Term Rewriting System R:
[X, Y]
p(0) -> 0
p(s(X)) -> X
leq(0, Y) -> true
leq(s(X), 0) -> false
leq(s(X), s(Y)) -> leq(X, Y)
if(true, X, Y) -> X
if(false, X, Y) -> Y
diff(X, Y) -> if(leq(X, Y), 0, s(diff(p(X), Y)))

Innermost Termination of R to be shown.

`   R`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

LEQ(s(X), s(Y)) -> LEQ(X, Y)
DIFF(X, Y) -> IF(leq(X, Y), 0, s(diff(p(X), Y)))
DIFF(X, Y) -> LEQ(X, Y)
DIFF(X, Y) -> DIFF(p(X), Y)
DIFF(X, Y) -> P(X)

Furthermore, R contains two SCCs.

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

Dependency Pair:

LEQ(s(X), s(Y)) -> LEQ(X, Y)

Rules:

p(0) -> 0
p(s(X)) -> X
leq(0, Y) -> true
leq(s(X), 0) -> false
leq(s(X), s(Y)) -> leq(X, Y)
if(true, X, Y) -> X
if(false, X, Y) -> Y
diff(X, Y) -> if(leq(X, Y), 0, s(diff(p(X), Y)))

Strategy:

innermost

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

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

LEQ(s(s(X'')), s(s(Y''))) -> LEQ(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`
`         ↳Remaining`

Dependency Pair:

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

Rules:

p(0) -> 0
p(s(X)) -> X
leq(0, Y) -> true
leq(s(X), 0) -> false
leq(s(X), s(Y)) -> leq(X, Y)
if(true, X, Y) -> X
if(false, X, Y) -> Y
diff(X, Y) -> if(leq(X, Y), 0, s(diff(p(X), Y)))

Strategy:

innermost

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

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

LEQ(s(s(s(X''''))), s(s(s(Y'''')))) -> LEQ(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`
`         ↳Remaining`

Dependency Pair:

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

Rules:

p(0) -> 0
p(s(X)) -> X
leq(0, Y) -> true
leq(s(X), 0) -> false
leq(s(X), s(Y)) -> leq(X, Y)
if(true, X, Y) -> X
if(false, X, Y) -> Y
diff(X, Y) -> if(leq(X, Y), 0, s(diff(p(X), Y)))

Strategy:

innermost

The following dependency pair can be strictly oriented:

LEQ(s(s(s(X''''))), s(s(s(Y'''')))) -> LEQ(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(LEQ(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`
`         ↳Remaining`

Dependency Pair:

Rules:

p(0) -> 0
p(s(X)) -> X
leq(0, Y) -> true
leq(s(X), 0) -> false
leq(s(X), s(Y)) -> leq(X, Y)
if(true, X, Y) -> X
if(false, X, Y) -> Y
diff(X, Y) -> if(leq(X, Y), 0, s(diff(p(X), Y)))

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`       →DP Problem 2`
`         ↳Remaining Obligation(s)`

The following remains to be proven:
Dependency Pair:

DIFF(X, Y) -> DIFF(p(X), Y)

Rules:

p(0) -> 0
p(s(X)) -> X
leq(0, Y) -> true
leq(s(X), 0) -> false
leq(s(X), s(Y)) -> leq(X, Y)
if(true, X, Y) -> X
if(false, X, Y) -> Y
diff(X, Y) -> if(leq(X, Y), 0, s(diff(p(X), Y)))

Strategy:

innermost

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