Termination w.r.t. Q of the following Term Rewriting System could be proven:

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.


↳ QTRS
  ↳ Overlay + Local Confluence

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.

The TRS is overlay and locally confluent. By [15] we can switch to innermost.

↳ QTRS
  ↳ Overlay + Local Confluence
    ↳ QTRS
      ↳ DependencyPairsProof

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))


Using Dependency Pairs [1,13] we result in the following initial DP problem:
Q DP problem:
The TRS P consists of the following rules:

MEM(x, union(y, z)) → MEM(x, z)
MEM(x, union(y, z)) → OR(mem(x, y), 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.

↳ QTRS
  ↳ Overlay + Local Confluence
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ EdgeDeletionProof

Q DP problem:
The TRS P consists of the following rules:

MEM(x, union(y, z)) → MEM(x, z)
MEM(x, union(y, z)) → OR(mem(x, y), 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.
We deleted some edges using various graph approximations

↳ QTRS
  ↳ Overlay + Local Confluence
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ EdgeDeletionProof
            ↳ QDP
              ↳ DependencyGraphProof

Q DP problem:
The TRS P consists of the following rules:

MEM(x, union(y, z)) → MEM(x, z)
MEM(x, union(y, z)) → OR(mem(x, y), 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.
The approximation of the Dependency Graph [13,14,18] contains 1 SCC with 1 less node.

↳ QTRS
  ↳ Overlay + Local Confluence
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ EdgeDeletionProof
            ↳ QDP
              ↳ DependencyGraphProof
                ↳ QDP
                  ↳ QDPOrderProof

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.
We use the reduction pair processor [13].


The following pairs can be oriented strictly and are deleted.


MEM(x, union(y, z)) → MEM(x, z)
MEM(x, union(y, z)) → MEM(x, y)
The remaining pairs can at least be oriented weakly.
none
Used ordering: Combined order from the following AFS and order.
MEM(x1, x2)  =  MEM(x2)
union(x1, x2)  =  union(x1, x2)

Lexicographic path order with status [19].
Quasi-Precedence:
[MEM1, union2]

Status:
union2: [2,1]
MEM1: [1]


The following usable rules [14] were oriented: none



↳ QTRS
  ↳ Overlay + Local Confluence
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ EdgeDeletionProof
            ↳ QDP
              ↳ DependencyGraphProof
                ↳ QDP
                  ↳ QDPOrderProof
                    ↳ QDP
                      ↳ PisEmptyProof

Q DP problem:
P is empty.
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.
The TRS P is empty. Hence, there is no (P,Q,R) chain.