Termination Proof using AProVE

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

Termination of R to be shown.

     ↳Dependency Pair Analysis

R contains the following Dependency Pairs:

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

Furthermore, R contains three SCCs.

     ↳DPs

       →DP Problem 1

         ↳Argument Filtering and Ordering

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳Inst

Dependency Pair:

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

Rules:

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

The following dependency pair can be strictly oriented:

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

There are no usable rules w.r.t. to the AFS that need to be oriented.
Used ordering: Homeomorphic Embedding Order with EMB
resulting in one new DP problem.
Used Argument Filtering System:

G(x₁, x₂) -> G(x₁, x₂)
e(x₁) -> e(x₁)

     ↳DPs

       →DP Problem 1

         ↳AFS

           →DP Problem 4

             ↳Dependency Graph

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳Inst

Dependency Pair:

Rules:

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

Using the Dependency Graph resulted in no new DP problems.

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳Argument Filtering and Ordering

       →DP Problem 3

         ↳Inst

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)

Rules:

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

The following dependency pairs can be strictly oriented:

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)

The following usable rule w.r.t. to the AFS can be oriented:

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

Used ordering: Homeomorphic Embedding Order with EMB
resulting in one new DP problem.
Used Argument Filtering System:

D(x₁, x₂) -> D(x₁, x₂)
c(x₁) -> c(x₁)
g(x₁, x₂) -> g(x₁, x₂)
e(x₁) -> x₁

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳AFS

           →DP Problem 5

             ↳Dependency Graph

       →DP Problem 3

         ↳Inst

Dependency Pair:

Rules:

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

Using the Dependency Graph resulted in no new DP problems.

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳Instantiation Transformation

Dependency Pair:

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

Rules:

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

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

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

one new Dependency Pair is created:

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

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳Inst

           →DP Problem 6

             ↳Instantiation Transformation

Dependency Pair:

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

Rules:

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

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

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

one new Dependency Pair is created:

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

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳Inst

           →DP Problem 6

             ↳Inst

...

               →DP Problem 7

                 ↳Instantiation Transformation

Dependency Pair:

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

Rules:

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

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

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

one new Dependency Pair is created:

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

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳Inst

           →DP Problem 6

             ↳Inst

...

               →DP Problem 8

                 ↳Instantiation Transformation

Dependency Pair:

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

Rules:

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

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

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

one new Dependency Pair is created:

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

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳Inst

           →DP Problem 6

             ↳Inst

...

               →DP Problem 9

                 ↳Instantiation Transformation

Dependency Pair:

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

Rules:

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

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

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

one new Dependency Pair is created:

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

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳Inst

           →DP Problem 6

             ↳Inst

...

               →DP Problem 10

                 ↳Remaining Obligation(s)

The following remains to be proven:
Dependency Pair:

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

Rules:

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

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