Termination Proof using AProVE

Term Rewriting System R:
[Z, X, Y]
fst(0, Z) -> nil
fst(s(X), cons(Y, Z)) -> cons(Y, fst(X, Z))
from(X) -> cons(X, from(s(X)))
add(0, X) -> X
add(s(X), Y) -> s(add(X, Y))
len(nil) -> 0
len(cons(X, Z)) -> s(len(Z))

Innermost Termination of R to be shown.

     ↳Dependency Pair Analysis

R contains the following Dependency Pairs:

FST(s(X), cons(Y, Z)) -> FST(X, Z)
FROM(X) -> FROM(s(X))
ADD(s(X), Y) -> ADD(X, Y)
LEN(cons(X, Z)) -> LEN(Z)

Furthermore, R contains four SCCs.

     ↳DPs

       →DP Problem 1

         ↳Forward Instantiation Transformation

       →DP Problem 2

         ↳Inst

       →DP Problem 3

         ↳Remaining

       →DP Problem 4

         ↳Remaining

Dependency Pair:

FST(s(X), cons(Y, Z)) -> FST(X, Z)

Rules:

fst(0, Z) -> nil
fst(s(X), cons(Y, Z)) -> cons(Y, fst(X, Z))
from(X) -> cons(X, from(s(X)))
add(0, X) -> X
add(s(X), Y) -> s(add(X, Y))
len(nil) -> 0
len(cons(X, Z)) -> s(len(Z))

Strategy:

innermost

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

FST(s(X), cons(Y, Z)) -> FST(X, Z)

one new Dependency Pair is created:

FST(s(s(X'')), cons(Y, cons(Y'', Z''))) -> FST(s(X''), cons(Y'', Z''))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

           →DP Problem 5

             ↳Forward Instantiation Transformation

       →DP Problem 2

         ↳Inst

       →DP Problem 3

         ↳Remaining

       →DP Problem 4

         ↳Remaining

Dependency Pair:

FST(s(s(X'')), cons(Y, cons(Y'', Z''))) -> FST(s(X''), cons(Y'', Z''))

Rules:

fst(0, Z) -> nil
fst(s(X), cons(Y, Z)) -> cons(Y, fst(X, Z))
from(X) -> cons(X, from(s(X)))
add(0, X) -> X
add(s(X), Y) -> s(add(X, Y))
len(nil) -> 0
len(cons(X, Z)) -> s(len(Z))

Strategy:

innermost

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

FST(s(s(X'')), cons(Y, cons(Y'', Z''))) -> FST(s(X''), cons(Y'', Z''))

one new Dependency Pair is created:

FST(s(s(s(X''''))), cons(Y, cons(Y''0, cons(Y'''', Z'''')))) -> FST(s(s(X'''')), cons(Y''0, cons(Y'''', Z'''')))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

           →DP Problem 5

             ↳FwdInst

...

               →DP Problem 6

                 ↳Argument Filtering and Ordering

       →DP Problem 2

         ↳Inst

       →DP Problem 3

         ↳Remaining

       →DP Problem 4

         ↳Remaining

Dependency Pair:

FST(s(s(s(X''''))), cons(Y, cons(Y''0, cons(Y'''', Z'''')))) -> FST(s(s(X'''')), cons(Y''0, cons(Y'''', Z'''')))

Rules:

fst(0, Z) -> nil
fst(s(X), cons(Y, Z)) -> cons(Y, fst(X, Z))
from(X) -> cons(X, from(s(X)))
add(0, X) -> X
add(s(X), Y) -> s(add(X, Y))
len(nil) -> 0
len(cons(X, Z)) -> s(len(Z))

Strategy:

innermost

The following dependency pair can be strictly oriented:

FST(s(s(s(X''''))), cons(Y, cons(Y''0, cons(Y'''', Z'''')))) -> FST(s(s(X'''')), cons(Y''0, cons(Y'''', Z'''')))

There are no usable rules for innermost w.r.t. to the AFS that need to be oriented.
Used ordering: Polynomial ordering with Polynomial interpretation:

POL(FST(x₁, x₂)) = 1 + x₁ + x₂

POL(cons(x₁, x₂)) = x₁ + x₂

POL(s(x₁)) = 1 + x₁

resulting in one new DP problem.
Used Argument Filtering System:

FST(x₁, x₂) -> FST(x₁, x₂)
s(x₁) -> s(x₁)
cons(x₁, x₂) -> cons(x₁, x₂)

     ↳DPs

       →DP Problem 1

         ↳FwdInst

           →DP Problem 5

             ↳FwdInst

...

               →DP Problem 7

                 ↳Dependency Graph

       →DP Problem 2

         ↳Inst

       →DP Problem 3

         ↳Remaining

       →DP Problem 4

         ↳Remaining

Dependency Pair:

Rules:

fst(0, Z) -> nil
fst(s(X), cons(Y, Z)) -> cons(Y, fst(X, Z))
from(X) -> cons(X, from(s(X)))
add(0, X) -> X
add(s(X), Y) -> s(add(X, Y))
len(nil) -> 0
len(cons(X, Z)) -> s(len(Z))

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳Instantiation Transformation

       →DP Problem 3

         ↳Remaining

       →DP Problem 4

         ↳Remaining

Dependency Pair:

FROM(X) -> FROM(s(X))

Rules:

fst(0, Z) -> nil
fst(s(X), cons(Y, Z)) -> cons(Y, fst(X, Z))
from(X) -> cons(X, from(s(X)))
add(0, X) -> X
add(s(X), Y) -> s(add(X, Y))
len(nil) -> 0
len(cons(X, Z)) -> s(len(Z))

Strategy:

innermost

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

FROM(X) -> FROM(s(X))

one new Dependency Pair is created:

FROM(s(X'')) -> FROM(s(s(X'')))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳Inst

           →DP Problem 8

             ↳Instantiation Transformation

       →DP Problem 3

         ↳Remaining

       →DP Problem 4

         ↳Remaining

Dependency Pair:

FROM(s(X'')) -> FROM(s(s(X'')))

Rules:

fst(0, Z) -> nil
fst(s(X), cons(Y, Z)) -> cons(Y, fst(X, Z))
from(X) -> cons(X, from(s(X)))
add(0, X) -> X
add(s(X), Y) -> s(add(X, Y))
len(nil) -> 0
len(cons(X, Z)) -> s(len(Z))

Strategy:

innermost

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

FROM(s(X'')) -> FROM(s(s(X'')))

one new Dependency Pair is created:

FROM(s(s(X''''))) -> FROM(s(s(s(X''''))))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳Inst

           →DP Problem 8

             ↳Inst

...

               →DP Problem 9

                 ↳Instantiation Transformation

       →DP Problem 3

         ↳Remaining

       →DP Problem 4

         ↳Remaining

Dependency Pair:

FROM(s(s(X''''))) -> FROM(s(s(s(X''''))))

Rules:

fst(0, Z) -> nil
fst(s(X), cons(Y, Z)) -> cons(Y, fst(X, Z))
from(X) -> cons(X, from(s(X)))
add(0, X) -> X
add(s(X), Y) -> s(add(X, Y))
len(nil) -> 0
len(cons(X, Z)) -> s(len(Z))

Strategy:

innermost

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

FROM(s(s(X''''))) -> FROM(s(s(s(X''''))))

one new Dependency Pair is created:

FROM(s(s(s(X'''''')))) -> FROM(s(s(s(s(X'''''')))))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳Inst

           →DP Problem 8

             ↳Inst

...

               →DP Problem 10

                 ↳Instantiation Transformation

       →DP Problem 3

         ↳Remaining

       →DP Problem 4

         ↳Remaining

Dependency Pair:

FROM(s(s(s(X'''''')))) -> FROM(s(s(s(s(X'''''')))))

Rules:

fst(0, Z) -> nil
fst(s(X), cons(Y, Z)) -> cons(Y, fst(X, Z))
from(X) -> cons(X, from(s(X)))
add(0, X) -> X
add(s(X), Y) -> s(add(X, Y))
len(nil) -> 0
len(cons(X, Z)) -> s(len(Z))

Strategy:

innermost

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

FROM(s(s(s(X'''''')))) -> FROM(s(s(s(s(X'''''')))))

one new Dependency Pair is created:

FROM(s(s(s(s(X''''''''))))) -> FROM(s(s(s(s(s(X''''''''))))))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳Inst

           →DP Problem 8

             ↳Inst

...

               →DP Problem 11

                 ↳Instantiation Transformation

       →DP Problem 3

         ↳Remaining

       →DP Problem 4

         ↳Remaining

Dependency Pair:

FROM(s(s(s(s(X''''''''))))) -> FROM(s(s(s(s(s(X''''''''))))))

Rules:

fst(0, Z) -> nil
fst(s(X), cons(Y, Z)) -> cons(Y, fst(X, Z))
from(X) -> cons(X, from(s(X)))
add(0, X) -> X
add(s(X), Y) -> s(add(X, Y))
len(nil) -> 0
len(cons(X, Z)) -> s(len(Z))

Strategy:

innermost

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

FROM(s(s(s(s(X''''''''))))) -> FROM(s(s(s(s(s(X''''''''))))))

one new Dependency Pair is created:

FROM(s(s(s(s(s(X'''''''''')))))) -> FROM(s(s(s(s(s(s(X'''''''''')))))))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳Inst

       →DP Problem 3

         ↳Remaining Obligation(s)

       →DP Problem 4

         ↳Remaining Obligation(s)

The following remains to be proven:

Dependency Pair:
FROM(s(s(s(s(s(X'''''''''')))))) -> FROM(s(s(s(s(s(s(X'''''''''')))))))

Rules:

fst(0, Z) -> nil
fst(s(X), cons(Y, Z)) -> cons(Y, fst(X, Z))
from(X) -> cons(X, from(s(X)))
add(0, X) -> X
add(s(X), Y) -> s(add(X, Y))
len(nil) -> 0
len(cons(X, Z)) -> s(len(Z))

Strategy:
innermost
Dependency Pair:
ADD(s(X), Y) -> ADD(X, Y)

Rules:

fst(0, Z) -> nil
fst(s(X), cons(Y, Z)) -> cons(Y, fst(X, Z))
from(X) -> cons(X, from(s(X)))
add(0, X) -> X
add(s(X), Y) -> s(add(X, Y))
len(nil) -> 0
len(cons(X, Z)) -> s(len(Z))

Strategy:
innermost
Dependency Pair:
LEN(cons(X, Z)) -> LEN(Z)

Rules:

fst(0, Z) -> nil
fst(s(X), cons(Y, Z)) -> cons(Y, fst(X, Z))
from(X) -> cons(X, from(s(X)))
add(0, X) -> X
add(s(X), Y) -> s(add(X, Y))
len(nil) -> 0
len(cons(X, Z)) -> s(len(Z))

Strategy:
innermost

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳Inst

       →DP Problem 3

         ↳Remaining Obligation(s)

       →DP Problem 4

         ↳Remaining Obligation(s)

The following remains to be proven:

Dependency Pair:
FROM(s(s(s(s(s(X'''''''''')))))) -> FROM(s(s(s(s(s(s(X'''''''''')))))))

Rules:

fst(0, Z) -> nil
fst(s(X), cons(Y, Z)) -> cons(Y, fst(X, Z))
from(X) -> cons(X, from(s(X)))
add(0, X) -> X
add(s(X), Y) -> s(add(X, Y))
len(nil) -> 0
len(cons(X, Z)) -> s(len(Z))

Strategy:
innermost
Dependency Pair:
ADD(s(X), Y) -> ADD(X, Y)

Rules:

fst(0, Z) -> nil
fst(s(X), cons(Y, Z)) -> cons(Y, fst(X, Z))
from(X) -> cons(X, from(s(X)))
add(0, X) -> X
add(s(X), Y) -> s(add(X, Y))
len(nil) -> 0
len(cons(X, Z)) -> s(len(Z))

Strategy:
innermost
Dependency Pair:
LEN(cons(X, Z)) -> LEN(Z)

Rules:

fst(0, Z) -> nil
fst(s(X), cons(Y, Z)) -> cons(Y, fst(X, Z))
from(X) -> cons(X, from(s(X)))
add(0, X) -> X
add(s(X), Y) -> s(add(X, Y))
len(nil) -> 0
len(cons(X, Z)) -> s(len(Z))

Strategy:
innermost

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳Inst

       →DP Problem 3

         ↳Remaining Obligation(s)

       →DP Problem 4

         ↳Remaining Obligation(s)

The following remains to be proven:

Dependency Pair:
FROM(s(s(s(s(s(X'''''''''')))))) -> FROM(s(s(s(s(s(s(X'''''''''')))))))

Rules:

fst(0, Z) -> nil
fst(s(X), cons(Y, Z)) -> cons(Y, fst(X, Z))
from(X) -> cons(X, from(s(X)))
add(0, X) -> X
add(s(X), Y) -> s(add(X, Y))
len(nil) -> 0
len(cons(X, Z)) -> s(len(Z))

Strategy:
innermost
Dependency Pair:
ADD(s(X), Y) -> ADD(X, Y)

Rules:

fst(0, Z) -> nil
fst(s(X), cons(Y, Z)) -> cons(Y, fst(X, Z))
from(X) -> cons(X, from(s(X)))
add(0, X) -> X
add(s(X), Y) -> s(add(X, Y))
len(nil) -> 0
len(cons(X, Z)) -> s(len(Z))

Strategy:
innermost
Dependency Pair:
LEN(cons(X, Z)) -> LEN(Z)

Rules:

fst(0, Z) -> nil
fst(s(X), cons(Y, Z)) -> cons(Y, fst(X, Z))
from(X) -> cons(X, from(s(X)))
add(0, X) -> X
add(s(X), Y) -> s(add(X, Y))
len(nil) -> 0
len(cons(X, Z)) -> s(len(Z))

Strategy:
innermost

Innermost Termination of R could not be shown.
Duration:
0:00 minutes

POL(FST(x₁, x₂))	= 1 + x₁ + x₂
POL(cons(x₁, x₂))	= x₁ + x₂
POL(s(x₁))	= 1 + x₁