(0) Obligation:

Q restricted rewrite system:
The TRS R consists of the following rules:

a__eq(0, 0) → true
a__eq(s(X), s(Y)) → a__eq(X, Y)
a__eq(X, Y) → false
a__inf(X) → cons(X, inf(s(X)))
a__take(0, X) → nil
a__take(s(X), cons(Y, L)) → cons(Y, take(X, L))
a__length(nil) → 0
a__length(cons(X, L)) → s(length(L))
mark(eq(X1, X2)) → a__eq(X1, X2)
mark(inf(X)) → a__inf(mark(X))
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(length(X)) → a__length(mark(X))
mark(0) → 0
mark(true) → true
mark(s(X)) → s(X)
mark(false) → false
mark(cons(X1, X2)) → cons(X1, X2)
mark(nil) → nil
a__eq(X1, X2) → eq(X1, X2)
a__inf(X) → inf(X)
a__take(X1, X2) → take(X1, X2)
a__length(X) → length(X)

Q is empty.

(1) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Combined order from the following AFS and order.
a__eq(x1, x2)  =  a__eq(x1, x2)
0  =  0
true  =  true
s(x1)  =  x1
false  =  false
a__inf(x1)  =  a__inf(x1)
cons(x1, x2)  =  cons(x1, x2)
inf(x1)  =  inf(x1)
a__take(x1, x2)  =  a__take(x1, x2)
nil  =  nil
take(x1, x2)  =  take(x1, x2)
a__length(x1)  =  a__length(x1)
length(x1)  =  length(x1)
mark(x1)  =  mark(x1)
eq(x1, x2)  =  eq(x1, x2)

Recursive path order with status [RPO].
Quasi-Precedence:
mark1 > [aeq2, true] > [0, false, inf1, nil, eq2]
mark1 > ainf1 > cons2 > length1 > [0, false, inf1, nil, eq2]
mark1 > atake2 > cons2 > length1 > [0, false, inf1, nil, eq2]
mark1 > atake2 > take2 > [0, false, inf1, nil, eq2]
mark1 > alength1 > length1 > [0, false, inf1, nil, eq2]

Status:
eq2: multiset
ainf1: [1]
true: multiset
mark1: [1]
take2: multiset
atake2: [1,2]
0: multiset
inf1: multiset
cons2: multiset
alength1: [1]
false: multiset
aeq2: [1,2]
length1: multiset
nil: multiset

With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

a__eq(0, 0) → true
a__eq(X, Y) → false
a__inf(X) → cons(X, inf(s(X)))
a__take(0, X) → nil
a__take(s(X), cons(Y, L)) → cons(Y, take(X, L))
a__length(nil) → 0
a__length(cons(X, L)) → s(length(L))
mark(eq(X1, X2)) → a__eq(X1, X2)
mark(inf(X)) → a__inf(mark(X))
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(length(X)) → a__length(mark(X))
mark(0) → 0
mark(true) → true
mark(s(X)) → s(X)
mark(false) → false
mark(cons(X1, X2)) → cons(X1, X2)
mark(nil) → nil
a__eq(X1, X2) → eq(X1, X2)
a__inf(X) → inf(X)
a__take(X1, X2) → take(X1, X2)
a__length(X) → length(X)


(2) Obligation:

Q restricted rewrite system:
The TRS R consists of the following rules:

a__eq(s(X), s(Y)) → a__eq(X, Y)

Q is empty.

(3) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(a__eq(x1, x2)) = x1 + x2   
POL(s(x1)) = 1 + x1   
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

a__eq(s(X), s(Y)) → a__eq(X, Y)


(4) Obligation:

Q restricted rewrite system:
R is empty.
Q is empty.

(5) RisEmptyProof (EQUIVALENT transformation)

The TRS R is empty. Hence, termination is trivially proven.

(6) TRUE

(7) RisEmptyProof (EQUIVALENT transformation)

The TRS R is empty. Hence, termination is trivially proven.

(8) TRUE

(9) RisEmptyProof (EQUIVALENT transformation)

The TRS R is empty. Hence, termination is trivially proven.

(10) TRUE