| Entry |
Value |
|
Name |
WF_FINITE |
|
Conclusion |
!r. (!x. ~r x x) /\
(!x y z. r x y /\ r y z ==> r x z) /\
(!x. FINITE {y | r y x})
==> WF r |
|
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))))
(\a. (\b. b = (\c. (\d. d) = (\d. d)))
(\e. (\f g.
(\h i.
(\j. j h i) =
(\k. k ((\d. d) = (\d. d)) ((\d. d) = (\d. d))))
f
g <=>
f)
((\l. l = (\m. (\d. d) = (\d. d)))
(\n. (\f g.
(\h i.
(\j. j h i) =
(\k. k ((\d. d) = (\d. d)) ((\d. d) = (\d. d))))
f
g <=>
f)
(a n)
e))
e))
(\p. (\h i. (\j. j h i) = (\k. k ((\d. d) = (\d. d)) ((\d. d) = (\d. d))))
((\q. q = (\r. (\d. d) = (\d. d)))
(\s. (\q. q = (\r. (\d. d) = (\d. d)))
(\t. (\f g.
(\h i.
(\j. j h i) =
(\k. k ((\d. d) = (\d. d)) ((\d. d) = (\d. d))))
f
g <=>
f)
(p s = p t)
(s = t))))
((\u. (\f g.
(\h i.
(\j. j h i) =
(\k. k ((\d. d) = (\d. d)) ((\d. d) = (\d. d))))
f
g <=>
f)
u
((\b. b = (\c. (\d. d) = (\d. d))) (\d. d)))
((\q. q = (\r. (\d. d) = (\d. d)))
(\v. (\w. (\b. b = (\c. (\d. d) = (\d. d)))
(\x. (\f g.
(\h i.
(\j. j h i) =
(\k. k ((\d. d) = (\d. d)) ((\d. d) = (\d. d))))
f
g <=>
f)
((\q. q = (\r. (\d. d) = (\d. d)))
(\y. (\f g.
(\h i.
(\j. j h i) =
(\k. k ((\d. d) = (\d. d))
((\d. d) = (\d. d))))
f
g <=>
f)
(w y)
x))
x))
(\z. v = p z))))) |
|
Classical Lemmas |
!a b a' b'. a,b = a',b' <=> a = a' /\ b = b'!a b. FST (a,b) = a!a b. SND (a,b) = b!e f. ?!fn. fn _0 = e /\ (!n. fn (SUC n) = f (fn n) n)!x s t. s SUBSET x INSERT t <=> s DELETE x SUBSET t!x s. x IN s ==> x INSERT (s DELETE x) = s!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.
(!x y. x IN s /\ y IN s /\ f x = f y ==> x = y)
==> (FINITE (IMAGE f s) <=> FINITE s)!f s. FINITE s ==> FINITE {y | ?x. x IN s /\ y = f x}!f s. FINITE s ==> FINITE (IMAGE f s)!f s.
INFINITE s /\ (!x y. x IN s /\ y IN s /\ f x = f y ==> x = y)
==> INFINITE (IMAGE f s)!f t s.
(!x y. x IN s /\ y IN s /\ f x = f y ==> x = y) /\ FINITE t
==> FINITE {x | x IN s /\ f x IN t}!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)!r. WF r <=> ~(?f. !n. r (f (SUC n)) (f n))!r x y. x,y IN relation_to_set r <=> r x y!r s. subrelation r s <=> (!x y. r x y ==> s x y)!r. subrelation successor r /\ transitive r ==> subrelation (<) r!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)!m n p. m + n = m + p <=> n = p!m n p. m <= n /\ n <= p ==> m <= p!m n. n <= MAX m n!m n. m <= MAX m n!m n. ~(m <= n) <=> n < m!m n. m <= n /\ n <= m <=> m = n!m n. m < n <=> (?d. n = m + SUC d)!m n. SUC m < SUC n <=> m < n!m n. SUC m <= n <=> m < n!m n. SUC m <= SUC n <=> m <= n!m n. SUC m = SUC n <=> m = n!m n. MAX m n = MAX n m!m n. m < n \/ n < m \/ m = n!m n. m <= n \/ n <= m!n. FINITE {m | m < n}!n. FINITE {m | m <= n}!n. ~(SUC n = _0)!x. FST x,SND x = x!s x. FINITE (x INSERT s) <=> FINITE s!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. FINITE s <=> (?a. !x. x IN s ==> x <= a)!s. FINITE s ==> (?a. ~(a IN s))INFINITE UNIV |
|
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 = y ==> y = x!a b a' b'. a,b = a',b' <=> a = a' /\ b = b'!a b. FST (a,b) = a!a b. SND (a,b) = b!e f. ?!fn. fn _0 = e /\ (!n. fn (SUC n) = f (fn n) n)!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. ?x. x = a!a. ?!x. x = a!x. ~(x IN {})!x. x = x!x. (@y. y = x) = x!x. x IN UNIV!y s f. y IN IMAGE f s <=> (?x. y = f x /\ x IN s)!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 /\ 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!a b. (a <=> b) ==> a ==> b!p q. (!x. p ==> q x) <=> p ==> (!x. q x)!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 x. x IN s ==> f x IN IMAGE f s!f s. IMAGE f s = {y | ?x. x IN s /\ y = f x}!f s.
(!x y. x IN s /\ y IN s /\ f x = f y ==> x = y)
==> (FINITE (IMAGE f s) <=> FINITE s)!f s. FINITE s ==> FINITE {y | ?x. x IN s /\ y = f x}!f s. FINITE s ==> FINITE (IMAGE f s)!f s.
INFINITE s /\ (!x y. x IN s /\ y IN s /\ f x = f y ==> x = y)
==> INFINITE (IMAGE f s)!f t s.
(!x y. x IN s /\ y IN s /\ f x = f y ==> x = y) /\ FINITE t
==> FINITE {x | x IN s /\ f x IN t}!f. ONE_ONE f <=> (!x1 x2. f x1 = f x2 ==> x1 = x2)!f. ONTO f <=> (!y. ?x. y = f 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. (!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 q. (?x. p x) \/ q <=> (?x. p x \/ q)!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. !y. p y <=> x = y)!p. (?!x. p x) <=> (?x. p x) /\ (!x x'. p x /\ p x' ==> x = x')!p. ~(!x. p x) <=> (?x. ~p x)!p. ~(?x. p x) <=> (!x. ~p x)!r. WF r <=> (!p. (?x. p x) ==> (?x. p x /\ (!y. r y x ==> ~p y)))!r. WF r <=> ~(?f. !n. r (f (SUC n)) (f n))!r. transitive r <=> (!x y z. r x y /\ r y z ==> r x z)!r x y. x,y IN relation_to_set r <=> r x y!r s. subrelation r s <=> (!x y. r x y ==> s x y)!r s. subrelation r s <=> relation_to_set r SUBSET relation_to_set s!p. (!x y. p x y) <=> (!y x. p x y)!r. (!x. ?y. r x y) <=> (?f. !x. r x (f x))!p. (!x. ?!y. p x y) <=> (?f. !x y. p x y <=> f x = y)!r. relation_to_set r = {x,y | r x y}!p f s. (!y. y IN IMAGE f s ==> p y) <=> (!x. x IN s ==> p (f x))!p. p _0 /\ (!n. p n ==> p (SUC n)) ==> (!n. p n)!r. subrelation successor r /\ transitive r ==> subrelation (<) r!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)!m n p. m + n = m + p <=> n = p!m n p. m <= n /\ n <= p ==> m <= p!m n. n <= MAX m n!m n. m <= MAX m n!m n. ~(m <= n) <=> n < m!m n. m <= n /\ n <= m <=> m = n!m n. m < n <=> (?d. n = m + SUC d)!m n. m < SUC n <=> m <= n!m n. m < SUC n <=> m = n \/ m < n!m n. SUC m < SUC n <=> m < n!m n. m <= SUC n <=> m = SUC n \/ m <= n!m n. SUC m <= n <=> m < n!m n. SUC m <= SUC n <=> m <= n!m n. SUC m = SUC n <=> m = n!m n. successor m n <=> SUC m = n!m n. m + SUC n = SUC (m + n)!m n. SUC m + n = SUC (m + n)!m n. MAX m n = MAX n m!m n. MAX m n = (if m <= n then n else m)!m n. m < n \/ n < m \/ m = n!m n. m <= n \/ n <= m!n. FINITE {m | m < n}!n. FINITE {m | m <= n}!m. ~(m < 0)!n. ~(SUC n = _0)!n. 0 < SUC n!n. n <= n!n. 0 <= n!m. m <= 0 <=> m = 0!m. m + 0 = m!n. 0 + n = n!n. {m | m < SUC n} = n INSERT {m | m < n}!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 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. INFINITE s <=> ~FINITE s!s. s SUBSET {} <=> s = {}!s. {} UNION s = s!s. s UNION {} = s!s. FINITE s <=> (?a. !x. x IN s ==> x <= a)!s. FINITE s ==> (?a. ~(a IN s))?f. ONE_ONE f /\ ~ONTO fFINITE {}INFINITE UNIVF <=> (!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))NUMERAL = (\n. n){} = {x | F}UNIV = {x | T}GSPEC (\x. F) = {}{m | m < 0} = {} |
|
Contained Package |
relation-natural-thm |
|
Comment |
Standard HOL library retrieved from OpenTheory |
