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

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

(0) Obligation:

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

and(x, false) → false
and(x, not(false)) → x
not(not(x)) → x
implies(false, y) → not(false)
implies(x, false) → not(x)
implies(not(x), not(y)) → implies(y, and(x, y))

Q is empty.

(1) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

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

implies(false, y) → not(false)
implies(x, false) → not(x)


(2) Obligation:

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

and(x, false) → false
and(x, not(false)) → x
not(not(x)) → x
implies(not(x), not(y)) → implies(y, and(x, y))

Q is empty.

(3) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

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

and(x, false) → false
and(x, not(false)) → x
not(not(x)) → x
implies(not(x), not(y)) → implies(y, and(x, y))


(4) Obligation:

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

(5) RisEmptyProof (EQUIVALENT transformation)

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

(6) TRUE

(7) RisEmptyProof (EQUIVALENT transformation)

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

(8) TRUE