Term Rewriting System R:
[x, y, z]
f(x, nil) -> g(nil, x)
f(x, g(y, z)) -> g(f(x, y), z)
++(x, nil) -> x
++(x, g(y, z)) -> g(++(x, y), z)
null(nil) -> true
null(g(x, y)) -> false
mem(nil, y) -> false
mem(g(x, y), z) -> or(=(y, z), mem(x, z))
mem(x, max(x)) -> not(null(x))
max(g(g(nil, x), y)) -> max'(x, y)
max(g(g(g(x, y), z), u)) -> max'(max(g(g(x, y), z)), u)

Innermost Termination of R to be shown.

`   R`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

F(x, g(y, z)) -> F(x, y)
++'(x, g(y, z)) -> ++'(x, y)
MEM(g(x, y), z) -> MEM(x, z)
MEM(x, max(x)) -> NULL(x)
MAX(g(g(g(x, y), z), u)) -> MAX(g(g(x, y), z))

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:

F(x, g(y, z)) -> F(x, y)

Rules:

f(x, nil) -> g(nil, x)
f(x, g(y, z)) -> g(f(x, y), z)
++(x, nil) -> x
++(x, g(y, z)) -> g(++(x, y), z)
null(nil) -> true
null(g(x, y)) -> false
mem(nil, y) -> false
mem(g(x, y), z) -> or(=(y, z), mem(x, z))
mem(x, max(x)) -> not(null(x))
max(g(g(nil, x), y)) -> max'(x, y)
max(g(g(g(x, y), z), u)) -> max'(max(g(g(x, y), z)), u)

Strategy:

innermost

The following dependency pair can be strictly oriented:

F(x, g(y, z)) -> F(x, 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(g(x1, x2)) =  1 + x1 POL(F(x1, x2)) =  x2

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:

f(x, nil) -> g(nil, x)
f(x, g(y, z)) -> g(f(x, y), z)
++(x, nil) -> x
++(x, g(y, z)) -> g(++(x, y), z)
null(nil) -> true
null(g(x, y)) -> false
mem(nil, y) -> false
mem(g(x, y), z) -> or(=(y, z), mem(x, z))
mem(x, max(x)) -> not(null(x))
max(g(g(nil, x), y)) -> max'(x, y)
max(g(g(g(x, y), z), u)) -> max'(max(g(g(x, y), z)), u)

Strategy:

innermost

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:

++'(x, g(y, z)) -> ++'(x, y)

Rules:

f(x, nil) -> g(nil, x)
f(x, g(y, z)) -> g(f(x, y), z)
++(x, nil) -> x
++(x, g(y, z)) -> g(++(x, y), z)
null(nil) -> true
null(g(x, y)) -> false
mem(nil, y) -> false
mem(g(x, y), z) -> or(=(y, z), mem(x, z))
mem(x, max(x)) -> not(null(x))
max(g(g(nil, x), y)) -> max'(x, y)
max(g(g(g(x, y), z), u)) -> max'(max(g(g(x, y), z)), u)

Strategy:

innermost

The following dependency pair can be strictly oriented:

++'(x, g(y, z)) -> ++'(x, 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(++'(x1, x2)) =  x2 POL(g(x1, x2)) =  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:

f(x, nil) -> g(nil, x)
f(x, g(y, z)) -> g(f(x, y), z)
++(x, nil) -> x
++(x, g(y, z)) -> g(++(x, y), z)
null(nil) -> true
null(g(x, y)) -> false
mem(nil, y) -> false
mem(g(x, y), z) -> or(=(y, z), mem(x, z))
mem(x, max(x)) -> not(null(x))
max(g(g(nil, x), y)) -> max'(x, y)
max(g(g(g(x, y), z), u)) -> max'(max(g(g(x, y), z)), u)

Strategy:

innermost

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:

MEM(g(x, y), z) -> MEM(x, z)

Rules:

f(x, nil) -> g(nil, x)
f(x, g(y, z)) -> g(f(x, y), z)
++(x, nil) -> x
++(x, g(y, z)) -> g(++(x, y), z)
null(nil) -> true
null(g(x, y)) -> false
mem(nil, y) -> false
mem(g(x, y), z) -> or(=(y, z), mem(x, z))
mem(x, max(x)) -> not(null(x))
max(g(g(nil, x), y)) -> max'(x, y)
max(g(g(g(x, y), z), u)) -> max'(max(g(g(x, y), z)), u)

Strategy:

innermost

The following dependency pair can be strictly oriented:

MEM(g(x, y), z) -> MEM(x, z)

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(MEM(x1, x2)) =  x1 POL(g(x1, x2)) =  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:

f(x, nil) -> g(nil, x)
f(x, g(y, z)) -> g(f(x, y), z)
++(x, nil) -> x
++(x, g(y, z)) -> g(++(x, y), z)
null(nil) -> true
null(g(x, y)) -> false
mem(nil, y) -> false
mem(g(x, y), z) -> or(=(y, z), mem(x, z))
mem(x, max(x)) -> not(null(x))
max(g(g(nil, x), y)) -> max'(x, y)
max(g(g(g(x, y), z), u)) -> max'(max(g(g(x, y), z)), u)

Strategy:

innermost

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 Pair:

MAX(g(g(g(x, y), z), u)) -> MAX(g(g(x, y), z))

Rules:

f(x, nil) -> g(nil, x)
f(x, g(y, z)) -> g(f(x, y), z)
++(x, nil) -> x
++(x, g(y, z)) -> g(++(x, y), z)
null(nil) -> true
null(g(x, y)) -> false
mem(nil, y) -> false
mem(g(x, y), z) -> or(=(y, z), mem(x, z))
mem(x, max(x)) -> not(null(x))
max(g(g(nil, x), y)) -> max'(x, y)
max(g(g(g(x, y), z), u)) -> max'(max(g(g(x, y), z)), u)

Strategy:

innermost

The following dependency pair can be strictly oriented:

MAX(g(g(g(x, y), z), u)) -> MAX(g(g(x, y), z))

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(MAX(x1)) =  1 + x1 POL(g(x1, x2)) =  1 + x1 POL(u) =  0

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`
`             ↳Dependency Graph`

Dependency Pair:

Rules:

f(x, nil) -> g(nil, x)
f(x, g(y, z)) -> g(f(x, y), z)
++(x, nil) -> x
++(x, g(y, z)) -> g(++(x, y), z)
null(nil) -> true
null(g(x, y)) -> false
mem(nil, y) -> false
mem(g(x, y), z) -> or(=(y, z), mem(x, z))
mem(x, max(x)) -> not(null(x))
max(g(g(nil, x), y)) -> max'(x, y)
max(g(g(g(x, y), z), u)) -> max'(max(g(g(x, y), z)), u)

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

Innermost Termination of R successfully shown.
Duration:
0:00 minutes