Termination Proof using AProVE

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

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:

FACT(s(x)) -> *'(s(x), fact(p(s(x))))
FACT(s(x)) -> FACT(p(s(x)))
FACT(s(x)) -> P(s(x))
*'(s(x), y) -> +'(*(x, y), y)
*'(s(x), y) -> *'(x, y)
+'(x, s(y)) -> +'(x, y)

Furthermore, R contains three SCCs.

     ↳OC

       →TRS2

         ↳DPs

           →DP Problem 1

             ↳Usable Rules (Innermost)

           →DP Problem 2

             ↳UsableRules

           →DP Problem 3

             ↳UsableRules

Dependency Pair:

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

Rules:

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

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 4

                 ↳Size-Change Principle

           →DP Problem 2

             ↳UsableRules

           →DP Problem 3

             ↳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

             ↳Usable Rules (Innermost)

           →DP Problem 3

             ↳UsableRules

Dependency Pair:

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

Rules:

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

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 5

                 ↳Size-Change Principle

           →DP Problem 3

             ↳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)

Dependency Pair:

FACT(s(x)) -> FACT(p(s(x)))

Rules:

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

Strategy:

innermost

As we are in the innermost case, we can delete all 6 non-usable-rules.

     ↳OC

       →TRS2

         ↳DPs

           →DP Problem 1

             ↳UsableRules

           →DP Problem 2

             ↳UsableRules

           →DP Problem 3

             ↳UsableRules

...

               →DP Problem 6

                 ↳Rewriting Transformation

Dependency Pair:

FACT(s(x)) -> FACT(p(s(x)))

Rule:

p(s(x)) -> x

Strategy:

innermost

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

FACT(s(x)) -> FACT(p(s(x)))

one new Dependency Pair is created:

FACT(s(x)) -> FACT(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 7

                 ↳Usable Rules (Innermost)

Dependency Pair:

FACT(s(x)) -> FACT(x)

Rule:

p(s(x)) -> 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 8

                 ↳Size-Change Principle

Dependency Pair:

FACT(s(x)) -> FACT(x)

Rule:

none

Strategy:

innermost

We number the DPs as follows:

FACT(s(x)) -> FACT(x)

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

{1}	,	{1}
1	>	1

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

{1}	,	{1}
1	>	1

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.

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