--- a/src/FOL/ex/Classical.thy Thu Jul 23 14:20:51 2015 +0200
+++ b/src/FOL/ex/Classical.thy Thu Jul 23 14:25:05 2015 +0200
@@ -3,14 +3,14 @@
Copyright 1994 University of Cambridge
*)
-section{*Classical Predicate Calculus Problems*}
+section\<open>Classical Predicate Calculus Problems\<close>
theory Classical imports FOL begin
lemma "(P --> Q | R) --> (P-->Q) | (P-->R)"
by blast
-text{*If and only if*}
+text\<open>If and only if\<close>
lemma "(P<->Q) <-> (Q<->P)"
by blast
@@ -19,7 +19,7 @@
by blast
-text{*Sample problems from
+text\<open>Sample problems from
F. J. Pelletier,
Seventy-Five Problems for Testing Automatic Theorem Provers,
J. Automated Reasoning 2 (1986), 191-216.
@@ -27,79 +27,79 @@
The hardest problems -- judging by experience with several theorem provers,
including matrix ones -- are 34 and 43.
-*}
+\<close>
-subsection{*Pelletier's examples*}
+subsection\<open>Pelletier's examples\<close>
-text{*1*}
+text\<open>1\<close>
lemma "(P-->Q) <-> (~Q --> ~P)"
by blast
-text{*2*}
+text\<open>2\<close>
lemma "~ ~ P <-> P"
by blast
-text{*3*}
+text\<open>3\<close>
lemma "~(P-->Q) --> (Q-->P)"
by blast
-text{*4*}
+text\<open>4\<close>
lemma "(~P-->Q) <-> (~Q --> P)"
by blast
-text{*5*}
+text\<open>5\<close>
lemma "((P|Q)-->(P|R)) --> (P|(Q-->R))"
by blast
-text{*6*}
+text\<open>6\<close>
lemma "P | ~ P"
by blast
-text{*7*}
+text\<open>7\<close>
lemma "P | ~ ~ ~ P"
by blast
-text{*8. Peirce's law*}
+text\<open>8. Peirce's law\<close>
lemma "((P-->Q) --> P) --> P"
by blast
-text{*9*}
+text\<open>9\<close>
lemma "((P|Q) & (~P|Q) & (P| ~Q)) --> ~ (~P | ~Q)"
by blast
-text{*10*}
+text\<open>10\<close>
lemma "(Q-->R) & (R-->P&Q) & (P-->Q|R) --> (P<->Q)"
by blast
-text{*11. Proved in each direction (incorrectly, says Pelletier!!) *}
+text\<open>11. Proved in each direction (incorrectly, says Pelletier!!)\<close>
lemma "P<->P"
by blast
-text{*12. "Dijkstra's law"*}
+text\<open>12. "Dijkstra's law"\<close>
lemma "((P <-> Q) <-> R) <-> (P <-> (Q <-> R))"
by blast
-text{*13. Distributive law*}
+text\<open>13. Distributive law\<close>
lemma "P | (Q & R) <-> (P | Q) & (P | R)"
by blast
-text{*14*}
+text\<open>14\<close>
lemma "(P <-> Q) <-> ((Q | ~P) & (~Q|P))"
by blast
-text{*15*}
+text\<open>15\<close>
lemma "(P --> Q) <-> (~P | Q)"
by blast
-text{*16*}
+text\<open>16\<close>
lemma "(P-->Q) | (Q-->P)"
by blast
-text{*17*}
+text\<open>17\<close>
lemma "((P & (Q-->R))-->S) <-> ((~P | Q | S) & (~P | ~R | S))"
by blast
-subsection{*Classical Logic: examples with quantifiers*}
+subsection\<open>Classical Logic: examples with quantifiers\<close>
lemma "(\<forall>x. P(x) & Q(x)) <-> (\<forall>x. P(x)) & (\<forall>x. Q(x))"
by blast
@@ -113,23 +113,23 @@
lemma "(\<forall>x. P(x)) | Q <-> (\<forall>x. P(x) | Q)"
by blast
-text{*Discussed in Avron, Gentzen-Type Systems, Resolution and Tableaux,
- JAR 10 (265-281), 1993. Proof is trivial!*}
+text\<open>Discussed in Avron, Gentzen-Type Systems, Resolution and Tableaux,
+ JAR 10 (265-281), 1993. Proof is trivial!\<close>
lemma "~((\<exists>x.~P(x)) & ((\<exists>x. P(x)) | (\<exists>x. P(x) & Q(x))) & ~ (\<exists>x. P(x)))"
by blast
-subsection{*Problems requiring quantifier duplication*}
+subsection\<open>Problems requiring quantifier duplication\<close>
-text{*Theorem B of Peter Andrews, Theorem Proving via General Matings,
- JACM 28 (1981).*}
+text\<open>Theorem B of Peter Andrews, Theorem Proving via General Matings,
+ JACM 28 (1981).\<close>
lemma "(\<exists>x. \<forall>y. P(x) <-> P(y)) --> ((\<exists>x. P(x)) <-> (\<forall>y. P(y)))"
by blast
-text{*Needs multiple instantiation of ALL.*}
+text\<open>Needs multiple instantiation of ALL.\<close>
lemma "(\<forall>x. P(x)-->P(f(x))) & P(d)-->P(f(f(f(d))))"
by blast
-text{*Needs double instantiation of the quantifier*}
+text\<open>Needs double instantiation of the quantifier\<close>
lemma "\<exists>x. P(x) --> P(a) & P(b)"
by blast
@@ -139,7 +139,7 @@
lemma "\<exists>x. (\<exists>y. P(y)) --> P(x)"
by blast
-text{*V. Lifschitz, What Is the Inverse Method?, JAR 5 (1989), 1--23. NOT PROVED*}
+text\<open>V. Lifschitz, What Is the Inverse Method?, JAR 5 (1989), 1--23. NOT PROVED\<close>
lemma "\<exists>x x'. \<forall>y. \<exists>z z'.
(~P(y,y) | P(x,x) | ~S(z,x)) &
(S(x,y) | ~S(y,z) | Q(z',z')) &
@@ -148,40 +148,40 @@
-subsection{*Hard examples with quantifiers*}
+subsection\<open>Hard examples with quantifiers\<close>
-text{*18*}
+text\<open>18\<close>
lemma "\<exists>y. \<forall>x. P(y)-->P(x)"
by blast
-text{*19*}
+text\<open>19\<close>
lemma "\<exists>x. \<forall>y z. (P(y)-->Q(z)) --> (P(x)-->Q(x))"
by blast
-text{*20*}
+text\<open>20\<close>
lemma "(\<forall>x y. \<exists>z. \<forall>w. (P(x)&Q(y)-->R(z)&S(w)))
--> (\<exists>x y. P(x) & Q(y)) --> (\<exists>z. R(z))"
by blast
-text{*21*}
+text\<open>21\<close>
lemma "(\<exists>x. P-->Q(x)) & (\<exists>x. Q(x)-->P) --> (\<exists>x. P<->Q(x))"
by blast
-text{*22*}
+text\<open>22\<close>
lemma "(\<forall>x. P <-> Q(x)) --> (P <-> (\<forall>x. Q(x)))"
by blast
-text{*23*}
+text\<open>23\<close>
lemma "(\<forall>x. P | Q(x)) <-> (P | (\<forall>x. Q(x)))"
by blast
-text{*24*}
+text\<open>24\<close>
lemma "~(\<exists>x. S(x)&Q(x)) & (\<forall>x. P(x) --> Q(x)|R(x)) &
(~(\<exists>x. P(x)) --> (\<exists>x. Q(x))) & (\<forall>x. Q(x)|R(x) --> S(x))
--> (\<exists>x. P(x)&R(x))"
by blast
-text{*25*}
+text\<open>25\<close>
lemma "(\<exists>x. P(x)) &
(\<forall>x. L(x) --> ~ (M(x) & R(x))) &
(\<forall>x. P(x) --> (M(x) & L(x))) &
@@ -189,13 +189,13 @@
--> (\<exists>x. Q(x)&P(x))"
by blast
-text{*26*}
+text\<open>26\<close>
lemma "((\<exists>x. p(x)) <-> (\<exists>x. q(x))) &
(\<forall>x. \<forall>y. p(x) & q(y) --> (r(x) <-> s(y)))
--> ((\<forall>x. p(x)-->r(x)) <-> (\<forall>x. q(x)-->s(x)))"
by blast
-text{*27*}
+text\<open>27\<close>
lemma "(\<exists>x. P(x) & ~Q(x)) &
(\<forall>x. P(x) --> R(x)) &
(\<forall>x. M(x) & L(x) --> P(x)) &
@@ -203,63 +203,63 @@
--> (\<forall>x. M(x) --> ~L(x))"
by blast
-text{*28. AMENDED*}
+text\<open>28. AMENDED\<close>
lemma "(\<forall>x. P(x) --> (\<forall>x. Q(x))) &
((\<forall>x. Q(x)|R(x)) --> (\<exists>x. Q(x)&S(x))) &
((\<exists>x. S(x)) --> (\<forall>x. L(x) --> M(x)))
--> (\<forall>x. P(x) & L(x) --> M(x))"
by blast
-text{*29. Essentially the same as Principia Mathematica *11.71*}
+text\<open>29. Essentially the same as Principia Mathematica *11.71\<close>
lemma "(\<exists>x. P(x)) & (\<exists>y. Q(y))
--> ((\<forall>x. P(x)-->R(x)) & (\<forall>y. Q(y)-->S(y)) <->
(\<forall>x y. P(x) & Q(y) --> R(x) & S(y)))"
by blast
-text{*30*}
+text\<open>30\<close>
lemma "(\<forall>x. P(x) | Q(x) --> ~ R(x)) &
(\<forall>x. (Q(x) --> ~ S(x)) --> P(x) & R(x))
--> (\<forall>x. S(x))"
by blast
-text{*31*}
+text\<open>31\<close>
lemma "~(\<exists>x. P(x) & (Q(x) | R(x))) &
(\<exists>x. L(x) & P(x)) &
(\<forall>x. ~ R(x) --> M(x))
--> (\<exists>x. L(x) & M(x))"
by blast
-text{*32*}
+text\<open>32\<close>
lemma "(\<forall>x. P(x) & (Q(x)|R(x))-->S(x)) &
(\<forall>x. S(x) & R(x) --> L(x)) &
(\<forall>x. M(x) --> R(x))
--> (\<forall>x. P(x) & M(x) --> L(x))"
by blast
-text{*33*}
+text\<open>33\<close>
lemma "(\<forall>x. P(a) & (P(x)-->P(b))-->P(c)) <->
(\<forall>x. (~P(a) | P(x) | P(c)) & (~P(a) | ~P(b) | P(c)))"
by blast
-text{*34 AMENDED (TWICE!!). Andrews's challenge*}
+text\<open>34 AMENDED (TWICE!!). Andrews's challenge\<close>
lemma "((\<exists>x. \<forall>y. p(x) <-> p(y)) <->
((\<exists>x. q(x)) <-> (\<forall>y. p(y)))) <->
((\<exists>x. \<forall>y. q(x) <-> q(y)) <->
((\<exists>x. p(x)) <-> (\<forall>y. q(y))))"
by blast
-text{*35*}
+text\<open>35\<close>
lemma "\<exists>x y. P(x,y) --> (\<forall>u v. P(u,v))"
by blast
-text{*36*}
+text\<open>36\<close>
lemma "(\<forall>x. \<exists>y. J(x,y)) &
(\<forall>x. \<exists>y. G(x,y)) &
(\<forall>x y. J(x,y) | G(x,y) --> (\<forall>z. J(y,z) | G(y,z) --> H(x,z)))
--> (\<forall>x. \<exists>y. H(x,y))"
by blast
-text{*37*}
+text\<open>37\<close>
lemma "(\<forall>z. \<exists>w. \<forall>x. \<exists>y.
(P(x,z)-->P(y,w)) & P(y,z) & (P(y,w) --> (\<exists>u. Q(u,w)))) &
(\<forall>x z. ~P(x,z) --> (\<exists>y. Q(y,z))) &
@@ -267,7 +267,7 @@
--> (\<forall>x. \<exists>y. R(x,y))"
by blast
-text{*38*}
+text\<open>38\<close>
lemma "(\<forall>x. p(a) & (p(x) --> (\<exists>y. p(y) & r(x,y))) -->
(\<exists>z. \<exists>w. p(z) & r(x,w) & r(w,z))) <->
(\<forall>x. (~p(a) | p(x) | (\<exists>z. \<exists>w. p(z) & r(x,w) & r(w,z))) &
@@ -275,25 +275,25 @@
(\<exists>z. \<exists>w. p(z) & r(x,w) & r(w,z))))"
by blast
-text{*39*}
+text\<open>39\<close>
lemma "~ (\<exists>x. \<forall>y. F(y,x) <-> ~F(y,y))"
by blast
-text{*40. AMENDED*}
+text\<open>40. AMENDED\<close>
lemma "(\<exists>y. \<forall>x. F(x,y) <-> F(x,x)) -->
~(\<forall>x. \<exists>y. \<forall>z. F(z,y) <-> ~ F(z,x))"
by blast
-text{*41*}
+text\<open>41\<close>
lemma "(\<forall>z. \<exists>y. \<forall>x. f(x,y) <-> f(x,z) & ~ f(x,x))
--> ~ (\<exists>z. \<forall>x. f(x,z))"
by blast
-text{*42*}
+text\<open>42\<close>
lemma "~ (\<exists>y. \<forall>x. p(x,y) <-> ~ (\<exists>z. p(x,z) & p(z,x)))"
by blast
-text{*43*}
+text\<open>43\<close>
lemma "(\<forall>x. \<forall>y. q(x,y) <-> (\<forall>z. p(z,x) <-> p(z,y)))
--> (\<forall>x. \<forall>y. q(x,y) <-> q(y,x))"
by blast
@@ -302,13 +302,13 @@
Deepen_tac alone requires 253 secs. Or
by (mini_tac @{context} 1 THEN Deepen_tac 5 1) *)
-text{*44*}
+text\<open>44\<close>
lemma "(\<forall>x. f(x) --> (\<exists>y. g(y) & h(x,y) & (\<exists>y. g(y) & ~ h(x,y)))) &
(\<exists>x. j(x) & (\<forall>y. g(y) --> h(x,y)))
--> (\<exists>x. j(x) & ~f(x))"
by blast
-text{*45*}
+text\<open>45\<close>
lemma "(\<forall>x. f(x) & (\<forall>y. g(y) & h(x,y) --> j(x,y))
--> (\<forall>y. g(y) & h(x,y) --> k(y))) &
~ (\<exists>y. l(y) & k(y)) &
@@ -318,7 +318,7 @@
by blast
-text{*46*}
+text\<open>46\<close>
lemma "(\<forall>x. f(x) & (\<forall>y. f(y) & h(y,x) --> g(y)) --> g(x)) &
((\<exists>x. f(x) & ~g(x)) -->
(\<exists>x. f(x) & ~g(x) & (\<forall>y. f(y) & ~g(y) --> j(x,y)))) &
@@ -327,42 +327,42 @@
by blast
-subsection{*Problems (mainly) involving equality or functions*}
+subsection\<open>Problems (mainly) involving equality or functions\<close>
-text{*48*}
+text\<open>48\<close>
lemma "(a=b | c=d) & (a=c | b=d) --> a=d | b=c"
by blast
-text{*49 NOT PROVED AUTOMATICALLY. Hard because it involves substitution
+text\<open>49 NOT PROVED AUTOMATICALLY. Hard because it involves substitution
for Vars
- the type constraint ensures that x,y,z have the same type as a,b,u. *}
+ the type constraint ensures that x,y,z have the same type as a,b,u.\<close>
lemma "(\<exists>x y::'a. \<forall>z. z=x | z=y) & P(a) & P(b) & a~=b
--> (\<forall>u::'a. P(u))"
apply safe
apply (rule_tac x = a in allE, assumption)
apply (rule_tac x = b in allE, assumption, fast)
- --{*blast's treatment of equality can't do it*}
+ --\<open>blast's treatment of equality can't do it\<close>
done
-text{*50. (What has this to do with equality?) *}
+text\<open>50. (What has this to do with equality?)\<close>
lemma "(\<forall>x. P(a,x) | (\<forall>y. P(x,y))) --> (\<exists>x. \<forall>y. P(x,y))"
by blast
-text{*51*}
+text\<open>51\<close>
lemma "(\<exists>z w. \<forall>x y. P(x,y) <-> (x=z & y=w)) -->
(\<exists>z. \<forall>x. \<exists>w. (\<forall>y. P(x,y) <-> y=w) <-> x=z)"
by blast
-text{*52*}
-text{*Almost the same as 51. *}
+text\<open>52\<close>
+text\<open>Almost the same as 51.\<close>
lemma "(\<exists>z w. \<forall>x y. P(x,y) <-> (x=z & y=w)) -->
(\<exists>w. \<forall>y. \<exists>z. (\<forall>x. P(x,y) <-> x=z) <-> y=w)"
by blast
-text{*55*}
+text\<open>55\<close>
-text{*Non-equational version, from Manthey and Bry, CADE-9 (Springer, 1988).
- fast DISCOVERS who killed Agatha. *}
+text\<open>Non-equational version, from Manthey and Bry, CADE-9 (Springer, 1988).
+ fast DISCOVERS who killed Agatha.\<close>
schematic_lemma "lives(agatha) & lives(butler) & lives(charles) &
(killed(agatha,agatha) | killed(butler,agatha) | killed(charles,agatha)) &
(\<forall>x y. killed(x,y) --> hates(x,y) & ~richer(x,y)) &
@@ -372,53 +372,53 @@
(\<forall>x. hates(agatha,x) --> hates(butler,x)) &
(\<forall>x. ~hates(x,agatha) | ~hates(x,butler) | ~hates(x,charles)) -->
killed(?who,agatha)"
-by fast --{*MUCH faster than blast*}
+by fast --\<open>MUCH faster than blast\<close>
-text{*56*}
+text\<open>56\<close>
lemma "(\<forall>x. (\<exists>y. P(y) & x=f(y)) --> P(x)) <-> (\<forall>x. P(x) --> P(f(x)))"
by blast
-text{*57*}
+text\<open>57\<close>
lemma "P(f(a,b), f(b,c)) & P(f(b,c), f(a,c)) &
(\<forall>x y z. P(x,y) & P(y,z) --> P(x,z)) --> P(f(a,b), f(a,c))"
by blast
-text{*58 NOT PROVED AUTOMATICALLY*}
+text\<open>58 NOT PROVED AUTOMATICALLY\<close>
lemma "(\<forall>x y. f(x)=g(y)) --> (\<forall>x y. f(f(x))=f(g(y)))"
by (slow elim: subst_context)
-text{*59*}
+text\<open>59\<close>
lemma "(\<forall>x. P(x) <-> ~P(f(x))) --> (\<exists>x. P(x) & ~P(f(x)))"
by blast
-text{*60*}
+text\<open>60\<close>
lemma "\<forall>x. P(x,f(x)) <-> (\<exists>y. (\<forall>z. P(z,y) --> P(z,f(x))) & P(x,y))"
by blast
-text{*62 as corrected in JAR 18 (1997), page 135*}
+text\<open>62 as corrected in JAR 18 (1997), page 135\<close>
lemma "(\<forall>x. p(a) & (p(x) --> p(f(x))) --> p(f(f(x)))) <->
(\<forall>x. (~p(a) | p(x) | p(f(f(x)))) &
(~p(a) | ~p(f(x)) | p(f(f(x)))))"
by blast
-text{*From Davis, Obvious Logical Inferences, IJCAI-81, 530-531
- fast indeed copes!*}
+text\<open>From Davis, Obvious Logical Inferences, IJCAI-81, 530-531
+ fast indeed copes!\<close>
lemma "(\<forall>x. F(x) & ~G(x) --> (\<exists>y. H(x,y) & J(y))) &
(\<exists>x. K(x) & F(x) & (\<forall>y. H(x,y) --> K(y))) &
(\<forall>x. K(x) --> ~G(x)) --> (\<exists>x. K(x) & J(x))"
by fast
-text{*From Rudnicki, Obvious Inferences, JAR 3 (1987), 383-393.
- It does seem obvious!*}
+text\<open>From Rudnicki, Obvious Inferences, JAR 3 (1987), 383-393.
+ It does seem obvious!\<close>
lemma "(\<forall>x. F(x) & ~G(x) --> (\<exists>y. H(x,y) & J(y))) &
(\<exists>x. K(x) & F(x) & (\<forall>y. H(x,y) --> K(y))) &
(\<forall>x. K(x) --> ~G(x)) --> (\<exists>x. K(x) --> ~G(x))"
by fast
-text{*Halting problem: Formulation of Li Dafa (AAR Newsletter 27, Oct 1994.)
- author U. Egly*}
+text\<open>Halting problem: Formulation of Li Dafa (AAR Newsletter 27, Oct 1994.)
+ author U. Egly\<close>
lemma "((\<exists>x. A(x) & (\<forall>y. C(y) --> (\<forall>z. D(x,y,z)))) -->
(\<exists>w. C(w) & (\<forall>y. C(y) --> (\<forall>z. D(w,y,z)))))
&
@@ -437,10 +437,10 @@
-->
~ (\<exists>x. A(x) & (\<forall>y. C(y) --> (\<forall>z. D(x,y,z))))"
by (blast 12)
- --{*Needed because the search for depths below 12 is very slow*}
+ --\<open>Needed because the search for depths below 12 is very slow\<close>
-text{*Halting problem II: credited to M. Bruschi by Li Dafa in JAR 18(1), p.105*}
+text\<open>Halting problem II: credited to M. Bruschi by Li Dafa in JAR 18(1), p.105\<close>
lemma "((\<exists>x. A(x) & (\<forall>y. C(y) --> (\<forall>z. D(x,y,z)))) -->
(\<exists>w. C(w) & (\<forall>y. C(y) --> (\<forall>z. D(w,y,z)))))
&
@@ -464,12 +464,12 @@
~ (\<exists>x. A(x) & (\<forall>y. C(y) --> (\<forall>z. D(x,y,z))))"
by blast
-text{* Challenge found on info-hol *}
+text\<open>Challenge found on info-hol\<close>
lemma "\<forall>x. \<exists>v w. \<forall>y z. P(x) & Q(y) --> (P(v) | R(w)) & (R(z) --> Q(v))"
by blast
-text{*Attributed to Lewis Carroll by S. G. Pulman. The first or last assumption
-can be deleted.*}
+text\<open>Attributed to Lewis Carroll by S. G. Pulman. The first or last assumption
+can be deleted.\<close>
lemma "(\<forall>x. honest(x) & industrious(x) --> healthy(x)) &
~ (\<exists>x. grocer(x) & healthy(x)) &
(\<forall>x. industrious(x) & grocer(x) --> honest(x)) &