Term Rewriting System R:
[X, Y, X1, X2, X3]
af(X) -> aif(mark(X), c, f(true))
af(X) -> f(X)
aif(true, X, Y) -> mark(X)
aif(false, X, Y) -> mark(Y)
aif(X1, X2, X3) -> if(X1, X2, X3)
mark(f(X)) -> af(mark(X))
mark(if(X1, X2, X3)) -> aif(mark(X1), mark(X2), X3)
mark(c) -> c
mark(true) -> true
mark(false) -> false

Termination of R to be shown.

`   R`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

AF(X) -> AIF(mark(X), c, f(true))
AF(X) -> MARK(X)
AIF(true, X, Y) -> MARK(X)
AIF(false, X, Y) -> MARK(Y)
MARK(f(X)) -> AF(mark(X))
MARK(f(X)) -> MARK(X)
MARK(if(X1, X2, X3)) -> AIF(mark(X1), mark(X2), X3)
MARK(if(X1, X2, X3)) -> MARK(X1)
MARK(if(X1, X2, X3)) -> MARK(X2)

Furthermore, R contains one SCC.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Polynomial Ordering`

Dependency Pairs:

MARK(if(X1, X2, X3)) -> MARK(X2)
MARK(if(X1, X2, X3)) -> MARK(X1)
AIF(false, X, Y) -> MARK(Y)
MARK(if(X1, X2, X3)) -> AIF(mark(X1), mark(X2), X3)
MARK(f(X)) -> MARK(X)
AF(X) -> MARK(X)
MARK(f(X)) -> AF(mark(X))
AIF(true, X, Y) -> MARK(X)
AF(X) -> AIF(mark(X), c, f(true))

Rules:

af(X) -> aif(mark(X), c, f(true))
af(X) -> f(X)
aif(true, X, Y) -> mark(X)
aif(false, X, Y) -> mark(Y)
aif(X1, X2, X3) -> if(X1, X2, X3)
mark(f(X)) -> af(mark(X))
mark(if(X1, X2, X3)) -> aif(mark(X1), mark(X2), X3)
mark(c) -> c
mark(true) -> true
mark(false) -> false

The following dependency pairs can be strictly oriented:

MARK(f(X)) -> MARK(X)
AF(X) -> MARK(X)

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

aif(true, X, Y) -> mark(X)
aif(false, X, Y) -> mark(Y)
aif(X1, X2, X3) -> if(X1, X2, X3)
mark(f(X)) -> af(mark(X))
mark(if(X1, X2, X3)) -> aif(mark(X1), mark(X2), X3)
mark(c) -> c
mark(true) -> true
mark(false) -> false
af(X) -> aif(mark(X), c, f(true))
af(X) -> f(X)

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(if(x1, x2, x3)) =  x1 + x2 + x3 POL(c) =  0 POL(MARK(x1)) =  x1 POL(false) =  0 POL(a__if(x1, x2, x3)) =  x1 + x2 + x3 POL(true) =  0 POL(A__F(x1)) =  1 + x1 POL(mark(x1)) =  x1 POL(f(x1)) =  1 + x1 POL(a__f(x1)) =  1 + x1 POL(A__IF(x1, x2, x3)) =  x2 + x3

resulting in one new DP problem.

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

Dependency Pairs:

MARK(if(X1, X2, X3)) -> MARK(X2)
MARK(if(X1, X2, X3)) -> MARK(X1)
AIF(false, X, Y) -> MARK(Y)
MARK(if(X1, X2, X3)) -> AIF(mark(X1), mark(X2), X3)
MARK(f(X)) -> AF(mark(X))
AIF(true, X, Y) -> MARK(X)
AF(X) -> AIF(mark(X), c, f(true))

Rules:

af(X) -> aif(mark(X), c, f(true))
af(X) -> f(X)
aif(true, X, Y) -> mark(X)
aif(false, X, Y) -> mark(Y)
aif(X1, X2, X3) -> if(X1, X2, X3)
mark(f(X)) -> af(mark(X))
mark(if(X1, X2, X3)) -> aif(mark(X1), mark(X2), X3)
mark(c) -> c
mark(true) -> true
mark(false) -> false

The following dependency pair can be strictly oriented:

AIF(false, X, Y) -> MARK(Y)

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

aif(true, X, Y) -> mark(X)
aif(false, X, Y) -> mark(Y)
aif(X1, X2, X3) -> if(X1, X2, X3)
mark(f(X)) -> af(mark(X))
mark(if(X1, X2, X3)) -> aif(mark(X1), mark(X2), X3)
mark(c) -> c
mark(true) -> true
mark(false) -> false
af(X) -> aif(mark(X), c, f(true))
af(X) -> f(X)

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(if(x1, x2, x3)) =  x1 + x2 + x3 POL(c) =  0 POL(MARK(x1)) =  x1 POL(false) =  1 POL(a__if(x1, x2, x3)) =  x1 + x2 + x3 POL(true) =  0 POL(A__F(x1)) =  x1 POL(mark(x1)) =  x1 POL(f(x1)) =  x1 POL(a__f(x1)) =  x1 POL(A__IF(x1, x2, x3)) =  x1 + x2 + x3

resulting in one new DP problem.

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

Dependency Pairs:

MARK(if(X1, X2, X3)) -> MARK(X2)
MARK(if(X1, X2, X3)) -> MARK(X1)
MARK(if(X1, X2, X3)) -> AIF(mark(X1), mark(X2), X3)
MARK(f(X)) -> AF(mark(X))
AIF(true, X, Y) -> MARK(X)
AF(X) -> AIF(mark(X), c, f(true))

Rules:

af(X) -> aif(mark(X), c, f(true))
af(X) -> f(X)
aif(true, X, Y) -> mark(X)
aif(false, X, Y) -> mark(Y)
aif(X1, X2, X3) -> if(X1, X2, X3)
mark(f(X)) -> af(mark(X))
mark(if(X1, X2, X3)) -> aif(mark(X1), mark(X2), X3)
mark(c) -> c
mark(true) -> true
mark(false) -> false

The following dependency pair can be strictly oriented:

MARK(f(X)) -> AF(mark(X))

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

aif(true, X, Y) -> mark(X)
aif(false, X, Y) -> mark(Y)
aif(X1, X2, X3) -> if(X1, X2, X3)
mark(f(X)) -> af(mark(X))
mark(if(X1, X2, X3)) -> aif(mark(X1), mark(X2), X3)
mark(c) -> c
mark(true) -> true
mark(false) -> false
af(X) -> aif(mark(X), c, f(true))
af(X) -> f(X)

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(if(x1, x2, x3)) =  x1 + x2 + x3 POL(c) =  0 POL(MARK(x1)) =  x1 POL(false) =  0 POL(a__if(x1, x2, x3)) =  x1 + x2 + x3 POL(true) =  0 POL(A__F(x1)) =  0 POL(mark(x1)) =  x1 POL(f(x1)) =  1 + x1 POL(a__f(x1)) =  1 + x1 POL(A__IF(x1, x2, x3)) =  x2

resulting in one new DP problem.

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

Dependency Pairs:

MARK(if(X1, X2, X3)) -> MARK(X2)
MARK(if(X1, X2, X3)) -> MARK(X1)
MARK(if(X1, X2, X3)) -> AIF(mark(X1), mark(X2), X3)
AIF(true, X, Y) -> MARK(X)
AF(X) -> AIF(mark(X), c, f(true))

Rules:

af(X) -> aif(mark(X), c, f(true))
af(X) -> f(X)
aif(true, X, Y) -> mark(X)
aif(false, X, Y) -> mark(Y)
aif(X1, X2, X3) -> if(X1, X2, X3)
mark(f(X)) -> af(mark(X))
mark(if(X1, X2, X3)) -> aif(mark(X1), mark(X2), X3)
mark(c) -> c
mark(true) -> true
mark(false) -> false

Using the Dependency Graph the DP problem was split into 1 DP problems.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Polo`
`           →DP Problem 2`
`             ↳Polo`
`             ...`
`               →DP Problem 5`
`                 ↳Narrowing Transformation`

Dependency Pairs:

MARK(if(X1, X2, X3)) -> MARK(X1)
AIF(true, X, Y) -> MARK(X)
MARK(if(X1, X2, X3)) -> AIF(mark(X1), mark(X2), X3)
MARK(if(X1, X2, X3)) -> MARK(X2)

Rules:

af(X) -> aif(mark(X), c, f(true))
af(X) -> f(X)
aif(true, X, Y) -> mark(X)
aif(false, X, Y) -> mark(Y)
aif(X1, X2, X3) -> if(X1, X2, X3)
mark(f(X)) -> af(mark(X))
mark(if(X1, X2, X3)) -> aif(mark(X1), mark(X2), X3)
mark(c) -> c
mark(true) -> true
mark(false) -> false

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

MARK(if(X1, X2, X3)) -> AIF(mark(X1), mark(X2), X3)
10 new Dependency Pairs are created:

MARK(if(f(X'), X2, X3)) -> AIF(af(mark(X')), mark(X2), X3)
MARK(if(if(X1'', X2'', X3''), X2, X3)) -> AIF(aif(mark(X1''), mark(X2''), X3''), mark(X2), X3)
MARK(if(c, X2, X3)) -> AIF(c, mark(X2), X3)
MARK(if(true, X2, X3)) -> AIF(true, mark(X2), X3)
MARK(if(false, X2, X3)) -> AIF(false, mark(X2), X3)
MARK(if(X1, f(X'), X3)) -> AIF(mark(X1), af(mark(X')), X3)
MARK(if(X1, if(X1'', X2'', X3''), X3)) -> AIF(mark(X1), aif(mark(X1''), mark(X2''), X3''), X3)
MARK(if(X1, c, X3)) -> AIF(mark(X1), c, X3)
MARK(if(X1, true, X3)) -> AIF(mark(X1), true, X3)
MARK(if(X1, false, X3)) -> AIF(mark(X1), false, X3)

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Polo`
`           →DP Problem 2`
`             ↳Polo`
`             ...`
`               →DP Problem 6`
`                 ↳Polynomial Ordering`

Dependency Pairs:

MARK(if(X1, false, X3)) -> AIF(mark(X1), false, X3)
MARK(if(X1, true, X3)) -> AIF(mark(X1), true, X3)
MARK(if(X1, c, X3)) -> AIF(mark(X1), c, X3)
MARK(if(X1, if(X1'', X2'', X3''), X3)) -> AIF(mark(X1), aif(mark(X1''), mark(X2''), X3''), X3)
MARK(if(X1, f(X'), X3)) -> AIF(mark(X1), af(mark(X')), X3)
MARK(if(true, X2, X3)) -> AIF(true, mark(X2), X3)
MARK(if(if(X1'', X2'', X3''), X2, X3)) -> AIF(aif(mark(X1''), mark(X2''), X3''), mark(X2), X3)
AIF(true, X, Y) -> MARK(X)
MARK(if(f(X'), X2, X3)) -> AIF(af(mark(X')), mark(X2), X3)
MARK(if(X1, X2, X3)) -> MARK(X2)
MARK(if(X1, X2, X3)) -> MARK(X1)

Rules:

af(X) -> aif(mark(X), c, f(true))
af(X) -> f(X)
aif(true, X, Y) -> mark(X)
aif(false, X, Y) -> mark(Y)
aif(X1, X2, X3) -> if(X1, X2, X3)
mark(f(X)) -> af(mark(X))
mark(if(X1, X2, X3)) -> aif(mark(X1), mark(X2), X3)
mark(c) -> c
mark(true) -> true
mark(false) -> false

The following dependency pair can be strictly oriented:

MARK(if(f(X'), X2, X3)) -> AIF(af(mark(X')), mark(X2), X3)

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

aif(true, X, Y) -> mark(X)
aif(false, X, Y) -> mark(Y)
aif(X1, X2, X3) -> if(X1, X2, X3)
mark(f(X)) -> af(mark(X))
mark(if(X1, X2, X3)) -> aif(mark(X1), mark(X2), X3)
mark(c) -> c
mark(true) -> true
mark(false) -> false
af(X) -> aif(mark(X), c, f(true))
af(X) -> f(X)

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(if(x1, x2, x3)) =  x1 + x2 + x3 POL(c) =  0 POL(MARK(x1)) =  x1 POL(false) =  0 POL(a__if(x1, x2, x3)) =  x1 + x2 + x3 POL(true) =  0 POL(mark(x1)) =  x1 POL(f(x1)) =  1 + x1 POL(a__f(x1)) =  1 + x1 POL(A__IF(x1, x2, x3)) =  x2

resulting in one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Polo`
`           →DP Problem 2`
`             ↳Polo`
`             ...`
`               →DP Problem 7`
`                 ↳Remaining Obligation(s)`

The following remains to be proven:
Dependency Pairs:

MARK(if(X1, false, X3)) -> AIF(mark(X1), false, X3)
MARK(if(X1, true, X3)) -> AIF(mark(X1), true, X3)
MARK(if(X1, c, X3)) -> AIF(mark(X1), c, X3)
MARK(if(X1, if(X1'', X2'', X3''), X3)) -> AIF(mark(X1), aif(mark(X1''), mark(X2''), X3''), X3)
MARK(if(X1, f(X'), X3)) -> AIF(mark(X1), af(mark(X')), X3)
MARK(if(true, X2, X3)) -> AIF(true, mark(X2), X3)
MARK(if(if(X1'', X2'', X3''), X2, X3)) -> AIF(aif(mark(X1''), mark(X2''), X3''), mark(X2), X3)
AIF(true, X, Y) -> MARK(X)
MARK(if(X1, X2, X3)) -> MARK(X2)
MARK(if(X1, X2, X3)) -> MARK(X1)

Rules:

af(X) -> aif(mark(X), c, f(true))
af(X) -> f(X)
aif(true, X, Y) -> mark(X)
aif(false, X, Y) -> mark(Y)
aif(X1, X2, X3) -> if(X1, X2, X3)
mark(f(X)) -> af(mark(X))
mark(if(X1, X2, X3)) -> aif(mark(X1), mark(X2), X3)
mark(c) -> c
mark(true) -> true
mark(false) -> false

Termination of R could not be shown.
Duration:
0:01 minutes