Termination Proof using AProVE

Term Rewriting System R:
[X, Y, X1, X2, X3]
a_{_fact}(X) -> a_{_if}(a_{_zero}(mark(X)), s(0), prod(X, fact(p(X))))
a_{_fact}(X) -> fact(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_{_prod}(0, X) -> 0
a_{_prod}(s(X), Y) -> a_{_add}(mark(Y), a_{_prod}(mark(X), mark(Y)))
a_{_prod}(X1, X2) -> prod(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_{_zero}(0) -> true
a_{_zero}(s(X)) -> false
a_{_zero}(X) -> zero(X)
a_{_p}(s(X)) -> mark(X)
a_{_p}(X) -> p(X)
mark(fact(X)) -> a_{_fact}(mark(X))
mark(if(X1, X2, X3)) -> a_{_if}(mark(X1), X2, X3)
mark(zero(X)) -> a_{_zero}(mark(X))
mark(prod(X1, X2)) -> a_{_prod}(mark(X1), mark(X2))
mark(p(X)) -> a_{_p}(mark(X))
mark(add(X1, X2)) -> a_{_add}(mark(X1), mark(X2))
mark(s(X)) -> s(mark(X))
mark(0) -> 0
mark(true) -> true
mark(false) -> false

Innermost Termination of R to be shown.

     ↳Dependency Pair Analysis

R contains the following Dependency Pairs:

A_{_FACT}(X) -> A_{_IF}(a_{_zero}(mark(X)), s(0), prod(X, fact(p(X))))
A_{_FACT}(X) -> A_{_ZERO}(mark(X))
A_{_FACT}(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_{_PROD}(s(X), Y) -> A_{_ADD}(mark(Y), a_{_prod}(mark(X), mark(Y)))
A_{_PROD}(s(X), Y) -> MARK(Y)
A_{_PROD}(s(X), Y) -> A_{_PROD}(mark(X), mark(Y))
A_{_PROD}(s(X), Y) -> MARK(X)
A_{_IF}(true, X, Y) -> MARK(X)
A_{_IF}(false, X, Y) -> MARK(Y)
A_{_P}(s(X)) -> MARK(X)
MARK(fact(X)) -> A_{_FACT}(mark(X))
MARK(fact(X)) -> MARK(X)
MARK(if(X1, X2, X3)) -> A_{_IF}(mark(X1), X2, X3)
MARK(if(X1, X2, X3)) -> MARK(X1)
MARK(zero(X)) -> A_{_ZERO}(mark(X))
MARK(zero(X)) -> MARK(X)
MARK(prod(X1, X2)) -> A_{_PROD}(mark(X1), mark(X2))
MARK(prod(X1, X2)) -> MARK(X1)
MARK(prod(X1, X2)) -> MARK(X2)
MARK(p(X)) -> A_{_P}(mark(X))
MARK(p(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(s(X)) -> MARK(X)

Furthermore, R contains one SCC.

     ↳DPs

       →DP Problem 1

         ↳Narrowing Transformation

Dependency Pairs:

A_{_PROD}(s(X), Y) -> MARK(X)
A_{_PROD}(s(X), Y) -> A_{_PROD}(mark(X), mark(Y))
A_{_PROD}(s(X), Y) -> MARK(Y)
A_{_ADD}(s(X), Y) -> MARK(Y)
MARK(s(X)) -> 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(p(X)) -> MARK(X)
A_{_P}(s(X)) -> MARK(X)
MARK(p(X)) -> A_{_P}(mark(X))
MARK(prod(X1, X2)) -> MARK(X2)
MARK(prod(X1, X2)) -> MARK(X1)
A_{_ADD}(0, X) -> MARK(X)
A_{_PROD}(s(X), Y) -> A_{_ADD}(mark(Y), a_{_prod}(mark(X), mark(Y)))
MARK(prod(X1, X2)) -> A_{_PROD}(mark(X1), mark(X2))
MARK(zero(X)) -> MARK(X)
MARK(if(X1, X2, X3)) -> MARK(X1)
A_{_IF}(false, X, Y) -> MARK(Y)
MARK(if(X1, X2, X3)) -> A_{_IF}(mark(X1), X2, X3)
MARK(fact(X)) -> MARK(X)
A_{_FACT}(X) -> MARK(X)
MARK(fact(X)) -> A_{_FACT}(mark(X))
A_{_IF}(true, X, Y) -> MARK(X)
A_{_FACT}(X) -> A_{_IF}(a_{_zero}(mark(X)), s(0), prod(X, fact(p(X))))

Rules:

a_{_fact}(X) -> a_{_if}(a_{_zero}(mark(X)), s(0), prod(X, fact(p(X))))
a_{_fact}(X) -> fact(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_{_prod}(0, X) -> 0
a_{_prod}(s(X), Y) -> a_{_add}(mark(Y), a_{_prod}(mark(X), mark(Y)))
a_{_prod}(X1, X2) -> prod(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_{_zero}(0) -> true
a_{_zero}(s(X)) -> false
a_{_zero}(X) -> zero(X)
a_{_p}(s(X)) -> mark(X)
a_{_p}(X) -> p(X)
mark(fact(X)) -> a_{_fact}(mark(X))
mark(if(X1, X2, X3)) -> a_{_if}(mark(X1), X2, X3)
mark(zero(X)) -> a_{_zero}(mark(X))
mark(prod(X1, X2)) -> a_{_prod}(mark(X1), mark(X2))
mark(p(X)) -> a_{_p}(mark(X))
mark(add(X1, X2)) -> a_{_add}(mark(X1), mark(X2))
mark(s(X)) -> s(mark(X))
mark(0) -> 0
mark(true) -> true
mark(false) -> false

Strategy:

innermost

On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

A_{_FACT}(X) -> A_{_IF}(a_{_zero}(mark(X)), s(0), prod(X, fact(p(X))))

11 new Dependency Pairs are created:

A_{_FACT}(X'') -> A_{_IF}(zero(mark(X'')), s(0), prod(X'', fact(p(X''))))
A_{_FACT}(fact(X'')) -> A_{_IF}(a_{_zero}(a_{_fact}(mark(X''))), s(0), prod(fact(X''), fact(p(fact(X'')))))
A_{_FACT}(if(X1', X2', X3')) -> A_{_IF}(a_{_zero}(a_{_if}(mark(X1'), X2', X3')), s(0), prod(if(X1', X2', X3'), fact(p(if(X1', X2', X3')))))
A_{_FACT}(zero(X'')) -> A_{_IF}(a_{_zero}(a_{_zero}(mark(X''))), s(0), prod(zero(X''), fact(p(zero(X'')))))
A_{_FACT}(prod(X1', X2')) -> A_{_IF}(a_{_zero}(a_{_prod}(mark(X1'), mark(X2'))), s(0), prod(prod(X1', X2'), fact(p(prod(X1', X2')))))
A_{_FACT}(p(X'')) -> A_{_IF}(a_{_zero}(a_{_p}(mark(X''))), s(0), prod(p(X''), fact(p(p(X'')))))
A_{_FACT}(add(X1', X2')) -> A_{_IF}(a_{_zero}(a_{_add}(mark(X1'), mark(X2'))), s(0), prod(add(X1', X2'), fact(p(add(X1', X2')))))
A_{_FACT}(s(X'')) -> A_{_IF}(a_{_zero}(s(mark(X''))), s(0), prod(s(X''), fact(p(s(X'')))))
A_{_FACT}(0) -> A_{_IF}(a_{_zero}(0), s(0), prod(0, fact(p(0))))
A_{_FACT}(true) -> A_{_IF}(a_{_zero}(true), s(0), prod(true, fact(p(true))))
A_{_FACT}(false) -> A_{_IF}(a_{_zero}(false), s(0), prod(false, fact(p(false))))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Remaining Obligation(s)

The following remains to be proven:
Dependency Pairs:

A_{_FACT}(0) -> A_{_IF}(a_{_zero}(0), s(0), prod(0, fact(p(0))))
A_{_FACT}(s(X'')) -> A_{_IF}(a_{_zero}(s(mark(X''))), s(0), prod(s(X''), fact(p(s(X'')))))
A_{_FACT}(add(X1', X2')) -> A_{_IF}(a_{_zero}(a_{_add}(mark(X1'), mark(X2'))), s(0), prod(add(X1', X2'), fact(p(add(X1', X2')))))
A_{_FACT}(p(X'')) -> A_{_IF}(a_{_zero}(a_{_p}(mark(X''))), s(0), prod(p(X''), fact(p(p(X'')))))
A_{_FACT}(prod(X1', X2')) -> A_{_IF}(a_{_zero}(a_{_prod}(mark(X1'), mark(X2'))), s(0), prod(prod(X1', X2'), fact(p(prod(X1', X2')))))
A_{_FACT}(zero(X'')) -> A_{_IF}(a_{_zero}(a_{_zero}(mark(X''))), s(0), prod(zero(X''), fact(p(zero(X'')))))
A_{_FACT}(if(X1', X2', X3')) -> A_{_IF}(a_{_zero}(a_{_if}(mark(X1'), X2', X3')), s(0), prod(if(X1', X2', X3'), fact(p(if(X1', X2', X3')))))
A_{_FACT}(fact(X'')) -> A_{_IF}(a_{_zero}(a_{_fact}(mark(X''))), s(0), prod(fact(X''), fact(p(fact(X'')))))
A_{_IF}(false, X, Y) -> MARK(Y)
A_{_PROD}(s(X), Y) -> A_{_PROD}(mark(X), mark(Y))
A_{_PROD}(s(X), Y) -> MARK(Y)
A_{_ADD}(s(X), Y) -> MARK(Y)
MARK(s(X)) -> 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(p(X)) -> MARK(X)
A_{_P}(s(X)) -> MARK(X)
MARK(p(X)) -> A_{_P}(mark(X))
MARK(prod(X1, X2)) -> MARK(X2)
MARK(prod(X1, X2)) -> MARK(X1)
A_{_ADD}(0, X) -> MARK(X)
A_{_PROD}(s(X), Y) -> A_{_ADD}(mark(Y), a_{_prod}(mark(X), mark(Y)))
MARK(prod(X1, X2)) -> A_{_PROD}(mark(X1), mark(X2))
MARK(zero(X)) -> MARK(X)
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(fact(X)) -> MARK(X)
A_{_FACT}(X) -> MARK(X)
MARK(fact(X)) -> A_{_FACT}(mark(X))
A_{_PROD}(s(X), Y) -> MARK(X)

Rules:

a_{_fact}(X) -> a_{_if}(a_{_zero}(mark(X)), s(0), prod(X, fact(p(X))))
a_{_fact}(X) -> fact(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_{_prod}(0, X) -> 0
a_{_prod}(s(X), Y) -> a_{_add}(mark(Y), a_{_prod}(mark(X), mark(Y)))
a_{_prod}(X1, X2) -> prod(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_{_zero}(0) -> true
a_{_zero}(s(X)) -> false
a_{_zero}(X) -> zero(X)
a_{_p}(s(X)) -> mark(X)
a_{_p}(X) -> p(X)
mark(fact(X)) -> a_{_fact}(mark(X))
mark(if(X1, X2, X3)) -> a_{_if}(mark(X1), X2, X3)
mark(zero(X)) -> a_{_zero}(mark(X))
mark(prod(X1, X2)) -> a_{_prod}(mark(X1), mark(X2))
mark(p(X)) -> a_{_p}(mark(X))
mark(add(X1, X2)) -> a_{_add}(mark(X1), mark(X2))
mark(s(X)) -> s(mark(X))
mark(0) -> 0
mark(true) -> true
mark(false) -> false

Strategy:

innermost

The Proof could not be continued due to a Timeout.
Innermost Termination of R could not be shown.
Duration:
1:03 minutes