INJECTIVE_LEFT

Entry	Value
Name	INJECTIVE_LEFT_INVERSE
Conclusion	!f. (!x y. f x = f y ==> x = y) <=> (?g. !x. g (f x) = x)
Constructive Proof	Yes
Axiom	(\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	N\|A
Constructive Lemmas	T !x. x = x !x. x IN UNIV !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 !f g. (!x. f x = g x) ==> f = g !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) !r. (!x. ?y. r x y) <=> (?f. !x. r x (f x)) F <=> (!p. p) T <=> (\p. p) = (\p. p) (~) = (\p. p ==> F) (/\) = (\p q. (\f. f p q) = (\f. f T T)) (==>) = (\p q. p /\ q <=> p) (!) = (\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