Termination Proof using AProVE

Term Rewriting System R:
[x, y]
+(x, 0) -> x
+(x, s(y)) -> s(+(x, y))
*(x, 0) -> 0
*(x, s(y)) -> +(*(x, y), x)
ge(x, 0) -> true
ge(0, s(y)) -> false
ge(s(x), s(y)) -> ge(x, y)
-(x, 0) -> x
-(s(x), s(y)) -> -(x, y)
fact(x) -> iffact(x, ge(x, s(s(0))))
iffact(x, true) -> *(x, fact(-(x, s(0))))
iffact(x, false) -> s(0)

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:

+'(x, s(y)) -> +'(x, y)
*'(x, s(y)) -> +'(*(x, y), x)
*'(x, s(y)) -> *'(x, y)
GE(s(x), s(y)) -> GE(x, y)
-'(s(x), s(y)) -> -'(x, y)
FACT(x) -> IFFACT(x, ge(x, s(s(0))))
FACT(x) -> GE(x, s(s(0)))
IFFACT(x, true) -> *'(x, fact(-(x, s(0))))
IFFACT(x, true) -> FACT(-(x, s(0)))
IFFACT(x, true) -> -'(x, s(0))

Furthermore, R contains five SCCs.

     ↳OC

       →TRS2

         ↳DPs

           →DP Problem 1

             ↳Usable Rules (Innermost)

           →DP Problem 2

             ↳UsableRules

           →DP Problem 3

             ↳UsableRules

           →DP Problem 4

             ↳UsableRules

           →DP Problem 5

             ↳UsableRules

Dependency Pair:

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

Rules:

+(x, 0) -> x
+(x, s(y)) -> s(+(x, y))
*(x, 0) -> 0
*(x, s(y)) -> +(*(x, y), x)
ge(x, 0) -> true
ge(0, s(y)) -> false
ge(s(x), s(y)) -> ge(x, y)
-(x, 0) -> x
-(s(x), s(y)) -> -(x, y)
fact(x) -> iffact(x, ge(x, s(s(0))))
iffact(x, true) -> *(x, fact(-(x, s(0))))
iffact(x, false) -> s(0)

Strategy:

innermost

As we are in the innermost case, we can delete all 12 non-usable-rules.

     ↳OC

       →TRS2

         ↳DPs

           →DP Problem 1

             ↳UsableRules

...

               →DP Problem 6

                 ↳Size-Change Principle

           →DP Problem 2

             ↳UsableRules

           →DP Problem 3

             ↳UsableRules

           →DP Problem 4

             ↳UsableRules

           →DP Problem 5

             ↳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

           →DP Problem 4

             ↳UsableRules

           →DP Problem 5

             ↳UsableRules

Dependency Pair:

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

Rules:

+(x, 0) -> x
+(x, s(y)) -> s(+(x, y))
*(x, 0) -> 0
*(x, s(y)) -> +(*(x, y), x)
ge(x, 0) -> true
ge(0, s(y)) -> false
ge(s(x), s(y)) -> ge(x, y)
-(x, 0) -> x
-(s(x), s(y)) -> -(x, y)
fact(x) -> iffact(x, ge(x, s(s(0))))
iffact(x, true) -> *(x, fact(-(x, s(0))))
iffact(x, false) -> s(0)

Strategy:

innermost

As we are in the innermost case, we can delete all 12 non-usable-rules.

     ↳OC

       →TRS2

         ↳DPs

           →DP Problem 1

             ↳UsableRules

           →DP Problem 2

             ↳UsableRules

...

               →DP Problem 7

                 ↳Size-Change Principle

           →DP Problem 3

             ↳UsableRules

           →DP Problem 4

             ↳UsableRules

           →DP Problem 5

             ↳UsableRules

Dependency Pair:

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

Rule:

none

Strategy:

innermost

We number the DPs as follows:

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

           →DP Problem 5

             ↳UsableRules

Dependency Pair:

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

Rules:

+(x, 0) -> x
+(x, s(y)) -> s(+(x, y))
*(x, 0) -> 0
*(x, s(y)) -> +(*(x, y), x)
ge(x, 0) -> true
ge(0, s(y)) -> false
ge(s(x), s(y)) -> ge(x, y)
-(x, 0) -> x
-(s(x), s(y)) -> -(x, y)
fact(x) -> iffact(x, ge(x, s(s(0))))
iffact(x, true) -> *(x, fact(-(x, s(0))))
iffact(x, false) -> s(0)

Strategy:

innermost

As we are in the innermost case, we can delete all 12 non-usable-rules.

     ↳OC

       →TRS2

         ↳DPs

           →DP Problem 1

             ↳UsableRules

           →DP Problem 2

             ↳UsableRules

           →DP Problem 3

             ↳UsableRules

...

               →DP Problem 8

                 ↳Size-Change Principle

           →DP Problem 4

             ↳UsableRules

           →DP Problem 5

             ↳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

             ↳UsableRules

           →DP Problem 3

             ↳UsableRules

           →DP Problem 4

             ↳Usable Rules (Innermost)

           →DP Problem 5

             ↳UsableRules

Dependency Pair:

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

Rules:

+(x, 0) -> x
+(x, s(y)) -> s(+(x, y))
*(x, 0) -> 0
*(x, s(y)) -> +(*(x, y), x)
ge(x, 0) -> true
ge(0, s(y)) -> false
ge(s(x), s(y)) -> ge(x, y)
-(x, 0) -> x
-(s(x), s(y)) -> -(x, y)
fact(x) -> iffact(x, ge(x, s(s(0))))
iffact(x, true) -> *(x, fact(-(x, s(0))))
iffact(x, false) -> s(0)

Strategy:

innermost

As we are in the innermost case, we can delete all 12 non-usable-rules.

     ↳OC

       →TRS2

         ↳DPs

           →DP Problem 1

             ↳UsableRules

           →DP Problem 2

             ↳UsableRules

           →DP Problem 3

             ↳UsableRules

           →DP Problem 4

             ↳UsableRules

...

               →DP Problem 9

                 ↳Size-Change Principle

           →DP Problem 5

             ↳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

             ↳UsableRules

           →DP Problem 5

             ↳Usable Rules (Innermost)

Dependency Pairs:

IFFACT(x, true) -> FACT(-(x, s(0)))
FACT(x) -> IFFACT(x, ge(x, s(s(0))))

Rules:

+(x, 0) -> x
+(x, s(y)) -> s(+(x, y))
*(x, 0) -> 0
*(x, s(y)) -> +(*(x, y), x)
ge(x, 0) -> true
ge(0, s(y)) -> false
ge(s(x), s(y)) -> ge(x, y)
-(x, 0) -> x
-(s(x), s(y)) -> -(x, y)
fact(x) -> iffact(x, ge(x, s(s(0))))
iffact(x, true) -> *(x, fact(-(x, s(0))))
iffact(x, false) -> s(0)

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 4

             ↳UsableRules

           →DP Problem 5

             ↳UsableRules

...

               →DP Problem 10

                 ↳Narrowing Transformation

Dependency Pairs:

IFFACT(x, true) -> FACT(-(x, s(0)))
FACT(x) -> IFFACT(x, ge(x, s(s(0))))

Rules:

-(s(x), s(y)) -> -(x, y)
-(x, 0) -> x
ge(x, 0) -> true
ge(0, s(y)) -> false
ge(s(x), s(y)) -> ge(x, y)

Strategy:

innermost

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

FACT(x) -> IFFACT(x, ge(x, s(s(0))))

two new Dependency Pairs are created:

FACT(0) -> IFFACT(0, false)
FACT(s(x'')) -> IFFACT(s(x''), ge(x'', s(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 5

             ↳UsableRules

...

               →DP Problem 11

                 ↳Narrowing Transformation

Dependency Pairs:

FACT(s(x'')) -> IFFACT(s(x''), ge(x'', s(0)))
IFFACT(x, true) -> FACT(-(x, s(0)))

Rules:

-(s(x), s(y)) -> -(x, y)
-(x, 0) -> x
ge(x, 0) -> true
ge(0, s(y)) -> false
ge(s(x), s(y)) -> ge(x, y)

Strategy:

innermost

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

IFFACT(x, true) -> FACT(-(x, s(0)))

one new Dependency Pair is created:

IFFACT(s(x''), true) -> FACT(-(x'', 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 5

             ↳UsableRules

...

               →DP Problem 12

                 ↳Rewriting Transformation

Dependency Pairs:

IFFACT(s(x''), true) -> FACT(-(x'', 0))
FACT(s(x'')) -> IFFACT(s(x''), ge(x'', s(0)))

Rules:

-(s(x), s(y)) -> -(x, y)
-(x, 0) -> x
ge(x, 0) -> true
ge(0, s(y)) -> false
ge(s(x), s(y)) -> ge(x, y)

Strategy:

innermost

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

IFFACT(s(x''), true) -> FACT(-(x'', 0))

one new Dependency Pair is created:

IFFACT(s(x''), true) -> 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 4

             ↳UsableRules

           →DP Problem 5

             ↳UsableRules

...

               →DP Problem 13

                 ↳Usable Rules (Innermost)

Dependency Pairs:

IFFACT(s(x''), true) -> FACT(x'')
FACT(s(x'')) -> IFFACT(s(x''), ge(x'', s(0)))

Rules:

-(s(x), s(y)) -> -(x, y)
-(x, 0) -> x
ge(x, 0) -> true
ge(0, s(y)) -> false
ge(s(x), s(y)) -> ge(x, y)

Strategy:

innermost

As we are in the innermost case, we can delete all 2 non-usable-rules.

     ↳OC

       →TRS2

         ↳DPs

           →DP Problem 1

             ↳UsableRules

           →DP Problem 2

             ↳UsableRules

           →DP Problem 3

             ↳UsableRules

           →DP Problem 4

             ↳UsableRules

           →DP Problem 5

             ↳UsableRules

...

               →DP Problem 14

                 ↳Size-Change Principle

Dependency Pairs:

IFFACT(s(x''), true) -> FACT(x'')
FACT(s(x'')) -> IFFACT(s(x''), ge(x'', s(0)))

Rules:

ge(x, 0) -> true
ge(0, s(y)) -> false
ge(s(x), s(y)) -> ge(x, y)

Strategy:

innermost

We number the DPs as follows:

IFFACT(s(x''), true) -> FACT(x'')
FACT(s(x'')) -> IFFACT(s(x''), ge(x'', s(0)))

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

{1}	,	{1}
1	>	1

{2}	,	{2}
1	=	1

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

{1}	,	{2}
1	>	1

{2}	,	{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:
0:06 minutes