Term Rewriting System R:
[x, y]
p(s(x)) -> x
s(p(x)) -> x
+(0, y) -> y
+(s(x), y) -> s(+(x, y))
+(p(x), y) -> p(+(x, y))
minus(0) -> 0
minus(s(x)) -> p(minus(x))
minus(p(x)) -> s(minus(x))
*(0, y) -> 0
*(s(x), y) -> +(*(x, y), y)
*(p(x), y) -> +(*(x, y), minus(y))

Innermost Termination of R to be shown.

`   R`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

+'(s(x), y) -> S(+(x, y))
+'(s(x), y) -> +'(x, y)
+'(p(x), y) -> P(+(x, y))
+'(p(x), y) -> +'(x, y)
MINUS(s(x)) -> P(minus(x))
MINUS(s(x)) -> MINUS(x)
MINUS(p(x)) -> S(minus(x))
MINUS(p(x)) -> MINUS(x)
*'(s(x), y) -> +'(*(x, y), y)
*'(s(x), y) -> *'(x, y)
*'(p(x), y) -> +'(*(x, y), minus(y))
*'(p(x), y) -> *'(x, y)
*'(p(x), y) -> MINUS(y)

Furthermore, R contains six SCCs.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Forward Instantiation Transformation`
`       →DP Problem 2`
`         ↳FwdInst`
`       →DP Problem 3`
`         ↳FwdInst`
`       →DP Problem 4`
`         ↳FwdInst`
`       →DP Problem 5`
`         ↳FwdInst`
`       →DP Problem 6`
`         ↳FwdInst`

Dependency Pair:

+'(p(x), y) -> +'(x, y)

Rules:

p(s(x)) -> x
s(p(x)) -> x
+(0, y) -> y
+(s(x), y) -> s(+(x, y))
+(p(x), y) -> p(+(x, y))
minus(0) -> 0
minus(s(x)) -> p(minus(x))
minus(p(x)) -> s(minus(x))
*(0, y) -> 0
*(s(x), y) -> +(*(x, y), y)
*(p(x), y) -> +(*(x, y), minus(y))

Strategy:

innermost

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

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

+'(p(p(x'')), y'') -> +'(p(x''), y'')

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`           →DP Problem 7`
`             ↳Polynomial Ordering`
`       →DP Problem 2`
`         ↳FwdInst`
`       →DP Problem 3`
`         ↳FwdInst`
`       →DP Problem 4`
`         ↳FwdInst`
`       →DP Problem 5`
`         ↳FwdInst`
`       →DP Problem 6`
`         ↳FwdInst`

Dependency Pair:

+'(p(p(x'')), y'') -> +'(p(x''), y'')

Rules:

p(s(x)) -> x
s(p(x)) -> x
+(0, y) -> y
+(s(x), y) -> s(+(x, y))
+(p(x), y) -> p(+(x, y))
minus(0) -> 0
minus(s(x)) -> p(minus(x))
minus(p(x)) -> s(minus(x))
*(0, y) -> 0
*(s(x), y) -> +(*(x, y), y)
*(p(x), y) -> +(*(x, y), minus(y))

Strategy:

innermost

The following dependency pair can be strictly oriented:

+'(p(p(x'')), y'') -> +'(p(x''), y'')

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

p(s(x)) -> x

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

resulting in one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`           →DP Problem 7`
`             ↳Polo`
`             ...`
`               →DP Problem 8`
`                 ↳Dependency Graph`
`       →DP Problem 2`
`         ↳FwdInst`
`       →DP Problem 3`
`         ↳FwdInst`
`       →DP Problem 4`
`         ↳FwdInst`
`       →DP Problem 5`
`         ↳FwdInst`
`       →DP Problem 6`
`         ↳FwdInst`

Dependency Pair:

Rules:

p(s(x)) -> x
s(p(x)) -> x
+(0, y) -> y
+(s(x), y) -> s(+(x, y))
+(p(x), y) -> p(+(x, y))
minus(0) -> 0
minus(s(x)) -> p(minus(x))
minus(p(x)) -> s(minus(x))
*(0, y) -> 0
*(s(x), y) -> +(*(x, y), y)
*(p(x), y) -> +(*(x, y), minus(y))

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`       →DP Problem 2`
`         ↳Forward Instantiation Transformation`
`       →DP Problem 3`
`         ↳FwdInst`
`       →DP Problem 4`
`         ↳FwdInst`
`       →DP Problem 5`
`         ↳FwdInst`
`       →DP Problem 6`
`         ↳FwdInst`

Dependency Pair:

MINUS(p(x)) -> MINUS(x)

Rules:

p(s(x)) -> x
s(p(x)) -> x
+(0, y) -> y
+(s(x), y) -> s(+(x, y))
+(p(x), y) -> p(+(x, y))
minus(0) -> 0
minus(s(x)) -> p(minus(x))
minus(p(x)) -> s(minus(x))
*(0, y) -> 0
*(s(x), y) -> +(*(x, y), y)
*(p(x), y) -> +(*(x, y), minus(y))

Strategy:

innermost

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

MINUS(p(x)) -> MINUS(x)
one new Dependency Pair is created:

MINUS(p(p(x''))) -> MINUS(p(x''))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`       →DP Problem 2`
`         ↳FwdInst`
`           →DP Problem 9`
`             ↳Polynomial Ordering`
`       →DP Problem 3`
`         ↳FwdInst`
`       →DP Problem 4`
`         ↳FwdInst`
`       →DP Problem 5`
`         ↳FwdInst`
`       →DP Problem 6`
`         ↳FwdInst`

Dependency Pair:

MINUS(p(p(x''))) -> MINUS(p(x''))

Rules:

p(s(x)) -> x
s(p(x)) -> x
+(0, y) -> y
+(s(x), y) -> s(+(x, y))
+(p(x), y) -> p(+(x, y))
minus(0) -> 0
minus(s(x)) -> p(minus(x))
minus(p(x)) -> s(minus(x))
*(0, y) -> 0
*(s(x), y) -> +(*(x, y), y)
*(p(x), y) -> +(*(x, y), minus(y))

Strategy:

innermost

The following dependency pair can be strictly oriented:

MINUS(p(p(x''))) -> MINUS(p(x''))

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

p(s(x)) -> x

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(MINUS(x1)) =  x1 POL(s(x1)) =  1 + x1 POL(p(x1)) =  1 + x1

resulting in one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`       →DP Problem 2`
`         ↳FwdInst`
`           →DP Problem 9`
`             ↳Polo`
`             ...`
`               →DP Problem 10`
`                 ↳Dependency Graph`
`       →DP Problem 3`
`         ↳FwdInst`
`       →DP Problem 4`
`         ↳FwdInst`
`       →DP Problem 5`
`         ↳FwdInst`
`       →DP Problem 6`
`         ↳FwdInst`

Dependency Pair:

Rules:

p(s(x)) -> x
s(p(x)) -> x
+(0, y) -> y
+(s(x), y) -> s(+(x, y))
+(p(x), y) -> p(+(x, y))
minus(0) -> 0
minus(s(x)) -> p(minus(x))
minus(p(x)) -> s(minus(x))
*(0, y) -> 0
*(s(x), y) -> +(*(x, y), y)
*(p(x), y) -> +(*(x, y), minus(y))

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`       →DP Problem 2`
`         ↳FwdInst`
`       →DP Problem 3`
`         ↳Forward Instantiation Transformation`
`       →DP Problem 4`
`         ↳FwdInst`
`       →DP Problem 5`
`         ↳FwdInst`
`       →DP Problem 6`
`         ↳FwdInst`

Dependency Pair:

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

Rules:

p(s(x)) -> x
s(p(x)) -> x
+(0, y) -> y
+(s(x), y) -> s(+(x, y))
+(p(x), y) -> p(+(x, y))
minus(0) -> 0
minus(s(x)) -> p(minus(x))
minus(p(x)) -> s(minus(x))
*(0, y) -> 0
*(s(x), y) -> +(*(x, y), y)
*(p(x), y) -> +(*(x, y), minus(y))

Strategy:

innermost

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

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

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

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`       →DP Problem 2`
`         ↳FwdInst`
`       →DP Problem 3`
`         ↳FwdInst`
`           →DP Problem 11`
`             ↳Polynomial Ordering`
`       →DP Problem 4`
`         ↳FwdInst`
`       →DP Problem 5`
`         ↳FwdInst`
`       →DP Problem 6`
`         ↳FwdInst`

Dependency Pair:

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

Rules:

p(s(x)) -> x
s(p(x)) -> x
+(0, y) -> y
+(s(x), y) -> s(+(x, y))
+(p(x), y) -> p(+(x, y))
minus(0) -> 0
minus(s(x)) -> p(minus(x))
minus(p(x)) -> s(minus(x))
*(0, y) -> 0
*(s(x), y) -> +(*(x, y), y)
*(p(x), y) -> +(*(x, y), minus(y))

Strategy:

innermost

The following dependency pair can be strictly oriented:

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

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

s(p(x)) -> x

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

resulting in one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`       →DP Problem 2`
`         ↳FwdInst`
`       →DP Problem 3`
`         ↳FwdInst`
`           →DP Problem 11`
`             ↳Polo`
`             ...`
`               →DP Problem 12`
`                 ↳Dependency Graph`
`       →DP Problem 4`
`         ↳FwdInst`
`       →DP Problem 5`
`         ↳FwdInst`
`       →DP Problem 6`
`         ↳FwdInst`

Dependency Pair:

Rules:

p(s(x)) -> x
s(p(x)) -> x
+(0, y) -> y
+(s(x), y) -> s(+(x, y))
+(p(x), y) -> p(+(x, y))
minus(0) -> 0
minus(s(x)) -> p(minus(x))
minus(p(x)) -> s(minus(x))
*(0, y) -> 0
*(s(x), y) -> +(*(x, y), y)
*(p(x), y) -> +(*(x, y), minus(y))

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`       →DP Problem 2`
`         ↳FwdInst`
`       →DP Problem 3`
`         ↳FwdInst`
`       →DP Problem 4`
`         ↳Forward Instantiation Transformation`
`       →DP Problem 5`
`         ↳FwdInst`
`       →DP Problem 6`
`         ↳FwdInst`

Dependency Pair:

MINUS(s(x)) -> MINUS(x)

Rules:

p(s(x)) -> x
s(p(x)) -> x
+(0, y) -> y
+(s(x), y) -> s(+(x, y))
+(p(x), y) -> p(+(x, y))
minus(0) -> 0
minus(s(x)) -> p(minus(x))
minus(p(x)) -> s(minus(x))
*(0, y) -> 0
*(s(x), y) -> +(*(x, y), y)
*(p(x), y) -> +(*(x, y), minus(y))

Strategy:

innermost

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

MINUS(s(x)) -> MINUS(x)
one new Dependency Pair is created:

MINUS(s(s(x''))) -> MINUS(s(x''))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`       →DP Problem 2`
`         ↳FwdInst`
`       →DP Problem 3`
`         ↳FwdInst`
`       →DP Problem 4`
`         ↳FwdInst`
`           →DP Problem 13`
`             ↳Polynomial Ordering`
`       →DP Problem 5`
`         ↳FwdInst`
`       →DP Problem 6`
`         ↳FwdInst`

Dependency Pair:

MINUS(s(s(x''))) -> MINUS(s(x''))

Rules:

p(s(x)) -> x
s(p(x)) -> x
+(0, y) -> y
+(s(x), y) -> s(+(x, y))
+(p(x), y) -> p(+(x, y))
minus(0) -> 0
minus(s(x)) -> p(minus(x))
minus(p(x)) -> s(minus(x))
*(0, y) -> 0
*(s(x), y) -> +(*(x, y), y)
*(p(x), y) -> +(*(x, y), minus(y))

Strategy:

innermost

The following dependency pair can be strictly oriented:

MINUS(s(s(x''))) -> MINUS(s(x''))

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

s(p(x)) -> x

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(MINUS(x1)) =  x1 POL(s(x1)) =  1 + x1 POL(p(x1)) =  1 + x1

resulting in one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`       →DP Problem 2`
`         ↳FwdInst`
`       →DP Problem 3`
`         ↳FwdInst`
`       →DP Problem 4`
`         ↳FwdInst`
`           →DP Problem 13`
`             ↳Polo`
`             ...`
`               →DP Problem 14`
`                 ↳Dependency Graph`
`       →DP Problem 5`
`         ↳FwdInst`
`       →DP Problem 6`
`         ↳FwdInst`

Dependency Pair:

Rules:

p(s(x)) -> x
s(p(x)) -> x
+(0, y) -> y
+(s(x), y) -> s(+(x, y))
+(p(x), y) -> p(+(x, y))
minus(0) -> 0
minus(s(x)) -> p(minus(x))
minus(p(x)) -> s(minus(x))
*(0, y) -> 0
*(s(x), y) -> +(*(x, y), y)
*(p(x), y) -> +(*(x, y), minus(y))

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`       →DP Problem 2`
`         ↳FwdInst`
`       →DP Problem 3`
`         ↳FwdInst`
`       →DP Problem 4`
`         ↳FwdInst`
`       →DP Problem 5`
`         ↳Forward Instantiation Transformation`
`       →DP Problem 6`
`         ↳FwdInst`

Dependency Pair:

*'(p(x), y) -> *'(x, y)

Rules:

p(s(x)) -> x
s(p(x)) -> x
+(0, y) -> y
+(s(x), y) -> s(+(x, y))
+(p(x), y) -> p(+(x, y))
minus(0) -> 0
minus(s(x)) -> p(minus(x))
minus(p(x)) -> s(minus(x))
*(0, y) -> 0
*(s(x), y) -> +(*(x, y), y)
*(p(x), y) -> +(*(x, y), minus(y))

Strategy:

innermost

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

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

*'(p(p(x'')), y'') -> *'(p(x''), y'')

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`       →DP Problem 2`
`         ↳FwdInst`
`       →DP Problem 3`
`         ↳FwdInst`
`       →DP Problem 4`
`         ↳FwdInst`
`       →DP Problem 5`
`         ↳FwdInst`
`           →DP Problem 15`
`             ↳Polynomial Ordering`
`       →DP Problem 6`
`         ↳FwdInst`

Dependency Pair:

*'(p(p(x'')), y'') -> *'(p(x''), y'')

Rules:

p(s(x)) -> x
s(p(x)) -> x
+(0, y) -> y
+(s(x), y) -> s(+(x, y))
+(p(x), y) -> p(+(x, y))
minus(0) -> 0
minus(s(x)) -> p(minus(x))
minus(p(x)) -> s(minus(x))
*(0, y) -> 0
*(s(x), y) -> +(*(x, y), y)
*(p(x), y) -> +(*(x, y), minus(y))

Strategy:

innermost

The following dependency pair can be strictly oriented:

*'(p(p(x'')), y'') -> *'(p(x''), y'')

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

p(s(x)) -> x

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

resulting in one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`       →DP Problem 2`
`         ↳FwdInst`
`       →DP Problem 3`
`         ↳FwdInst`
`       →DP Problem 4`
`         ↳FwdInst`
`       →DP Problem 5`
`         ↳FwdInst`
`           →DP Problem 15`
`             ↳Polo`
`             ...`
`               →DP Problem 16`
`                 ↳Dependency Graph`
`       →DP Problem 6`
`         ↳FwdInst`

Dependency Pair:

Rules:

p(s(x)) -> x
s(p(x)) -> x
+(0, y) -> y
+(s(x), y) -> s(+(x, y))
+(p(x), y) -> p(+(x, y))
minus(0) -> 0
minus(s(x)) -> p(minus(x))
minus(p(x)) -> s(minus(x))
*(0, y) -> 0
*(s(x), y) -> +(*(x, y), y)
*(p(x), y) -> +(*(x, y), minus(y))

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`       →DP Problem 2`
`         ↳FwdInst`
`       →DP Problem 3`
`         ↳FwdInst`
`       →DP Problem 4`
`         ↳FwdInst`
`       →DP Problem 5`
`         ↳FwdInst`
`       →DP Problem 6`
`         ↳Forward Instantiation Transformation`

Dependency Pair:

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

Rules:

p(s(x)) -> x
s(p(x)) -> x
+(0, y) -> y
+(s(x), y) -> s(+(x, y))
+(p(x), y) -> p(+(x, y))
minus(0) -> 0
minus(s(x)) -> p(minus(x))
minus(p(x)) -> s(minus(x))
*(0, y) -> 0
*(s(x), y) -> +(*(x, y), y)
*(p(x), y) -> +(*(x, y), minus(y))

Strategy:

innermost

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

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

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

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`       →DP Problem 2`
`         ↳FwdInst`
`       →DP Problem 3`
`         ↳FwdInst`
`       →DP Problem 4`
`         ↳FwdInst`
`       →DP Problem 5`
`         ↳FwdInst`
`       →DP Problem 6`
`         ↳FwdInst`
`           →DP Problem 17`
`             ↳Polynomial Ordering`

Dependency Pair:

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

Rules:

p(s(x)) -> x
s(p(x)) -> x
+(0, y) -> y
+(s(x), y) -> s(+(x, y))
+(p(x), y) -> p(+(x, y))
minus(0) -> 0
minus(s(x)) -> p(minus(x))
minus(p(x)) -> s(minus(x))
*(0, y) -> 0
*(s(x), y) -> +(*(x, y), y)
*(p(x), y) -> +(*(x, y), minus(y))

Strategy:

innermost

The following dependency pair can be strictly oriented:

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

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

s(p(x)) -> x

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

resulting in one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`       →DP Problem 2`
`         ↳FwdInst`
`       →DP Problem 3`
`         ↳FwdInst`
`       →DP Problem 4`
`         ↳FwdInst`
`       →DP Problem 5`
`         ↳FwdInst`
`       →DP Problem 6`
`         ↳FwdInst`
`           →DP Problem 17`
`             ↳Polo`
`             ...`
`               →DP Problem 18`
`                 ↳Dependency Graph`

Dependency Pair:

Rules:

p(s(x)) -> x
s(p(x)) -> x
+(0, y) -> y
+(s(x), y) -> s(+(x, y))
+(p(x), y) -> p(+(x, y))
minus(0) -> 0
minus(s(x)) -> p(minus(x))
minus(p(x)) -> s(minus(x))
*(0, y) -> 0
*(s(x), y) -> +(*(x, y), y)
*(p(x), y) -> +(*(x, y), minus(y))

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

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