Term Rewriting System R:
[X, Y]
+(X, 0) -> X
+(X, s(Y)) -> s(+(X, Y))
double(X) -> +(X, X)
f(0, s(0), X) -> f(X, double(X), X)
g(X, Y) -> X
g(X, Y) -> Y

Innermost Termination of R to be shown.

`   R`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

+'(X, s(Y)) -> +'(X, Y)
DOUBLE(X) -> +'(X, X)
F(0, s(0), X) -> F(X, double(X), X)
F(0, s(0), X) -> DOUBLE(X)

Furthermore, R contains two SCCs.

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

Dependency Pair:

+'(X, s(Y)) -> +'(X, Y)

Rules:

+(X, 0) -> X
+(X, s(Y)) -> s(+(X, Y))
double(X) -> +(X, X)
f(0, s(0), X) -> f(X, double(X), X)
g(X, Y) -> X
g(X, Y) -> Y

Strategy:

innermost

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

+'(X, s(Y)) -> +'(X, Y)
one new Dependency Pair is created:

+'(X'', s(s(Y''))) -> +'(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`
`         ↳Rw`

Dependency Pair:

+'(X'', s(s(Y''))) -> +'(X'', s(Y''))

Rules:

+(X, 0) -> X
+(X, s(Y)) -> s(+(X, Y))
double(X) -> +(X, X)
f(0, s(0), X) -> f(X, double(X), X)
g(X, Y) -> X
g(X, Y) -> Y

Strategy:

innermost

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

+'(X'', s(s(Y''))) -> +'(X'', s(Y''))
one new Dependency Pair is created:

+'(X'''', s(s(s(Y'''')))) -> +'(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`
`         ↳Rw`

Dependency Pair:

+'(X'''', s(s(s(Y'''')))) -> +'(X'''', s(s(Y'''')))

Rules:

+(X, 0) -> X
+(X, s(Y)) -> s(+(X, Y))
double(X) -> +(X, X)
f(0, s(0), X) -> f(X, double(X), X)
g(X, Y) -> X
g(X, Y) -> Y

Strategy:

innermost

The following dependency pair can be strictly oriented:

+'(X'''', s(s(s(Y'''')))) -> +'(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(s(x1)) =  1 + x1 POL(+'(x1, x2)) =  1 + x2

resulting in one new DP problem.

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

Dependency Pair:

Rules:

+(X, 0) -> X
+(X, s(Y)) -> s(+(X, Y))
double(X) -> +(X, X)
f(0, s(0), X) -> f(X, double(X), X)
g(X, Y) -> X
g(X, Y) -> Y

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

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

Dependency Pair:

F(0, s(0), X) -> F(X, double(X), X)

Rules:

+(X, 0) -> X
+(X, s(Y)) -> s(+(X, Y))
double(X) -> +(X, X)
f(0, s(0), X) -> f(X, double(X), X)
g(X, Y) -> X
g(X, Y) -> Y

Strategy:

innermost

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

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

F(0, s(0), X) -> F(X, +(X, X), X)

The transformation is resulting in one new DP problem:

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

Dependency Pair:

F(0, s(0), X) -> F(X, +(X, X), X)

Rules:

+(X, 0) -> X
+(X, s(Y)) -> s(+(X, Y))
double(X) -> +(X, X)
f(0, s(0), X) -> f(X, double(X), X)
g(X, Y) -> X
g(X, Y) -> Y

Strategy:

innermost

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

F(0, s(0), X) -> F(X, +(X, X), X)
two new Dependency Pairs are created:

F(0, s(0), 0) -> F(0, 0, 0)
F(0, s(0), s(Y')) -> F(s(Y'), s(+(s(Y'), Y')), s(Y'))

The transformation is resulting in no new DP problems.

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