(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
g(tt, x, y) → g(f(x, y), s(x), s(y))
f(s(x), y) → f(x, y)
f(x, s(y)) → f(x, y)
f(0, 0) → tt
Q is empty.
(1) AAECC Innermost (EQUIVALENT transformation)
We have applied [NOC,AAECCNOC] to switch to innermost. The TRS R 1 is
f(s(x), y) → f(x, y)
f(x, s(y)) → f(x, y)
f(0, 0) → tt
The TRS R 2 is
g(tt, x, y) → g(f(x, y), s(x), s(y))
The signature Sigma is {
g}
(2) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
g(tt, x, y) → g(f(x, y), s(x), s(y))
f(s(x), y) → f(x, y)
f(x, s(y)) → f(x, y)
f(0, 0) → tt
The set Q consists of the following terms:
g(tt, x0, x1)
f(s(x0), x1)
f(x0, s(x1))
f(0, 0)
(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, y) → G(f(x, y), s(x), s(y))
G(tt, x, y) → F(x, y)
F(s(x), y) → F(x, y)
F(x, s(y)) → F(x, y)
The TRS R consists of the following rules:
g(tt, x, y) → g(f(x, y), s(x), s(y))
f(s(x), y) → f(x, y)
f(x, s(y)) → f(x, y)
f(0, 0) → tt
The set Q consists of the following terms:
g(tt, x0, x1)
f(s(x0), x1)
f(x0, s(x1))
f(0, 0)
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 1 less node.
(6) Complex Obligation (AND)
(7) Obligation:
Q DP problem:
The TRS P consists of the following rules:
F(x, s(y)) → F(x, y)
F(s(x), y) → F(x, y)
The TRS R consists of the following rules:
g(tt, x, y) → g(f(x, y), s(x), s(y))
f(s(x), y) → f(x, y)
f(x, s(y)) → f(x, y)
f(0, 0) → tt
The set Q consists of the following terms:
g(tt, x0, x1)
f(s(x0), x1)
f(x0, s(x1))
f(0, 0)
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(x, s(y)) → F(x, y)
F(s(x), y) → F(x, y)
R is empty.
The set Q consists of the following terms:
g(tt, x0, x1)
f(s(x0), x1)
f(x0, s(x1))
f(0, 0)
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, x1)
f(s(x0), x1)
f(x0, s(x1))
f(0, 0)
(11) Obligation:
Q DP problem:
The TRS P consists of the following rules:
F(x, s(y)) → F(x, y)
F(s(x), y) → F(x, 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(x, s(y)) → F(x, y)
The graph contains the following edges 1 >= 1, 2 > 2
- F(s(x), y) → F(x, y)
The graph contains the following edges 1 > 1, 2 >= 2
(13) YES
(14) Obligation:
Q DP problem:
The TRS P consists of the following rules:
G(tt, x, y) → G(f(x, y), s(x), s(y))
The TRS R consists of the following rules:
g(tt, x, y) → g(f(x, y), s(x), s(y))
f(s(x), y) → f(x, y)
f(x, s(y)) → f(x, y)
f(0, 0) → tt
The set Q consists of the following terms:
g(tt, x0, x1)
f(s(x0), x1)
f(x0, s(x1))
f(0, 0)
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, y) → G(f(x, y), s(x), s(y))
The TRS R consists of the following rules:
g(tt, x, y) → g(f(x, y), s(x), s(y))
f(s(x), y) → f(x, y)
f(x, s(y)) → f(x, y)
f(0, 0) → tt
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), s(zr2))[zr0 / s(zr0), zr2 / s(zr2)]n[zr0 / 0, zr2 / 0] → G(tt, s(s(zr0)), s(s(zr2)))[zr0 / s(zr0), zr2 / s(zr2)]n[zr0 / 0, zr2 / 0]
This rule is correct for the QDP as the following derivation shows:
G(tt, s(zr0), s(zr2))[zr0 / s(zr0), zr2 / s(zr2)]n[zr0 / 0, zr2 / 0] → G(tt, s(s(zr0)), s(s(zr2)))[zr0 / s(zr0), zr2 / s(zr2)]n[zr0 / 0, zr2 / 0]
by Equivalence by Domain Renaming of the lhs with [zl0 / zr0, zl2 / zr2]
intermediate steps: Equiv DR (rhs) - Equiv DR (lhs) - Equiv IPS (rhs) - Equiv IPS (lhs)
G(tt, s(zl1), s(zl3))[zr1 / s(zr1), zr3 / s(zr3), zl1 / s(zl1), zl3 / s(zl3)]n[zr1 / 0, zr3 / 0, zl1 / 0, zl3 / 0] → G(tt, s(s(zr1)), s(s(zr3)))[zr1 / s(zr1), zr3 / s(zr3), zl1 / s(zl1), zl3 / s(zl3)]n[zr1 / 0, zr3 / 0, zl1 / 0, zl3 / 0]
by Narrowing at position: [0]
intermediate steps: Equiv IPS (rhs) - Equiv IPS (lhs) - Equiv IPS (rhs) - Equiv IPS (lhs) - Instantiation - Instantiation - Equiv DR (rhs) - Equiv DR (lhs) - Instantiation - Equiv DR (rhs) - Equiv DR (lhs) - Equiv IPS (rhs) - Equiv IPS (lhs)
G(tt, s(zl1), s(zs1))[zr1 / s(zr1), zl1 / s(zl1), zs1 / s(zs1)]n[zr1 / x0, zl1 / x0, zs1 / y0] → G(f(x0, y0), s(s(zr1)), s(s(zs1)))[zr1 / s(zr1), zl1 / s(zl1), zs1 / s(zs1)]n[zr1 / x0, zl1 / x0, zs1 / y0]
by Narrowing at position: [0]
intermediate steps: Equiv IPS (rhs) - Equiv IPS (lhs) - Instantiate mu - Equiv IPS (rhs) - Equiv IPS (lhs) - Instantiate Sigma - Instantiation - Equiv DR (rhs) - Equiv DR (lhs) - Instantiation - Equiv DR (rhs) - Equiv DR (lhs)
G(tt, s(zs1), x1)[zs1 / s(zs1)]n[zs1 / y1] → G(f(y1, x1), s(s(zs1)), s(x1))[zs1 / s(zs1)]n[zs1 / y1]
by Narrowing at position: [0]
intermediate steps: Instantiate mu - Instantiate Sigma - Instantiation - Instantiation
G(tt, x, y)[ ]n[ ] → G(f(x, y), s(x), s(y))[ ]n[ ]
by Rule from TRS P
intermediate steps: Equiv IPS (rhs) - Equiv IPS (rhs) - Instantiation - Instantiation - Equiv DR (lhs) - Instantiation - Equiv DR (lhs)
f(s(x), y)[x / s(x)]n[ ] → f(x, y)[ ]n[ ]
by PatternCreation I with delta: [ ], theta: [ ], sigma: [x / s(x)]
f(s(x), y)[ ]n[ ] → f(x, y)[ ]n[ ]
by Rule from TRS R
intermediate steps: Equiv IPS (rhs) - Equiv IPS (lhs) - Equiv IPS (rhs) - Equiv IPS (lhs) - Instantiation - Instantiation - Equiv DR (lhs) - Instantiation - Equiv DR (lhs)
f(x, s(y))[y / s(y)]n[ ] → f(x, y)[ ]n[ ]
by PatternCreation I with delta: [ ], theta: [ ], sigma: [y / s(y)]
f(x, s(y))[ ]n[ ] → f(x, y)[ ]n[ ]
by Rule from TRS R
intermediate steps: Equiv IPS (rhs) - Equiv IPS (lhs) - Equiv IPS (rhs) - Equiv IPS (lhs)
f(0, 0)[ ]n[ ] → tt[ ]n[ ]
by Rule from TRS R
(18) NO