(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
f(tt, x) → f(eq(toOne(x), s(0)), s(x))
eq(s(x), s(y)) → eq(x, y)
eq(0, 0) → tt
toOne(s(s(x))) → toOne(s(x))
toOne(s(0)) → s(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(tt, x) → F(eq(toOne(x), s(0)), s(x))
F(tt, x) → EQ(toOne(x), s(0))
F(tt, x) → TOONE(x)
EQ(s(x), s(y)) → EQ(x, y)
TOONE(s(s(x))) → TOONE(s(x))
The TRS R consists of the following rules:
f(tt, x) → f(eq(toOne(x), s(0)), s(x))
eq(s(x), s(y)) → eq(x, y)
eq(0, 0) → tt
toOne(s(s(x))) → toOne(s(x))
toOne(s(0)) → s(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 3 SCCs with 2 less nodes.
(4) Complex Obligation (AND)
(5) Obligation:
Q DP problem:
The TRS P consists of the following rules:
TOONE(s(s(x))) → TOONE(s(x))
The TRS R consists of the following rules:
f(tt, x) → f(eq(toOne(x), s(0)), s(x))
eq(s(x), s(y)) → eq(x, y)
eq(0, 0) → tt
toOne(s(s(x))) → toOne(s(x))
toOne(s(0)) → s(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:
- TOONE(s(s(x))) → TOONE(s(x))
The graph contains the following edges 1 > 1
(7) YES
(8) Obligation:
Q DP problem:
The TRS P consists of the following rules:
EQ(s(x), s(y)) → EQ(x, y)
The TRS R consists of the following rules:
f(tt, x) → f(eq(toOne(x), s(0)), s(x))
eq(s(x), s(y)) → eq(x, y)
eq(0, 0) → tt
toOne(s(s(x))) → toOne(s(x))
toOne(s(0)) → s(0)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(9) 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:
- EQ(s(x), s(y)) → EQ(x, y)
The graph contains the following edges 1 > 1, 2 > 2
(10) YES
(11) Obligation:
Q DP problem:
The TRS P consists of the following rules:
F(tt, x) → F(eq(toOne(x), s(0)), s(x))
The TRS R consists of the following rules:
f(tt, x) → f(eq(toOne(x), s(0)), s(x))
eq(s(x), s(y)) → eq(x, y)
eq(0, 0) → tt
toOne(s(s(x))) → toOne(s(x))
toOne(s(0)) → s(0)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(12) NonLoopProof (EQUIVALENT transformation)
By Theorem 8 [NONLOOP] we deduce infiniteness of the QDP.
We apply the theorem with m = 1, b = 1,
σ' = [ ], and μ' = [ ] on the rule
F(tt, s(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)
F(tt, s(s(zl1)))[zl1 / s(zl1)]n[zl1 / 0] → F(tt, s(s(s(zr1))))[zr1 / s(zr1)]n[zr1 / 0]
by Rewrite t
intermediate steps: Equivalent (Remove Unused)
F(tt, s(s(zl1)))[zl1 / s(zl1), zr1 / s(zr1)]n[zl1 / 0, zr1 / 0] → F(eq(s(0), s(0)), s(s(s(zr1))))[zl1 / s(zl1), zr1 / s(zr1)]n[zl1 / 0, zr1 / 0]
by Narrowing at position: [0,0]
intermediate steps: Equivalent (Add Unused) - Equivalent (Add Unused) - Instantiation - Equivalent (Domain Renaming) - Instantiation - Equivalent (Domain Renaming)
F(tt, s(s(zs1)))[zs1 / s(zs1)]n[zs1 / y0] → F(eq(toOne(s(y0)), s(0)), s(s(s(zs1))))[zs1 / s(zs1)]n[zs1 / y0]
by Narrowing at position: [0,0]
intermediate steps: Instantiate mu - Instantiate Sigma - Instantiation - Instantiation
F(tt, x)[ ]n[ ] → F(eq(toOne(x), s(0)), s(x))[ ]n[ ]
by OriginalRule from TRS P
intermediate steps: Equivalent (Add Unused) - Equivalent (Add Unused) - Equivalent (Domain Renaming) - Instantiation - Equivalent (Domain Renaming)
toOne(s(s(x)))[x / s(x)]n[ ] → toOne(s(x))[ ]n[ ]
by PatternCreation I
toOne(s(s(x)))[ ]n[ ] → toOne(s(x))[ ]n[ ]
by OriginalRule from TRS R
intermediate steps: Equivalent (Add Unused) - Equivalent (Add Unused)
toOne(s(0))[ ]n[ ] → s(0)[ ]n[ ]
by OriginalRule from TRS R
(13) NO