Termination w.r.t. Q proof of TRS/SK902.32.trs

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:

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.

↳ QTRS

  ↳ DirectTerminationProof

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.

We use [23] with the following order to prove termination.

Lexicographic path order with status [19].
Precedence:

and₂ > if₃
and₂ > false
or₂ > if₃
or₂ > true
implies₂ > if₃
implies₂ > true
=₂ > not₁ > if₃
=₂ > not₁ > false
=₂ > not₁ > true

Status:

trivial