(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) QTRSRRRProof (EQUIVALENT transformation)
Used ordering:
Knuth-Bendix order [KBO] with precedence:
length1 > s > eq > false > nil > 0 > take2 > inf1 > cons > true
and weight map:
eq=1
true=1
false=1
cons=2
0=2
nil=1
s=3
inf_1=1
length_1=1
take_2=0
The variable weight is 1With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
eq → true
eq → false
inf(X) → cons
take(0, X) → nil
take(s, cons) → cons
length(nil) → 0
length(cons) → s
(2) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
eq → eq
Q is empty.
(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 → eq
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(5) NonLoopProof (EQUIVALENT transformation)
By Theorem 8 [NONLOOP] we deduce infiniteness of the QDP.
We apply the theorem with m = 1, b = 0,
σ' = [ ], and μ' = [ ] on the rule
EQ[ ]n[ ] → EQ[ ]n[ ]
This rule is correct for the QDP as the following derivation shows:
EQ[ ]n[ ] → EQ[ ]n[ ]
by OriginalRule from TRS P
(6) NO