| Entry |
Value |
|
Name |
ALL_MP |
|
Conclusion |
!p q l. ALL (\x. p x ==> q x) l /\ ALL p l ==> ALL q l |
|
Constructive Proof |
Yes |
|
Axiom |
N|A |
|
Classical Lemmas |
N|A |
|
Constructive Lemmas |
T!x y s. x IN y INSERT s <=> x = y \/ x IN s!h t. set_of_list (CONS h t) = h INSERT set_of_list t!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!p q r. p ==> q ==> r <=> p /\ q ==> r!t. (!x. 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. t ==> t!f y. (\x. f x) y = f y!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. ALL p []!p. p [] /\ (!h t. p t ==> p (CONS h t)) ==> (!l. p l)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 |
