Term Rewriting System R:
[X, Y, Z]
and(true, X) -> X
and(false, Y) -> false
if(true, X, Y) -> X
if(false, X, Y) -> Y
first(0, X) -> nil
first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z))
from(X) -> cons(X, from(s(X)))

Termination of R to be shown.

`   R`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

FIRST(s(X), cons(Y, Z)) -> FIRST(X, Z)
FROM(X) -> FROM(s(X))

Furthermore, R contains three SCCs.

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

Dependency Pair:

Rules:

and(true, X) -> X
and(false, Y) -> false
if(true, X, Y) -> X
if(false, X, Y) -> Y
first(0, X) -> nil
first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z))
from(X) -> cons(X, from(s(X)))

The following dependency pair can be strictly oriented:

Additionally, the following rules can be oriented:

and(true, X) -> X
and(false, Y) -> false
if(true, X, Y) -> X
if(false, X, Y) -> Y
first(0, X) -> nil
first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z))
from(X) -> cons(X, from(s(X)))

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(from(x1)) =  0 POL(and(x1, x2)) =  x2 POL(if(x1, x2, x3)) =  x2 + x3 POL(0) =  0 POL(first(x1, x2)) =  0 POL(false) =  0 POL(cons(x1, x2)) =  0 POL(true) =  0 POL(nil) =  0 POL(s(x1)) =  1 + x1 POL(ADD(x1, x2)) =  x1 POL(add(x1, x2)) =  x1 + x2

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:

and(true, X) -> X
and(false, Y) -> false
if(true, X, Y) -> X
if(false, X, Y) -> Y
first(0, X) -> nil
first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z))
from(X) -> cons(X, from(s(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:

FIRST(s(X), cons(Y, Z)) -> FIRST(X, Z)

Rules:

and(true, X) -> X
and(false, Y) -> false
if(true, X, Y) -> X
if(false, X, Y) -> Y
first(0, X) -> nil
first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z))
from(X) -> cons(X, from(s(X)))

The following dependency pair can be strictly oriented:

FIRST(s(X), cons(Y, Z)) -> FIRST(X, Z)

Additionally, the following rules can be oriented:

and(true, X) -> X
and(false, Y) -> false
if(true, X, Y) -> X
if(false, X, Y) -> Y
first(0, X) -> nil
first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z))
from(X) -> cons(X, from(s(X)))

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(from(x1)) =  0 POL(and(x1, x2)) =  x2 POL(if(x1, x2, x3)) =  x2 + x3 POL(0) =  0 POL(first(x1, x2)) =  0 POL(false) =  0 POL(cons(x1, x2)) =  0 POL(FIRST(x1, x2)) =  x1 POL(true) =  0 POL(nil) =  0 POL(s(x1)) =  1 + x1 POL(add(x1, x2)) =  x1 + x2

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:

and(true, X) -> X
and(false, Y) -> false
if(true, X, Y) -> X
if(false, X, Y) -> Y
first(0, X) -> nil
first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z))
from(X) -> cons(X, from(s(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:

FROM(X) -> FROM(s(X))

Rules:

and(true, X) -> X
and(false, Y) -> false
if(true, X, Y) -> X
if(false, X, Y) -> Y