Termination Proof using AProVE

Term Rewriting System R:
[X, Y, L, X1, X2]
eq(n_₀, n_₀) -> true
eq(n_{_s}(X), n_{_s}(Y)) -> eq(activate(X), activate(Y))
eq(X, Y) -> false
inf(X) -> cons(X, n_{_inf}(n_{_s}(X)))
inf(X) -> n_{_inf}(X)
take(0, X) -> nil
take(s(X), cons(Y, L)) -> cons(activate(Y), n_{_take}(activate(X), activate(L)))
take(X1, X2) -> n_{_take}(X1, X2)
length(nil) -> 0
length(cons(X, L)) -> s(n_{_length}(activate(L)))
length(X) -> n_{_length}(X)
0 -> n_₀
s(X) -> n_{_s}(X)
activate(n_₀) -> 0
activate(n_{_s}(X)) -> s(X)
activate(n_{_inf}(X)) -> inf(activate(X))
activate(n_{_take}(X1, X2)) -> take(activate(X1), activate(X2))
activate(n_{_length}(X)) -> length(activate(X))
activate(X) -> X

Innermost Termination of R to be shown.

     ↳Dependency Pair Analysis

R contains the following Dependency Pairs:

EQ(n_{_s}(X), n_{_s}(Y)) -> EQ(activate(X), activate(Y))
EQ(n_{_s}(X), n_{_s}(Y)) -> ACTIVATE(X)
EQ(n_{_s}(X), n_{_s}(Y)) -> ACTIVATE(Y)
TAKE(s(X), cons(Y, L)) -> ACTIVATE(Y)
TAKE(s(X), cons(Y, L)) -> ACTIVATE(X)
TAKE(s(X), cons(Y, L)) -> ACTIVATE(L)
LENGTH(nil) -> 0'
LENGTH(cons(X, L)) -> S(n_{_length}(activate(L)))
LENGTH(cons(X, L)) -> ACTIVATE(L)
ACTIVATE(n_₀) -> 0'
ACTIVATE(n_{_s}(X)) -> S(X)
ACTIVATE(n_{_inf}(X)) -> INF(activate(X))
ACTIVATE(n_{_inf}(X)) -> ACTIVATE(X)
ACTIVATE(n_{_take}(X1, X2)) -> TAKE(activate(X1), activate(X2))
ACTIVATE(n_{_take}(X1, X2)) -> ACTIVATE(X1)
ACTIVATE(n_{_take}(X1, X2)) -> ACTIVATE(X2)
ACTIVATE(n_{_length}(X)) -> LENGTH(activate(X))
ACTIVATE(n_{_length}(X)) -> ACTIVATE(X)

Furthermore, R contains two SCCs.

     ↳DPs

       →DP Problem 1

         ↳Narrowing Transformation

       →DP Problem 2

         ↳Remaining

Dependency Pairs:

ACTIVATE(n_{_length}(X)) -> ACTIVATE(X)
LENGTH(cons(X, L)) -> ACTIVATE(L)
ACTIVATE(n_{_length}(X)) -> LENGTH(activate(X))
ACTIVATE(n_{_take}(X1, X2)) -> ACTIVATE(X2)
ACTIVATE(n_{_take}(X1, X2)) -> ACTIVATE(X1)
ACTIVATE(n_{_inf}(X)) -> ACTIVATE(X)

Rules:

eq(n_₀, n_₀) -> true
eq(n_{_s}(X), n_{_s}(Y)) -> eq(activate(X), activate(Y))
eq(X, Y) -> false
inf(X) -> cons(X, n_{_inf}(n_{_s}(X)))
inf(X) -> n_{_inf}(X)
take(0, X) -> nil
take(s(X), cons(Y, L)) -> cons(activate(Y), n_{_take}(activate(X), activate(L)))
take(X1, X2) -> n_{_take}(X1, X2)
length(nil) -> 0
length(cons(X, L)) -> s(n_{_length}(activate(L)))
length(X) -> n_{_length}(X)
0 -> n_₀
s(X) -> n_{_s}(X)
activate(n_₀) -> 0
activate(n_{_s}(X)) -> s(X)
activate(n_{_inf}(X)) -> inf(activate(X))
activate(n_{_take}(X1, X2)) -> take(activate(X1), activate(X2))
activate(n_{_length}(X)) -> length(activate(X))
activate(X) -> X

Strategy:

innermost

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

ACTIVATE(n_{_length}(X)) -> LENGTH(activate(X))

six new Dependency Pairs are created:

ACTIVATE(n_{_length}(n_₀)) -> LENGTH(0)
ACTIVATE(n_{_length}(n_{_s}(X''))) -> LENGTH(s(X''))
ACTIVATE(n_{_length}(n_{_inf}(X''))) -> LENGTH(inf(activate(X'')))
ACTIVATE(n_{_length}(n_{_take}(X1', X2'))) -> LENGTH(take(activate(X1'), activate(X2')))
ACTIVATE(n_{_length}(n_{_length}(X''))) -> LENGTH(length(activate(X'')))
ACTIVATE(n_{_length}(X'')) -> LENGTH(X'')

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 3

             ↳Rewriting Transformation

       →DP Problem 2

         ↳Remaining

Dependency Pairs:

ACTIVATE(n_{_length}(X'')) -> LENGTH(X'')
ACTIVATE(n_{_length}(n_{_length}(X''))) -> LENGTH(length(activate(X'')))
ACTIVATE(n_{_length}(n_{_take}(X1', X2'))) -> LENGTH(take(activate(X1'), activate(X2')))
ACTIVATE(n_{_length}(n_{_inf}(X''))) -> LENGTH(inf(activate(X'')))
ACTIVATE(n_{_length}(n_{_s}(X''))) -> LENGTH(s(X''))
LENGTH(cons(X, L)) -> ACTIVATE(L)
ACTIVATE(n_{_length}(n_₀)) -> LENGTH(0)
ACTIVATE(n_{_take}(X1, X2)) -> ACTIVATE(X2)
ACTIVATE(n_{_take}(X1, X2)) -> ACTIVATE(X1)
ACTIVATE(n_{_inf}(X)) -> ACTIVATE(X)
ACTIVATE(n_{_length}(X)) -> ACTIVATE(X)

Rules:

eq(n_₀, n_₀) -> true
eq(n_{_s}(X), n_{_s}(Y)) -> eq(activate(X), activate(Y))
eq(X, Y) -> false
inf(X) -> cons(X, n_{_inf}(n_{_s}(X)))
inf(X) -> n_{_inf}(X)
take(0, X) -> nil
take(s(X), cons(Y, L)) -> cons(activate(Y), n_{_take}(activate(X), activate(L)))
take(X1, X2) -> n_{_take}(X1, X2)
length(nil) -> 0
length(cons(X, L)) -> s(n_{_length}(activate(L)))
length(X) -> n_{_length}(X)
0 -> n_₀
s(X) -> n_{_s}(X)
activate(n_₀) -> 0
activate(n_{_s}(X)) -> s(X)
activate(n_{_inf}(X)) -> inf(activate(X))
activate(n_{_take}(X1, X2)) -> take(activate(X1), activate(X2))
activate(n_{_length}(X)) -> length(activate(X))
activate(X) -> X

Strategy:

innermost

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

ACTIVATE(n_{_length}(n_₀)) -> LENGTH(0)

one new Dependency Pair is created:

ACTIVATE(n_{_length}(n_₀)) -> LENGTH(n_₀)

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 3

             ↳Rw

...

               →DP Problem 4

                 ↳Rewriting Transformation

       →DP Problem 2

         ↳Remaining

Dependency Pairs:

ACTIVATE(n_{_length}(n_{_length}(X''))) -> LENGTH(length(activate(X'')))
ACTIVATE(n_{_length}(n_{_take}(X1', X2'))) -> LENGTH(take(activate(X1'), activate(X2')))
ACTIVATE(n_{_length}(n_{_inf}(X''))) -> LENGTH(inf(activate(X'')))
ACTIVATE(n_{_length}(n_{_s}(X''))) -> LENGTH(s(X''))
ACTIVATE(n_{_length}(X)) -> ACTIVATE(X)
ACTIVATE(n_{_take}(X1, X2)) -> ACTIVATE(X2)
ACTIVATE(n_{_take}(X1, X2)) -> ACTIVATE(X1)
ACTIVATE(n_{_inf}(X)) -> ACTIVATE(X)
LENGTH(cons(X, L)) -> ACTIVATE(L)
ACTIVATE(n_{_length}(X'')) -> LENGTH(X'')

Rules:

eq(n_₀, n_₀) -> true
eq(n_{_s}(X), n_{_s}(Y)) -> eq(activate(X), activate(Y))
eq(X, Y) -> false
inf(X) -> cons(X, n_{_inf}(n_{_s}(X)))
inf(X) -> n_{_inf}(X)
take(0, X) -> nil
take(s(X), cons(Y, L)) -> cons(activate(Y), n_{_take}(activate(X), activate(L)))
take(X1, X2) -> n_{_take}(X1, X2)
length(nil) -> 0
length(cons(X, L)) -> s(n_{_length}(activate(L)))
length(X) -> n_{_length}(X)
0 -> n_₀
s(X) -> n_{_s}(X)
activate(n_₀) -> 0
activate(n_{_s}(X)) -> s(X)
activate(n_{_inf}(X)) -> inf(activate(X))
activate(n_{_take}(X1, X2)) -> take(activate(X1), activate(X2))
activate(n_{_length}(X)) -> length(activate(X))
activate(X) -> X

Strategy:

innermost

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

ACTIVATE(n_{_length}(n_{_s}(X''))) -> LENGTH(s(X''))

one new Dependency Pair is created:

ACTIVATE(n_{_length}(n_{_s}(X''))) -> LENGTH(n_{_s}(X''))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

       →DP Problem 2

         ↳Remaining Obligation(s)

The following remains to be proven:

Dependency Pairs:
ACTIVATE(n_{_length}(X'')) -> LENGTH(X'')
ACTIVATE(n_{_length}(n_{_take}(X1', X2'))) -> LENGTH(take(activate(X1'), activate(X2')))
ACTIVATE(n_{_length}(n_{_inf}(X''))) -> LENGTH(inf(activate(X'')))
ACTIVATE(n_{_length}(X)) -> ACTIVATE(X)
ACTIVATE(n_{_take}(X1, X2)) -> ACTIVATE(X2)
ACTIVATE(n_{_take}(X1, X2)) -> ACTIVATE(X1)
ACTIVATE(n_{_inf}(X)) -> ACTIVATE(X)
LENGTH(cons(X, L)) -> ACTIVATE(L)
ACTIVATE(n_{_length}(n_{_length}(X''))) -> LENGTH(length(activate(X'')))

Rules:

eq(n_₀, n_₀) -> true
eq(n_{_s}(X), n_{_s}(Y)) -> eq(activate(X), activate(Y))
eq(X, Y) -> false
inf(X) -> cons(X, n_{_inf}(n_{_s}(X)))
inf(X) -> n_{_inf}(X)
take(0, X) -> nil
take(s(X), cons(Y, L)) -> cons(activate(Y), n_{_take}(activate(X), activate(L)))
take(X1, X2) -> n_{_take}(X1, X2)
length(nil) -> 0
length(cons(X, L)) -> s(n_{_length}(activate(L)))
length(X) -> n_{_length}(X)
0 -> n_₀
s(X) -> n_{_s}(X)
activate(n_₀) -> 0
activate(n_{_s}(X)) -> s(X)
activate(n_{_inf}(X)) -> inf(activate(X))
activate(n_{_take}(X1, X2)) -> take(activate(X1), activate(X2))
activate(n_{_length}(X)) -> length(activate(X))
activate(X) -> X

Strategy:
innermost
Dependency Pair:
EQ(n_{_s}(X), n_{_s}(Y)) -> EQ(activate(X), activate(Y))

Rules:

eq(n_₀, n_₀) -> true
eq(n_{_s}(X), n_{_s}(Y)) -> eq(activate(X), activate(Y))
eq(X, Y) -> false
inf(X) -> cons(X, n_{_inf}(n_{_s}(X)))
inf(X) -> n_{_inf}(X)
take(0, X) -> nil
take(s(X), cons(Y, L)) -> cons(activate(Y), n_{_take}(activate(X), activate(L)))
take(X1, X2) -> n_{_take}(X1, X2)
length(nil) -> 0
length(cons(X, L)) -> s(n_{_length}(activate(L)))
length(X) -> n_{_length}(X)
0 -> n_₀
s(X) -> n_{_s}(X)
activate(n_₀) -> 0
activate(n_{_s}(X)) -> s(X)
activate(n_{_inf}(X)) -> inf(activate(X))
activate(n_{_take}(X1, X2)) -> take(activate(X1), activate(X2))
activate(n_{_length}(X)) -> length(activate(X))
activate(X) -> X

Strategy:
innermost

     ↳DPs

       →DP Problem 1

         ↳Nar

       →DP Problem 2

         ↳Remaining Obligation(s)

The following remains to be proven:

Dependency Pairs:
ACTIVATE(n_{_length}(X'')) -> LENGTH(X'')
ACTIVATE(n_{_length}(n_{_take}(X1', X2'))) -> LENGTH(take(activate(X1'), activate(X2')))
ACTIVATE(n_{_length}(n_{_inf}(X''))) -> LENGTH(inf(activate(X'')))
ACTIVATE(n_{_length}(X)) -> ACTIVATE(X)
ACTIVATE(n_{_take}(X1, X2)) -> ACTIVATE(X2)
ACTIVATE(n_{_take}(X1, X2)) -> ACTIVATE(X1)
ACTIVATE(n_{_inf}(X)) -> ACTIVATE(X)
LENGTH(cons(X, L)) -> ACTIVATE(L)
ACTIVATE(n_{_length}(n_{_length}(X''))) -> LENGTH(length(activate(X'')))

Rules:

eq(n_₀, n_₀) -> true
eq(n_{_s}(X), n_{_s}(Y)) -> eq(activate(X), activate(Y))
eq(X, Y) -> false
inf(X) -> cons(X, n_{_inf}(n_{_s}(X)))
inf(X) -> n_{_inf}(X)
take(0, X) -> nil
take(s(X), cons(Y, L)) -> cons(activate(Y), n_{_take}(activate(X), activate(L)))
take(X1, X2) -> n_{_take}(X1, X2)
length(nil) -> 0
length(cons(X, L)) -> s(n_{_length}(activate(L)))
length(X) -> n_{_length}(X)
0 -> n_₀
s(X) -> n_{_s}(X)
activate(n_₀) -> 0
activate(n_{_s}(X)) -> s(X)
activate(n_{_inf}(X)) -> inf(activate(X))
activate(n_{_take}(X1, X2)) -> take(activate(X1), activate(X2))
activate(n_{_length}(X)) -> length(activate(X))
activate(X) -> X

Strategy:
innermost
Dependency Pair:
EQ(n_{_s}(X), n_{_s}(Y)) -> EQ(activate(X), activate(Y))

Rules:

eq(n_₀, n_₀) -> true
eq(n_{_s}(X), n_{_s}(Y)) -> eq(activate(X), activate(Y))
eq(X, Y) -> false
inf(X) -> cons(X, n_{_inf}(n_{_s}(X)))
inf(X) -> n_{_inf}(X)
take(0, X) -> nil
take(s(X), cons(Y, L)) -> cons(activate(Y), n_{_take}(activate(X), activate(L)))
take(X1, X2) -> n_{_take}(X1, X2)
length(nil) -> 0
length(cons(X, L)) -> s(n_{_length}(activate(L)))
length(X) -> n_{_length}(X)
0 -> n_₀
s(X) -> n_{_s}(X)
activate(n_₀) -> 0
activate(n_{_s}(X)) -> s(X)
activate(n_{_inf}(X)) -> inf(activate(X))
activate(n_{_take}(X1, X2)) -> take(activate(X1), activate(X2))
activate(n_{_length}(X)) -> length(activate(X))
activate(X) -> X

Strategy:
innermost

Innermost Termination of R could not be shown.
Duration:
0:02 minutes