Termination Proof using AProVE

Term Rewriting System R:
[Y, X]
minus(n_₀, Y) -> 0
minus(n_{_s}(X), n_{_s}(Y)) -> minus(activate(X), activate(Y))
geq(X, n_₀) -> true
geq(n_₀, n_{_s}(Y)) -> false
geq(n_{_s}(X), n_{_s}(Y)) -> geq(activate(X), activate(Y))
div(0, n_{_s}(Y)) -> 0
div(s(X), n_{_s}(Y)) -> if(geq(X, activate(Y)), n_{_s}(div(minus(X, activate(Y)), n_{_s}(activate(Y)))), n_₀)
if(true, X, Y) -> activate(X)
if(false, X, Y) -> activate(Y)
0 -> n_₀
s(X) -> n_{_s}(X)
activate(n_₀) -> 0
activate(n_{_s}(X)) -> s(X)
activate(X) -> X

Innermost Termination of R to be shown.

     ↳Dependency Pair Analysis

R contains the following Dependency Pairs:

MINUS(n_₀, Y) -> 0'
MINUS(n_{_s}(X), n_{_s}(Y)) -> MINUS(activate(X), activate(Y))
MINUS(n_{_s}(X), n_{_s}(Y)) -> ACTIVATE(X)
MINUS(n_{_s}(X), n_{_s}(Y)) -> ACTIVATE(Y)
GEQ(n_{_s}(X), n_{_s}(Y)) -> GEQ(activate(X), activate(Y))
GEQ(n_{_s}(X), n_{_s}(Y)) -> ACTIVATE(X)
GEQ(n_{_s}(X), n_{_s}(Y)) -> ACTIVATE(Y)
DIV(s(X), n_{_s}(Y)) -> IF(geq(X, activate(Y)), n_{_s}(div(minus(X, activate(Y)), n_{_s}(activate(Y)))), n_₀)
DIV(s(X), n_{_s}(Y)) -> GEQ(X, activate(Y))
DIV(s(X), n_{_s}(Y)) -> ACTIVATE(Y)
DIV(s(X), n_{_s}(Y)) -> DIV(minus(X, activate(Y)), n_{_s}(activate(Y)))
DIV(s(X), n_{_s}(Y)) -> MINUS(X, activate(Y))
IF(true, X, Y) -> ACTIVATE(X)
IF(false, X, Y) -> ACTIVATE(Y)
ACTIVATE(n_₀) -> 0'
ACTIVATE(n_{_s}(X)) -> S(X)

Furthermore, R contains two SCCs.

     ↳DPs

       →DP Problem 1

         ↳Narrowing Transformation

       →DP Problem 2

         ↳Remaining

Dependency Pair:

MINUS(n_{_s}(X), n_{_s}(Y)) -> MINUS(activate(X), activate(Y))

Rules:

minus(n_₀, Y) -> 0
minus(n_{_s}(X), n_{_s}(Y)) -> minus(activate(X), activate(Y))
geq(X, n_₀) -> true
geq(n_₀, n_{_s}(Y)) -> false
geq(n_{_s}(X), n_{_s}(Y)) -> geq(activate(X), activate(Y))
div(0, n_{_s}(Y)) -> 0
div(s(X), n_{_s}(Y)) -> if(geq(X, activate(Y)), n_{_s}(div(minus(X, activate(Y)), n_{_s}(activate(Y)))), n_₀)
if(true, X, Y) -> activate(X)
if(false, X, Y) -> activate(Y)
0 -> n_₀
s(X) -> n_{_s}(X)
activate(n_₀) -> 0
activate(n_{_s}(X)) -> s(X)
activate(X) -> X

Strategy:

innermost

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

MINUS(n_{_s}(X), n_{_s}(Y)) -> MINUS(activate(X), activate(Y))

six new Dependency Pairs are created:

MINUS(n_{_s}(n_₀), n_{_s}(Y)) -> MINUS(0, activate(Y))
MINUS(n_{_s}(n_{_s}(X'')), n_{_s}(Y)) -> MINUS(s(X''), activate(Y))
MINUS(n_{_s}(X''), n_{_s}(Y)) -> MINUS(X'', activate(Y))
MINUS(n_{_s}(X), n_{_s}(n_₀)) -> MINUS(activate(X), 0)
MINUS(n_{_s}(X), n_{_s}(n_{_s}(X''))) -> MINUS(activate(X), s(X''))
MINUS(n_{_s}(X), n_{_s}(Y')) -> MINUS(activate(X), Y')

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 3

             ↳Rewriting Transformation

       →DP Problem 2

         ↳Remaining

Dependency Pairs:

MINUS(n_{_s}(X), n_{_s}(Y')) -> MINUS(activate(X), Y')
MINUS(n_{_s}(X), n_{_s}(n_{_s}(X''))) -> MINUS(activate(X), s(X''))
MINUS(n_{_s}(X), n_{_s}(n_₀)) -> MINUS(activate(X), 0)
MINUS(n_{_s}(X''), n_{_s}(Y)) -> MINUS(X'', activate(Y))
MINUS(n_{_s}(n_{_s}(X'')), n_{_s}(Y)) -> MINUS(s(X''), activate(Y))
MINUS(n_{_s}(n_₀), n_{_s}(Y)) -> MINUS(0, activate(Y))

Rules:

minus(n_₀, Y) -> 0
minus(n_{_s}(X), n_{_s}(Y)) -> minus(activate(X), activate(Y))
geq(X, n_₀) -> true
geq(n_₀, n_{_s}(Y)) -> false
geq(n_{_s}(X), n_{_s}(Y)) -> geq(activate(X), activate(Y))
div(0, n_{_s}(Y)) -> 0
div(s(X), n_{_s}(Y)) -> if(geq(X, activate(Y)), n_{_s}(div(minus(X, activate(Y)), n_{_s}(activate(Y)))), n_₀)
if(true, X, Y) -> activate(X)
if(false, X, Y) -> activate(Y)
0 -> n_₀
s(X) -> n_{_s}(X)
activate(n_₀) -> 0
activate(n_{_s}(X)) -> s(X)
activate(X) -> X

Strategy:

innermost

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

MINUS(n_{_s}(n_₀), n_{_s}(Y)) -> MINUS(0, activate(Y))

one new Dependency Pair is created:

MINUS(n_{_s}(n_₀), n_{_s}(Y)) -> MINUS(n_₀, activate(Y))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 3

             ↳Rw

...

               →DP Problem 4

                 ↳Rewriting Transformation

       →DP Problem 2

         ↳Remaining

Dependency Pairs:

MINUS(n_{_s}(X), n_{_s}(n_{_s}(X''))) -> MINUS(activate(X), s(X''))
MINUS(n_{_s}(X), n_{_s}(n_₀)) -> MINUS(activate(X), 0)
MINUS(n_{_s}(X''), n_{_s}(Y)) -> MINUS(X'', activate(Y))
MINUS(n_{_s}(n_{_s}(X'')), n_{_s}(Y)) -> MINUS(s(X''), activate(Y))
MINUS(n_{_s}(X), n_{_s}(Y')) -> MINUS(activate(X), Y')

Rules:

minus(n_₀, Y) -> 0
minus(n_{_s}(X), n_{_s}(Y)) -> minus(activate(X), activate(Y))
geq(X, n_₀) -> true
geq(n_₀, n_{_s}(Y)) -> false
geq(n_{_s}(X), n_{_s}(Y)) -> geq(activate(X), activate(Y))
div(0, n_{_s}(Y)) -> 0
div(s(X), n_{_s}(Y)) -> if(geq(X, activate(Y)), n_{_s}(div(minus(X, activate(Y)), n_{_s}(activate(Y)))), n_₀)
if(true, X, Y) -> activate(X)
if(false, X, Y) -> activate(Y)
0 -> n_₀
s(X) -> n_{_s}(X)
activate(n_₀) -> 0
activate(n_{_s}(X)) -> s(X)
activate(X) -> X

Strategy:

innermost

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

MINUS(n_{_s}(n_{_s}(X'')), n_{_s}(Y)) -> MINUS(s(X''), activate(Y))

one new Dependency Pair is created:

MINUS(n_{_s}(n_{_s}(X'')), n_{_s}(Y)) -> MINUS(n_{_s}(X''), activate(Y))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 3

             ↳Rw

...

               →DP Problem 5

                 ↳Rewriting Transformation

       →DP Problem 2

         ↳Remaining

Dependency Pairs:

MINUS(n_{_s}(n_{_s}(X'')), n_{_s}(Y)) -> MINUS(n_{_s}(X''), activate(Y))
MINUS(n_{_s}(X), n_{_s}(Y')) -> MINUS(activate(X), Y')
MINUS(n_{_s}(X), n_{_s}(n_₀)) -> MINUS(activate(X), 0)
MINUS(n_{_s}(X''), n_{_s}(Y)) -> MINUS(X'', activate(Y))
MINUS(n_{_s}(X), n_{_s}(n_{_s}(X''))) -> MINUS(activate(X), s(X''))

Rules:

minus(n_₀, Y) -> 0
minus(n_{_s}(X), n_{_s}(Y)) -> minus(activate(X), activate(Y))
geq(X, n_₀) -> true
geq(n_₀, n_{_s}(Y)) -> false
geq(n_{_s}(X), n_{_s}(Y)) -> geq(activate(X), activate(Y))
div(0, n_{_s}(Y)) -> 0
div(s(X), n_{_s}(Y)) -> if(geq(X, activate(Y)), n_{_s}(div(minus(X, activate(Y)), n_{_s}(activate(Y)))), n_₀)
if(true, X, Y) -> activate(X)
if(false, X, Y) -> activate(Y)
0 -> n_₀
s(X) -> n_{_s}(X)
activate(n_₀) -> 0
activate(n_{_s}(X)) -> s(X)
activate(X) -> X

Strategy:

innermost

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

MINUS(n_{_s}(X), n_{_s}(n_₀)) -> MINUS(activate(X), 0)

one new Dependency Pair is created:

MINUS(n_{_s}(X), n_{_s}(n_₀)) -> MINUS(activate(X), n_₀)

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 3

             ↳Rw

...

               →DP Problem 6

                 ↳Rewriting Transformation

       →DP Problem 2

         ↳Remaining

Dependency Pairs:

MINUS(n_{_s}(X), n_{_s}(Y')) -> MINUS(activate(X), Y')
MINUS(n_{_s}(X), n_{_s}(n_{_s}(X''))) -> MINUS(activate(X), s(X''))
MINUS(n_{_s}(X''), n_{_s}(Y)) -> MINUS(X'', activate(Y))
MINUS(n_{_s}(n_{_s}(X'')), n_{_s}(Y)) -> MINUS(n_{_s}(X''), activate(Y))

Rules:

minus(n_₀, Y) -> 0
minus(n_{_s}(X), n_{_s}(Y)) -> minus(activate(X), activate(Y))
geq(X, n_₀) -> true
geq(n_₀, n_{_s}(Y)) -> false
geq(n_{_s}(X), n_{_s}(Y)) -> geq(activate(X), activate(Y))
div(0, n_{_s}(Y)) -> 0
div(s(X), n_{_s}(Y)) -> if(geq(X, activate(Y)), n_{_s}(div(minus(X, activate(Y)), n_{_s}(activate(Y)))), n_₀)
if(true, X, Y) -> activate(X)
if(false, X, Y) -> activate(Y)
0 -> n_₀
s(X) -> n_{_s}(X)
activate(n_₀) -> 0
activate(n_{_s}(X)) -> s(X)
activate(X) -> X

Strategy:

innermost

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

MINUS(n_{_s}(X), n_{_s}(n_{_s}(X''))) -> MINUS(activate(X), s(X''))

one new Dependency Pair is created:

MINUS(n_{_s}(X), n_{_s}(n_{_s}(X''))) -> MINUS(activate(X), n_{_s}(X''))

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:
MINUS(n_{_s}(X), n_{_s}(n_{_s}(X''))) -> MINUS(activate(X), n_{_s}(X''))
MINUS(n_{_s}(n_{_s}(X'')), n_{_s}(Y)) -> MINUS(n_{_s}(X''), activate(Y))
MINUS(n_{_s}(X''), n_{_s}(Y)) -> MINUS(X'', activate(Y))
MINUS(n_{_s}(X), n_{_s}(Y')) -> MINUS(activate(X), Y')

Rules:

minus(n_₀, Y) -> 0
minus(n_{_s}(X), n_{_s}(Y)) -> minus(activate(X), activate(Y))
geq(X, n_₀) -> true
geq(n_₀, n_{_s}(Y)) -> false
geq(n_{_s}(X), n_{_s}(Y)) -> geq(activate(X), activate(Y))
div(0, n_{_s}(Y)) -> 0
div(s(X), n_{_s}(Y)) -> if(geq(X, activate(Y)), n_{_s}(div(minus(X, activate(Y)), n_{_s}(activate(Y)))), n_₀)
if(true, X, Y) -> activate(X)
if(false, X, Y) -> activate(Y)
0 -> n_₀
s(X) -> n_{_s}(X)
activate(n_₀) -> 0
activate(n_{_s}(X)) -> s(X)
activate(X) -> X

Strategy:
innermost
Dependency Pair:
GEQ(n_{_s}(X), n_{_s}(Y)) -> GEQ(activate(X), activate(Y))

Rules:

minus(n_₀, Y) -> 0
minus(n_{_s}(X), n_{_s}(Y)) -> minus(activate(X), activate(Y))
geq(X, n_₀) -> true
geq(n_₀, n_{_s}(Y)) -> false
geq(n_{_s}(X), n_{_s}(Y)) -> geq(activate(X), activate(Y))
div(0, n_{_s}(Y)) -> 0
div(s(X), n_{_s}(Y)) -> if(geq(X, activate(Y)), n_{_s}(div(minus(X, activate(Y)), n_{_s}(activate(Y)))), n_₀)
if(true, X, Y) -> activate(X)
if(false, X, Y) -> activate(Y)
0 -> n_₀
s(X) -> n_{_s}(X)
activate(n_₀) -> 0
activate(n_{_s}(X)) -> s(X)
activate(X) -> X

Strategy:
innermost

     ↳DPs

       →DP Problem 1

         ↳Nar

       →DP Problem 2

         ↳Remaining Obligation(s)

The following remains to be proven:

Dependency Pairs:
MINUS(n_{_s}(X), n_{_s}(n_{_s}(X''))) -> MINUS(activate(X), n_{_s}(X''))
MINUS(n_{_s}(n_{_s}(X'')), n_{_s}(Y)) -> MINUS(n_{_s}(X''), activate(Y))
MINUS(n_{_s}(X''), n_{_s}(Y)) -> MINUS(X'', activate(Y))
MINUS(n_{_s}(X), n_{_s}(Y')) -> MINUS(activate(X), Y')

Rules:

minus(n_₀, Y) -> 0
minus(n_{_s}(X), n_{_s}(Y)) -> minus(activate(X), activate(Y))
geq(X, n_₀) -> true
geq(n_₀, n_{_s}(Y)) -> false
geq(n_{_s}(X), n_{_s}(Y)) -> geq(activate(X), activate(Y))
div(0, n_{_s}(Y)) -> 0
div(s(X), n_{_s}(Y)) -> if(geq(X, activate(Y)), n_{_s}(div(minus(X, activate(Y)), n_{_s}(activate(Y)))), n_₀)
if(true, X, Y) -> activate(X)
if(false, X, Y) -> activate(Y)
0 -> n_₀
s(X) -> n_{_s}(X)
activate(n_₀) -> 0
activate(n_{_s}(X)) -> s(X)
activate(X) -> X

Strategy:
innermost
Dependency Pair:
GEQ(n_{_s}(X), n_{_s}(Y)) -> GEQ(activate(X), activate(Y))

Rules:

minus(n_₀, Y) -> 0
minus(n_{_s}(X), n_{_s}(Y)) -> minus(activate(X), activate(Y))
geq(X, n_₀) -> true
geq(n_₀, n_{_s}(Y)) -> false
geq(n_{_s}(X), n_{_s}(Y)) -> geq(activate(X), activate(Y))
div(0, n_{_s}(Y)) -> 0
div(s(X), n_{_s}(Y)) -> if(geq(X, activate(Y)), n_{_s}(div(minus(X, activate(Y)), n_{_s}(activate(Y)))), n_₀)
if(true, X, Y) -> activate(X)
if(false, X, Y) -> activate(Y)
0 -> n_₀
s(X) -> n_{_s}(X)
activate(n_₀) -> 0
activate(n_{_s}(X)) -> s(X)
activate(X) -> X

Strategy:
innermost

Innermost Termination of R could not be shown.
Duration:
0:01 minutes