Termination Proof using AProVE

Term Rewriting System R:
[x, y, z]
app(app(times, x), app(app(plus, y), app(s, z))) -> app(app(plus, app(app(times, x), app(app(plus, y), app(app(times, app(s, z)), 0)))), app(app(times, x), app(s, z)))
app(app(times, x), 0) -> 0
app(app(times, x), app(s, y)) -> app(app(plus, app(app(times, x), y)), x)
app(app(plus, x), 0) -> x
app(app(plus, x), app(s, y)) -> app(s, app(app(plus, x), y))

Innermost Termination of R to be shown.

     ↳Removing Redundant Rules for Innermost Termination

Removing the following rules from R which left hand sides contain non normal subterms

app(app(times, x), app(app(plus, y), app(s, z))) -> app(app(plus, app(app(times, x), app(app(plus, y), app(app(times, app(s, z)), 0)))), app(app(times, x), app(s, z)))

     ↳RRRI

       →TRS2

         ↳Dependency Pair Analysis

R contains the following Dependency Pairs:

APP(app(times, x), app(s, y)) -> APP(app(plus, app(app(times, x), y)), x)
APP(app(times, x), app(s, y)) -> APP(plus, app(app(times, x), y))
APP(app(times, x), app(s, y)) -> APP(app(times, x), y)
APP(app(plus, x), app(s, y)) -> APP(s, app(app(plus, x), y))
APP(app(plus, x), app(s, y)) -> APP(app(plus, x), y)

Furthermore, R contains two SCCs.

     ↳RRRI

       →TRS2

         ↳DPs

           →DP Problem 1

             ↳Non-Overlappingness Check

           →DP Problem 2

             ↳NOC

Dependency Pair:

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

Rules:

app(app(times, x), 0) -> 0
app(app(times, x), app(s, y)) -> app(app(plus, app(app(times, x), y)), x)
app(app(plus, x), 0) -> x
app(app(plus, x), app(s, y)) -> app(s, app(app(plus, x), y))

R does not overlap into P. Moreover, R is locally confluent (all critical pairs are trivially joinable).Hence we can switch to innermost.
The transformation is resulting in one subcycle:

     ↳RRRI

       →TRS2

         ↳DPs

           →DP Problem 1

             ↳NOC

...

               →DP Problem 3

                 ↳Usable Rules (Innermost)

           →DP Problem 2

             ↳NOC

Dependency Pair:

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

Rules:

app(app(times, x), 0) -> 0
app(app(times, x), app(s, y)) -> app(app(plus, app(app(times, x), y)), x)
app(app(plus, x), 0) -> x
app(app(plus, x), app(s, y)) -> app(s, app(app(plus, x), y))

Strategy:

innermost

As we are in the innermost case, we can delete all 4 non-usable-rules.

     ↳RRRI

       →TRS2

         ↳DPs

           →DP Problem 1

             ↳NOC

...

               →DP Problem 4

                 ↳A-Transformation

           →DP Problem 2

             ↳NOC

Dependency Pair:

APP(app(plus, x), app(s, y)) -> APP(app(plus, 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.

     ↳RRRI

       →TRS2

         ↳DPs

           →DP Problem 1

             ↳NOC

...

               →DP Problem 5

                 ↳Size-Change Principle

           →DP Problem 2

             ↳NOC

Dependency Pair:

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

Rule:

none

Strategy:

innermost

We number the DPs as follows:

PLUS(x, s(y)) -> PLUS(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.

     ↳RRRI

       →TRS2

         ↳DPs

           →DP Problem 1

             ↳NOC

           →DP Problem 2

             ↳Non-Overlappingness Check

Dependency Pair:

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

Rules:

app(app(times, x), 0) -> 0
app(app(times, x), app(s, y)) -> app(app(plus, app(app(times, x), y)), x)
app(app(plus, x), 0) -> x
app(app(plus, x), app(s, y)) -> app(s, app(app(plus, x), y))

R does not overlap into P. Moreover, R is locally confluent (all critical pairs are trivially joinable).Hence we can switch to innermost.
The transformation is resulting in one subcycle:

     ↳RRRI

       →TRS2

         ↳DPs

           →DP Problem 1

             ↳NOC

           →DP Problem 2

             ↳NOC

...

               →DP Problem 6

                 ↳Usable Rules (Innermost)

Dependency Pair:

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

Rules:

app(app(times, x), 0) -> 0
app(app(times, x), app(s, y)) -> app(app(plus, app(app(times, x), y)), x)
app(app(plus, x), 0) -> x
app(app(plus, x), app(s, y)) -> app(s, app(app(plus, x), y))

Strategy:

innermost

As we are in the innermost case, we can delete all 4 non-usable-rules.

     ↳RRRI

       →TRS2

         ↳DPs

           →DP Problem 1

             ↳NOC

           →DP Problem 2

             ↳NOC

...

               →DP Problem 7

                 ↳A-Transformation

Dependency Pair:

APP(app(times, x), app(s, y)) -> APP(app(times, 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.

     ↳RRRI

       →TRS2

         ↳DPs

           →DP Problem 1

             ↳NOC

           →DP Problem 2

             ↳NOC

...

               →DP Problem 8

                 ↳Size-Change Principle

Dependency Pair:

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

Rule:

none

Strategy:

innermost

We number the DPs as follows:

TIMES(x, s(y)) -> TIMES(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.

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