(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
*(*(x, y), z) → *(x, *(y, z))
*(+(x, y), z) → +(*(x, z), *(y, z))
*(x, +(y, f(z))) → *(g(x, z), +(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:
*1(*(x, y), z) → *1(x, *(y, z))
*1(*(x, y), z) → *1(y, z)
*1(+(x, y), z) → *1(x, z)
*1(+(x, y), z) → *1(y, z)
*1(x, +(y, f(z))) → *1(g(x, z), +(y, y))
The TRS R consists of the following rules:
*(*(x, y), z) → *(x, *(y, z))
*(+(x, y), z) → +(*(x, z), *(y, z))
*(x, +(y, f(z))) → *(g(x, z), +(y, y))
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.
(4) Complex Obligation (AND)
(5) Obligation:
Q DP problem:
The TRS P consists of the following rules:
*1(x, +(y, f(z))) → *1(g(x, z), +(y, y))
The TRS R consists of the following rules:
*(*(x, y), z) → *(x, *(y, z))
*(+(x, y), z) → +(*(x, z), *(y, z))
*(x, +(y, f(z))) → *(g(x, z), +(y, y))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(6) NonLoopProof (EQUIVALENT transformation)
By Theorem 8 [NONLOOP] we deduce infiniteness of the QDP.
We apply the theorem with m = 1, b = 0,
σ' = [ ], and μ' = [x0 / x0', x2 / g(x2, x0)] on the rule
*1(g(x2, x0), +(f(x0'), f(x0')))[ ]n[x1 / f(x0')] → *1(g(x2, x0), +(f(x0'), f(x0')))[ ]n[x1 / f(x0'), x0 / x0', x2 / g(x2, x0)]
This rule is correct for the QDP as the following derivation shows:
intermediate steps: Equivalent (Simplify mu) - Instantiate mu - Equivalent (Simplify mu) - Instantiate mu - Instantiation
*1(x, +(y, f(z)))[ ]n[ ] → *1(g(x, z), +(y, y))[ ]n[ ]
by OriginalRule from TRS P
(7) NO
(8) Obligation:
Q DP problem:
The TRS P consists of the following rules:
*1(*(x, y), z) → *1(y, z)
*1(*(x, y), z) → *1(x, *(y, z))
*1(+(x, y), z) → *1(x, z)
*1(+(x, y), z) → *1(y, z)
The TRS R consists of the following rules:
*(*(x, y), z) → *(x, *(y, z))
*(+(x, y), z) → +(*(x, z), *(y, z))
*(x, +(y, f(z))) → *(g(x, z), +(y, y))
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:
- *1(*(x, y), z) → *1(y, z)
The graph contains the following edges 1 > 1, 2 >= 2
- *1(*(x, y), z) → *1(x, *(y, z))
The graph contains the following edges 1 > 1
- *1(+(x, y), z) → *1(x, z)
The graph contains the following edges 1 > 1, 2 >= 2
- *1(+(x, y), z) → *1(y, z)
The graph contains the following edges 1 > 1, 2 >= 2
(10) YES