author | paulson |
Fri, 29 Oct 2004 15:16:02 +0200 | |
changeset 15270 | 8b3f707a78a7 |
parent 5143 | b94cd208f073 |
child 17456 | bcf7544875b2 |
permissions | -rw-r--r-- |
1459 | 1 |
(* Title: set/set |
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ID: $Id$ |
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For set.thy. |
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Modified version of |
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1459 | 7 |
Title: HOL/set |
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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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val [prem] = goal Set.thy "[| P(a) |] ==> a : {x. P(x)}"; |
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by (rtac (mem_Collect_iff RS iffD2) 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_iff RS iffD1]) 1); |
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qed "CollectD"; |
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val CollectE = make_elim CollectD; |
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val [prem] = goal Set.thy "[| !!x. x:A <-> x:B |] ==> A = B"; |
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by (rtac (set_extension RS iffD2) 1); |
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by (rtac (prem RS allI) 1); |
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qed "set_ext"; |
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(*** Bounded quantifiers ***) |
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val prems = goalw Set.thy [Ball_def] |
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"[| !!x. x:A ==> P(x) |] ==> ALL 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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"[| ALL x:A. P(x); x:A |] ==> P(x)"; |
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by (rtac (minor RS (major RS spec RS mp)) 1); |
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qed "bspec"; |
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val major::prems = goalw Set.thy [Ball_def] |
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"[| ALL 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 ALL 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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val prems = goalw Set.thy [Bex_def] |
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"[| P(x); x:A |] ==> EX 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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"[| EX x:A. ~P(x) ==> P(a); a:A |] ==> EX 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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"[| EX 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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val prems = goal Set.thy |
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"(ALL 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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(** Congruence rules **) |
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val prems = goal Set.thy |
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"[| A=A'; !!x. x:A' ==> P(x) <-> P'(x) |] ==> \ |
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\ (ALL x:A. P(x)) <-> (ALL x:A'. P'(x))"; |
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by (resolve_tac (prems RL [ssubst,iffD2]) 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=A'; !!x. x:A' ==> P(x) <-> P'(x) |] ==> \ |
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\ (EX x:A. P(x)) <-> (EX x:A'. P'(x))"; |
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by (resolve_tac (prems RL [ssubst,iffD2]) 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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(*** Rules for 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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(*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" |
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(fn _=> [ (REPEAT (ares_tac [subsetI] 1)) ]); |
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Goal "[| A<=B; B<=C |] ==> A<=C"; |
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by (rtac subsetI 1); |
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by (REPEAT (eresolve_tac [asm_rl, subsetD] 1)); |
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qed "subset_trans"; |
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(*** Rules for 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"; |
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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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(* Equality rules from ZF set theory -- are they appropriate here? *) |
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val prems = goal Set.thy "A = B ==> A<=B"; |
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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"; |
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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 |] ==> 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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Goal "{x. x:A} = A"; |
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by (REPEAT (ares_tac [equalityI,subsetI,CollectI] 1 ORELSE etac CollectD 1)); |
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qed "trivial_set"; |
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(*** Rules for 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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(*** Rules for small 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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757 | 214 |
qed "IntE"; |
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(*** Rules for 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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(*** Empty sets ***) |
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Goalw [empty_def] "{x. False} = {}"; |
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by (rtac refl 1); |
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qed "empty_eq"; |
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val [prem] = goalw Set.thy [empty_def] "a : {} ==> P"; |
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by (rtac (prem RS CollectD RS FalseE) 1); |
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qed "emptyD"; |
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val emptyE = make_elim emptyD; |
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val [prem] = goal Set.thy "~ A={} ==> (EX x. x:A)"; |
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by (rtac (prem RS swap) 1); |
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by (rtac equalityI 1); |
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by (ALLGOALS (fast_tac (FOL_cs addSIs [subsetI] addSEs [emptyD]))); |
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qed "not_emptyD"; |
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(*** Singleton sets ***) |
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5062 | 255 |
Goalw [singleton_def] "a : {a}"; |
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by (rtac CollectI 1); |
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by (rtac refl 1); |
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qed "singletonI"; |
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val [major] = goalw Set.thy [singleton_def] "b : {a} ==> b=a"; |
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by (rtac (major RS CollectD) 1); |
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qed "singletonD"; |
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val singletonE = make_elim singletonD; |
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(*** Unions of families ***) |
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(*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)); |
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qed "UN_I"; |
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val major::prems = goalw Set.thy [UNION_def] |
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"[| b : (UN x:A. B(x)); !!x.[| x:A; b: B(x) |] ==> R |] ==> R"; |
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by (rtac (major RS CollectD RS bexE) 1); |
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by (REPEAT (ares_tac prems 1)); |
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qed "UN_E"; |
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val prems = goal Set.thy |
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"[| A=B; !!x. x:B ==> C(x) = D(x) |] ==> \ |
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\ (UN x:A. C(x)) = (UN x:B. D(x))"; |
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by (REPEAT (etac UN_E 1 |
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ORELSE ares_tac ([UN_I,equalityI,subsetI] @ |
|
1459 | 285 |
(prems RL [equalityD1,equalityD2] RL [subsetD])) 1)); |
757 | 286 |
qed "UN_cong"; |
0 | 287 |
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(*** Intersections of families -- INTER x:A. B(x) is Inter(B)``A ) *) |
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val prems = goalw Set.thy [INTER_def] |
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"(!!x. x:A ==> b: B(x)) ==> b : (INT x:A. B(x))"; |
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by (REPEAT (ares_tac ([CollectI,ballI] @ prems) 1)); |
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qed "INT_I"; |
0 | 294 |
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val major::prems = goalw Set.thy [INTER_def] |
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"[| b : (INT x:A. B(x)); a:A |] ==> b: B(a)"; |
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by (rtac (major RS CollectD RS bspec) 1); |
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by (resolve_tac prems 1); |
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757 | 299 |
qed "INT_D"; |
0 | 300 |
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(*"Classical" elimination rule -- does not require proving X:C *) |
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val major::prems = goalw Set.thy [INTER_def] |
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"[| b : (INT x:A. B(x)); b: B(a) ==> R; ~ a:A ==> R |] ==> R"; |
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by (rtac (major RS CollectD RS ballE) 1); |
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by (REPEAT (eresolve_tac prems 1)); |
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757 | 306 |
qed "INT_E"; |
0 | 307 |
|
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val prems = goal Set.thy |
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"[| A=B; !!x. x:B ==> C(x) = D(x) |] ==> \ |
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\ (INT x:A. C(x)) = (INT x:B. D(x))"; |
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by (REPEAT_FIRST (resolve_tac [INT_I,equalityI,subsetI])); |
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312 |
by (REPEAT (dtac INT_D 1 |
|
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ORELSE ares_tac (prems RL [equalityD1,equalityD2] RL [subsetD]) 1)); |
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757 | 314 |
qed "INT_cong"; |
0 | 315 |
|
316 |
(*** Rules for Unions ***) |
|
317 |
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318 |
(*The order of the premises presupposes that C is rigid; A may be flexible*) |
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319 |
val prems = goalw Set.thy [Union_def] |
|
320 |
"[| X:C; A:X |] ==> A : Union(C)"; |
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by (REPEAT (resolve_tac (prems @ [UN_I]) 1)); |
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757 | 322 |
qed "UnionI"; |
0 | 323 |
|
324 |
val major::prems = goalw Set.thy [Union_def] |
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325 |
"[| A : Union(C); !!X.[| A:X; X:C |] ==> R |] ==> R"; |
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326 |
by (rtac (major RS UN_E) 1); |
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327 |
by (REPEAT (ares_tac prems 1)); |
|
757 | 328 |
qed "UnionE"; |
0 | 329 |
|
330 |
(*** Rules for Inter ***) |
|
331 |
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332 |
val prems = goalw Set.thy [Inter_def] |
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333 |
"[| !!X. X:C ==> A:X |] ==> A : Inter(C)"; |
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334 |
by (REPEAT (ares_tac ([INT_I] @ prems) 1)); |
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757 | 335 |
qed "InterI"; |
0 | 336 |
|
337 |
(*A "destruct" rule -- every X in C contains A as an element, but |
|
338 |
A:X can hold when X:C does not! This rule is analogous to "spec". *) |
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339 |
val major::prems = goalw Set.thy [Inter_def] |
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"[| A : Inter(C); X:C |] ==> A:X"; |
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341 |
by (rtac (major RS INT_D) 1); |
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by (resolve_tac prems 1); |
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757 | 343 |
qed "InterD"; |
0 | 344 |
|
345 |
(*"Classical" elimination rule -- does not require proving X:C *) |
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346 |
val major::prems = goalw Set.thy [Inter_def] |
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347 |
"[| A : Inter(C); A:X ==> R; ~ X:C ==> R |] ==> R"; |
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by (rtac (major RS INT_E) 1); |
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349 |
by (REPEAT (eresolve_tac prems 1)); |
|
757 | 350 |
qed "InterE"; |