| Entry |
Value |
|
Name |
FINITE_FUNSPACE |
|
Conclusion |
!d s t.
FINITE s /\ FINITE t
==> FINITE {f | (!x. x IN s ==> f x IN t) /\ (!x. ~(x IN s) ==> f x = d)} |
|
Constructive Proof |
No |
|
Axiom |
!t. t \/ ~t
(\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 |
!a b a' b'. a,b = a',b' <=> a = a' /\ b = b'!x y s t. x,y IN s CROSS t <=> x IN s /\ y IN t!a b. FST (a,b) = a!a b. SND (a,b) = b!x s t. s SUBSET x INSERT t <=> s DELETE x SUBSET t!x s. x IN s ==> x INSERT (s DELETE x) = s!a. FINITE {a}!d a s t.
{f | (!x. x IN a INSERT s ==> f x IN t) /\
(!x. ~(x IN a INSERT s) ==> f x = d)} =
IMAGE (\(b,g) x. if x = a then b else g x)
(t CROSS {f | (!x. x IN s ==> f x IN t) /\ (!x. ~(x IN s) ==> f x = d)})!t1 t2. ~(t1 /\ t2) <=> ~t1 \/ ~t2!t1 t2. ~(t1 ==> t2) <=> t1 /\ ~t2!t1 t2. ~(t1 \/ t2) <=> ~t1 /\ ~t2!t. ~ ~t <=> t!t. (t <=> T) \/ (t <=> F)!f s. FINITE s ==> FINITE {y | ?x. x IN s /\ y = f x}!f s. FINITE s ==> FINITE (IMAGE f s)!p c x y. p (if c then x else y) <=> (c ==> p x) /\ (~c ==> p y)!p q. (!x. p x \/ q) <=> (!x. p x) \/ q!p q. (!x. p x) \/ q <=> (!x. p x \/ q)!p. (?x. ~p x) <=> ~(!x. p x)!p. ~(!x. p x) <=> (?x. ~p x)!p. ~(?x. p x) <=> (!x. ~p x)!f s t.
FINITE s /\ (!x. x IN s ==> FINITE (t x))
==> FINITE {f x y | x IN s /\ y IN t x}!f. ?fn. !a b. fn (a,b) = f a b!p a b. a,b IN {x,y | p x y} <=> p a b!p. (?x. p x) <=> (?a b. p (a,b))!p. p {} /\ (!x s. p s /\ ~(x IN s) /\ FINITE s ==> p (x INSERT s))
==> (!s. FINITE s ==> p s)!x. FST x,SND x = x!s x. FINITE (x INSERT s) <=> FINITE s!s t a.
{x,y | x IN a INSERT s /\ y IN t x} =
IMAGE ((,) a) (t a) UNION {x,y | x IN s /\ y IN t x}!s t u. s SUBSET t UNION u <=> s DIFF t SUBSET u!s t. FINITE (s UNION t) <=> FINITE s /\ FINITE t!s t. FINITE t /\ s SUBSET t ==> FINITE s!s t. FINITE s /\ FINITE t ==> FINITE {x,y | x IN s /\ y IN t}!s t. FINITE s /\ FINITE t ==> FINITE (s CROSS t) |
|
Constructive Lemmas |
T!x y s. x IN y INSERT s <=> x = y \/ x IN s!t1 t2. (if F then t1 else t2) = t2!t1 t2. (if T then t1 else t2) = t1!x y. x = y <=> y = x!x y. x IN {y} <=> x = y!x y. x = y ==> y = x!a b a' b'. a,b = a',b' <=> a = a' /\ b = b'!x y s t. x,y IN s CROSS t <=> x IN s /\ y IN t!a b. FST (a,b) = a!a b. SND (a,b) = b!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 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!a. FINITE {a}!x. ~(x IN {})!x. x = x!x. (@y. y = x) = x!d a s t.
{f | (!x. x IN a INSERT s ==> f x IN t) /\
(!x. ~(x IN a INSERT s) ==> f x = d)} =
IMAGE (\(b,g) x. if x = a then b else g x)
(t CROSS {f | (!x. x IN s ==> f x IN t) /\ (!x. ~(x IN s) ==> f x = d)})!y s f. y IN IMAGE f s <=> (?x. y = f x /\ x IN s)!d t.
{f | (!x. x IN {} ==> f x IN t) /\ (!x. ~(x IN {}) ==> f x = d)} =
{(\x. d)}!p1 p2 q1 q2. (p1 ==> p2) /\ (q1 ==> q2) ==> p1 /\ q1 ==> p2 /\ q2!p1 p2 q1 q2. (p1 ==> p2) /\ (q1 ==> q2) ==> p1 \/ q1 ==> p2 \/ q2!t1 t2 t3. (t1 /\ t2) /\ t3 <=> t1 /\ t2 /\ t3!p q r. (p \/ q) /\ r <=> p /\ r \/ q /\ r!p q r. p ==> q ==> r <=> p /\ q ==> r!t1 t2 t3. (t1 \/ t2) \/ t3 <=> t1 \/ t2 \/ t3!t1 t2. ~(t1 /\ t2) <=> ~t1 \/ ~t2!t1 t2. ~(t1 ==> t2) <=> t1 /\ ~t2!t1 t2. ~(t1 \/ t2) <=> ~t1 /\ ~t2!t1 t2. t1 /\ t2 <=> t2 /\ t1!t1 t2. t1 \/ t2 <=> t2 \/ t1!p q. (!x. p ==> q x) <=> p ==> (!x. q x)!p q. p /\ (?x. q x) <=> (?x. p /\ q x)!p q. p \/ (?x. q x) <=> (?x. p \/ 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!f s. IMAGE f s = {y | ?x. x IN s /\ y = f x}!f s. FINITE s ==> FINITE {y | ?x. x IN s /\ y = f x}!f s. FINITE s ==> FINITE (IMAGE f s)!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. (!y. p y <=> y = x) ==> (@) p = x!p x. p x ==> p ((@) p)!p c x y. p (if c then x else y) <=> (c ==> p x) /\ (~c ==> p y)!p q. (!x. p x \/ q) <=> (!x. p x) \/ q!p q. (?x. p x /\ q) <=> (?x. p x) /\ q!p q. (?x. p x) /\ q <=> (?x. p x /\ q)!p q. (!x. p x) \/ q <=> (!x. p x \/ q)!p q. (?x. p x) \/ q <=> (?x. p x \/ q)!p f q. (!z. z IN {f x | p x} ==> q z) <=> (!x. p x ==> q (f 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 q. (?x. p x) \/ (?x. q x) <=> (?x. p 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)!f s t.
FINITE s /\ (!x. x IN s ==> FINITE (t x))
==> FINITE {f x y | x IN s /\ y IN t x}!f. ?fn. !a b. fn (a,b) = f a b!p a b. a,b IN {x,y | p x y} <=> p a b!p f q. (!z. z IN {f x y | p x y} ==> q z) <=> (!x y. p x y ==> q (f x y))!p. (!x y. p x y) <=> (!y x. p x y)!r. (!x. ?y. r x y) <=> (?f. !x. r x (f x))!p f q.
(!z. z IN {f w x y | p w x y} ==> q z) <=>
(!w x y. p w x y ==> q (f w x y))!t. {x,y | x IN {} /\ y IN t x} = {}!p. (?x. p x) <=> (?a b. p (a,b))!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)!x. ?a b. x = a,b!x. FST x,SND x = x!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 a.
{x,y | x IN a INSERT s /\ y IN t x} =
IMAGE ((,) a) (t a) UNION {x,y | x IN s /\ y IN t 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 u. (s UNION t) UNION u = s UNION t UNION u!s t. (!x. x IN s <=> x IN t) <=> s = t!s t. FINITE (s UNION t) <=> FINITE s /\ FINITE 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. FINITE s /\ FINITE t ==> FINITE (s UNION t)!s t. s SUBSET t UNION s!s t. s SUBSET s UNION t!s t. s CROSS t = {x,y | x IN s /\ y IN t}!s t. FINITE s /\ FINITE t ==> FINITE {x,y | x IN s /\ y IN t}!s t. FINITE s /\ FINITE t ==> FINITE (s CROSS 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)COND = (\t t1 t2. @x. ((t <=> T) ==> x = t1) /\ ((t <=> F) ==> x = t2))(/\) = (\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)(?) = (\p. p ((@) p))(?!) = (\p. (?) p /\ (!x y. p x /\ p y ==> x = y)){} = {x | F}GSPEC (\x. F) = {} |
|
Contained Package |
set-finite-thm |
|
Comment |
Standard HOL library retrieved from OpenTheory |
