Termination Proof using AProVE

Term Rewriting System R:
[X, Y, X1, X2, X3]
a_{_p}(0) -> 0
a_{_p}(s(X)) -> mark(X)
a_{_p}(X) -> p(X)
a_{_leq}(0, Y) -> true
a_{_leq}(s(X), 0) -> false
a_{_leq}(s(X), s(Y)) -> a_{_leq}(mark(X), mark(Y))
a_{_leq}(X1, X2) -> leq(X1, X2)
a_{_if}(true, X, Y) -> mark(X)
a_{_if}(false, X, Y) -> mark(Y)
a_{_if}(X1, X2, X3) -> if(X1, X2, X3)
a_{_diff}(X, Y) -> a_{_if}(a_{_leq}(mark(X), mark(Y)), 0, s(diff(p(X), Y)))
a_{_diff}(X1, X2) -> diff(X1, X2)
mark(p(X)) -> a_{_p}(mark(X))
mark(leq(X1, X2)) -> a_{_leq}(mark(X1), mark(X2))
mark(if(X1, X2, X3)) -> a_{_if}(mark(X1), X2, X3)
mark(diff(X1, X2)) -> a_{_diff}(mark(X1), mark(X2))
mark(0) -> 0
mark(s(X)) -> s(mark(X))
mark(true) -> true
mark(false) -> false

Innermost Termination of R to be shown.

     ↳Dependency Pair Analysis

R contains the following Dependency Pairs:

A_{_P}(s(X)) -> MARK(X)
A_{_LEQ}(s(X), s(Y)) -> A_{_LEQ}(mark(X), mark(Y))
A_{_LEQ}(s(X), s(Y)) -> MARK(X)
A_{_LEQ}(s(X), s(Y)) -> MARK(Y)
A_{_IF}(true, X, Y) -> MARK(X)
A_{_IF}(false, X, Y) -> MARK(Y)
A_{_DIFF}(X, Y) -> A_{_IF}(a_{_leq}(mark(X), mark(Y)), 0, s(diff(p(X), Y)))
A_{_DIFF}(X, Y) -> A_{_LEQ}(mark(X), mark(Y))
A_{_DIFF}(X, Y) -> MARK(X)
A_{_DIFF}(X, Y) -> MARK(Y)
MARK(p(X)) -> A_{_P}(mark(X))
MARK(p(X)) -> MARK(X)
MARK(leq(X1, X2)) -> A_{_LEQ}(mark(X1), mark(X2))
MARK(leq(X1, X2)) -> MARK(X1)
MARK(leq(X1, X2)) -> MARK(X2)
MARK(if(X1, X2, X3)) -> A_{_IF}(mark(X1), X2, X3)
MARK(if(X1, X2, X3)) -> MARK(X1)
MARK(diff(X1, X2)) -> A_{_DIFF}(mark(X1), mark(X2))
MARK(diff(X1, X2)) -> MARK(X1)
MARK(diff(X1, X2)) -> MARK(X2)
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_{_DIFF}(X, Y) -> MARK(Y)
A_{_DIFF}(X, Y) -> MARK(X)
A_{_LEQ}(s(X), s(Y)) -> MARK(Y)
A_{_DIFF}(X, Y) -> A_{_LEQ}(mark(X), mark(Y))
MARK(s(X)) -> MARK(X)
MARK(diff(X1, X2)) -> MARK(X2)
MARK(diff(X1, X2)) -> MARK(X1)
A_{_IF}(false, X, Y) -> MARK(Y)
A_{_DIFF}(X, Y) -> A_{_IF}(a_{_leq}(mark(X), mark(Y)), 0, s(diff(p(X), Y)))
MARK(diff(X1, X2)) -> A_{_DIFF}(mark(X1), mark(X2))
MARK(if(X1, X2, X3)) -> MARK(X1)
A_{_IF}(true, X, Y) -> MARK(X)
MARK(if(X1, X2, X3)) -> A_{_IF}(mark(X1), X2, X3)
MARK(leq(X1, X2)) -> MARK(X2)
MARK(leq(X1, X2)) -> MARK(X1)
A_{_LEQ}(s(X), s(Y)) -> MARK(X)
A_{_LEQ}(s(X), s(Y)) -> A_{_LEQ}(mark(X), mark(Y))
MARK(leq(X1, X2)) -> A_{_LEQ}(mark(X1), mark(X2))
MARK(p(X)) -> MARK(X)
MARK(p(X)) -> A_{_P}(mark(X))
A_{_P}(s(X)) -> MARK(X)

Rules:

a_{_p}(0) -> 0
a_{_p}(s(X)) -> mark(X)
a_{_p}(X) -> p(X)
a_{_leq}(0, Y) -> true
a_{_leq}(s(X), 0) -> false
a_{_leq}(s(X), s(Y)) -> a_{_leq}(mark(X), mark(Y))
a_{_leq}(X1, X2) -> leq(X1, X2)
a_{_if}(true, X, Y) -> mark(X)
a_{_if}(false, X, Y) -> mark(Y)
a_{_if}(X1, X2, X3) -> if(X1, X2, X3)
a_{_diff}(X, Y) -> a_{_if}(a_{_leq}(mark(X), mark(Y)), 0, s(diff(p(X), Y)))
a_{_diff}(X1, X2) -> diff(X1, X2)
mark(p(X)) -> a_{_p}(mark(X))
mark(leq(X1, X2)) -> a_{_leq}(mark(X1), mark(X2))
mark(if(X1, X2, X3)) -> a_{_if}(mark(X1), X2, X3)
mark(diff(X1, X2)) -> a_{_diff}(mark(X1), mark(X2))
mark(0) -> 0
mark(s(X)) -> s(mark(X))
mark(true) -> true
mark(false) -> false

Strategy:

innermost

Innermost Termination of R could not be shown.
Duration:
0:06 minutes