(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
f(g(x), s(0)) → f(g(x), g(x))
g(s(x)) → s(g(x))
g(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:
F(g(x), s(0)) → F(g(x), g(x))
G(s(x)) → G(x)
The TRS R consists of the following rules:
f(g(x), s(0)) → f(g(x), g(x))
g(s(x)) → s(g(x))
g(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 2 SCCs.
(4) Complex Obligation (AND)
(5) Obligation:
Q DP problem:
The TRS P consists of the following rules:
G(s(x)) → G(x)
The TRS R consists of the following rules:
f(g(x), s(0)) → f(g(x), g(x))
g(s(x)) → s(g(x))
g(0) → 0
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:
- G(s(x)) → G(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(g(x), s(0)) → F(g(x), g(x))
The TRS R consists of the following rules:
f(g(x), s(0)) → f(g(x), g(x))
g(s(x)) → s(g(x))
g(0) → 0
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(g(zl0), s(0))[zl0 / g(zl0)]n[zl0 / s(0)] → F(g(zl0), s(0))[zl0 / g(zl0)]n[zl0 / s(0)]
This rule is correct for the QDP as the following derivation shows:
intermediate steps: Equivalent (Domain Renaming) - Equivalent (Domain Renaming) - Equivalent (Remove Unused)
F(g(zl1), s(0))[zl1 / g(zl1), zr2 / g(zr2), zr3 / g(zr3)]n[zl1 / s(0), x0 / 0, zr2 / s(0), zr3 / 0] → F(g(zr2), s(0))[zl1 / g(zl1), zr2 / g(zr2), zr3 / g(zr3)]n[zl1 / s(0), x0 / 0, zr2 / s(0), zr3 / 0]
by Narrowing at position: [1,0]
intermediate steps: Equivalent (Add Unused) - Instantiate mu - Equivalent (Add Unused) - Equivalent (Domain Renaming) - Instantiation - Equivalent (Domain Renaming) - Equivalent (Remove Unused)
F(g(x0), s(0))[x0 / g(x0), zt1 / g(zt1)]n[x0 / s(y0), zt1 / y0] → F(g(x0), s(g(zt1)))[x0 / g(x0), zt1 / g(zt1)]n[x0 / s(y0), zt1 / y0]
by Narrowing at position: [1]
intermediate steps: Equivalent (Add Unused) - Instantiate mu - Equivalent (Add Unused) - Instantiate Sigma - Instantiation
F(g(x), s(0))[ ]n[ ] → F(g(x), g(x))[ ]n[ ]
by OriginalRule from TRS P
intermediate steps: Equivalent (Add Unused) - Equivalent (Add Unused) - Equivalent (Domain Renaming) - Equivalent (Domain Renaming) - Instantiation - Equivalent (Domain Renaming) - Equivalent (Remove Unused)
g(z)[x / g(x), z / g(z)]n[z / s(x)] → s(g(x))[x / g(x)]n[ ]
by PatternCreation II
g(s(x))[ ]n[ ] → s(g(x))[ ]n[ ]
by OriginalRule from TRS R
intermediate steps: Equivalent (Add Unused) - Equivalent (Add Unused) - Equivalent (Domain Renaming) - Equivalent (Domain Renaming) - Equivalent (Domain Renaming)
g(z)[z / g(z)]n[z / 0] → 0[ ]n[ ]
by PatternCreation II
g(0)[ ]n[ ] → 0[ ]n[ ]
by OriginalRule from TRS R
(10) NO