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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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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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val CollectI = result();
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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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val CollectD = result();
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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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val set_ext = result();
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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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val Collect_cong = result();
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val CollectE = make_elim CollectD;
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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) |] ==> ! x:A. P(x)";
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by (REPEAT (ares_tac (prems @ [allI,impI]) 1));
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val ballI = result();
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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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val bspec = result();
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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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val ballE = result();
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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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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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val bexI = result();
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val bexCI = prove_goal 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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val bexE = result();
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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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"(! x:A. True) = True";
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by (REPEAT (ares_tac [TrueI,ballI,iffI] 1));
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val ball_rew = result();
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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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val ball_cong = result();
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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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val bex_cong = result();
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(*** 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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val subsetI = result();
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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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val subsetD = result();
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(*The same, with reversed premises for use with etac -- cf rev_mp*)
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val rev_subsetD = prove_goal 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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val subsetCE = result();
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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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val subset_refl = prove_goal 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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val subset_trans = result();
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(*** 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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val subset_antisym = result();
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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::'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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val equalityD1 = result();
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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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val equalityD2 = result();
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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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val equalityE = result();
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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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val equalityCE = result();
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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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val setup_induction = result();
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(*** 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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val ComplI = result();
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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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val ComplD = result();
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val ComplE = make_elim ComplD;
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(*** 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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val UnI1 = result();
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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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val UnI2 = result();
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(*Classical introduction rule: no commitment to A vs B*)
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val UnCI = prove_goal 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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val UnE = result();
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(*** 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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val IntI = result();
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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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val IntD1 = result();
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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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val IntD2 = result();
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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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val IntE = result();
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(*** Set difference ***)
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val DiffI = prove_goalw 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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val DiffD1 = prove_goalw 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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val DiffD2 = prove_goalw 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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val DiffE = prove_goal 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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val Diff_iff = prove_goal Set.thy "(c : A-B) = (c:A & c~:B)"
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(fn _ => [ (fast_tac (HOL_cs addSIs [DiffI] addSEs [DiffE]) 1) ]);
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(*** The empty set -- {} ***)
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val emptyE = prove_goalw Set.thy [empty_def] "a:{} ==> P"
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(fn [prem] => [rtac (prem RS CollectD RS FalseE) 1]);
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val empty_subsetI = prove_goal Set.thy "{} <= A"
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(fn _ => [ (REPEAT (ares_tac [equalityI,subsetI,emptyE] 1)) ]);
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val equals0I = prove_goal 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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val equals0D = prove_goal 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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(*** Augmenting a set -- insert ***)
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val insertI1 = prove_goalw 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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val insertI2 = prove_goalw 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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val insertE = prove_goalw 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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val insert_iff = prove_goal Set.thy "a : insert(b,A) = (a=b | a:A)"
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(fn _ => [fast_tac (HOL_cs addIs [insertI1,insertI2] addSEs [insertE]) 1]);
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(*Classical introduction rule*)
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val insertCI = prove_goal 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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(*** Singletons, using insert ***)
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val singletonI = prove_goal Set.thy "a : {a}"
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(fn _=> [ (rtac insertI1 1) ]);
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val singletonE = prove_goal 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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goalw Set.thy [insert_def] "!!a. b : {a} ==> b=a";
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by(fast_tac (HOL_cs addSEs [emptyE,CollectE,UnE]) 1);
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val singletonD = result();
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val singletonE = make_elim singletonD;
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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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val singleton_inject = result();
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(*** Unions of families -- UNION x:A. B(x) is Union(B``A) ***)
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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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val UN_I = result();
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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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val UN_E = result();
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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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340 |
by (REPEAT (etac UN_E 1
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341 |
ORELSE ares_tac ([UN_I,equalityI,subsetI] @
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342 |
(prems RL [equalityD1,equalityD2] RL [subsetD])) 1));
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343 |
val UN_cong = result();
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344 |
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345 |
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346 |
(*** Intersections of families -- INTER x:A. B(x) is Inter(B``A) *)
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347 |
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348 |
val prems = goalw Set.thy [INTER_def]
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349 |
"(!!x. x:A ==> b: B(x)) ==> b : (INT x:A. B(x))";
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350 |
by (REPEAT (ares_tac ([CollectI,ballI] @ prems) 1));
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351 |
val INT_I = result();
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352 |
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353 |
val major::prems = goalw Set.thy [INTER_def]
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354 |
"[| b : (INT x:A. B(x)); a:A |] ==> b: B(a)";
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355 |
by (rtac (major RS CollectD RS bspec) 1);
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356 |
by (resolve_tac prems 1);
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357 |
val INT_D = result();
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358 |
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359 |
(*"Classical" elimination -- by the Excluded Middle on a:A *)
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360 |
val major::prems = goalw Set.thy [INTER_def]
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5
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361 |
"[| b : (INT x:A. B(x)); b: B(a) ==> R; a~:A ==> R |] ==> R";
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0
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362 |
by (rtac (major RS CollectD RS ballE) 1);
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363 |
by (REPEAT (eresolve_tac prems 1));
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364 |
val INT_E = result();
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365 |
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366 |
val prems = goal Set.thy
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|
367 |
"[| A=B; !!x. x:B ==> C(x) = D(x) |] ==> \
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|
368 |
\ (INT x:A. C(x)) = (INT x:B. D(x))";
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369 |
by (REPEAT_FIRST (resolve_tac [INT_I,equalityI,subsetI]));
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|
370 |
by (REPEAT (dtac INT_D 1
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|
371 |
ORELSE ares_tac (prems RL [equalityD1,equalityD2] RL [subsetD]) 1));
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|
372 |
val INT_cong = result();
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373 |
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374 |
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|
375 |
(*** Unions over a type; UNION1(B) = Union(range(B)) ***)
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376 |
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|
377 |
(*The order of the premises presupposes that A is rigid; b may be flexible*)
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|
378 |
val prems = goalw Set.thy [UNION1_def]
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|
379 |
"b: B(x) ==> b: (UN x. B(x))";
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|
380 |
by (REPEAT (resolve_tac (prems @ [TrueI, CollectI RS UN_I]) 1));
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|
381 |
val UN1_I = result();
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|
382 |
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|
383 |
val major::prems = goalw Set.thy [UNION1_def]
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|
384 |
"[| b : (UN x. B(x)); !!x. b: B(x) ==> R |] ==> R";
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|
385 |
by (rtac (major RS UN_E) 1);
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|
386 |
by (REPEAT (ares_tac prems 1));
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|
387 |
val UN1_E = result();
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|
388 |
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|
389 |
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|
390 |
(*** Intersections over a type; INTER1(B) = Inter(range(B)) *)
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|
391 |
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|
392 |
val prems = goalw Set.thy [INTER1_def]
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|
393 |
"(!!x. b: B(x)) ==> b : (INT x. B(x))";
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|
394 |
by (REPEAT (ares_tac (INT_I::prems) 1));
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|
395 |
val INT1_I = result();
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|
396 |
|
|
397 |
val [major] = goalw Set.thy [INTER1_def]
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|
398 |
"b : (INT x. B(x)) ==> b: B(a)";
|
|
399 |
by (rtac (TrueI RS (CollectI RS (major RS INT_D))) 1);
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|
400 |
val INT1_D = result();
|
|
401 |
|
|
402 |
(*** Unions ***)
|
|
403 |
|
|
404 |
(*The order of the premises presupposes that C is rigid; A may be flexible*)
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|
405 |
val prems = goalw Set.thy [Union_def]
|
|
406 |
"[| X:C; A:X |] ==> A : Union(C)";
|
|
407 |
by (REPEAT (resolve_tac (prems @ [UN_I]) 1));
|
|
408 |
val UnionI = result();
|
|
409 |
|
|
410 |
val major::prems = goalw Set.thy [Union_def]
|
|
411 |
"[| A : Union(C); !!X.[| A:X; X:C |] ==> R |] ==> R";
|
|
412 |
by (rtac (major RS UN_E) 1);
|
|
413 |
by (REPEAT (ares_tac prems 1));
|
|
414 |
val UnionE = result();
|
|
415 |
|
|
416 |
(*** Inter ***)
|
|
417 |
|
|
418 |
val prems = goalw Set.thy [Inter_def]
|
|
419 |
"[| !!X. X:C ==> A:X |] ==> A : Inter(C)";
|
|
420 |
by (REPEAT (ares_tac ([INT_I] @ prems) 1));
|
|
421 |
val InterI = result();
|
|
422 |
|
|
423 |
(*A "destruct" rule -- every X in C contains A as an element, but
|
|
424 |
A:X can hold when X:C does not! This rule is analogous to "spec". *)
|
|
425 |
val major::prems = goalw Set.thy [Inter_def]
|
|
426 |
"[| A : Inter(C); X:C |] ==> A:X";
|
|
427 |
by (rtac (major RS INT_D) 1);
|
|
428 |
by (resolve_tac prems 1);
|
|
429 |
val InterD = result();
|
|
430 |
|
|
431 |
(*"Classical" elimination rule -- does not require proving X:C *)
|
|
432 |
val major::prems = goalw Set.thy [Inter_def]
|
5
|
433 |
"[| A : Inter(C); A:X ==> R; X~:C ==> R |] ==> R";
|
0
|
434 |
by (rtac (major RS INT_E) 1);
|
|
435 |
by (REPEAT (eresolve_tac prems 1));
|
|
436 |
val InterE = result();
|