERMS(N) → A__SQR(mark(N))
A__SQR(s(X)) → A__DBL(mark(X))

The TRS R consists of the following rules:

a__terms(N) → cons(recip(a__sqr(mark(N))), terms(s(N)))
a__sqr(0) → 0
a__sqr(s(X)) → s(a__add(a__sqr(mark(X)), a__dbl(mark(X))))
a__dbl(0) → 0
a__dbl(s(X)) → s(s(a__dbl(mark(X))))
a__add(0, X) → mark(X)
a__add(s(X), Y) → s(a__add(mark(X), mark(Y)))
a__first(0, X) → nil
a__first(s(X), cons(Y, Z)) → cons(mark(Y), first(X, Z))
a__half(0) → 0
a__half(s(0)) → 0
a__half(s(s(X))) → s(a__half(mark(X)))
a__half(dbl(X)) → mark(X)
mark(terms(X)) → a__terms(mark(X))
mark(sqr(X)) → a__sqr(mark(X))
mark(add(X1, X2)) → a__add(mark(X1), mark(X2))
mark(dbl(X)) → a__dbl(mark(X))
mark(first(X1, X2)) → a__first(mark(X1), mark(X2))
mark(half(X)) → a__half(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(recip(X)) → recip(mark(X))
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(nil) → nil
a__terms(X) → terms(X)
a__sqr(X) → sqr(X)
a__add(X1, X2) → add(X1, X2)
a__dbl(X) → dbl(X)
a__first(X1, X2) → first(X1, X2)
a__half(X) → half(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We deleted some edges using various graph approximations

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ EdgeDeletionProof

        ↳ QDP

          ↳ QDPOrderProof

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

A__SQR(s(X)) → MARK(X)
MARK(first(X1, X2)) → A__FIRST(mark(X1), mark(X2))
MARK(half(X)) → A__HALF(mark(X))
MARK(sqr(X)) → A__SQR(mark(X))
MARK(dbl(X)) → A__DBL(mark(X))
MARK(add(X1, X2)) → MARK(X2)
MARK(first(X1, X2)) → MARK(X2)
MARK(recip(X)) → MARK(X)
MARK(terms(X)) → MARK(X)
A__SQR(s(X)) → A__SQR(mark(X))
A__FIRST(s(X), cons(Y, Z)) → MARK(Y)
MARK(first(X1, X2)) → MARK(X1)
A__ADD(0, X) → MARK(X)
MARK(terms(X)) → A__TERMS(mark(X))
A__ADD(s(X), Y) → A__ADD(mark(X), mark(Y))
A__DBL(s(X)) → MARK(X)
MARK(add(X1, X2)) → A__ADD(mark(X1), mark(X2))
MARK(cons(X1, X2)) → MARK(X1)
MARK(dbl(X)) → MARK(X)
A__HALF(s(s(X))) → A__HALF(mark(X))
MARK(sqr(X)) → MARK(X)
A__SQR(s(X)) → A__ADD(a__sqr(mark(X)), a__dbl(mark(X)))
MARK(half(X)) → MARK(X)
A__ADD(s(X), Y) → MARK(X)
A__HALF(dbl(X)) → MARK(X)
A__TERMS(N) → MARK(N)
MARK(add(X1, X2)) → MARK(X1)
MARK(s(X)) → MARK(X)
A__HALF(s(s(X))) → MARK(X)
A__ADD(s(X), Y) → MARK(Y)
A__TERMS(N) → A__SQR(mark(N))
A__DBL(s(X)) → A__DBL(mark(X))
A__SQR(s(X)) → A__DBL(mark(X))

The TRS R consists of the following rules:

a__terms(N) → cons(recip(a__sqr(mark(N))), terms(s(N)))
a__sqr(0) → 0
a__sqr(s(X)) → s(a__add(a__sqr(mark(X)), a__dbl(mark(X))))
a__dbl(0) → 0
a__dbl(s(X)) → s(s(a__dbl(mark(X))))
a__add(0, X) → mark(X)
a__add(s(X), Y) → s(a__add(mark(X), mark(Y)))
a__first(0, X) → nil
a__first(s(X), cons(Y, Z)) → cons(mark(Y), first(X, Z))
a__half(0) → 0
a__half(s(0)) → 0
a__half(s(s(X))) → s(a__half(mark(X)))
a__half(dbl(X)) → mark(X)
mark(terms(X)) → a__terms(mark(X))
mark(sqr(X)) → a__sqr(mark(X))
mark(add(X1, X2)) → a__add(mark(X1), mark(X2))
mark(dbl(X)) → a__dbl(mark(X))
mark(first(X1, X2)) → a__first(mark(X1), mark(X2))
mark(half(X)) → a__half(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(recip(X)) → recip(mark(X))
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(nil) → nil
a__terms(X) → terms(X)
a__sqr(X) → sqr(X)
a__add(X1, X2) → add(X1, X2)
a__dbl(X) → dbl(X)
a__first(X1, X2) → first(X1, X2)
a__half(X) → half(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].

The following pairs can be oriented strictly and are deleted.

A__SQR(s(X)) → MARK(X)
MARK(first(X1, X2)) → A__FIRST(mark(X1), mark(X2))
MARK(half(X)) → A__HALF(mark(X))
MARK(dbl(X)) → A__DBL(mark(X))
MARK(add(X1, X2)) → MARK(X2)
MARK(first(X1, X2)) → MARK(X2)
MARK(recip(X)) → MARK(X)
MARK(terms(X)) → MARK(X)
A__SQR(s(X)) → A__SQR(mark(X))
A__FIRST(s(X), cons(Y, Z)) → MARK(Y)
MARK(first(X1, X2)) → MARK(X1)
A__ADD(0, X) → MARK(X)
MARK(terms(X)) → A__TERMS(mark(X))
A__ADD(s(X), Y) → A__ADD(mark(X), mark(Y))
A__DBL(s(X)) → MARK(X)
MARK(cons(X1, X2)) → MARK(X1)
MARK(dbl(X)) → MARK(X)
A__HALF(s(s(X))) → A__HALF(mark(X))
MARK(sqr(X)) → MARK(X)
A__SQR(s(X)) → A__ADD(a__sqr(mark(X)), a__dbl(mark(X)))
MARK(half(X)) → MARK(X)
A__ADD(s(X), Y) → MARK(X)
A__HALF(dbl(X)) → MARK(X)
A__TERMS(N) → MARK(N)
MARK(add(X1, X2)) → MARK(X1)
MARK(s(X)) → MARK(X)
A__HALF(s(s(X))) → MARK(X)
A__ADD(s(X), Y) → MARK(Y)
A__DBL(s(X)) → A__DBL(mark(X))
A__SQR(s(X)) → A__DBL(mark(X))

The remaining pairs can at least be oriented weakly.

MARK(sqr(X)) → A__SQR(mark(X))
MARK(add(X1, X2)) → A__ADD(mark(X1), mark(X2))
A__TERMS(N) → A__SQR(mark(N))

Used ordering: Combined order from the following AFS and order.
A__SQR(x1) = A__SQR(x1)
s(x1) = s(x1)
MARK(x1) = x1
first(x1, x2) = first(x1, x2)
A__FIRST(x1, x2) = A__FIRST(x1, x2)
mark(x1) = x1
half(x1) = half(x1)
A__HALF(x1) = A__HALF(x1)
sqr(x1) = sqr(x1)
dbl(x1) = dbl(x1)
A__DBL(x1) = A__DBL(x1)
add(x1, x2) = add(x1, x2)
recip(x1) = recip(x1)
terms(x1) = terms(x1)
cons(x1, x2) = cons(x1)
A__ADD(x1, x2) = A__ADD(x1, x2)
0 = 0
A__TERMS(x1) = A__TERMS(x1)
a__sqr(x1) = a__sqr(x1)
a__dbl(x1) = a__dbl(x1)
nil = nil
a__add(x1, x2) = a__add(x1, x2)
a__half(x1) = a__half(x1)
a__first(x1, x2) = a__first(x1, x2)
a__terms(x1) = a__terms(x1)

Lexicographic Path Order [19].
Precedence:

[first₂, a_{_{first_2]}} > A_{_FIRST₂} > [recip₁, nil]
[first₂, a_{_{first_2]}} > cons₁ > [recip₁, nil]
[half₁, a_{_{half_1]}} > s₁ > A_{_DBL₁} > [recip₁, nil]
[half₁, a_{_{half_1]}} > s₁ > cons₁ > [recip₁, nil]
[half₁, a_{_{half_1]}} > s₁ > 0 > [recip₁, nil]
[half₁, a_{_{half_1]}} > A_{_HALF₁} > [recip₁, nil]
[terms₁, a_{_{terms_1]}} > [A_{_SQR₁}, sqr₁, A_{_TERMS₁}, a_{_{sqr_1]}} > [dbl₁, a_{_{dbl_1]}} > s₁ > A_{_DBL₁} > [recip₁, nil]
[terms₁, a_{_{terms_1]}} > [A_{_SQR₁}, sqr₁, A_{_TERMS₁}, a_{_{sqr_1]}} > [dbl₁, a_{_{dbl_1]}} > s₁ > cons₁ > [recip₁, nil]
[terms₁, a_{_{terms_1]}} > [A_{_SQR₁}, sqr₁, A_{_TERMS₁}, a_{_{sqr_1]}} > [dbl₁, a_{_{dbl_1]}} > s₁ > 0 > [recip₁, nil]
[terms₁, a_{_{terms_1]}} > [A_{_SQR₁}, sqr₁, A_{_TERMS₁}, a_{_{sqr_1]}} > [add₂, A_{_ADD₂}, a_{_{add_2]}} > s₁ > A_{_DBL₁} > [recip₁, nil]
[terms₁, a_{_{terms_1]}} > [A_{_SQR₁}, sqr₁, A_{_TERMS₁}, a_{_{sqr_1]}} > [add₂, A_{_ADD₂}, a_{_{add_2]}} > s₁ > cons₁ > [recip₁, nil]
[terms₁, a_{_{terms_1]}} > [A_{_SQR₁}, sqr₁, A_{_TERMS₁}, a_{_{sqr_1]}} > [add₂, A_{_ADD₂}, a_{_{add_2]}} > s₁ > 0 > [recip₁, nil]

The following usable rules [14] were oriented:

mark(nil) → nil
a__add(X1, X2) → add(X1, X2)
mark(dbl(X)) → a__dbl(mark(X))
a__half(0) → 0
a__dbl(s(X)) → s(s(a__dbl(mark(X))))
a__half(s(0)) → 0
a__first(0, X) → nil
a__sqr(s(X)) → s(a__add(a__sqr(mark(X)), a__dbl(mark(X))))
a__add(0, X) → mark(X)
mark(add(X1, X2)) → a__add(mark(X1), mark(X2))
mark(half(X)) → a__half(mark(X))
a__half(dbl(X)) → mark(X)
a__terms(X) → terms(X)
mark(first(X1, X2)) → a__first(mark(X1), mark(X2))
a__sqr(X) → sqr(X)
a__terms(N) → cons(recip(a__sqr(mark(N))), terms(s(N)))
mark(terms(X)) → a__terms(mark(X))
a__sqr(0) → 0
mark(cons(X1, X2)) → cons(mark(X1), X2)
a__dbl(0) → 0
mark(0) → 0
a__dbl(X) → dbl(X)
mark(sqr(X)) → a__sqr(mark(X))
mark(s(X)) → s(mark(X))
a__half(s(s(X))) → s(a__half(mark(X)))
a__first(X1, X2) → first(X1, X2)
a__add(s(X), Y) → s(a__add(mark(X), mark(Y)))
a__first(s(X), cons(Y, Z)) → cons(mark(Y), first(X, Z))
mark(recip(X)) → recip(mark(X))
a__half(X) → half(X)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ EdgeDeletionProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

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

MARK(sqr(X)) → A__SQR(mark(X))
A__TERMS(N) → A__SQR(mark(N))
MARK(add(X1, X2)) → A__ADD(mark(X1), mark(X2))

The TRS R consists of the following rules:

a__terms(N) → cons(recip(a__sqr(mark(N))), terms(s(N)))
a__sqr(0) → 0
a__sqr(s(X)) → s(a__add(a__sqr(mark(X)), a__dbl(mark(X))))
a__dbl(0) → 0
a__dbl(s(X)) → s(s(a__dbl(mark(X))))
a__add(0, X) → mark(X)
a__add(s(X), Y) → s(a__add(mark(X), mark(Y)))
a__first(0, X) → nil
a__first(s(X), cons(Y, Z)) → cons(mark(Y), first(X, Z))
a__half(0) → 0
a__half(s(0)) → 0
a__half(s(s(X))) → s(a__half(mark(X)))
a__half(dbl(X)) → mark(X)
mark(terms(X)) → a__terms(mark(X))
mark(sqr(X)) → a__sqr(mark(X))
mark(add(X1, X2)) → a__add(mark(X1), mark(X2))
mark(dbl(X)) → a__dbl(mark(X))
mark(first(X1, X2)) → a__first(mark(X1), mark(X2))
mark(half(X)) → a__half(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(recip(X)) → recip(mark(X))
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(nil) → nil
a__terms(X) → terms(X)
a__sqr(X) → sqr(X)
a__add(X1, X2) → add(X1, X2)
a__dbl(X) → dbl(X)
a__first(X1, X2) → first(X1, X2)
a__half(X) → half(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [13,14,18] contains 0 SCCs with 3 less nodes.