src/HOL/Set.ML
author berghofe
Tue, 30 May 2000 18:02:49 +0200
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the is now defined using primrec, avoiding explicit use of arbitrary.
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(*  Title:      HOL/set
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    ID:         $Id$
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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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Set theory for higher-order logic.  A set is simply a predicate.
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*)
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section "Relating predicates and sets";
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Addsimps [Collect_mem_eq];
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AddIffs  [mem_Collect_eq];
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Goal "P(a) ==> a : {x. P(x)}";
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by (Asm_simp_tac 1);
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qed "CollectI";
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Goal "a : {x. P(x)} ==> P(a)";
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by (Asm_full_simp_tac 1);
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qed "CollectD";
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bind_thm ("CollectE", make_elim CollectD);
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val [prem] = Goal "[| !!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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by (rtac Collect_mem_eq 1);
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by (rtac Collect_mem_eq 1);
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qed "set_ext";
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val [prem] = Goal "[| !!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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AddSIs [CollectI];
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AddSEs [CollectE];
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section "Bounded quantifiers";
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val prems = Goalw [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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bind_thms ("strip", [impI, allI, ballI]);
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Goalw [Ball_def] "[| ! x:A. P(x);  x:A |] ==> P(x)";
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by (Blast_tac 1);
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qed "bspec";
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val major::prems = Goalw [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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AddXDs [bspec];
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(* gives better instantiation for bound: *)
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claset_ref() := claset() addWrapper ("bspec", fn tac2 =>
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			 (dtac bspec THEN' atac) APPEND' tac2);
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(*Normally the best argument order: P(x) constrains the choice of x:A*)
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Goalw [Bex_def] "[| P(x);  x:A |] ==> ? x:A. P(x)";
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by (Blast_tac 1);
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qed "bexI";
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(*The best argument order when there is only one x:A*)
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Goalw [Bex_def] "[| x:A;  P(x) |] ==> ? x:A. P(x)";
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by (Blast_tac 1);
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qed "rev_bexI";
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val prems = Goal 
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   "[| ! x:A. ~P(x) ==> P(a);  a:A |] ==> ? x:A. P(x)";
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by (rtac classical 1);
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by (REPEAT (ares_tac (prems@[bexI,ballI,notI,notE]) 1))  ;
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qed "bexCI";
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val major::prems = Goalw [Bex_def]
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    "[| ? x:A. P(x);  !!x. [| x:A; P(x) |] ==> Q  |] ==> Q";
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by (rtac (major RS exE) 1);
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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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(*Trival rewrite rule*)
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Goal "(! x:A. P) = ((? x. x:A) --> P)";
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by (simp_tac (simpset() addsimps [Ball_def]) 1);
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qed "ball_triv";
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(*Dual form for existentials*)
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Goal "(? x:A. P) = ((? x. x:A) & P)";
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by (simp_tac (simpset() addsimps [Bex_def]) 1);
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qed "bex_triv";
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Addsimps [ball_triv, bex_triv];
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(** Congruence rules **)
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val prems = Goalw [Ball_def]
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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 (asm_simp_tac (simpset() addsimps prems) 1);
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qed "ball_cong";
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val prems = Goalw [Bex_def]
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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 (asm_simp_tac (simpset() addcongs [conj_cong] addsimps prems) 1);
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qed "bex_cong";
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Addcongs [ball_cong,bex_cong];
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section "Subsets";
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val prems = Goalw [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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(*Map the type ('a set => anything) to just 'a.
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  For overloading constants whose first argument has type "'a set" *)
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fun overload_1st_set s = Blast.overloaded (s, HOLogic.dest_setT o domain_type);
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(*While (:) is not, its type must be kept
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  for overloading of = to work.*)
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Blast.overloaded ("op :", domain_type);
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overload_1st_set "Ball";		(*need UNION, INTER also?*)
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overload_1st_set "Bex";
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(*Image: retain the type of the set being expressed*)
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Blast.overloaded ("image", domain_type);
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(*Rule in Modus Ponens style*)
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Goalw [subset_def] "[| A <= B;  c:A |] ==> c:B";
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by (Blast_tac 1);
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qed "subsetD";
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AddXIs [subsetD];
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(*The same, with reversed premises for use with etac -- cf rev_mp*)
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Goal "[| c:A;  A <= B |] ==> c:B";
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by (REPEAT (ares_tac [subsetD] 1)) ;
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qed "rev_subsetD";
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AddXIs [rev_subsetD];
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(*Converts A<=B to x:A ==> x:B*)
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fun impOfSubs th = th RSN (2, rev_subsetD);
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Goal "[| A <= B; c ~: B |] ==> c ~: A";
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by (REPEAT (eresolve_tac [asm_rl, contrapos, subsetD] 1)) ;
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qed "contra_subsetD";
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Goal "[| c ~: B;  A <= B |] ==> c ~: A";
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by (REPEAT (eresolve_tac [asm_rl, contrapos, subsetD] 1)) ;
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qed "rev_contra_subsetD";
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(*Classical elimination rule*)
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val major::prems = Goalw [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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AddSIs [subsetI];
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AddEs  [subsetD, subsetCE];
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Goal "A <= (A::'a set)";
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by (Fast_tac 1);
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qed "subset_refl";		(*Blast_tac would try order_refl and fail*)
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Goal "[| A<=B;  B<=C |] ==> A<=(C::'a set)";
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by (Blast_tac 1);
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qed "subset_trans";
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section "Equality";
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(*Anti-symmetry of the subset relation*)
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Goal "[| A <= B;  B <= A |] ==> A = (B::'a set)";
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by (rtac set_ext 1);
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by (blast_tac (claset() addIs [subsetD]) 1);
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qed "subset_antisym";
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val equalityI = subset_antisym;
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AddSIs [equalityI];
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(* Equality rules from ZF set theory -- are they appropriate here? *)
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Goal "A = B ==> A<=(B::'a set)";
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by (etac ssubst 1);
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by (rtac subset_refl 1);
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qed "equalityD1";
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Goal "A = B ==> B<=(A::'a set)";
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by (etac ssubst 1);
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by (rtac subset_refl 1);
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qed "equalityD2";
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val prems = Goal
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    "[| A = B;  [| A<=B; B<=(A::'a set) |] ==> P |]  ==>  P";
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by (resolve_tac prems 1);
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by (REPEAT (resolve_tac (prems RL [equalityD1,equalityD2]) 1));
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qed "equalityE";
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val major::prems = Goal
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    "[| A = B;  [| c:A; c:B |] ==> P;  [| c~:A; c~:B |] ==> P |]  ==>  P";
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by (rtac (major RS equalityE) 1);
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by (REPEAT (contr_tac 1 ORELSE eresolve_tac ([asm_rl,subsetCE]@prems) 1));
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qed "equalityCE";
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(*Lemma for creating induction formulae -- for "pattern matching" on p
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  To make the induction hypotheses usable, apply "spec" or "bspec" to
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  put universal quantifiers over the free variables in p. *)
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val prems = Goal 
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    "[| p:A;  !!z. z:A ==> p=z --> R |] ==> R";
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by (rtac mp 1);
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by (REPEAT (resolve_tac (refl::prems) 1));
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qed "setup_induction";
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Goal "A = B ==> (x : A) = (x : B)";
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by (Asm_simp_tac 1);
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qed "eqset_imp_iff";
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section "The universal set -- UNIV";
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Goalw [UNIV_def] "x : UNIV";
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by (rtac CollectI 1);
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by (rtac TrueI 1);
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qed "UNIV_I";
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Addsimps [UNIV_I];
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AddIs    [UNIV_I];  (*unsafe makes it less likely to cause problems*)
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Goal "A <= UNIV";
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by (rtac subsetI 1);
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by (rtac UNIV_I 1);
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qed "subset_UNIV";
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(** Eta-contracting these two rules (to remove P) causes them to be ignored
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    because of their interaction with congruence rules. **)
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Goalw [Ball_def] "Ball UNIV P = All P";
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by (Simp_tac 1);
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qed "ball_UNIV";
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Goalw [Bex_def] "Bex UNIV P = Ex P";
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by (Simp_tac 1);
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qed "bex_UNIV";
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Addsimps [ball_UNIV, bex_UNIV];
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section "The empty set -- {}";
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Goalw [empty_def] "(c : {}) = False";
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by (Blast_tac 1) ;
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qed "empty_iff";
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Addsimps [empty_iff];
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Goal "a:{} ==> P";
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by (Full_simp_tac 1);
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qed "emptyE";
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AddSEs [emptyE];
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Goal "{} <= A";
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by (Blast_tac 1) ;
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qed "empty_subsetI";
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(*One effect is to delete the ASSUMPTION {} <= A*)
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AddIffs [empty_subsetI];
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val [prem]= Goal "[| !!y. y:A ==> False |] ==> A={}";
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by (blast_tac (claset() addIs [prem RS FalseE]) 1) ;
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qed "equals0I";
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(*Use for reasoning about disjointness: A Int B = {} *)
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Goal "A={} ==> a ~: A";
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by (Blast_tac 1) ;
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qed "equals0D";
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AddDs [equals0D, sym RS equals0D];
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Goalw [Ball_def] "Ball {} P = True";
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by (Simp_tac 1);
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qed "ball_empty";
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Goalw [Bex_def] "Bex {} P = False";
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by (Simp_tac 1);
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qed "bex_empty";
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Addsimps [ball_empty, bex_empty];
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Goal "UNIV ~= {}";
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by (blast_tac (claset() addEs [equalityE]) 1);
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qed "UNIV_not_empty";
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AddIffs [UNIV_not_empty];
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section "The Powerset operator -- Pow";
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Goalw [Pow_def] "(A : Pow(B)) = (A <= B)";
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by (Asm_simp_tac 1);
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qed "Pow_iff";
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AddIffs [Pow_iff]; 
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Goalw [Pow_def] "A <= B ==> A : Pow(B)";
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by (etac CollectI 1);
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   321
qed "PowI";
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Goalw [Pow_def] "A : Pow(B)  ==>  A<=B";
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   324
by (etac CollectD 1);
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qed "PowD";
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   326
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val Pow_bottom = empty_subsetI RS PowI;        (* {}: Pow(B) *)
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val Pow_top = subset_refl RS PowI;             (* A : Pow(A) *)
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1f3f5c44e159 Re-ordering of rules to assist blast_tac
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parents: 2721
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   331
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section "Set complement";
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   334
Goalw [Compl_def] "(c : -A) = (c~:A)";
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   335
by (Blast_tac 1);
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qed "Compl_iff";
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0bc87b063447 Tidying of proofs. New theorems are enterred immediately into the
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Addsimps [Compl_iff];
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parents: 2031
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5490
85855f65d0c6 From Compl(A) to -A
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   340
val prems = Goalw [Compl_def] "[| c:A ==> False |] ==> c : -A";
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   341
by (REPEAT (ares_tac (prems @ [CollectI,notI]) 1));
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qed "ComplI";
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   343
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(*This form, with negated conclusion, works well with the Classical prover.
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parents:
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  Negated assumptions behave like formulae on the right side of the notional
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   346
  turnstile...*)
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85855f65d0c6 From Compl(A) to -A
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Goalw [Compl_def] "c : -A ==> c~:A";
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by (etac CollectD 1);
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qed "ComplD";
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val ComplE = make_elim ComplD;
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AddSIs [ComplI];
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AddSEs [ComplE];
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581165679095 Added more _iff rewrites for Compl, Un, Int
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parents: 1618
diff changeset
   355
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parents:
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   356
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   357
section "Binary union -- Un";
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   359
Goalw [Un_def] "(c : A Un B) = (c:A | c:B)";
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   360
by (Blast_tac 1);
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   361
qed "Un_iff";
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Addsimps [Un_iff];
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parents: 2031
diff changeset
   363
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parents: 5069
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   364
Goal "c:A ==> c : A Un B";
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0bc87b063447 Tidying of proofs. New theorems are enterred immediately into the
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parents: 2031
diff changeset
   365
by (Asm_simp_tac 1);
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parents:
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qed "UnI1";
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parents:
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   367
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parents: 5069
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   368
Goal "c:B ==> c : A Un B";
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parents: 2031
diff changeset
   369
by (Asm_simp_tac 1);
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qed "UnI2";
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parents:
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   371
ff1574a81019 new version of HOL with curried function application
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parents:
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   372
(*Classical introduction rule: no commitment to A vs B*)
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parents: 6443
diff changeset
   373
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972b5f62f476 getting rid of qed_goal
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   374
val prems = Goal "(c~:B ==> c:A) ==> c : A Un B";
7007
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parents: 6443
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   375
by (Simp_tac 1);
b46ccfee8e59 qed_goal -> Goal
paulson
parents: 6443
diff changeset
   376
by (REPEAT (ares_tac (prems@[disjCI]) 1)) ;
b46ccfee8e59 qed_goal -> Goal
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parents: 6443
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   377
qed "UnCI";
923
ff1574a81019 new version of HOL with curried function application
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parents:
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   378
5316
7a8975451a89 even more tidying of Goal commands
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   379
val major::prems = Goalw [Un_def]
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clasohm
parents:
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   380
    "[| c : A Un B;  c:A ==> P;  c:B ==> P |] ==> P";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   381
by (rtac (major RS CollectD RS disjE) 1);
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   382
by (REPEAT (eresolve_tac prems 1));
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   383
qed "UnE";
ff1574a81019 new version of HOL with curried function application
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parents:
diff changeset
   384
2499
0bc87b063447 Tidying of proofs. New theorems are enterred immediately into the
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parents: 2031
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   385
AddSIs [UnCI];
0bc87b063447 Tidying of proofs. New theorems are enterred immediately into the
paulson
parents: 2031
diff changeset
   386
AddSEs [UnE];
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581165679095 Added more _iff rewrites for Compl, Un, Int
paulson
parents: 1618
diff changeset
   387
923
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clasohm
parents:
diff changeset
   388
1548
afe750876848 Added 'section' commands
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parents: 1531
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   389
section "Binary intersection -- Int";
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   390
7031
972b5f62f476 getting rid of qed_goal
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parents: 7007
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   391
Goalw [Int_def] "(c : A Int B) = (c:A & c:B)";
972b5f62f476 getting rid of qed_goal
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parents: 7007
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   392
by (Blast_tac 1);
972b5f62f476 getting rid of qed_goal
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parents: 7007
diff changeset
   393
qed "Int_iff";
2499
0bc87b063447 Tidying of proofs. New theorems are enterred immediately into the
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parents: 2031
diff changeset
   394
Addsimps [Int_iff];
0bc87b063447 Tidying of proofs. New theorems are enterred immediately into the
paulson
parents: 2031
diff changeset
   395
5143
b94cd208f073 Removal of leading "\!\!..." from most Goal commands
paulson
parents: 5069
diff changeset
   396
Goal "[| c:A;  c:B |] ==> c : A Int B";
2499
0bc87b063447 Tidying of proofs. New theorems are enterred immediately into the
paulson
parents: 2031
diff changeset
   397
by (Asm_simp_tac 1);
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   398
qed "IntI";
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clasohm
parents:
diff changeset
   399
5143
b94cd208f073 Removal of leading "\!\!..." from most Goal commands
paulson
parents: 5069
diff changeset
   400
Goal "c : A Int B ==> c:A";
2499
0bc87b063447 Tidying of proofs. New theorems are enterred immediately into the
paulson
parents: 2031
diff changeset
   401
by (Asm_full_simp_tac 1);
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   402
qed "IntD1";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   403
5143
b94cd208f073 Removal of leading "\!\!..." from most Goal commands
paulson
parents: 5069
diff changeset
   404
Goal "c : A Int B ==> c:B";
2499
0bc87b063447 Tidying of proofs. New theorems are enterred immediately into the
paulson
parents: 2031
diff changeset
   405
by (Asm_full_simp_tac 1);
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   406
qed "IntD2";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   407
5316
7a8975451a89 even more tidying of Goal commands
paulson
parents: 5305
diff changeset
   408
val [major,minor] = Goal
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   409
    "[| c : A Int B;  [| c:A; c:B |] ==> P |] ==> P";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   410
by (rtac minor 1);
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   411
by (rtac (major RS IntD1) 1);
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   412
by (rtac (major RS IntD2) 1);
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   413
qed "IntE";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   414
2499
0bc87b063447 Tidying of proofs. New theorems are enterred immediately into the
paulson
parents: 2031
diff changeset
   415
AddSIs [IntI];
0bc87b063447 Tidying of proofs. New theorems are enterred immediately into the
paulson
parents: 2031
diff changeset
   416
AddSEs [IntE];
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   417
1548
afe750876848 Added 'section' commands
nipkow
parents: 1531
diff changeset
   418
section "Set difference";
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   419
7031
972b5f62f476 getting rid of qed_goal
paulson
parents: 7007
diff changeset
   420
Goalw [set_diff_def] "(c : A-B) = (c:A & c~:B)";
972b5f62f476 getting rid of qed_goal
paulson
parents: 7007
diff changeset
   421
by (Blast_tac 1);
972b5f62f476 getting rid of qed_goal
paulson
parents: 7007
diff changeset
   422
qed "Diff_iff";
2499
0bc87b063447 Tidying of proofs. New theorems are enterred immediately into the
paulson
parents: 2031
diff changeset
   423
Addsimps [Diff_iff];
0bc87b063447 Tidying of proofs. New theorems are enterred immediately into the
paulson
parents: 2031
diff changeset
   424
7007
b46ccfee8e59 qed_goal -> Goal
paulson
parents: 6443
diff changeset
   425
Goal "[| c : A;  c ~: B |] ==> c : A - B";
b46ccfee8e59 qed_goal -> Goal
paulson
parents: 6443
diff changeset
   426
by (Asm_simp_tac 1) ;
b46ccfee8e59 qed_goal -> Goal
paulson
parents: 6443
diff changeset
   427
qed "DiffI";
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   428
7007
b46ccfee8e59 qed_goal -> Goal
paulson
parents: 6443
diff changeset
   429
Goal "c : A - B ==> c : A";
b46ccfee8e59 qed_goal -> Goal
paulson
parents: 6443
diff changeset
   430
by (Asm_full_simp_tac 1) ;
b46ccfee8e59 qed_goal -> Goal
paulson
parents: 6443
diff changeset
   431
qed "DiffD1";
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   432
7007
b46ccfee8e59 qed_goal -> Goal
paulson
parents: 6443
diff changeset
   433
Goal "[| c : A - B;  c : B |] ==> P";
b46ccfee8e59 qed_goal -> Goal
paulson
parents: 6443
diff changeset
   434
by (Asm_full_simp_tac 1) ;
b46ccfee8e59 qed_goal -> Goal
paulson
parents: 6443
diff changeset
   435
qed "DiffD2";
2499
0bc87b063447 Tidying of proofs. New theorems are enterred immediately into the
paulson
parents: 2031
diff changeset
   436
7031
972b5f62f476 getting rid of qed_goal
paulson
parents: 7007
diff changeset
   437
val prems = Goal "[| c : A - B;  [| c:A; c~:B |] ==> P |] ==> P";
7007
b46ccfee8e59 qed_goal -> Goal
paulson
parents: 6443
diff changeset
   438
by (resolve_tac prems 1);
b46ccfee8e59 qed_goal -> Goal
paulson
parents: 6443
diff changeset
   439
by (REPEAT (ares_tac (prems RL [DiffD1, DiffD2 RS notI]) 1)) ;
b46ccfee8e59 qed_goal -> Goal
paulson
parents: 6443
diff changeset
   440
qed "DiffE";
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   441
2499
0bc87b063447 Tidying of proofs. New theorems are enterred immediately into the
paulson
parents: 2031
diff changeset
   442
AddSIs [DiffI];
0bc87b063447 Tidying of proofs. New theorems are enterred immediately into the
paulson
parents: 2031
diff changeset
   443
AddSEs [DiffE];
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   444
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   445
1548
afe750876848 Added 'section' commands
nipkow
parents: 1531
diff changeset
   446
section "Augmenting a set -- insert";
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   447
7031
972b5f62f476 getting rid of qed_goal
paulson
parents: 7007
diff changeset
   448
Goalw [insert_def] "a : insert b A = (a=b | a:A)";
972b5f62f476 getting rid of qed_goal
paulson
parents: 7007
diff changeset
   449
by (Blast_tac 1);
972b5f62f476 getting rid of qed_goal
paulson
parents: 7007
diff changeset
   450
qed "insert_iff";
2499
0bc87b063447 Tidying of proofs. New theorems are enterred immediately into the
paulson
parents: 2031
diff changeset
   451
Addsimps [insert_iff];
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   452
7031
972b5f62f476 getting rid of qed_goal
paulson
parents: 7007
diff changeset
   453
Goal "a : insert a B";
7007
b46ccfee8e59 qed_goal -> Goal
paulson
parents: 6443
diff changeset
   454
by (Simp_tac 1);
b46ccfee8e59 qed_goal -> Goal
paulson
parents: 6443
diff changeset
   455
qed "insertI1";
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   456
7007
b46ccfee8e59 qed_goal -> Goal
paulson
parents: 6443
diff changeset
   457
Goal "!!a. a : B ==> a : insert b B";
b46ccfee8e59 qed_goal -> Goal
paulson
parents: 6443
diff changeset
   458
by (Asm_simp_tac 1);
b46ccfee8e59 qed_goal -> Goal
paulson
parents: 6443
diff changeset
   459
qed "insertI2";
b46ccfee8e59 qed_goal -> Goal
paulson
parents: 6443
diff changeset
   460
b46ccfee8e59 qed_goal -> Goal
paulson
parents: 6443
diff changeset
   461
val major::prems = Goalw [insert_def]
b46ccfee8e59 qed_goal -> Goal
paulson
parents: 6443
diff changeset
   462
    "[| a : insert b A;  a=b ==> P;  a:A ==> P |] ==> P";
b46ccfee8e59 qed_goal -> Goal
paulson
parents: 6443
diff changeset
   463
by (rtac (major RS UnE) 1);
b46ccfee8e59 qed_goal -> Goal
paulson
parents: 6443
diff changeset
   464
by (REPEAT (eresolve_tac (prems @ [CollectE]) 1));
b46ccfee8e59 qed_goal -> Goal
paulson
parents: 6443
diff changeset
   465
qed "insertE";
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   466
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   467
(*Classical introduction rule*)
7031
972b5f62f476 getting rid of qed_goal
paulson
parents: 7007
diff changeset
   468
val prems = Goal "(a~:B ==> a=b) ==> a: insert b B";
7007
b46ccfee8e59 qed_goal -> Goal
paulson
parents: 6443
diff changeset
   469
by (Simp_tac 1);
b46ccfee8e59 qed_goal -> Goal
paulson
parents: 6443
diff changeset
   470
by (REPEAT (ares_tac (prems@[disjCI]) 1)) ;
b46ccfee8e59 qed_goal -> Goal
paulson
parents: 6443
diff changeset
   471
qed "insertCI";
2499
0bc87b063447 Tidying of proofs. New theorems are enterred immediately into the
paulson
parents: 2031
diff changeset
   472
0bc87b063447 Tidying of proofs. New theorems are enterred immediately into the
paulson
parents: 2031
diff changeset
   473
AddSIs [insertCI]; 
0bc87b063447 Tidying of proofs. New theorems are enterred immediately into the
paulson
parents: 2031
diff changeset
   474
AddSEs [insertE];
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   475
7496
93ae11d887ff added theorems subset_insertD, singleton_insert_inj_eq, subset_singletonD
oheimb
parents: 7441
diff changeset
   476
Goal "A <= insert x B ==> A <= B & x ~: A | (? B'. A = insert x B' & B' <= B)";
93ae11d887ff added theorems subset_insertD, singleton_insert_inj_eq, subset_singletonD
oheimb
parents: 7441
diff changeset
   477
by (case_tac "x:A" 1);
93ae11d887ff added theorems subset_insertD, singleton_insert_inj_eq, subset_singletonD
oheimb
parents: 7441
diff changeset
   478
by  (Fast_tac 2);
7499
23e090051cb8 isatool expandshort;
wenzelm
parents: 7496
diff changeset
   479
by (rtac disjI2 1);
7496
93ae11d887ff added theorems subset_insertD, singleton_insert_inj_eq, subset_singletonD
oheimb
parents: 7441
diff changeset
   480
by (res_inst_tac [("x","A-{x}")] exI 1);
93ae11d887ff added theorems subset_insertD, singleton_insert_inj_eq, subset_singletonD
oheimb
parents: 7441
diff changeset
   481
by (Fast_tac 1);
93ae11d887ff added theorems subset_insertD, singleton_insert_inj_eq, subset_singletonD
oheimb
parents: 7441
diff changeset
   482
qed "subset_insertD";
93ae11d887ff added theorems subset_insertD, singleton_insert_inj_eq, subset_singletonD
oheimb
parents: 7441
diff changeset
   483
1548
afe750876848 Added 'section' commands
nipkow
parents: 1531
diff changeset
   484
section "Singletons, using insert";
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   485
7007
b46ccfee8e59 qed_goal -> Goal
paulson
parents: 6443
diff changeset
   486
Goal "a : {a}";
b46ccfee8e59 qed_goal -> Goal
paulson
parents: 6443
diff changeset
   487
by (rtac insertI1 1) ;
b46ccfee8e59 qed_goal -> Goal
paulson
parents: 6443
diff changeset
   488
qed "singletonI";
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   489
5143
b94cd208f073 Removal of leading "\!\!..." from most Goal commands
paulson
parents: 5069
diff changeset
   490
Goal "b : {a} ==> b=a";
2891
d8f254ad1ab9 Calls Blast_tac
paulson
parents: 2881
diff changeset
   491
by (Blast_tac 1);
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   492
qed "singletonD";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   493
1776
d7e77cb8ce5c moved mem_simps and the corresponding update of !simpset from Fun.ML to Set.ML,
oheimb
parents: 1762
diff changeset
   494
bind_thm ("singletonE", make_elim singletonD);
d7e77cb8ce5c moved mem_simps and the corresponding update of !simpset from Fun.ML to Set.ML,
oheimb
parents: 1762
diff changeset
   495
7007
b46ccfee8e59 qed_goal -> Goal
paulson
parents: 6443
diff changeset
   496
Goal "(b : {a}) = (b=a)";
b46ccfee8e59 qed_goal -> Goal
paulson
parents: 6443
diff changeset
   497
by (Blast_tac 1);
b46ccfee8e59 qed_goal -> Goal
paulson
parents: 6443
diff changeset
   498
qed "singleton_iff";
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   499
5143
b94cd208f073 Removal of leading "\!\!..." from most Goal commands
paulson
parents: 5069
diff changeset
   500
Goal "{a}={b} ==> a=b";
4089
96fba19bcbe2 isatool fixclasimp;
wenzelm
parents: 4059
diff changeset
   501
by (blast_tac (claset() addEs [equalityE]) 1);
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   502
qed "singleton_inject";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   503
2858
1f3f5c44e159 Re-ordering of rules to assist blast_tac
paulson
parents: 2721
diff changeset
   504
(*Redundant? But unlike insertCI, it proves the subgoal immediately!*)
1f3f5c44e159 Re-ordering of rules to assist blast_tac
paulson
parents: 2721
diff changeset
   505
AddSIs [singletonI];   
2499
0bc87b063447 Tidying of proofs. New theorems are enterred immediately into the
paulson
parents: 2031
diff changeset
   506
AddSDs [singleton_inject];
3718
d78cf498a88c Minor tidying to use Clarify_tac, etc.
paulson
parents: 3582
diff changeset
   507
AddSEs [singletonE];
2499
0bc87b063447 Tidying of proofs. New theorems are enterred immediately into the
paulson
parents: 2031
diff changeset
   508
7969
7a20317850ab improved singleton_insert_inj_eq
oheimb
parents: 7717
diff changeset
   509
Goal "{b} = insert a A = (a = b & A <= {b})";
8326
0e329578b0ef tidied the proofs of singleton_insert_inj_eq, singleton_insert_inj_eq' and
paulson
parents: 8053
diff changeset
   510
by (blast_tac (claset() addSEs [equalityE]) 1);
7496
93ae11d887ff added theorems subset_insertD, singleton_insert_inj_eq, subset_singletonD
oheimb
parents: 7441
diff changeset
   511
qed "singleton_insert_inj_eq";
93ae11d887ff added theorems subset_insertD, singleton_insert_inj_eq, subset_singletonD
oheimb
parents: 7441
diff changeset
   512
8326
0e329578b0ef tidied the proofs of singleton_insert_inj_eq, singleton_insert_inj_eq' and
paulson
parents: 8053
diff changeset
   513
Goal "(insert a A = {b}) = (a = b & A <= {b})";
0e329578b0ef tidied the proofs of singleton_insert_inj_eq, singleton_insert_inj_eq' and
paulson
parents: 8053
diff changeset
   514
by (blast_tac (claset() addSEs [equalityE]) 1);
7969
7a20317850ab improved singleton_insert_inj_eq
oheimb
parents: 7717
diff changeset
   515
qed "singleton_insert_inj_eq'";
7a20317850ab improved singleton_insert_inj_eq
oheimb
parents: 7717
diff changeset
   516
8326
0e329578b0ef tidied the proofs of singleton_insert_inj_eq, singleton_insert_inj_eq' and
paulson
parents: 8053
diff changeset
   517
AddIffs [singleton_insert_inj_eq, singleton_insert_inj_eq'];
0e329578b0ef tidied the proofs of singleton_insert_inj_eq, singleton_insert_inj_eq' and
paulson
parents: 8053
diff changeset
   518
7496
93ae11d887ff added theorems subset_insertD, singleton_insert_inj_eq, subset_singletonD
oheimb
parents: 7441
diff changeset
   519
Goal "A <= {x} ==> A={} | A = {x}";
93ae11d887ff added theorems subset_insertD, singleton_insert_inj_eq, subset_singletonD
oheimb
parents: 7441
diff changeset
   520
by (Fast_tac 1);
93ae11d887ff added theorems subset_insertD, singleton_insert_inj_eq, subset_singletonD
oheimb
parents: 7441
diff changeset
   521
qed "subset_singletonD";
93ae11d887ff added theorems subset_insertD, singleton_insert_inj_eq, subset_singletonD
oheimb
parents: 7441
diff changeset
   522
5069
3ea049f7979d isatool fixgoal;
wenzelm
parents: 4830
diff changeset
   523
Goal "{x. x=a} = {a}";
4423
a129b817b58a expandshort;
wenzelm
parents: 4240
diff changeset
   524
by (Blast_tac 1);
3582
b87c86b6c291 Added {x.x=a} = a to !simpset.
nipkow
parents: 3469
diff changeset
   525
qed "singleton_conv";
b87c86b6c291 Added {x.x=a} = a to !simpset.
nipkow
parents: 3469
diff changeset
   526
Addsimps [singleton_conv];
1531
e5eb247ad13c Added a constant UNIV == {x.True}
nipkow
parents: 1465
diff changeset
   527
5600
34b3366b83ac new singleton_conv2
nipkow
parents: 5521
diff changeset
   528
Goal "{x. a=x} = {a}";
6301
08245f5a436d expandshort
paulson
parents: 6291
diff changeset
   529
by (Blast_tac 1);
5600
34b3366b83ac new singleton_conv2
nipkow
parents: 5521
diff changeset
   530
qed "singleton_conv2";
34b3366b83ac new singleton_conv2
nipkow
parents: 5521
diff changeset
   531
Addsimps [singleton_conv2];
34b3366b83ac new singleton_conv2
nipkow
parents: 5521
diff changeset
   532
1531
e5eb247ad13c Added a constant UNIV == {x.True}
nipkow
parents: 1465
diff changeset
   533
1548
afe750876848 Added 'section' commands
nipkow
parents: 1531
diff changeset
   534
section "Unions of families -- UNION x:A. B(x) is Union(B``A)";
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   535
5069
3ea049f7979d isatool fixgoal;
wenzelm
parents: 4830
diff changeset
   536
Goalw [UNION_def] "(b: (UN x:A. B(x))) = (EX x:A. b: B(x))";
2891
d8f254ad1ab9 Calls Blast_tac
paulson
parents: 2881
diff changeset
   537
by (Blast_tac 1);
2499
0bc87b063447 Tidying of proofs. New theorems are enterred immediately into the
paulson
parents: 2031
diff changeset
   538
qed "UN_iff";
0bc87b063447 Tidying of proofs. New theorems are enterred immediately into the
paulson
parents: 2031
diff changeset
   539
0bc87b063447 Tidying of proofs. New theorems are enterred immediately into the
paulson
parents: 2031
diff changeset
   540
Addsimps [UN_iff];
0bc87b063447 Tidying of proofs. New theorems are enterred immediately into the
paulson
parents: 2031
diff changeset
   541
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   542
(*The order of the premises presupposes that A is rigid; b may be flexible*)
5143
b94cd208f073 Removal of leading "\!\!..." from most Goal commands
paulson
parents: 5069
diff changeset
   543
Goal "[| a:A;  b: B(a) |] ==> b: (UN x:A. B(x))";
4477
b3e5857d8d99 New Auto_tac (by Oheimb), and new syntax (without parens), and expandshort
paulson
parents: 4469
diff changeset
   544
by Auto_tac;
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   545
qed "UN_I";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   546
5316
7a8975451a89 even more tidying of Goal commands
paulson
parents: 5305
diff changeset
   547
val major::prems = Goalw [UNION_def]
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   548
    "[| b : (UN x:A. B(x));  !!x.[| x:A;  b: B(x) |] ==> R |] ==> R";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   549
by (rtac (major RS CollectD RS bexE) 1);
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   550
by (REPEAT (ares_tac prems 1));
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   551
qed "UN_E";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   552
2499
0bc87b063447 Tidying of proofs. New theorems are enterred immediately into the
paulson
parents: 2031
diff changeset
   553
AddIs  [UN_I];
0bc87b063447 Tidying of proofs. New theorems are enterred immediately into the
paulson
parents: 2031
diff changeset
   554
AddSEs [UN_E];
0bc87b063447 Tidying of proofs. New theorems are enterred immediately into the
paulson
parents: 2031
diff changeset
   555
6291
2c3f72d9f5d1 simpler proofs of congruence rules
paulson
parents: 6171
diff changeset
   556
val prems = Goalw [UNION_def]
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   557
    "[| A=B;  !!x. x:B ==> C(x) = D(x) |] ==> \
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   558
\    (UN x:A. C(x)) = (UN x:B. D(x))";
6291
2c3f72d9f5d1 simpler proofs of congruence rules
paulson
parents: 6171
diff changeset
   559
by (asm_simp_tac (simpset() addsimps prems) 1);
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   560
qed "UN_cong";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   561
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   562
1548
afe750876848 Added 'section' commands
nipkow
parents: 1531
diff changeset
   563
section "Intersections of families -- INTER x:A. B(x) is Inter(B``A)";
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   564
5069
3ea049f7979d isatool fixgoal;
wenzelm
parents: 4830
diff changeset
   565
Goalw [INTER_def] "(b: (INT x:A. B(x))) = (ALL x:A. b: B(x))";
4477
b3e5857d8d99 New Auto_tac (by Oheimb), and new syntax (without parens), and expandshort
paulson
parents: 4469
diff changeset
   566
by Auto_tac;
2499
0bc87b063447 Tidying of proofs. New theorems are enterred immediately into the
paulson
parents: 2031
diff changeset
   567
qed "INT_iff";
0bc87b063447 Tidying of proofs. New theorems are enterred immediately into the
paulson
parents: 2031
diff changeset
   568
0bc87b063447 Tidying of proofs. New theorems are enterred immediately into the
paulson
parents: 2031
diff changeset
   569
Addsimps [INT_iff];
0bc87b063447 Tidying of proofs. New theorems are enterred immediately into the
paulson
parents: 2031
diff changeset
   570
5316
7a8975451a89 even more tidying of Goal commands
paulson
parents: 5305
diff changeset
   571
val prems = Goalw [INTER_def]
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   572
    "(!!x. x:A ==> b: B(x)) ==> b : (INT x:A. B(x))";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   573
by (REPEAT (ares_tac ([CollectI,ballI] @ prems) 1));
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   574
qed "INT_I";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   575
5143
b94cd208f073 Removal of leading "\!\!..." from most Goal commands
paulson
parents: 5069
diff changeset
   576
Goal "[| b : (INT x:A. B(x));  a:A |] ==> b: B(a)";
4477
b3e5857d8d99 New Auto_tac (by Oheimb), and new syntax (without parens), and expandshort
paulson
parents: 4469
diff changeset
   577
by Auto_tac;
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   578
qed "INT_D";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   579
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   580
(*"Classical" elimination -- by the Excluded Middle on a:A *)
5316
7a8975451a89 even more tidying of Goal commands
paulson
parents: 5305
diff changeset
   581
val major::prems = Goalw [INTER_def]
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   582
    "[| b : (INT x:A. B(x));  b: B(a) ==> R;  a~:A ==> R |] ==> R";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   583
by (rtac (major RS CollectD RS ballE) 1);
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   584
by (REPEAT (eresolve_tac prems 1));
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   585
qed "INT_E";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   586
2499
0bc87b063447 Tidying of proofs. New theorems are enterred immediately into the
paulson
parents: 2031
diff changeset
   587
AddSIs [INT_I];
0bc87b063447 Tidying of proofs. New theorems are enterred immediately into the
paulson
parents: 2031
diff changeset
   588
AddEs  [INT_D, INT_E];
0bc87b063447 Tidying of proofs. New theorems are enterred immediately into the
paulson
parents: 2031
diff changeset
   589
6291
2c3f72d9f5d1 simpler proofs of congruence rules
paulson
parents: 6171
diff changeset
   590
val prems = Goalw [INTER_def]
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   591
    "[| A=B;  !!x. x:B ==> C(x) = D(x) |] ==> \
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   592
\    (INT x:A. C(x)) = (INT x:B. D(x))";
6291
2c3f72d9f5d1 simpler proofs of congruence rules
paulson
parents: 6171
diff changeset
   593
by (asm_simp_tac (simpset() addsimps prems) 1);
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   594
qed "INT_cong";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   595
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   596
1548
afe750876848 Added 'section' commands
nipkow
parents: 1531
diff changeset
   597
section "Union";
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   598
5069
3ea049f7979d isatool fixgoal;
wenzelm
parents: 4830
diff changeset
   599
Goalw [Union_def] "(A : Union(C)) = (EX X:C. A:X)";
2891
d8f254ad1ab9 Calls Blast_tac
paulson
parents: 2881
diff changeset
   600
by (Blast_tac 1);
2499
0bc87b063447 Tidying of proofs. New theorems are enterred immediately into the
paulson
parents: 2031
diff changeset
   601
qed "Union_iff";
0bc87b063447 Tidying of proofs. New theorems are enterred immediately into the
paulson
parents: 2031
diff changeset
   602
0bc87b063447 Tidying of proofs. New theorems are enterred immediately into the
paulson
parents: 2031
diff changeset
   603
Addsimps [Union_iff];
0bc87b063447 Tidying of proofs. New theorems are enterred immediately into the
paulson
parents: 2031
diff changeset
   604
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   605
(*The order of the premises presupposes that C is rigid; A may be flexible*)
5143
b94cd208f073 Removal of leading "\!\!..." from most Goal commands
paulson
parents: 5069
diff changeset
   606
Goal "[| X:C;  A:X |] ==> A : Union(C)";
4477
b3e5857d8d99 New Auto_tac (by Oheimb), and new syntax (without parens), and expandshort
paulson
parents: 4469
diff changeset
   607
by Auto_tac;
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   608
qed "UnionI";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   609
5316
7a8975451a89 even more tidying of Goal commands
paulson
parents: 5305
diff changeset
   610
val major::prems = Goalw [Union_def]
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   611
    "[| A : Union(C);  !!X.[| A:X;  X:C |] ==> R |] ==> R";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   612
by (rtac (major RS UN_E) 1);
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   613
by (REPEAT (ares_tac prems 1));
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   614
qed "UnionE";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   615
2499
0bc87b063447 Tidying of proofs. New theorems are enterred immediately into the
paulson
parents: 2031
diff changeset
   616
AddIs  [UnionI];
0bc87b063447 Tidying of proofs. New theorems are enterred immediately into the
paulson
parents: 2031
diff changeset
   617
AddSEs [UnionE];
0bc87b063447 Tidying of proofs. New theorems are enterred immediately into the
paulson
parents: 2031
diff changeset
   618
0bc87b063447 Tidying of proofs. New theorems are enterred immediately into the
paulson
parents: 2031
diff changeset
   619
1548
afe750876848 Added 'section' commands
nipkow
parents: 1531
diff changeset
   620
section "Inter";
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   621
5069
3ea049f7979d isatool fixgoal;
wenzelm
parents: 4830
diff changeset
   622
Goalw [Inter_def] "(A : Inter(C)) = (ALL X:C. A:X)";
2891
d8f254ad1ab9 Calls Blast_tac
paulson
parents: 2881
diff changeset
   623
by (Blast_tac 1);
2499
0bc87b063447 Tidying of proofs. New theorems are enterred immediately into the
paulson
parents: 2031
diff changeset
   624
qed "Inter_iff";
0bc87b063447 Tidying of proofs. New theorems are enterred immediately into the
paulson
parents: 2031
diff changeset
   625
0bc87b063447 Tidying of proofs. New theorems are enterred immediately into the
paulson
parents: 2031
diff changeset
   626
Addsimps [Inter_iff];
0bc87b063447 Tidying of proofs. New theorems are enterred immediately into the
paulson
parents: 2031
diff changeset
   627
5316
7a8975451a89 even more tidying of Goal commands
paulson
parents: 5305
diff changeset
   628
val prems = Goalw [Inter_def]
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   629
    "[| !!X. X:C ==> A:X |] ==> A : Inter(C)";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   630
by (REPEAT (ares_tac ([INT_I] @ prems) 1));
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   631
qed "InterI";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   632
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   633
(*A "destruct" rule -- every X in C contains A as an element, but
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   634
  A:X can hold when X:C does not!  This rule is analogous to "spec". *)
5143
b94cd208f073 Removal of leading "\!\!..." from most Goal commands
paulson
parents: 5069
diff changeset
   635
Goal "[| A : Inter(C);  X:C |] ==> A:X";
4477
b3e5857d8d99 New Auto_tac (by Oheimb), and new syntax (without parens), and expandshort
paulson
parents: 4469
diff changeset
   636
by Auto_tac;
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   637
qed "InterD";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   638
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   639
(*"Classical" elimination rule -- does not require proving X:C *)
5316
7a8975451a89 even more tidying of Goal commands
paulson
parents: 5305
diff changeset
   640
val major::prems = Goalw [Inter_def]
2721
f08042e18c7d New version of InterE, like its ZF counterpart
paulson
parents: 2608
diff changeset
   641
    "[| A : Inter(C);  X~:C ==> R;  A:X ==> R |] ==> R";
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   642
by (rtac (major RS INT_E) 1);
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   643
by (REPEAT (eresolve_tac prems 1));
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   644
qed "InterE";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   645
2499
0bc87b063447 Tidying of proofs. New theorems are enterred immediately into the
paulson
parents: 2031
diff changeset
   646
AddSIs [InterI];
0bc87b063447 Tidying of proofs. New theorems are enterred immediately into the
paulson
parents: 2031
diff changeset
   647
AddEs  [InterD, InterE];
0bc87b063447 Tidying of proofs. New theorems are enterred immediately into the
paulson
parents: 2031
diff changeset
   648
0bc87b063447 Tidying of proofs. New theorems are enterred immediately into the
paulson
parents: 2031
diff changeset
   649
2912
3fac3e8d5d3e moved inj and surj from Set to Fun and Inv -> inv.
nipkow
parents: 2891
diff changeset
   650
(*** Image of a set under a function ***)
3fac3e8d5d3e moved inj and surj from Set to Fun and Inv -> inv.
nipkow
parents: 2891
diff changeset
   651
3fac3e8d5d3e moved inj and surj from Set to Fun and Inv -> inv.
nipkow
parents: 2891
diff changeset
   652
(*Frequently b does not have the syntactic form of f(x).*)
5316
7a8975451a89 even more tidying of Goal commands
paulson
parents: 5305
diff changeset
   653
Goalw [image_def] "[| b=f(x);  x:A |] ==> b : f``A";
7a8975451a89 even more tidying of Goal commands
paulson
parents: 5305
diff changeset
   654
by (Blast_tac 1);
2912
3fac3e8d5d3e moved inj and surj from Set to Fun and Inv -> inv.
nipkow
parents: 2891
diff changeset
   655
qed "image_eqI";
3909
e48e2fb8b895 Added image_eqI to simpset.
nipkow
parents: 3905
diff changeset
   656
Addsimps [image_eqI];
2912
3fac3e8d5d3e moved inj and surj from Set to Fun and Inv -> inv.
nipkow
parents: 2891
diff changeset
   657
3fac3e8d5d3e moved inj and surj from Set to Fun and Inv -> inv.
nipkow
parents: 2891
diff changeset
   658
bind_thm ("imageI", refl RS image_eqI);
3fac3e8d5d3e moved inj and surj from Set to Fun and Inv -> inv.
nipkow
parents: 2891
diff changeset
   659
8025
61dde9078e24 new theorem rev_image_eqI
paulson
parents: 8005
diff changeset
   660
(*This version's more effective when we already have the required x*)
61dde9078e24 new theorem rev_image_eqI
paulson
parents: 8005
diff changeset
   661
Goalw [image_def] "[| x:A;  b=f(x) |] ==> b : f``A";
61dde9078e24 new theorem rev_image_eqI
paulson
parents: 8005
diff changeset
   662
by (Blast_tac 1);
61dde9078e24 new theorem rev_image_eqI
paulson
parents: 8005
diff changeset
   663
qed "rev_image_eqI";
61dde9078e24 new theorem rev_image_eqI
paulson
parents: 8005
diff changeset
   664
2912
3fac3e8d5d3e moved inj and surj from Set to Fun and Inv -> inv.
nipkow
parents: 2891
diff changeset
   665
(*The eta-expansion gives variable-name preservation.*)
5316
7a8975451a89 even more tidying of Goal commands
paulson
parents: 5305
diff changeset
   666
val major::prems = Goalw [image_def]
3842
b55686a7b22c fixed dots;
wenzelm
parents: 3718
diff changeset
   667
    "[| b : (%x. f(x))``A;  !!x.[| b=f(x);  x:A |] ==> P |] ==> P"; 
2912
3fac3e8d5d3e moved inj and surj from Set to Fun and Inv -> inv.
nipkow
parents: 2891
diff changeset
   668
by (rtac (major RS CollectD RS bexE) 1);
3fac3e8d5d3e moved inj and surj from Set to Fun and Inv -> inv.
nipkow
parents: 2891
diff changeset
   669
by (REPEAT (ares_tac prems 1));
3fac3e8d5d3e moved inj and surj from Set to Fun and Inv -> inv.
nipkow
parents: 2891
diff changeset
   670
qed "imageE";
3fac3e8d5d3e moved inj and surj from Set to Fun and Inv -> inv.
nipkow
parents: 2891
diff changeset
   671
3fac3e8d5d3e moved inj and surj from Set to Fun and Inv -> inv.
nipkow
parents: 2891
diff changeset
   672
AddIs  [image_eqI];
3fac3e8d5d3e moved inj and surj from Set to Fun and Inv -> inv.
nipkow
parents: 2891
diff changeset
   673
AddSEs [imageE]; 
3fac3e8d5d3e moved inj and surj from Set to Fun and Inv -> inv.
nipkow
parents: 2891
diff changeset
   674
5069
3ea049f7979d isatool fixgoal;
wenzelm
parents: 4830
diff changeset
   675
Goal "f``(A Un B) = f``A Un f``B";
2935
998cb95fdd43 Yet more fast_tac->blast_tac, and other tidying
paulson
parents: 2912
diff changeset
   676
by (Blast_tac 1);
2912
3fac3e8d5d3e moved inj and surj from Set to Fun and Inv -> inv.
nipkow
parents: 2891
diff changeset
   677
qed "image_Un";
3fac3e8d5d3e moved inj and surj from Set to Fun and Inv -> inv.
nipkow
parents: 2891
diff changeset
   678
5069
3ea049f7979d isatool fixgoal;
wenzelm
parents: 4830
diff changeset
   679
Goal "(z : f``A) = (EX x:A. z = f x)";
3960
7a38fae985f9 New rewrite rules image_iff
paulson
parents: 3919
diff changeset
   680
by (Blast_tac 1);
7a38fae985f9 New rewrite rules image_iff
paulson
parents: 3919
diff changeset
   681
qed "image_iff";
7a38fae985f9 New rewrite rules image_iff
paulson
parents: 3919
diff changeset
   682
4523
16f5efe9812d New rule: image_subset
paulson
parents: 4510
diff changeset
   683
(*This rewrite rule would confuse users if made default.*)
5069
3ea049f7979d isatool fixgoal;
wenzelm
parents: 4830
diff changeset
   684
Goal "(f``A <= B) = (ALL x:A. f(x): B)";
4523
16f5efe9812d New rule: image_subset
paulson
parents: 4510
diff changeset
   685
by (Blast_tac 1);
16f5efe9812d New rule: image_subset
paulson
parents: 4510
diff changeset
   686
qed "image_subset_iff";
16f5efe9812d New rule: image_subset
paulson
parents: 4510
diff changeset
   687
16f5efe9812d New rule: image_subset
paulson
parents: 4510
diff changeset
   688
(*Replaces the three steps subsetI, imageE, hyp_subst_tac, but breaks too
16f5efe9812d New rule: image_subset
paulson
parents: 4510
diff changeset
   689
  many existing proofs.*)
5316
7a8975451a89 even more tidying of Goal commands
paulson
parents: 5305
diff changeset
   690
val prems = Goal "(!!x. x:A ==> f(x) : B) ==> f``A <= B";
4510
a37f515a0b25 New theorem image_subsetI
paulson
parents: 4477
diff changeset
   691
by (blast_tac (claset() addIs prems) 1);
a37f515a0b25 New theorem image_subsetI
paulson
parents: 4477
diff changeset
   692
qed "image_subsetI";
a37f515a0b25 New theorem image_subsetI
paulson
parents: 4477
diff changeset
   693
2912
3fac3e8d5d3e moved inj and surj from Set to Fun and Inv -> inv.
nipkow
parents: 2891
diff changeset
   694
3fac3e8d5d3e moved inj and surj from Set to Fun and Inv -> inv.
nipkow
parents: 2891
diff changeset
   695
(*** Range of a function -- just a translation for image! ***)
3fac3e8d5d3e moved inj and surj from Set to Fun and Inv -> inv.
nipkow
parents: 2891
diff changeset
   696
5143
b94cd208f073 Removal of leading "\!\!..." from most Goal commands
paulson
parents: 5069
diff changeset
   697
Goal "b=f(x) ==> b : range(f)";
2912
3fac3e8d5d3e moved inj and surj from Set to Fun and Inv -> inv.
nipkow
parents: 2891
diff changeset
   698
by (EVERY1 [etac image_eqI, rtac UNIV_I]);
3fac3e8d5d3e moved inj and surj from Set to Fun and Inv -> inv.
nipkow
parents: 2891
diff changeset
   699
bind_thm ("range_eqI", UNIV_I RSN (2,image_eqI));
3fac3e8d5d3e moved inj and surj from Set to Fun and Inv -> inv.
nipkow
parents: 2891
diff changeset
   700
3fac3e8d5d3e moved inj and surj from Set to Fun and Inv -> inv.
nipkow
parents: 2891
diff changeset
   701
bind_thm ("rangeI", UNIV_I RS imageI);
3fac3e8d5d3e moved inj and surj from Set to Fun and Inv -> inv.
nipkow
parents: 2891
diff changeset
   702
5316
7a8975451a89 even more tidying of Goal commands
paulson
parents: 5305
diff changeset
   703
val [major,minor] = Goal 
3842
b55686a7b22c fixed dots;
wenzelm
parents: 3718
diff changeset
   704
    "[| b : range(%x. f(x));  !!x. b=f(x) ==> P |] ==> P"; 
2912
3fac3e8d5d3e moved inj and surj from Set to Fun and Inv -> inv.
nipkow
parents: 2891
diff changeset
   705
by (rtac (major RS imageE) 1);
3fac3e8d5d3e moved inj and surj from Set to Fun and Inv -> inv.
nipkow
parents: 2891
diff changeset
   706
by (etac minor 1);
3fac3e8d5d3e moved inj and surj from Set to Fun and Inv -> inv.
nipkow
parents: 2891
diff changeset
   707
qed "rangeE";
3fac3e8d5d3e moved inj and surj from Set to Fun and Inv -> inv.
nipkow
parents: 2891
diff changeset
   708
1776
d7e77cb8ce5c moved mem_simps and the corresponding update of !simpset from Fun.ML to Set.ML,
oheimb
parents: 1762
diff changeset
   709
d7e77cb8ce5c moved mem_simps and the corresponding update of !simpset from Fun.ML to Set.ML,
oheimb
parents: 1762
diff changeset
   710
(*** Set reasoning tools ***)
d7e77cb8ce5c moved mem_simps and the corresponding update of !simpset from Fun.ML to Set.ML,
oheimb
parents: 1762
diff changeset
   711
d7e77cb8ce5c moved mem_simps and the corresponding update of !simpset from Fun.ML to Set.ML,
oheimb
parents: 1762
diff changeset
   712
3912
4ed64ad7fb42 New rewrite rules for simplifying conditionals
paulson
parents: 3909
diff changeset
   713
(** Rewrite rules for boolean case-splitting: faster than 
4830
bd73675adbed Added a few lemmas.
nipkow
parents: 4770
diff changeset
   714
	addsplits[split_if]
3912
4ed64ad7fb42 New rewrite rules for simplifying conditionals
paulson
parents: 3909
diff changeset
   715
**)
4ed64ad7fb42 New rewrite rules for simplifying conditionals
paulson
parents: 3909
diff changeset
   716
4830
bd73675adbed Added a few lemmas.
nipkow
parents: 4770
diff changeset
   717
bind_thm ("split_if_eq1", read_instantiate [("P", "%x. x = ?b")] split_if);
bd73675adbed Added a few lemmas.
nipkow
parents: 4770
diff changeset
   718
bind_thm ("split_if_eq2", read_instantiate [("P", "%x. ?a = x")] split_if);
3912
4ed64ad7fb42 New rewrite rules for simplifying conditionals
paulson
parents: 3909
diff changeset
   719
5237
aebc63048f2d Deleted the redundant rule mem_if
paulson
parents: 5148
diff changeset
   720
(*Split ifs on either side of the membership relation.
aebc63048f2d Deleted the redundant rule mem_if
paulson
parents: 5148
diff changeset
   721
	Not for Addsimps -- can cause goals to blow up!*)
4830
bd73675adbed Added a few lemmas.
nipkow
parents: 4770
diff changeset
   722
bind_thm ("split_if_mem1", 
6394
3d9fd50fcc43 Theory.sign_of;
wenzelm
parents: 6301
diff changeset
   723
    read_instantiate_sg (Theory.sign_of Set.thy) [("P", "%x. x : ?b")] split_if);
4830
bd73675adbed Added a few lemmas.
nipkow
parents: 4770
diff changeset
   724
bind_thm ("split_if_mem2", 
6394
3d9fd50fcc43 Theory.sign_of;
wenzelm
parents: 6301
diff changeset
   725
    read_instantiate_sg (Theory.sign_of Set.thy) [("P", "%x. ?a : x")] split_if);
3912
4ed64ad7fb42 New rewrite rules for simplifying conditionals
paulson
parents: 3909
diff changeset
   726
4830
bd73675adbed Added a few lemmas.
nipkow
parents: 4770
diff changeset
   727
val split_ifs = [if_bool_eq_conj, split_if_eq1, split_if_eq2,
bd73675adbed Added a few lemmas.
nipkow
parents: 4770
diff changeset
   728
		  split_if_mem1, split_if_mem2];
3912
4ed64ad7fb42 New rewrite rules for simplifying conditionals
paulson
parents: 3909
diff changeset
   729
4ed64ad7fb42 New rewrite rules for simplifying conditionals
paulson
parents: 3909
diff changeset
   730
4089
96fba19bcbe2 isatool fixclasimp;
wenzelm
parents: 4059
diff changeset
   731
(*Each of these has ALREADY been added to simpset() above.*)
2024
909153d8318f Rationalized the rewriting of membership for {} and insert
paulson
parents: 1985
diff changeset
   732
val mem_simps = [insert_iff, empty_iff, Un_iff, Int_iff, Compl_iff, Diff_iff, 
4159
4aff9b7e5597 UNIV now a constant; UNION1, INTER1 now translations and no longer have
paulson
parents: 4135
diff changeset
   733
                 mem_Collect_eq, UN_iff, Union_iff, INT_iff, Inter_iff];
1776
d7e77cb8ce5c moved mem_simps and the corresponding update of !simpset from Fun.ML to Set.ML,
oheimb
parents: 1762
diff changeset
   734
d7e77cb8ce5c moved mem_simps and the corresponding update of !simpset from Fun.ML to Set.ML,
oheimb
parents: 1762
diff changeset
   735
val mksimps_pairs = ("Ball",[bspec]) :: mksimps_pairs;
d7e77cb8ce5c moved mem_simps and the corresponding update of !simpset from Fun.ML to Set.ML,
oheimb
parents: 1762
diff changeset
   736
6291
2c3f72d9f5d1 simpler proofs of congruence rules
paulson
parents: 6171
diff changeset
   737
simpset_ref() := simpset() setmksimps (mksimps mksimps_pairs);
3222
726a9b069947 Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents: 2935
diff changeset
   738
5256
e6983301ce5e Renamed equals0D to equals0E
paulson
parents: 5237
diff changeset
   739
Addsimps[subset_UNIV, subset_refl];
3222
726a9b069947 Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents: 2935
diff changeset
   740
726a9b069947 Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents: 2935
diff changeset
   741
8001
14c8843cd35b new psubset lemma
paulson
parents: 7969
diff changeset
   742
(*** The 'proper subset' relation (<) ***)
3222
726a9b069947 Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents: 2935
diff changeset
   743
5069
3ea049f7979d isatool fixgoal;
wenzelm
parents: 4830
diff changeset
   744
Goalw [psubset_def] "!!A::'a set. [| A <= B; A ~= B |] ==> A<B";
3222
726a9b069947 Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents: 2935
diff changeset
   745
by (Blast_tac 1);
726a9b069947 Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents: 2935
diff changeset
   746
qed "psubsetI";
8913
0bc13d5e60b8 psubsetI is a safe rule
paulson
parents: 8839
diff changeset
   747
AddSIs [psubsetI];
3222
726a9b069947 Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents: 2935
diff changeset
   748
5148
74919e8f221c More tidying and removal of "\!\!... from Goal commands
paulson
parents: 5143
diff changeset
   749
Goalw [psubset_def] "A < insert x B ==> (x ~: A) & A<=B | x:A & A-{x}<B";
4477
b3e5857d8d99 New Auto_tac (by Oheimb), and new syntax (without parens), and expandshort
paulson
parents: 4469
diff changeset
   750
by Auto_tac;
3222
726a9b069947 Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents: 2935
diff changeset
   751
qed "psubset_insertD";
4059
59c1422c9da5 New Blast_tac (and minor tidying...)
paulson
parents: 3960
diff changeset
   752
59c1422c9da5 New Blast_tac (and minor tidying...)
paulson
parents: 3960
diff changeset
   753
bind_thm ("psubset_eq", psubset_def RS meta_eq_to_obj_eq);
6443
6d5d3ecedf50 lemmas about proper subset relation;
wenzelm
parents: 6394
diff changeset
   754
6d5d3ecedf50 lemmas about proper subset relation;
wenzelm
parents: 6394
diff changeset
   755
bind_thm ("psubset_imp_subset", psubset_eq RS iffD1 RS conjunct1);
6d5d3ecedf50 lemmas about proper subset relation;
wenzelm
parents: 6394
diff changeset
   756
6d5d3ecedf50 lemmas about proper subset relation;
wenzelm
parents: 6394
diff changeset
   757
Goal"[| (A::'a set) < B; B <= C |] ==> A < C";
6d5d3ecedf50 lemmas about proper subset relation;
wenzelm
parents: 6394
diff changeset
   758
by (auto_tac (claset(), simpset() addsimps [psubset_eq]));
6d5d3ecedf50 lemmas about proper subset relation;
wenzelm
parents: 6394
diff changeset
   759
qed "psubset_subset_trans";
6d5d3ecedf50 lemmas about proper subset relation;
wenzelm
parents: 6394
diff changeset
   760
6d5d3ecedf50 lemmas about proper subset relation;
wenzelm
parents: 6394
diff changeset
   761
Goal"[| (A::'a set) <= B; B < C|] ==> A < C";
6d5d3ecedf50 lemmas about proper subset relation;
wenzelm
parents: 6394
diff changeset
   762
by (auto_tac (claset(), simpset() addsimps [psubset_eq]));
6d5d3ecedf50 lemmas about proper subset relation;
wenzelm
parents: 6394
diff changeset
   763
qed "subset_psubset_trans";
7717
e7ecfa617443 Added attribute rulify_prems (useful for modifying premises of introduction
berghofe
parents: 7658
diff changeset
   764
8001
14c8843cd35b new psubset lemma
paulson
parents: 7969
diff changeset
   765
Goalw [psubset_def] "A < B ==> EX b. b : (B - A)";
14c8843cd35b new psubset lemma
paulson
parents: 7969
diff changeset
   766
by (Blast_tac 1);
14c8843cd35b new psubset lemma
paulson
parents: 7969
diff changeset
   767
qed "psubset_imp_ex_mem";
14c8843cd35b new psubset lemma
paulson
parents: 7969
diff changeset
   768
7717
e7ecfa617443 Added attribute rulify_prems (useful for modifying premises of introduction
berghofe
parents: 7658
diff changeset
   769
e7ecfa617443 Added attribute rulify_prems (useful for modifying premises of introduction
berghofe
parents: 7658
diff changeset
   770
(* attributes *)
e7ecfa617443 Added attribute rulify_prems (useful for modifying premises of introduction
berghofe
parents: 7658
diff changeset
   771
e7ecfa617443 Added attribute rulify_prems (useful for modifying premises of introduction
berghofe
parents: 7658
diff changeset
   772
local
e7ecfa617443 Added attribute rulify_prems (useful for modifying premises of introduction
berghofe
parents: 7658
diff changeset
   773
e7ecfa617443 Added attribute rulify_prems (useful for modifying premises of introduction
berghofe
parents: 7658
diff changeset
   774
fun gen_rulify_prems x =
e7ecfa617443 Added attribute rulify_prems (useful for modifying premises of introduction
berghofe
parents: 7658
diff changeset
   775
  Attrib.no_args (Drule.rule_attribute (fn _ => (standard o
e7ecfa617443 Added attribute rulify_prems (useful for modifying premises of introduction
berghofe
parents: 7658
diff changeset
   776
    rule_by_tactic (REPEAT (ALLGOALS (resolve_tac [allI, ballI, impI])))))) x;
e7ecfa617443 Added attribute rulify_prems (useful for modifying premises of introduction
berghofe
parents: 7658
diff changeset
   777
e7ecfa617443 Added attribute rulify_prems (useful for modifying premises of introduction
berghofe
parents: 7658
diff changeset
   778
in
e7ecfa617443 Added attribute rulify_prems (useful for modifying premises of introduction
berghofe
parents: 7658
diff changeset
   779
e7ecfa617443 Added attribute rulify_prems (useful for modifying premises of introduction
berghofe
parents: 7658
diff changeset
   780
val rulify_prems_attrib_setup =
e7ecfa617443 Added attribute rulify_prems (useful for modifying premises of introduction
berghofe
parents: 7658
diff changeset
   781
 [Attrib.add_attributes
e7ecfa617443 Added attribute rulify_prems (useful for modifying premises of introduction
berghofe
parents: 7658
diff changeset
   782
  [("rulify_prems", (gen_rulify_prems, gen_rulify_prems), "put theorem into standard rule form")]];
e7ecfa617443 Added attribute rulify_prems (useful for modifying premises of introduction
berghofe
parents: 7658
diff changeset
   783
e7ecfa617443 Added attribute rulify_prems (useful for modifying premises of introduction
berghofe
parents: 7658
diff changeset
   784
end;