Termination Proof using AProVE

Term Rewriting System R:
[X, Y, X1, X2]
fact(X) -> if(zero(X), n_{_s}(n_₀), n_{_prod}(X, n_{_fact}(n_{_p}(X))))
fact(X) -> n_{_fact}(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
p(X) -> n_{_p}(X)
s(X) -> n_{_s}(X)
0 -> n_₀
activate(n_{_s}(X)) -> s(activate(X))
activate(n_₀) -> 0
activate(n_{_prod}(X1, X2)) -> prod(activate(X1), activate(X2))
activate(n_{_fact}(X)) -> fact(activate(X))
activate(n_{_p}(X)) -> p(activate(X))
activate(X) -> X

Innermost Termination of R to be shown.

     ↳Dependency Pair Analysis

R contains the following Dependency Pairs:

FACT(X) -> IF(zero(X), n_{_s}(n_₀), n_{_prod}(X, n_{_fact}(n_{_p}(X))))
FACT(X) -> ZERO(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(activate(X))
ACTIVATE(n_{_s}(X)) -> ACTIVATE(X)
ACTIVATE(n_₀) -> 0'
ACTIVATE(n_{_prod}(X1, X2)) -> PROD(activate(X1), activate(X2))
ACTIVATE(n_{_prod}(X1, X2)) -> ACTIVATE(X1)
ACTIVATE(n_{_prod}(X1, X2)) -> ACTIVATE(X2)
ACTIVATE(n_{_fact}(X)) -> FACT(activate(X))
ACTIVATE(n_{_fact}(X)) -> ACTIVATE(X)
ACTIVATE(n_{_p}(X)) -> P(activate(X))
ACTIVATE(n_{_p}(X)) -> ACTIVATE(X)

Furthermore, R contains one SCC.

     ↳DPs

       →DP Problem 1

         ↳Narrowing Transformation

Dependency Pairs:

IF(false, X, Y) -> ACTIVATE(Y)
ACTIVATE(n_{_p}(X)) -> ACTIVATE(X)
ACTIVATE(n_{_fact}(X)) -> ACTIVATE(X)
ACTIVATE(n_{_fact}(X)) -> FACT(activate(X))
ACTIVATE(n_{_prod}(X1, X2)) -> ACTIVATE(X2)
ACTIVATE(n_{_prod}(X1, X2)) -> ACTIVATE(X1)
ACTIVATE(n_{_s}(X)) -> ACTIVATE(X)
IF(true, X, Y) -> ACTIVATE(X)
FACT(X) -> IF(zero(X), n_{_s}(n_₀), n_{_prod}(X, n_{_fact}(n_{_p}(X))))

Rules:

fact(X) -> if(zero(X), n_{_s}(n_₀), n_{_prod}(X, n_{_fact}(n_{_p}(X))))
fact(X) -> n_{_fact}(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
p(X) -> n_{_p}(X)
s(X) -> n_{_s}(X)
0 -> n_₀
activate(n_{_s}(X)) -> s(activate(X))
activate(n_₀) -> 0
activate(n_{_prod}(X1, X2)) -> prod(activate(X1), activate(X2))
activate(n_{_fact}(X)) -> fact(activate(X))
activate(n_{_p}(X)) -> p(activate(X))
activate(X) -> X

Strategy:

innermost

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

FACT(X) -> IF(zero(X), n_{_s}(n_₀), n_{_prod}(X, n_{_fact}(n_{_p}(X))))

two new Dependency Pairs are created:

FACT(0) -> IF(true, n_{_s}(n_₀), n_{_prod}(0, n_{_fact}(n_{_p}(0))))
FACT(s(X'')) -> IF(false, n_{_s}(n_₀), n_{_prod}(s(X''), n_{_fact}(n_{_p}(s(X'')))))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Polynomial Ordering

Dependency Pairs:

ACTIVATE(n_{_p}(X)) -> ACTIVATE(X)
ACTIVATE(n_{_fact}(X)) -> ACTIVATE(X)
ACTIVATE(n_{_prod}(X1, X2)) -> ACTIVATE(X2)
ACTIVATE(n_{_prod}(X1, X2)) -> ACTIVATE(X1)
ACTIVATE(n_{_s}(X)) -> ACTIVATE(X)

Rules:

fact(X) -> if(zero(X), n_{_s}(n_₀), n_{_prod}(X, n_{_fact}(n_{_p}(X))))
fact(X) -> n_{_fact}(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
p(X) -> n_{_p}(X)
s(X) -> n_{_s}(X)
0 -> n_₀
activate(n_{_s}(X)) -> s(activate(X))
activate(n_₀) -> 0
activate(n_{_prod}(X1, X2)) -> prod(activate(X1), activate(X2))
activate(n_{_fact}(X)) -> fact(activate(X))
activate(n_{_p}(X)) -> p(activate(X))
activate(X) -> X

Strategy:

innermost

The following dependency pair can be strictly oriented:

ACTIVATE(n_{_s}(X)) -> ACTIVATE(X)

There are no usable rules for innermost w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(n__s(x₁)) = 1 + x₁

POL(ACTIVATE(x₁)) = x₁

POL(n__fact(x₁)) = x₁

POL(n__prod(x₁, x₂)) = x₁ + x₂

POL(n__p(x₁)) = x₁

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Polo

...

               →DP Problem 3

                 ↳Polynomial Ordering

Dependency Pairs:

ACTIVATE(n_{_p}(X)) -> ACTIVATE(X)
ACTIVATE(n_{_fact}(X)) -> ACTIVATE(X)
ACTIVATE(n_{_prod}(X1, X2)) -> ACTIVATE(X2)
ACTIVATE(n_{_prod}(X1, X2)) -> ACTIVATE(X1)

Rules:

fact(X) -> if(zero(X), n_{_s}(n_₀), n_{_prod}(X, n_{_fact}(n_{_p}(X))))
fact(X) -> n_{_fact}(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
p(X) -> n_{_p}(X)
s(X) -> n_{_s}(X)
0 -> n_₀
activate(n_{_s}(X)) -> s(activate(X))
activate(n_₀) -> 0
activate(n_{_prod}(X1, X2)) -> prod(activate(X1), activate(X2))
activate(n_{_fact}(X)) -> fact(activate(X))
activate(n_{_p}(X)) -> p(activate(X))
activate(X) -> X

Strategy:

innermost

The following dependency pair can be strictly oriented:

ACTIVATE(n_{_p}(X)) -> ACTIVATE(X)

There are no usable rules for innermost w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(ACTIVATE(x₁)) = x₁

POL(n__fact(x₁)) = x₁

POL(n__prod(x₁, x₂)) = x₁ + x₂

POL(n__p(x₁)) = 1 + x₁

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Polo

...

               →DP Problem 4

                 ↳Polynomial Ordering

Dependency Pairs:

ACTIVATE(n_{_fact}(X)) -> ACTIVATE(X)
ACTIVATE(n_{_prod}(X1, X2)) -> ACTIVATE(X2)
ACTIVATE(n_{_prod}(X1, X2)) -> ACTIVATE(X1)

Rules:

fact(X) -> if(zero(X), n_{_s}(n_₀), n_{_prod}(X, n_{_fact}(n_{_p}(X))))
fact(X) -> n_{_fact}(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
p(X) -> n_{_p}(X)
s(X) -> n_{_s}(X)
0 -> n_₀
activate(n_{_s}(X)) -> s(activate(X))
activate(n_₀) -> 0
activate(n_{_prod}(X1, X2)) -> prod(activate(X1), activate(X2))
activate(n_{_fact}(X)) -> fact(activate(X))
activate(n_{_p}(X)) -> p(activate(X))
activate(X) -> X

Strategy:

innermost

The following dependency pair can be strictly oriented:

ACTIVATE(n_{_fact}(X)) -> ACTIVATE(X)

There are no usable rules for innermost w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(ACTIVATE(x₁)) = x₁

POL(n__fact(x₁)) = 1 + x₁

POL(n__prod(x₁, x₂)) = x₁ + x₂

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Polo

...

               →DP Problem 5

                 ↳Polynomial Ordering

Dependency Pairs:

ACTIVATE(n_{_prod}(X1, X2)) -> ACTIVATE(X2)
ACTIVATE(n_{_prod}(X1, X2)) -> ACTIVATE(X1)

Rules:

fact(X) -> if(zero(X), n_{_s}(n_₀), n_{_prod}(X, n_{_fact}(n_{_p}(X))))
fact(X) -> n_{_fact}(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
p(X) -> n_{_p}(X)
s(X) -> n_{_s}(X)
0 -> n_₀
activate(n_{_s}(X)) -> s(activate(X))
activate(n_₀) -> 0
activate(n_{_prod}(X1, X2)) -> prod(activate(X1), activate(X2))
activate(n_{_fact}(X)) -> fact(activate(X))
activate(n_{_p}(X)) -> p(activate(X))
activate(X) -> X

Strategy:

innermost

The following dependency pairs can be strictly oriented:

ACTIVATE(n_{_prod}(X1, X2)) -> ACTIVATE(X2)
ACTIVATE(n_{_prod}(X1, X2)) -> ACTIVATE(X1)

There are no usable rules for innermost w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(ACTIVATE(x₁)) = x₁

POL(n__prod(x₁, x₂)) = 1 + x₁ + x₂

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Polo

...

               →DP Problem 6

                 ↳Dependency Graph

Dependency Pair:

Rules:

fact(X) -> if(zero(X), n_{_s}(n_₀), n_{_prod}(X, n_{_fact}(n_{_p}(X))))
fact(X) -> n_{_fact}(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
p(X) -> n_{_p}(X)
s(X) -> n_{_s}(X)
0 -> n_₀
activate(n_{_s}(X)) -> s(activate(X))
activate(n_₀) -> 0
activate(n_{_prod}(X1, X2)) -> prod(activate(X1), activate(X2))
activate(n_{_fact}(X)) -> fact(activate(X))
activate(n_{_p}(X)) -> p(activate(X))
activate(X) -> X

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

Innermost Termination of R successfully shown.
Duration:
0:00 minutes

POL(n__s(x₁))	= 1 + x₁
POL(ACTIVATE(x₁))	= x₁
POL(n__fact(x₁))	= x₁
POL(n__prod(x₁, x₂))	= x₁ + x₂
POL(n__p(x₁))	= x₁