(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
not(x) → if(x, false, true)
and(x, y) → if(x, y, false)
or(x, y) → if(x, true, y)
implies(x, y) → if(x, y, true)
=(x, x) → true
=(x, y) → if(x, y, not(y))
if(true, x, y) → x
if(false, x, y) → y
if(x, x, if(x, false, true)) → true
=(x, y) → if(x, y, if(y, false, true))
Q is empty.
(1) QTRSRRRProof (EQUIVALENT transformation)
Used ordering:
Recursive Path Order [RPO].
Precedence:
and2 > if3
and2 > false
or2 > if3
or2 > true
implies2 > if3
implies2 > true
=2 > not1 > if3
=2 > not1 > false
=2 > not1 > true
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
not(x) → if(x, false, true)
and(x, y) → if(x, y, false)
or(x, y) → if(x, true, y)
implies(x, y) → if(x, y, true)
=(x, x) → true
=(x, y) → if(x, y, not(y))
if(true, x, y) → x
if(false, x, y) → y
if(x, x, if(x, false, true)) → true
=(x, y) → if(x, y, if(y, false, true))
(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) TRUE