(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
eq → true
eq → eq
eq → false
inf(X) → cons
take(0, X) → nil
take(s, cons) → cons
length(nil) → 0
length(cons) → s
Q is empty.
(1) AAECC Innermost (EQUIVALENT transformation)
We have applied [NOC,AAECCNOC] to switch to innermost. The TRS R 1 is
length(nil) → 0
length(cons) → s
inf(X) → cons
take(0, X) → nil
take(s, cons) → cons
The TRS R 2 is
eq → true
eq → eq
eq → false
The signature Sigma is {
eq,
true,
false}
(2) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
eq → true
eq → eq
eq → false
inf(X) → cons
take(0, X) → nil
take(s, cons) → cons
length(nil) → 0
length(cons) → s
The set Q consists of the following terms:
eq
inf(x0)
take(0, x0)
take(s, cons)
length(nil)
length(cons)
(3) DependencyPairsProof (EQUIVALENT transformation)
Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
(4) Obligation:
Q DP problem:
The TRS P consists of the following rules:
EQ → EQ
The TRS R consists of the following rules:
eq → true
eq → eq
eq → false
inf(X) → cons
take(0, X) → nil
take(s, cons) → cons
length(nil) → 0
length(cons) → s
The set Q consists of the following terms:
eq
inf(x0)
take(0, x0)
take(s, cons)
length(nil)
length(cons)
We have to consider all minimal (P,Q,R)-chains.