Termination Proof using AProVE

Term Rewriting System R:
[y, x, f]
app(app(add, 0), y) -> y
app(app(add, app(s, x)), y) -> app(s, app(app(add, x), y))
app(app(mult, 0), y) -> 0
app(app(mult, app(s, x)), y) -> app(app(add, app(app(mult, x), y)), y)
app(app(app(rec, f), x), 0) -> x
app(app(app(rec, f), x), app(s, y)) -> app(app(f, app(s, y)), app(app(app(rec, f), x), y))
fact -> app(app(rec, mult), app(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:

APP(app(add, app(s, x)), y) -> APP(s, app(app(add, x), y))
APP(app(add, app(s, x)), y) -> APP(app(add, x), y)
APP(app(add, app(s, x)), y) -> APP(add, x)
APP(app(mult, app(s, x)), y) -> APP(app(add, app(app(mult, x), y)), y)
APP(app(mult, app(s, x)), y) -> APP(add, app(app(mult, x), y))
APP(app(mult, app(s, x)), y) -> APP(app(mult, x), y)
APP(app(mult, app(s, x)), y) -> APP(mult, x)
APP(app(app(rec, f), x), app(s, y)) -> APP(app(f, app(s, y)), app(app(app(rec, f), x), y))
APP(app(app(rec, f), x), app(s, y)) -> APP(f, app(s, y))
APP(app(app(rec, f), x), app(s, y)) -> APP(app(app(rec, f), x), y)
FACT -> APP(app(rec, mult), app(s, 0))
FACT -> APP(rec, mult)
FACT -> APP(s, 0)

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

Rules:

app(app(add, 0), y) -> y
app(app(add, app(s, x)), y) -> app(s, app(app(add, x), y))
app(app(mult, 0), y) -> 0
app(app(mult, app(s, x)), y) -> app(app(add, app(app(mult, x), y)), y)
app(app(app(rec, f), x), 0) -> x
app(app(app(rec, f), x), app(s, y)) -> app(app(f, app(s, y)), app(app(app(rec, f), x), y))
fact -> app(app(rec, mult), app(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 4

                 ↳A-Transformation

           →DP Problem 2

             ↳UsableRules

           →DP Problem 3

             ↳UsableRules

Dependency Pair:

APP(app(add, app(s, x)), y) -> APP(app(add, 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:

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

Rule:

none

Strategy:

innermost

We number the DPs as follows:

ADD(s(x), y) -> ADD(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(mult, app(s, x)), y) -> APP(app(mult, x), y)

Rules:

app(app(add, 0), y) -> y
app(app(add, app(s, x)), y) -> app(s, app(app(add, x), y))
app(app(mult, 0), y) -> 0
app(app(mult, app(s, x)), y) -> app(app(add, app(app(mult, x), y)), y)
app(app(app(rec, f), x), 0) -> x
app(app(app(rec, f), x), app(s, y)) -> app(app(f, app(s, y)), app(app(app(rec, f), x), y))
fact -> app(app(rec, mult), app(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 6

                 ↳A-Transformation

           →DP Problem 3

             ↳UsableRules

Dependency Pair:

APP(app(mult, app(s, x)), y) -> APP(app(mult, 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:

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

Rule:

none

Strategy:

innermost

We number the DPs as follows:

MULT(s(x), y) -> MULT(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 Pairs:

APP(app(app(rec, f), x), app(s, y)) -> APP(app(app(rec, f), x), y)
APP(app(app(rec, f), x), app(s, y)) -> APP(f, app(s, y))
APP(app(app(rec, f), x), app(s, y)) -> APP(app(f, app(s, y)), app(app(app(rec, f), x), y))

Rules:

app(app(add, 0), y) -> y
app(app(add, app(s, x)), y) -> app(s, app(app(add, x), y))
app(app(mult, 0), y) -> 0
app(app(mult, app(s, x)), y) -> app(app(add, app(app(mult, x), y)), y)
app(app(app(rec, f), x), 0) -> x
app(app(app(rec, f), x), app(s, y)) -> app(app(f, app(s, y)), app(app(app(rec, f), x), y))
fact -> app(app(rec, mult), app(s, 0))

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

                 ↳Narrowing Transformation

Dependency Pairs:

Rules:

app(app(app(rec, f), x), 0) -> x
app(app(app(rec, f), x), app(s, y)) -> app(app(f, app(s, y)), app(app(app(rec, f), x), y))
app(app(mult, app(s, x)), y) -> app(app(add, app(app(mult, x), y)), y)
app(app(add, app(s, x)), y) -> app(s, app(app(add, x), y))
app(app(mult, 0), y) -> 0
app(app(add, 0), y) -> y

Strategy:

innermost

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

APP(app(app(rec, f), x), app(s, y)) -> APP(app(f, app(s, y)), app(app(app(rec, f), x), y))

seven new Dependency Pairs are created:

APP(app(app(rec, app(app(rec, f''), x'')), x), app(s, y'')) -> APP(app(app(f'', app(s, y'')), app(app(app(rec, f''), x''), y'')), app(app(app(rec, app(app(rec, f''), x'')), x), y''))
APP(app(app(rec, app(mult, app(s, x''))), x), app(s, y'')) -> APP(app(app(add, app(app(mult, x''), app(s, y''))), app(s, y'')), app(app(app(rec, app(mult, app(s, x''))), x), y''))
APP(app(app(rec, app(add, app(s, x''))), x), app(s, y'')) -> APP(app(s, app(app(add, x''), app(s, y''))), app(app(app(rec, app(add, app(s, x''))), x), y''))
APP(app(app(rec, app(mult, 0)), x), app(s, y'')) -> APP(0, app(app(app(rec, app(mult, 0)), x), y''))
APP(app(app(rec, app(add, 0)), x), app(s, y'')) -> APP(app(s, y''), app(app(app(rec, app(add, 0)), x), y''))
APP(app(app(rec, f''), x''), app(s, 0)) -> APP(app(f'', app(s, 0)), x'')
APP(app(app(rec, f''), x''), app(s, app(s, y''))) -> APP(app(f'', app(s, app(s, y''))), app(app(f'', app(s, y'')), app(app(app(rec, f''), x''), 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 9

                 ↳Remaining Obligation(s)

The following remains to be proven:
Dependency Pairs:

APP(app(app(rec, f''), x''), app(s, app(s, y''))) -> APP(app(f'', app(s, app(s, y''))), app(app(f'', app(s, y'')), app(app(app(rec, f''), x''), y'')))
APP(app(app(rec, f''), x''), app(s, 0)) -> APP(app(f'', app(s, 0)), x'')
APP(app(app(rec, app(mult, app(s, x''))), x), app(s, y'')) -> APP(app(app(add, app(app(mult, x''), app(s, y''))), app(s, y'')), app(app(app(rec, app(mult, app(s, x''))), x), y''))
APP(app(app(rec, app(app(rec, f''), x'')), x), app(s, y'')) -> APP(app(app(f'', app(s, y'')), app(app(app(rec, f''), x''), y'')), app(app(app(rec, app(app(rec, f''), x'')), x), y''))
APP(app(app(rec, f), x), app(s, y)) -> APP(f, app(s, y))
APP(app(app(rec, f), x), app(s, y)) -> APP(app(app(rec, f), x), y)

Rules:

app(app(app(rec, f), x), 0) -> x
app(app(app(rec, f), x), app(s, y)) -> app(app(f, app(s, y)), app(app(app(rec, f), x), y))
app(app(mult, app(s, x)), y) -> app(app(add, app(app(mult, x), y)), y)
app(app(add, app(s, x)), y) -> app(s, app(app(add, x), y))
app(app(mult, 0), y) -> 0
app(app(add, 0), y) -> y

Strategy:

innermost

The Proof could not be continued due to a Timeout.
Termination of R could not be shown.
Duration:
1:00 minutes