Entry Value
Name BIJECTIVE_LEFT_RIGHT_INVERSE
Conclusion !f. (!x y. f x = f y ==> x = y) /\ (!y. ?x. f x = y) <=> (?g. (!y. f (g y) = y) /\ (!x. g (f x) = x))
Constructive Proof No
Axiom 
!t. t \/ ~t 
(\a. a = (\b. (\c. c) = (\c. c))) (\d. (\e. d e) = d) 
(\a. a = (\b. (\c. c) = (\c. c)))
(\d. (\e. e = (\f. (\c. c) = (\c. c)))
     (\g. (\h i.
               (\j k.
                    (\l. l j k) =
                    (\m. m ((\c. c) = (\c. c)) ((\c. c) = (\c. c))))
               h
               i <=>
               h)
          (d g)
          (d ((@) d))))
Classical Lemmas
  • !t1 t2. ~(t1 ==> t2) <=> t1 /\ ~t2
  • !p q. p ==> (?x. q x) <=> (?x. p ==> q x)
  • !t. ~ ~t <=> t
  • !t. (t <=> T) \/ (t <=> F)
  • !f s t. (!y. y IN t ==> (?x. x IN s /\ f x = y)) <=> (?g. !y. y IN t ==> g y IN s /\ f (g y) = y)
  • !f s t. (!x. x IN s ==> f x IN t) ==> ((!x y. x IN s /\ y IN s /\ f x = f y ==> x = y) /\ (!y. y IN t ==> (?x. x IN s /\ f x = y)) <=> (?g. (!y. y IN t ==> g y IN s) /\ (!y. y IN t ==> f (g y) = y) /\ (!x. x IN s ==> g (f x) = x)))
  • !p. ~(?x. p x) <=> (!x. ~p x)
    		•
    Constructive Lemmas
  • T
  • !x y z. x = y /\ y = z ==> x = z
  • !x y. x = y ==> y = x
  • !x. x = x
  • !x. x IN UNIV
  • !t1 t2. ~(t1 ==> t2) <=> t1 /\ ~t2
  • !p q. p /\ (!x. q x) <=> (!x. p /\ q x)
  • !p q. p /\ (?x. q x) <=> (?x. p /\ q x)
  • !p q. p ==> (?x. q x) <=> (?x. p ==> q x)
  • !t. (!x. t) <=> t
  • !t. ~ ~t <=> t
  • !t. F /\ t <=> F
  • !t. T /\ t <=> t
  • !t. t /\ F <=> F
  • !t. t /\ T <=> t
  • !t. t /\ t <=> t
  • !t. (F <=> t) <=> ~t
  • !t. (T <=> t) <=> t
  • !t. (t <=> F) <=> ~t
  • !t. (t <=> T) <=> t
  • !t. F ==> t <=> T
  • !t. T ==> t <=> t
  • !t. t ==> F <=> ~t
  • !t. t ==> T <=> T
  • !t. F \/ t <=> t
  • !t. T \/ t <=> T
  • !t. t \/ F <=> t
  • !t. t \/ T <=> T
  • !t. t \/ t <=> t
  • !t. t ==> t
  • !t. (t <=> T) \/ (t <=> F)
  • !f y. (\x. f x) y = f y
  • !f g. (!x. f x = g x) ==> f = g
  • !f s t. (!y. y IN t ==> (?x. x IN s /\ f x = y)) <=> (?g. !y. y IN t ==> g y IN s /\ f (g y) = y)
  • !f s t. (!x. x IN s ==> f x IN t) ==> ((!x y. x IN s /\ y IN s /\ f x = f y ==> x = y) /\ (!y. y IN t ==> (?x. x IN s /\ f x = y)) <=> (?g. (!y. y IN t ==> g y IN s) /\ (!y. y IN t ==> f (g y) = y) /\ (!x. x IN s ==> g (f x) = x)))
  • !f s. (!x y. x IN s /\ y IN s /\ f x = f y ==> x = y) <=> (?g. !x. x IN s ==> g (f x) = x)
  • !t. (\x. t x) = t
  • !p a. (?x. a = x /\ p x) <=> p a
  • !p x. x IN GSPEC p <=> p x
  • !p x. x IN {y | p y} <=> p x
  • !p x. p x ==> p ((@) p)
  • !p. ~(?x. p x) <=> (!x. ~p x)
  • !r. (!x. ?y. r x y) <=> (?f. !x. r x (f x))
  • F <=> (!p. p)
  • T <=> (\p. p) = (\p. p)
  • ~F <=> T
  • ~T <=> F
  • (~) = (\p. p ==> F)
  • (/\) = (\p q. (\f. f p q) = (\f. f T T))
  • (==>) = (\p q. p /\ q <=> p)
  • (\/) = (\p q. !r. (p ==> r) ==> (q ==> r) ==> r)
  • (!) = (\p. p = (\x. T))
  • (?) = (\p. !q. (!x. p x ==> q) ==> q)
  • (?) = (\p. p ((@) p))
  • UNIV = {x | T}
    		•
    Contained Package set-thm
    Comment Standard HOL library retrieved from OpenTheory
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