Termination Proof using AProVE

Term Rewriting System R:
[N, X, Y, X1, X2, Z]
a_{_terms}(N) -> cons(recip(a_{_sqr}(mark(N))), terms(s(N)))
a_{_terms}(X) -> terms(X)
a_{_sqr}(0) -> 0
a_{_sqr}(s(X)) -> s(a_{_add}(a_{_sqr}(mark(X)), a_{_dbl}(mark(X))))
a_{_sqr}(X) -> sqr(X)
a_{_dbl}(0) -> 0
a_{_dbl}(s(X)) -> s(s(a_{_dbl}(mark(X))))
a_{_dbl}(X) -> dbl(X)
a_{_add}(0, X) -> mark(X)
a_{_add}(s(X), Y) -> s(a_{_add}(mark(X), mark(Y)))
a_{_add}(X1, X2) -> add(X1, X2)
a_{_first}(0, X) -> nil
a_{_first}(s(X), cons(Y, Z)) -> cons(mark(Y), first(X, Z))
a_{_first}(X1, X2) -> first(X1, X2)
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)
a_{_half}(X) -> half(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

Innermost Termination of R to be shown.

     ↳Dependency Pair Analysis

R contains the following Dependency Pairs:

A_{_TERMS}(N) -> A_{_SQR}(mark(N))
A_{_TERMS}(N) -> MARK(N)
A_{_SQR}(s(X)) -> A_{_ADD}(a_{_sqr}(mark(X)), a_{_dbl}(mark(X)))
A_{_SQR}(s(X)) -> A_{_SQR}(mark(X))
A_{_SQR}(s(X)) -> MARK(X)
A_{_SQR}(s(X)) -> A_{_DBL}(mark(X))
A_{_DBL}(s(X)) -> A_{_DBL}(mark(X))
A_{_DBL}(s(X)) -> MARK(X)
A_{_ADD}(0, X) -> MARK(X)
A_{_ADD}(s(X), Y) -> A_{_ADD}(mark(X), mark(Y))
A_{_ADD}(s(X), Y) -> MARK(X)
A_{_ADD}(s(X), Y) -> MARK(Y)
A_{_FIRST}(s(X), cons(Y, Z)) -> MARK(Y)
A_{_HALF}(s(s(X))) -> A_{_HALF}(mark(X))
A_{_HALF}(s(s(X))) -> MARK(X)
A_{_HALF}(dbl(X)) -> MARK(X)
MARK(terms(X)) -> A_{_TERMS}(mark(X))
MARK(terms(X)) -> MARK(X)
MARK(sqr(X)) -> A_{_SQR}(mark(X))
MARK(sqr(X)) -> MARK(X)
MARK(add(X1, X2)) -> A_{_ADD}(mark(X1), mark(X2))
MARK(add(X1, X2)) -> MARK(X1)
MARK(add(X1, X2)) -> MARK(X2)
MARK(dbl(X)) -> A_{_DBL}(mark(X))
MARK(dbl(X)) -> MARK(X)
MARK(first(X1, X2)) -> A_{_FIRST}(mark(X1), mark(X2))
MARK(first(X1, X2)) -> MARK(X1)
MARK(first(X1, X2)) -> MARK(X2)
MARK(half(X)) -> A_{_HALF}(mark(X))
MARK(half(X)) -> MARK(X)
MARK(cons(X1, X2)) -> MARK(X1)
MARK(recip(X)) -> MARK(X)
MARK(s(X)) -> MARK(X)

Furthermore, R contains one SCC.

     ↳DPs

       →DP Problem 1

         ↳Remaining Obligation(s)

The following remains to be proven:
Dependency Pairs:

A_{_SQR}(s(X)) -> A_{_DBL}(mark(X))
A_{_ADD}(s(X), Y) -> MARK(Y)
A_{_HALF}(dbl(X)) -> MARK(X)
MARK(s(X)) -> MARK(X)
MARK(recip(X)) -> MARK(X)
MARK(cons(X1, X2)) -> MARK(X1)
MARK(half(X)) -> MARK(X)
A_{_HALF}(s(s(X))) -> MARK(X)
A_{_HALF}(s(s(X))) -> A_{_HALF}(mark(X))
MARK(half(X)) -> A_{_HALF}(mark(X))
MARK(first(X1, X2)) -> MARK(X2)
MARK(first(X1, X2)) -> MARK(X1)
A_{_FIRST}(s(X), cons(Y, Z)) -> MARK(Y)
MARK(first(X1, X2)) -> A_{_FIRST}(mark(X1), mark(X2))
MARK(dbl(X)) -> MARK(X)
A_{_DBL}(s(X)) -> MARK(X)
A_{_DBL}(s(X)) -> A_{_DBL}(mark(X))
MARK(dbl(X)) -> A_{_DBL}(mark(X))
MARK(add(X1, X2)) -> MARK(X2)
MARK(add(X1, X2)) -> MARK(X1)
A_{_ADD}(s(X), Y) -> MARK(X)
A_{_ADD}(s(X), Y) -> A_{_ADD}(mark(X), mark(Y))
MARK(add(X1, X2)) -> A_{_ADD}(mark(X1), mark(X2))
MARK(sqr(X)) -> MARK(X)
A_{_SQR}(s(X)) -> MARK(X)
A_{_SQR}(s(X)) -> A_{_SQR}(mark(X))
MARK(sqr(X)) -> A_{_SQR}(mark(X))
MARK(terms(X)) -> MARK(X)
A_{_TERMS}(N) -> MARK(N)
MARK(terms(X)) -> A_{_TERMS}(mark(X))
A_{_ADD}(0, X) -> MARK(X)
A_{_SQR}(s(X)) -> A_{_ADD}(a_{_sqr}(mark(X)), a_{_dbl}(mark(X)))
A_{_TERMS}(N) -> A_{_SQR}(mark(N))

Rules:

a_{_terms}(N) -> cons(recip(a_{_sqr}(mark(N))), terms(s(N)))
a_{_terms}(X) -> terms(X)
a_{_sqr}(0) -> 0
a_{_sqr}(s(X)) -> s(a_{_add}(a_{_sqr}(mark(X)), a_{_dbl}(mark(X))))
a_{_sqr}(X) -> sqr(X)
a_{_dbl}(0) -> 0
a_{_dbl}(s(X)) -> s(s(a_{_dbl}(mark(X))))
a_{_dbl}(X) -> dbl(X)
a_{_add}(0, X) -> mark(X)
a_{_add}(s(X), Y) -> s(a_{_add}(mark(X), mark(Y)))
a_{_add}(X1, X2) -> add(X1, X2)
a_{_first}(0, X) -> nil
a_{_first}(s(X), cons(Y, Z)) -> cons(mark(Y), first(X, Z))
a_{_first}(X1, X2) -> first(X1, X2)
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)
a_{_half}(X) -> half(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

Strategy:

innermost

Innermost Termination of R could not be shown.
Duration:
0:45 minutes