diff -r 805de10ca485 -r ad6106d7b4bb src/HOL/ex/set.thy --- a/src/HOL/ex/set.thy Thu Mar 14 16:00:29 2002 +0100 +++ b/src/HOL/ex/set.thy Thu Mar 14 16:48:34 2002 +0100 @@ -1,4 +1,177 @@ +(* Title: HOL/ex/set.thy + ID: \$Id\$ + Author: Tobias Nipkow and Lawrence C Paulson + Copyright 1991 University of Cambridge + +Set Theory examples: Cantor's Theorem, Schroeder-Berstein Theorem, etc. +*) theory set = Main: +text{*These two are cited in Benzmueller and Kohlhase's system description +of LEO, CADE-15, 1998 (pages 139-143) as theorems LEO could not prove.*} + +lemma "(X = Y Un Z) = (Y<=X & Z<=X & (ALL V. Y<=V & Z<=V --> X<=V))" +by blast + +lemma "(X = Y Int Z) = (X<=Y & X<=Z & (ALL V. V<=Y & V<=Z --> V<=X))" +by blast + +text{*trivial example of term synthesis: apparently hard for some provers!*} +lemma "a ~= b ==> a:?X & b ~: ?X" +by blast + +(** Examples for the Blast_tac paper **) + +text{*Union-image, called Un_Union_image on equalities.ML*} +lemma "(UN x:C. f(x) Un g(x)) = Union(f`C) Un Union(g`C)" +by blast + +text{*Inter-image, called Int_Inter_image on equalities.ML*} +lemma "(INT x:C. f(x) Int g(x)) = Inter(f`C) Int Inter(g`C)" +by blast + +text{*Singleton I. Nice demonstration of blast_tac--and its limitations. +For some unfathomable reason, UNIV_I increases the search space greatly*} +lemma "!!S::'a set set. ALL x:S. ALL y:S. x<=y ==> EX z. S <= {z}" +by (blast del: UNIV_I) + +text{*Singleton II. variant of the benchmark above*} +lemma "ALL x:S. Union(S) <= x ==> EX z. S <= {z}" +by (blast del: UNIV_I) + +text{* A unique fixpoint theorem --- fast/best/meson all fail *} + +lemma "EX! x. f(g(x))=x ==> EX! y. g(f(y))=y" +apply (erule ex1E, rule ex1I, erule arg_cong) +apply (rule subst, assumption, erule allE, rule arg_cong, erule mp) +apply (erule arg_cong) +done + + + +text{* Cantor's Theorem: There is no surjection from a set to its powerset. *} + +text{*requires best-first search because it is undirectional*} +lemma cantor1: "~ (EX f:: 'a=>'a set. ALL S. EX x. f(x) = S)" +by best + +text{*This form displays the diagonal term*} +lemma "ALL f:: 'a=>'a set. ALL x. f(x) ~= ?S(f)" +by best + +text{*This form exploits the set constructs*} +lemma "?S ~: range(f :: 'a=>'a set)" +by (rule notI, erule rangeE, best) + +text{*Or just this!*} +lemma "?S ~: range(f :: 'a=>'a set)" +by best + +text{* The Schroeder-Berstein Theorem *} + +lemma disj_lemma: "[| -(f`X) = g`(-X); f(a)=g(b); a:X |] ==> b:X" +by blast + +lemma surj_if_then_else: + "-(f`X) = g`(-X) ==> surj(%z. if z:X then f(z) else g(z))" +by (simp add: surj_def, blast) + +lemma bij_if_then_else: + "[| inj_on f X; inj_on g (-X); -(f`X) = g`(-X); + h = (%z. if z:X then f(z) else g(z)) |] + ==> inj(h) & surj(h)" +apply (unfold inj_on_def) +apply (simp add: surj_if_then_else) +apply (blast dest: disj_lemma sym) +done + +lemma decomposition: "EX X. X = - (g`(- (f`X)))" +apply (rule exI) +apply (rule lfp_unfold) +apply (rule monoI, blast) +done + +text{*Schroeder-Bernstein Theorem*} +lemma "[| inj (f:: 'a=>'b); inj (g:: 'b=>'a) |] + ==> EX h:: 'a=>'b. inj(h) & surj(h)" +apply (rule decomposition [THEN exE]) +apply (rule exI) +apply (rule bij_if_then_else) + apply (rule_tac [4] refl) + apply (rule_tac [2] inj_on_inv) + apply (erule subset_inj_on [OF subset_UNIV]) + txt{*tricky variable instantiations!*} + apply (erule ssubst, subst double_complement) + apply (rule subsetI, erule imageE, erule ssubst, rule rangeI) +apply (erule ssubst, subst double_complement, erule inv_image_comp [symmetric]) +done + + +text{*Set variable instantiation examples from +W. W. Bledsoe and Guohui Feng, SET-VAR. +JAR 11 (3), 1993, pages 293-314. + +Isabelle can prove the easy examples without any special mechanisms, but it +can't prove the hard ones. +*} + +text{*Example 1, page 295.*} +lemma "(EX A. (ALL x:A. x <= (0::int)))" +by force + +text{*Example 2*} +lemma "D : F --> (EX G. (ALL A:G. EX B:F. A <= B))"; +by force + +text{*Example 3*} +lemma "P(a) --> (EX A. (ALL x:A. P(x)) & (EX y. y:A))"; +by force + +text{*Example 4*} +lemma "a (EX A. a~:A & b:A & c~: A)" +by force + +text{*Example 5, page 298.*} +lemma "P(f(b)) --> (EX s A. (ALL x:A. P(x)) & f(s):A)"; +by force + +text{*Example 6*} +lemma "P(f(b)) --> (EX s A. (ALL x:A. P(x)) & f(s):A)"; +by force + +text{*Example 7*} +lemma "EX A. a ~: A" +by force + +text{*Example 8*} +lemma "(ALL u v. u < (0::int) --> u ~= abs v) --> (EX A::int set. (ALL y. abs y ~: A) & -2 : A)" +by force text{*not blast, which can't simplify -2<0*} + +text{*Example 9 omitted (requires the reals)*} + +text{*The paper has no Example 10!*} + +text{*Example 11: needs a hint*} +lemma "(ALL A. 0:A & (ALL x:A. Suc(x):A) --> n:A) & + P(0) & (ALL x. P(x) --> P(Suc(x))) --> P(n)" +apply clarify +apply (drule_tac x="{x. P x}" in spec) +by force + +text{*Example 12*} +lemma "(ALL A. (0,0):A & (ALL x y. (x,y):A --> (Suc(x),Suc(y)):A) --> (n,m):A) + & P(n) --> P(m)" +by auto + +text{*Example EO1: typo in article, and with the obvious fix it seems + to require arithmetic reasoning.*} +lemma "(ALL x. (EX u. x=2*u) = (~(EX v. Suc x = 2*v))) --> + (EX A. ALL x. (x : A) = (Suc x ~: A))" +apply clarify +apply (rule_tac x="{x. EX u. x = 2*u}" in exI, auto) +apply (case_tac v, auto) +apply (drule_tac x="Suc v" and P="%x. ?a(x) ~= ?b(x)" in spec, force) +done + end