Termination Proof using AProVE

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

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

APP(app(+, x), app(s, y)) -> APP(app(+, x), y)

Rules:

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

                 ↳A-Transformation

           →DP Problem 2

             ↳UsableRules

           →DP Problem 3

             ↳UsableRules

Dependency Pair:

APP(app(+, x), app(s, y)) -> APP(app(+, x), y)

Rule:

none

Strategy:

innermost

We have an applicative DP problem with proper arity. Thus we can use the A-Transformation to obtain one new DP problem which consists of the A-transformed TRSs.

     ↳OC

       →TRS2

         ↳DPs

           →DP Problem 1

             ↳UsableRules

...

               →DP Problem 5

                 ↳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:

APP(app(*, app(s, x)), y) -> APP(app(*, x), y)

Rules:

app(p, app(s, x)) -> x
app(fact, 0) -> app(s, 0)
app(fact, app(s, x)) -> app(app(*, app(s, x)), app(fact, app(p, app(s, x))))
app(app(*, 0), y) -> 0
app(app(*, app(s, x)), y) -> app(app(+, app(app(*, x), y)), y)
app(app(+, x), 0) -> x
app(app(+, x), app(s, y)) -> app(s, app(app(+, 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 6

                 ↳A-Transformation

           →DP Problem 3

             ↳UsableRules

Dependency Pair:

APP(app(*, app(s, x)), y) -> APP(app(*, x), y)

Rule:

none

Strategy:

innermost

We have an applicative DP problem with proper arity. Thus we can use the A-Transformation to obtain one new DP problem which consists of the A-transformed TRSs.

     ↳OC

       →TRS2

         ↳DPs

           →DP Problem 1

             ↳UsableRules

           →DP Problem 2

             ↳UsableRules

...

               →DP Problem 7

                 ↳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:

APP(fact, app(s, x)) -> APP(fact, app(p, app(s, x)))

Rules:

app(p, app(s, x)) -> x
app(fact, 0) -> app(s, 0)
app(fact, app(s, x)) -> app(app(*, app(s, x)), app(fact, app(p, app(s, x))))
app(app(*, 0), y) -> 0
app(app(*, app(s, x)), y) -> app(app(+, app(app(*, x), y)), y)
app(app(+, x), 0) -> x
app(app(+, x), app(s, y)) -> app(s, app(app(+, 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 8

                 ↳A-Transformation

Dependency Pair:

APP(fact, app(s, x)) -> APP(fact, app(p, app(s, x)))

Rule:

app(p, app(s, x)) -> x

Strategy:

innermost

We have an applicative DP problem with proper arity. Thus we can use the A-Transformation to obtain one new DP problem which consists of the A-transformed TRSs.

     ↳OC

       →TRS2

         ↳DPs

           →DP Problem 1

             ↳UsableRules

           →DP Problem 2

             ↳UsableRules

           →DP Problem 3

             ↳UsableRules

...

               →DP Problem 9

                 ↳Modular Removal of Rules

Dependency Pair:

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

Rule:

p(s(x)) -> x

Strategy:

innermost

We have the following set of usable rules:

p(s(x)) -> x

To remove rules and DPs from this DP problem we used the following monotonic and C_E-compatible order: Polynomial ordering.
Polynomial interpretation:

POL(FACT(x₁)) = 1 + x₁

POL(s(x₁)) = 1 + x₁

POL(p(x₁)) = x₁

We have the following set D of usable symbols: {FACT, s, p}
No Dependency Pairs can be deleted.
The following rules can be deleted as the lhs is strictly greater than the corresponding rhs:

p(s(x)) -> x

The result of this processor delivers one new DP problem.

     ↳OC

       →TRS2

         ↳DPs

           →DP Problem 1

             ↳UsableRules

           →DP Problem 2

             ↳UsableRules

           →DP Problem 3

             ↳UsableRules

...

               →DP Problem 10

                 ↳Dependency Graph

Dependency Pair:

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

Rule:

none

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

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

POL(FACT(x₁))	= 1 + x₁
POL(s(x₁))	= 1 + x₁
POL(p(x₁))	= x₁