Termination Proof using AProVE

Term Rewriting System R:
[X, Y, X1, X2]
fact(X) -> if(zero(X), n_{_s}(0), n_{_prod}(X, fact(p(X))))
add(0, X) -> X
add(s(X), Y) -> s(add(X, Y))
prod(0, X) -> 0
prod(s(X), Y) -> add(Y, prod(X, Y))
prod(X1, X2) -> n_{_prod}(X1, X2)
if(true, X, Y) -> activate(X)
if(false, X, Y) -> activate(Y)
zero(0) -> true
zero(s(X)) -> false
p(s(X)) -> X
s(X) -> n_{_s}(X)
activate(n_{_s}(X)) -> s(X)
activate(n_{_prod}(X1, X2)) -> prod(X1, X2)
activate(X) -> X

Termination of R to be shown.

     ↳Dependency Pair Analysis

R contains the following Dependency Pairs:

FACT(X) -> IF(zero(X), n_{_s}(0), n_{_prod}(X, fact(p(X))))
FACT(X) -> ZERO(X)
FACT(X) -> FACT(p(X))
FACT(X) -> P(X)
ADD(s(X), Y) -> S(add(X, Y))
ADD(s(X), Y) -> ADD(X, Y)
PROD(s(X), Y) -> ADD(Y, prod(X, Y))
PROD(s(X), Y) -> PROD(X, Y)
IF(true, X, Y) -> ACTIVATE(X)
IF(false, X, Y) -> ACTIVATE(Y)
ACTIVATE(n_{_s}(X)) -> S(X)
ACTIVATE(n_{_prod}(X1, X2)) -> PROD(X1, X2)

Furthermore, R contains three SCCs.

     ↳DPs

       →DP Problem 1

         ↳Remaining Obligation(s)

       →DP Problem 2

         ↳Remaining Obligation(s)

       →DP Problem 3

         ↳Remaining Obligation(s)

The following remains to be proven:

Dependency Pair:
ADD(s(X), Y) -> ADD(X, Y)

Rules:

fact(X) -> if(zero(X), n_{_s}(0), n_{_prod}(X, fact(p(X))))
add(0, X) -> X
add(s(X), Y) -> s(add(X, Y))
prod(0, X) -> 0
prod(s(X), Y) -> add(Y, prod(X, Y))
prod(X1, X2) -> n_{_prod}(X1, X2)
if(true, X, Y) -> activate(X)
if(false, X, Y) -> activate(Y)
zero(0) -> true
zero(s(X)) -> false
p(s(X)) -> X
s(X) -> n_{_s}(X)
activate(n_{_s}(X)) -> s(X)
activate(n_{_prod}(X1, X2)) -> prod(X1, X2)
activate(X) -> X
Dependency Pair:
PROD(s(X), Y) -> PROD(X, Y)

Rules:

fact(X) -> if(zero(X), n_{_s}(0), n_{_prod}(X, fact(p(X))))
add(0, X) -> X
add(s(X), Y) -> s(add(X, Y))
prod(0, X) -> 0
prod(s(X), Y) -> add(Y, prod(X, Y))
prod(X1, X2) -> n_{_prod}(X1, X2)
if(true, X, Y) -> activate(X)
if(false, X, Y) -> activate(Y)
zero(0) -> true
zero(s(X)) -> false
p(s(X)) -> X
s(X) -> n_{_s}(X)
activate(n_{_s}(X)) -> s(X)
activate(n_{_prod}(X1, X2)) -> prod(X1, X2)
activate(X) -> X
Dependency Pair:
FACT(X) -> FACT(p(X))

Rules:

fact(X) -> if(zero(X), n_{_s}(0), n_{_prod}(X, fact(p(X))))
add(0, X) -> X
add(s(X), Y) -> s(add(X, Y))
prod(0, X) -> 0
prod(s(X), Y) -> add(Y, prod(X, Y))
prod(X1, X2) -> n_{_prod}(X1, X2)
if(true, X, Y) -> activate(X)
if(false, X, Y) -> activate(Y)
zero(0) -> true
zero(s(X)) -> false
p(s(X)) -> X
s(X) -> n_{_s}(X)
activate(n_{_s}(X)) -> s(X)
activate(n_{_prod}(X1, X2)) -> prod(X1, X2)
activate(X) -> X

     ↳DPs

       →DP Problem 1

         ↳Remaining Obligation(s)

       →DP Problem 2

         ↳Remaining Obligation(s)

       →DP Problem 3

         ↳Remaining Obligation(s)

The following remains to be proven:

Dependency Pair:
ADD(s(X), Y) -> ADD(X, Y)

Rules:

fact(X) -> if(zero(X), n_{_s}(0), n_{_prod}(X, fact(p(X))))
add(0, X) -> X
add(s(X), Y) -> s(add(X, Y))
prod(0, X) -> 0
prod(s(X), Y) -> add(Y, prod(X, Y))
prod(X1, X2) -> n_{_prod}(X1, X2)
if(true, X, Y) -> activate(X)
if(false, X, Y) -> activate(Y)
zero(0) -> true
zero(s(X)) -> false
p(s(X)) -> X
s(X) -> n_{_s}(X)
activate(n_{_s}(X)) -> s(X)
activate(n_{_prod}(X1, X2)) -> prod(X1, X2)
activate(X) -> X
Dependency Pair:
PROD(s(X), Y) -> PROD(X, Y)

Rules:

fact(X) -> if(zero(X), n_{_s}(0), n_{_prod}(X, fact(p(X))))
add(0, X) -> X
add(s(X), Y) -> s(add(X, Y))
prod(0, X) -> 0
prod(s(X), Y) -> add(Y, prod(X, Y))
prod(X1, X2) -> n_{_prod}(X1, X2)
if(true, X, Y) -> activate(X)
if(false, X, Y) -> activate(Y)
zero(0) -> true
zero(s(X)) -> false
p(s(X)) -> X
s(X) -> n_{_s}(X)
activate(n_{_s}(X)) -> s(X)
activate(n_{_prod}(X1, X2)) -> prod(X1, X2)
activate(X) -> X
Dependency Pair:
FACT(X) -> FACT(p(X))

Rules:

fact(X) -> if(zero(X), n_{_s}(0), n_{_prod}(X, fact(p(X))))
add(0, X) -> X
add(s(X), Y) -> s(add(X, Y))
prod(0, X) -> 0
prod(s(X), Y) -> add(Y, prod(X, Y))
prod(X1, X2) -> n_{_prod}(X1, X2)
if(true, X, Y) -> activate(X)
if(false, X, Y) -> activate(Y)
zero(0) -> true
zero(s(X)) -> false
p(s(X)) -> X
s(X) -> n_{_s}(X)
activate(n_{_s}(X)) -> s(X)
activate(n_{_prod}(X1, X2)) -> prod(X1, X2)
activate(X) -> X

     ↳DPs

       →DP Problem 1

         ↳Remaining Obligation(s)

       →DP Problem 2

         ↳Remaining Obligation(s)

       →DP Problem 3

         ↳Remaining Obligation(s)

The following remains to be proven:

Dependency Pair:
ADD(s(X), Y) -> ADD(X, Y)

Rules:

fact(X) -> if(zero(X), n_{_s}(0), n_{_prod}(X, fact(p(X))))
add(0, X) -> X
add(s(X), Y) -> s(add(X, Y))
prod(0, X) -> 0
prod(s(X), Y) -> add(Y, prod(X, Y))
prod(X1, X2) -> n_{_prod}(X1, X2)
if(true, X, Y) -> activate(X)
if(false, X, Y) -> activate(Y)
zero(0) -> true
zero(s(X)) -> false
p(s(X)) -> X
s(X) -> n_{_s}(X)
activate(n_{_s}(X)) -> s(X)
activate(n_{_prod}(X1, X2)) -> prod(X1, X2)
activate(X) -> X
Dependency Pair:
PROD(s(X), Y) -> PROD(X, Y)

Rules:

fact(X) -> if(zero(X), n_{_s}(0), n_{_prod}(X, fact(p(X))))
add(0, X) -> X
add(s(X), Y) -> s(add(X, Y))
prod(0, X) -> 0
prod(s(X), Y) -> add(Y, prod(X, Y))
prod(X1, X2) -> n_{_prod}(X1, X2)
if(true, X, Y) -> activate(X)
if(false, X, Y) -> activate(Y)
zero(0) -> true
zero(s(X)) -> false
p(s(X)) -> X
s(X) -> n_{_s}(X)
activate(n_{_s}(X)) -> s(X)
activate(n_{_prod}(X1, X2)) -> prod(X1, X2)
activate(X) -> X
Dependency Pair:
FACT(X) -> FACT(p(X))

Rules:

fact(X) -> if(zero(X), n_{_s}(0), n_{_prod}(X, fact(p(X))))
add(0, X) -> X
add(s(X), Y) -> s(add(X, Y))
prod(0, X) -> 0
prod(s(X), Y) -> add(Y, prod(X, Y))
prod(X1, X2) -> n_{_prod}(X1, X2)
if(true, X, Y) -> activate(X)
if(false, X, Y) -> activate(Y)
zero(0) -> true
zero(s(X)) -> false
p(s(X)) -> X
s(X) -> n_{_s}(X)
activate(n_{_s}(X)) -> s(X)
activate(n_{_prod}(X1, X2)) -> prod(X1, X2)
activate(X) -> X

Termination of R could not be shown.
Duration:
0:00 minutes