(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
f(tt, x) → f(isNat(x), s(s(x)))
isNat(s(x)) → isNat(x)
isNat(0) → tt
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(tt, x) → F(isNat(x), s(s(x)))
F(tt, x) → ISNAT(x)
ISNAT(s(x)) → ISNAT(x)
The TRS R consists of the following rules:
f(tt, x) → f(isNat(x), s(s(x)))
isNat(s(x)) → isNat(x)
isNat(0) → tt
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:
ISNAT(s(x)) → ISNAT(x)
The TRS R consists of the following rules:
f(tt, x) → f(isNat(x), s(s(x)))
isNat(s(x)) → isNat(x)
isNat(0) → tt
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:
- ISNAT(s(x)) → ISNAT(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(tt, x) → F(isNat(x), s(s(x)))
The TRS R consists of the following rules:
f(tt, x) → f(isNat(x), s(s(x)))
isNat(s(x)) → isNat(x)
isNat(0) → tt
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 = 2,
σ' = [ ], and μ' = [ ] on the rule
F(tt, s(zr0))[zr0 / s(zr0)]n[zr0 / 0] → F(tt, s(s(s(zr0))))[zr0 / s(zr0)]n[zr0 / 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(tt, s(zl1))[zl1 / s(zl1), zr1 / s(zr1)]n[zl1 / 0, zr1 / 0] → F(tt, s(s(s(zr1))))[zl1 / s(zl1), zr1 / s(zr1)]n[zl1 / 0, zr1 / 0]
by Narrowing at position: [0]
intermediate steps: Equivalent (Add Unused) - Equivalent (Add Unused) - Instantiation - Equivalent (Domain Renaming) - Instantiation - Equivalent (Domain Renaming)
F(tt, s(zs1))[zs1 / s(zs1)]n[zs1 / y0] → F(isNat(y0), s(s(s(zs1))))[zs1 / s(zs1)]n[zs1 / y0]
by Narrowing at position: [0]
intermediate steps: Instantiate mu - Instantiate Sigma - Instantiation - Instantiation
F(tt, x)[ ]n[ ] → F(isNat(x), s(s(x)))[ ]n[ ]
by OriginalRule from TRS P
intermediate steps: Equivalent (Add Unused) - Equivalent (Add Unused) - Equivalent (Domain Renaming) - Instantiation - Equivalent (Domain Renaming)
isNat(s(x))[x / s(x)]n[ ] → isNat(x)[ ]n[ ]
by PatternCreation I
isNat(s(x))[ ]n[ ] → isNat(x)[ ]n[ ]
by OriginalRule from TRS R
intermediate steps: Equivalent (Add Unused) - Equivalent (Add Unused)
isNat(0)[ ]n[ ] → tt[ ]n[ ]
by OriginalRule from TRS R
(10) NO