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

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

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

A_{_ADD}(s(X), Y) -> A_{_ADD}(mark(X), mark(Y))

20 new Dependency Pairs are created:

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

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Narrowing Transformation

Dependency Pairs:

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

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

A_{_PROD}(s(X), Y) -> A_{_PROD}(mark(X), mark(Y))

20 new Dependency Pairs are created:

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

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 3

                 ↳Narrowing Transformation

Dependency Pairs:

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

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

MARK(if(X1, X2, X3)) -> A_{_IF}(mark(X1), X2, X3)

10 new Dependency Pairs are created:

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

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 4

                 ↳Narrowing Transformation

Dependency Pairs:

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

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

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

MARK(prod(X1, X2)) -> A_{_PROD}(mark(X1), mark(X2))

20 new Dependency Pairs are created:

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

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 5

                 ↳Narrowing Transformation

Dependency Pairs:

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

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

MARK(p(X)) -> A_{_P}(mark(X))

10 new Dependency Pairs are created:

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

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 6

                 ↳Narrowing Transformation

Dependency Pairs:

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

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

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

MARK(add(X1, X2)) -> A_{_ADD}(mark(X1), mark(X2))

20 new Dependency Pairs are created:

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

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 7

                 ↳Remaining Obligation(s)

The following remains to be proven:
Dependency Pairs:

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

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