Term Rewriting System R:
[x, y]
-(x, 0) -> x
-(0, s(y)) -> 0
-(s(x), s(y)) -> -(x, y)
f(0) -> 0
f(s(x)) -> -(s(x), g(f(x)))
g(0) -> s(0)
g(s(x)) -> -(s(x), f(g(x)))

Innermost Termination of R to be shown.

`   R`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

-'(s(x), s(y)) -> -'(x, y)
F(s(x)) -> -'(s(x), g(f(x)))
F(s(x)) -> G(f(x))
F(s(x)) -> F(x)
G(s(x)) -> -'(s(x), f(g(x)))
G(s(x)) -> F(g(x))
G(s(x)) -> G(x)

Furthermore, R contains two SCCs.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Forward Instantiation Transformation`
`       →DP Problem 2`
`         ↳Nar`

Dependency Pair:

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

Rules:

-(x, 0) -> x
-(0, s(y)) -> 0
-(s(x), s(y)) -> -(x, y)
f(0) -> 0
f(s(x)) -> -(s(x), g(f(x)))
g(0) -> s(0)
g(s(x)) -> -(s(x), f(g(x)))

Strategy:

innermost

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

-'(s(x), s(y)) -> -'(x, y)
one new Dependency Pair is created:

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

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`           →DP Problem 3`
`             ↳Forward Instantiation Transformation`
`       →DP Problem 2`
`         ↳Nar`

Dependency Pair:

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

Rules:

-(x, 0) -> x
-(0, s(y)) -> 0
-(s(x), s(y)) -> -(x, y)
f(0) -> 0
f(s(x)) -> -(s(x), g(f(x)))
g(0) -> s(0)
g(s(x)) -> -(s(x), f(g(x)))

Strategy:

innermost

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

-'(s(s(x'')), s(s(y''))) -> -'(s(x''), s(y''))
one new Dependency Pair is created:

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

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`           →DP Problem 3`
`             ↳FwdInst`
`             ...`
`               →DP Problem 4`
`                 ↳Polynomial Ordering`
`       →DP Problem 2`
`         ↳Nar`

Dependency Pair:

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

Rules:

-(x, 0) -> x
-(0, s(y)) -> 0
-(s(x), s(y)) -> -(x, y)
f(0) -> 0
f(s(x)) -> -(s(x), g(f(x)))
g(0) -> s(0)
g(s(x)) -> -(s(x), f(g(x)))

Strategy:

innermost

The following dependency pair can be strictly oriented:

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

There are no usable rules for innermost w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(-'(x1, x2)) =  1 + x1 POL(s(x1)) =  1 + x1

resulting in one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`           →DP Problem 3`
`             ↳FwdInst`
`             ...`
`               →DP Problem 5`
`                 ↳Dependency Graph`
`       →DP Problem 2`
`         ↳Nar`

Dependency Pair:

Rules:

-(x, 0) -> x
-(0, s(y)) -> 0
-(s(x), s(y)) -> -(x, y)
f(0) -> 0
f(s(x)) -> -(s(x), g(f(x)))
g(0) -> s(0)
g(s(x)) -> -(s(x), f(g(x)))

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`       →DP Problem 2`
`         ↳Narrowing Transformation`

Dependency Pairs:

G(s(x)) -> G(x)
F(s(x)) -> F(x)
G(s(x)) -> F(g(x))
F(s(x)) -> G(f(x))

Rules:

-(x, 0) -> x
-(0, s(y)) -> 0
-(s(x), s(y)) -> -(x, y)
f(0) -> 0
f(s(x)) -> -(s(x), g(f(x)))
g(0) -> s(0)
g(s(x)) -> -(s(x), f(g(x)))

Strategy:

innermost

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

F(s(x)) -> G(f(x))
two new Dependency Pairs are created:

F(s(0)) -> G(0)
F(s(s(x''))) -> G(-(s(x''), g(f(x''))))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`       →DP Problem 2`
`         ↳Nar`
`           →DP Problem 6`
`             ↳Narrowing Transformation`

Dependency Pairs:

F(s(s(x''))) -> G(-(s(x''), g(f(x''))))
F(s(x)) -> F(x)
G(s(x)) -> F(g(x))
G(s(x)) -> G(x)

Rules:

-(x, 0) -> x
-(0, s(y)) -> 0
-(s(x), s(y)) -> -(x, y)
f(0) -> 0
f(s(x)) -> -(s(x), g(f(x)))
g(0) -> s(0)
g(s(x)) -> -(s(x), f(g(x)))

Strategy:

innermost

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

G(s(x)) -> F(g(x))
two new Dependency Pairs are created:

G(s(0)) -> F(s(0))
G(s(s(x''))) -> F(-(s(x''), f(g(x''))))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`       →DP Problem 2`
`         ↳Nar`
`           →DP Problem 6`
`             ↳Nar`
`             ...`
`               →DP Problem 7`
`                 ↳Narrowing Transformation`

Dependency Pairs:

G(s(s(x''))) -> F(-(s(x''), f(g(x''))))
F(s(x)) -> F(x)
G(s(0)) -> F(s(0))
G(s(x)) -> G(x)
F(s(s(x''))) -> G(-(s(x''), g(f(x''))))

Rules:

-(x, 0) -> x
-(0, s(y)) -> 0
-(s(x), s(y)) -> -(x, y)
f(0) -> 0
f(s(x)) -> -(s(x), g(f(x)))
g(0) -> s(0)
g(s(x)) -> -(s(x), f(g(x)))

Strategy:

innermost

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

F(s(s(x''))) -> G(-(s(x''), g(f(x''))))
two new Dependency Pairs are created:

F(s(s(0))) -> G(-(s(0), g(0)))
F(s(s(s(x')))) -> G(-(s(s(x')), g(-(s(x'), g(f(x'))))))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`       →DP Problem 2`
`         ↳Nar`
`           →DP Problem 6`
`             ↳Nar`
`             ...`
`               →DP Problem 8`
`                 ↳Rewriting Transformation`

Dependency Pairs:

F(s(s(s(x')))) -> G(-(s(s(x')), g(-(s(x'), g(f(x'))))))
G(s(0)) -> F(s(0))
G(s(x)) -> G(x)
F(s(s(0))) -> G(-(s(0), g(0)))
F(s(x)) -> F(x)
G(s(s(x''))) -> F(-(s(x''), f(g(x''))))

Rules:

-(x, 0) -> x
-(0, s(y)) -> 0
-(s(x), s(y)) -> -(x, y)
f(0) -> 0
f(s(x)) -> -(s(x), g(f(x)))
g(0) -> s(0)
g(s(x)) -> -(s(x), f(g(x)))

Strategy:

innermost

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

F(s(s(0))) -> G(-(s(0), g(0)))
one new Dependency Pair is created:

F(s(s(0))) -> G(-(s(0), s(0)))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`       →DP Problem 2`
`         ↳Nar`
`           →DP Problem 6`
`             ↳Nar`
`             ...`
`               →DP Problem 9`
`                 ↳Rewriting Transformation`

Dependency Pairs:

G(s(s(x''))) -> F(-(s(x''), f(g(x''))))
F(s(s(0))) -> G(-(s(0), s(0)))
F(s(x)) -> F(x)
G(s(0)) -> F(s(0))
G(s(x)) -> G(x)
F(s(s(s(x')))) -> G(-(s(s(x')), g(-(s(x'), g(f(x'))))))

Rules:

-(x, 0) -> x
-(0, s(y)) -> 0
-(s(x), s(y)) -> -(x, y)
f(0) -> 0
f(s(x)) -> -(s(x), g(f(x)))
g(0) -> s(0)
g(s(x)) -> -(s(x), f(g(x)))

Strategy:

innermost

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

F(s(s(0))) -> G(-(s(0), s(0)))
one new Dependency Pair is created:

F(s(s(0))) -> G(-(0, 0))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`       →DP Problem 2`
`         ↳Nar`
`           →DP Problem 6`
`             ↳Nar`
`             ...`
`               →DP Problem 10`
`                 ↳Rewriting Transformation`

Dependency Pairs:

F(s(s(0))) -> G(-(0, 0))
G(s(0)) -> F(s(0))
G(s(x)) -> G(x)
F(s(s(s(x')))) -> G(-(s(s(x')), g(-(s(x'), g(f(x'))))))
F(s(x)) -> F(x)
G(s(s(x''))) -> F(-(s(x''), f(g(x''))))

Rules:

-(x, 0) -> x
-(0, s(y)) -> 0
-(s(x), s(y)) -> -(x, y)
f(0) -> 0
f(s(x)) -> -(s(x), g(f(x)))
g(0) -> s(0)
g(s(x)) -> -(s(x), f(g(x)))

Strategy:

innermost

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

F(s(s(0))) -> G(-(0, 0))
one new Dependency Pair is created:

F(s(s(0))) -> G(0)

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`       →DP Problem 2`
`         ↳Nar`
`           →DP Problem 6`
`             ↳Nar`
`             ...`
`               →DP Problem 11`
`                 ↳Narrowing Transformation`

Dependency Pairs:

G(s(s(x''))) -> F(-(s(x''), f(g(x''))))
G(s(x)) -> G(x)
F(s(s(s(x')))) -> G(-(s(s(x')), g(-(s(x'), g(f(x'))))))
F(s(x)) -> F(x)
G(s(0)) -> F(s(0))

Rules:

-(x, 0) -> x
-(0, s(y)) -> 0
-(s(x), s(y)) -> -(x, y)
f(0) -> 0
f(s(x)) -> -(s(x), g(f(x)))
g(0) -> s(0)
g(s(x)) -> -(s(x), f(g(x)))

Strategy:

innermost

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

G(s(s(x''))) -> F(-(s(x''), f(g(x''))))
two new Dependency Pairs are created:

G(s(s(0))) -> F(-(s(0), f(s(0))))
G(s(s(s(x')))) -> F(-(s(s(x')), f(-(s(x'), f(g(x'))))))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`       →DP Problem 2`
`         ↳Nar`
`           →DP Problem 6`
`             ↳Nar`
`             ...`
`               →DP Problem 12`
`                 ↳Rewriting Transformation`

Dependency Pairs:

G(s(s(s(x')))) -> F(-(s(s(x')), f(-(s(x'), f(g(x'))))))
G(s(s(0))) -> F(-(s(0), f(s(0))))
F(s(s(s(x')))) -> G(-(s(s(x')), g(-(s(x'), g(f(x'))))))
F(s(x)) -> F(x)
G(s(0)) -> F(s(0))
G(s(x)) -> G(x)

Rules:

-(x, 0) -> x
-(0, s(y)) -> 0
-(s(x), s(y)) -> -(x, y)
f(0) -> 0
f(s(x)) -> -(s(x), g(f(x)))
g(0) -> s(0)
g(s(x)) -> -(s(x), f(g(x)))

Strategy:

innermost

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

G(s(s(0))) -> F(-(s(0), f(s(0))))
one new Dependency Pair is created:

G(s(s(0))) -> F(-(s(0), -(s(0), g(f(0)))))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`       →DP Problem 2`
`         ↳Nar`
`           →DP Problem 6`
`             ↳Nar`
`             ...`
`               →DP Problem 13`
`                 ↳Rewriting Transformation`

Dependency Pairs:

G(s(s(0))) -> F(-(s(0), -(s(0), g(f(0)))))
G(s(0)) -> F(s(0))
G(s(x)) -> G(x)
F(s(s(s(x')))) -> G(-(s(s(x')), g(-(s(x'), g(f(x'))))))
F(s(x)) -> F(x)
G(s(s(s(x')))) -> F(-(s(s(x')), f(-(s(x'), f(g(x'))))))

Rules:

-(x, 0) -> x
-(0, s(y)) -> 0
-(s(x), s(y)) -> -(x, y)
f(0) -> 0
f(s(x)) -> -(s(x), g(f(x)))
g(0) -> s(0)
g(s(x)) -> -(s(x), f(g(x)))

Strategy:

innermost

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

G(s(s(0))) -> F(-(s(0), -(s(0), g(f(0)))))
one new Dependency Pair is created:

G(s(s(0))) -> F(-(s(0), -(s(0), g(0))))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`       →DP Problem 2`
`         ↳Nar`
`           →DP Problem 6`
`             ↳Nar`
`             ...`
`               →DP Problem 14`
`                 ↳Rewriting Transformation`

Dependency Pairs:

G(s(s(0))) -> F(-(s(0), -(s(0), g(0))))
G(s(s(s(x')))) -> F(-(s(s(x')), f(-(s(x'), f(g(x'))))))
G(s(x)) -> G(x)
F(s(s(s(x')))) -> G(-(s(s(x')), g(-(s(x'), g(f(x'))))))
F(s(x)) -> F(x)
G(s(0)) -> F(s(0))

Rules:

-(x, 0) -> x
-(0, s(y)) -> 0
-(s(x), s(y)) -> -(x, y)
f(0) -> 0
f(s(x)) -> -(s(x), g(f(x)))
g(0) -> s(0)
g(s(x)) -> -(s(x), f(g(x)))

Strategy:

innermost

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

G(s(s(0))) -> F(-(s(0), -(s(0), g(0))))
one new Dependency Pair is created:

G(s(s(0))) -> F(-(s(0), -(s(0), s(0))))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`       →DP Problem 2`
`         ↳Nar`
`           →DP Problem 6`
`             ↳Nar`
`             ...`
`               →DP Problem 15`
`                 ↳Rewriting Transformation`

Dependency Pairs:

G(s(s(0))) -> F(-(s(0), -(s(0), s(0))))
G(s(0)) -> F(s(0))
G(s(x)) -> G(x)
F(s(s(s(x')))) -> G(-(s(s(x')), g(-(s(x'), g(f(x'))))))
F(s(x)) -> F(x)
G(s(s(s(x')))) -> F(-(s(s(x')), f(-(s(x'), f(g(x'))))))

Rules:

-(x, 0) -> x
-(0, s(y)) -> 0
-(s(x), s(y)) -> -(x, y)
f(0) -> 0
f(s(x)) -> -(s(x), g(f(x)))
g(0) -> s(0)
g(s(x)) -> -(s(x), f(g(x)))

Strategy:

innermost

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

G(s(s(0))) -> F(-(s(0), -(s(0), s(0))))
one new Dependency Pair is created:

G(s(s(0))) -> F(-(s(0), -(0, 0)))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`       →DP Problem 2`
`         ↳Nar`
`           →DP Problem 6`
`             ↳Nar`
`             ...`
`               →DP Problem 16`
`                 ↳Rewriting Transformation`

Dependency Pairs:

G(s(s(0))) -> F(-(s(0), -(0, 0)))
G(s(s(s(x')))) -> F(-(s(s(x')), f(-(s(x'), f(g(x'))))))
G(s(x)) -> G(x)
F(s(s(s(x')))) -> G(-(s(s(x')), g(-(s(x'), g(f(x'))))))
F(s(x)) -> F(x)
G(s(0)) -> F(s(0))

Rules:

-(x, 0) -> x
-(0, s(y)) -> 0
-(s(x), s(y)) -> -(x, y)
f(0) -> 0
f(s(x)) -> -(s(x), g(f(x)))
g(0) -> s(0)
g(s(x)) -> -(s(x), f(g(x)))

Strategy:

innermost

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

G(s(s(0))) -> F(-(s(0), -(0, 0)))
one new Dependency Pair is created:

G(s(s(0))) -> F(-(s(0), 0))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`       →DP Problem 2`
`         ↳Nar`
`           →DP Problem 6`
`             ↳Nar`
`             ...`
`               →DP Problem 17`
`                 ↳Rewriting Transformation`

Dependency Pairs:

G(s(s(0))) -> F(-(s(0), 0))
G(s(0)) -> F(s(0))
G(s(x)) -> G(x)
F(s(s(s(x')))) -> G(-(s(s(x')), g(-(s(x'), g(f(x'))))))
F(s(x)) -> F(x)
G(s(s(s(x')))) -> F(-(s(s(x')), f(-(s(x'), f(g(x'))))))

Rules:

-(x, 0) -> x
-(0, s(y)) -> 0
-(s(x), s(y)) -> -(x, y)
f(0) -> 0
f(s(x)) -> -(s(x), g(f(x)))
g(0) -> s(0)
g(s(x)) -> -(s(x), f(g(x)))

Strategy:

innermost

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

G(s(s(0))) -> F(-(s(0), 0))
one new Dependency Pair is created:

G(s(s(0))) -> F(s(0))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`       →DP Problem 2`
`         ↳Nar`
`           →DP Problem 6`
`             ↳Nar`
`             ...`
`               →DP Problem 18`
`                 ↳Forward Instantiation Transformation`

Dependency Pairs:

G(s(s(0))) -> F(s(0))
G(s(s(s(x')))) -> F(-(s(s(x')), f(-(s(x'), f(g(x'))))))
G(s(x)) -> G(x)
F(s(s(s(x')))) -> G(-(s(s(x')), g(-(s(x'), g(f(x'))))))
F(s(x)) -> F(x)
G(s(0)) -> F(s(0))

Rules:

-(x, 0) -> x
-(0, s(y)) -> 0
-(s(x), s(y)) -> -(x, y)
f(0) -> 0
f(s(x)) -> -(s(x), g(f(x)))
g(0) -> s(0)
g(s(x)) -> -(s(x), f(g(x)))

Strategy:

innermost

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

F(s(x)) -> F(x)
two new Dependency Pairs are created:

F(s(s(x''))) -> F(s(x''))
F(s(s(s(s(x'''))))) -> F(s(s(s(x'''))))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`       →DP Problem 2`
`         ↳Nar`
`           →DP Problem 6`
`             ↳Nar`
`             ...`
`               →DP Problem 19`
`                 ↳Forward Instantiation Transformation`

Dependency Pairs:

F(s(s(s(s(x'''))))) -> F(s(s(s(x'''))))
F(s(s(x''))) -> F(s(x''))
G(s(x)) -> G(x)
F(s(s(s(x')))) -> G(-(s(s(x')), g(-(s(x'), g(f(x'))))))
G(s(s(s(x')))) -> F(-(s(s(x')), f(-(s(x'), f(g(x'))))))

Rules:

-(x, 0) -> x
-(0, s(y)) -> 0
-(s(x), s(y)) -> -(x, y)
f(0) -> 0
f(s(x)) -> -(s(x), g(f(x)))
g(0) -> s(0)
g(s(x)) -> -(s(x), f(g(x)))

Strategy:

innermost

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

G(s(x)) -> G(x)
two new Dependency Pairs are created:

G(s(s(x''))) -> G(s(x''))
G(s(s(s(s(x'''))))) -> G(s(s(s(x'''))))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`       →DP Problem 2`
`         ↳Nar`
`           →DP Problem 6`
`             ↳Nar`
`             ...`
`               →DP Problem 20`
`                 ↳Forward Instantiation Transformation`

Dependency Pairs:

G(s(s(s(s(x'''))))) -> G(s(s(s(x'''))))
G(s(s(x''))) -> G(s(x''))
F(s(s(x''))) -> F(s(x''))
G(s(s(s(x')))) -> F(-(s(s(x')), f(-(s(x'), f(g(x'))))))
F(s(s(s(x')))) -> G(-(s(s(x')), g(-(s(x'), g(f(x'))))))
F(s(s(s(s(x'''))))) -> F(s(s(s(x'''))))

Rules:

-(x, 0) -> x
-(0, s(y)) -> 0
-(s(x), s(y)) -> -(x, y)
f(0) -> 0
f(s(x)) -> -(s(x), g(f(x)))
g(0) -> s(0)
g(s(x)) -> -(s(x), f(g(x)))

Strategy:

innermost

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

F(s(s(x''))) -> F(s(x''))
three new Dependency Pairs are created:

F(s(s(s(s(x''''))))) -> F(s(s(s(x''''))))
F(s(s(s(x'''')))) -> F(s(s(x'''')))
F(s(s(s(s(s(x''''')))))) -> F(s(s(s(s(x''''')))))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`       →DP Problem 2`
`         ↳Nar`
`           →DP Problem 6`
`             ↳Nar`
`             ...`
`               →DP Problem 21`
`                 ↳Forward Instantiation Transformation`

Dependency Pairs:

F(s(s(s(s(s(x''''')))))) -> F(s(s(s(s(x''''')))))
F(s(s(s(x'''')))) -> F(s(s(x'''')))
F(s(s(s(s(x''''))))) -> F(s(s(s(x''''))))
F(s(s(s(s(x'''))))) -> F(s(s(s(x'''))))
G(s(s(x''))) -> G(s(x''))
F(s(s(s(x')))) -> G(-(s(s(x')), g(-(s(x'), g(f(x'))))))
G(s(s(s(x')))) -> F(-(s(s(x')), f(-(s(x'), f(g(x'))))))
G(s(s(s(s(x'''))))) -> G(s(s(s(x'''))))

Rules:

-(x, 0) -> x
-(0, s(y)) -> 0
-(s(x), s(y)) -> -(x, y)
f(0) -> 0
f(s(x)) -> -(s(x), g(f(x)))
g(0) -> s(0)
g(s(x)) -> -(s(x), f(g(x)))

Strategy:

innermost

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

G(s(s(x''))) -> G(s(x''))
three new Dependency Pairs are created:

G(s(s(s(s(x''''))))) -> G(s(s(s(x''''))))
G(s(s(s(x'''')))) -> G(s(s(x'''')))
G(s(s(s(s(s(x''''')))))) -> G(s(s(s(s(x''''')))))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`       →DP Problem 2`
`         ↳Nar`
`           →DP Problem 6`
`             ↳Nar`
`             ...`
`               →DP Problem 22`
`                 ↳Polynomial Ordering`

Dependency Pairs:

G(s(s(s(s(s(x''''')))))) -> G(s(s(s(s(x''''')))))
G(s(s(s(x'''')))) -> G(s(s(x'''')))
G(s(s(s(s(x''''))))) -> G(s(s(s(x''''))))
G(s(s(s(s(x'''))))) -> G(s(s(s(x'''))))
F(s(s(s(x'''')))) -> F(s(s(x'''')))
F(s(s(s(s(x''''))))) -> F(s(s(s(x''''))))
F(s(s(s(s(x'''))))) -> F(s(s(s(x'''))))
G(s(s(s(x')))) -> F(-(s(s(x')), f(-(s(x'), f(g(x'))))))
F(s(s(s(x')))) -> G(-(s(s(x')), g(-(s(x'), g(f(x'))))))
F(s(s(s(s(s(x''''')))))) -> F(s(s(s(s(x''''')))))

Rules:

-(x, 0) -> x
-(0, s(y)) -> 0
-(s(x), s(y)) -> -(x, y)
f(0) -> 0
f(s(x)) -> -(s(x), g(f(x)))
g(0) -> s(0)
g(s(x)) -> -(s(x), f(g(x)))

Strategy:

innermost

The following dependency pairs can be strictly oriented:

G(s(s(s(s(s(x''''')))))) -> G(s(s(s(s(x''''')))))
G(s(s(s(x'''')))) -> G(s(s(x'''')))
G(s(s(s(s(x''''))))) -> G(s(s(s(x''''))))
G(s(s(s(s(x'''))))) -> G(s(s(s(x'''))))
F(s(s(s(x'''')))) -> F(s(s(x'''')))
F(s(s(s(s(x''''))))) -> F(s(s(s(x''''))))
F(s(s(s(s(x'''))))) -> F(s(s(s(x'''))))
G(s(s(s(x')))) -> F(-(s(s(x')), f(-(s(x'), f(g(x'))))))
F(s(s(s(x')))) -> G(-(s(s(x')), g(-(s(x'), g(f(x'))))))
F(s(s(s(s(s(x''''')))))) -> F(s(s(s(s(x''''')))))

Additionally, the following usable rules for innermost w.r.t. to the implicit AFS can be oriented:

-(x, 0) -> x
-(0, s(y)) -> 0
-(s(x), s(y)) -> -(x, y)

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(0) =  0 POL(g(x1)) =  0 POL(G(x1)) =  x1 POL(s(x1)) =  1 + x1 POL(-(x1, x2)) =  x1 POL(f(x1)) =  0 POL(F(x1)) =  x1

resulting in one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`       →DP Problem 2`
`         ↳Nar`
`           →DP Problem 6`
`             ↳Nar`
`             ...`
`               →DP Problem 23`
`                 ↳Dependency Graph`

Dependency Pair:

Rules:

-(x, 0) -> x
-(0, s(y)) -> 0
-(s(x), s(y)) -> -(x, y)
f(0) -> 0
f(s(x)) -> -(s(x), g(f(x)))
g(0) -> s(0)
g(s(x)) -> -(s(x), f(g(x)))

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

Innermost Termination of R successfully shown.
Duration:
0:01 minutes