author | paulson |
Tue, 01 Jul 1997 10:37:03 +0200 | |
changeset 3469 | 61d927bd57ec |
parent 3420 | 02dc9c5b035f |
child 3582 | b87c86b6c291 |
permissions | -rw-r--r-- |
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(* Title: HOL/set |
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ID: $Id$ |
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Author: Lawrence C Paulson, Cambridge University Computer Laboratory |
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Copyright 1991 University of Cambridge |
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1985
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Set theory for higher-order logic. A set is simply a predicate. |
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*) |
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open Set; |
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section "Relating predicates and sets"; |
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Addsimps [Collect_mem_eq]; |
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AddIffs [mem_Collect_eq]; |
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goal Set.thy "!!a. P(a) ==> a : {x.P(x)}"; |
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by (Asm_simp_tac 1); |
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qed "CollectI"; |
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val prems = goal Set.thy "!!a. a : {x.P(x)} ==> P(a)"; |
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by (Asm_full_simp_tac 1); |
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qed "CollectD"; |
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||
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val [prem] = goal Set.thy "[| !!x. (x:A) = (x:B) |] ==> A = B"; |
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by (rtac (prem RS ext RS arg_cong RS box_equals) 1); |
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by (rtac Collect_mem_eq 1); |
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by (rtac Collect_mem_eq 1); |
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qed "set_ext"; |
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val [prem] = goal Set.thy "[| !!x. P(x)=Q(x) |] ==> {x. P(x)} = {x. Q(x)}"; |
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by (rtac (prem RS ext RS arg_cong) 1); |
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qed "Collect_cong"; |
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val CollectE = make_elim CollectD; |
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||
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AddSIs [CollectI]; |
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AddSEs [CollectE]; |
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section "Bounded quantifiers"; |
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val prems = goalw Set.thy [Ball_def] |
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"[| !!x. x:A ==> P(x) |] ==> ! x:A. P(x)"; |
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by (REPEAT (ares_tac (prems @ [allI,impI]) 1)); |
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qed "ballI"; |
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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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AddSIs [ballI]; |
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AddEs [ballE]; |
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val prems = goalw Set.thy [Bex_def] |
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"[| P(x); x:A |] ==> ? x:A. P(x)"; |
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by (REPEAT (ares_tac (prems @ [exI,conjI]) 1)); |
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qed "bexI"; |
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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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AddIs [bexI]; |
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AddSEs [bexE]; |
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|
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(*Trival rewrite rule*) |
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goal Set.thy "(! x:A.P) = ((? x. x:A) --> P)"; |
|
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by (simp_tac (!simpset addsimps [Ball_def]) 1); |
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qed "ball_triv"; |
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(*Dual form for existentials*) |
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goal Set.thy "(? x:A.P) = ((? x. x:A) & P)"; |
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by (simp_tac (!simpset addsimps [Bex_def]) 1); |
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qed "bex_triv"; |
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Addsimps [ball_triv, bex_triv]; |
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(** Congruence rules **) |
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||
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val prems = goal Set.thy |
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"[| A=B; !!x. x:B ==> P(x) = Q(x) |] ==> \ |
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\ (! x:A. P(x)) = (! x:B. Q(x))"; |
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by (resolve_tac (prems RL [ssubst]) 1); |
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by (REPEAT (ares_tac [ballI,iffI] 1 |
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ORELSE eresolve_tac ([make_elim bspec, mp] @ (prems RL [iffE])) 1)); |
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qed "ball_cong"; |
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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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||
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Blast.declConsts (["op <="], [subsetI]); (*overloading of <=*) |
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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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||
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(*Converts A<=B to x:A ==> x:B*) |
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fun impOfSubs th = th RSN (2, rev_subsetD); |
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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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AddSIs [subsetI]; |
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AddEs [subsetD, subsetCE]; |
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qed_goal "subset_refl" Set.thy "A <= (A::'a set)" |
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(fn _=> [Blast_tac 1]); |
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val prems = goal Set.thy "!!B. [| A<=B; B<=C |] ==> A<=(C::'a set)"; |
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by (Blast_tac 1); |
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qed "subset_trans"; |
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section "Equality"; |
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(*Anti-symmetry of the subset relation*) |
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val prems = goal Set.thy "[| A <= B; B <= A |] ==> A = (B::'a set)"; |
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by (rtac (iffI RS set_ext) 1); |
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by (REPEAT (ares_tac (prems RL [subsetD]) 1)); |
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qed "subset_antisym"; |
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val equalityI = subset_antisym; |
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||
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Blast.declConsts (["op ="], [equalityI]); (*overloading of equality*) |
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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 "The empty set -- {}"; |
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qed_goalw "empty_iff" Set.thy [empty_def] "(c : {}) = False" |
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(fn _ => [ (Blast_tac 1) ]); |
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Addsimps [empty_iff]; |
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||
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qed_goal "emptyE" Set.thy "!!a. a:{} ==> P" |
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(fn _ => [Full_simp_tac 1]); |
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AddSEs [emptyE]; |
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qed_goal "empty_subsetI" Set.thy "{} <= A" |
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(fn _ => [ (Blast_tac 1) ]); |
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qed_goal "equals0I" Set.thy "[| !!y. y:A ==> False |] ==> A={}" |
|
223 |
(fn [prem]=> |
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[ (blast_tac (!claset addIs [prem RS FalseE]) 1) ]); |
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qed_goal "equals0D" Set.thy "!!a. [| A={}; a:A |] ==> P" |
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(fn _ => [ (Blast_tac 1) ]); |
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|
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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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qed "bex_empty"; |
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Addsimps [ball_empty, bex_empty]; |
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||
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section "The Powerset operator -- Pow"; |
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qed_goalw "Pow_iff" Set.thy [Pow_def] "(A : Pow(B)) = (A <= B)" |
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(fn _ => [ (Asm_simp_tac 1) ]); |
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AddIffs [Pow_iff]; |
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||
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qed_goalw "PowI" Set.thy [Pow_def] "!!A B. A <= B ==> A : Pow(B)" |
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(fn _ => [ (etac CollectI 1) ]); |
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||
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qed_goalw "PowD" Set.thy [Pow_def] "!!A B. A : Pow(B) ==> A<=B" |
|
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(fn _=> [ (etac CollectD 1) ]); |
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val Pow_bottom = empty_subsetI RS PowI; (* {}: Pow(B) *) |
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val Pow_top = subset_refl RS PowI; (* A : Pow(A) *) |
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255 |
||
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section "Set complement -- Compl"; |
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qed_goalw "Compl_iff" Set.thy [Compl_def] "(c : Compl(A)) = (c~:A)" |
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(fn _ => [ (Blast_tac 1) ]); |
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Addsimps [Compl_iff]; |
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|
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val prems = goalw Set.thy [Compl_def] |
264 |
"[| 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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268 |
(*This form, with negated conclusion, works well with the Classical prover. |
|
269 |
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); |
274 |
qed "ComplD"; |
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val ComplE = make_elim ComplD; |
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AddSIs [ComplI]; |
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AddSEs [ComplE]; |
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section "Binary union -- Un"; |
923 | 283 |
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qed_goalw "Un_iff" Set.thy [Un_def] "(c : A Un B) = (c:A | c:B)" |
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(fn _ => [ Blast_tac 1 ]); |
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|
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Addsimps [Un_iff]; |
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goal Set.thy "!!c. c:A ==> c : A Un B"; |
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by (Asm_simp_tac 1); |
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qed "UnI1"; |
292 |
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goal Set.thy "!!c. c:B ==> c : A Un B"; |
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by (Asm_simp_tac 1); |
923 | 295 |
qed "UnI2"; |
296 |
||
297 |
(*Classical introduction rule: no commitment to A vs B*) |
|
298 |
qed_goal "UnCI" Set.thy "(c~:B ==> c:A) ==> c : A Un B" |
|
299 |
(fn prems=> |
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|
300 |
[ (Simp_tac 1), |
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
301 |
(REPEAT (ares_tac (prems@[disjCI]) 1)) ]); |
923 | 302 |
|
303 |
val major::prems = goalw Set.thy [Un_def] |
|
304 |
"[| c : A Un B; c:A ==> P; c:B ==> P |] ==> P"; |
|
305 |
by (rtac (major RS CollectD RS disjE) 1); |
|
306 |
by (REPEAT (eresolve_tac prems 1)); |
|
307 |
qed "UnE"; |
|
308 |
||
2499
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
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parents:
2031
diff
changeset
|
309 |
AddSIs [UnCI]; |
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
310 |
AddSEs [UnE]; |
1640 | 311 |
|
923 | 312 |
|
1548 | 313 |
section "Binary intersection -- Int"; |
923 | 314 |
|
2499
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
315 |
qed_goalw "Int_iff" Set.thy [Int_def] "(c : A Int B) = (c:A & c:B)" |
2891 | 316 |
(fn _ => [ (Blast_tac 1) ]); |
2499
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
317 |
|
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
318 |
Addsimps [Int_iff]; |
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
319 |
|
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
320 |
goal Set.thy "!!c. [| c:A; c:B |] ==> c : A Int B"; |
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
321 |
by (Asm_simp_tac 1); |
923 | 322 |
qed "IntI"; |
323 |
||
2499
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
324 |
goal Set.thy "!!c. c : A Int B ==> c:A"; |
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
325 |
by (Asm_full_simp_tac 1); |
923 | 326 |
qed "IntD1"; |
327 |
||
2499
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
328 |
goal Set.thy "!!c. c : A Int B ==> c:B"; |
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
329 |
by (Asm_full_simp_tac 1); |
923 | 330 |
qed "IntD2"; |
331 |
||
332 |
val [major,minor] = goal Set.thy |
|
333 |
"[| c : A Int B; [| c:A; c:B |] ==> P |] ==> P"; |
|
334 |
by (rtac minor 1); |
|
335 |
by (rtac (major RS IntD1) 1); |
|
336 |
by (rtac (major RS IntD2) 1); |
|
337 |
qed "IntE"; |
|
338 |
||
2499
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
339 |
AddSIs [IntI]; |
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
340 |
AddSEs [IntE]; |
923 | 341 |
|
1548 | 342 |
section "Set difference"; |
923 | 343 |
|
2499
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
344 |
qed_goalw "Diff_iff" Set.thy [set_diff_def] "(c : A-B) = (c:A & c~:B)" |
2891 | 345 |
(fn _ => [ (Blast_tac 1) ]); |
923 | 346 |
|
2499
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
347 |
Addsimps [Diff_iff]; |
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
348 |
|
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
349 |
qed_goal "DiffI" Set.thy "!!c. [| c : A; c ~: B |] ==> c : A - B" |
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
350 |
(fn _=> [ Asm_simp_tac 1 ]); |
923 | 351 |
|
2499
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
352 |
qed_goal "DiffD1" Set.thy "!!c. c : A - B ==> c : A" |
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
353 |
(fn _=> [ (Asm_full_simp_tac 1) ]); |
923 | 354 |
|
2499
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
355 |
qed_goal "DiffD2" Set.thy "!!c. [| c : A - B; c : B |] ==> P" |
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
356 |
(fn _=> [ (Asm_full_simp_tac 1) ]); |
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
357 |
|
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
358 |
qed_goal "DiffE" Set.thy "[| c : A - B; [| c:A; c~:B |] ==> P |] ==> P" |
923 | 359 |
(fn prems=> |
360 |
[ (resolve_tac prems 1), |
|
361 |
(REPEAT (ares_tac (prems RL [DiffD1, DiffD2 RS notI]) 1)) ]); |
|
362 |
||
2499
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
363 |
AddSIs [DiffI]; |
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
364 |
AddSEs [DiffE]; |
923 | 365 |
|
366 |
||
1548 | 367 |
section "Augmenting a set -- insert"; |
923 | 368 |
|
2499
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
369 |
qed_goalw "insert_iff" Set.thy [insert_def] "a : insert b A = (a=b | a:A)" |
2891 | 370 |
(fn _ => [Blast_tac 1]); |
2499
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
371 |
|
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
372 |
Addsimps [insert_iff]; |
923 | 373 |
|
2499
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
374 |
qed_goal "insertI1" Set.thy "a : insert a B" |
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
375 |
(fn _ => [Simp_tac 1]); |
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
376 |
|
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
377 |
qed_goal "insertI2" Set.thy "!!a. a : B ==> a : insert b B" |
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
378 |
(fn _=> [Asm_simp_tac 1]); |
923 | 379 |
|
380 |
qed_goalw "insertE" Set.thy [insert_def] |
|
381 |
"[| a : insert b A; a=b ==> P; a:A ==> P |] ==> P" |
|
382 |
(fn major::prems=> |
|
383 |
[ (rtac (major RS UnE) 1), |
|
384 |
(REPEAT (eresolve_tac (prems @ [CollectE]) 1)) ]); |
|
385 |
||
386 |
(*Classical introduction rule*) |
|
387 |
qed_goal "insertCI" Set.thy "(a~:B ==> a=b) ==> a: insert b B" |
|
2499
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
388 |
(fn prems=> |
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
389 |
[ (Simp_tac 1), |
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
390 |
(REPEAT (ares_tac (prems@[disjCI]) 1)) ]); |
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
391 |
|
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
392 |
AddSIs [insertCI]; |
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
393 |
AddSEs [insertE]; |
923 | 394 |
|
1548 | 395 |
section "Singletons, using insert"; |
923 | 396 |
|
397 |
qed_goal "singletonI" Set.thy "a : {a}" |
|
398 |
(fn _=> [ (rtac insertI1 1) ]); |
|
399 |
||
2499
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
400 |
goal Set.thy "!!a. b : {a} ==> b=a"; |
2891 | 401 |
by (Blast_tac 1); |
923 | 402 |
qed "singletonD"; |
403 |
||
1776
d7e77cb8ce5c
moved mem_simps and the corresponding update of !simpset from Fun.ML to Set.ML,
oheimb
parents:
1762
diff
changeset
|
404 |
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
|
405 |
|
2499
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
406 |
qed_goal "singleton_iff" thy "(b : {a}) = (b=a)" |
2891 | 407 |
(fn _ => [Blast_tac 1]); |
923 | 408 |
|
2499
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
409 |
goal Set.thy "!!a b. {a}={b} ==> a=b"; |
2935 | 410 |
by (blast_tac (!claset addEs [equalityE]) 1); |
923 | 411 |
qed "singleton_inject"; |
412 |
||
2858 | 413 |
(*Redundant? But unlike insertCI, it proves the subgoal immediately!*) |
414 |
AddSIs [singletonI]; |
|
415 |
||
2499
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
416 |
AddSDs [singleton_inject]; |
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
417 |
|
1531 | 418 |
|
1548 | 419 |
section "The universal set -- UNIV"; |
1531 | 420 |
|
1882 | 421 |
qed_goal "UNIV_I" Set.thy "x : UNIV" |
422 |
(fn _ => [rtac ComplI 1, etac emptyE 1]); |
|
423 |
||
1531 | 424 |
qed_goal "subset_UNIV" Set.thy "A <= UNIV" |
1882 | 425 |
(fn _ => [rtac subsetI 1, rtac UNIV_I 1]); |
1531 | 426 |
|
427 |
||
1548 | 428 |
section "Unions of families -- UNION x:A. B(x) is Union(B``A)"; |
923 | 429 |
|
2499
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
430 |
goalw Set.thy [UNION_def] "(b: (UN x:A. B(x))) = (EX x:A. b: B(x))"; |
2891 | 431 |
by (Blast_tac 1); |
2499
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
432 |
qed "UN_iff"; |
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
433 |
|
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
434 |
Addsimps [UN_iff]; |
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
435 |
|
923 | 436 |
(*The order of the premises presupposes that A is rigid; b may be flexible*) |
2499
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
437 |
goal Set.thy "!!b. [| a:A; b: B(a) |] ==> b: (UN x:A. B(x))"; |
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
438 |
by (Auto_tac()); |
923 | 439 |
qed "UN_I"; |
440 |
||
441 |
val major::prems = goalw Set.thy [UNION_def] |
|
442 |
"[| b : (UN x:A. B(x)); !!x.[| x:A; b: B(x) |] ==> R |] ==> R"; |
|
443 |
by (rtac (major RS CollectD RS bexE) 1); |
|
444 |
by (REPEAT (ares_tac prems 1)); |
|
445 |
qed "UN_E"; |
|
446 |
||
2499
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
447 |
AddIs [UN_I]; |
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
448 |
AddSEs [UN_E]; |
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
449 |
|
923 | 450 |
val prems = goal Set.thy |
451 |
"[| A=B; !!x. x:B ==> C(x) = D(x) |] ==> \ |
|
452 |
\ (UN x:A. C(x)) = (UN x:B. D(x))"; |
|
453 |
by (REPEAT (etac UN_E 1 |
|
454 |
ORELSE ares_tac ([UN_I,equalityI,subsetI] @ |
|
1465 | 455 |
(prems RL [equalityD1,equalityD2] RL [subsetD])) 1)); |
923 | 456 |
qed "UN_cong"; |
457 |
||
458 |
||
1548 | 459 |
section "Intersections of families -- INTER x:A. B(x) is Inter(B``A)"; |
923 | 460 |
|
2499
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
461 |
goalw Set.thy [INTER_def] "(b: (INT x:A. B(x))) = (ALL x:A. b: B(x))"; |
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
462 |
by (Auto_tac()); |
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
463 |
qed "INT_iff"; |
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
464 |
|
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
465 |
Addsimps [INT_iff]; |
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
466 |
|
923 | 467 |
val prems = goalw Set.thy [INTER_def] |
468 |
"(!!x. x:A ==> b: B(x)) ==> b : (INT x:A. B(x))"; |
|
469 |
by (REPEAT (ares_tac ([CollectI,ballI] @ prems) 1)); |
|
470 |
qed "INT_I"; |
|
471 |
||
2499
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
472 |
goal Set.thy "!!b. [| b : (INT x:A. B(x)); a:A |] ==> b: B(a)"; |
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
473 |
by (Auto_tac()); |
923 | 474 |
qed "INT_D"; |
475 |
||
476 |
(*"Classical" elimination -- by the Excluded Middle on a:A *) |
|
477 |
val major::prems = goalw Set.thy [INTER_def] |
|
478 |
"[| b : (INT x:A. B(x)); b: B(a) ==> R; a~:A ==> R |] ==> R"; |
|
479 |
by (rtac (major RS CollectD RS ballE) 1); |
|
480 |
by (REPEAT (eresolve_tac prems 1)); |
|
481 |
qed "INT_E"; |
|
482 |
||
2499
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
483 |
AddSIs [INT_I]; |
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
484 |
AddEs [INT_D, INT_E]; |
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
485 |
|
923 | 486 |
val prems = goal Set.thy |
487 |
"[| A=B; !!x. x:B ==> C(x) = D(x) |] ==> \ |
|
488 |
\ (INT x:A. C(x)) = (INT x:B. D(x))"; |
|
489 |
by (REPEAT_FIRST (resolve_tac [INT_I,equalityI,subsetI])); |
|
490 |
by (REPEAT (dtac INT_D 1 |
|
491 |
ORELSE ares_tac (prems RL [equalityD1,equalityD2] RL [subsetD]) 1)); |
|
492 |
qed "INT_cong"; |
|
493 |
||
494 |
||
1548 | 495 |
section "Unions over a type; UNION1(B) = Union(range(B))"; |
923 | 496 |
|
2499
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
497 |
goalw Set.thy [UNION1_def] "(b: (UN x. B(x))) = (EX x. b: B(x))"; |
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
498 |
by (Simp_tac 1); |
2891 | 499 |
by (Blast_tac 1); |
2499
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
500 |
qed "UN1_iff"; |
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
501 |
|
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
502 |
Addsimps [UN1_iff]; |
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
503 |
|
923 | 504 |
(*The order of the premises presupposes that A is rigid; b may be flexible*) |
2499
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
505 |
goal Set.thy "!!b. b: B(x) ==> b: (UN x. B(x))"; |
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
506 |
by (Auto_tac()); |
923 | 507 |
qed "UN1_I"; |
508 |
||
509 |
val major::prems = goalw Set.thy [UNION1_def] |
|
510 |
"[| b : (UN x. B(x)); !!x. b: B(x) ==> R |] ==> R"; |
|
511 |
by (rtac (major RS UN_E) 1); |
|
512 |
by (REPEAT (ares_tac prems 1)); |
|
513 |
qed "UN1_E"; |
|
514 |
||
2499
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
515 |
AddIs [UN1_I]; |
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
516 |
AddSEs [UN1_E]; |
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
517 |
|
923 | 518 |
|
1548 | 519 |
section "Intersections over a type; INTER1(B) = Inter(range(B))"; |
923 | 520 |
|
2499
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
521 |
goalw Set.thy [INTER1_def] "(b: (INT x. B(x))) = (ALL x. b: B(x))"; |
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
522 |
by (Simp_tac 1); |
2891 | 523 |
by (Blast_tac 1); |
2499
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
524 |
qed "INT1_iff"; |
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
525 |
|
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
526 |
Addsimps [INT1_iff]; |
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
527 |
|
923 | 528 |
val prems = goalw Set.thy [INTER1_def] |
529 |
"(!!x. b: B(x)) ==> b : (INT x. B(x))"; |
|
530 |
by (REPEAT (ares_tac (INT_I::prems) 1)); |
|
531 |
qed "INT1_I"; |
|
532 |
||
2499
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
533 |
goal Set.thy "!!b. b : (INT x. B(x)) ==> b: B(a)"; |
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
534 |
by (Asm_full_simp_tac 1); |
923 | 535 |
qed "INT1_D"; |
536 |
||
2499
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
537 |
AddSIs [INT1_I]; |
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
538 |
AddDs [INT1_D]; |
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
539 |
|
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
540 |
|
1548 | 541 |
section "Union"; |
923 | 542 |
|
2499
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
543 |
goalw Set.thy [Union_def] "(A : Union(C)) = (EX X:C. A:X)"; |
2891 | 544 |
by (Blast_tac 1); |
2499
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
545 |
qed "Union_iff"; |
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
546 |
|
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
547 |
Addsimps [Union_iff]; |
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
548 |
|
923 | 549 |
(*The order of the premises presupposes that C is rigid; A may be flexible*) |
2499
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
550 |
goal Set.thy "!!X. [| X:C; A:X |] ==> A : Union(C)"; |
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
551 |
by (Auto_tac()); |
923 | 552 |
qed "UnionI"; |
553 |
||
554 |
val major::prems = goalw Set.thy [Union_def] |
|
555 |
"[| A : Union(C); !!X.[| A:X; X:C |] ==> R |] ==> R"; |
|
556 |
by (rtac (major RS UN_E) 1); |
|
557 |
by (REPEAT (ares_tac prems 1)); |
|
558 |
qed "UnionE"; |
|
559 |
||
2499
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
560 |
AddIs [UnionI]; |
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
561 |
AddSEs [UnionE]; |
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
562 |
|
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
563 |
|
1548 | 564 |
section "Inter"; |
923 | 565 |
|
2499
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
566 |
goalw Set.thy [Inter_def] "(A : Inter(C)) = (ALL X:C. A:X)"; |
2891 | 567 |
by (Blast_tac 1); |
2499
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
568 |
qed "Inter_iff"; |
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
569 |
|
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
570 |
Addsimps [Inter_iff]; |
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
571 |
|
923 | 572 |
val prems = goalw Set.thy [Inter_def] |
573 |
"[| !!X. X:C ==> A:X |] ==> A : Inter(C)"; |
|
574 |
by (REPEAT (ares_tac ([INT_I] @ prems) 1)); |
|
575 |
qed "InterI"; |
|
576 |
||
577 |
(*A "destruct" rule -- every X in C contains A as an element, but |
|
578 |
A:X can hold when X:C does not! This rule is analogous to "spec". *) |
|
2499
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
579 |
goal Set.thy "!!X. [| A : Inter(C); X:C |] ==> A:X"; |
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
580 |
by (Auto_tac()); |
923 | 581 |
qed "InterD"; |
582 |
||
583 |
(*"Classical" elimination rule -- does not require proving X:C *) |
|
584 |
val major::prems = goalw Set.thy [Inter_def] |
|
2721 | 585 |
"[| A : Inter(C); X~:C ==> R; A:X ==> R |] ==> R"; |
923 | 586 |
by (rtac (major RS INT_E) 1); |
587 |
by (REPEAT (eresolve_tac prems 1)); |
|
588 |
qed "InterE"; |
|
589 |
||
2499
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
590 |
AddSIs [InterI]; |
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
591 |
AddEs [InterD, InterE]; |
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
592 |
|
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
593 |
|
2912 | 594 |
(*** Image of a set under a function ***) |
595 |
||
596 |
(*Frequently b does not have the syntactic form of f(x).*) |
|
597 |
val prems = goalw thy [image_def] "[| b=f(x); x:A |] ==> b : f``A"; |
|
598 |
by (REPEAT (resolve_tac (prems @ [CollectI,bexI,prem]) 1)); |
|
599 |
qed "image_eqI"; |
|
600 |
||
601 |
bind_thm ("imageI", refl RS image_eqI); |
|
602 |
||
603 |
(*The eta-expansion gives variable-name preservation.*) |
|
604 |
val major::prems = goalw thy [image_def] |
|
605 |
"[| b : (%x.f(x))``A; !!x.[| b=f(x); x:A |] ==> P |] ==> P"; |
|
606 |
by (rtac (major RS CollectD RS bexE) 1); |
|
607 |
by (REPEAT (ares_tac prems 1)); |
|
608 |
qed "imageE"; |
|
609 |
||
610 |
AddIs [image_eqI]; |
|
611 |
AddSEs [imageE]; |
|
612 |
||
613 |
goalw thy [o_def] "(f o g)``r = f``(g``r)"; |
|
2935 | 614 |
by (Blast_tac 1); |
2912 | 615 |
qed "image_compose"; |
616 |
||
617 |
goal thy "f``(A Un B) = f``A Un f``B"; |
|
2935 | 618 |
by (Blast_tac 1); |
2912 | 619 |
qed "image_Un"; |
620 |
||
621 |
||
622 |
(*** Range of a function -- just a translation for image! ***) |
|
623 |
||
624 |
goal thy "!!b. b=f(x) ==> b : range(f)"; |
|
625 |
by (EVERY1 [etac image_eqI, rtac UNIV_I]); |
|
626 |
bind_thm ("range_eqI", UNIV_I RSN (2,image_eqI)); |
|
627 |
||
628 |
bind_thm ("rangeI", UNIV_I RS imageI); |
|
629 |
||
630 |
val [major,minor] = goal thy |
|
631 |
"[| b : range(%x.f(x)); !!x. b=f(x) ==> P |] ==> P"; |
|
632 |
by (rtac (major RS imageE) 1); |
|
633 |
by (etac minor 1); |
|
634 |
qed "rangeE"; |
|
635 |
||
636 |
AddIs [rangeI]; |
|
637 |
AddSEs [rangeE]; |
|
638 |
||
1776
d7e77cb8ce5c
moved mem_simps and the corresponding update of !simpset from Fun.ML to Set.ML,
oheimb
parents:
1762
diff
changeset
|
639 |
|
d7e77cb8ce5c
moved mem_simps and the corresponding update of !simpset from Fun.ML to Set.ML,
oheimb
parents:
1762
diff
changeset
|
640 |
(*** Set reasoning tools ***) |
d7e77cb8ce5c
moved mem_simps and the corresponding update of !simpset from Fun.ML to Set.ML,
oheimb
parents:
1762
diff
changeset
|
641 |
|
d7e77cb8ce5c
moved mem_simps and the corresponding update of !simpset from Fun.ML to Set.ML,
oheimb
parents:
1762
diff
changeset
|
642 |
|
2499
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
643 |
(*Each of these has ALREADY been added to !simpset above.*) |
2024
909153d8318f
Rationalized the rewriting of membership for {} and insert
paulson
parents:
1985
diff
changeset
|
644 |
val mem_simps = [insert_iff, empty_iff, Un_iff, Int_iff, Compl_iff, Diff_iff, |
2499
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
645 |
mem_Collect_eq, |
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
646 |
UN_iff, UN1_iff, Union_iff, |
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
647 |
INT_iff, INT1_iff, Inter_iff]; |
1776
d7e77cb8ce5c
moved mem_simps and the corresponding update of !simpset from Fun.ML to Set.ML,
oheimb
parents:
1762
diff
changeset
|
648 |
|
1937 | 649 |
(*Not for Addsimps -- it can cause goals to blow up!*) |
650 |
goal Set.thy "(a : (if Q then x else y)) = ((Q --> a:x) & (~Q --> a : y))"; |
|
651 |
by (simp_tac (!simpset setloop split_tac [expand_if]) 1); |
|
652 |
qed "mem_if"; |
|
653 |
||
1776
d7e77cb8ce5c
moved mem_simps and the corresponding update of !simpset from Fun.ML to Set.ML,
oheimb
parents:
1762
diff
changeset
|
654 |
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
|
655 |
|
2499
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
656 |
simpset := !simpset addcongs [ball_cong,bex_cong] |
1776
d7e77cb8ce5c
moved mem_simps and the corresponding update of !simpset from Fun.ML to Set.ML,
oheimb
parents:
1762
diff
changeset
|
657 |
setmksimps (mksimps mksimps_pairs); |
3222
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents:
2935
diff
changeset
|
658 |
|
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents:
2935
diff
changeset
|
659 |
Addsimps[subset_UNIV, empty_subsetI, subset_refl]; |
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents:
2935
diff
changeset
|
660 |
|
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents:
2935
diff
changeset
|
661 |
|
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents:
2935
diff
changeset
|
662 |
(*** < ***) |
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents:
2935
diff
changeset
|
663 |
|
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents:
2935
diff
changeset
|
664 |
goalw Set.thy [psubset_def] "!!A::'a set. [| A <= B; A ~= B |] ==> A<B"; |
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents:
2935
diff
changeset
|
665 |
by (Blast_tac 1); |
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents:
2935
diff
changeset
|
666 |
qed "psubsetI"; |
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents:
2935
diff
changeset
|
667 |
|
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents:
2935
diff
changeset
|
668 |
goalw Set.thy [psubset_def] |
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents:
2935
diff
changeset
|
669 |
"!!x. A < insert x B ==> (x ~: A) & A<=B | x:A & A-{x}<B"; |
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents:
2935
diff
changeset
|
670 |
by (Auto_tac()); |
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents:
2935
diff
changeset
|
671 |
qed "psubset_insertD"; |