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

0 QTRS
↳1 QTRSRRRProof (⇔)
↳2 QTRS
↳3 QTRSRRRProof (⇔)
↳4 QTRS
↳5 QTRSRRRProof (⇔)
↳6 QTRS
  ↳7 RisEmptyProof (⇔)
    ↳8 TRUE
  ↳9 RisEmptyProof (⇔)
    ↳10 TRUE

(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:
Polynomial interpretation [POLO]:

POL(=(x1, x2)) = 1 + 2·x1 + 2·x2   
POL(and(x1, x2)) = 1 + 2·x1 + 2·x2   
POL(false) = 0   
POL(if(x1, x2, x3)) = x1 + x2 + x3   
POL(implies(x1, x2)) = 2 + 2·x1 + x2   
POL(not(x1)) = 1 + x1   
POL(or(x1, x2)) = 2 + x1 + 2·x2   
POL(true) = 1   
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

and(x, y) → if(x, y, false)
or(x, y) → if(x, true, y)
implies(x, y) → if(x, y, true)
if(true, x, y) → x


(2) Obligation:

Q restricted rewrite system:
The TRS R consists of the following rules:

not(x) → if(x, false, true)
=(x, x) → true
=(x, y) → if(x, y, not(y))
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.

(3) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(=(x1, x2)) = 1 + 2·x1 + 2·x2   
POL(false) = 0   
POL(if(x1, x2, x3)) = x1 + x2 + x3   
POL(not(x1)) = x1   
POL(true) = 0   
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

=(x, x) → true
=(x, y) → if(x, y, not(y))
=(x, y) → if(x, y, if(y, false, true))


(4) Obligation:

Q restricted rewrite system:
The TRS R consists of the following rules:

not(x) → if(x, false, true)
if(false, x, y) → y
if(x, x, if(x, false, true)) → true

Q is empty.

(5) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(false) = 1   
POL(if(x1, x2, x3)) = x1 + x2 + x3   
POL(not(x1)) = 2 + x1   
POL(true) = 0   
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)
if(false, x, y) → y
if(x, x, if(x, false, true)) → true


(6) Obligation:

Q restricted rewrite system:
R is empty.
Q is empty.

(7) RisEmptyProof (EQUIVALENT transformation)

The TRS R is empty. Hence, termination is trivially proven.

(8) TRUE

(9) RisEmptyProof (EQUIVALENT transformation)

The TRS R is empty. Hence, termination is trivially proven.

(10) TRUE