Term Rewriting System R:
[X, Y, X1, X2]
fact(X) -> if(zero(X), ns(0), nprod(X, fact(p(X))))
prod(0, X) -> 0
prod(s(X), Y) -> add(Y, prod(X, Y))
prod(X1, X2) -> nprod(X1, X2)
if(true, X, Y) -> activate(X)
if(false, X, Y) -> activate(Y)
zero(0) -> true
zero(s(X)) -> false
p(s(X)) -> X
s(X) -> ns(X)
activate(ns(X)) -> s(X)
activate(nprod(X1, X2)) -> prod(X1, X2)
activate(X) -> X

Termination of R to be shown.

`   R`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

FACT(X) -> IF(zero(X), ns(0), nprod(X, fact(p(X))))
FACT(X) -> ZERO(X)
FACT(X) -> FACT(p(X))
FACT(X) -> P(X)
PROD(s(X), Y) -> ADD(Y, prod(X, Y))
PROD(s(X), Y) -> PROD(X, Y)
IF(true, X, Y) -> ACTIVATE(X)
IF(false, X, Y) -> ACTIVATE(Y)
ACTIVATE(ns(X)) -> S(X)
ACTIVATE(nprod(X1, X2)) -> PROD(X1, X2)

Furthermore, R contains three SCCs.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Polynomial Ordering`
`       →DP Problem 2`
`         ↳Polo`
`       →DP Problem 3`
`         ↳Remaining`

Dependency Pair:

Rules:

fact(X) -> if(zero(X), ns(0), nprod(X, fact(p(X))))
prod(0, X) -> 0
prod(s(X), Y) -> add(Y, prod(X, Y))
prod(X1, X2) -> nprod(X1, X2)
if(true, X, Y) -> activate(X)
if(false, X, Y) -> activate(Y)
zero(0) -> true
zero(s(X)) -> false
p(s(X)) -> X
s(X) -> ns(X)
activate(ns(X)) -> s(X)
activate(nprod(X1, X2)) -> prod(X1, X2)
activate(X) -> X

The following dependency pair can be strictly oriented:

There are no usable rules using the Ce-refinement that need to be oriented.

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

resulting in one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Polo`
`           →DP Problem 4`
`             ↳Dependency Graph`
`       →DP Problem 2`
`         ↳Polo`
`       →DP Problem 3`
`         ↳Remaining`

Dependency Pair:

Rules:

fact(X) -> if(zero(X), ns(0), nprod(X, fact(p(X))))
prod(0, X) -> 0
prod(s(X), Y) -> add(Y, prod(X, Y))
prod(X1, X2) -> nprod(X1, X2)
if(true, X, Y) -> activate(X)
if(false, X, Y) -> activate(Y)
zero(0) -> true
zero(s(X)) -> false
p(s(X)) -> X
s(X) -> ns(X)
activate(ns(X)) -> s(X)
activate(nprod(X1, X2)) -> prod(X1, X2)
activate(X) -> X

Using the Dependency Graph resulted in no new DP problems.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Polo`
`       →DP Problem 2`
`         ↳Polynomial Ordering`
`       →DP Problem 3`
`         ↳Remaining`

Dependency Pair:

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

Rules:

fact(X) -> if(zero(X), ns(0), nprod(X, fact(p(X))))
prod(0, X) -> 0
prod(s(X), Y) -> add(Y, prod(X, Y))
prod(X1, X2) -> nprod(X1, X2)
if(true, X, Y) -> activate(X)
if(false, X, Y) -> activate(Y)
zero(0) -> true
zero(s(X)) -> false
p(s(X)) -> X
s(X) -> ns(X)
activate(ns(X)) -> s(X)
activate(nprod(X1, X2)) -> prod(X1, X2)
activate(X) -> X

The following dependency pair can be strictly oriented:

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

There are no usable rules using the Ce-refinement that need to be oriented.

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

resulting in one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Polo`
`       →DP Problem 2`
`         ↳Polo`
`           →DP Problem 5`
`             ↳Dependency Graph`
`       →DP Problem 3`
`         ↳Remaining`

Dependency Pair:

Rules:

fact(X) -> if(zero(X), ns(0), nprod(X, fact(p(X))))
prod(0, X) -> 0
prod(s(X), Y) -> add(Y, prod(X, Y))
prod(X1, X2) -> nprod(X1, X2)
if(true, X, Y) -> activate(X)
if(false, X, Y) -> activate(Y)
zero(0) -> true
zero(s(X)) -> false
p(s(X)) -> X
s(X) -> ns(X)
activate(ns(X)) -> s(X)
activate(nprod(X1, X2)) -> prod(X1, X2)
activate(X) -> X

Using the Dependency Graph resulted in no new DP problems.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Polo`
`       →DP Problem 2`
`         ↳Polo`
`       →DP Problem 3`
`         ↳Remaining Obligation(s)`

The following remains to be proven:
Dependency Pair:

FACT(X) -> FACT(p(X))

Rules:

fact(X) -> if(zero(X), ns(0), nprod(X, fact(p(X))))
prod(0, X) -> 0
prod(s(X), Y) -> add(Y, prod(X, Y))
prod(X1, X2) -> nprod(X1, X2)
if(true, X, Y) -> activate(X)
if(false, X, Y) -> activate(Y)
zero(0) -> true
zero(s(X)) -> false
p(s(X)) -> X
s(X) -> ns(X)
activate(ns(X)) -> s(X)
activate(nprod(X1, X2)) -> prod(X1, X2)
activate(X) -> X

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