author | paulson |
Wed, 15 Jul 1998 10:15:13 +0200 | |
changeset 5143 | b94cd208f073 |
parent 5069 | 3ea049f7979d |
child 5148 | 74919e8f221c |
permissions | -rw-r--r-- |
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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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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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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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||
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val prems = goal Set.thy "!!a. 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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||
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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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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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||
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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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||
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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 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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||
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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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||
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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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||
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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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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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||
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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=> |
72 |
[ (rtac classical 1), |
|
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(REPEAT (ares_tac (prems@[bexI,ballI,notI,notE]) 1)) ]); |
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||
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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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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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83 |
|
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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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|
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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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|
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Addsimps [ball_triv, bex_triv]; |
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|
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(** Congruence rules **) |
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||
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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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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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||
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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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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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||
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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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Blast.overloaded ("op <=", domain_type); (*The <= relation is overloaded*) |
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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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seq (fn a => Blast.overloaded (a, HOLogic.dest_setT o domain_type)) |
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["Ball", "Bex"]; |
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(*need UNION, INTER also?*) |
|
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||
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(*Image: retain the type of the set being expressed*) |
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Blast.overloaded ("op ``", domain_type o domain_type); |
|
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|
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(*Rule in Modus Ponens style*) |
133 |
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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||
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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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||
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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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||
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qed_goal "contra_subsetD" Set.thy "!!c. [| A <= B; c ~: B |] ==> c ~: A" |
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(fn prems=> [ (REPEAT (eresolve_tac [asm_rl, contrapos, subsetD] 1)) ]); |
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||
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qed_goal "rev_contra_subsetD" Set.thy "!!c. [| c ~: B; A <= B |] ==> c ~: A" |
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(fn prems=> [ (REPEAT (eresolve_tac [asm_rl, contrapos, subsetD] 1)) ]); |
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||
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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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AddSIs [subsetI]; |
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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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(fn _=> [Fast_tac 1]); (*Blast_tac would try order_refl and fail*) |
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|
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val prems = goal Set.thy "!!B. [| 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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|
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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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by (rtac (iffI RS set_ext) 1); |
|
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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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||
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AddSIs [equalityI]; |
182 |
||
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(* Equality rules from ZF set theory -- are they appropriate here? *) |
184 |
val prems = goal Set.thy "A = B ==> A<=(B::'a set)"; |
|
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by (resolve_tac (prems RL [subst]) 1); |
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by (rtac subset_refl 1); |
|
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qed "equalityD1"; |
|
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||
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val prems = goal Set.thy "A = B ==> B<=(A::'a set)"; |
|
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by (resolve_tac (prems RL [subst]) 1); |
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by (rtac subset_refl 1); |
|
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qed "equalityD2"; |
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||
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val prems = goal Set.thy |
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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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199 |
||
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val major::prems = goal Set.thy |
|
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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"; |
|
205 |
||
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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 Set.thy |
|
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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)); |
|
213 |
qed "setup_induction"; |
|
214 |
||
215 |
||
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section "The universal set -- UNIV"; |
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qed_goalw "UNIV_I" Set.thy [UNIV_def] "x : UNIV" |
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(fn _ => [rtac CollectI 1, rtac TrueI 1]); |
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Addsimps [UNIV_I]; |
222 |
AddIs [UNIV_I]; (*unsafe makes it less likely to cause problems*) |
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qed_goal "subset_UNIV" Set.thy "A <= UNIV" |
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(fn _ => [rtac subsetI 1, rtac UNIV_I 1]); |
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|
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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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229 |
|
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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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|
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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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237 |
Addsimps [ball_UNIV, bex_UNIV]; |
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|
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239 |
|
2858 | 240 |
section "The empty set -- {}"; |
241 |
||
242 |
qed_goalw "empty_iff" Set.thy [empty_def] "(c : {}) = False" |
|
2891 | 243 |
(fn _ => [ (Blast_tac 1) ]); |
2858 | 244 |
|
245 |
Addsimps [empty_iff]; |
|
246 |
||
247 |
qed_goal "emptyE" Set.thy "!!a. a:{} ==> P" |
|
248 |
(fn _ => [Full_simp_tac 1]); |
|
249 |
||
250 |
AddSEs [emptyE]; |
|
251 |
||
252 |
qed_goal "empty_subsetI" Set.thy "{} <= A" |
|
2891 | 253 |
(fn _ => [ (Blast_tac 1) ]); |
2858 | 254 |
|
255 |
qed_goal "equals0I" Set.thy "[| !!y. y:A ==> False |] ==> A={}" |
|
256 |
(fn [prem]=> |
|
4089 | 257 |
[ (blast_tac (claset() addIs [prem RS FalseE]) 1) ]); |
2858 | 258 |
|
259 |
qed_goal "equals0D" Set.thy "!!a. [| A={}; a:A |] ==> P" |
|
2891 | 260 |
(fn _ => [ (Blast_tac 1) ]); |
2858 | 261 |
|
5069 | 262 |
Goalw [Ball_def] "Ball {} P = True"; |
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by (Simp_tac 1); |
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264 |
qed "ball_empty"; |
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|
265 |
|
5069 | 266 |
Goalw [Bex_def] "Bex {} P = False"; |
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267 |
by (Simp_tac 1); |
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|
268 |
qed "bex_empty"; |
4aff9b7e5597
UNIV now a constant; UNION1, INTER1 now translations and no longer have
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parents:
4135
diff
changeset
|
269 |
Addsimps [ball_empty, bex_empty]; |
4aff9b7e5597
UNIV now a constant; UNION1, INTER1 now translations and no longer have
paulson
parents:
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diff
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|
270 |
|
5069 | 271 |
Goal "UNIV ~= {}"; |
4159
4aff9b7e5597
UNIV now a constant; UNION1, INTER1 now translations and no longer have
paulson
parents:
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diff
changeset
|
272 |
by (blast_tac (claset() addEs [equalityE]) 1); |
4aff9b7e5597
UNIV now a constant; UNION1, INTER1 now translations and no longer have
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parents:
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diff
changeset
|
273 |
qed "UNIV_not_empty"; |
4aff9b7e5597
UNIV now a constant; UNION1, INTER1 now translations and no longer have
paulson
parents:
4135
diff
changeset
|
274 |
AddIffs [UNIV_not_empty]; |
4aff9b7e5597
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parents:
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|
275 |
|
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parents:
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|
276 |
|
2858 | 277 |
|
278 |
section "The Powerset operator -- Pow"; |
|
279 |
||
280 |
qed_goalw "Pow_iff" Set.thy [Pow_def] "(A : Pow(B)) = (A <= B)" |
|
281 |
(fn _ => [ (Asm_simp_tac 1) ]); |
|
282 |
||
283 |
AddIffs [Pow_iff]; |
|
284 |
||
285 |
qed_goalw "PowI" Set.thy [Pow_def] "!!A B. A <= B ==> A : Pow(B)" |
|
286 |
(fn _ => [ (etac CollectI 1) ]); |
|
287 |
||
288 |
qed_goalw "PowD" Set.thy [Pow_def] "!!A B. A : Pow(B) ==> A<=B" |
|
289 |
(fn _=> [ (etac CollectD 1) ]); |
|
290 |
||
291 |
val Pow_bottom = empty_subsetI RS PowI; (* {}: Pow(B) *) |
|
292 |
val Pow_top = subset_refl RS PowI; (* A : Pow(A) *) |
|
293 |
||
294 |
||
1548 | 295 |
section "Set complement -- Compl"; |
923 | 296 |
|
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Tidying of proofs. New theorems are enterred immediately into the
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|
297 |
qed_goalw "Compl_iff" Set.thy [Compl_def] "(c : Compl(A)) = (c~:A)" |
2891 | 298 |
(fn _ => [ (Blast_tac 1) ]); |
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|
299 |
|
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
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parents:
2031
diff
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|
300 |
Addsimps [Compl_iff]; |
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parents:
2031
diff
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|
301 |
|
923 | 302 |
val prems = goalw Set.thy [Compl_def] |
303 |
"[| c:A ==> False |] ==> c : Compl(A)"; |
|
304 |
by (REPEAT (ares_tac (prems @ [CollectI,notI]) 1)); |
|
305 |
qed "ComplI"; |
|
306 |
||
307 |
(*This form, with negated conclusion, works well with the Classical prover. |
|
308 |
Negated assumptions behave like formulae on the right side of the notional |
|
309 |
turnstile...*) |
|
310 |
val major::prems = goalw Set.thy [Compl_def] |
|
2499
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|
311 |
"c : Compl(A) ==> c~:A"; |
923 | 312 |
by (rtac (major RS CollectD) 1); |
313 |
qed "ComplD"; |
|
314 |
||
315 |
val ComplE = make_elim ComplD; |
|
316 |
||
2499
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Tidying of proofs. New theorems are enterred immediately into the
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2031
diff
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|
317 |
AddSIs [ComplI]; |
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
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parents:
2031
diff
changeset
|
318 |
AddSEs [ComplE]; |
1640 | 319 |
|
923 | 320 |
|
1548 | 321 |
section "Binary union -- Un"; |
923 | 322 |
|
2499
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Tidying of proofs. New theorems are enterred immediately into the
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diff
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|
323 |
qed_goalw "Un_iff" Set.thy [Un_def] "(c : A Un B) = (c:A | c:B)" |
2891 | 324 |
(fn _ => [ Blast_tac 1 ]); |
2499
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Tidying of proofs. New theorems are enterred immediately into the
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2031
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|
325 |
|
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
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parents:
2031
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|
326 |
Addsimps [Un_iff]; |
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parents:
2031
diff
changeset
|
327 |
|
5143
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Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5069
diff
changeset
|
328 |
Goal "c:A ==> c : A Un B"; |
2499
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
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parents:
2031
diff
changeset
|
329 |
by (Asm_simp_tac 1); |
923 | 330 |
qed "UnI1"; |
331 |
||
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5069
diff
changeset
|
332 |
Goal "c:B ==> c : A Un B"; |
2499
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
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parents:
2031
diff
changeset
|
333 |
by (Asm_simp_tac 1); |
923 | 334 |
qed "UnI2"; |
335 |
||
336 |
(*Classical introduction rule: no commitment to A vs B*) |
|
337 |
qed_goal "UnCI" Set.thy "(c~:B ==> c:A) ==> c : A Un B" |
|
338 |
(fn prems=> |
|
2499
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
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2031
diff
changeset
|
339 |
[ (Simp_tac 1), |
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
340 |
(REPEAT (ares_tac (prems@[disjCI]) 1)) ]); |
923 | 341 |
|
342 |
val major::prems = goalw Set.thy [Un_def] |
|
343 |
"[| c : A Un B; c:A ==> P; c:B ==> P |] ==> P"; |
|
344 |
by (rtac (major RS CollectD RS disjE) 1); |
|
345 |
by (REPEAT (eresolve_tac prems 1)); |
|
346 |
qed "UnE"; |
|
347 |
||
2499
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
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parents:
2031
diff
changeset
|
348 |
AddSIs [UnCI]; |
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
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parents:
2031
diff
changeset
|
349 |
AddSEs [UnE]; |
1640 | 350 |
|
923 | 351 |
|
1548 | 352 |
section "Binary intersection -- Int"; |
923 | 353 |
|
2499
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
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parents:
2031
diff
changeset
|
354 |
qed_goalw "Int_iff" Set.thy [Int_def] "(c : A Int B) = (c:A & c:B)" |
2891 | 355 |
(fn _ => [ (Blast_tac 1) ]); |
2499
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
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parents:
2031
diff
changeset
|
356 |
|
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
357 |
Addsimps [Int_iff]; |
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
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parents:
2031
diff
changeset
|
358 |
|
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5069
diff
changeset
|
359 |
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
|
360 |
by (Asm_simp_tac 1); |
923 | 361 |
qed "IntI"; |
362 |
||
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5069
diff
changeset
|
363 |
Goal "c : A Int B ==> c:A"; |
2499
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
364 |
by (Asm_full_simp_tac 1); |
923 | 365 |
qed "IntD1"; |
366 |
||
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5069
diff
changeset
|
367 |
Goal "c : A Int B ==> c:B"; |
2499
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
368 |
by (Asm_full_simp_tac 1); |
923 | 369 |
qed "IntD2"; |
370 |
||
371 |
val [major,minor] = goal Set.thy |
|
372 |
"[| c : A Int B; [| c:A; c:B |] ==> P |] ==> P"; |
|
373 |
by (rtac minor 1); |
|
374 |
by (rtac (major RS IntD1) 1); |
|
375 |
by (rtac (major RS IntD2) 1); |
|
376 |
qed "IntE"; |
|
377 |
||
2499
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
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parents:
2031
diff
changeset
|
378 |
AddSIs [IntI]; |
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
379 |
AddSEs [IntE]; |
923 | 380 |
|
1548 | 381 |
section "Set difference"; |
923 | 382 |
|
2499
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
383 |
qed_goalw "Diff_iff" Set.thy [set_diff_def] "(c : A-B) = (c:A & c~:B)" |
2891 | 384 |
(fn _ => [ (Blast_tac 1) ]); |
923 | 385 |
|
2499
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
386 |
Addsimps [Diff_iff]; |
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
387 |
|
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
388 |
qed_goal "DiffI" Set.thy "!!c. [| c : A; c ~: B |] ==> c : A - B" |
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
389 |
(fn _=> [ Asm_simp_tac 1 ]); |
923 | 390 |
|
2499
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
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parents:
2031
diff
changeset
|
391 |
qed_goal "DiffD1" Set.thy "!!c. c : A - B ==> c : A" |
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
392 |
(fn _=> [ (Asm_full_simp_tac 1) ]); |
923 | 393 |
|
2499
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
394 |
qed_goal "DiffD2" Set.thy "!!c. [| c : A - B; c : B |] ==> P" |
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
395 |
(fn _=> [ (Asm_full_simp_tac 1) ]); |
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
396 |
|
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
397 |
qed_goal "DiffE" Set.thy "[| c : A - B; [| c:A; c~:B |] ==> P |] ==> P" |
923 | 398 |
(fn prems=> |
399 |
[ (resolve_tac prems 1), |
|
400 |
(REPEAT (ares_tac (prems RL [DiffD1, DiffD2 RS notI]) 1)) ]); |
|
401 |
||
2499
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
402 |
AddSIs [DiffI]; |
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
403 |
AddSEs [DiffE]; |
923 | 404 |
|
405 |
||
1548 | 406 |
section "Augmenting a set -- insert"; |
923 | 407 |
|
2499
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
408 |
qed_goalw "insert_iff" Set.thy [insert_def] "a : insert b A = (a=b | a:A)" |
2891 | 409 |
(fn _ => [Blast_tac 1]); |
2499
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
410 |
|
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
411 |
Addsimps [insert_iff]; |
923 | 412 |
|
2499
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
413 |
qed_goal "insertI1" Set.thy "a : insert a B" |
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
414 |
(fn _ => [Simp_tac 1]); |
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
415 |
|
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
416 |
qed_goal "insertI2" Set.thy "!!a. a : B ==> a : insert b B" |
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
417 |
(fn _=> [Asm_simp_tac 1]); |
923 | 418 |
|
419 |
qed_goalw "insertE" Set.thy [insert_def] |
|
420 |
"[| a : insert b A; a=b ==> P; a:A ==> P |] ==> P" |
|
421 |
(fn major::prems=> |
|
422 |
[ (rtac (major RS UnE) 1), |
|
423 |
(REPEAT (eresolve_tac (prems @ [CollectE]) 1)) ]); |
|
424 |
||
425 |
(*Classical introduction rule*) |
|
426 |
qed_goal "insertCI" Set.thy "(a~:B ==> a=b) ==> a: insert b B" |
|
2499
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
427 |
(fn prems=> |
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
428 |
[ (Simp_tac 1), |
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
429 |
(REPEAT (ares_tac (prems@[disjCI]) 1)) ]); |
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
430 |
|
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
431 |
AddSIs [insertCI]; |
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
432 |
AddSEs [insertE]; |
923 | 433 |
|
1548 | 434 |
section "Singletons, using insert"; |
923 | 435 |
|
436 |
qed_goal "singletonI" Set.thy "a : {a}" |
|
437 |
(fn _=> [ (rtac insertI1 1) ]); |
|
438 |
||
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5069
diff
changeset
|
439 |
Goal "b : {a} ==> b=a"; |
2891 | 440 |
by (Blast_tac 1); |
923 | 441 |
qed "singletonD"; |
442 |
||
1776
d7e77cb8ce5c
moved mem_simps and the corresponding update of !simpset from Fun.ML to Set.ML,
oheimb
parents:
1762
diff
changeset
|
443 |
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
|
444 |
|
2499
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
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parents:
2031
diff
changeset
|
445 |
qed_goal "singleton_iff" thy "(b : {a}) = (b=a)" |
2891 | 446 |
(fn _ => [Blast_tac 1]); |
923 | 447 |
|
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5069
diff
changeset
|
448 |
Goal "{a}={b} ==> a=b"; |
4089 | 449 |
by (blast_tac (claset() addEs [equalityE]) 1); |
923 | 450 |
qed "singleton_inject"; |
451 |
||
2858 | 452 |
(*Redundant? But unlike insertCI, it proves the subgoal immediately!*) |
453 |
AddSIs [singletonI]; |
|
2499
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
454 |
AddSDs [singleton_inject]; |
3718 | 455 |
AddSEs [singletonE]; |
2499
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
456 |
|
5069 | 457 |
Goal "{x. x=a} = {a}"; |
4423 | 458 |
by (Blast_tac 1); |
3582 | 459 |
qed "singleton_conv"; |
460 |
Addsimps [singleton_conv]; |
|
1531 | 461 |
|
462 |
||
1548 | 463 |
section "Unions of families -- UNION x:A. B(x) is Union(B``A)"; |
923 | 464 |
|
5069 | 465 |
Goalw [UNION_def] "(b: (UN x:A. B(x))) = (EX x:A. b: B(x))"; |
2891 | 466 |
by (Blast_tac 1); |
2499
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
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2031
diff
changeset
|
467 |
qed "UN_iff"; |
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
468 |
|
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
469 |
Addsimps [UN_iff]; |
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
470 |
|
923 | 471 |
(*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
|
472 |
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
|
473 |
by Auto_tac; |
923 | 474 |
qed "UN_I"; |
475 |
||
476 |
val major::prems = goalw Set.thy [UNION_def] |
|
477 |
"[| b : (UN x:A. B(x)); !!x.[| x:A; b: B(x) |] ==> R |] ==> R"; |
|
478 |
by (rtac (major RS CollectD RS bexE) 1); |
|
479 |
by (REPEAT (ares_tac prems 1)); |
|
480 |
qed "UN_E"; |
|
481 |
||
2499
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
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parents:
2031
diff
changeset
|
482 |
AddIs [UN_I]; |
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
483 |
AddSEs [UN_E]; |
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
484 |
|
923 | 485 |
val prems = goal Set.thy |
486 |
"[| A=B; !!x. x:B ==> C(x) = D(x) |] ==> \ |
|
487 |
\ (UN x:A. C(x)) = (UN x:B. D(x))"; |
|
488 |
by (REPEAT (etac UN_E 1 |
|
489 |
ORELSE ares_tac ([UN_I,equalityI,subsetI] @ |
|
1465 | 490 |
(prems RL [equalityD1,equalityD2] RL [subsetD])) 1)); |
923 | 491 |
qed "UN_cong"; |
492 |
||
493 |
||
1548 | 494 |
section "Intersections of families -- INTER x:A. B(x) is Inter(B``A)"; |
923 | 495 |
|
5069 | 496 |
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
|
497 |
by Auto_tac; |
2499
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
498 |
qed "INT_iff"; |
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
499 |
|
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
500 |
Addsimps [INT_iff]; |
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
501 |
|
923 | 502 |
val prems = goalw Set.thy [INTER_def] |
503 |
"(!!x. x:A ==> b: B(x)) ==> b : (INT x:A. B(x))"; |
|
504 |
by (REPEAT (ares_tac ([CollectI,ballI] @ prems) 1)); |
|
505 |
qed "INT_I"; |
|
506 |
||
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5069
diff
changeset
|
507 |
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
|
508 |
by Auto_tac; |
923 | 509 |
qed "INT_D"; |
510 |
||
511 |
(*"Classical" elimination -- by the Excluded Middle on a:A *) |
|
512 |
val major::prems = goalw Set.thy [INTER_def] |
|
513 |
"[| b : (INT x:A. B(x)); b: B(a) ==> R; a~:A ==> R |] ==> R"; |
|
514 |
by (rtac (major RS CollectD RS ballE) 1); |
|
515 |
by (REPEAT (eresolve_tac prems 1)); |
|
516 |
qed "INT_E"; |
|
517 |
||
2499
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
518 |
AddSIs [INT_I]; |
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
519 |
AddEs [INT_D, INT_E]; |
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
520 |
|
923 | 521 |
val prems = goal Set.thy |
522 |
"[| A=B; !!x. x:B ==> C(x) = D(x) |] ==> \ |
|
523 |
\ (INT x:A. C(x)) = (INT x:B. D(x))"; |
|
524 |
by (REPEAT_FIRST (resolve_tac [INT_I,equalityI,subsetI])); |
|
525 |
by (REPEAT (dtac INT_D 1 |
|
526 |
ORELSE ares_tac (prems RL [equalityD1,equalityD2] RL [subsetD]) 1)); |
|
527 |
qed "INT_cong"; |
|
528 |
||
529 |
||
1548 | 530 |
section "Union"; |
923 | 531 |
|
5069 | 532 |
Goalw [Union_def] "(A : Union(C)) = (EX X:C. A:X)"; |
2891 | 533 |
by (Blast_tac 1); |
2499
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
534 |
qed "Union_iff"; |
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
535 |
|
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
536 |
Addsimps [Union_iff]; |
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
537 |
|
923 | 538 |
(*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
|
539 |
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
|
540 |
by Auto_tac; |
923 | 541 |
qed "UnionI"; |
542 |
||
543 |
val major::prems = goalw Set.thy [Union_def] |
|
544 |
"[| A : Union(C); !!X.[| A:X; X:C |] ==> R |] ==> R"; |
|
545 |
by (rtac (major RS UN_E) 1); |
|
546 |
by (REPEAT (ares_tac prems 1)); |
|
547 |
qed "UnionE"; |
|
548 |
||
2499
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
549 |
AddIs [UnionI]; |
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
550 |
AddSEs [UnionE]; |
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
551 |
|
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
552 |
|
1548 | 553 |
section "Inter"; |
923 | 554 |
|
5069 | 555 |
Goalw [Inter_def] "(A : Inter(C)) = (ALL X:C. A:X)"; |
2891 | 556 |
by (Blast_tac 1); |
2499
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
557 |
qed "Inter_iff"; |
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
558 |
|
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
559 |
Addsimps [Inter_iff]; |
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
560 |
|
923 | 561 |
val prems = goalw Set.thy [Inter_def] |
562 |
"[| !!X. X:C ==> A:X |] ==> A : Inter(C)"; |
|
563 |
by (REPEAT (ares_tac ([INT_I] @ prems) 1)); |
|
564 |
qed "InterI"; |
|
565 |
||
566 |
(*A "destruct" rule -- every X in C contains A as an element, but |
|
567 |
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
|
568 |
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
|
569 |
by Auto_tac; |
923 | 570 |
qed "InterD"; |
571 |
||
572 |
(*"Classical" elimination rule -- does not require proving X:C *) |
|
573 |
val major::prems = goalw Set.thy [Inter_def] |
|
2721 | 574 |
"[| A : Inter(C); X~:C ==> R; A:X ==> R |] ==> R"; |
923 | 575 |
by (rtac (major RS INT_E) 1); |
576 |
by (REPEAT (eresolve_tac prems 1)); |
|
577 |
qed "InterE"; |
|
578 |
||
2499
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
579 |
AddSIs [InterI]; |
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
580 |
AddEs [InterD, InterE]; |
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
581 |
|
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
582 |
|
2912 | 583 |
(*** Image of a set under a function ***) |
584 |
||
585 |
(*Frequently b does not have the syntactic form of f(x).*) |
|
586 |
val prems = goalw thy [image_def] "[| b=f(x); x:A |] ==> b : f``A"; |
|
587 |
by (REPEAT (resolve_tac (prems @ [CollectI,bexI,prem]) 1)); |
|
588 |
qed "image_eqI"; |
|
3909 | 589 |
Addsimps [image_eqI]; |
2912 | 590 |
|
591 |
bind_thm ("imageI", refl RS image_eqI); |
|
592 |
||
593 |
(*The eta-expansion gives variable-name preservation.*) |
|
594 |
val major::prems = goalw thy [image_def] |
|
3842 | 595 |
"[| b : (%x. f(x))``A; !!x.[| b=f(x); x:A |] ==> P |] ==> P"; |
2912 | 596 |
by (rtac (major RS CollectD RS bexE) 1); |
597 |
by (REPEAT (ares_tac prems 1)); |
|
598 |
qed "imageE"; |
|
599 |
||
600 |
AddIs [image_eqI]; |
|
601 |
AddSEs [imageE]; |
|
602 |
||
5069 | 603 |
Goalw [o_def] "(f o g)``r = f``(g``r)"; |
2935 | 604 |
by (Blast_tac 1); |
2912 | 605 |
qed "image_compose"; |
606 |
||
5069 | 607 |
Goal "f``(A Un B) = f``A Un f``B"; |
2935 | 608 |
by (Blast_tac 1); |
2912 | 609 |
qed "image_Un"; |
610 |
||
5069 | 611 |
Goal "(z : f``A) = (EX x:A. z = f x)"; |
3960 | 612 |
by (Blast_tac 1); |
613 |
qed "image_iff"; |
|
614 |
||
4523 | 615 |
(*This rewrite rule would confuse users if made default.*) |
5069 | 616 |
Goal "(f``A <= B) = (ALL x:A. f(x): B)"; |
4523 | 617 |
by (Blast_tac 1); |
618 |
qed "image_subset_iff"; |
|
619 |
||
620 |
(*Replaces the three steps subsetI, imageE, hyp_subst_tac, but breaks too |
|
621 |
many existing proofs.*) |
|
4510 | 622 |
val prems = goal thy "(!!x. x:A ==> f(x) : B) ==> f``A <= B"; |
623 |
by (blast_tac (claset() addIs prems) 1); |
|
624 |
qed "image_subsetI"; |
|
625 |
||
2912 | 626 |
|
627 |
(*** Range of a function -- just a translation for image! ***) |
|
628 |
||
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5069
diff
changeset
|
629 |
Goal "b=f(x) ==> b : range(f)"; |
2912 | 630 |
by (EVERY1 [etac image_eqI, rtac UNIV_I]); |
631 |
bind_thm ("range_eqI", UNIV_I RSN (2,image_eqI)); |
|
632 |
||
633 |
bind_thm ("rangeI", UNIV_I RS imageI); |
|
634 |
||
635 |
val [major,minor] = goal thy |
|
3842 | 636 |
"[| b : range(%x. f(x)); !!x. b=f(x) ==> P |] ==> P"; |
2912 | 637 |
by (rtac (major RS imageE) 1); |
638 |
by (etac minor 1); |
|
639 |
qed "rangeE"; |
|
640 |
||
1776
d7e77cb8ce5c
moved mem_simps and the corresponding update of !simpset from Fun.ML to Set.ML,
oheimb
parents:
1762
diff
changeset
|
641 |
|
d7e77cb8ce5c
moved mem_simps and the corresponding update of !simpset from Fun.ML to Set.ML,
oheimb
parents:
1762
diff
changeset
|
642 |
(*** Set reasoning tools ***) |
d7e77cb8ce5c
moved mem_simps and the corresponding update of !simpset from Fun.ML to Set.ML,
oheimb
parents:
1762
diff
changeset
|
643 |
|
d7e77cb8ce5c
moved mem_simps and the corresponding update of !simpset from Fun.ML to Set.ML,
oheimb
parents:
1762
diff
changeset
|
644 |
|
3912 | 645 |
(** Rewrite rules for boolean case-splitting: faster than |
4830 | 646 |
addsplits[split_if] |
3912 | 647 |
**) |
648 |
||
4830 | 649 |
bind_thm ("split_if_eq1", read_instantiate [("P", "%x. x = ?b")] split_if); |
650 |
bind_thm ("split_if_eq2", read_instantiate [("P", "%x. ?a = x")] split_if); |
|
3912 | 651 |
|
4830 | 652 |
bind_thm ("split_if_mem1", |
653 |
read_instantiate_sg (sign_of Set.thy) [("P", "%x. x : ?b")] split_if); |
|
654 |
bind_thm ("split_if_mem2", |
|
655 |
read_instantiate_sg (sign_of Set.thy) [("P", "%x. ?a : x")] split_if); |
|
3912 | 656 |
|
4830 | 657 |
val split_ifs = [if_bool_eq_conj, split_if_eq1, split_if_eq2, |
658 |
split_if_mem1, split_if_mem2]; |
|
3912 | 659 |
|
660 |
||
4089 | 661 |
(*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
|
662 |
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
|
663 |
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
|
664 |
|
1937 | 665 |
(*Not for Addsimps -- it can cause goals to blow up!*) |
5069 | 666 |
Goal "(a : (if Q then x else y)) = ((Q --> a:x) & (~Q --> a : y))"; |
4686 | 667 |
by (Simp_tac 1); |
1937 | 668 |
qed "mem_if"; |
669 |
||
1776
d7e77cb8ce5c
moved mem_simps and the corresponding update of !simpset from Fun.ML to Set.ML,
oheimb
parents:
1762
diff
changeset
|
670 |
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
|
671 |
|
4089 | 672 |
simpset_ref() := simpset() addcongs [ball_cong,bex_cong] |
1776
d7e77cb8ce5c
moved mem_simps and the corresponding update of !simpset from Fun.ML to Set.ML,
oheimb
parents:
1762
diff
changeset
|
673 |
setmksimps (mksimps mksimps_pairs); |
3222
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents:
2935
diff
changeset
|
674 |
|
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents:
2935
diff
changeset
|
675 |
Addsimps[subset_UNIV, empty_subsetI, subset_refl]; |
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents:
2935
diff
changeset
|
676 |
|
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents:
2935
diff
changeset
|
677 |
|
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents:
2935
diff
changeset
|
678 |
(*** < ***) |
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents:
2935
diff
changeset
|
679 |
|
5069 | 680 |
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
|
681 |
by (Blast_tac 1); |
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents:
2935
diff
changeset
|
682 |
qed "psubsetI"; |
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents:
2935
diff
changeset
|
683 |
|
5069 | 684 |
Goalw [psubset_def] |
3222
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents:
2935
diff
changeset
|
685 |
"!!x. 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
|
686 |
by Auto_tac; |
3222
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents:
2935
diff
changeset
|
687 |
qed "psubset_insertD"; |
4059 | 688 |
|
689 |
bind_thm ("psubset_eq", psubset_def RS meta_eq_to_obj_eq); |