author | berghofe |
Thu, 23 May 1996 15:16:12 +0200 | |
changeset 1762 | 6e481897a811 |
parent 1760 | 6f41a494f3b1 |
child 1776 | d7e77cb8ce5c |
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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For set.thy. 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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||
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section "Relating predicates and sets"; |
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val [prem] = goal Set.thy "P(a) ==> a : {x.P(x)}"; |
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by (rtac (mem_Collect_eq RS ssubst) 1); |
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by (rtac prem 1); |
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qed "CollectI"; |
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||
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val prems = goal Set.thy "[| a : {x.P(x)} |] ==> P(a)"; |
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by (resolve_tac (prems RL [mem_Collect_eq RS subst]) 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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val CollectE = make_elim CollectD; |
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||
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section "Bounded quantifiers"; |
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val prems = goalw Set.thy [Ball_def] |
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"[| !!x. x:A ==> P(x) |] ==> ! x:A. P(x)"; |
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by (REPEAT (ares_tac (prems @ [allI,impI]) 1)); |
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qed "ballI"; |
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val [major,minor] = goalw Set.thy [Ball_def] |
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"[| ! x:A. P(x); x:A |] ==> P(x)"; |
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by (rtac (minor RS (major RS spec RS mp)) 1); |
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qed "bspec"; |
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||
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val major::prems = goalw Set.thy [Ball_def] |
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"[| ! x:A. P(x); P(x) ==> Q; x~:A ==> Q |] ==> Q"; |
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by (rtac (major RS spec RS impCE) 1); |
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by (REPEAT (eresolve_tac prems 1)); |
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qed "ballE"; |
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(*Takes assumptions ! x:A.P(x) and a:A; creates assumption P(a)*) |
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fun ball_tac i = etac ballE i THEN contr_tac (i+1); |
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||
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val prems = goalw Set.thy [Bex_def] |
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"[| P(x); x:A |] ==> ? x:A. P(x)"; |
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by (REPEAT (ares_tac (prems @ [exI,conjI]) 1)); |
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qed "bexI"; |
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qed_goal "bexCI" Set.thy |
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"[| ! x:A. ~P(x) ==> P(a); a:A |] ==> ? x:A.P(x)" |
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(fn prems=> |
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[ (rtac classical 1), |
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(REPEAT (ares_tac (prems@[bexI,ballI,notI,notE]) 1)) ]); |
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val major::prems = goalw Set.thy [Bex_def] |
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"[| ? x:A. P(x); !!x. [| x:A; P(x) |] ==> Q |] ==> Q"; |
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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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(*Trival rewrite rule; (! x:A.P)=P holds only if A is nonempty!*) |
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goal Set.thy "(! x:A. True) = True"; |
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by (REPEAT (ares_tac [TrueI,ballI,iffI] 1)); |
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qed "ball_rew"; |
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Addsimps [ball_rew]; |
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(** Congruence rules **) |
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val prems = goal Set.thy |
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"[| A=B; !!x. x:B ==> P(x) = Q(x) |] ==> \ |
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\ (! x:A. P(x)) = (! x:B. Q(x))"; |
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by (resolve_tac (prems RL [ssubst]) 1); |
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by (REPEAT (ares_tac [ballI,iffI] 1 |
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ORELSE eresolve_tac ([make_elim bspec, mp] @ (prems RL [iffE])) 1)); |
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qed "ball_cong"; |
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val prems = goal Set.thy |
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"[| A=B; !!x. x:B ==> P(x) = Q(x) |] ==> \ |
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\ (? x:A. P(x)) = (? x:B. Q(x))"; |
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by (resolve_tac (prems RL [ssubst]) 1); |
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by (REPEAT (etac bexE 1 |
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ORELSE ares_tac ([bexI,iffI] @ (prems RL [iffD1,iffD2])) 1)); |
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qed "bex_cong"; |
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section "Subsets"; |
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val prems = goalw Set.thy [subset_def] "(!!x.x:A ==> x:B) ==> A <= B"; |
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by (REPEAT (ares_tac (prems @ [ballI]) 1)); |
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qed "subsetI"; |
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(*Rule in Modus Ponens style*) |
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val major::prems = goalw Set.thy [subset_def] "[| A <= B; c:A |] ==> c:B"; |
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by (rtac (major RS bspec) 1); |
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by (resolve_tac prems 1); |
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qed "subsetD"; |
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(*The same, with reversed premises for use with etac -- cf rev_mp*) |
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qed_goal "rev_subsetD" Set.thy "[| c:A; A <= B |] ==> c:B" |
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(fn prems=> [ (REPEAT (resolve_tac (prems@[subsetD]) 1)) ]); |
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(*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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qed_goal "subset_refl" Set.thy "A <= (A::'a set)" |
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(fn _=> [ (REPEAT (ares_tac [subsetI] 1)) ]); |
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val prems = goal Set.thy "[| A<=B; B<=C |] ==> A<=(C::'a set)"; |
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by (cut_facts_tac prems 1); |
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by (REPEAT (ares_tac [subsetI] 1 ORELSE set_mp_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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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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AddSIs [equalityI]; |
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(* Equality rules from ZF set theory -- are they appropriate here? *) |
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val prems = goal Set.thy "A = B ==> A<=(B::'a set)"; |
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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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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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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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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"; |
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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)); |
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qed "setup_induction"; |
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section "Set complement -- Compl"; |
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val prems = goalw Set.thy [Compl_def] |
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"[| c:A ==> False |] ==> c : Compl(A)"; |
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by (REPEAT (ares_tac (prems @ [CollectI,notI]) 1)); |
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qed "ComplI"; |
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(*This form, with negated conclusion, works well with the Classical prover. |
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Negated assumptions behave like formulae on the right side of the notional |
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turnstile...*) |
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val major::prems = goalw Set.thy [Compl_def] |
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"[| c : Compl(A) |] ==> c~:A"; |
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by (rtac (major RS CollectD) 1); |
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qed "ComplD"; |
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val ComplE = make_elim ComplD; |
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qed_goal "Compl_iff" Set.thy "(c : Compl(A)) = (c~:A)" |
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(fn _ => [ (fast_tac (!claset addSIs [ComplI] addSEs [ComplE]) 1) ]); |
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section "Binary union -- Un"; |
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val prems = goalw Set.thy [Un_def] "c:A ==> c : A Un B"; |
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by (REPEAT (resolve_tac (prems @ [CollectI,disjI1]) 1)); |
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qed "UnI1"; |
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val prems = goalw Set.thy [Un_def] "c:B ==> c : A Un B"; |
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by (REPEAT (resolve_tac (prems @ [CollectI,disjI2]) 1)); |
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qed "UnI2"; |
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(*Classical introduction rule: no commitment to A vs B*) |
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qed_goal "UnCI" Set.thy "(c~:B ==> c:A) ==> c : A Un B" |
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(fn prems=> |
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[ (rtac classical 1), |
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(REPEAT (ares_tac (prems@[UnI1,notI]) 1)), |
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(REPEAT (ares_tac (prems@[UnI2,notE]) 1)) ]); |
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val major::prems = goalw Set.thy [Un_def] |
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"[| c : A Un B; c:A ==> P; c:B ==> P |] ==> P"; |
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by (rtac (major RS CollectD RS disjE) 1); |
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by (REPEAT (eresolve_tac prems 1)); |
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qed "UnE"; |
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||
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qed_goal "Un_iff" Set.thy "(c : A Un B) = (c:A | c:B)" |
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(fn _ => [ (fast_tac (!claset addSIs [UnCI] addSEs [UnE]) 1) ]); |
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section "Binary intersection -- Int"; |
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val prems = goalw Set.thy [Int_def] |
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"[| c:A; c:B |] ==> c : A Int B"; |
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by (REPEAT (resolve_tac (prems @ [CollectI,conjI]) 1)); |
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qed "IntI"; |
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val [major] = goalw Set.thy [Int_def] "c : A Int B ==> c:A"; |
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by (rtac (major RS CollectD RS conjunct1) 1); |
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qed "IntD1"; |
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val [major] = goalw Set.thy [Int_def] "c : A Int B ==> c:B"; |
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by (rtac (major RS CollectD RS conjunct2) 1); |
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qed "IntD2"; |
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val [major,minor] = goal Set.thy |
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"[| c : A Int B; [| c:A; c:B |] ==> P |] ==> P"; |
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by (rtac minor 1); |
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by (rtac (major RS IntD1) 1); |
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by (rtac (major RS IntD2) 1); |
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qed "IntE"; |
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qed_goal "Int_iff" Set.thy "(c : A Int B) = (c:A & c:B)" |
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(fn _ => [ (fast_tac (!claset addSIs [IntI] addSEs [IntE]) 1) ]); |
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section "Set difference"; |
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qed_goalw "DiffI" Set.thy [set_diff_def] |
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"[| c : A; c ~: B |] ==> c : A - B" |
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(fn prems=> [ (REPEAT (resolve_tac (prems @ [CollectI,conjI]) 1)) ]); |
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qed_goalw "DiffD1" Set.thy [set_diff_def] |
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"c : A - B ==> c : A" |
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(fn [major]=> [ (rtac (major RS CollectD RS conjunct1) 1) ]); |
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qed_goalw "DiffD2" Set.thy [set_diff_def] |
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"[| c : A - B; c : B |] ==> P" |
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(fn [major,minor]=> |
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[rtac (minor RS (major RS CollectD RS conjunct2 RS notE)) 1]); |
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qed_goal "DiffE" Set.thy |
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"[| c : A - B; [| c:A; c~:B |] ==> P |] ==> P" |
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(fn prems=> |
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[ (resolve_tac prems 1), |
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(REPEAT (ares_tac (prems RL [DiffD1, DiffD2 RS notI]) 1)) ]); |
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qed_goal "Diff_iff" Set.thy "(c : A-B) = (c:A & c~:B)" |
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(fn _ => [ (fast_tac (!claset addSIs [DiffI] addSEs [DiffE]) 1) ]); |
923 | 272 |
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section "The empty set -- {}"; |
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qed_goalw "emptyE" Set.thy [empty_def] "a:{} ==> P" |
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(fn [prem] => [rtac (prem RS CollectD RS FalseE) 1]); |
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qed_goal "empty_subsetI" Set.thy "{} <= A" |
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(fn _ => [ (REPEAT (ares_tac [equalityI,subsetI,emptyE] 1)) ]); |
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qed_goal "equals0I" Set.thy "[| !!y. y:A ==> False |] ==> A={}" |
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(fn prems=> |
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[ (REPEAT (ares_tac (prems@[empty_subsetI,subsetI,equalityI]) 1 |
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ORELSE eresolve_tac (prems RL [FalseE]) 1)) ]); |
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qed_goal "equals0D" Set.thy "[| A={}; a:A |] ==> P" |
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(fn [major,minor]=> |
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[ (rtac (minor RS (major RS equalityD1 RS subsetD RS emptyE)) 1) ]); |
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||
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qed_goal "empty_iff" Set.thy "(c : {}) = False" |
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(fn _ => [ (fast_tac (!claset addSEs [emptyE]) 1) ]); |
1640 | 292 |
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923 | 293 |
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section "Augmenting a set -- insert"; |
923 | 295 |
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qed_goalw "insertI1" Set.thy [insert_def] "a : insert a B" |
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(fn _ => [rtac (CollectI RS UnI1) 1, rtac refl 1]); |
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299 |
qed_goalw "insertI2" Set.thy [insert_def] "a : B ==> a : insert b B" |
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(fn [prem]=> [ (rtac (prem RS UnI2) 1) ]); |
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qed_goalw "insertE" Set.thy [insert_def] |
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"[| a : insert b A; a=b ==> P; a:A ==> P |] ==> P" |
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(fn major::prems=> |
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[ (rtac (major RS UnE) 1), |
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(REPEAT (eresolve_tac (prems @ [CollectE]) 1)) ]); |
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qed_goal "insert_iff" Set.thy "a : insert b A = (a=b | a:A)" |
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|
309 |
(fn _ => [fast_tac (!claset addIs [insertI1,insertI2] addSEs [insertE]) 1]); |
923 | 310 |
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(*Classical introduction rule*) |
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qed_goal "insertCI" Set.thy "(a~:B ==> a=b) ==> a: insert b B" |
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(fn [prem]=> |
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[ (rtac (disjCI RS (insert_iff RS iffD2)) 1), |
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(etac prem 1) ]); |
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||
1548 | 317 |
section "Singletons, using insert"; |
923 | 318 |
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qed_goal "singletonI" Set.thy "a : {a}" |
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(fn _=> [ (rtac insertI1 1) ]); |
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322 |
qed_goal "singletonE" Set.thy "[| a: {b}; a=b ==> P |] ==> P" |
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(fn major::prems=> |
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[ (rtac (major RS insertE) 1), |
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(REPEAT (eresolve_tac (prems @ [emptyE]) 1)) ]); |
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327 |
goalw Set.thy [insert_def] "!!a. b : {a} ==> b=a"; |
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328 |
by (fast_tac (!claset addSEs [emptyE,CollectE,UnE]) 1); |
923 | 329 |
qed "singletonD"; |
330 |
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331 |
val singletonE = make_elim singletonD; |
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333 |
val [major] = goal Set.thy "{a}={b} ==> a=b"; |
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by (rtac (major RS equalityD1 RS subsetD RS singletonD) 1); |
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by (rtac singletonI 1); |
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qed "singleton_inject"; |
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||
1531 | 338 |
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1548 | 339 |
section "The universal set -- UNIV"; |
1531 | 340 |
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qed_goal "subset_UNIV" Set.thy "A <= UNIV" |
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(fn _ => [rtac subsetI 1, rtac ComplI 1, etac emptyE 1]); |
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344 |
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1548 | 345 |
section "Unions of families -- UNION x:A. B(x) is Union(B``A)"; |
923 | 346 |
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347 |
(*The order of the premises presupposes that A is rigid; b may be flexible*) |
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val prems = goalw Set.thy [UNION_def] |
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"[| a:A; b: B(a) |] ==> b: (UN x:A. B(x))"; |
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by (REPEAT (resolve_tac (prems @ [bexI,CollectI]) 1)); |
|
351 |
qed "UN_I"; |
|
352 |
||
353 |
val major::prems = goalw Set.thy [UNION_def] |
|
354 |
"[| b : (UN x:A. B(x)); !!x.[| x:A; b: B(x) |] ==> R |] ==> R"; |
|
355 |
by (rtac (major RS CollectD RS bexE) 1); |
|
356 |
by (REPEAT (ares_tac prems 1)); |
|
357 |
qed "UN_E"; |
|
358 |
||
359 |
val prems = goal Set.thy |
|
360 |
"[| A=B; !!x. x:B ==> C(x) = D(x) |] ==> \ |
|
361 |
\ (UN x:A. C(x)) = (UN x:B. D(x))"; |
|
362 |
by (REPEAT (etac UN_E 1 |
|
363 |
ORELSE ares_tac ([UN_I,equalityI,subsetI] @ |
|
1465 | 364 |
(prems RL [equalityD1,equalityD2] RL [subsetD])) 1)); |
923 | 365 |
qed "UN_cong"; |
366 |
||
367 |
||
1548 | 368 |
section "Intersections of families -- INTER x:A. B(x) is Inter(B``A)"; |
923 | 369 |
|
370 |
val prems = goalw Set.thy [INTER_def] |
|
371 |
"(!!x. x:A ==> b: B(x)) ==> b : (INT x:A. B(x))"; |
|
372 |
by (REPEAT (ares_tac ([CollectI,ballI] @ prems) 1)); |
|
373 |
qed "INT_I"; |
|
374 |
||
375 |
val major::prems = goalw Set.thy [INTER_def] |
|
376 |
"[| b : (INT x:A. B(x)); a:A |] ==> b: B(a)"; |
|
377 |
by (rtac (major RS CollectD RS bspec) 1); |
|
378 |
by (resolve_tac prems 1); |
|
379 |
qed "INT_D"; |
|
380 |
||
381 |
(*"Classical" elimination -- by the Excluded Middle on a:A *) |
|
382 |
val major::prems = goalw Set.thy [INTER_def] |
|
383 |
"[| b : (INT x:A. B(x)); b: B(a) ==> R; a~:A ==> R |] ==> R"; |
|
384 |
by (rtac (major RS CollectD RS ballE) 1); |
|
385 |
by (REPEAT (eresolve_tac prems 1)); |
|
386 |
qed "INT_E"; |
|
387 |
||
388 |
val prems = goal Set.thy |
|
389 |
"[| A=B; !!x. x:B ==> C(x) = D(x) |] ==> \ |
|
390 |
\ (INT x:A. C(x)) = (INT x:B. D(x))"; |
|
391 |
by (REPEAT_FIRST (resolve_tac [INT_I,equalityI,subsetI])); |
|
392 |
by (REPEAT (dtac INT_D 1 |
|
393 |
ORELSE ares_tac (prems RL [equalityD1,equalityD2] RL [subsetD]) 1)); |
|
394 |
qed "INT_cong"; |
|
395 |
||
396 |
||
1548 | 397 |
section "Unions over a type; UNION1(B) = Union(range(B))"; |
923 | 398 |
|
399 |
(*The order of the premises presupposes that A is rigid; b may be flexible*) |
|
400 |
val prems = goalw Set.thy [UNION1_def] |
|
401 |
"b: B(x) ==> b: (UN x. B(x))"; |
|
402 |
by (REPEAT (resolve_tac (prems @ [TrueI, CollectI RS UN_I]) 1)); |
|
403 |
qed "UN1_I"; |
|
404 |
||
405 |
val major::prems = goalw Set.thy [UNION1_def] |
|
406 |
"[| b : (UN x. B(x)); !!x. b: B(x) ==> R |] ==> R"; |
|
407 |
by (rtac (major RS UN_E) 1); |
|
408 |
by (REPEAT (ares_tac prems 1)); |
|
409 |
qed "UN1_E"; |
|
410 |
||
411 |
||
1548 | 412 |
section "Intersections over a type; INTER1(B) = Inter(range(B))"; |
923 | 413 |
|
414 |
val prems = goalw Set.thy [INTER1_def] |
|
415 |
"(!!x. b: B(x)) ==> b : (INT x. B(x))"; |
|
416 |
by (REPEAT (ares_tac (INT_I::prems) 1)); |
|
417 |
qed "INT1_I"; |
|
418 |
||
419 |
val [major] = goalw Set.thy [INTER1_def] |
|
420 |
"b : (INT x. B(x)) ==> b: B(a)"; |
|
421 |
by (rtac (TrueI RS (CollectI RS (major RS INT_D))) 1); |
|
422 |
qed "INT1_D"; |
|
423 |
||
1548 | 424 |
section "Union"; |
923 | 425 |
|
426 |
(*The order of the premises presupposes that C is rigid; A may be flexible*) |
|
427 |
val prems = goalw Set.thy [Union_def] |
|
428 |
"[| X:C; A:X |] ==> A : Union(C)"; |
|
429 |
by (REPEAT (resolve_tac (prems @ [UN_I]) 1)); |
|
430 |
qed "UnionI"; |
|
431 |
||
432 |
val major::prems = goalw Set.thy [Union_def] |
|
433 |
"[| A : Union(C); !!X.[| A:X; X:C |] ==> R |] ==> R"; |
|
434 |
by (rtac (major RS UN_E) 1); |
|
435 |
by (REPEAT (ares_tac prems 1)); |
|
436 |
qed "UnionE"; |
|
437 |
||
1548 | 438 |
section "Inter"; |
923 | 439 |
|
440 |
val prems = goalw Set.thy [Inter_def] |
|
441 |
"[| !!X. X:C ==> A:X |] ==> A : Inter(C)"; |
|
442 |
by (REPEAT (ares_tac ([INT_I] @ prems) 1)); |
|
443 |
qed "InterI"; |
|
444 |
||
445 |
(*A "destruct" rule -- every X in C contains A as an element, but |
|
446 |
A:X can hold when X:C does not! This rule is analogous to "spec". *) |
|
447 |
val major::prems = goalw Set.thy [Inter_def] |
|
448 |
"[| A : Inter(C); X:C |] ==> A:X"; |
|
449 |
by (rtac (major RS INT_D) 1); |
|
450 |
by (resolve_tac prems 1); |
|
451 |
qed "InterD"; |
|
452 |
||
453 |
(*"Classical" elimination rule -- does not require proving X:C *) |
|
454 |
val major::prems = goalw Set.thy [Inter_def] |
|
455 |
"[| A : Inter(C); A:X ==> R; X~:C ==> R |] ==> R"; |
|
456 |
by (rtac (major RS INT_E) 1); |
|
457 |
by (REPEAT (eresolve_tac prems 1)); |
|
458 |
qed "InterE"; |
|
459 |
||
1548 | 460 |
section "The Powerset operator -- Pow"; |
923 | 461 |
|
462 |
qed_goalw "PowI" Set.thy [Pow_def] "!!A B. A <= B ==> A : Pow(B)" |
|
463 |
(fn _ => [ (etac CollectI 1) ]); |
|
464 |
||
465 |
qed_goalw "PowD" Set.thy [Pow_def] "!!A B. A : Pow(B) ==> A<=B" |
|
466 |
(fn _=> [ (etac CollectD 1) ]); |
|
467 |
||
468 |
val Pow_bottom = empty_subsetI RS PowI; (* {}: Pow(B) *) |
|
469 |
val Pow_top = subset_refl RS PowI; (* A : Pow(A) *) |