(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
+(1, x) → +(+(0, 1), x)
+(0, x) → x
Q is empty.
(1) DependencyPairsProof (EQUIVALENT transformation)
Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
(2) Obligation:
Q DP problem:
The TRS P consists of the following rules:
+1(1, x) → +1(+(0, 1), x)
+1(1, x) → +1(0, 1)
The TRS R consists of the following rules:
+(1, x) → +(+(0, 1), x)
+(0, x) → x
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(3) DependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
(4) Obligation:
Q DP problem:
The TRS P consists of the following rules:
+1(1, x) → +1(+(0, 1), x)
The TRS R consists of the following rules:
+(1, x) → +(+(0, 1), x)
+(0, x) → x
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
+1(1, x0)[ ]n[ ] → +1(1, x0)[ ]n[ ]
This rule is correct for the QDP as the following derivation shows:
+1(1, x0)[ ]n[ ] → +1(1, x0)[ ]n[ ]
by Narrowing at position: [0]
intermediate steps: Instantiation
+1(1, x)[ ]n[ ] → +1(+(0, 1), x)[ ]n[ ]
by OriginalRule from TRS P
intermediate steps: Instantiation - Instantiation
+(0, x)[ ]n[ ] → x[ ]n[ ]
by OriginalRule from TRS R
(6) NO