Entry Value
Name ALL_F
Conclusion !l. ALL (\x. F) l <=> NULL l
Constructive Proof No
Axiom 
!t. t \/ ~t 
(\a. a = (\b. (\c. c) = (\c. c))) (\d. (\e. d e) = d)
Classical Lemmas
  • !x s. ~(x INSERT s = {})
  • !t. ~ ~t <=> t
  • !t. (t <=> T) \/ (t <=> F)
  • !p. (?x. ~p x) <=> ~(!x. p x)
  • !p. ~(!x. p x) <=> (?x. ~p x)
  • !p. ~(?x. p x) <=> (!x. ~p x)
  • !l. set_of_list l = {} <=> NULL l
  • !s t. ~(s = t) <=> (?x. x IN t <=> ~(x IN s))
  • !s. (?x. x IN s) <=> ~(s = {})
    		•
    Constructive Lemmas
  • T
  • !x y s. x IN y INSERT s <=> x = y \/ x IN s
  • !x y. x = y <=> y = x
  • !x y. x = y ==> y = x
  • !h t. ~NULL (CONS h t)
  • !h t. set_of_list (CONS h t) = h INSERT set_of_list t
  • !x s. ~(x INSERT s = {})
  • !x s. x INSERT s = {y | y = x \/ y IN s}
  • !x. ~(x IN {})
  • !x. x = x
  • !p1 p2 q1 q2. (p1 ==> p2) /\ (q1 ==> q2) ==> p1 /\ q1 ==> p2 /\ q2
  • !p1 p2 q1 q2. (p1 ==> p2) /\ (q1 ==> q2) ==> p1 \/ q1 ==> p2 \/ q2
  • !t. (!x. 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 g. (!x. f x = g x) ==> f = g
  • !t. (\x. t x) = t
  • !p h t. ALL p (CONS h t) <=> p h /\ ALL p 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 q. (!x. p x ==> q x) ==> (?x. p x) ==> (?x. q x)
  • !p l. ALL p l <=> (!x. x IN set_of_list l ==> p x)
  • !p. (?x. ~p x) <=> ~(!x. p x)
  • !p. ~(!x. p x) <=> (?x. ~p x)
  • !p. ~(?x. p x) <=> (!x. ~p x)
  • !p. ALL p []
  • !p. p [] /\ (!h t. p t ==> p (CONS h t)) ==> (!l. p l)
  • !l. set_of_list l = {} <=> NULL l
  • !s t. (!x. x IN s <=> x IN t) <=> s = t
  • !s t. ~(s = t) <=> (?x. x IN t <=> ~(x IN s))
  • !s t. (!x. x IN s <=> x IN t) ==> s = t
  • !s. (?x. x IN s) <=> ~(s = {})
  • NULL []
  • 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)
  • {} = {x | F}
  • set_of_list [] = {}
    		•
    Contained Package list-set-thm
    Comment Standard HOL library retrieved from OpenTheory
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