(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
*(X, +(Y, 1)) → +(*(X, +(Y, *(1, 0))), X)
*(X, 1) → X
*(X, 0) → X
*(X, 0) → 0
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(X, +(Y, 1)) → *1(X, +(Y, *(1, 0)))
*1(X, +(Y, 1)) → *1(1, 0)
The TRS R consists of the following rules:
*(X, +(Y, 1)) → +(*(X, +(Y, *(1, 0))), X)
*(X, 1) → X
*(X, 0) → X
*(X, 0) → 0
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(X, +(Y, 1)) → *1(X, +(Y, *(1, 0)))
The TRS R consists of the following rules:
*(X, +(Y, 1)) → +(*(X, +(Y, *(1, 0))), X)
*(X, 1) → X
*(X, 0) → X
*(X, 0) → 0
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(x1, +(x0, 1))[ ]n[ ] → *1(x1, +(x0, 1))[ ]n[ ]
This rule is correct for the QDP as the following derivation shows:
*1(x1, +(x0, 1))[ ]n[ ] → *1(x1, +(x0, 1))[ ]n[ ]
by Narrowing at position: [1,1]
intermediate steps: Instantiation
*1(X, +(Y, 1))[ ]n[ ] → *1(X, +(Y, *(1, 0)))[ ]n[ ]
by OriginalRule from TRS P
intermediate steps: Instantiation - Instantiation
*(X, 0)[ ]n[ ] → X[ ]n[ ]
by OriginalRule from TRS R
(6) NO