(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
f(f(f(a, x), y), z) → f(f(x, z), f(y, z))
f(f(b, x), y) → x
f(c, y) → y
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(f(f(a, x), y), z) → F(f(x, z), f(y, z))
F(f(f(a, x), y), z) → F(x, z)
F(f(f(a, x), y), z) → F(y, z)
The TRS R consists of the following rules:
f(f(f(a, x), y), z) → f(f(x, z), f(y, z))
f(f(b, x), y) → x
f(c, y) → y
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(3) 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(f(f(a, c), c), zr1)[zr1 / f(c, zr1)]n[zr1 / f(f(a, c), c)] → F(f(f(a, c), c), f(c, zr1))[zr1 / f(c, zr1)]n[zr1 / f(f(a, c), c)]
This rule is correct for the QDP as the following derivation shows:
intermediate steps: Equivalent (Domain Renaming) - Equivalent (Domain Renaming) - Equivalent (Simplify mu) - Instantiate mu - Instantiate mu - Equivalent (Remove Unused) - Instantiate mu - Equivalent (Domain Renaming) - Instantiation - Equivalent (Domain Renaming)
F(f(f(a, c), x1), x0)[x0 / f(c, x0)]n[x0 / y0] → F(y0, f(x1, x0))[x0 / f(c, x0)]n[x0 / y0]
by Narrowing at position: [0]
intermediate steps: Instantiate mu - Instantiate Sigma - Instantiation - Instantiation
F(f(f(a, x), y), z)[ ]n[ ] → F(f(x, z), f(y, z))[ ]n[ ]
by OriginalRule from TRS P
intermediate steps: Equivalent (Add Unused) - Equivalent (Add Unused) - Equivalent (Domain Renaming) - Equivalent (Domain Renaming) - Instantiation - Equivalent (Domain Renaming)
f(c, y)[y / f(c, y)]n[ ] → y[ ]n[ ]
by PatternCreation I
f(c, y)[ ]n[ ] → y[ ]n[ ]
by OriginalRule from TRS R
(4) NO