(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
g(tt, x) → g(h(f(x, x)), s(x))
h(f(0, y)) → tt
f(s(x), y) → f(x, s(y))
Q is empty.
(1) AAECC Innermost (EQUIVALENT transformation)
We have applied [NOC,AAECCNOC] to switch to innermost. The TRS R 1 is
h(f(0, y)) → tt
f(s(x), y) → f(x, s(y))
The TRS R 2 is
g(tt, x) → g(h(f(x, x)), s(x))
The signature Sigma is {
g}
(2) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
g(tt, x) → g(h(f(x, x)), s(x))
h(f(0, y)) → tt
f(s(x), y) → f(x, s(y))
The set Q consists of the following terms:
g(tt, x0)
h(f(0, x0))
f(s(x0), x1)
(3) DependencyPairsProof (EQUIVALENT transformation)
Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
(4) Obligation:
Q DP problem:
The TRS P consists of the following rules:
G(tt, x) → G(h(f(x, x)), s(x))
G(tt, x) → H(f(x, x))
G(tt, x) → F(x, x)
F(s(x), y) → F(x, s(y))
The TRS R consists of the following rules:
g(tt, x) → g(h(f(x, x)), s(x))
h(f(0, y)) → tt
f(s(x), y) → f(x, s(y))
The set Q consists of the following terms:
g(tt, x0)
h(f(0, x0))
f(s(x0), x1)
We have to consider all minimal (P,Q,R)-chains.
(5) DependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 2 less nodes.
(6) Complex Obligation (AND)
(7) Obligation:
Q DP problem:
The TRS P consists of the following rules:
F(s(x), y) → F(x, s(y))
The TRS R consists of the following rules:
g(tt, x) → g(h(f(x, x)), s(x))
h(f(0, y)) → tt
f(s(x), y) → f(x, s(y))
The set Q consists of the following terms:
g(tt, x0)
h(f(0, x0))
f(s(x0), x1)
We have to consider all minimal (P,Q,R)-chains.
(8) UsableRulesProof (EQUIVALENT transformation)
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
(9) Obligation:
Q DP problem:
The TRS P consists of the following rules:
F(s(x), y) → F(x, s(y))
R is empty.
The set Q consists of the following terms:
g(tt, x0)
h(f(0, x0))
f(s(x0), x1)
We have to consider all minimal (P,Q,R)-chains.
(10) QReductionProof (EQUIVALENT transformation)
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
g(tt, x0)
h(f(0, x0))
f(s(x0), x1)
(11) Obligation:
Q DP problem:
The TRS P consists of the following rules:
F(s(x), y) → F(x, s(y))
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(12) 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:
- F(s(x), y) → F(x, s(y))
The graph contains the following edges 1 > 1
(13) YES
(14) Obligation:
Q DP problem:
The TRS P consists of the following rules:
G(tt, x) → G(h(f(x, x)), s(x))
The TRS R consists of the following rules:
g(tt, x) → g(h(f(x, x)), s(x))
h(f(0, y)) → tt
f(s(x), y) → f(x, s(y))
The set Q consists of the following terms:
g(tt, x0)
h(f(0, x0))
f(s(x0), x1)
We have to consider all minimal (P,Q,R)-chains.
(15) MNOCProof (EQUIVALENT transformation)
We use the modular non-overlap check [FROCOS05] to decrease Q to the empty set.
(16) Obligation:
Q DP problem:
The TRS P consists of the following rules:
G(tt, x) → G(h(f(x, x)), s(x))
The TRS R consists of the following rules:
g(tt, x) → g(h(f(x, x)), s(x))
h(f(0, y)) → tt
f(s(x), y) → f(x, s(y))
Q is empty.
We have to consider all (P,Q,R)-chains.
(17) NonLoopProof (COMPLETE 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
G(tt, s(zr0))[zr0 / s(zr0)]n[zr0 / 0] → G(tt, s(s(zr0)))[zr0 / s(zr0)]n[zr0 / 0]
This rule is correct for the QDP as the following derivation shows:
G(tt, s(zr0))[zr0 / s(zr0)]n[zr0 / 0] → G(tt, s(s(zr0)))[zr0 / s(zr0)]n[zr0 / 0]
by Equivalence by Domain Renaming of the lhs with [zl0 / zr0]
intermediate steps: Equiv DR (rhs) - Equiv DR (lhs) - Equiv IPS (rhs) - Equiv IPS (lhs)
G(tt, s(zl1))[zr2 / s(s(zr2)), zr3 / s(zr3), zl1 / s(zl1)]n[zr2 / 0, zr3 / 0, zl1 / 0] → G(tt, s(s(zr3)))[zr2 / s(s(zr2)), zr3 / s(zr3), zl1 / s(zl1)]n[zr2 / 0, zr3 / 0, zl1 / 0]
by Narrowing at position: [0]
intermediate steps: Equiv IPS (rhs) - Equiv IPS (lhs) - Equiv IPS (rhs) - Equiv IPS (lhs) - Instantiation - Equiv DR (rhs) - Equiv DR (lhs) - Instantiation - Equiv DR (rhs) - Equiv DR (lhs) - Equiv IPS (lhs)
G(tt, s(zs1))[zt1 / s(s(zt1)), zs1 / s(zs1)]n[zt1 / y1, zs1 / y1] → G(h(f(y1, s(s(zt1)))), s(s(zs1)))[zt1 / s(s(zt1)), zs1 / s(zs1)]n[zt1 / y1, zs1 / y1]
by Narrowing at position: [0,0]
intermediate steps: Equiv IPS (rhs) - Equiv IPS (lhs) - Instantiate mu - Equiv IPS (rhs) - Equiv IPS (lhs) - Instantiate Sigma - Instantiation - Instantiation
G(tt, x)[ ]n[ ] → G(h(f(x, x)), s(x))[ ]n[ ]
by Rule from TRS P
intermediate steps: Equiv IPS (rhs) - Equiv IPS (lhs) - Equiv IPS (rhs) - Equiv IPS (lhs) - Equiv DR (lhs) - Equiv IPS (lhs) - Equiv Sµ (lhs) - Instantiate mu - Instantiate Sigma - Equiv Sµ (lhs) - Equiv IPS (rhs) - Instantiate mu - Equiv Sµ (rhs) - Instantiation - Instantiation - Equiv DR (rhs) - Equiv DR (lhs) - Instantiation - Equiv DR (rhs) - Equiv DR (lhs)
f(s(x), y)[x / s(x)]n[ ] → f(x, s(y))[y / s(y)]n[ ]
by PatternCreation I with delta: [ ], theta: [y / s(y)], sigma: [x / s(x)]
f(s(x), y)[ ]n[ ] → f(x, s(y))[ ]n[ ]
by Rule from TRS R
intermediate steps: Equiv IPS (rhs) - Equiv IPS (lhs) - Instantiate mu - Equiv IPS (rhs) - Equiv IPS (lhs) - Instantiate Sigma - Instantiation - Instantiation
h(f(0, y))[ ]n[ ] → tt[ ]n[ ]
by Rule from TRS R
(18) NO