(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
or(true, y) → true
or(x, true) → true
or(false, false) → false
mem(x, nil) → false
mem(x, set(y)) → =(x, y)
mem(x, union(y, z)) → or(mem(x, y), mem(x, z))
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:
MEM(x, union(y, z)) → OR(mem(x, y), mem(x, z))
MEM(x, union(y, z)) → MEM(x, y)
MEM(x, union(y, z)) → MEM(x, z)
The TRS R consists of the following rules:
or(true, y) → true
or(x, true) → true
or(false, false) → false
mem(x, nil) → false
mem(x, set(y)) → =(x, y)
mem(x, union(y, z)) → or(mem(x, y), mem(x, z))
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:
MEM(x, union(y, z)) → MEM(x, z)
MEM(x, union(y, z)) → MEM(x, y)
The TRS R consists of the following rules:
or(true, y) → true
or(x, true) → true
or(false, false) → false
mem(x, nil) → false
mem(x, set(y)) → =(x, y)
mem(x, union(y, z)) → or(mem(x, y), mem(x, z))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(5) QDPSizeChangeProof (EQUIVALENT transformation)
We used the following order and afs together with the size-change analysis [AAECC05] to show that there are no infinite chains for this DP problem.
Order:Homeomorphic Embedding Order
AFS:
union(x1, x2) = union(x1, x2)
From the DPs we obtained the following set of size-change graphs:
- MEM(x, union(y, z)) → MEM(x, z) (allowed arguments on rhs = {1, 2})
The graph contains the following edges 1 >= 1, 2 > 2
- MEM(x, union(y, z)) → MEM(x, y) (allowed arguments on rhs = {1, 2})
The graph contains the following edges 1 >= 1, 2 > 2
We oriented the following set of usable rules [AAECC05,FROCOS05].
none
(6) TRUE