clasohm@1465
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(* Title: HOL/set
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clasohm@923
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ID: $Id$
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clasohm@1465
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Author: Lawrence C Paulson, Cambridge University Computer Laboratory
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Copyright 1991 University of Cambridge
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paulson@1985
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Set theory for higher-order logic. A set is simply a predicate.
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*)
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open Set;
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nipkow@1548
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section "Relating predicates and sets";
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nipkow@1548
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paulson@2499
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AddIffs [mem_Collect_eq];
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paulson@2499
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paulson@2499
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goal Set.thy "!!a. P(a) ==> a : {x.P(x)}";
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paulson@2499
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by (Asm_simp_tac 1);
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qed "CollectI";
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clasohm@923
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paulson@2499
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val prems = goal Set.thy "!!a. a : {x.P(x)} ==> P(a)";
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paulson@2499
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by (Asm_full_simp_tac 1);
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clasohm@923
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qed "CollectD";
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val [prem] = goal Set.thy "[| !!x. (x:A) = (x:B) |] ==> A = B";
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by (rtac (prem RS ext RS arg_cong RS box_equals) 1);
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clasohm@923
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by (rtac Collect_mem_eq 1);
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clasohm@923
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by (rtac Collect_mem_eq 1);
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qed "set_ext";
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val [prem] = goal Set.thy "[| !!x. P(x)=Q(x) |] ==> {x. P(x)} = {x. Q(x)}";
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by (rtac (prem RS ext RS arg_cong) 1);
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qed "Collect_cong";
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val CollectE = make_elim CollectD;
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paulson@2499
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AddSIs [CollectI];
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paulson@2499
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AddSEs [CollectE];
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paulson@2499
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paulson@2499
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nipkow@1548
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section "Bounded quantifiers";
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val prems = goalw Set.thy [Ball_def]
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"[| !!x. x:A ==> P(x) |] ==> ! x:A. P(x)";
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by (REPEAT (ares_tac (prems @ [allI,impI]) 1));
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qed "ballI";
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val [major,minor] = goalw Set.thy [Ball_def]
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"[| ! x:A. P(x); x:A |] ==> P(x)";
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by (rtac (minor RS (major RS spec RS mp)) 1);
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qed "bspec";
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val major::prems = goalw Set.thy [Ball_def]
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"[| ! x:A. P(x); P(x) ==> Q; x~:A ==> Q |] ==> Q";
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by (rtac (major RS spec RS impCE) 1);
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by (REPEAT (eresolve_tac prems 1));
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qed "ballE";
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(*Takes assumptions ! x:A.P(x) and a:A; creates assumption P(a)*)
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fun ball_tac i = etac ballE i THEN contr_tac (i+1);
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AddSIs [ballI];
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AddEs [ballE];
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paulson@2499
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val prems = goalw Set.thy [Bex_def]
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"[| P(x); x:A |] ==> ? x:A. P(x)";
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by (REPEAT (ares_tac (prems @ [exI,conjI]) 1));
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qed "bexI";
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qed_goal "bexCI" Set.thy
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"[| ! x:A. ~P(x) ==> P(a); a:A |] ==> ? x:A.P(x)"
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(fn prems=>
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[ (rtac classical 1),
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(REPEAT (ares_tac (prems@[bexI,ballI,notI,notE]) 1)) ]);
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val major::prems = goalw Set.thy [Bex_def]
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"[| ? x:A. P(x); !!x. [| x:A; P(x) |] ==> Q |] ==> Q";
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clasohm@923
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by (rtac (major RS exE) 1);
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clasohm@923
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by (REPEAT (eresolve_tac (prems @ [asm_rl,conjE]) 1));
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qed "bexE";
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AddIs [bexI];
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AddSEs [bexE];
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paulson@2499
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(*Trival rewrite rule; (! x:A.P)=P holds only if A is nonempty!*)
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paulson@1882
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goalw Set.thy [Ball_def] "(! x:A. True) = True";
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paulson@1882
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by (Simp_tac 1);
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qed "ball_True";
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paulson@1816
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paulson@1882
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(*Dual form for existentials*)
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goalw Set.thy [Bex_def] "(? x:A. False) = False";
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by (Simp_tac 1);
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qed "bex_False";
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paulson@1882
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paulson@1882
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Addsimps [ball_True, bex_False];
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clasohm@923
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(** Congruence rules **)
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val prems = goal Set.thy
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"[| A=B; !!x. x:B ==> P(x) = Q(x) |] ==> \
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\ (! x:A. P(x)) = (! x:B. Q(x))";
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by (resolve_tac (prems RL [ssubst]) 1);
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clasohm@923
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by (REPEAT (ares_tac [ballI,iffI] 1
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ORELSE eresolve_tac ([make_elim bspec, mp] @ (prems RL [iffE])) 1));
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qed "ball_cong";
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val prems = goal Set.thy
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"[| A=B; !!x. x:B ==> P(x) = Q(x) |] ==> \
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\ (? x:A. P(x)) = (? x:B. Q(x))";
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by (resolve_tac (prems RL [ssubst]) 1);
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clasohm@923
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by (REPEAT (etac bexE 1
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ORELSE ares_tac ([bexI,iffI] @ (prems RL [iffD1,iffD2])) 1));
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qed "bex_cong";
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section "Subsets";
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val prems = goalw Set.thy [subset_def] "(!!x.x:A ==> x:B) ==> A <= B";
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by (REPEAT (ares_tac (prems @ [ballI]) 1));
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qed "subsetI";
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paulson@2881
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Blast.declConsts (["op <="], [subsetI]); (*overloading of <=*)
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paulson@2881
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(*Rule in Modus Ponens style*)
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val major::prems = goalw Set.thy [subset_def] "[| A <= B; c:A |] ==> c:B";
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by (rtac (major RS bspec) 1);
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by (resolve_tac prems 1);
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qed "subsetD";
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(*The same, with reversed premises for use with etac -- cf rev_mp*)
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qed_goal "rev_subsetD" Set.thy "[| c:A; A <= B |] ==> c:B"
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(fn prems=> [ (REPEAT (resolve_tac (prems@[subsetD]) 1)) ]);
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clasohm@923
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paulson@1920
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(*Converts A<=B to x:A ==> x:B*)
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paulson@1920
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fun impOfSubs th = th RSN (2, rev_subsetD);
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paulson@1920
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paulson@1841
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qed_goal "contra_subsetD" Set.thy "!!c. [| A <= B; c ~: B |] ==> c ~: A"
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paulson@1841
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(fn prems=> [ (REPEAT (eresolve_tac [asm_rl, contrapos, subsetD] 1)) ]);
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paulson@1841
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paulson@1841
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qed_goal "rev_contra_subsetD" Set.thy "!!c. [| c ~: B; A <= B |] ==> c ~: A"
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paulson@1841
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(fn prems=> [ (REPEAT (eresolve_tac [asm_rl, contrapos, subsetD] 1)) ]);
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paulson@1841
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(*Classical elimination rule*)
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val major::prems = goalw Set.thy [subset_def]
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"[| A <= B; c~:A ==> P; c:B ==> P |] ==> P";
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by (rtac (major RS ballE) 1);
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by (REPEAT (eresolve_tac prems 1));
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qed "subsetCE";
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(*Takes assumptions A<=B; c:A and creates the assumption c:B *)
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fun set_mp_tac i = etac subsetCE i THEN mp_tac i;
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paulson@2499
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AddSIs [subsetI];
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paulson@2499
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AddEs [subsetD, subsetCE];
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qed_goal "subset_refl" Set.thy "A <= (A::'a set)"
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paulson@2891
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(fn _=> [Blast_tac 1]);
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paulson@2499
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paulson@2499
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val prems = goal Set.thy "!!B. [| A<=B; B<=C |] ==> A<=(C::'a set)";
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paulson@2891
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by (Blast_tac 1);
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qed "subset_trans";
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nipkow@1548
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section "Equality";
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(*Anti-symmetry of the subset relation*)
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val prems = goal Set.thy "[| A <= B; B <= A |] ==> A = (B::'a set)";
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clasohm@923
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by (rtac (iffI RS set_ext) 1);
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clasohm@923
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by (REPEAT (ares_tac (prems RL [subsetD]) 1));
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qed "subset_antisym";
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val equalityI = subset_antisym;
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paulson@2881
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Blast.declConsts (["op ="], [equalityI]); (*overloading of equality*)
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berghofe@1762
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AddSIs [equalityI];
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berghofe@1762
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clasohm@923
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(* Equality rules from ZF set theory -- are they appropriate here? *)
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val prems = goal Set.thy "A = B ==> A<=(B::'a set)";
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clasohm@923
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by (resolve_tac (prems RL [subst]) 1);
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clasohm@923
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by (rtac subset_refl 1);
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clasohm@923
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qed "equalityD1";
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clasohm@923
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val prems = goal Set.thy "A = B ==> B<=(A::'a set)";
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clasohm@923
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by (resolve_tac (prems RL [subst]) 1);
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clasohm@923
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by (rtac subset_refl 1);
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clasohm@923
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qed "equalityD2";
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val prems = goal Set.thy
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clasohm@923
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"[| A = B; [| A<=B; B<=(A::'a set) |] ==> P |] ==> P";
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clasohm@923
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by (resolve_tac prems 1);
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clasohm@923
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by (REPEAT (resolve_tac (prems RL [equalityD1,equalityD2]) 1));
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qed "equalityE";
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clasohm@923
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clasohm@923
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val major::prems = goal Set.thy
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clasohm@923
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"[| A = B; [| c:A; c:B |] ==> P; [| c~:A; c~:B |] ==> P |] ==> P";
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clasohm@923
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by (rtac (major RS equalityE) 1);
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clasohm@923
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by (REPEAT (contr_tac 1 ORELSE eresolve_tac ([asm_rl,subsetCE]@prems) 1));
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clasohm@923
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qed "equalityCE";
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clasohm@923
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clasohm@923
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(*Lemma for creating induction formulae -- for "pattern matching" on p
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clasohm@923
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To make the induction hypotheses usable, apply "spec" or "bspec" to
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clasohm@923
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put universal quantifiers over the free variables in p. *)
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clasohm@923
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val prems = goal Set.thy
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clasohm@923
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"[| p:A; !!z. z:A ==> p=z --> R |] ==> R";
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clasohm@923
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by (rtac mp 1);
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clasohm@923
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by (REPEAT (resolve_tac (refl::prems) 1));
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clasohm@923
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qed "setup_induction";
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clasohm@923
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clasohm@923
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paulson@2858
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section "The empty set -- {}";
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paulson@2858
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paulson@2858
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qed_goalw "empty_iff" Set.thy [empty_def] "(c : {}) = False"
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paulson@2891
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(fn _ => [ (Blast_tac 1) ]);
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paulson@2858
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paulson@2858
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Addsimps [empty_iff];
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paulson@2858
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paulson@2858
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qed_goal "emptyE" Set.thy "!!a. a:{} ==> P"
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paulson@2858
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(fn _ => [Full_simp_tac 1]);
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paulson@2858
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paulson@2858
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AddSEs [emptyE];
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paulson@2858
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paulson@2858
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qed_goal "empty_subsetI" Set.thy "{} <= A"
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paulson@2891
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(fn _ => [ (Blast_tac 1) ]);
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paulson@2858
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paulson@2858
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qed_goal "equals0I" Set.thy "[| !!y. y:A ==> False |] ==> A={}"
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paulson@2858
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(fn [prem]=>
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paulson@2935
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[ (blast_tac (!claset addIs [prem RS FalseE]) 1) ]);
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paulson@2858
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paulson@2858
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qed_goal "equals0D" Set.thy "!!a. [| A={}; a:A |] ==> P"
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paulson@2891
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(fn _ => [ (Blast_tac 1) ]);
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paulson@2858
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paulson@2858
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goal Set.thy "Ball {} P = True";
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paulson@2858
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by (simp_tac (HOL_ss addsimps [mem_Collect_eq, Ball_def, empty_def]) 1);
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paulson@2858
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qed "ball_empty";
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paulson@2858
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paulson@2858
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goal Set.thy "Bex {} P = False";
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paulson@2858
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by (simp_tac (HOL_ss addsimps [mem_Collect_eq, Bex_def, empty_def]) 1);
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paulson@2858
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qed "bex_empty";
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paulson@2858
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Addsimps [ball_empty, bex_empty];
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paulson@2858
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paulson@2858
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goalw Set.thy [Ball_def] "(!x:A.False) = (A = {})";
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paulson@2891
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by(Blast_tac 1);
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paulson@2858
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qed "ball_False";
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paulson@2858
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Addsimps [ball_False];
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paulson@2858
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paulson@2858
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242 |
(* The dual is probably not helpful:
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paulson@2858
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goal Set.thy "(? x:A.True) = (A ~= {})";
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paulson@2891
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by(Blast_tac 1);
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paulson@2858
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qed "bex_True";
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paulson@2858
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Addsimps [bex_True];
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paulson@2858
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247 |
*)
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paulson@2858
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248 |
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paulson@2858
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249 |
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paulson@2858
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250 |
section "The Powerset operator -- Pow";
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paulson@2858
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paulson@2858
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252 |
qed_goalw "Pow_iff" Set.thy [Pow_def] "(A : Pow(B)) = (A <= B)"
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paulson@2858
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(fn _ => [ (Asm_simp_tac 1) ]);
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paulson@2858
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254 |
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paulson@2858
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255 |
AddIffs [Pow_iff];
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paulson@2858
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256 |
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paulson@2858
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qed_goalw "PowI" Set.thy [Pow_def] "!!A B. A <= B ==> A : Pow(B)"
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paulson@2858
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(fn _ => [ (etac CollectI 1) ]);
|
paulson@2858
|
259 |
|
paulson@2858
|
260 |
qed_goalw "PowD" Set.thy [Pow_def] "!!A B. A : Pow(B) ==> A<=B"
|
paulson@2858
|
261 |
(fn _=> [ (etac CollectD 1) ]);
|
paulson@2858
|
262 |
|
paulson@2858
|
263 |
val Pow_bottom = empty_subsetI RS PowI; (* {}: Pow(B) *)
|
paulson@2858
|
264 |
val Pow_top = subset_refl RS PowI; (* A : Pow(A) *)
|
paulson@2858
|
265 |
|
paulson@2858
|
266 |
|
nipkow@1548
|
267 |
section "Set complement -- Compl";
|
clasohm@923
|
268 |
|
paulson@2499
|
269 |
qed_goalw "Compl_iff" Set.thy [Compl_def] "(c : Compl(A)) = (c~:A)"
|
paulson@2891
|
270 |
(fn _ => [ (Blast_tac 1) ]);
|
paulson@2499
|
271 |
|
paulson@2499
|
272 |
Addsimps [Compl_iff];
|
paulson@2499
|
273 |
|
clasohm@923
|
274 |
val prems = goalw Set.thy [Compl_def]
|
clasohm@923
|
275 |
"[| c:A ==> False |] ==> c : Compl(A)";
|
clasohm@923
|
276 |
by (REPEAT (ares_tac (prems @ [CollectI,notI]) 1));
|
clasohm@923
|
277 |
qed "ComplI";
|
clasohm@923
|
278 |
|
clasohm@923
|
279 |
(*This form, with negated conclusion, works well with the Classical prover.
|
clasohm@923
|
280 |
Negated assumptions behave like formulae on the right side of the notional
|
clasohm@923
|
281 |
turnstile...*)
|
clasohm@923
|
282 |
val major::prems = goalw Set.thy [Compl_def]
|
paulson@2499
|
283 |
"c : Compl(A) ==> c~:A";
|
clasohm@923
|
284 |
by (rtac (major RS CollectD) 1);
|
clasohm@923
|
285 |
qed "ComplD";
|
clasohm@923
|
286 |
|
clasohm@923
|
287 |
val ComplE = make_elim ComplD;
|
clasohm@923
|
288 |
|
paulson@2499
|
289 |
AddSIs [ComplI];
|
paulson@2499
|
290 |
AddSEs [ComplE];
|
paulson@1640
|
291 |
|
clasohm@923
|
292 |
|
nipkow@1548
|
293 |
section "Binary union -- Un";
|
clasohm@923
|
294 |
|
paulson@2499
|
295 |
qed_goalw "Un_iff" Set.thy [Un_def] "(c : A Un B) = (c:A | c:B)"
|
paulson@2891
|
296 |
(fn _ => [ Blast_tac 1 ]);
|
paulson@2499
|
297 |
|
paulson@2499
|
298 |
Addsimps [Un_iff];
|
paulson@2499
|
299 |
|
paulson@2499
|
300 |
goal Set.thy "!!c. c:A ==> c : A Un B";
|
paulson@2499
|
301 |
by (Asm_simp_tac 1);
|
clasohm@923
|
302 |
qed "UnI1";
|
clasohm@923
|
303 |
|
paulson@2499
|
304 |
goal Set.thy "!!c. c:B ==> c : A Un B";
|
paulson@2499
|
305 |
by (Asm_simp_tac 1);
|
clasohm@923
|
306 |
qed "UnI2";
|
clasohm@923
|
307 |
|
clasohm@923
|
308 |
(*Classical introduction rule: no commitment to A vs B*)
|
clasohm@923
|
309 |
qed_goal "UnCI" Set.thy "(c~:B ==> c:A) ==> c : A Un B"
|
clasohm@923
|
310 |
(fn prems=>
|
paulson@2499
|
311 |
[ (Simp_tac 1),
|
paulson@2499
|
312 |
(REPEAT (ares_tac (prems@[disjCI]) 1)) ]);
|
clasohm@923
|
313 |
|
clasohm@923
|
314 |
val major::prems = goalw Set.thy [Un_def]
|
clasohm@923
|
315 |
"[| c : A Un B; c:A ==> P; c:B ==> P |] ==> P";
|
clasohm@923
|
316 |
by (rtac (major RS CollectD RS disjE) 1);
|
clasohm@923
|
317 |
by (REPEAT (eresolve_tac prems 1));
|
clasohm@923
|
318 |
qed "UnE";
|
clasohm@923
|
319 |
|
paulson@2499
|
320 |
AddSIs [UnCI];
|
paulson@2499
|
321 |
AddSEs [UnE];
|
paulson@1640
|
322 |
|
clasohm@923
|
323 |
|
nipkow@1548
|
324 |
section "Binary intersection -- Int";
|
clasohm@923
|
325 |
|
paulson@2499
|
326 |
qed_goalw "Int_iff" Set.thy [Int_def] "(c : A Int B) = (c:A & c:B)"
|
paulson@2891
|
327 |
(fn _ => [ (Blast_tac 1) ]);
|
paulson@2499
|
328 |
|
paulson@2499
|
329 |
Addsimps [Int_iff];
|
paulson@2499
|
330 |
|
paulson@2499
|
331 |
goal Set.thy "!!c. [| c:A; c:B |] ==> c : A Int B";
|
paulson@2499
|
332 |
by (Asm_simp_tac 1);
|
clasohm@923
|
333 |
qed "IntI";
|
clasohm@923
|
334 |
|
paulson@2499
|
335 |
goal Set.thy "!!c. c : A Int B ==> c:A";
|
paulson@2499
|
336 |
by (Asm_full_simp_tac 1);
|
clasohm@923
|
337 |
qed "IntD1";
|
clasohm@923
|
338 |
|
paulson@2499
|
339 |
goal Set.thy "!!c. c : A Int B ==> c:B";
|
paulson@2499
|
340 |
by (Asm_full_simp_tac 1);
|
clasohm@923
|
341 |
qed "IntD2";
|
clasohm@923
|
342 |
|
clasohm@923
|
343 |
val [major,minor] = goal Set.thy
|
clasohm@923
|
344 |
"[| c : A Int B; [| c:A; c:B |] ==> P |] ==> P";
|
clasohm@923
|
345 |
by (rtac minor 1);
|
clasohm@923
|
346 |
by (rtac (major RS IntD1) 1);
|
clasohm@923
|
347 |
by (rtac (major RS IntD2) 1);
|
clasohm@923
|
348 |
qed "IntE";
|
clasohm@923
|
349 |
|
paulson@2499
|
350 |
AddSIs [IntI];
|
paulson@2499
|
351 |
AddSEs [IntE];
|
clasohm@923
|
352 |
|
nipkow@1548
|
353 |
section "Set difference";
|
clasohm@923
|
354 |
|
paulson@2499
|
355 |
qed_goalw "Diff_iff" Set.thy [set_diff_def] "(c : A-B) = (c:A & c~:B)"
|
paulson@2891
|
356 |
(fn _ => [ (Blast_tac 1) ]);
|
clasohm@923
|
357 |
|
paulson@2499
|
358 |
Addsimps [Diff_iff];
|
paulson@2499
|
359 |
|
paulson@2499
|
360 |
qed_goal "DiffI" Set.thy "!!c. [| c : A; c ~: B |] ==> c : A - B"
|
paulson@2499
|
361 |
(fn _=> [ Asm_simp_tac 1 ]);
|
clasohm@923
|
362 |
|
paulson@2499
|
363 |
qed_goal "DiffD1" Set.thy "!!c. c : A - B ==> c : A"
|
paulson@2499
|
364 |
(fn _=> [ (Asm_full_simp_tac 1) ]);
|
clasohm@923
|
365 |
|
paulson@2499
|
366 |
qed_goal "DiffD2" Set.thy "!!c. [| c : A - B; c : B |] ==> P"
|
paulson@2499
|
367 |
(fn _=> [ (Asm_full_simp_tac 1) ]);
|
paulson@2499
|
368 |
|
paulson@2499
|
369 |
qed_goal "DiffE" Set.thy "[| c : A - B; [| c:A; c~:B |] ==> P |] ==> P"
|
clasohm@923
|
370 |
(fn prems=>
|
clasohm@923
|
371 |
[ (resolve_tac prems 1),
|
clasohm@923
|
372 |
(REPEAT (ares_tac (prems RL [DiffD1, DiffD2 RS notI]) 1)) ]);
|
clasohm@923
|
373 |
|
paulson@2499
|
374 |
AddSIs [DiffI];
|
paulson@2499
|
375 |
AddSEs [DiffE];
|
clasohm@923
|
376 |
|
clasohm@923
|
377 |
|
nipkow@1548
|
378 |
section "Augmenting a set -- insert";
|
clasohm@923
|
379 |
|
paulson@2499
|
380 |
qed_goalw "insert_iff" Set.thy [insert_def] "a : insert b A = (a=b | a:A)"
|
paulson@2891
|
381 |
(fn _ => [Blast_tac 1]);
|
paulson@2499
|
382 |
|
paulson@2499
|
383 |
Addsimps [insert_iff];
|
clasohm@923
|
384 |
|
paulson@2499
|
385 |
qed_goal "insertI1" Set.thy "a : insert a B"
|
paulson@2499
|
386 |
(fn _ => [Simp_tac 1]);
|
paulson@2499
|
387 |
|
paulson@2499
|
388 |
qed_goal "insertI2" Set.thy "!!a. a : B ==> a : insert b B"
|
paulson@2499
|
389 |
(fn _=> [Asm_simp_tac 1]);
|
clasohm@923
|
390 |
|
clasohm@923
|
391 |
qed_goalw "insertE" Set.thy [insert_def]
|
clasohm@923
|
392 |
"[| a : insert b A; a=b ==> P; a:A ==> P |] ==> P"
|
clasohm@923
|
393 |
(fn major::prems=>
|
clasohm@923
|
394 |
[ (rtac (major RS UnE) 1),
|
clasohm@923
|
395 |
(REPEAT (eresolve_tac (prems @ [CollectE]) 1)) ]);
|
clasohm@923
|
396 |
|
clasohm@923
|
397 |
(*Classical introduction rule*)
|
clasohm@923
|
398 |
qed_goal "insertCI" Set.thy "(a~:B ==> a=b) ==> a: insert b B"
|
paulson@2499
|
399 |
(fn prems=>
|
paulson@2499
|
400 |
[ (Simp_tac 1),
|
paulson@2499
|
401 |
(REPEAT (ares_tac (prems@[disjCI]) 1)) ]);
|
paulson@2499
|
402 |
|
paulson@2499
|
403 |
AddSIs [insertCI];
|
paulson@2499
|
404 |
AddSEs [insertE];
|
clasohm@923
|
405 |
|
nipkow@1548
|
406 |
section "Singletons, using insert";
|
clasohm@923
|
407 |
|
clasohm@923
|
408 |
qed_goal "singletonI" Set.thy "a : {a}"
|
clasohm@923
|
409 |
(fn _=> [ (rtac insertI1 1) ]);
|
clasohm@923
|
410 |
|
paulson@2499
|
411 |
goal Set.thy "!!a. b : {a} ==> b=a";
|
paulson@2891
|
412 |
by (Blast_tac 1);
|
clasohm@923
|
413 |
qed "singletonD";
|
clasohm@923
|
414 |
|
oheimb@1776
|
415 |
bind_thm ("singletonE", make_elim singletonD);
|
oheimb@1776
|
416 |
|
paulson@2499
|
417 |
qed_goal "singleton_iff" thy "(b : {a}) = (b=a)"
|
paulson@2891
|
418 |
(fn _ => [Blast_tac 1]);
|
clasohm@923
|
419 |
|
paulson@2499
|
420 |
goal Set.thy "!!a b. {a}={b} ==> a=b";
|
paulson@2935
|
421 |
by (blast_tac (!claset addEs [equalityE]) 1);
|
clasohm@923
|
422 |
qed "singleton_inject";
|
clasohm@923
|
423 |
|
paulson@2858
|
424 |
(*Redundant? But unlike insertCI, it proves the subgoal immediately!*)
|
paulson@2858
|
425 |
AddSIs [singletonI];
|
paulson@2858
|
426 |
|
paulson@2499
|
427 |
AddSDs [singleton_inject];
|
paulson@2499
|
428 |
|
nipkow@1531
|
429 |
|
nipkow@1548
|
430 |
section "The universal set -- UNIV";
|
nipkow@1531
|
431 |
|
paulson@1882
|
432 |
qed_goal "UNIV_I" Set.thy "x : UNIV"
|
paulson@1882
|
433 |
(fn _ => [rtac ComplI 1, etac emptyE 1]);
|
paulson@1882
|
434 |
|
nipkow@1531
|
435 |
qed_goal "subset_UNIV" Set.thy "A <= UNIV"
|
paulson@1882
|
436 |
(fn _ => [rtac subsetI 1, rtac UNIV_I 1]);
|
nipkow@1531
|
437 |
|
nipkow@1531
|
438 |
|
nipkow@1548
|
439 |
section "Unions of families -- UNION x:A. B(x) is Union(B``A)";
|
clasohm@923
|
440 |
|
paulson@2499
|
441 |
goalw Set.thy [UNION_def] "(b: (UN x:A. B(x))) = (EX x:A. b: B(x))";
|
paulson@2891
|
442 |
by (Blast_tac 1);
|
paulson@2499
|
443 |
qed "UN_iff";
|
paulson@2499
|
444 |
|
paulson@2499
|
445 |
Addsimps [UN_iff];
|
paulson@2499
|
446 |
|
clasohm@923
|
447 |
(*The order of the premises presupposes that A is rigid; b may be flexible*)
|
paulson@2499
|
448 |
goal Set.thy "!!b. [| a:A; b: B(a) |] ==> b: (UN x:A. B(x))";
|
paulson@2499
|
449 |
by (Auto_tac());
|
clasohm@923
|
450 |
qed "UN_I";
|
clasohm@923
|
451 |
|
clasohm@923
|
452 |
val major::prems = goalw Set.thy [UNION_def]
|
clasohm@923
|
453 |
"[| b : (UN x:A. B(x)); !!x.[| x:A; b: B(x) |] ==> R |] ==> R";
|
clasohm@923
|
454 |
by (rtac (major RS CollectD RS bexE) 1);
|
clasohm@923
|
455 |
by (REPEAT (ares_tac prems 1));
|
clasohm@923
|
456 |
qed "UN_E";
|
clasohm@923
|
457 |
|
paulson@2499
|
458 |
AddIs [UN_I];
|
paulson@2499
|
459 |
AddSEs [UN_E];
|
paulson@2499
|
460 |
|
clasohm@923
|
461 |
val prems = goal Set.thy
|
clasohm@923
|
462 |
"[| A=B; !!x. x:B ==> C(x) = D(x) |] ==> \
|
clasohm@923
|
463 |
\ (UN x:A. C(x)) = (UN x:B. D(x))";
|
clasohm@923
|
464 |
by (REPEAT (etac UN_E 1
|
clasohm@923
|
465 |
ORELSE ares_tac ([UN_I,equalityI,subsetI] @
|
clasohm@1465
|
466 |
(prems RL [equalityD1,equalityD2] RL [subsetD])) 1));
|
clasohm@923
|
467 |
qed "UN_cong";
|
clasohm@923
|
468 |
|
clasohm@923
|
469 |
|
nipkow@1548
|
470 |
section "Intersections of families -- INTER x:A. B(x) is Inter(B``A)";
|
clasohm@923
|
471 |
|
paulson@2499
|
472 |
goalw Set.thy [INTER_def] "(b: (INT x:A. B(x))) = (ALL x:A. b: B(x))";
|
paulson@2499
|
473 |
by (Auto_tac());
|
paulson@2499
|
474 |
qed "INT_iff";
|
paulson@2499
|
475 |
|
paulson@2499
|
476 |
Addsimps [INT_iff];
|
paulson@2499
|
477 |
|
clasohm@923
|
478 |
val prems = goalw Set.thy [INTER_def]
|
clasohm@923
|
479 |
"(!!x. x:A ==> b: B(x)) ==> b : (INT x:A. B(x))";
|
clasohm@923
|
480 |
by (REPEAT (ares_tac ([CollectI,ballI] @ prems) 1));
|
clasohm@923
|
481 |
qed "INT_I";
|
clasohm@923
|
482 |
|
paulson@2499
|
483 |
goal Set.thy "!!b. [| b : (INT x:A. B(x)); a:A |] ==> b: B(a)";
|
paulson@2499
|
484 |
by (Auto_tac());
|
clasohm@923
|
485 |
qed "INT_D";
|
clasohm@923
|
486 |
|
clasohm@923
|
487 |
(*"Classical" elimination -- by the Excluded Middle on a:A *)
|
clasohm@923
|
488 |
val major::prems = goalw Set.thy [INTER_def]
|
clasohm@923
|
489 |
"[| b : (INT x:A. B(x)); b: B(a) ==> R; a~:A ==> R |] ==> R";
|
clasohm@923
|
490 |
by (rtac (major RS CollectD RS ballE) 1);
|
clasohm@923
|
491 |
by (REPEAT (eresolve_tac prems 1));
|
clasohm@923
|
492 |
qed "INT_E";
|
clasohm@923
|
493 |
|
paulson@2499
|
494 |
AddSIs [INT_I];
|
paulson@2499
|
495 |
AddEs [INT_D, INT_E];
|
paulson@2499
|
496 |
|
clasohm@923
|
497 |
val prems = goal Set.thy
|
clasohm@923
|
498 |
"[| A=B; !!x. x:B ==> C(x) = D(x) |] ==> \
|
clasohm@923
|
499 |
\ (INT x:A. C(x)) = (INT x:B. D(x))";
|
clasohm@923
|
500 |
by (REPEAT_FIRST (resolve_tac [INT_I,equalityI,subsetI]));
|
clasohm@923
|
501 |
by (REPEAT (dtac INT_D 1
|
clasohm@923
|
502 |
ORELSE ares_tac (prems RL [equalityD1,equalityD2] RL [subsetD]) 1));
|
clasohm@923
|
503 |
qed "INT_cong";
|
clasohm@923
|
504 |
|
clasohm@923
|
505 |
|
nipkow@1548
|
506 |
section "Unions over a type; UNION1(B) = Union(range(B))";
|
clasohm@923
|
507 |
|
paulson@2499
|
508 |
goalw Set.thy [UNION1_def] "(b: (UN x. B(x))) = (EX x. b: B(x))";
|
paulson@2499
|
509 |
by (Simp_tac 1);
|
paulson@2891
|
510 |
by (Blast_tac 1);
|
paulson@2499
|
511 |
qed "UN1_iff";
|
paulson@2499
|
512 |
|
paulson@2499
|
513 |
Addsimps [UN1_iff];
|
paulson@2499
|
514 |
|
clasohm@923
|
515 |
(*The order of the premises presupposes that A is rigid; b may be flexible*)
|
paulson@2499
|
516 |
goal Set.thy "!!b. b: B(x) ==> b: (UN x. B(x))";
|
paulson@2499
|
517 |
by (Auto_tac());
|
clasohm@923
|
518 |
qed "UN1_I";
|
clasohm@923
|
519 |
|
clasohm@923
|
520 |
val major::prems = goalw Set.thy [UNION1_def]
|
clasohm@923
|
521 |
"[| b : (UN x. B(x)); !!x. b: B(x) ==> R |] ==> R";
|
clasohm@923
|
522 |
by (rtac (major RS UN_E) 1);
|
clasohm@923
|
523 |
by (REPEAT (ares_tac prems 1));
|
clasohm@923
|
524 |
qed "UN1_E";
|
clasohm@923
|
525 |
|
paulson@2499
|
526 |
AddIs [UN1_I];
|
paulson@2499
|
527 |
AddSEs [UN1_E];
|
paulson@2499
|
528 |
|
clasohm@923
|
529 |
|
nipkow@1548
|
530 |
section "Intersections over a type; INTER1(B) = Inter(range(B))";
|
clasohm@923
|
531 |
|
paulson@2499
|
532 |
goalw Set.thy [INTER1_def] "(b: (INT x. B(x))) = (ALL x. b: B(x))";
|
paulson@2499
|
533 |
by (Simp_tac 1);
|
paulson@2891
|
534 |
by (Blast_tac 1);
|
paulson@2499
|
535 |
qed "INT1_iff";
|
paulson@2499
|
536 |
|
paulson@2499
|
537 |
Addsimps [INT1_iff];
|
paulson@2499
|
538 |
|
clasohm@923
|
539 |
val prems = goalw Set.thy [INTER1_def]
|
clasohm@923
|
540 |
"(!!x. b: B(x)) ==> b : (INT x. B(x))";
|
clasohm@923
|
541 |
by (REPEAT (ares_tac (INT_I::prems) 1));
|
clasohm@923
|
542 |
qed "INT1_I";
|
clasohm@923
|
543 |
|
paulson@2499
|
544 |
goal Set.thy "!!b. b : (INT x. B(x)) ==> b: B(a)";
|
paulson@2499
|
545 |
by (Asm_full_simp_tac 1);
|
clasohm@923
|
546 |
qed "INT1_D";
|
clasohm@923
|
547 |
|
paulson@2499
|
548 |
AddSIs [INT1_I];
|
paulson@2499
|
549 |
AddDs [INT1_D];
|
paulson@2499
|
550 |
|
paulson@2499
|
551 |
|
nipkow@1548
|
552 |
section "Union";
|
clasohm@923
|
553 |
|
paulson@2499
|
554 |
goalw Set.thy [Union_def] "(A : Union(C)) = (EX X:C. A:X)";
|
paulson@2891
|
555 |
by (Blast_tac 1);
|
paulson@2499
|
556 |
qed "Union_iff";
|
paulson@2499
|
557 |
|
paulson@2499
|
558 |
Addsimps [Union_iff];
|
paulson@2499
|
559 |
|
clasohm@923
|
560 |
(*The order of the premises presupposes that C is rigid; A may be flexible*)
|
paulson@2499
|
561 |
goal Set.thy "!!X. [| X:C; A:X |] ==> A : Union(C)";
|
paulson@2499
|
562 |
by (Auto_tac());
|
clasohm@923
|
563 |
qed "UnionI";
|
clasohm@923
|
564 |
|
clasohm@923
|
565 |
val major::prems = goalw Set.thy [Union_def]
|
clasohm@923
|
566 |
"[| A : Union(C); !!X.[| A:X; X:C |] ==> R |] ==> R";
|
clasohm@923
|
567 |
by (rtac (major RS UN_E) 1);
|
clasohm@923
|
568 |
by (REPEAT (ares_tac prems 1));
|
clasohm@923
|
569 |
qed "UnionE";
|
clasohm@923
|
570 |
|
paulson@2499
|
571 |
AddIs [UnionI];
|
paulson@2499
|
572 |
AddSEs [UnionE];
|
paulson@2499
|
573 |
|
paulson@2499
|
574 |
|
nipkow@1548
|
575 |
section "Inter";
|
clasohm@923
|
576 |
|
paulson@2499
|
577 |
goalw Set.thy [Inter_def] "(A : Inter(C)) = (ALL X:C. A:X)";
|
paulson@2891
|
578 |
by (Blast_tac 1);
|
paulson@2499
|
579 |
qed "Inter_iff";
|
paulson@2499
|
580 |
|
paulson@2499
|
581 |
Addsimps [Inter_iff];
|
paulson@2499
|
582 |
|
clasohm@923
|
583 |
val prems = goalw Set.thy [Inter_def]
|
clasohm@923
|
584 |
"[| !!X. X:C ==> A:X |] ==> A : Inter(C)";
|
clasohm@923
|
585 |
by (REPEAT (ares_tac ([INT_I] @ prems) 1));
|
clasohm@923
|
586 |
qed "InterI";
|
clasohm@923
|
587 |
|
clasohm@923
|
588 |
(*A "destruct" rule -- every X in C contains A as an element, but
|
clasohm@923
|
589 |
A:X can hold when X:C does not! This rule is analogous to "spec". *)
|
paulson@2499
|
590 |
goal Set.thy "!!X. [| A : Inter(C); X:C |] ==> A:X";
|
paulson@2499
|
591 |
by (Auto_tac());
|
clasohm@923
|
592 |
qed "InterD";
|
clasohm@923
|
593 |
|
clasohm@923
|
594 |
(*"Classical" elimination rule -- does not require proving X:C *)
|
clasohm@923
|
595 |
val major::prems = goalw Set.thy [Inter_def]
|
paulson@2721
|
596 |
"[| A : Inter(C); X~:C ==> R; A:X ==> R |] ==> R";
|
clasohm@923
|
597 |
by (rtac (major RS INT_E) 1);
|
clasohm@923
|
598 |
by (REPEAT (eresolve_tac prems 1));
|
clasohm@923
|
599 |
qed "InterE";
|
clasohm@923
|
600 |
|
paulson@2499
|
601 |
AddSIs [InterI];
|
paulson@2499
|
602 |
AddEs [InterD, InterE];
|
paulson@2499
|
603 |
|
paulson@2499
|
604 |
|
nipkow@2912
|
605 |
(*** Image of a set under a function ***)
|
nipkow@2912
|
606 |
|
nipkow@2912
|
607 |
(*Frequently b does not have the syntactic form of f(x).*)
|
nipkow@2912
|
608 |
val prems = goalw thy [image_def] "[| b=f(x); x:A |] ==> b : f``A";
|
nipkow@2912
|
609 |
by (REPEAT (resolve_tac (prems @ [CollectI,bexI,prem]) 1));
|
nipkow@2912
|
610 |
qed "image_eqI";
|
nipkow@2912
|
611 |
|
nipkow@2912
|
612 |
bind_thm ("imageI", refl RS image_eqI);
|
nipkow@2912
|
613 |
|
nipkow@2912
|
614 |
(*The eta-expansion gives variable-name preservation.*)
|
nipkow@2912
|
615 |
val major::prems = goalw thy [image_def]
|
nipkow@2912
|
616 |
"[| b : (%x.f(x))``A; !!x.[| b=f(x); x:A |] ==> P |] ==> P";
|
nipkow@2912
|
617 |
by (rtac (major RS CollectD RS bexE) 1);
|
nipkow@2912
|
618 |
by (REPEAT (ares_tac prems 1));
|
nipkow@2912
|
619 |
qed "imageE";
|
nipkow@2912
|
620 |
|
nipkow@2912
|
621 |
AddIs [image_eqI];
|
nipkow@2912
|
622 |
AddSEs [imageE];
|
nipkow@2912
|
623 |
|
nipkow@2912
|
624 |
goalw thy [o_def] "(f o g)``r = f``(g``r)";
|
paulson@2935
|
625 |
by (Blast_tac 1);
|
nipkow@2912
|
626 |
qed "image_compose";
|
nipkow@2912
|
627 |
|
nipkow@2912
|
628 |
goal thy "f``(A Un B) = f``A Un f``B";
|
paulson@2935
|
629 |
by (Blast_tac 1);
|
nipkow@2912
|
630 |
qed "image_Un";
|
nipkow@2912
|
631 |
|
nipkow@2912
|
632 |
|
nipkow@2912
|
633 |
(*** Range of a function -- just a translation for image! ***)
|
nipkow@2912
|
634 |
|
nipkow@2912
|
635 |
goal thy "!!b. b=f(x) ==> b : range(f)";
|
nipkow@2912
|
636 |
by (EVERY1 [etac image_eqI, rtac UNIV_I]);
|
nipkow@2912
|
637 |
bind_thm ("range_eqI", UNIV_I RSN (2,image_eqI));
|
nipkow@2912
|
638 |
|
nipkow@2912
|
639 |
bind_thm ("rangeI", UNIV_I RS imageI);
|
nipkow@2912
|
640 |
|
nipkow@2912
|
641 |
val [major,minor] = goal thy
|
nipkow@2912
|
642 |
"[| b : range(%x.f(x)); !!x. b=f(x) ==> P |] ==> P";
|
nipkow@2912
|
643 |
by (rtac (major RS imageE) 1);
|
nipkow@2912
|
644 |
by (etac minor 1);
|
nipkow@2912
|
645 |
qed "rangeE";
|
nipkow@2912
|
646 |
|
nipkow@2912
|
647 |
AddIs [rangeI];
|
nipkow@2912
|
648 |
AddSEs [rangeE];
|
nipkow@2912
|
649 |
|
oheimb@1776
|
650 |
|
oheimb@1776
|
651 |
(*** Set reasoning tools ***)
|
oheimb@1776
|
652 |
|
oheimb@1776
|
653 |
|
paulson@2499
|
654 |
(*Each of these has ALREADY been added to !simpset above.*)
|
paulson@2024
|
655 |
val mem_simps = [insert_iff, empty_iff, Un_iff, Int_iff, Compl_iff, Diff_iff,
|
paulson@2499
|
656 |
mem_Collect_eq,
|
paulson@2499
|
657 |
UN_iff, UN1_iff, Union_iff,
|
paulson@2499
|
658 |
INT_iff, INT1_iff, Inter_iff];
|
oheimb@1776
|
659 |
|
paulson@1937
|
660 |
(*Not for Addsimps -- it can cause goals to blow up!*)
|
paulson@1937
|
661 |
goal Set.thy "(a : (if Q then x else y)) = ((Q --> a:x) & (~Q --> a : y))";
|
paulson@1937
|
662 |
by (simp_tac (!simpset setloop split_tac [expand_if]) 1);
|
paulson@1937
|
663 |
qed "mem_if";
|
paulson@1937
|
664 |
|
oheimb@1776
|
665 |
val mksimps_pairs = ("Ball",[bspec]) :: mksimps_pairs;
|
oheimb@1776
|
666 |
|
paulson@2499
|
667 |
simpset := !simpset addcongs [ball_cong,bex_cong]
|
oheimb@1776
|
668 |
setmksimps (mksimps mksimps_pairs);
|