Entry Value
Name ALL_APPEND
Conclusion !p l1 l2. ALL p (APPEND l1 l2) <=> ALL p l1 /\ ALL p l2
Constructive Proof No
Axiom 
!t. t \/ ~t 
(\a. a = (\b. (\c. c) = (\c. c))) (\d. (\e. d e) = d)
Classical Lemmas
  • !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)
  • !p l1 l2. EX p (APPEND l1 l2) <=> EX p l1 \/ EX p l2
  • !p l. ~ALL p l <=> EX (\x. ~p x) l
  • !p l. ~ALL (\x. ~p x) l <=> EX p l
  • !p l. ~EX p l <=> ALL (\x. ~p x) l
  • !p l. ~EX (\x. ~p x) l <=> ALL p l
  • !p l. EX p l <=> (?x. x IN set_of_list l /\ p x)
  • !p. (?x. ~p x) <=> ~(!x. p x)
  • !p. ~(!x. p x) <=> (?x. ~p x)
  • !p. ~(?x. p x) <=> (!x. ~p x)
    		•
    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
  • !h t. set_of_list (CONS h t) = h INSERT set_of_list t
  • !x s t. x INSERT s UNION t = x INSERT (s UNION t)
  • !x s. x INSERT s = {y | y = x \/ y IN s}
  • !x s. {x} UNION s = x INSERT 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 /\ r \/ 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
  • !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 h t. ALL p (CONS h t) <=> p h /\ ALL p t
  • !p h t. EX p (CONS h t) <=> p h \/ EX p 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 q. (!x. p x ==> q x) ==> (?x. p x) ==> (?x. q x)
  • !p l1 l2. EX p (APPEND l1 l2) <=> EX p l1 \/ EX p l2
  • !p l. ~ALL p l <=> EX (\x. ~p x) l
  • !p l. ~ALL (\x. ~p x) l <=> EX p l
  • !p l. ~EX p l <=> ALL (\x. ~p x) l
  • !p l. ~EX (\x. ~p x) l <=> ALL p l
  • !p l. ALL p l <=> (!x. x IN set_of_list l ==> p x)
  • !p l. EX p l <=> (?x. x IN set_of_list l /\ p x)
  • !p. ~EX p []
  • !p. (?x. ~p x) <=> ~(!x. p x)
  • !p. ~(!x. p x) <=> (?x. ~p x)
  • !p. ~(?x. p x) <=> (!x. ~p x)
  • !p. ALL p []
  • !p. (!x y. p x y) <=> (!y x. p x y)
  • !p. p [] /\ (!h t. p t ==> p (CONS h t)) ==> (!l. p l)
  • !l h t. APPEND (CONS h t) l = CONS h (APPEND t l)
  • !l1 l2. set_of_list (APPEND l1 l2) = set_of_list l1 UNION set_of_list l2
  • !l. APPEND [] l = l
  • !s t x. x IN s UNION t <=> x IN s \/ x IN t
  • !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. 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. {} UNION s = s
  • !s. s UNION {} = s
  • 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}
  • set_of_list [] = {}
    		•
    Contained Package list-append-thm
    Comment Standard HOL library retrieved from OpenTheory
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