Term Rewriting System R:
[x, y]
even(0) -> true
even(s(0)) -> false
even(s(s(x))) -> even(x)
half(0) -> 0
half(s(s(x))) -> s(half(x))
plus(0, y) -> y
plus(s(x), y) -> s(plus(x, y))
times(0, y) -> 0
times(s(x), y) -> iftimes(even(s(x)), s(x), y)
iftimes(true, s(x), y) -> plus(times(half(s(x)), y), times(half(s(x)), y))
iftimes(false, s(x), y) -> plus(y, times(x, y))

Termination of R to be shown.

`   R`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

EVEN(s(s(x))) -> EVEN(x)
HALF(s(s(x))) -> HALF(x)
PLUS(s(x), y) -> PLUS(x, y)
TIMES(s(x), y) -> IFTIMES(even(s(x)), s(x), y)
TIMES(s(x), y) -> EVEN(s(x))
IFTIMES(true, s(x), y) -> PLUS(times(half(s(x)), y), times(half(s(x)), y))
IFTIMES(true, s(x), y) -> TIMES(half(s(x)), y)
IFTIMES(true, s(x), y) -> HALF(s(x))
IFTIMES(false, s(x), y) -> PLUS(y, times(x, y))
IFTIMES(false, s(x), y) -> TIMES(x, y)

Furthermore, R contains four SCCs.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Polynomial Ordering`
`       →DP Problem 2`
`         ↳Polo`
`       →DP Problem 3`
`         ↳Polo`
`       →DP Problem 4`
`         ↳Polo`

Dependency Pair:

EVEN(s(s(x))) -> EVEN(x)

Rules:

even(0) -> true
even(s(0)) -> false
even(s(s(x))) -> even(x)
half(0) -> 0
half(s(s(x))) -> s(half(x))
plus(0, y) -> y
plus(s(x), y) -> s(plus(x, y))
times(0, y) -> 0
times(s(x), y) -> iftimes(even(s(x)), s(x), y)
iftimes(true, s(x), y) -> plus(times(half(s(x)), y), times(half(s(x)), y))
iftimes(false, s(x), y) -> plus(y, times(x, y))

The following dependency pair can be strictly oriented:

EVEN(s(s(x))) -> EVEN(x)

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

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

resulting in one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Polo`
`           →DP Problem 5`
`             ↳Dependency Graph`
`       →DP Problem 2`
`         ↳Polo`
`       →DP Problem 3`
`         ↳Polo`
`       →DP Problem 4`
`         ↳Polo`

Dependency Pair:

Rules:

even(0) -> true
even(s(0)) -> false
even(s(s(x))) -> even(x)
half(0) -> 0
half(s(s(x))) -> s(half(x))
plus(0, y) -> y
plus(s(x), y) -> s(plus(x, y))
times(0, y) -> 0
times(s(x), y) -> iftimes(even(s(x)), s(x), y)
iftimes(true, s(x), y) -> plus(times(half(s(x)), y), times(half(s(x)), y))
iftimes(false, s(x), y) -> plus(y, times(x, y))

Using the Dependency Graph resulted in no new DP problems.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Polo`
`       →DP Problem 2`
`         ↳Polynomial Ordering`
`       →DP Problem 3`
`         ↳Polo`
`       →DP Problem 4`
`         ↳Polo`

Dependency Pair:

HALF(s(s(x))) -> HALF(x)

Rules:

even(0) -> true
even(s(0)) -> false
even(s(s(x))) -> even(x)
half(0) -> 0
half(s(s(x))) -> s(half(x))
plus(0, y) -> y
plus(s(x), y) -> s(plus(x, y))
times(0, y) -> 0
times(s(x), y) -> iftimes(even(s(x)), s(x), y)
iftimes(true, s(x), y) -> plus(times(half(s(x)), y), times(half(s(x)), y))
iftimes(false, s(x), y) -> plus(y, times(x, y))

The following dependency pair can be strictly oriented:

HALF(s(s(x))) -> HALF(x)

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

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

resulting in one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Polo`
`       →DP Problem 2`
`         ↳Polo`
`           →DP Problem 6`
`             ↳Dependency Graph`
`       →DP Problem 3`
`         ↳Polo`
`       →DP Problem 4`
`         ↳Polo`

Dependency Pair:

Rules:

even(0) -> true
even(s(0)) -> false
even(s(s(x))) -> even(x)
half(0) -> 0
half(s(s(x))) -> s(half(x))
plus(0, y) -> y
plus(s(x), y) -> s(plus(x, y))
times(0, y) -> 0
times(s(x), y) -> iftimes(even(s(x)), s(x), y)
iftimes(true, s(x), y) -> plus(times(half(s(x)), y), times(half(s(x)), y))
iftimes(false, s(x), y) -> plus(y, times(x, y))

Using the Dependency Graph resulted in no new DP problems.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Polo`
`       →DP Problem 2`
`         ↳Polo`
`       →DP Problem 3`
`         ↳Polynomial Ordering`
`       →DP Problem 4`
`         ↳Polo`

Dependency Pair:

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

Rules:

even(0) -> true
even(s(0)) -> false
even(s(s(x))) -> even(x)
half(0) -> 0
half(s(s(x))) -> s(half(x))
plus(0, y) -> y
plus(s(x), y) -> s(plus(x, y))
times(0, y) -> 0
times(s(x), y) -> iftimes(even(s(x)), s(x), y)
iftimes(true, s(x), y) -> plus(times(half(s(x)), y), times(half(s(x)), y))
iftimes(false, s(x), y) -> plus(y, times(x, y))

The following dependency pair can be strictly oriented:

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

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

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

resulting in one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Polo`
`       →DP Problem 2`
`         ↳Polo`
`       →DP Problem 3`
`         ↳Polo`
`           →DP Problem 7`
`             ↳Dependency Graph`
`       →DP Problem 4`
`         ↳Polo`

Dependency Pair:

Rules:

even(0) -> true
even(s(0)) -> false
even(s(s(x))) -> even(x)
half(0) -> 0
half(s(s(x))) -> s(half(x))
plus(0, y) -> y
plus(s(x), y) -> s(plus(x, y))
times(0, y) -> 0
times(s(x), y) -> iftimes(even(s(x)), s(x), y)
iftimes(true, s(x), y) -> plus(times(half(s(x)), y), times(half(s(x)), y))
iftimes(false, s(x), y) -> plus(y, times(x, y))

Using the Dependency Graph resulted in no new DP problems.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Polo`
`       →DP Problem 2`
`         ↳Polo`
`       →DP Problem 3`
`         ↳Polo`
`       →DP Problem 4`
`         ↳Polynomial Ordering`

Dependency Pairs:

IFTIMES(false, s(x), y) -> TIMES(x, y)
IFTIMES(true, s(x), y) -> TIMES(half(s(x)), y)
TIMES(s(x), y) -> IFTIMES(even(s(x)), s(x), y)

Rules:

even(0) -> true
even(s(0)) -> false
even(s(s(x))) -> even(x)
half(0) -> 0
half(s(s(x))) -> s(half(x))
plus(0, y) -> y
plus(s(x), y) -> s(plus(x, y))
times(0, y) -> 0
times(s(x), y) -> iftimes(even(s(x)), s(x), y)
iftimes(true, s(x), y) -> plus(times(half(s(x)), y), times(half(s(x)), y))
iftimes(false, s(x), y) -> plus(y, times(x, y))

The following dependency pair can be strictly oriented:

IFTIMES(false, s(x), y) -> TIMES(x, y)

Additionally, the following usable rules using the Ce-refinement can be oriented:

half(0) -> 0
half(s(s(x))) -> s(half(x))
even(0) -> true
even(s(0)) -> false
even(s(s(x))) -> even(x)

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(IF_TIMES(x1, x2, x3)) =  x2 POL(TIMES(x1, x2)) =  x1 POL(0) =  0 POL(false) =  0 POL(even(x1)) =  0 POL(true) =  0 POL(s(x1)) =  1 + x1 POL(half(x1)) =  x1

resulting in one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Polo`
`       →DP Problem 2`
`         ↳Polo`
`       →DP Problem 3`
`         ↳Polo`
`       →DP Problem 4`
`         ↳Polo`
`           →DP Problem 8`
`             ↳Remaining Obligation(s)`

The following remains to be proven:
Dependency Pairs:

IFTIMES(true, s(x), y) -> TIMES(half(s(x)), y)
TIMES(s(x), y) -> IFTIMES(even(s(x)), s(x), y)

Rules:

even(0) -> true
even(s(0)) -> false
even(s(s(x))) -> even(x)
half(0) -> 0
half(s(s(x))) -> s(half(x))
plus(0, y) -> y
plus(s(x), y) -> s(plus(x, y))
times(0, y) -> 0
times(s(x), y) -> iftimes(even(s(x)), s(x), y)
iftimes(true, s(x), y) -> plus(times(half(s(x)), y), times(half(s(x)), y))
iftimes(false, s(x), y) -> plus(y, times(x, y))

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