(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) 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:
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
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:
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
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:
- 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
(7) YES
(8) 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 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 = 1,
σ' = [ ], and μ' = [ ] on the rule
G(tt, s(zr2), s(zr0))[zr2 / s(zr2), zr0 / s(zr0)]n[zr2 / 0, zr0 / 0] → G(tt, s(s(zr2)), s(s(zr0)))[zr0 / s(zr0), zr2 / s(zr2)]n[zr0 / 0, zr2 / 0]
This rule is correct for the QDP as the following derivation shows:
intermediate steps: Equivalent (Domain Renaming) - Equivalent (Domain Renaming) - Equivalent (Remove Unused)
G(tt, s(zl1), s(zl3))[zl1 / s(zl1), zl3 / s(zl3), zr1 / s(zr1), zr3 / s(zr3)]n[zl1 / 0, zl3 / 0, zr1 / 0, zr3 / 0] → G(tt, s(s(zr3)), s(s(zr1)))[zl1 / s(zl1), zl3 / s(zl3), zr1 / s(zr1), zr3 / s(zr3)]n[zl1 / 0, zl3 / 0, zr1 / 0, zr3 / 0]
by Narrowing at position: [0]
intermediate steps: Equivalent (Add Unused) - Equivalent (Add Unused) - Instantiation - Instantiation - Equivalent (Domain Renaming) - Instantiation - Equivalent (Domain Renaming) - Equivalent (Remove Unused)
G(tt, s(zl1), s(zs1))[zl1 / s(zl1), zs1 / s(zs1), zr1 / s(zr1)]n[zl1 / x1, zs1 / y0, zr1 / x1] → G(f(x1, y0), s(s(zr1)), s(s(zs1)))[zl1 / s(zl1), zs1 / s(zs1), zr1 / s(zr1)]n[zl1 / x1, zs1 / y0, zr1 / x1]
by Narrowing at position: [0]
intermediate steps: Equivalent (Add Unused) - Instantiate mu - Equivalent (Add Unused) - Instantiate Sigma - Instantiation - Equivalent (Domain Renaming) - Instantiation - Equivalent (Domain Renaming)
G(tt, s(zs1), x0)[zs1 / s(zs1)]n[zs1 / y1] → G(f(y1, x0), s(s(zs1)), s(x0))[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 OriginalRule from TRS P
intermediate steps: Equivalent (Add Unused) - Equivalent (Add Unused) - Instantiation - Equivalent (Domain Renaming) - Instantiation - Equivalent (Domain Renaming)
f(s(x), y)[x / s(x)]n[ ] → f(x, y)[ ]n[ ]
by PatternCreation I
f(s(x), y)[ ]n[ ] → f(x, y)[ ]n[ ]
by OriginalRule from TRS R
intermediate steps: Equivalent (Add Unused) - Equivalent (Add Unused) - Instantiation - Equivalent (Domain Renaming) - Instantiation - Equivalent (Domain Renaming)
f(x, s(y))[y / s(y)]n[ ] → f(x, y)[ ]n[ ]
by PatternCreation I
f(x, s(y))[ ]n[ ] → f(x, y)[ ]n[ ]
by OriginalRule from TRS R
intermediate steps: Equivalent (Add Unused) - Equivalent (Add Unused)
f(0, 0)[ ]n[ ] → tt[ ]n[ ]
by OriginalRule from TRS R
(10) NO