(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
f(s(x), x) → f(s(x), round(x))
round(0) → 0
round(0) → s(0)
round(s(0)) → s(0)
round(s(s(x))) → s(s(round(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:
F(s(x), x) → F(s(x), round(x))
F(s(x), x) → ROUND(x)
ROUND(s(s(x))) → ROUND(x)
The TRS R consists of the following rules:
f(s(x), x) → f(s(x), round(x))
round(0) → 0
round(0) → s(0)
round(s(0)) → s(0)
round(s(s(x))) → s(s(round(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 2 SCCs with 1 less node.
(4) Complex Obligation (AND)
(5) Obligation:
Q DP problem:
The TRS P consists of the following rules:
ROUND(s(s(x))) → ROUND(x)
The TRS R consists of the following rules:
f(s(x), x) → f(s(x), round(x))
round(0) → 0
round(0) → s(0)
round(s(0)) → s(0)
round(s(s(x))) → s(s(round(x)))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(6) QDPSizeChangeProof (EQUIVALENT transformation)
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem.
From the DPs we obtained the following set of size-change graphs:
- ROUND(s(s(x))) → ROUND(x)
The graph contains the following edges 1 > 1
(7) YES
(8) Obligation:
Q DP problem:
The TRS P consists of the following rules:
F(s(x), x) → F(s(x), round(x))
The TRS R consists of the following rules:
f(s(x), x) → f(s(x), round(x))
round(0) → 0
round(0) → s(0)
round(s(0)) → s(0)
round(s(s(x))) → s(s(round(x)))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(9) 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
F(s(0), 0)[ ]n[ ] → F(s(0), 0)[ ]n[ ]
This rule is correct for the QDP as the following derivation shows:
F(s(0), 0)[ ]n[ ] → F(s(0), 0)[ ]n[ ]
by Narrowing at position: [1]
intermediate steps: Instantiation - Instantiation
F(s(x), x)[ ]n[ ] → F(s(x), round(x))[ ]n[ ]
by OriginalRule from TRS P
round(0)[ ]n[ ] → 0[ ]n[ ]
by OriginalRule from TRS R
(10) NO