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cvc5-cvc5-1.2.0.src.theory.sets.rewrites Maven / Gradle / Ivy

; Equality

(define-cond-rule sets-eq-singleton-emp ((x ?Set) (y ?))
  (set.is_empty x)
  (= x (set.singleton y))
  false)

(define-rule sets-member-singleton ((x ?) (y ?))
  (set.member x (set.singleton y))
  (= x y))

(define-cond-rule sets-member-emp ((x ?) (y ?Set))
  (set.is_empty y)
  (set.member x y)
  false)
  
(define-rule sets-subset-elim ((x ?Set) (y ?Set))
  (set.subset x y)
  (= (set.union x y) y))

(define-rule sets-union-comm ((x ?Set) (y ?Set))
  (set.union x y)
  (set.union y x))
(define-rule sets-inter-comm ((x ?Set) (y ?Set))
  (set.inter x y)
  (set.inter y x))

(define-cond-rule sets-inter-emp1 ((x ?Set) (y ?Set))
  (set.is_empty x)
  (set.inter x y)
  x)
(define-cond-rule sets-inter-emp2 ((x ?Set) (y ?Set))
  (set.is_empty y)
  (set.inter x y)
  y)
(define-cond-rule sets-minus-emp1 ((x ?Set) (y ?Set))
  (set.is_empty x)
  (set.minus x y)
  x)
(define-cond-rule sets-minus-emp2 ((x ?Set) (y ?Set))
  (set.is_empty y)
  (set.minus x y)
  x)
(define-cond-rule sets-union-emp1 ((x ?Set) (y ?Set))
  (set.is_empty x)
  (set.union x y)
  y)
(define-cond-rule sets-union-emp2 ((x ?Set) (y ?Set))
  (set.is_empty y)
  (set.union x y)
  x)

(define-rule sets-inter-member ((x ?) (y ?Set) (z ?Set))
  (set.member x (set.inter y z))
  (and (set.member x y) (set.member x z)))
(define-rule sets-minus-member ((x ?) (y ?Set) (z ?Set))
  (set.member x (set.minus y z))
  (and (set.member x y) (not (set.member x z))))
(define-rule sets-union-member ((x ?) (y ?Set) (z ?Set))
  (set.member x (set.union y z))
  (or (set.member x y) (set.member x z)))

(define-rule sets-choose-singleton ((x ?))
  (set.choose (set.singleton x))
  x)

(define-rule sets-card-singleton ((x ?))
  (set.card (set.singleton x))
  1)

(define-rule sets-card-union ((s ?Set) (t ?Set))
  (set.card (set.union s t))
  (- (+ (set.card s) (set.card t)) (set.card (set.inter s t))))

(define-rule sets-card-minus ((s ?Set) (t ?Set))
  (set.card (set.minus s t))
  (- (set.card s) (set.card (set.inter s t))))

(define-cond-rule sets-card-emp ((x ?Set))
  (set.is_empty x)
  (set.card x)
  0)

; (set.complement S) ---> (set.minus (as set.universe (Set Int)) S)