Termination Proof using AProVE

Term Rewriting System R:
[X, Z, N, Y, X1, X2]
a_{_from}(X) -> cons(mark(X), from(s(X)))
a_{_from}(X) -> from(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_{_2ndspos}(X1, X2) -> 2ndspos(X1, X2)
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_{_2ndsneg}(X1, X2) -> 2ndsneg(X1, X2)
a_{_pi}(X) -> a_{_2ndspos}(mark(X), a_{_from}(0))
a_{_pi}(X) -> pi(X)
a_{_plus}(0, Y) -> mark(Y)
a_{_plus}(s(X), Y) -> s(a_{_plus}(mark(X), mark(Y)))
a_{_plus}(X1, X2) -> plus(X1, X2)
a_{_times}(0, Y) -> 0
a_{_times}(s(X), Y) -> a_{_plus}(mark(Y), a_{_times}(mark(X), mark(Y)))
a_{_times}(X1, X2) -> times(X1, X2)
a_{_square}(X) -> a_{_times}(mark(X), mark(X))
a_{_square}(X) -> square(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))

Termination of R to be shown.

     ↳Dependency Pair Analysis

R contains the following Dependency Pairs:

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)

Furthermore, R contains one SCC.

     ↳DPs

       →DP Problem 1

         ↳Remaining Obligation(s)

The following remains to be proven:
Dependency Pairs:

A_{_PI}(X) -> A_{_FROM}(0)
A_{_PI}(X) -> MARK(X)
A_{_2NDSNEG}(s(N), cons2(X, cons(Y, Z))) -> MARK(Z)
A_{_2NDSNEG}(s(N), cons2(X, cons(Y, Z))) -> MARK(N)
A_{_2NDSPOS}(s(N), cons2(X, cons(Y, Z))) -> MARK(Z)
A_{_2NDSPOS}(s(N), cons2(X, cons(Y, Z))) -> MARK(N)
A_{_2NDSNEG}(s(N), cons2(X, cons(Y, Z))) -> A_{_2NDSPOS}(mark(N), mark(Z))
A_{_2NDSNEG}(s(N), cons(X, Z)) -> MARK(Z)
A_{_2NDSNEG}(s(N), cons(X, Z)) -> MARK(N)
A_{_2NDSPOS}(s(N), cons2(X, cons(Y, Z))) -> A_{_2NDSNEG}(mark(N), mark(Z))
A_{_2NDSPOS}(s(N), cons(X, Z)) -> MARK(Z)
A_{_PLUS}(s(X), Y) -> MARK(Y)
A_{_SQUARE}(X) -> MARK(X)
A_{_TIMES}(s(X), Y) -> MARK(X)
A_{_TIMES}(s(X), Y) -> A_{_TIMES}(mark(X), mark(Y))
MARK(rcons(X1, X2)) -> MARK(X2)
MARK(rcons(X1, X2)) -> MARK(X1)
MARK(cons2(X1, X2)) -> MARK(X2)
MARK(cons(X1, X2)) -> MARK(X1)
MARK(negrecip(X)) -> MARK(X)
MARK(posrecip(X)) -> MARK(X)
MARK(s(X)) -> MARK(X)
MARK(square(X)) -> MARK(X)
A_{_TIMES}(s(X), Y) -> MARK(Y)
A_{_SQUARE}(X) -> A_{_TIMES}(mark(X), mark(X))
MARK(square(X)) -> A_{_SQUARE}(mark(X))
MARK(times(X1, X2)) -> MARK(X2)
MARK(times(X1, X2)) -> MARK(X1)
A_{_PLUS}(s(X), Y) -> MARK(X)
A_{_PLUS}(s(X), Y) -> A_{_PLUS}(mark(X), mark(Y))
A_{_TIMES}(s(X), Y) -> A_{_PLUS}(mark(Y), a_{_times}(mark(X), mark(Y)))
MARK(times(X1, X2)) -> A_{_TIMES}(mark(X1), mark(X2))
MARK(plus(X1, X2)) -> MARK(X2)
MARK(plus(X1, X2)) -> MARK(X1)
A_{_PLUS}(0, Y) -> MARK(Y)
MARK(plus(X1, X2)) -> A_{_PLUS}(mark(X1), mark(X2))
MARK(pi(X)) -> MARK(X)
A_{_2NDSPOS}(s(N), cons(X, Z)) -> MARK(N)
A_{_PI}(X) -> A_{_2NDSPOS}(mark(X), a_{_from}(0))
MARK(pi(X)) -> A_{_PI}(mark(X))
MARK(2ndsneg(X1, X2)) -> MARK(X2)
MARK(2ndsneg(X1, X2)) -> MARK(X1)
A_{_2NDSNEG}(s(N), cons2(X, cons(Y, Z))) -> MARK(Y)
A_{_2NDSNEG}(s(N), cons(X, Z)) -> A_{_2NDSNEG}(s(mark(N)), cons2(X, mark(Z)))
MARK(2ndsneg(X1, X2)) -> A_{_2NDSNEG}(mark(X1), mark(X2))
MARK(2ndspos(X1, X2)) -> MARK(X2)
MARK(2ndspos(X1, X2)) -> MARK(X1)
A_{_2NDSPOS}(s(N), cons2(X, cons(Y, Z))) -> MARK(Y)
A_{_2NDSPOS}(s(N), cons(X, Z)) -> A_{_2NDSPOS}(s(mark(N)), cons2(X, mark(Z)))
MARK(2ndspos(X1, X2)) -> A_{_2NDSPOS}(mark(X1), mark(X2))
MARK(from(X)) -> MARK(X)
MARK(from(X)) -> A_{_FROM}(mark(X))
A_{_FROM}(X) -> MARK(X)

Rules:

a_{_from}(X) -> cons(mark(X), from(s(X)))
a_{_from}(X) -> from(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_{_2ndspos}(X1, X2) -> 2ndspos(X1, X2)
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_{_2ndsneg}(X1, X2) -> 2ndsneg(X1, X2)
a_{_pi}(X) -> a_{_2ndspos}(mark(X), a_{_from}(0))
a_{_pi}(X) -> pi(X)
a_{_plus}(0, Y) -> mark(Y)
a_{_plus}(s(X), Y) -> s(a_{_plus}(mark(X), mark(Y)))
a_{_plus}(X1, X2) -> plus(X1, X2)
a_{_times}(0, Y) -> 0
a_{_times}(s(X), Y) -> a_{_plus}(mark(Y), a_{_times}(mark(X), mark(Y)))
a_{_times}(X1, X2) -> times(X1, X2)
a_{_square}(X) -> a_{_times}(mark(X), mark(X))
a_{_square}(X) -> square(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))

The Proof could not be continued due to a Timeout.
Termination of R could not be shown.
Duration:
1:00 minutes