(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) Overlay + Local Confluence (EQUIVALENT transformation)
The TRS is overlay and locally confluent. By [NOC] we can switch to innermost.
(2) 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))
The set Q consists of the following terms:
or(true, x0)
or(x0, true)
or(false, false)
mem(x0, nil)
mem(x0, set(x1))
mem(x0, union(x1, x2))
(3) DependencyPairsProof (EQUIVALENT transformation)
Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
(4) 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))
The set Q consists of the following terms:
or(true, x0)
or(x0, true)
or(false, false)
mem(x0, nil)
mem(x0, set(x1))
mem(x0, union(x1, x2))
We have to consider all minimal (P,Q,R)-chains.
(5) DependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
(6) 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))
The set Q consists of the following terms:
or(true, x0)
or(x0, true)
or(false, false)
mem(x0, nil)
mem(x0, set(x1))
mem(x0, union(x1, x2))
We have to consider all minimal (P,Q,R)-chains.
(7) UsableRulesProof (EQUIVALENT transformation)
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
(8) 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)
R is empty.
The set Q consists of the following terms:
or(true, x0)
or(x0, true)
or(false, false)
mem(x0, nil)
mem(x0, set(x1))
mem(x0, union(x1, x2))
We have to consider all minimal (P,Q,R)-chains.
(9) QReductionProof (EQUIVALENT transformation)
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
or(true, x0)
or(x0, true)
or(false, false)
mem(x0, nil)
mem(x0, set(x1))
mem(x0, union(x1, x2))
(10) 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)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(11) 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:
- MEM(x, union(y, z)) → MEM(x, z)
The graph contains the following edges 1 >= 1, 2 > 2
- MEM(x, union(y, z)) → MEM(x, y)
The graph contains the following edges 1 >= 1, 2 > 2
(12) TRUE