(* Title: FOLP/ex/Classical.thy ID: $Id: Classical.thy,v 1.1 2008/03/18 21:19:19 wenzelm Exp $ Author: Lawrence C Paulson, Cambridge University Computer Laboratory Copyright 1993 University of Cambridge Classical First-Order Logic. *) theory Classical imports FOLP begin lemma "?p : (P --> Q | R) --> (P-->Q) | (P-->R)" by (tactic "fast_tac FOLP_cs 1") (*If and only if*) lemma "?p : (P<->Q) <-> (Q<->P)" by (tactic "fast_tac FOLP_cs 1") lemma "?p : ~ (P <-> ~P)" by (tactic "fast_tac FOLP_cs 1") (*Sample problems 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. The hardest problems -- judging by experience with several theorem provers, including matrix ones -- are 34 and 43. *) text "Pelletier's examples" (*1*) lemma "?p : (P-->Q) <-> (~Q --> ~P)" by (tactic "fast_tac FOLP_cs 1") (*2*) lemma "?p : ~ ~ P <-> P" by (tactic "fast_tac FOLP_cs 1") (*3*) lemma "?p : ~(P-->Q) --> (Q-->P)" by (tactic "fast_tac FOLP_cs 1") (*4*) lemma "?p : (~P-->Q) <-> (~Q --> P)" by (tactic "fast_tac FOLP_cs 1") (*5*) lemma "?p : ((P|Q)-->(P|R)) --> (P|(Q-->R))" by (tactic "fast_tac FOLP_cs 1") (*6*) lemma "?p : P | ~ P" by (tactic "fast_tac FOLP_cs 1") (*7*) lemma "?p : P | ~ ~ ~ P" by (tactic "fast_tac FOLP_cs 1") (*8. Peirce's law*) lemma "?p : ((P-->Q) --> P) --> P" by (tactic "fast_tac FOLP_cs 1") (*9*) lemma "?p : ((P|Q) & (~P|Q) & (P| ~Q)) --> ~ (~P | ~Q)" by (tactic "fast_tac FOLP_cs 1") (*10*) lemma "?p : (Q-->R) & (R-->P&Q) & (P-->Q|R) --> (P<->Q)" by (tactic "fast_tac FOLP_cs 1") (*11. Proved in each direction (incorrectly, says Pelletier!!) *) lemma "?p : P<->P" by (tactic "fast_tac FOLP_cs 1") (*12. "Dijkstra's law"*) lemma "?p : ((P <-> Q) <-> R) <-> (P <-> (Q <-> R))" by (tactic "fast_tac FOLP_cs 1") (*13. Distributive law*) lemma "?p : P | (Q & R) <-> (P | Q) & (P | R)" by (tactic "fast_tac FOLP_cs 1") (*14*) lemma "?p : (P <-> Q) <-> ((Q | ~P) & (~Q|P))" by (tactic "fast_tac FOLP_cs 1") (*15*) lemma "?p : (P --> Q) <-> (~P | Q)" by (tactic "fast_tac FOLP_cs 1") (*16*) lemma "?p : (P-->Q) | (Q-->P)" by (tactic "fast_tac FOLP_cs 1") (*17*) lemma "?p : ((P & (Q-->R))-->S) <-> ((~P | Q | S) & (~P | ~R | S))" by (tactic "fast_tac FOLP_cs 1") text "Classical Logic: examples with quantifiers" lemma "?p : (ALL x. P(x) & Q(x)) <-> (ALL x. P(x)) & (ALL x. Q(x))" by (tactic "fast_tac FOLP_cs 1") lemma "?p : (EX x. P-->Q(x)) <-> (P --> (EX x. Q(x)))" by (tactic "fast_tac FOLP_cs 1") lemma "?p : (EX x. P(x)-->Q) <-> (ALL x. P(x)) --> Q" by (tactic "fast_tac FOLP_cs 1") lemma "?p : (ALL x. P(x)) | Q <-> (ALL x. P(x) | Q)" by (tactic "fast_tac FOLP_cs 1") text "Problems requiring quantifier duplication" (*Needs multiple instantiation of ALL.*) lemma "?p : (ALL x. P(x)-->P(f(x))) & P(d)-->P(f(f(f(d))))" by (tactic "best_tac FOLP_dup_cs 1") (*Needs double instantiation of the quantifier*) lemma "?p : EX x. P(x) --> P(a) & P(b)" by (tactic "best_tac FOLP_dup_cs 1") lemma "?p : EX z. P(z) --> (ALL x. P(x))" by (tactic "best_tac FOLP_dup_cs 1") text "Hard examples with quantifiers" text "Problem 18" lemma "?p : EX y. ALL x. P(y)-->P(x)" by (tactic "best_tac FOLP_dup_cs 1") text "Problem 19" lemma "?p : EX x. ALL y z. (P(y)-->Q(z)) --> (P(x)-->Q(x))" by (tactic "best_tac FOLP_dup_cs 1") text "Problem 20" lemma "?p : (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 (tactic "fast_tac FOLP_cs 1") text "Problem 21" lemma "?p : (EX x. P-->Q(x)) & (EX x. Q(x)-->P) --> (EX x. P<->Q(x))"; by (tactic "best_tac FOLP_dup_cs 1") text "Problem 22" lemma "?p : (ALL x. P <-> Q(x)) --> (P <-> (ALL x. Q(x)))" by (tactic "fast_tac FOLP_cs 1") text "Problem 23" lemma "?p : (ALL x. P | Q(x)) <-> (P | (ALL x. Q(x)))" by (tactic "best_tac FOLP_dup_cs 1") text "Problem 24" lemma "?p : ~(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 "fast_tac FOLP_cs 1") text "Problem 25" lemma "?p : (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))" oops text "Problem 26" lemma "?u : ((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 "fast_tac FOLP_cs 1") text "Problem 27" lemma "?p : (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 "fast_tac FOLP_cs 1") text "Problem 28. AMENDED" lemma "?p : (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 "fast_tac FOLP_cs 1") text "Problem 29. Essentially the same as Principia Mathematica *11.71" lemma "?p : (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 "fast_tac FOLP_cs 1") text "Problem 30" lemma "?p : (ALL x. P(x) | Q(x) --> ~ R(x)) & (ALL x. (Q(x) --> ~ S(x)) --> P(x) & R(x)) --> (ALL x. S(x))" by (tactic "fast_tac FOLP_cs 1") text "Problem 31" lemma "?p : ~(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 (tactic "fast_tac FOLP_cs 1") text "Problem 32" lemma "?p : (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 (tactic "best_tac FOLP_dup_cs 1") text "Problem 33" lemma "?p : (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 (tactic "best_tac FOLP_dup_cs 1") text "Problem 35" lemma "?p : EX x y. P(x,y) --> (ALL u v. P(u,v))" by (tactic "best_tac FOLP_dup_cs 1") text "Problem 36" lemma "?p : (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 (tactic "fast_tac FOLP_cs 1") text "Problem 37" lemma "?p : (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 "fast_tac FOLP_cs 1") text "Problem 39" lemma "?p : ~ (EX x. ALL y. F(y,x) <-> ~F(y,y))" by (tactic "fast_tac FOLP_cs 1") text "Problem 40. AMENDED" lemma "?p : (EX y. ALL x. F(x,y) <-> F(x,x)) --> ~(ALL x. EX y. ALL z. F(z,y) <-> ~ F(z,x))" by (tactic "fast_tac FOLP_cs 1") text "Problem 41" lemma "?p : (ALL z. EX y. ALL x. f(x,y) <-> f(x,z) & ~ f(x,x)) --> ~ (EX z. ALL x. f(x,z))" by (tactic "best_tac FOLP_dup_cs 1") text "Problem 44" lemma "?p : (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 (tactic "fast_tac FOLP_cs 1") text "Problems (mainly) involving equality or functions" text "Problem 48" lemma "?p : (a=b | c=d) & (a=c | b=d) --> a=d | b=c" by (tactic "fast_tac FOLP_cs 1") text "Problem 50" (*What has this to do with equality?*) lemma "?p : (ALL x. P(a,x) | (ALL y. P(x,y))) --> (EX x. ALL y. P(x,y))" by (tactic "best_tac FOLP_dup_cs 1") text "Problem 56" lemma "?p : (ALL x. (EX y. P(y) & x=f(y)) --> P(x)) <-> (ALL x. P(x) --> P(f(x)))" by (tactic "fast_tac FOLP_cs 1") text "Problem 57" lemma "?p : 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 (tactic "fast_tac FOLP_cs 1") text "Problem 58 NOT PROVED AUTOMATICALLY" lemma notes f_cong = subst_context [where t = f] shows "?p : (ALL x y. f(x)=g(y)) --> (ALL x y. f(f(x))=f(g(y)))" by (tactic {* fast_tac (FOLP_cs addSIs [@{thm f_cong}]) 1 *}) text "Problem 59" lemma "?p : (ALL x. P(x) <-> ~P(f(x))) --> (EX x. P(x) & ~P(f(x)))" by (tactic "best_tac FOLP_dup_cs 1") text "Problem 60" lemma "?p : ALL x. P(x,f(x)) <-> (EX y. (ALL z. P(z,y) --> P(z,f(x))) & P(x,y))" by (tactic "fast_tac FOLP_cs 1") end
lemma
(P --> Q | R) --> (P --> Q) | (P --> R)
lemma
(P <-> Q) <-> Q <-> P
lemma
~ (P <-> ~ P)
lemma
(P --> Q) <-> ~ Q --> ~ P
lemma
~ ~ P <-> P
lemma
~ (P --> Q) --> Q --> P
lemma
(~ P --> Q) <-> ~ Q --> P
lemma
(P | Q --> P | R) --> P | (Q --> R)
lemma
P | ~ P
lemma
P | ~ ~ ~ P
lemma
((P --> Q) --> P) --> P
lemma
(P | Q) & (~ P | Q) & (P | ~ Q) --> ~ (~ P | ~ Q)
lemma
(Q --> R) & (R --> P & Q) & (P --> Q | R) --> P <-> Q
lemma
P <-> P
lemma
((P <-> Q) <-> R) <-> P <-> Q <-> R
lemma
P | Q & R <-> (P | Q) & (P | R)
lemma
(P <-> Q) <-> (Q | ~ P) & (~ Q | P)
lemma
(P --> Q) <-> ~ P | Q
lemma
(P --> Q) | (Q --> P)
lemma
(P & (Q --> R) --> S) <-> (~ P | Q | S) & (~ P | ~ R | S)
lemma
(ALL x. P(x) & Q(x)) <-> (ALL x. P(x)) & (ALL x. Q(x))
lemma
(EX x. P --> Q(x)) <-> P --> (EX x. Q(x))
lemma
(EX x. P(x) --> Q) <-> (ALL x. P(x)) --> Q
lemma
(ALL x. P(x)) | Q <-> (ALL x. P(x) | Q)
lemma
(ALL x. P(x) --> P(f(x))) & P(d) --> P(f(f(f(d))))
lemma
EX x. P(x) --> P(a) & P(b)
lemma
EX z. P(z) --> (ALL x. P(x))
lemma
EX y. ALL x. P(y) --> P(x)
lemma
EX x. ALL y z. (P(y) --> Q(z)) --> P(x) --> Q(x)
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))
lemma
(EX x. P --> Q(x)) & (EX x. Q(x) --> P) --> (EX x. P <-> Q(x))
lemma
(ALL x. P <-> Q(x)) --> P <-> (ALL x. Q(x))
lemma
(ALL x. P | Q(x)) <-> P | (ALL x. Q(x))
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))
lemma
((EX x. p(x)) <-> (EX x. q(x))) & (ALL x y. p(x) & q(y) --> r(x) <-> s(y)) -->
(ALL x. p(x) --> r(x)) <-> (ALL x. q(x) --> s(x))
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))
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))
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))
lemma
(ALL x. P(x) | Q(x) --> ~ R(x)) & (ALL x. (Q(x) --> ~ S(x)) --> P(x) & R(x)) -->
(ALL x. S(x))
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))
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))
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)))
lemma
EX x y. P(x, y) --> (ALL u v. P(u, v))
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))
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))
lemma
~ (EX x. ALL y. F(y, x) <-> ~ F(y, y))
lemma
(EX y. ALL x. F(x, y) <-> F(x, x)) -->
~ (ALL x. EX y. ALL z. F(z, y) <-> ~ F(z, x))
lemma
(ALL z. EX y. ALL x. f(x, y) <-> f(x, z) & ~ f(x, x)) -->
~ (EX z. ALL x. f(x, z))
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))
lemma
(a = b | c = d) & (a = c | b = d) --> a = d | b = c
lemma
(ALL x. P(a, x) | (ALL y. P(x, y))) --> (EX x. ALL y. P(x, y))
lemma
(ALL x. (EX y. P(y) & x = f(y)) --> P(x)) <-> (ALL x. P(x) --> P(f(x)))
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))
lemma
(ALL x y. f(x) = g(y)) --> (ALL x y. f(f(x)) = f(g(y)))
lemma
(ALL x. P(x) <-> ~ P(f(x))) --> (EX x. P(x) & ~ P(f(x)))
lemma
ALL x. P(x, f(x)) <-> (EX y. (ALL z. P(z, y) --> P(z, f(x))) & P(x, y))