| Entry |
Value |
|
Name |
FINITE_SING |
|
Conclusion |
!a. FINITE {a} |
|
Constructive Proof |
No |
|
Axiom |
!t. t \/ ~t
(\a. a = (\b. (\c. c) = (\c. c))) (\d. (\e. d e) = d) |
|
Classical Lemmas |
!x s t. s SUBSET x INSERT t <=> s DELETE x SUBSET t!x s. x IN s ==> x INSERT (s DELETE x) = s!t. (t <=> T) \/ (t <=> F)!s x. FINITE (x INSERT s) <=> FINITE s!s t u. s SUBSET t UNION u <=> s DIFF t SUBSET u!s t. FINITE t /\ s SUBSET t ==> FINITE 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!x s t. s SUBSET x INSERT t <=> s DELETE x SUBSET t!x s t. x INSERT s UNION t = x INSERT (s UNION t)!x s. s DIFF {x} = s DELETE x!x s. x INSERT s = {y | y = x \/ y IN s}!x s. {x} UNION s = x INSERT s!x s. FINITE s ==> FINITE (x INSERT s)!x s. x IN s ==> x INSERT (s DELETE x) = 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!t1 t2 t3. (t1 \/ t2) \/ t3 <=> t1 \/ t2 \/ t3!t1 t2. t1 \/ t2 <=> t2 \/ t1!p q. (!x. p ==> q x) <=> p ==> (!x. q x)!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. 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 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. (!x y. p x y) <=> (!y x. p x y)!p. p {} /\ (!x s. p s ==> p (x INSERT s)) ==> (!a. FINITE a ==> p a)!s x y. x IN s DELETE y <=> x IN s /\ ~(x = y)!s x. FINITE (x INSERT s) <=> FINITE s!s x. s DELETE x = {y | y IN s /\ ~(y = x)}!s t x. x IN s DIFF t <=> x IN s /\ ~(x IN t)!s t x. x IN s UNION t <=> x IN s \/ x IN t!s t u. s SUBSET t UNION u <=> s DIFF t SUBSET u!s t. (!x. x IN s <=> x IN t) <=> s = t!s t. s SUBSET t <=> (!x. x IN s ==> x IN t)!s t. s DIFF t = {x | x IN s /\ ~(x IN t)}!s t. s UNION t = {x | x IN s \/ x IN t}!s t. s UNION t = t UNION s!s t. (!x. x IN s <=> x IN t) ==> s = t!s t. FINITE t /\ s SUBSET t ==> FINITE s!s t. s SUBSET t UNION s!s t. s SUBSET s UNION t!s. s SUBSET {} <=> s = {}!s. {} UNION s = s!s. s UNION {} = sFINITE {}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} |
|
Contained Package |
set-finite-thm |
|
Comment |
Standard HOL library retrieved from OpenTheory |
