FINITE_TRANSITIVITY

Entry	Value
Name	FINITE_TRANSITIVITY_CHAIN
Conclusion	!r s. FINITE s /\ (!x. ~r x x) /\ (!x y z. r x y /\ r y z ==> r x z) /\ (!x. x IN s ==> (?y. y IN s /\ r x y)) ==> s = {}
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 INSERT s = {}) !x s. x IN s ==> x INSERT (s DELETE x) = 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) !p. p {} /\ (!x s. p s /\ ~(x IN s) /\ FINITE s ==> p (x INSERT s)) ==> (!s. FINITE s ==> p s) !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. ~(x INSERT s = {}) !x s. x IN s <=> x INSERT s = s !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 !p q r. p \/ q ==> r <=> (p ==> r) /\ (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. (?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 g. (!x. f x = g x) ==> f = g !t. (\x. t x) = t !p a. (!x. x = a ==> p x) <=> p a !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 q. (!x. p x ==> q x) ==> (?x. p x) ==> (?x. q x) !p. (?x. ~p x) <=> ~(!x. p x) !p. ~(!x. p x) <=> (?x. ~p x) !p. ~(?x. p x) <=> (!x. ~p 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) !p. p {} /\ (!x s. p s /\ ~(x IN s) /\ FINITE s ==> p (x INSERT s)) ==> (!s. FINITE s ==> p s) !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 {} = s FINITE {} 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

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