(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) QTRSRRRProof (EQUIVALENT transformation)
Used ordering:
Combined order from the following AFS and order.
or(
x1,
x2) =
or(
x1,
x2)
true =
true
false =
false
mem(
x1,
x2) =
mem(
x1,
x2)
nil =
nil
set(
x1) =
x1
=(
x1,
x2) =
=(
x1,
x2)
union(
x1,
x2) =
union(
x1,
x2)
Recursive path order with status [RPO].
Quasi-Precedence:
mem2 > or2
mem2 > [false, nil]
mem2 > =2
Status:
or2: multiset
true: multiset
false: multiset
mem2: multiset
nil: multiset
=2: multiset
union2: multiset
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
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))
(2) Obligation:
Q restricted rewrite system:
R is empty.
Q is empty.
(3) RisEmptyProof (EQUIVALENT transformation)
The TRS R is empty. Hence, termination is trivially proven.
(4) YES