# Theory Hard_Quantifiers

Up to index of Isabelle/Sequents/Sequents-LK

theory Hard_Quantifiers
imports LK
`(*  Title:      Sequents/LK/Hard_Quantifiers.thy    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory    Copyright   1992  University of CambridgeHard 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_Quantifiersimports LKbeginlemma "|- (ALL x. P(x) & Q(x)) <-> (ALL x. P(x))  &  (ALL x. Q(x))"  by fastlemma "|- (EX x. P-->Q(x))  <->  (P --> (EX x. Q(x)))"  by fastlemma "|- (EX x. P(x)-->Q)  <->  (ALL x. P(x)) --> Q"  by fastlemma "|- (ALL x. P(x)) | Q  <->  (ALL x. P(x) | Q)"  by fasttext "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_duplemma "|- EX z. P(z) --> (ALL x. P(x))"  by best_duptext "Hard examples with quantifiers"text "Problem 18"lemma "|- EX y. ALL x. P(y)-->P(x)"  by best_duptext "Problem 19"lemma "|- EX x. ALL y z. (P(y)-->Q(z)) --> (P(x)-->Q(x))"  by best_duptext "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 fasttext "Problem 21"lemma "|- (EX x. P-->Q(x)) & (EX x. Q(x)-->P) --> (EX x. P<->Q(x))"  by best_duptext "Problem 22"lemma "|- (ALL x. P <-> Q(x))  -->  (P <-> (ALL x. Q(x)))"  by fasttext "Problem 23"lemma "|- (ALL x. P | Q(x))  <->  (P | (ALL x. Q(x)))"  by besttext "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 besttext "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 fasttext "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 fasttext "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 besttext "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 fasttext "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_duptext "Problem 35"lemma "|- EX x y. P(x,y) -->  (ALL u v. P(u,v))"  by best_duptext "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 fasttext "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 fasttext "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 fasttext "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 fasttext "Problem 42"lemma "|- ~ (EX y. ALL x. p(x,y) <-> ~ (EX z. p(x,z) & p(z,x)))"  oopstext "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)))"  oopstext "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 fasttext "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 besttext "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_duptext "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 fasttext "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_duptext "Problem 60"lemma "|- ALL x. P(x,f(x)) <-> (EX y. (ALL z. P(z,y) --> P(z,f(x))) & P(x,y))"  by fasttext "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`