Termination Proof using AProVE

Term Rewriting System R:
[x, y]
-(x, 0) -> x
-(s(x), s(y)) -> -(x, y)
+(0, y) -> y
+(s(x), y) -> s(+(x, y))
*(x, 0) -> 0
*(x, s(y)) -> +(x, *(x, y))
f(s(x)) -> f(-(*(s(s(0)), s(x)), s(s(x))))

Termination of R to be shown.

     ↳Overlay and local confluence Check

The TRS is overlay and locally confluent (all critical pairs are trivially joinable).Hence, we can switch to innermost.

     ↳OC

       →TRS2

         ↳Dependency Pair Analysis

R contains the following Dependency Pairs:

-'(s(x), s(y)) -> -'(x, y)
+'(s(x), y) -> +'(x, y)
*'(x, s(y)) -> +'(x, *(x, y))
*'(x, s(y)) -> *'(x, y)
F(s(x)) -> F(-(*(s(s(0)), s(x)), s(s(x))))
F(s(x)) -> -'(*(s(s(0)), s(x)), s(s(x)))
F(s(x)) -> *'(s(s(0)), s(x))

Furthermore, R contains four SCCs.

     ↳OC

       →TRS2

         ↳DPs

           →DP Problem 1

             ↳Usable Rules (Innermost)

           →DP Problem 2

             ↳UsableRules

           →DP Problem 3

             ↳UsableRules

           →DP Problem 4

             ↳UsableRules

Dependency Pair:

-'(s(x), s(y)) -> -'(x, y)

Rules:

-(x, 0) -> x
-(s(x), s(y)) -> -(x, y)
+(0, y) -> y
+(s(x), y) -> s(+(x, y))
*(x, 0) -> 0
*(x, s(y)) -> +(x, *(x, y))
f(s(x)) -> f(-(*(s(s(0)), s(x)), s(s(x))))

Strategy:

innermost

As we are in the innermost case, we can delete all 7 non-usable-rules.

     ↳OC

       →TRS2

         ↳DPs

           →DP Problem 1

             ↳UsableRules

...

               →DP Problem 5

                 ↳Size-Change Principle

           →DP Problem 2

             ↳UsableRules

           →DP Problem 3

             ↳UsableRules

           →DP Problem 4

             ↳UsableRules

Dependency Pair:

-'(s(x), s(y)) -> -'(x, y)

Rule:

none

Strategy:

innermost

We number the DPs as follows:

-'(s(x), s(y)) -> -'(x, y)

and get the following Size-Change Graph(s):

{1}	,	{1}
1	>	1
2	>	2

which lead(s) to this/these maximal multigraph(s):

{1}	,	{1}
1	>	1
2	>	2

D_P: empty set
Oriented Rules: none

We used the order Homeomorphic Embedding Order with Non-Strict Precedence.

trivial

with Argument Filtering System:

s(x₁) -> s(x₁)

We obtain no new DP problems.

     ↳OC

       →TRS2

         ↳DPs

           →DP Problem 1

             ↳UsableRules

           →DP Problem 2

             ↳Usable Rules (Innermost)

           →DP Problem 3

             ↳UsableRules

           →DP Problem 4

             ↳UsableRules

Dependency Pair:

+'(s(x), y) -> +'(x, y)

Rules:

-(x, 0) -> x
-(s(x), s(y)) -> -(x, y)
+(0, y) -> y
+(s(x), y) -> s(+(x, y))
*(x, 0) -> 0
*(x, s(y)) -> +(x, *(x, y))
f(s(x)) -> f(-(*(s(s(0)), s(x)), s(s(x))))

Strategy:

innermost

As we are in the innermost case, we can delete all 7 non-usable-rules.

     ↳OC

       →TRS2

         ↳DPs

           →DP Problem 1

             ↳UsableRules

           →DP Problem 2

             ↳UsableRules

...

               →DP Problem 6

                 ↳Size-Change Principle

           →DP Problem 3

             ↳UsableRules

           →DP Problem 4

             ↳UsableRules

Dependency Pair:

+'(s(x), y) -> +'(x, y)

Rule:

none

Strategy:

innermost

We number the DPs as follows:

+'(s(x), y) -> +'(x, y)

and get the following Size-Change Graph(s):

{1}	,	{1}
1	>	1
2	=	2

which lead(s) to this/these maximal multigraph(s):

{1}	,	{1}
1	>	1
2	=	2

D_P: empty set
Oriented Rules: none

We used the order Homeomorphic Embedding Order with Non-Strict Precedence.

trivial

with Argument Filtering System:

s(x₁) -> s(x₁)

We obtain no new DP problems.

     ↳OC

       →TRS2

         ↳DPs

           →DP Problem 1

             ↳UsableRules

           →DP Problem 2

             ↳UsableRules

           →DP Problem 3

             ↳Usable Rules (Innermost)

           →DP Problem 4

             ↳UsableRules

Dependency Pair:

*'(x, s(y)) -> *'(x, y)

Rules:

-(x, 0) -> x
-(s(x), s(y)) -> -(x, y)
+(0, y) -> y
+(s(x), y) -> s(+(x, y))
*(x, 0) -> 0
*(x, s(y)) -> +(x, *(x, y))
f(s(x)) -> f(-(*(s(s(0)), s(x)), s(s(x))))

Strategy:

innermost

As we are in the innermost case, we can delete all 7 non-usable-rules.

     ↳OC

       →TRS2

         ↳DPs

           →DP Problem 1

             ↳UsableRules

           →DP Problem 2

             ↳UsableRules

           →DP Problem 3

             ↳UsableRules

...

               →DP Problem 7

                 ↳Size-Change Principle

           →DP Problem 4

             ↳UsableRules

Dependency Pair:

*'(x, s(y)) -> *'(x, y)

Rule:

none

Strategy:

innermost

We number the DPs as follows:

*'(x, s(y)) -> *'(x, y)

and get the following Size-Change Graph(s):

{1}	,	{1}
1	=	1
2	>	2

which lead(s) to this/these maximal multigraph(s):

{1}	,	{1}
1	=	1
2	>	2

D_P: empty set
Oriented Rules: none

We used the order Homeomorphic Embedding Order with Non-Strict Precedence.

trivial

with Argument Filtering System:

s(x₁) -> s(x₁)

We obtain no new DP problems.

     ↳OC

       →TRS2

         ↳DPs

           →DP Problem 1

             ↳UsableRules

           →DP Problem 2

             ↳UsableRules

           →DP Problem 3

             ↳UsableRules

           →DP Problem 4

             ↳Usable Rules (Innermost)

Dependency Pair:

F(s(x)) -> F(-(*(s(s(0)), s(x)), s(s(x))))

Rules:

-(x, 0) -> x
-(s(x), s(y)) -> -(x, y)
+(0, y) -> y
+(s(x), y) -> s(+(x, y))
*(x, 0) -> 0
*(x, s(y)) -> +(x, *(x, y))
f(s(x)) -> f(-(*(s(s(0)), s(x)), s(s(x))))

Strategy:

innermost

As we are in the innermost case, we can delete all 1 non-usable-rules.

     ↳OC

       →TRS2

         ↳DPs

           →DP Problem 1

             ↳UsableRules

           →DP Problem 2

             ↳UsableRules

           →DP Problem 3

             ↳UsableRules

           →DP Problem 4

             ↳UsableRules

...

               →DP Problem 8

                 ↳Rewriting Transformation

Dependency Pair:

F(s(x)) -> F(-(*(s(s(0)), s(x)), s(s(x))))

Rules:

+(0, y) -> y
+(s(x), y) -> s(+(x, y))
-(s(x), s(y)) -> -(x, y)
-(x, 0) -> x
*(x, s(y)) -> +(x, *(x, y))
*(x, 0) -> 0

Strategy:

innermost

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

F(s(x)) -> F(-(*(s(s(0)), s(x)), s(s(x))))

one new Dependency Pair is created:

F(s(x)) -> F(-(+(s(s(0)), *(s(s(0)), x)), s(s(x))))

The transformation is resulting in one new DP problem:

     ↳OC

       →TRS2

         ↳DPs

           →DP Problem 1

             ↳UsableRules

           →DP Problem 2

             ↳UsableRules

           →DP Problem 3

             ↳UsableRules

           →DP Problem 4

             ↳UsableRules

...

               →DP Problem 9

                 ↳Rewriting Transformation

Dependency Pair:

F(s(x)) -> F(-(+(s(s(0)), *(s(s(0)), x)), s(s(x))))

Rules:

+(0, y) -> y
+(s(x), y) -> s(+(x, y))
-(s(x), s(y)) -> -(x, y)
-(x, 0) -> x
*(x, s(y)) -> +(x, *(x, y))
*(x, 0) -> 0

Strategy:

innermost

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

F(s(x)) -> F(-(+(s(s(0)), *(s(s(0)), x)), s(s(x))))

one new Dependency Pair is created:

F(s(x)) -> F(-(s(+(s(0), *(s(s(0)), x))), s(s(x))))

The transformation is resulting in one new DP problem:

     ↳OC

       →TRS2

         ↳DPs

           →DP Problem 1

             ↳UsableRules

           →DP Problem 2

             ↳UsableRules

           →DP Problem 3

             ↳UsableRules

           →DP Problem 4

             ↳UsableRules

...

               →DP Problem 10

                 ↳Rewriting Transformation

Dependency Pair:

F(s(x)) -> F(-(s(+(s(0), *(s(s(0)), x))), s(s(x))))

Rules:

+(0, y) -> y
+(s(x), y) -> s(+(x, y))
-(s(x), s(y)) -> -(x, y)
-(x, 0) -> x
*(x, s(y)) -> +(x, *(x, y))
*(x, 0) -> 0

Strategy:

innermost

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

F(s(x)) -> F(-(s(+(s(0), *(s(s(0)), x))), s(s(x))))

one new Dependency Pair is created:

F(s(x)) -> F(-(+(s(0), *(s(s(0)), x)), s(x)))

The transformation is resulting in one new DP problem:

     ↳OC

       →TRS2

         ↳DPs

           →DP Problem 1

             ↳UsableRules

           →DP Problem 2

             ↳UsableRules

           →DP Problem 3

             ↳UsableRules

           →DP Problem 4

             ↳UsableRules

...

               →DP Problem 11

                 ↳Rewriting Transformation

Dependency Pair:

F(s(x)) -> F(-(+(s(0), *(s(s(0)), x)), s(x)))

Rules:

+(0, y) -> y
+(s(x), y) -> s(+(x, y))
-(s(x), s(y)) -> -(x, y)
-(x, 0) -> x
*(x, s(y)) -> +(x, *(x, y))
*(x, 0) -> 0

Strategy:

innermost

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

F(s(x)) -> F(-(+(s(0), *(s(s(0)), x)), s(x)))

one new Dependency Pair is created:

F(s(x)) -> F(-(s(+(0, *(s(s(0)), x))), s(x)))

The transformation is resulting in one new DP problem:

     ↳OC

       →TRS2

         ↳DPs

           →DP Problem 1

             ↳UsableRules

           →DP Problem 2

             ↳UsableRules

           →DP Problem 3

             ↳UsableRules

           →DP Problem 4

             ↳UsableRules

...

               →DP Problem 12

                 ↳Rewriting Transformation

Dependency Pair:

F(s(x)) -> F(-(s(+(0, *(s(s(0)), x))), s(x)))

Rules:

+(0, y) -> y
+(s(x), y) -> s(+(x, y))
-(s(x), s(y)) -> -(x, y)
-(x, 0) -> x
*(x, s(y)) -> +(x, *(x, y))
*(x, 0) -> 0

Strategy:

innermost

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

F(s(x)) -> F(-(s(+(0, *(s(s(0)), x))), s(x)))

one new Dependency Pair is created:

F(s(x)) -> F(-(+(0, *(s(s(0)), x)), x))

The transformation is resulting in one new DP problem:

     ↳OC

       →TRS2

         ↳DPs

           →DP Problem 1

             ↳UsableRules

           →DP Problem 2

             ↳UsableRules

           →DP Problem 3

             ↳UsableRules

           →DP Problem 4

             ↳UsableRules

...

               →DP Problem 13

                 ↳Rewriting Transformation

Dependency Pair:

F(s(x)) -> F(-(+(0, *(s(s(0)), x)), x))

Rules:

+(0, y) -> y
+(s(x), y) -> s(+(x, y))
-(s(x), s(y)) -> -(x, y)
-(x, 0) -> x
*(x, s(y)) -> +(x, *(x, y))
*(x, 0) -> 0

Strategy:

innermost

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

F(s(x)) -> F(-(+(0, *(s(s(0)), x)), x))

one new Dependency Pair is created:

F(s(x)) -> F(-(*(s(s(0)), x), x))

The transformation is resulting in one new DP problem:

     ↳OC

       →TRS2

         ↳DPs

           →DP Problem 1

             ↳UsableRules

           →DP Problem 2

             ↳UsableRules

           →DP Problem 3

             ↳UsableRules

           →DP Problem 4

             ↳UsableRules

...

               →DP Problem 14

                 ↳Narrowing Transformation

Dependency Pair:

F(s(x)) -> F(-(*(s(s(0)), x), x))

Rules:

+(0, y) -> y
+(s(x), y) -> s(+(x, y))
-(s(x), s(y)) -> -(x, y)
-(x, 0) -> x
*(x, s(y)) -> +(x, *(x, y))
*(x, 0) -> 0

Strategy:

innermost

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

F(s(x)) -> F(-(*(s(s(0)), x), x))

three new Dependency Pairs are created:

F(s(0)) -> F(*(s(s(0)), 0))
F(s(s(y'))) -> F(-(+(s(s(0)), *(s(s(0)), y')), s(y')))
F(s(0)) -> F(-(0, 0))

The transformation is resulting in one new DP problem:

     ↳OC

       →TRS2

         ↳DPs

           →DP Problem 1

             ↳UsableRules

           →DP Problem 2

             ↳UsableRules

           →DP Problem 3

             ↳UsableRules

           →DP Problem 4

             ↳UsableRules

...

               →DP Problem 15

                 ↳Rewriting Transformation

Dependency Pairs:

F(s(0)) -> F(-(0, 0))
F(s(s(y'))) -> F(-(+(s(s(0)), *(s(s(0)), y')), s(y')))

Rules:

+(0, y) -> y
+(s(x), y) -> s(+(x, y))
-(s(x), s(y)) -> -(x, y)
-(x, 0) -> x
*(x, s(y)) -> +(x, *(x, y))
*(x, 0) -> 0

Strategy:

innermost

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

F(s(s(y'))) -> F(-(+(s(s(0)), *(s(s(0)), y')), s(y')))

one new Dependency Pair is created:

F(s(s(y'))) -> F(-(s(+(s(0), *(s(s(0)), y'))), s(y')))

The transformation is resulting in one new DP problem:

     ↳OC

       →TRS2

         ↳DPs

           →DP Problem 1

             ↳UsableRules

           →DP Problem 2

             ↳UsableRules

           →DP Problem 3

             ↳UsableRules

           →DP Problem 4

             ↳UsableRules

...

               →DP Problem 16

                 ↳Rewriting Transformation

Dependency Pairs:

F(s(s(y'))) -> F(-(s(+(s(0), *(s(s(0)), y'))), s(y')))
F(s(0)) -> F(-(0, 0))

Rules:

+(0, y) -> y
+(s(x), y) -> s(+(x, y))
-(s(x), s(y)) -> -(x, y)
-(x, 0) -> x
*(x, s(y)) -> +(x, *(x, y))
*(x, 0) -> 0

Strategy:

innermost

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

F(s(0)) -> F(-(0, 0))

one new Dependency Pair is created:

F(s(0)) -> F(0)

The transformation is resulting in one new DP problem:

     ↳OC

       →TRS2

         ↳DPs

           →DP Problem 1

             ↳UsableRules

           →DP Problem 2

             ↳UsableRules

           →DP Problem 3

             ↳UsableRules

           →DP Problem 4

             ↳UsableRules

...

               →DP Problem 17

                 ↳Rewriting Transformation

Dependency Pair:

F(s(s(y'))) -> F(-(s(+(s(0), *(s(s(0)), y'))), s(y')))

Rules:

+(0, y) -> y
+(s(x), y) -> s(+(x, y))
-(s(x), s(y)) -> -(x, y)
-(x, 0) -> x
*(x, s(y)) -> +(x, *(x, y))
*(x, 0) -> 0

Strategy:

innermost

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

F(s(s(y'))) -> F(-(s(+(s(0), *(s(s(0)), y'))), s(y')))

one new Dependency Pair is created:

F(s(s(y'))) -> F(-(+(s(0), *(s(s(0)), y')), y'))

The transformation is resulting in one new DP problem:

     ↳OC

       →TRS2

         ↳DPs

           →DP Problem 1

             ↳UsableRules

           →DP Problem 2

             ↳UsableRules

           →DP Problem 3

             ↳UsableRules

           →DP Problem 4

             ↳UsableRules

...

               →DP Problem 18

                 ↳Rewriting Transformation

Dependency Pair:

F(s(s(y'))) -> F(-(+(s(0), *(s(s(0)), y')), y'))

Rules:

+(0, y) -> y
+(s(x), y) -> s(+(x, y))
-(s(x), s(y)) -> -(x, y)
-(x, 0) -> x
*(x, s(y)) -> +(x, *(x, y))
*(x, 0) -> 0

Strategy:

innermost

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

F(s(s(y'))) -> F(-(+(s(0), *(s(s(0)), y')), y'))

one new Dependency Pair is created:

F(s(s(y'))) -> F(-(s(+(0, *(s(s(0)), y'))), y'))

The transformation is resulting in one new DP problem:

     ↳OC

       →TRS2

         ↳DPs

           →DP Problem 1

             ↳UsableRules

           →DP Problem 2

             ↳UsableRules

           →DP Problem 3

             ↳UsableRules

           →DP Problem 4

             ↳UsableRules

...

               →DP Problem 19

                 ↳Rewriting Transformation

Dependency Pair:

F(s(s(y'))) -> F(-(s(+(0, *(s(s(0)), y'))), y'))

Rules:

+(0, y) -> y
+(s(x), y) -> s(+(x, y))
-(s(x), s(y)) -> -(x, y)
-(x, 0) -> x
*(x, s(y)) -> +(x, *(x, y))
*(x, 0) -> 0

Strategy:

innermost

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

F(s(s(y'))) -> F(-(s(+(0, *(s(s(0)), y'))), y'))

one new Dependency Pair is created:

F(s(s(y'))) -> F(-(s(*(s(s(0)), y')), y'))

The transformation is resulting in one new DP problem:

     ↳OC

       →TRS2

         ↳DPs

           →DP Problem 1

             ↳UsableRules

           →DP Problem 2

             ↳UsableRules

           →DP Problem 3

             ↳UsableRules

           →DP Problem 4

             ↳UsableRules

...

               →DP Problem 20

                 ↳Remaining Obligation(s)

The following remains to be proven:
Dependency Pair:

F(s(s(y'))) -> F(-(s(*(s(s(0)), y')), y'))

Rules:

+(0, y) -> y
+(s(x), y) -> s(+(x, y))
-(s(x), s(y)) -> -(x, y)
-(x, 0) -> x
*(x, s(y)) -> +(x, *(x, y))
*(x, 0) -> 0

Strategy:

innermost

Termination of R could not be shown.
Duration:
0:27 minutes