src/HOL/Set.ML
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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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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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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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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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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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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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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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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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goalw Set.thy [Ball_def] "(! x:A. True) = True";
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by (Simp_tac 1);
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qed "ball_True";
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(*Dual form for existentials*)
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goalw Set.thy [Bex_def] "(? x:A. False) = False";
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by (Simp_tac 1);
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qed "bex_False";
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Addsimps [ball_True, bex_False];
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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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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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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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(*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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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) ]);
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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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qed_goal "empty_iff" Set.thy "(c : {}) = False"
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 (fn _ => [ (fast_tac (!claset addSEs [emptyE]) 1) ]);
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goal Set.thy "Ball {} P = True";
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by (simp_tac (HOL_ss addsimps [mem_Collect_eq, Ball_def, empty_def]) 1);
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qed "ball_empty";
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goal Set.thy "Bex {} P = False";
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by (simp_tac (HOL_ss addsimps [mem_Collect_eq, Bex_def, empty_def]) 1);
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parents: 1776
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   311
qed "bex_empty";
b03dba9116d4 New rewrites for vacuous quantification
paulson
parents: 1776
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   312
Addsimps [ball_empty, bex_empty];
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   313
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   314
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   315
section "Augmenting a set -- insert";
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   316
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   317
qed_goalw "insertI1" Set.thy [insert_def] "a : insert a B"
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parents:
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   318
 (fn _ => [rtac (CollectI RS UnI1) 1, rtac refl 1]);
ff1574a81019 new version of HOL with curried function application
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parents:
diff changeset
   319
ff1574a81019 new version of HOL with curried function application
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parents:
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   320
qed_goalw "insertI2" Set.thy [insert_def] "a : B ==> a : insert b B"
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parents:
diff changeset
   321
 (fn [prem]=> [ (rtac (prem RS UnI2) 1) ]);
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parents:
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   322
ff1574a81019 new version of HOL with curried function application
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parents:
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   323
qed_goalw "insertE" Set.thy [insert_def]
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parents:
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   324
    "[| a : insert b A;  a=b ==> P;  a:A ==> P |] ==> P"
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parents:
diff changeset
   325
 (fn major::prems=>
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parents:
diff changeset
   326
  [ (rtac (major RS UnE) 1),
ff1574a81019 new version of HOL with curried function application
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parents:
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   327
    (REPEAT (eresolve_tac (prems @ [CollectE]) 1)) ]);
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parents:
diff changeset
   328
ff1574a81019 new version of HOL with curried function application
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parents:
diff changeset
   329
qed_goal "insert_iff" Set.thy "a : insert b A = (a=b | a:A)"
1760
6f41a494f3b1 Replaced fast_tac by Fast_tac (which uses default claset)
berghofe
parents: 1640
diff changeset
   330
 (fn _ => [fast_tac (!claset addIs [insertI1,insertI2] addSEs [insertE]) 1]);
923
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parents:
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   331
ff1574a81019 new version of HOL with curried function application
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parents:
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   332
(*Classical introduction rule*)
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parents:
diff changeset
   333
qed_goal "insertCI" Set.thy "(a~:B ==> a=b) ==> a: insert b B"
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parents:
diff changeset
   334
 (fn [prem]=>
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parents:
diff changeset
   335
  [ (rtac (disjCI RS (insert_iff RS iffD2)) 1),
ff1574a81019 new version of HOL with curried function application
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parents:
diff changeset
   336
    (etac prem 1) ]);
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parents:
diff changeset
   337
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parents: 1531
diff changeset
   338
section "Singletons, using insert";
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parents:
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   339
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parents:
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   340
qed_goal "singletonI" Set.thy "a : {a}"
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parents:
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   341
 (fn _=> [ (rtac insertI1 1) ]);
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parents:
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   342
ff1574a81019 new version of HOL with curried function application
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parents:
diff changeset
   343
goalw Set.thy [insert_def] "!!a. b : {a} ==> b=a";
1760
6f41a494f3b1 Replaced fast_tac by Fast_tac (which uses default claset)
berghofe
parents: 1640
diff changeset
   344
by (fast_tac (!claset addSEs [emptyE,CollectE,UnE]) 1);
923
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parents:
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   345
qed "singletonD";
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parents:
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   346
1776
d7e77cb8ce5c moved mem_simps and the corresponding update of !simpset from Fun.ML to Set.ML,
oheimb
parents: 1762
diff changeset
   347
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
   348
d7e77cb8ce5c moved mem_simps and the corresponding update of !simpset from Fun.ML to Set.ML,
oheimb
parents: 1762
diff changeset
   349
qed_goal "singleton_iff" thy "(b : {a}) = (b=a)" (fn _ => [
d7e77cb8ce5c moved mem_simps and the corresponding update of !simpset from Fun.ML to Set.ML,
oheimb
parents: 1762
diff changeset
   350
	rtac iffI 1,
d7e77cb8ce5c moved mem_simps and the corresponding update of !simpset from Fun.ML to Set.ML,
oheimb
parents: 1762
diff changeset
   351
	etac singletonD 1,
d7e77cb8ce5c moved mem_simps and the corresponding update of !simpset from Fun.ML to Set.ML,
oheimb
parents: 1762
diff changeset
   352
	hyp_subst_tac 1,
d7e77cb8ce5c moved mem_simps and the corresponding update of !simpset from Fun.ML to Set.ML,
oheimb
parents: 1762
diff changeset
   353
	rtac singletonI 1]);
923
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   354
ff1574a81019 new version of HOL with curried function application
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parents:
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   355
val [major] = goal Set.thy "{a}={b} ==> a=b";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   356
by (rtac (major RS equalityD1 RS subsetD RS singletonD) 1);
ff1574a81019 new version of HOL with curried function application
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parents:
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   357
by (rtac singletonI 1);
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clasohm
parents:
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   358
qed "singleton_inject";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   359
1531
e5eb247ad13c Added a constant UNIV == {x.True}
nipkow
parents: 1465
diff changeset
   360
1548
afe750876848 Added 'section' commands
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parents: 1531
diff changeset
   361
section "The universal set -- UNIV";
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e5eb247ad13c Added a constant UNIV == {x.True}
nipkow
parents: 1465
diff changeset
   362
1882
67f49e8c4355 Proved bex_False
paulson
parents: 1841
diff changeset
   363
qed_goal "UNIV_I" Set.thy "x : UNIV"
67f49e8c4355 Proved bex_False
paulson
parents: 1841
diff changeset
   364
  (fn _ => [rtac ComplI 1, etac emptyE 1]);
67f49e8c4355 Proved bex_False
paulson
parents: 1841
diff changeset
   365
1531
e5eb247ad13c Added a constant UNIV == {x.True}
nipkow
parents: 1465
diff changeset
   366
qed_goal "subset_UNIV" Set.thy "A <= UNIV"
1882
67f49e8c4355 Proved bex_False
paulson
parents: 1841
diff changeset
   367
  (fn _ => [rtac subsetI 1, rtac UNIV_I 1]);
1531
e5eb247ad13c Added a constant UNIV == {x.True}
nipkow
parents: 1465
diff changeset
   368
e5eb247ad13c Added a constant UNIV == {x.True}
nipkow
parents: 1465
diff changeset
   369
1548
afe750876848 Added 'section' commands
nipkow
parents: 1531
diff changeset
   370
section "Unions of families -- UNION x:A. B(x) is Union(B``A)";
923
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parents:
diff changeset
   371
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   372
(*The order of the premises presupposes that A is rigid; b may be flexible*)
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   373
val prems = goalw Set.thy [UNION_def]
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   374
    "[| a:A;  b: B(a) |] ==> b: (UN x:A. B(x))";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   375
by (REPEAT (resolve_tac (prems @ [bexI,CollectI]) 1));
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   376
qed "UN_I";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   377
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   378
val major::prems = goalw Set.thy [UNION_def]
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   379
    "[| b : (UN x:A. B(x));  !!x.[| x:A;  b: B(x) |] ==> R |] ==> R";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   380
by (rtac (major RS CollectD RS bexE) 1);
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   381
by (REPEAT (ares_tac prems 1));
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   382
qed "UN_E";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   383
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   384
val prems = goal Set.thy
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   385
    "[| A=B;  !!x. x:B ==> C(x) = D(x) |] ==> \
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   386
\    (UN x:A. C(x)) = (UN x:B. D(x))";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   387
by (REPEAT (etac UN_E 1
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   388
     ORELSE ares_tac ([UN_I,equalityI,subsetI] @ 
1465
5d7a7e439cec expanded tabs
clasohm
parents: 923
diff changeset
   389
                      (prems RL [equalityD1,equalityD2] RL [subsetD])) 1));
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   390
qed "UN_cong";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   391
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   392
1548
afe750876848 Added 'section' commands
nipkow
parents: 1531
diff changeset
   393
section "Intersections of families -- INTER x:A. B(x) is Inter(B``A)";
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   394
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   395
val prems = goalw Set.thy [INTER_def]
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   396
    "(!!x. x:A ==> b: B(x)) ==> b : (INT x:A. B(x))";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   397
by (REPEAT (ares_tac ([CollectI,ballI] @ prems) 1));
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   398
qed "INT_I";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   399
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   400
val major::prems = goalw Set.thy [INTER_def]
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   401
    "[| b : (INT x:A. B(x));  a:A |] ==> b: B(a)";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   402
by (rtac (major RS CollectD RS bspec) 1);
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   403
by (resolve_tac prems 1);
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   404
qed "INT_D";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   405
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   406
(*"Classical" elimination -- by the Excluded Middle on a:A *)
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   407
val major::prems = goalw Set.thy [INTER_def]
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   408
    "[| b : (INT x:A. B(x));  b: B(a) ==> R;  a~:A ==> R |] ==> R";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   409
by (rtac (major RS CollectD RS ballE) 1);
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   410
by (REPEAT (eresolve_tac prems 1));
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   411
qed "INT_E";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   412
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   413
val prems = goal Set.thy
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   414
    "[| A=B;  !!x. x:B ==> C(x) = D(x) |] ==> \
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   415
\    (INT x:A. C(x)) = (INT x:B. D(x))";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   416
by (REPEAT_FIRST (resolve_tac [INT_I,equalityI,subsetI]));
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   417
by (REPEAT (dtac INT_D 1
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   418
     ORELSE ares_tac (prems RL [equalityD1,equalityD2] RL [subsetD]) 1));
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   419
qed "INT_cong";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   420
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   421
1548
afe750876848 Added 'section' commands
nipkow
parents: 1531
diff changeset
   422
section "Unions over a type; UNION1(B) = Union(range(B))";
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   423
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   424
(*The order of the premises presupposes that A is rigid; b may be flexible*)
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   425
val prems = goalw Set.thy [UNION1_def]
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   426
    "b: B(x) ==> b: (UN x. B(x))";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   427
by (REPEAT (resolve_tac (prems @ [TrueI, CollectI RS UN_I]) 1));
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   428
qed "UN1_I";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   429
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   430
val major::prems = goalw Set.thy [UNION1_def]
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   431
    "[| b : (UN x. B(x));  !!x. b: B(x) ==> R |] ==> R";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   432
by (rtac (major RS UN_E) 1);
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   433
by (REPEAT (ares_tac prems 1));
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   434
qed "UN1_E";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   435
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   436
1548
afe750876848 Added 'section' commands
nipkow
parents: 1531
diff changeset
   437
section "Intersections over a type; INTER1(B) = Inter(range(B))";
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   438
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   439
val prems = goalw Set.thy [INTER1_def]
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   440
    "(!!x. b: B(x)) ==> b : (INT x. B(x))";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   441
by (REPEAT (ares_tac (INT_I::prems) 1));
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   442
qed "INT1_I";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   443
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   444
val [major] = goalw Set.thy [INTER1_def]
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   445
    "b : (INT x. B(x)) ==> b: B(a)";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   446
by (rtac (TrueI RS (CollectI RS (major RS INT_D))) 1);
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   447
qed "INT1_D";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   448
1548
afe750876848 Added 'section' commands
nipkow
parents: 1531
diff changeset
   449
section "Union";
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   450
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   451
(*The order of the premises presupposes that C is rigid; A may be flexible*)
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   452
val prems = goalw Set.thy [Union_def]
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   453
    "[| X:C;  A:X |] ==> A : Union(C)";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   454
by (REPEAT (resolve_tac (prems @ [UN_I]) 1));
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   455
qed "UnionI";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   456
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   457
val major::prems = goalw Set.thy [Union_def]
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   458
    "[| A : Union(C);  !!X.[| A:X;  X:C |] ==> R |] ==> R";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   459
by (rtac (major RS UN_E) 1);
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   460
by (REPEAT (ares_tac prems 1));
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   461
qed "UnionE";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   462
1548
afe750876848 Added 'section' commands
nipkow
parents: 1531
diff changeset
   463
section "Inter";
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   464
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   465
val prems = goalw Set.thy [Inter_def]
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   466
    "[| !!X. X:C ==> A:X |] ==> A : Inter(C)";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   467
by (REPEAT (ares_tac ([INT_I] @ prems) 1));
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   468
qed "InterI";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   469
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   470
(*A "destruct" rule -- every X in C contains A as an element, but
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   471
  A:X can hold when X:C does not!  This rule is analogous to "spec". *)
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   472
val major::prems = goalw Set.thy [Inter_def]
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   473
    "[| A : Inter(C);  X:C |] ==> A:X";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   474
by (rtac (major RS INT_D) 1);
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   475
by (resolve_tac prems 1);
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   476
qed "InterD";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   477
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   478
(*"Classical" elimination rule -- does not require proving X:C *)
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   479
val major::prems = goalw Set.thy [Inter_def]
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   480
    "[| A : Inter(C);  A:X ==> R;  X~:C ==> R |] ==> R";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   481
by (rtac (major RS INT_E) 1);
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   482
by (REPEAT (eresolve_tac prems 1));
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   483
qed "InterE";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   484
1548
afe750876848 Added 'section' commands
nipkow
parents: 1531
diff changeset
   485
section "The Powerset operator -- Pow";
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   486
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   487
qed_goalw "PowI" Set.thy [Pow_def] "!!A B. A <= B ==> A : Pow(B)"
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   488
 (fn _ => [ (etac CollectI 1) ]);
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   489
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   490
qed_goalw "PowD" Set.thy [Pow_def] "!!A B. A : Pow(B)  ==>  A<=B"
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   491
 (fn _=> [ (etac CollectD 1) ]);
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   492
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   493
val Pow_bottom = empty_subsetI RS PowI;        (* {}: Pow(B) *)
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   494
val Pow_top = subset_refl RS PowI;             (* A : Pow(A) *)
1776
d7e77cb8ce5c moved mem_simps and the corresponding update of !simpset from Fun.ML to Set.ML,
oheimb
parents: 1762
diff changeset
   495
d7e77cb8ce5c moved mem_simps and the corresponding update of !simpset from Fun.ML to Set.ML,
oheimb
parents: 1762
diff changeset
   496
d7e77cb8ce5c moved mem_simps and the corresponding update of !simpset from Fun.ML to Set.ML,
oheimb
parents: 1762
diff changeset
   497
d7e77cb8ce5c moved mem_simps and the corresponding update of !simpset from Fun.ML to Set.ML,
oheimb
parents: 1762
diff changeset
   498
(*** Set reasoning tools ***)
d7e77cb8ce5c moved mem_simps and the corresponding update of !simpset from Fun.ML to Set.ML,
oheimb
parents: 1762
diff changeset
   499
d7e77cb8ce5c moved mem_simps and the corresponding update of !simpset from Fun.ML to Set.ML,
oheimb
parents: 1762
diff changeset
   500
d7e77cb8ce5c moved mem_simps and the corresponding update of !simpset from Fun.ML to Set.ML,
oheimb
parents: 1762
diff changeset
   501
val mem_simps = [ Un_iff, Int_iff, Compl_iff, Diff_iff, singleton_iff,
d7e77cb8ce5c moved mem_simps and the corresponding update of !simpset from Fun.ML to Set.ML,
oheimb
parents: 1762
diff changeset
   502
		  mem_Collect_eq];
d7e77cb8ce5c moved mem_simps and the corresponding update of !simpset from Fun.ML to Set.ML,
oheimb
parents: 1762
diff changeset
   503
d7e77cb8ce5c moved mem_simps and the corresponding update of !simpset from Fun.ML to Set.ML,
oheimb
parents: 1762
diff changeset
   504
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
   505
d7e77cb8ce5c moved mem_simps and the corresponding update of !simpset from Fun.ML to Set.ML,
oheimb
parents: 1762
diff changeset
   506
simpset := !simpset addsimps mem_simps
d7e77cb8ce5c moved mem_simps and the corresponding update of !simpset from Fun.ML to Set.ML,
oheimb
parents: 1762
diff changeset
   507
                    addcongs [ball_cong,bex_cong]
d7e77cb8ce5c moved mem_simps and the corresponding update of !simpset from Fun.ML to Set.ML,
oheimb
parents: 1762
diff changeset
   508
                    setmksimps (mksimps mksimps_pairs);