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theory Hard_Quantifiers(* Title: LK/Hard_Quantifiers.thy ID: $Id: Hard_Quantifiers.thy,v 1.3 2007/08/07 18:19:55 wenzelm Exp $ Author: Lawrence C Paulson, Cambridge University Computer Laboratory Copyright 1992 University of Cambridge Hard examples with quantifiers. Can be read to test the LK system. From F. J. Pelletier, Seventy-Five Problems for Testing Automatic Theorem Provers, J. Automated Reasoning 2 (1986), 191-216. Errata, JAR 4 (1988), 236-236. Uses pc_tac rather than fast_tac when the former is significantly faster. *) theory Hard_Quantifiers imports LK begin lemma "|- (ALL x. P(x) & Q(x)) <-> (ALL x. P(x)) & (ALL x. Q(x))" by fast lemma "|- (EX x. P-->Q(x)) <-> (P --> (EX x. Q(x)))" by fast lemma "|- (EX x. P(x)-->Q) <-> (ALL x. P(x)) --> Q" by fast lemma "|- (ALL x. P(x)) | Q <-> (ALL x. P(x) | Q)" by fast text "Problems requiring quantifier duplication" (*Not provable by fast: needs multiple instantiation of ALL*) lemma "|- (ALL x. P(x)-->P(f(x))) & P(d)-->P(f(f(f(d))))" by best_dup (*Needs double instantiation of the quantifier*) lemma "|- EX x. P(x) --> P(a) & P(b)" by fast_dup lemma "|- EX z. P(z) --> (ALL x. P(x))" by best_dup text "Hard examples with quantifiers" text "Problem 18" lemma "|- EX y. ALL x. P(y)-->P(x)" by best_dup text "Problem 19" lemma "|- EX x. ALL y z. (P(y)-->Q(z)) --> (P(x)-->Q(x))" by best_dup text "Problem 20" lemma "|- (ALL x y. EX z. ALL w. (P(x)&Q(y)-->R(z)&S(w))) --> (EX x y. P(x) & Q(y)) --> (EX z. R(z))" by fast text "Problem 21" lemma "|- (EX x. P-->Q(x)) & (EX x. Q(x)-->P) --> (EX x. P<->Q(x))" by best_dup text "Problem 22" lemma "|- (ALL x. P <-> Q(x)) --> (P <-> (ALL x. Q(x)))" by fast text "Problem 23" lemma "|- (ALL x. P | Q(x)) <-> (P | (ALL x. Q(x)))" by best text "Problem 24" lemma "|- ~(EX x. S(x)&Q(x)) & (ALL x. P(x) --> Q(x)|R(x)) & ~(EX x. P(x)) --> (EX x. Q(x)) & (ALL x. Q(x)|R(x) --> S(x)) --> (EX x. P(x)&R(x))" by (tactic "pc_tac LK_pack 1") text "Problem 25" lemma "|- (EX x. P(x)) & (ALL x. L(x) --> ~ (M(x) & R(x))) & (ALL x. P(x) --> (M(x) & L(x))) & ((ALL x. P(x)-->Q(x)) | (EX x. P(x)&R(x))) --> (EX x. Q(x)&P(x))" by best text "Problem 26" lemma "|- ((EX x. p(x)) <-> (EX x. q(x))) & (ALL x. ALL y. p(x) & q(y) --> (r(x) <-> s(y))) --> ((ALL x. p(x)-->r(x)) <-> (ALL x. q(x)-->s(x)))" by (tactic "pc_tac LK_pack 1") text "Problem 27" lemma "|- (EX x. P(x) & ~Q(x)) & (ALL x. P(x) --> R(x)) & (ALL x. M(x) & L(x) --> P(x)) & ((EX x. R(x) & ~ Q(x)) --> (ALL x. L(x) --> ~ R(x))) --> (ALL x. M(x) --> ~L(x))" by (tactic "pc_tac LK_pack 1") text "Problem 28. AMENDED" lemma "|- (ALL x. P(x) --> (ALL x. Q(x))) & ((ALL x. Q(x)|R(x)) --> (EX x. Q(x)&S(x))) & ((EX x. S(x)) --> (ALL x. L(x) --> M(x))) --> (ALL x. P(x) & L(x) --> M(x))" by (tactic "pc_tac LK_pack 1") text "Problem 29. Essentially the same as Principia Mathematica *11.71" lemma "|- (EX x. P(x)) & (EX y. Q(y)) --> ((ALL x. P(x)-->R(x)) & (ALL y. Q(y)-->S(y)) <-> (ALL x y. P(x) & Q(y) --> R(x) & S(y)))" by (tactic "pc_tac LK_pack 1") text "Problem 30" lemma "|- (ALL x. P(x) | Q(x) --> ~ R(x)) & (ALL x. (Q(x) --> ~ S(x)) --> P(x) & R(x)) --> (ALL x. S(x))" by fast text "Problem 31" lemma "|- ~(EX x. P(x) & (Q(x) | R(x))) & (EX x. L(x) & P(x)) & (ALL x. ~ R(x) --> M(x)) --> (EX x. L(x) & M(x))" by fast text "Problem 32" lemma "|- (ALL x. P(x) & (Q(x)|R(x))-->S(x)) & (ALL x. S(x) & R(x) --> L(x)) & (ALL x. M(x) --> R(x)) --> (ALL x. P(x) & M(x) --> L(x))" by best text "Problem 33" lemma "|- (ALL x. P(a) & (P(x)-->P(b))-->P(c)) <-> (ALL x. (~P(a) | P(x) | P(c)) & (~P(a) | ~P(b) | P(c)))" by fast text "Problem 34 AMENDED (TWICE!!)" (*Andrews's challenge*) lemma "|- ((EX x. ALL y. p(x) <-> p(y)) <-> ((EX x. q(x)) <-> (ALL y. p(y)))) <-> ((EX x. ALL y. q(x) <-> q(y)) <-> ((EX x. p(x)) <-> (ALL y. q(y))))" by best_dup text "Problem 35" lemma "|- EX x y. P(x,y) --> (ALL u v. P(u,v))" by best_dup text "Problem 36" lemma "|- (ALL x. EX y. J(x,y)) & (ALL x. EX y. G(x,y)) & (ALL x y. J(x,y) | G(x,y) --> (ALL z. J(y,z) | G(y,z) --> H(x,z))) --> (ALL x. EX y. H(x,y))" by fast text "Problem 37" lemma "|- (ALL z. EX w. ALL x. EX y. (P(x,z)-->P(y,w)) & P(y,z) & (P(y,w) --> (EX u. Q(u,w)))) & (ALL x z. ~P(x,z) --> (EX y. Q(y,z))) & ((EX x y. Q(x,y)) --> (ALL x. R(x,x))) --> (ALL x. EX y. R(x,y))" by (tactic "pc_tac LK_pack 1") text "Problem 38" lemma "|- (ALL x. p(a) & (p(x) --> (EX y. p(y) & r(x,y))) --> (EX z. EX w. p(z) & r(x,w) & r(w,z))) <-> (ALL x. (~p(a) | p(x) | (EX z. EX w. p(z) & r(x,w) & r(w,z))) & (~p(a) | ~(EX y. p(y) & r(x,y)) | (EX z. EX w. p(z) & r(x,w) & r(w,z))))" by (tactic "pc_tac LK_pack 1") text "Problem 39" lemma "|- ~ (EX x. ALL y. F(y,x) <-> ~F(y,y))" by fast text "Problem 40. AMENDED" lemma "|- (EX y. ALL x. F(x,y) <-> F(x,x)) --> ~(ALL x. EX y. ALL z. F(z,y) <-> ~ F(z,x))" by fast text "Problem 41" lemma "|- (ALL z. EX y. ALL x. f(x,y) <-> f(x,z) & ~ f(x,x)) --> ~ (EX z. ALL x. f(x,z))" by fast text "Problem 42" lemma "|- ~ (EX y. ALL x. p(x,y) <-> ~ (EX z. p(x,z) & p(z,x)))" oops text "Problem 43" lemma "|- (ALL x. ALL y. q(x,y) <-> (ALL z. p(z,x) <-> p(z,y))) --> (ALL x. (ALL y. q(x,y) <-> q(y,x)))" oops text "Problem 44" lemma "|- (ALL x. f(x) --> (EX y. g(y) & h(x,y) & (EX y. g(y) & ~ h(x,y)))) & (EX x. j(x) & (ALL y. g(y) --> h(x,y))) --> (EX x. j(x) & ~f(x))" by fast text "Problem 45" lemma "|- (ALL x. f(x) & (ALL y. g(y) & h(x,y) --> j(x,y)) --> (ALL y. g(y) & h(x,y) --> k(y))) & ~ (EX y. l(y) & k(y)) & (EX x. f(x) & (ALL y. h(x,y) --> l(y)) & (ALL y. g(y) & h(x,y) --> j(x,y))) --> (EX x. f(x) & ~ (EX y. g(y) & h(x,y)))" by best text "Problems (mainly) involving equality or functions" text "Problem 48" lemma "|- (a=b | c=d) & (a=c | b=d) --> a=d | b=c" by (tactic {* fast_tac (LK_pack add_safes @{thms subst}) 1 *}) text "Problem 50" lemma "|- (ALL x. P(a,x) | (ALL y. P(x,y))) --> (EX x. ALL y. P(x,y))" by best_dup text "Problem 51" lemma "|- (EX z w. ALL x y. P(x,y) <-> (x=z & y=w)) --> (EX z. ALL x. EX w. (ALL y. P(x,y) <-> y=w) <-> x=z)" by (tactic {* fast_tac (LK_pack add_safes @{thms subst}) 1 *}) text "Problem 52" (*Almost the same as 51. *) lemma "|- (EX z w. ALL x y. P(x,y) <-> (x=z & y=w)) --> (EX w. ALL y. EX z. (ALL x. P(x,y) <-> x=z) <-> y=w)" by (tactic {* fast_tac (LK_pack add_safes @{thms subst}) 1 *}) text "Problem 56" lemma "|- (ALL x.(EX y. P(y) & x=f(y)) --> P(x)) <-> (ALL x. P(x) --> P(f(x)))" by (tactic {* best_tac (LK_pack add_unsafes [@{thm symL}, @{thm subst}]) 1 *}) (*requires tricker to orient the equality properly*) text "Problem 57" lemma "|- P(f(a,b), f(b,c)) & P(f(b,c), f(a,c)) & (ALL x y z. P(x,y) & P(y,z) --> P(x,z)) --> P(f(a,b), f(a,c))" by fast text "Problem 58!" lemma "|- (ALL x y. f(x)=g(y)) --> (ALL x y. f(f(x))=f(g(y)))" by (tactic {* fast_tac (LK_pack add_safes @{thms subst}) 1 *}) text "Problem 59" (*Unification works poorly here -- the abstraction %sobj prevents efficient operation of the occurs check*) lemma "|- (ALL x. P(x) <-> ~P(f(x))) --> (EX x. P(x) & ~P(f(x)))" by best_dup text "Problem 60" lemma "|- ALL x. P(x,f(x)) <-> (EX y. (ALL z. P(z,y) --> P(z,f(x))) & P(x,y))" by fast text "Problem 62 as corrected in JAR 18 (1997), page 135" lemma "|- (ALL x. p(a) & (p(x) --> p(f(x))) --> p(f(f(x)))) <-> (ALL x. (~p(a) | p(x) | p(f(f(x)))) & (~p(a) | ~p(f(x)) | p(f(f(x)))))" by fast (*18 June 92: loaded in 372 secs*) (*19 June 92: loaded in 166 secs except #34, using repeat_goal_tac*) (*29 June 92: loaded in 370 secs*) (*18 September 2005: loaded in 1.809 secs*) end
lemma
|- (∀x. P(x) ∧ Q(x)) <-> (∀x. P(x)) ∧ (∀x. Q(x))
lemma
|- (∃x. P --> Q(x)) <-> P --> (∃x. Q(x))
lemma
|- (∃x. P(x) --> Q) <-> (∀x. P(x)) --> Q
lemma
|- (∀x. P(x)) ∨ Q <-> (∀x. P(x) ∨ Q)
lemma
|- (∀x. P(x) --> P(f(x))) ∧ P(d) --> P(f(f(f(d))))
lemma
|- ∃x. P(x) --> P(a) ∧ P(b)
lemma
|- ∃z. P(z) --> (∀x. P(x))
lemma
|- ∃y. ∀x. P(y) --> P(x)
lemma
|- ∃x. ∀y z. (P(y) --> Q(z)) --> P(x) --> Q(x)
lemma
|- (∀x y. ∃z. ∀w. P(x) ∧ Q(y) --> R(z) ∧ S(w)) -->
(∃x y. P(x) ∧ Q(y)) --> (∃z. R(z))
lemma
|- (∃x. P --> Q(x)) ∧ (∃x. Q(x) --> P) --> (∃x. P <-> Q(x))
lemma
|- (∀x. P <-> Q(x)) --> P <-> (∀x. Q(x))
lemma
|- (∀x. P ∨ Q(x)) <-> P ∨ (∀x. Q(x))
lemma
|- ¬ (∃x. S(x) ∧ Q(x)) ∧ (∀x. P(x) --> Q(x) ∨ R(x)) ∧ ¬ (∃x. P(x)) -->
(∃x. Q(x)) ∧ (∀x. Q(x) ∨ R(x) --> S(x)) --> (∃x. P(x) ∧ R(x))
lemma
|- (∃x. P(x)) ∧
(∀x. L(x) --> ¬ (M(x) ∧ R(x))) ∧
(∀x. P(x) --> M(x) ∧ L(x)) ∧ ((∀x. P(x) --> Q(x)) ∨ (∃x. P(x) ∧ R(x))) -->
(∃x. Q(x) ∧ P(x))
lemma
|- ((∃x. p(x)) <-> (∃x. q(x))) ∧ (∀x y. p(x) ∧ q(y) --> r(x) <-> s(y)) -->
(∀x. p(x) --> r(x)) <-> (∀x. q(x) --> s(x))
lemma
|- (∃x. P(x) ∧ ¬ Q(x)) ∧
(∀x. P(x) --> R(x)) ∧
(∀x. M(x) ∧ L(x) --> P(x)) ∧
((∃x. R(x) ∧ ¬ Q(x)) --> (∀x. L(x) --> ¬ R(x))) -->
(∀x. M(x) --> ¬ L(x))
lemma
|- (∀x. P(x) --> (∀x. Q(x))) ∧
((∀x. Q(x) ∨ R(x)) --> (∃x. Q(x) ∧ S(x))) ∧
((∃x. S(x)) --> (∀x. L(x) --> M(x))) -->
(∀x. P(x) ∧ L(x) --> M(x))
lemma
|- (∃x. P(x)) ∧ (∃y. Q(y)) -->
(∀x. P(x) --> R(x)) ∧ (∀y. Q(y) --> S(y)) <->
(∀x y. P(x) ∧ Q(y) --> R(x) ∧ S(y))
lemma
|- (∀x. P(x) ∨ Q(x) --> ¬ R(x)) ∧ (∀x. (Q(x) --> ¬ S(x)) --> P(x) ∧ R(x)) -->
(∀x. S(x))
lemma
|- ¬ (∃x. P(x) ∧ (Q(x) ∨ R(x))) ∧ (∃x. L(x) ∧ P(x)) ∧ (∀x. ¬ R(x) --> M(x)) -->
(∃x. L(x) ∧ M(x))
lemma
|- (∀x. P(x) ∧ (Q(x) ∨ R(x)) --> S(x)) ∧
(∀x. S(x) ∧ R(x) --> L(x)) ∧ (∀x. M(x) --> R(x)) -->
(∀x. P(x) ∧ M(x) --> L(x))
lemma
|- (∀x. P(a) ∧ (P(x) --> P(b)) --> P(c)) <->
(∀x. (¬ P(a) ∨ P(x) ∨ P(c)) ∧ (¬ P(a) ∨ ¬ P(b) ∨ P(c)))
lemma
|- ((∃x. ∀y. p(x) <-> p(y)) <-> (∃x. q(x)) <-> (∀y. p(y))) <->
(∃x. ∀y. q(x) <-> q(y)) <-> (∃x. p(x)) <-> (∀y. q(y))
lemma
|- ∃x y. P(x, y) --> (∀u v. P(u, v))
lemma
|- (∀x. ∃y. J(x, y)) ∧
(∀x. ∃y. G(x, y)) ∧
(∀x y. J(x, y) ∨ G(x, y) --> (∀z. J(y, z) ∨ G(y, z) --> H(x, z))) -->
(∀x. ∃y. H(x, y))
lemma
|- (∀z. ∃w. ∀x. ∃y. (P(x, z) --> P(y, w)) ∧
P(y, z) ∧ (P(y, w) --> (∃u. Q(u, w)))) ∧
(∀x z. ¬ P(x, z) --> (∃y. Q(y, z))) ∧
((∃x y. Q(x, y)) --> (∀x. R(x, x))) -->
(∀x. ∃y. R(x, y))
lemma
|- (∀x. p(a) ∧ (p(x) --> (∃y. p(y) ∧ r(x, y))) -->
(∃z w. p(z) ∧ r(x, w) ∧ r(w, z))) <->
(∀x. (¬ p(a) ∨ p(x) ∨ (∃z w. p(z) ∧ r(x, w) ∧ r(w, z))) ∧
(¬ p(a) ∨ ¬ (∃y. p(y) ∧ r(x, y)) ∨ (∃z w. p(z) ∧ r(x, w) ∧ r(w, z))))
lemma
|- ¬ (∃x. ∀y. F(y, x) <-> ¬ F(y, y))
lemma
|- (∃y. ∀x. F(x, y) <-> F(x, x)) --> ¬ (∀x. ∃y. ∀z. F(z, y) <-> ¬ F(z, x))
lemma
|- (∀z. ∃y. ∀x. f(x, y) <-> f(x, z) ∧ ¬ f(x, x)) --> ¬ (∃z. ∀x. f(x, z))
lemma
|- (∀x. f(x) --> (∃y. g(y) ∧ h(x, y) ∧ (∃y. g(y) ∧ ¬ h(x, y)))) ∧
(∃x. j(x) ∧ (∀y. g(y) --> h(x, y))) -->
(∃x. j(x) ∧ ¬ f(x))
lemma
|- (∀x. f(x) ∧ (∀y. g(y) ∧ h(x, y) --> j(x, y)) -->
(∀y. g(y) ∧ h(x, y) --> k(y))) ∧
¬ (∃y. l(y) ∧ k(y)) ∧
(∃x. f(x) ∧ (∀y. h(x, y) --> l(y)) ∧ (∀y. g(y) ∧ h(x, y) --> j(x, y))) -->
(∃x. f(x) ∧ ¬ (∃y. g(y) ∧ h(x, y)))
lemma
|- (a = b ∨ c = d) ∧ (a = c ∨ b = d) --> a = d ∨ b = c
lemma
|- (∀x. P(a, x) ∨ (∀y. P(x, y))) --> (∃x. ∀y. P(x, y))
lemma
|- (∃z w. ∀x y. P(x, y) <-> x = z ∧ y = w) -->
(∃z. ∀x. ∃w. (∀y. P(x, y) <-> y = w) <-> x = z)
lemma
|- (∃z w. ∀x y. P(x, y) <-> x = z ∧ y = w) -->
(∃w. ∀y. ∃z. (∀x. P(x, y) <-> x = z) <-> y = w)
lemma
|- (∀x. (∃y. P(y) ∧ x = f(y)) --> P(x)) <-> (∀x. P(x) --> P(f(x)))
lemma
|- P(f(a, b), f(b, c)) ∧
P(f(b, c), f(a, c)) ∧ (∀x y z. P(x, y) ∧ P(y, z) --> P(x, z)) -->
P(f(a, b), f(a, c))
lemma
|- (∀x y. f(x) = g(y)) --> (∀x y. f(f(x)) = f(g(y)))
lemma
|- (∀x. P(x) <-> ¬ P(f(x))) --> (∃x. P(x) ∧ ¬ P(f(x)))
lemma
|- ∀x. P(x, f(x)) <-> (∃y. (∀z. P(z, y) --> P(z, f(x))) ∧ P(x, y))
lemma
|- (∀x. p(a) ∧ (p(x) --> p(f(x))) --> p(f(f(x)))) <->
(∀x. (¬ p(a) ∨ p(x) ∨ p(f(f(x)))) ∧ (¬ p(a) ∨ ¬ p(f(x)) ∨ p(f(f(x)))))