Termination w.r.t. Q of the given QTRS could be proven:

(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 with status [RPO].
Quasi-Precedence:

[not₁, =₂] > if₃ > [false, true]
and₂ > if₃ > [false, true]
or₂ > if₃ > [false, true]
implies₂ > if₃ > [false, true]

Status:

if₃: [3,2,1]
=₂: [2,1]
not₁: [1]
or₂: [2,1]
true: multiset
false: multiset
and₂: multiset
implies₂: [1,2]

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