(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))
:(z, +(x, f(y))) → :(g(z, y), +(x, a))
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(z, +(x, f(y))) → :1(g(z, y), +(x, a))
The TRS R consists of the following rules:
:(:(x, y), z) → :(x, :(y, z))
:(+(x, y), z) → +(:(x, z), :(y, z))
:(z, +(x, f(y))) → :(g(z, y), +(x, a))
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 1 SCC with 1 less node.
(4) 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))
:(z, +(x, f(y))) → :(g(z, y), +(x, a))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(5) 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
(6) YES