Termination Proof using AProVE

Term Rewriting System R:
[z, x, y]
app(app(h, z), app(e, x)) -> app(app(h, app(c, z)), app(app(d, z), x))
app(app(d, z), app(app(g, 0), 0)) -> app(e, 0)
app(app(d, z), app(app(g, x), y)) -> app(app(g, app(e, x)), app(app(d, z), y))
app(app(d, app(c, z)), app(app(g, app(app(g, x), y)), 0)) -> app(app(g, app(app(d, app(c, z)), app(app(g, x), y))), app(app(d, z), app(app(g, x), y)))
app(app(g, app(e, x)), app(e, y)) -> app(e, app(app(g, x), y))

Innermost Termination of R to be shown.

     ↳Dependency Pair Analysis

R contains the following Dependency Pairs:

APP(app(h, z), app(e, x)) -> APP(app(h, app(c, z)), app(app(d, z), x))
APP(app(h, z), app(e, x)) -> APP(h, app(c, z))
APP(app(h, z), app(e, x)) -> APP(c, z)
APP(app(h, z), app(e, x)) -> APP(app(d, z), x)
APP(app(h, z), app(e, x)) -> APP(d, z)
APP(app(d, z), app(app(g, 0), 0)) -> APP(e, 0)
APP(app(d, z), app(app(g, x), y)) -> APP(app(g, app(e, x)), app(app(d, z), y))
APP(app(d, z), app(app(g, x), y)) -> APP(g, app(e, x))
APP(app(d, z), app(app(g, x), y)) -> APP(e, x)
APP(app(d, z), app(app(g, x), y)) -> APP(app(d, z), y)
APP(app(d, app(c, z)), app(app(g, app(app(g, x), y)), 0)) -> APP(app(g, app(app(d, app(c, z)), app(app(g, x), y))), app(app(d, z), app(app(g, x), y)))
APP(app(d, app(c, z)), app(app(g, app(app(g, x), y)), 0)) -> APP(g, app(app(d, app(c, z)), app(app(g, x), y)))
APP(app(d, app(c, z)), app(app(g, app(app(g, x), y)), 0)) -> APP(app(d, app(c, z)), app(app(g, x), y))
APP(app(d, app(c, z)), app(app(g, app(app(g, x), y)), 0)) -> APP(app(d, z), app(app(g, x), y))
APP(app(d, app(c, z)), app(app(g, app(app(g, x), y)), 0)) -> APP(d, z)
APP(app(g, app(e, x)), app(e, y)) -> APP(e, app(app(g, x), y))
APP(app(g, app(e, x)), app(e, y)) -> APP(app(g, x), y)
APP(app(g, app(e, x)), app(e, y)) -> APP(g, x)

Furthermore, R contains three SCCs.

     ↳DPs

       →DP Problem 1

         ↳Usable Rules (Innermost)

       →DP Problem 2

         ↳UsableRules

       →DP Problem 3

         ↳UsableRules

Dependency Pair:

APP(app(g, app(e, x)), app(e, y)) -> APP(app(g, x), y)

Rules:

app(app(h, z), app(e, x)) -> app(app(h, app(c, z)), app(app(d, z), x))
app(app(d, z), app(app(g, 0), 0)) -> app(e, 0)
app(app(d, z), app(app(g, x), y)) -> app(app(g, app(e, x)), app(app(d, z), y))
app(app(d, app(c, z)), app(app(g, app(app(g, x), y)), 0)) -> app(app(g, app(app(d, app(c, z)), app(app(g, x), y))), app(app(d, z), app(app(g, x), y)))
app(app(g, app(e, x)), app(e, y)) -> app(e, app(app(g, x), y))

Strategy:

innermost

As we are in the innermost case, we can delete all 5 non-usable-rules.

     ↳DPs

       →DP Problem 1

         ↳UsableRules

           →DP Problem 4

             ↳A-Transformation

       →DP Problem 2

         ↳UsableRules

       →DP Problem 3

         ↳UsableRules

Dependency Pair:

APP(app(g, app(e, x)), app(e, y)) -> APP(app(g, 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.

     ↳DPs

       →DP Problem 1

         ↳UsableRules

           →DP Problem 4

             ↳ATrans

...

               →DP Problem 5

                 ↳Size-Change Principle

       →DP Problem 2

         ↳UsableRules

       →DP Problem 3

         ↳UsableRules

Dependency Pair:

G(e(x), e(y)) -> G(x, y)

Rule:

none

Strategy:

innermost

We number the DPs as follows:

G(e(x), e(y)) -> G(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:

e(x₁) -> e(x₁)

We obtain no new DP problems.

     ↳DPs

       →DP Problem 1

         ↳UsableRules

       →DP Problem 2

         ↳Usable Rules (Innermost)

       →DP Problem 3

         ↳UsableRules

Dependency Pairs:

APP(app(d, app(c, z)), app(app(g, app(app(g, x), y)), 0)) -> APP(app(d, z), app(app(g, x), y))
APP(app(d, app(c, z)), app(app(g, app(app(g, x), y)), 0)) -> APP(app(d, app(c, z)), app(app(g, x), y))
APP(app(d, z), app(app(g, x), y)) -> APP(app(d, z), y)

Rules:

app(app(h, z), app(e, x)) -> app(app(h, app(c, z)), app(app(d, z), x))
app(app(d, z), app(app(g, 0), 0)) -> app(e, 0)
app(app(d, z), app(app(g, x), y)) -> app(app(g, app(e, x)), app(app(d, z), y))
app(app(d, app(c, z)), app(app(g, app(app(g, x), y)), 0)) -> app(app(g, app(app(d, app(c, z)), app(app(g, x), y))), app(app(d, z), app(app(g, x), y)))
app(app(g, app(e, x)), app(e, y)) -> app(e, app(app(g, x), y))

Strategy:

innermost

As we are in the innermost case, we can delete all 5 non-usable-rules.

     ↳DPs

       →DP Problem 1

         ↳UsableRules

       →DP Problem 2

         ↳UsableRules

           →DP Problem 6

             ↳A-Transformation

       →DP Problem 3

         ↳UsableRules

Dependency Pairs:

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.

     ↳DPs

       →DP Problem 1

         ↳UsableRules

       →DP Problem 2

         ↳UsableRules

           →DP Problem 6

             ↳ATrans

...

               →DP Problem 7

                 ↳Size-Change Principle

       →DP Problem 3

         ↳UsableRules

Dependency Pairs:

D(c(z), g(g(x, y), 0)) -> D(z, g(x, y))
D(c(z), g(g(x, y), 0)) -> D(c(z), g(x, y))
D(z, g(x, y)) -> D(z, y)

Rule:

none

Strategy:

innermost

We number the DPs as follows:

D(c(z), g(g(x, y), 0)) -> D(z, g(x, y))
D(c(z), g(g(x, y), 0)) -> D(c(z), g(x, y))
D(z, g(x, y)) -> D(z, y)

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

{1, 2, 3}	,	{1, 2, 3}
1	>	1
2	>	2

{1, 2, 3}	,	{1, 2, 3}
1	=	1
2	>	2

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

{1, 2, 3}	,	{1, 2, 3}
1	=	1
2	>	2

{1, 2, 3}	,	{1, 2, 3}
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:

c(x₁) -> c(x₁)
g(x₁, x₂) -> g(x₁, x₂)

We obtain no new DP problems.

     ↳DPs

       →DP Problem 1

         ↳UsableRules

       →DP Problem 2

         ↳UsableRules

       →DP Problem 3

         ↳Usable Rules (Innermost)

Dependency Pair:

APP(app(h, z), app(e, x)) -> APP(app(h, app(c, z)), app(app(d, z), x))

Rules:

app(app(h, z), app(e, x)) -> app(app(h, app(c, z)), app(app(d, z), x))
app(app(d, z), app(app(g, 0), 0)) -> app(e, 0)
app(app(d, z), app(app(g, x), y)) -> app(app(g, app(e, x)), app(app(d, z), y))
app(app(d, app(c, z)), app(app(g, app(app(g, x), y)), 0)) -> app(app(g, app(app(d, app(c, z)), app(app(g, x), y))), app(app(d, z), app(app(g, x), y)))
app(app(g, app(e, x)), app(e, y)) -> app(e, app(app(g, x), y))

Strategy:

innermost

As we are in the innermost case, we can delete all 1 non-usable-rules.

     ↳DPs

       →DP Problem 1

         ↳UsableRules

       →DP Problem 2

         ↳UsableRules

       →DP Problem 3

         ↳UsableRules

           →DP Problem 8

             ↳A-Transformation

Dependency Pair:

APP(app(h, z), app(e, x)) -> APP(app(h, app(c, z)), app(app(d, z), x))

Rules:

app(app(g, app(e, x)), app(e, y)) -> app(e, app(app(g, x), y))
app(app(d, z), app(app(g, 0), 0)) -> app(e, 0)
app(app(d, z), app(app(g, x), y)) -> app(app(g, app(e, x)), app(app(d, z), y))
app(app(d, app(c, z)), app(app(g, app(app(g, x), y)), 0)) -> app(app(g, app(app(d, app(c, z)), app(app(g, x), y))), app(app(d, z), app(app(g, x), y)))

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.

     ↳DPs

       →DP Problem 1

         ↳UsableRules

       →DP Problem 2

         ↳UsableRules

       →DP Problem 3

         ↳UsableRules

           →DP Problem 8

             ↳ATrans

...

               →DP Problem 9

                 ↳Narrowing Transformation

Dependency Pair:

H(z, e(x)) -> H(c(z), d(z, x))

Rules:

g(e(x), e(y)) -> e(g(x, y))
d(z, g(0, 0)) -> e(0)
d(z, g(x, y)) -> g(e(x), d(z, y))
d(c(z), g(g(x, y), 0)) -> g(d(c(z), g(x, y)), d(z, g(x, y)))

Strategy:

innermost

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

H(z, e(x)) -> H(c(z), d(z, x))

three new Dependency Pairs are created:

H(z'', e(g(0, 0))) -> H(c(z''), e(0))
H(z'', e(g(x'', y'))) -> H(c(z''), g(e(x''), d(z'', y')))
H(c(z''), e(g(g(x'', y'), 0))) -> H(c(c(z'')), g(d(c(z''), g(x'', y')), d(z'', g(x'', y'))))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳UsableRules

       →DP Problem 2

         ↳UsableRules

       →DP Problem 3

         ↳UsableRules

           →DP Problem 8

             ↳ATrans

...

               →DP Problem 10

                 ↳Instantiation Transformation

Dependency Pairs:

H(c(z''), e(g(g(x'', y'), 0))) -> H(c(c(z'')), g(d(c(z''), g(x'', y')), d(z'', g(x'', y'))))
H(z'', e(g(x'', y'))) -> H(c(z''), g(e(x''), d(z'', y')))

Rules:

g(e(x), e(y)) -> e(g(x, y))
d(z, g(0, 0)) -> e(0)
d(z, g(x, y)) -> g(e(x), d(z, y))
d(c(z), g(g(x, y), 0)) -> g(d(c(z), g(x, y)), d(z, g(x, y)))

Strategy:

innermost

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

H(z'', e(g(x'', y'))) -> H(c(z''), g(e(x''), d(z'', y')))

two new Dependency Pairs are created:

H(c(z''''), e(g(x''', y''))) -> H(c(c(z'''')), g(e(x'''), d(c(z''''), y'')))
H(c(c(z'''')), e(g(x''', y''))) -> H(c(c(c(z''''))), g(e(x'''), d(c(c(z'''')), y'')))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳UsableRules

       →DP Problem 2

         ↳UsableRules

       →DP Problem 3

         ↳UsableRules

           →DP Problem 8

             ↳ATrans

...

               →DP Problem 11

                 ↳Instantiation Transformation

Dependency Pairs:

H(c(c(z'''')), e(g(x''', y''))) -> H(c(c(c(z''''))), g(e(x'''), d(c(c(z'''')), y'')))
H(c(z''''), e(g(x''', y''))) -> H(c(c(z'''')), g(e(x'''), d(c(z''''), y'')))
H(c(z''), e(g(g(x'', y'), 0))) -> H(c(c(z'')), g(d(c(z''), g(x'', y')), d(z'', g(x'', y'))))

Rules:

g(e(x), e(y)) -> e(g(x, y))
d(z, g(0, 0)) -> e(0)
d(z, g(x, y)) -> g(e(x), d(z, y))
d(c(z), g(g(x, y), 0)) -> g(d(c(z), g(x, y)), d(z, g(x, y)))

Strategy:

innermost

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

H(c(z''), e(g(g(x'', y'), 0))) -> H(c(c(z'')), g(d(c(z''), g(x'', y')), d(z'', g(x'', y'))))

three new Dependency Pairs are created:

H(c(c(z'''')), e(g(g(x''', y''), 0))) -> H(c(c(c(z''''))), g(d(c(c(z'''')), g(x''', y'')), d(c(z''''), g(x''', y''))))
H(c(c(z'''''')), e(g(g(x''', y''), 0))) -> H(c(c(c(z''''''))), g(d(c(c(z'''''')), g(x''', y'')), d(c(z''''''), g(x''', y''))))
H(c(c(c(z''''''))), e(g(g(x''', y''), 0))) -> H(c(c(c(c(z'''''')))), g(d(c(c(c(z''''''))), g(x''', y'')), d(c(c(z'''''')), g(x''', y''))))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳UsableRules

       →DP Problem 2

         ↳UsableRules

       →DP Problem 3

         ↳UsableRules

           →DP Problem 8

             ↳ATrans

...

               →DP Problem 12

                 ↳Remaining Obligation(s)

The following remains to be proven:
Dependency Pairs:

H(c(c(c(z''''''))), e(g(g(x''', y''), 0))) -> H(c(c(c(c(z'''''')))), g(d(c(c(c(z''''''))), g(x''', y'')), d(c(c(z'''''')), g(x''', y''))))
H(c(c(z'''''')), e(g(g(x''', y''), 0))) -> H(c(c(c(z''''''))), g(d(c(c(z'''''')), g(x''', y'')), d(c(z''''''), g(x''', y''))))
H(c(c(z'''')), e(g(g(x''', y''), 0))) -> H(c(c(c(z''''))), g(d(c(c(z'''')), g(x''', y'')), d(c(z''''), g(x''', y''))))
H(c(z''''), e(g(x''', y''))) -> H(c(c(z'''')), g(e(x'''), d(c(z''''), y'')))
H(c(c(z'''')), e(g(x''', y''))) -> H(c(c(c(z''''))), g(e(x'''), d(c(c(z'''')), y'')))

Rules:

g(e(x), e(y)) -> e(g(x, y))
d(z, g(0, 0)) -> e(0)
d(z, g(x, y)) -> g(e(x), d(z, y))
d(c(z), g(g(x, y), 0)) -> g(d(c(z), g(x, y)), d(z, g(x, y)))

Strategy:

innermost

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