Termination Proof using AProVE

Term Rewriting System R:
[X, Y]
f(X) -> if(X, c, n_{_f}(n_{_true}))
f(X) -> n_{_f}(X)
if(true, X, Y) -> X
if(false, X, Y) -> activate(Y)
true -> n_{_true}
activate(n_{_f}(X)) -> f(activate(X))
activate(n_{_true}) -> true
activate(X) -> X

Innermost Termination of R to be shown.

     ↳Dependency Pair Analysis

R contains the following Dependency Pairs:

F(X) -> IF(X, c, n_{_f}(n_{_true}))
IF(false, X, Y) -> ACTIVATE(Y)
ACTIVATE(n_{_f}(X)) -> F(activate(X))
ACTIVATE(n_{_f}(X)) -> ACTIVATE(X)
ACTIVATE(n_{_true}) -> TRUE

Furthermore, R contains one SCC.

     ↳DPs

       →DP Problem 1

         ↳Narrowing Transformation

Dependency Pairs:

ACTIVATE(n_{_f}(X)) -> ACTIVATE(X)
ACTIVATE(n_{_f}(X)) -> F(activate(X))
IF(false, X, Y) -> ACTIVATE(Y)
F(X) -> IF(X, c, n_{_f}(n_{_true}))

Rules:

f(X) -> if(X, c, n_{_f}(n_{_true}))
f(X) -> n_{_f}(X)
if(true, X, Y) -> X
if(false, X, Y) -> activate(Y)
true -> n_{_true}
activate(n_{_f}(X)) -> f(activate(X))
activate(n_{_true}) -> true
activate(X) -> X

Strategy:

innermost

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

ACTIVATE(n_{_f}(X)) -> F(activate(X))

three new Dependency Pairs are created:

ACTIVATE(n_{_f}(n_{_f}(X''))) -> F(f(activate(X'')))
ACTIVATE(n_{_f}(n_{_true})) -> F(true)
ACTIVATE(n_{_f}(X'')) -> F(X'')

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Rewriting Transformation

Dependency Pairs:

ACTIVATE(n_{_f}(X'')) -> F(X'')
ACTIVATE(n_{_f}(n_{_true})) -> F(true)
IF(false, X, Y) -> ACTIVATE(Y)
F(X) -> IF(X, c, n_{_f}(n_{_true}))
ACTIVATE(n_{_f}(n_{_f}(X''))) -> F(f(activate(X'')))
ACTIVATE(n_{_f}(X)) -> ACTIVATE(X)

Rules:

f(X) -> if(X, c, n_{_f}(n_{_true}))
f(X) -> n_{_f}(X)
if(true, X, Y) -> X
if(false, X, Y) -> activate(Y)
true -> n_{_true}
activate(n_{_f}(X)) -> f(activate(X))
activate(n_{_true}) -> true
activate(X) -> X

Strategy:

innermost

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

ACTIVATE(n_{_f}(n_{_true})) -> F(true)

one new Dependency Pair is created:

ACTIVATE(n_{_f}(n_{_true})) -> F(n_{_true})

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Rw

...

               →DP Problem 3

                 ↳Instantiation Transformation

Dependency Pairs:

ACTIVATE(n_{_f}(n_{_true})) -> F(n_{_true})
ACTIVATE(n_{_f}(n_{_f}(X''))) -> F(f(activate(X'')))
ACTIVATE(n_{_f}(X)) -> ACTIVATE(X)
IF(false, X, Y) -> ACTIVATE(Y)
F(X) -> IF(X, c, n_{_f}(n_{_true}))
ACTIVATE(n_{_f}(X'')) -> F(X'')

Rules:

f(X) -> if(X, c, n_{_f}(n_{_true}))
f(X) -> n_{_f}(X)
if(true, X, Y) -> X
if(false, X, Y) -> activate(Y)
true -> n_{_true}
activate(n_{_f}(X)) -> f(activate(X))
activate(n_{_true}) -> true
activate(X) -> X

Strategy:

innermost

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

IF(false, X, Y) -> ACTIVATE(Y)

one new Dependency Pair is created:

IF(false, c, n_{_f}(n_{_true})) -> ACTIVATE(n_{_f}(n_{_true}))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Rw

...

               →DP Problem 4

                 ↳Forward Instantiation Transformation

Dependency Pairs:

ACTIVATE(n_{_f}(X'')) -> F(X'')
ACTIVATE(n_{_f}(n_{_f}(X''))) -> F(f(activate(X'')))
ACTIVATE(n_{_f}(X)) -> ACTIVATE(X)
IF(false, c, n_{_f}(n_{_true})) -> ACTIVATE(n_{_f}(n_{_true}))
F(X) -> IF(X, c, n_{_f}(n_{_true}))
ACTIVATE(n_{_f}(n_{_true})) -> F(n_{_true})

Rules:

f(X) -> if(X, c, n_{_f}(n_{_true}))
f(X) -> n_{_f}(X)
if(true, X, Y) -> X
if(false, X, Y) -> activate(Y)
true -> n_{_true}
activate(n_{_f}(X)) -> f(activate(X))
activate(n_{_true}) -> true
activate(X) -> X

Strategy:

innermost

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

F(X) -> IF(X, c, n_{_f}(n_{_true}))

one new Dependency Pair is created:

F(false) -> IF(false, c, n_{_f}(n_{_true}))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Rw

...

               →DP Problem 5

                 ↳Forward Instantiation Transformation

Dependency Pairs:

ACTIVATE(n_{_f}(n_{_f}(X''))) -> F(f(activate(X'')))
ACTIVATE(n_{_f}(X)) -> ACTIVATE(X)
IF(false, c, n_{_f}(n_{_true})) -> ACTIVATE(n_{_f}(n_{_true}))
F(false) -> IF(false, c, n_{_f}(n_{_true}))
ACTIVATE(n_{_f}(X'')) -> F(X'')

Rules:

f(X) -> if(X, c, n_{_f}(n_{_true}))
f(X) -> n_{_f}(X)
if(true, X, Y) -> X
if(false, X, Y) -> activate(Y)
true -> n_{_true}
activate(n_{_f}(X)) -> f(activate(X))
activate(n_{_true}) -> true
activate(X) -> X

Strategy:

innermost

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

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

three new Dependency Pairs are created:

ACTIVATE(n_{_f}(n_{_f}(X''))) -> ACTIVATE(n_{_f}(X''))
ACTIVATE(n_{_f}(n_{_f}(n_{_f}(X'''')))) -> ACTIVATE(n_{_f}(n_{_f}(X'''')))
ACTIVATE(n_{_f}(n_{_f}(X''''))) -> ACTIVATE(n_{_f}(X''''))

The transformation is resulting in two new DP problems:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Rw

...

               →DP Problem 6

                 ↳Instantiation Transformation

Dependency Pairs:

ACTIVATE(n_{_f}(X'')) -> F(X'')
IF(false, c, n_{_f}(n_{_true})) -> ACTIVATE(n_{_f}(n_{_true}))
F(false) -> IF(false, c, n_{_f}(n_{_true}))

Rules:

f(X) -> if(X, c, n_{_f}(n_{_true}))
f(X) -> n_{_f}(X)
if(true, X, Y) -> X
if(false, X, Y) -> activate(Y)
true -> n_{_true}
activate(n_{_f}(X)) -> f(activate(X))
activate(n_{_true}) -> true
activate(X) -> X

Strategy:

innermost

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

ACTIVATE(n_{_f}(X'')) -> F(X'')

one new Dependency Pair is created:

ACTIVATE(n_{_f}(n_{_true})) -> F(n_{_true})

The transformation is resulting in no new DP problems.

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Rw

...

               →DP Problem 7

                 ↳Argument Filtering and Ordering

Dependency Pairs:

ACTIVATE(n_{_f}(n_{_f}(X''''))) -> ACTIVATE(n_{_f}(X''''))
ACTIVATE(n_{_f}(n_{_f}(n_{_f}(X'''')))) -> ACTIVATE(n_{_f}(n_{_f}(X'''')))
ACTIVATE(n_{_f}(n_{_f}(X''))) -> ACTIVATE(n_{_f}(X''))

Rules:

f(X) -> if(X, c, n_{_f}(n_{_true}))
f(X) -> n_{_f}(X)
if(true, X, Y) -> X
if(false, X, Y) -> activate(Y)
true -> n_{_true}
activate(n_{_f}(X)) -> f(activate(X))
activate(n_{_true}) -> true
activate(X) -> X

Strategy:

innermost

The following dependency pairs can be strictly oriented:

ACTIVATE(n_{_f}(n_{_f}(X''''))) -> ACTIVATE(n_{_f}(X''''))
ACTIVATE(n_{_f}(n_{_f}(n_{_f}(X'''')))) -> ACTIVATE(n_{_f}(n_{_f}(X'''')))
ACTIVATE(n_{_f}(n_{_f}(X''))) -> ACTIVATE(n_{_f}(X''))

There are no usable rules for innermost w.r.t. to the AFS that need to be oriented.
Used ordering: Lexicographic Path Order with Non-Strict Precedence with Quasi Precedence:

trivial

resulting in one new DP problem.
Used Argument Filtering System:

ACTIVATE(x₁) -> ACTIVATE(x₁)
n_{_f}(x₁) -> n_{_f}(x₁)

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Rw

...

               →DP Problem 8

                 ↳Dependency Graph

Dependency Pair:

Rules:

f(X) -> if(X, c, n_{_f}(n_{_true}))
f(X) -> n_{_f}(X)
if(true, X, Y) -> X
if(false, X, Y) -> activate(Y)
true -> n_{_true}
activate(n_{_f}(X)) -> f(activate(X))
activate(n_{_true}) -> true
activate(X) -> X

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

Innermost Termination of R successfully shown.
Duration:
0:02 minutes