src/HOL/Set.ML
 author paulson Thu Apr 04 11:41:35 1996 +0200 (1996-04-04) changeset 1640 581165679095 parent 1618 372880456b5b child 1760 6f41a494f3b1 permissions -rw-r--r--
Added more _iff rewrites for Compl, Un, Int
 clasohm@1465 ` 1` ```(* Title: HOL/set ``` clasohm@923 ` 2` ``` ID: \$Id\$ ``` clasohm@1465 ` 3` ``` Author: Lawrence C Paulson, Cambridge University Computer Laboratory ``` clasohm@923 ` 4` ``` Copyright 1991 University of Cambridge ``` clasohm@923 ` 5` clasohm@923 ` 6` ```For set.thy. Set theory for higher-order logic. A set is simply a predicate. ``` clasohm@923 ` 7` ```*) ``` clasohm@923 ` 8` clasohm@923 ` 9` ```open Set; ``` clasohm@923 ` 10` nipkow@1548 ` 11` ```section "Relating predicates and sets"; ``` nipkow@1548 ` 12` nipkow@1548 ` 13` ```val [prem] = goal Set.thy "P(a) ==> a : {x.P(x)}"; ``` clasohm@923 ` 14` ```by (rtac (mem_Collect_eq RS ssubst) 1); ``` clasohm@923 ` 15` ```by (rtac prem 1); ``` clasohm@923 ` 16` ```qed "CollectI"; ``` clasohm@923 ` 17` clasohm@923 ` 18` ```val prems = goal Set.thy "[| a : {x.P(x)} |] ==> P(a)"; ``` clasohm@923 ` 19` ```by (resolve_tac (prems RL [mem_Collect_eq RS subst]) 1); ``` clasohm@923 ` 20` ```qed "CollectD"; ``` clasohm@923 ` 21` clasohm@923 ` 22` ```val [prem] = goal Set.thy "[| !!x. (x:A) = (x:B) |] ==> A = B"; ``` clasohm@923 ` 23` ```by (rtac (prem RS ext RS arg_cong RS box_equals) 1); ``` clasohm@923 ` 24` ```by (rtac Collect_mem_eq 1); ``` clasohm@923 ` 25` ```by (rtac Collect_mem_eq 1); ``` clasohm@923 ` 26` ```qed "set_ext"; ``` clasohm@923 ` 27` clasohm@923 ` 28` ```val [prem] = goal Set.thy "[| !!x. P(x)=Q(x) |] ==> {x. P(x)} = {x. Q(x)}"; ``` clasohm@923 ` 29` ```by (rtac (prem RS ext RS arg_cong) 1); ``` clasohm@923 ` 30` ```qed "Collect_cong"; ``` clasohm@923 ` 31` clasohm@923 ` 32` ```val CollectE = make_elim CollectD; ``` clasohm@923 ` 33` nipkow@1548 ` 34` ```section "Bounded quantifiers"; ``` clasohm@923 ` 35` clasohm@923 ` 36` ```val prems = goalw Set.thy [Ball_def] ``` clasohm@923 ` 37` ``` "[| !!x. x:A ==> P(x) |] ==> ! x:A. P(x)"; ``` clasohm@923 ` 38` ```by (REPEAT (ares_tac (prems @ [allI,impI]) 1)); ``` clasohm@923 ` 39` ```qed "ballI"; ``` clasohm@923 ` 40` clasohm@923 ` 41` ```val [major,minor] = goalw Set.thy [Ball_def] ``` clasohm@923 ` 42` ``` "[| ! x:A. P(x); x:A |] ==> P(x)"; ``` clasohm@923 ` 43` ```by (rtac (minor RS (major RS spec RS mp)) 1); ``` clasohm@923 ` 44` ```qed "bspec"; ``` clasohm@923 ` 45` clasohm@923 ` 46` ```val major::prems = goalw Set.thy [Ball_def] ``` clasohm@923 ` 47` ``` "[| ! x:A. P(x); P(x) ==> Q; x~:A ==> Q |] ==> Q"; ``` clasohm@923 ` 48` ```by (rtac (major RS spec RS impCE) 1); ``` clasohm@923 ` 49` ```by (REPEAT (eresolve_tac prems 1)); ``` clasohm@923 ` 50` ```qed "ballE"; ``` clasohm@923 ` 51` clasohm@923 ` 52` ```(*Takes assumptions ! x:A.P(x) and a:A; creates assumption P(a)*) ``` clasohm@923 ` 53` ```fun ball_tac i = etac ballE i THEN contr_tac (i+1); ``` clasohm@923 ` 54` clasohm@923 ` 55` ```val prems = goalw Set.thy [Bex_def] ``` clasohm@923 ` 56` ``` "[| P(x); x:A |] ==> ? x:A. P(x)"; ``` clasohm@923 ` 57` ```by (REPEAT (ares_tac (prems @ [exI,conjI]) 1)); ``` clasohm@923 ` 58` ```qed "bexI"; ``` clasohm@923 ` 59` clasohm@923 ` 60` ```qed_goal "bexCI" Set.thy ``` clasohm@923 ` 61` ``` "[| ! x:A. ~P(x) ==> P(a); a:A |] ==> ? x:A.P(x)" ``` clasohm@923 ` 62` ``` (fn prems=> ``` clasohm@923 ` 63` ``` [ (rtac classical 1), ``` clasohm@923 ` 64` ``` (REPEAT (ares_tac (prems@[bexI,ballI,notI,notE]) 1)) ]); ``` clasohm@923 ` 65` clasohm@923 ` 66` ```val major::prems = goalw Set.thy [Bex_def] ``` clasohm@923 ` 67` ``` "[| ? x:A. P(x); !!x. [| x:A; P(x) |] ==> Q |] ==> Q"; ``` clasohm@923 ` 68` ```by (rtac (major RS exE) 1); ``` clasohm@923 ` 69` ```by (REPEAT (eresolve_tac (prems @ [asm_rl,conjE]) 1)); ``` clasohm@923 ` 70` ```qed "bexE"; ``` clasohm@923 ` 71` clasohm@923 ` 72` ```(*Trival rewrite rule; (! x:A.P)=P holds only if A is nonempty!*) ``` paulson@1618 ` 73` ```goal Set.thy "(! x:A. True) = True"; ``` clasohm@923 ` 74` ```by (REPEAT (ares_tac [TrueI,ballI,iffI] 1)); ``` clasohm@923 ` 75` ```qed "ball_rew"; ``` paulson@1618 ` 76` ```Addsimps [ball_rew]; ``` clasohm@923 ` 77` clasohm@923 ` 78` ```(** Congruence rules **) ``` clasohm@923 ` 79` clasohm@923 ` 80` ```val prems = goal Set.thy ``` clasohm@923 ` 81` ``` "[| A=B; !!x. x:B ==> P(x) = Q(x) |] ==> \ ``` clasohm@923 ` 82` ```\ (! x:A. P(x)) = (! x:B. Q(x))"; ``` clasohm@923 ` 83` ```by (resolve_tac (prems RL [ssubst]) 1); ``` clasohm@923 ` 84` ```by (REPEAT (ares_tac [ballI,iffI] 1 ``` clasohm@923 ` 85` ``` ORELSE eresolve_tac ([make_elim bspec, mp] @ (prems RL [iffE])) 1)); ``` clasohm@923 ` 86` ```qed "ball_cong"; ``` clasohm@923 ` 87` clasohm@923 ` 88` ```val prems = goal Set.thy ``` clasohm@923 ` 89` ``` "[| A=B; !!x. x:B ==> P(x) = Q(x) |] ==> \ ``` clasohm@923 ` 90` ```\ (? x:A. P(x)) = (? x:B. Q(x))"; ``` clasohm@923 ` 91` ```by (resolve_tac (prems RL [ssubst]) 1); ``` clasohm@923 ` 92` ```by (REPEAT (etac bexE 1 ``` clasohm@923 ` 93` ``` ORELSE ares_tac ([bexI,iffI] @ (prems RL [iffD1,iffD2])) 1)); ``` clasohm@923 ` 94` ```qed "bex_cong"; ``` clasohm@923 ` 95` nipkow@1548 ` 96` ```section "Subsets"; ``` clasohm@923 ` 97` clasohm@923 ` 98` ```val prems = goalw Set.thy [subset_def] "(!!x.x:A ==> x:B) ==> A <= B"; ``` clasohm@923 ` 99` ```by (REPEAT (ares_tac (prems @ [ballI]) 1)); ``` clasohm@923 ` 100` ```qed "subsetI"; ``` clasohm@923 ` 101` clasohm@923 ` 102` ```(*Rule in Modus Ponens style*) ``` clasohm@923 ` 103` ```val major::prems = goalw Set.thy [subset_def] "[| A <= B; c:A |] ==> c:B"; ``` clasohm@923 ` 104` ```by (rtac (major RS bspec) 1); ``` clasohm@923 ` 105` ```by (resolve_tac prems 1); ``` clasohm@923 ` 106` ```qed "subsetD"; ``` clasohm@923 ` 107` clasohm@923 ` 108` ```(*The same, with reversed premises for use with etac -- cf rev_mp*) ``` clasohm@923 ` 109` ```qed_goal "rev_subsetD" Set.thy "[| c:A; A <= B |] ==> c:B" ``` clasohm@923 ` 110` ``` (fn prems=> [ (REPEAT (resolve_tac (prems@[subsetD]) 1)) ]); ``` clasohm@923 ` 111` clasohm@923 ` 112` ```(*Classical elimination rule*) ``` clasohm@923 ` 113` ```val major::prems = goalw Set.thy [subset_def] ``` clasohm@923 ` 114` ``` "[| A <= B; c~:A ==> P; c:B ==> P |] ==> P"; ``` clasohm@923 ` 115` ```by (rtac (major RS ballE) 1); ``` clasohm@923 ` 116` ```by (REPEAT (eresolve_tac prems 1)); ``` clasohm@923 ` 117` ```qed "subsetCE"; ``` clasohm@923 ` 118` clasohm@923 ` 119` ```(*Takes assumptions A<=B; c:A and creates the assumption c:B *) ``` clasohm@923 ` 120` ```fun set_mp_tac i = etac subsetCE i THEN mp_tac i; ``` clasohm@923 ` 121` clasohm@923 ` 122` ```qed_goal "subset_refl" Set.thy "A <= (A::'a set)" ``` clasohm@923 ` 123` ``` (fn _=> [ (REPEAT (ares_tac [subsetI] 1)) ]); ``` clasohm@923 ` 124` clasohm@923 ` 125` ```val prems = goal Set.thy "[| A<=B; B<=C |] ==> A<=(C::'a set)"; ``` clasohm@923 ` 126` ```by (cut_facts_tac prems 1); ``` clasohm@923 ` 127` ```by (REPEAT (ares_tac [subsetI] 1 ORELSE set_mp_tac 1)); ``` clasohm@923 ` 128` ```qed "subset_trans"; ``` clasohm@923 ` 129` clasohm@923 ` 130` nipkow@1548 ` 131` ```section "Equality"; ``` clasohm@923 ` 132` clasohm@923 ` 133` ```(*Anti-symmetry of the subset relation*) ``` clasohm@923 ` 134` ```val prems = goal Set.thy "[| A <= B; B <= A |] ==> A = (B::'a set)"; ``` clasohm@923 ` 135` ```by (rtac (iffI RS set_ext) 1); ``` clasohm@923 ` 136` ```by (REPEAT (ares_tac (prems RL [subsetD]) 1)); ``` clasohm@923 ` 137` ```qed "subset_antisym"; ``` clasohm@923 ` 138` ```val equalityI = subset_antisym; ``` clasohm@923 ` 139` clasohm@923 ` 140` ```(* Equality rules from ZF set theory -- are they appropriate here? *) ``` clasohm@923 ` 141` ```val prems = goal Set.thy "A = B ==> A<=(B::'a set)"; ``` clasohm@923 ` 142` ```by (resolve_tac (prems RL [subst]) 1); ``` clasohm@923 ` 143` ```by (rtac subset_refl 1); ``` clasohm@923 ` 144` ```qed "equalityD1"; ``` clasohm@923 ` 145` clasohm@923 ` 146` ```val prems = goal Set.thy "A = B ==> B<=(A::'a set)"; ``` clasohm@923 ` 147` ```by (resolve_tac (prems RL [subst]) 1); ``` clasohm@923 ` 148` ```by (rtac subset_refl 1); ``` clasohm@923 ` 149` ```qed "equalityD2"; ``` clasohm@923 ` 150` clasohm@923 ` 151` ```val prems = goal Set.thy ``` clasohm@923 ` 152` ``` "[| A = B; [| A<=B; B<=(A::'a set) |] ==> P |] ==> P"; ``` clasohm@923 ` 153` ```by (resolve_tac prems 1); ``` clasohm@923 ` 154` ```by (REPEAT (resolve_tac (prems RL [equalityD1,equalityD2]) 1)); ``` clasohm@923 ` 155` ```qed "equalityE"; ``` clasohm@923 ` 156` clasohm@923 ` 157` ```val major::prems = goal Set.thy ``` clasohm@923 ` 158` ``` "[| A = B; [| c:A; c:B |] ==> P; [| c~:A; c~:B |] ==> P |] ==> P"; ``` clasohm@923 ` 159` ```by (rtac (major RS equalityE) 1); ``` clasohm@923 ` 160` ```by (REPEAT (contr_tac 1 ORELSE eresolve_tac ([asm_rl,subsetCE]@prems) 1)); ``` clasohm@923 ` 161` ```qed "equalityCE"; ``` clasohm@923 ` 162` clasohm@923 ` 163` ```(*Lemma for creating induction formulae -- for "pattern matching" on p ``` clasohm@923 ` 164` ``` To make the induction hypotheses usable, apply "spec" or "bspec" to ``` clasohm@923 ` 165` ``` put universal quantifiers over the free variables in p. *) ``` clasohm@923 ` 166` ```val prems = goal Set.thy ``` clasohm@923 ` 167` ``` "[| p:A; !!z. z:A ==> p=z --> R |] ==> R"; ``` clasohm@923 ` 168` ```by (rtac mp 1); ``` clasohm@923 ` 169` ```by (REPEAT (resolve_tac (refl::prems) 1)); ``` clasohm@923 ` 170` ```qed "setup_induction"; ``` clasohm@923 ` 171` clasohm@923 ` 172` nipkow@1548 ` 173` ```section "Set complement -- Compl"; ``` clasohm@923 ` 174` clasohm@923 ` 175` ```val prems = goalw Set.thy [Compl_def] ``` clasohm@923 ` 176` ``` "[| c:A ==> False |] ==> c : Compl(A)"; ``` clasohm@923 ` 177` ```by (REPEAT (ares_tac (prems @ [CollectI,notI]) 1)); ``` clasohm@923 ` 178` ```qed "ComplI"; ``` clasohm@923 ` 179` clasohm@923 ` 180` ```(*This form, with negated conclusion, works well with the Classical prover. ``` clasohm@923 ` 181` ``` Negated assumptions behave like formulae on the right side of the notional ``` clasohm@923 ` 182` ``` turnstile...*) ``` clasohm@923 ` 183` ```val major::prems = goalw Set.thy [Compl_def] ``` clasohm@923 ` 184` ``` "[| c : Compl(A) |] ==> c~:A"; ``` clasohm@923 ` 185` ```by (rtac (major RS CollectD) 1); ``` clasohm@923 ` 186` ```qed "ComplD"; ``` clasohm@923 ` 187` clasohm@923 ` 188` ```val ComplE = make_elim ComplD; ``` clasohm@923 ` 189` paulson@1640 ` 190` ```qed_goal "Compl_iff" Set.thy "(c : Compl(A)) = (c~:A)" ``` paulson@1640 ` 191` ``` (fn _ => [ (fast_tac (HOL_cs addSIs [ComplI] addSEs [ComplE]) 1) ]); ``` paulson@1640 ` 192` clasohm@923 ` 193` nipkow@1548 ` 194` ```section "Binary union -- Un"; ``` clasohm@923 ` 195` clasohm@923 ` 196` ```val prems = goalw Set.thy [Un_def] "c:A ==> c : A Un B"; ``` clasohm@923 ` 197` ```by (REPEAT (resolve_tac (prems @ [CollectI,disjI1]) 1)); ``` clasohm@923 ` 198` ```qed "UnI1"; ``` clasohm@923 ` 199` clasohm@923 ` 200` ```val prems = goalw Set.thy [Un_def] "c:B ==> c : A Un B"; ``` clasohm@923 ` 201` ```by (REPEAT (resolve_tac (prems @ [CollectI,disjI2]) 1)); ``` clasohm@923 ` 202` ```qed "UnI2"; ``` clasohm@923 ` 203` clasohm@923 ` 204` ```(*Classical introduction rule: no commitment to A vs B*) ``` clasohm@923 ` 205` ```qed_goal "UnCI" Set.thy "(c~:B ==> c:A) ==> c : A Un B" ``` clasohm@923 ` 206` ``` (fn prems=> ``` clasohm@923 ` 207` ``` [ (rtac classical 1), ``` clasohm@923 ` 208` ``` (REPEAT (ares_tac (prems@[UnI1,notI]) 1)), ``` clasohm@923 ` 209` ``` (REPEAT (ares_tac (prems@[UnI2,notE]) 1)) ]); ``` clasohm@923 ` 210` clasohm@923 ` 211` ```val major::prems = goalw Set.thy [Un_def] ``` clasohm@923 ` 212` ``` "[| c : A Un B; c:A ==> P; c:B ==> P |] ==> P"; ``` clasohm@923 ` 213` ```by (rtac (major RS CollectD RS disjE) 1); ``` clasohm@923 ` 214` ```by (REPEAT (eresolve_tac prems 1)); ``` clasohm@923 ` 215` ```qed "UnE"; ``` clasohm@923 ` 216` paulson@1640 ` 217` ```qed_goal "Un_iff" Set.thy "(c : A Un B) = (c:A | c:B)" ``` paulson@1640 ` 218` ``` (fn _ => [ (fast_tac (HOL_cs addSIs [UnCI] addSEs [UnE]) 1) ]); ``` paulson@1640 ` 219` clasohm@923 ` 220` nipkow@1548 ` 221` ```section "Binary intersection -- Int"; ``` clasohm@923 ` 222` clasohm@923 ` 223` ```val prems = goalw Set.thy [Int_def] ``` clasohm@923 ` 224` ``` "[| c:A; c:B |] ==> c : A Int B"; ``` clasohm@923 ` 225` ```by (REPEAT (resolve_tac (prems @ [CollectI,conjI]) 1)); ``` clasohm@923 ` 226` ```qed "IntI"; ``` clasohm@923 ` 227` clasohm@923 ` 228` ```val [major] = goalw Set.thy [Int_def] "c : A Int B ==> c:A"; ``` clasohm@923 ` 229` ```by (rtac (major RS CollectD RS conjunct1) 1); ``` clasohm@923 ` 230` ```qed "IntD1"; ``` clasohm@923 ` 231` clasohm@923 ` 232` ```val [major] = goalw Set.thy [Int_def] "c : A Int B ==> c:B"; ``` clasohm@923 ` 233` ```by (rtac (major RS CollectD RS conjunct2) 1); ``` clasohm@923 ` 234` ```qed "IntD2"; ``` clasohm@923 ` 235` clasohm@923 ` 236` ```val [major,minor] = goal Set.thy ``` clasohm@923 ` 237` ``` "[| c : A Int B; [| c:A; c:B |] ==> P |] ==> P"; ``` clasohm@923 ` 238` ```by (rtac minor 1); ``` clasohm@923 ` 239` ```by (rtac (major RS IntD1) 1); ``` clasohm@923 ` 240` ```by (rtac (major RS IntD2) 1); ``` clasohm@923 ` 241` ```qed "IntE"; ``` clasohm@923 ` 242` paulson@1640 ` 243` ```qed_goal "Int_iff" Set.thy "(c : A Int B) = (c:A & c:B)" ``` paulson@1640 ` 244` ``` (fn _ => [ (fast_tac (HOL_cs addSIs [IntI] addSEs [IntE]) 1) ]); ``` paulson@1640 ` 245` clasohm@923 ` 246` nipkow@1548 ` 247` ```section "Set difference"; ``` clasohm@923 ` 248` clasohm@923 ` 249` ```qed_goalw "DiffI" Set.thy [set_diff_def] ``` clasohm@923 ` 250` ``` "[| c : A; c ~: B |] ==> c : A - B" ``` clasohm@923 ` 251` ``` (fn prems=> [ (REPEAT (resolve_tac (prems @ [CollectI,conjI]) 1)) ]); ``` clasohm@923 ` 252` clasohm@923 ` 253` ```qed_goalw "DiffD1" Set.thy [set_diff_def] ``` clasohm@923 ` 254` ``` "c : A - B ==> c : A" ``` clasohm@923 ` 255` ``` (fn [major]=> [ (rtac (major RS CollectD RS conjunct1) 1) ]); ``` clasohm@923 ` 256` clasohm@923 ` 257` ```qed_goalw "DiffD2" Set.thy [set_diff_def] ``` clasohm@923 ` 258` ``` "[| c : A - B; c : B |] ==> P" ``` clasohm@923 ` 259` ``` (fn [major,minor]=> ``` clasohm@923 ` 260` ``` [rtac (minor RS (major RS CollectD RS conjunct2 RS notE)) 1]); ``` clasohm@923 ` 261` clasohm@923 ` 262` ```qed_goal "DiffE" Set.thy ``` clasohm@923 ` 263` ``` "[| c : A - B; [| c:A; c~:B |] ==> P |] ==> P" ``` clasohm@923 ` 264` ``` (fn prems=> ``` clasohm@923 ` 265` ``` [ (resolve_tac prems 1), ``` clasohm@923 ` 266` ``` (REPEAT (ares_tac (prems RL [DiffD1, DiffD2 RS notI]) 1)) ]); ``` clasohm@923 ` 267` clasohm@923 ` 268` ```qed_goal "Diff_iff" Set.thy "(c : A-B) = (c:A & c~:B)" ``` clasohm@923 ` 269` ``` (fn _ => [ (fast_tac (HOL_cs addSIs [DiffI] addSEs [DiffE]) 1) ]); ``` clasohm@923 ` 270` nipkow@1548 ` 271` ```section "The empty set -- {}"; ``` clasohm@923 ` 272` clasohm@923 ` 273` ```qed_goalw "emptyE" Set.thy [empty_def] "a:{} ==> P" ``` clasohm@923 ` 274` ``` (fn [prem] => [rtac (prem RS CollectD RS FalseE) 1]); ``` clasohm@923 ` 275` clasohm@923 ` 276` ```qed_goal "empty_subsetI" Set.thy "{} <= A" ``` clasohm@923 ` 277` ``` (fn _ => [ (REPEAT (ares_tac [equalityI,subsetI,emptyE] 1)) ]); ``` clasohm@923 ` 278` clasohm@923 ` 279` ```qed_goal "equals0I" Set.thy "[| !!y. y:A ==> False |] ==> A={}" ``` clasohm@923 ` 280` ``` (fn prems=> ``` clasohm@923 ` 281` ``` [ (REPEAT (ares_tac (prems@[empty_subsetI,subsetI,equalityI]) 1 ``` clasohm@923 ` 282` ``` ORELSE eresolve_tac (prems RL [FalseE]) 1)) ]); ``` clasohm@923 ` 283` clasohm@923 ` 284` ```qed_goal "equals0D" Set.thy "[| A={}; a:A |] ==> P" ``` clasohm@923 ` 285` ``` (fn [major,minor]=> ``` clasohm@923 ` 286` ``` [ (rtac (minor RS (major RS equalityD1 RS subsetD RS emptyE)) 1) ]); ``` clasohm@923 ` 287` paulson@1640 ` 288` ```qed_goal "empty_iff" Set.thy "(c : {}) = False" ``` paulson@1640 ` 289` ``` (fn _ => [ (fast_tac (HOL_cs addSEs [emptyE]) 1) ]); ``` paulson@1640 ` 290` clasohm@923 ` 291` nipkow@1548 ` 292` ```section "Augmenting a set -- insert"; ``` clasohm@923 ` 293` clasohm@923 ` 294` ```qed_goalw "insertI1" Set.thy [insert_def] "a : insert a B" ``` clasohm@923 ` 295` ``` (fn _ => [rtac (CollectI RS UnI1) 1, rtac refl 1]); ``` clasohm@923 ` 296` clasohm@923 ` 297` ```qed_goalw "insertI2" Set.thy [insert_def] "a : B ==> a : insert b B" ``` clasohm@923 ` 298` ``` (fn [prem]=> [ (rtac (prem RS UnI2) 1) ]); ``` clasohm@923 ` 299` clasohm@923 ` 300` ```qed_goalw "insertE" Set.thy [insert_def] ``` clasohm@923 ` 301` ``` "[| a : insert b A; a=b ==> P; a:A ==> P |] ==> P" ``` clasohm@923 ` 302` ``` (fn major::prems=> ``` clasohm@923 ` 303` ``` [ (rtac (major RS UnE) 1), ``` clasohm@923 ` 304` ``` (REPEAT (eresolve_tac (prems @ [CollectE]) 1)) ]); ``` clasohm@923 ` 305` clasohm@923 ` 306` ```qed_goal "insert_iff" Set.thy "a : insert b A = (a=b | a:A)" ``` clasohm@923 ` 307` ``` (fn _ => [fast_tac (HOL_cs addIs [insertI1,insertI2] addSEs [insertE]) 1]); ``` clasohm@923 ` 308` clasohm@923 ` 309` ```(*Classical introduction rule*) ``` clasohm@923 ` 310` ```qed_goal "insertCI" Set.thy "(a~:B ==> a=b) ==> a: insert b B" ``` clasohm@923 ` 311` ``` (fn [prem]=> ``` clasohm@923 ` 312` ``` [ (rtac (disjCI RS (insert_iff RS iffD2)) 1), ``` clasohm@923 ` 313` ``` (etac prem 1) ]); ``` clasohm@923 ` 314` nipkow@1548 ` 315` ```section "Singletons, using insert"; ``` clasohm@923 ` 316` clasohm@923 ` 317` ```qed_goal "singletonI" Set.thy "a : {a}" ``` clasohm@923 ` 318` ``` (fn _=> [ (rtac insertI1 1) ]); ``` clasohm@923 ` 319` clasohm@923 ` 320` ```qed_goal "singletonE" Set.thy "[| a: {b}; a=b ==> P |] ==> P" ``` clasohm@923 ` 321` ``` (fn major::prems=> ``` clasohm@923 ` 322` ``` [ (rtac (major RS insertE) 1), ``` clasohm@923 ` 323` ``` (REPEAT (eresolve_tac (prems @ [emptyE]) 1)) ]); ``` clasohm@923 ` 324` clasohm@923 ` 325` ```goalw Set.thy [insert_def] "!!a. b : {a} ==> b=a"; ``` paulson@1552 ` 326` ```by (fast_tac (HOL_cs addSEs [emptyE,CollectE,UnE]) 1); ``` clasohm@923 ` 327` ```qed "singletonD"; ``` clasohm@923 ` 328` clasohm@923 ` 329` ```val singletonE = make_elim singletonD; ``` clasohm@923 ` 330` clasohm@923 ` 331` ```val [major] = goal Set.thy "{a}={b} ==> a=b"; ``` clasohm@923 ` 332` ```by (rtac (major RS equalityD1 RS subsetD RS singletonD) 1); ``` clasohm@923 ` 333` ```by (rtac singletonI 1); ``` clasohm@923 ` 334` ```qed "singleton_inject"; ``` clasohm@923 ` 335` nipkow@1531 ` 336` nipkow@1548 ` 337` ```section "The universal set -- UNIV"; ``` nipkow@1531 ` 338` nipkow@1531 ` 339` ```qed_goal "subset_UNIV" Set.thy "A <= UNIV" ``` nipkow@1531 ` 340` ``` (fn _ => [rtac subsetI 1, rtac ComplI 1, etac emptyE 1]); ``` nipkow@1531 ` 341` nipkow@1531 ` 342` nipkow@1548 ` 343` ```section "Unions of families -- UNION x:A. B(x) is Union(B``A)"; ``` clasohm@923 ` 344` clasohm@923 ` 345` ```(*The order of the premises presupposes that A is rigid; b may be flexible*) ``` clasohm@923 ` 346` ```val prems = goalw Set.thy [UNION_def] ``` clasohm@923 ` 347` ``` "[| a:A; b: B(a) |] ==> b: (UN x:A. B(x))"; ``` clasohm@923 ` 348` ```by (REPEAT (resolve_tac (prems @ [bexI,CollectI]) 1)); ``` clasohm@923 ` 349` ```qed "UN_I"; ``` clasohm@923 ` 350` clasohm@923 ` 351` ```val major::prems = goalw Set.thy [UNION_def] ``` clasohm@923 ` 352` ``` "[| b : (UN x:A. B(x)); !!x.[| x:A; b: B(x) |] ==> R |] ==> R"; ``` clasohm@923 ` 353` ```by (rtac (major RS CollectD RS bexE) 1); ``` clasohm@923 ` 354` ```by (REPEAT (ares_tac prems 1)); ``` clasohm@923 ` 355` ```qed "UN_E"; ``` clasohm@923 ` 356` clasohm@923 ` 357` ```val prems = goal Set.thy ``` clasohm@923 ` 358` ``` "[| A=B; !!x. x:B ==> C(x) = D(x) |] ==> \ ``` clasohm@923 ` 359` ```\ (UN x:A. C(x)) = (UN x:B. D(x))"; ``` clasohm@923 ` 360` ```by (REPEAT (etac UN_E 1 ``` clasohm@923 ` 361` ``` ORELSE ares_tac ([UN_I,equalityI,subsetI] @ ``` clasohm@1465 ` 362` ``` (prems RL [equalityD1,equalityD2] RL [subsetD])) 1)); ``` clasohm@923 ` 363` ```qed "UN_cong"; ``` clasohm@923 ` 364` clasohm@923 ` 365` nipkow@1548 ` 366` ```section "Intersections of families -- INTER x:A. B(x) is Inter(B``A)"; ``` clasohm@923 ` 367` clasohm@923 ` 368` ```val prems = goalw Set.thy [INTER_def] ``` clasohm@923 ` 369` ``` "(!!x. x:A ==> b: B(x)) ==> b : (INT x:A. B(x))"; ``` clasohm@923 ` 370` ```by (REPEAT (ares_tac ([CollectI,ballI] @ prems) 1)); ``` clasohm@923 ` 371` ```qed "INT_I"; ``` clasohm@923 ` 372` clasohm@923 ` 373` ```val major::prems = goalw Set.thy [INTER_def] ``` clasohm@923 ` 374` ``` "[| b : (INT x:A. B(x)); a:A |] ==> b: B(a)"; ``` clasohm@923 ` 375` ```by (rtac (major RS CollectD RS bspec) 1); ``` clasohm@923 ` 376` ```by (resolve_tac prems 1); ``` clasohm@923 ` 377` ```qed "INT_D"; ``` clasohm@923 ` 378` clasohm@923 ` 379` ```(*"Classical" elimination -- by the Excluded Middle on a:A *) ``` clasohm@923 ` 380` ```val major::prems = goalw Set.thy [INTER_def] ``` clasohm@923 ` 381` ``` "[| b : (INT x:A. B(x)); b: B(a) ==> R; a~:A ==> R |] ==> R"; ``` clasohm@923 ` 382` ```by (rtac (major RS CollectD RS ballE) 1); ``` clasohm@923 ` 383` ```by (REPEAT (eresolve_tac prems 1)); ``` clasohm@923 ` 384` ```qed "INT_E"; ``` clasohm@923 ` 385` clasohm@923 ` 386` ```val prems = goal Set.thy ``` clasohm@923 ` 387` ``` "[| A=B; !!x. x:B ==> C(x) = D(x) |] ==> \ ``` clasohm@923 ` 388` ```\ (INT x:A. C(x)) = (INT x:B. D(x))"; ``` clasohm@923 ` 389` ```by (REPEAT_FIRST (resolve_tac [INT_I,equalityI,subsetI])); ``` clasohm@923 ` 390` ```by (REPEAT (dtac INT_D 1 ``` clasohm@923 ` 391` ``` ORELSE ares_tac (prems RL [equalityD1,equalityD2] RL [subsetD]) 1)); ``` clasohm@923 ` 392` ```qed "INT_cong"; ``` clasohm@923 ` 393` clasohm@923 ` 394` nipkow@1548 ` 395` ```section "Unions over a type; UNION1(B) = Union(range(B))"; ``` clasohm@923 ` 396` clasohm@923 ` 397` ```(*The order of the premises presupposes that A is rigid; b may be flexible*) ``` clasohm@923 ` 398` ```val prems = goalw Set.thy [UNION1_def] ``` clasohm@923 ` 399` ``` "b: B(x) ==> b: (UN x. B(x))"; ``` clasohm@923 ` 400` ```by (REPEAT (resolve_tac (prems @ [TrueI, CollectI RS UN_I]) 1)); ``` clasohm@923 ` 401` ```qed "UN1_I"; ``` clasohm@923 ` 402` clasohm@923 ` 403` ```val major::prems = goalw Set.thy [UNION1_def] ``` clasohm@923 ` 404` ``` "[| b : (UN x. B(x)); !!x. b: B(x) ==> R |] ==> R"; ``` clasohm@923 ` 405` ```by (rtac (major RS UN_E) 1); ``` clasohm@923 ` 406` ```by (REPEAT (ares_tac prems 1)); ``` clasohm@923 ` 407` ```qed "UN1_E"; ``` clasohm@923 ` 408` clasohm@923 ` 409` nipkow@1548 ` 410` ```section "Intersections over a type; INTER1(B) = Inter(range(B))"; ``` clasohm@923 ` 411` clasohm@923 ` 412` ```val prems = goalw Set.thy [INTER1_def] ``` clasohm@923 ` 413` ``` "(!!x. b: B(x)) ==> b : (INT x. B(x))"; ``` clasohm@923 ` 414` ```by (REPEAT (ares_tac (INT_I::prems) 1)); ``` clasohm@923 ` 415` ```qed "INT1_I"; ``` clasohm@923 ` 416` clasohm@923 ` 417` ```val [major] = goalw Set.thy [INTER1_def] ``` clasohm@923 ` 418` ``` "b : (INT x. B(x)) ==> b: B(a)"; ``` clasohm@923 ` 419` ```by (rtac (TrueI RS (CollectI RS (major RS INT_D))) 1); ``` clasohm@923 ` 420` ```qed "INT1_D"; ``` clasohm@923 ` 421` nipkow@1548 ` 422` ```section "Union"; ``` clasohm@923 ` 423` clasohm@923 ` 424` ```(*The order of the premises presupposes that C is rigid; A may be flexible*) ``` clasohm@923 ` 425` ```val prems = goalw Set.thy [Union_def] ``` clasohm@923 ` 426` ``` "[| X:C; A:X |] ==> A : Union(C)"; ``` clasohm@923 ` 427` ```by (REPEAT (resolve_tac (prems @ [UN_I]) 1)); ``` clasohm@923 ` 428` ```qed "UnionI"; ``` clasohm@923 ` 429` clasohm@923 ` 430` ```val major::prems = goalw Set.thy [Union_def] ``` clasohm@923 ` 431` ``` "[| A : Union(C); !!X.[| A:X; X:C |] ==> R |] ==> R"; ``` clasohm@923 ` 432` ```by (rtac (major RS UN_E) 1); ``` clasohm@923 ` 433` ```by (REPEAT (ares_tac prems 1)); ``` clasohm@923 ` 434` ```qed "UnionE"; ``` clasohm@923 ` 435` nipkow@1548 ` 436` ```section "Inter"; ``` clasohm@923 ` 437` clasohm@923 ` 438` ```val prems = goalw Set.thy [Inter_def] ``` clasohm@923 ` 439` ``` "[| !!X. X:C ==> A:X |] ==> A : Inter(C)"; ``` clasohm@923 ` 440` ```by (REPEAT (ares_tac ([INT_I] @ prems) 1)); ``` clasohm@923 ` 441` ```qed "InterI"; ``` clasohm@923 ` 442` clasohm@923 ` 443` ```(*A "destruct" rule -- every X in C contains A as an element, but ``` clasohm@923 ` 444` ``` A:X can hold when X:C does not! This rule is analogous to "spec". *) ``` clasohm@923 ` 445` ```val major::prems = goalw Set.thy [Inter_def] ``` clasohm@923 ` 446` ``` "[| A : Inter(C); X:C |] ==> A:X"; ``` clasohm@923 ` 447` ```by (rtac (major RS INT_D) 1); ``` clasohm@923 ` 448` ```by (resolve_tac prems 1); ``` clasohm@923 ` 449` ```qed "InterD"; ``` clasohm@923 ` 450` clasohm@923 ` 451` ```(*"Classical" elimination rule -- does not require proving X:C *) ``` clasohm@923 ` 452` ```val major::prems = goalw Set.thy [Inter_def] ``` clasohm@923 ` 453` ``` "[| A : Inter(C); A:X ==> R; X~:C ==> R |] ==> R"; ``` clasohm@923 ` 454` ```by (rtac (major RS INT_E) 1); ``` clasohm@923 ` 455` ```by (REPEAT (eresolve_tac prems 1)); ``` clasohm@923 ` 456` ```qed "InterE"; ``` clasohm@923 ` 457` nipkow@1548 ` 458` ```section "The Powerset operator -- Pow"; ``` clasohm@923 ` 459` clasohm@923 ` 460` ```qed_goalw "PowI" Set.thy [Pow_def] "!!A B. A <= B ==> A : Pow(B)" ``` clasohm@923 ` 461` ``` (fn _ => [ (etac CollectI 1) ]); ``` clasohm@923 ` 462` clasohm@923 ` 463` ```qed_goalw "PowD" Set.thy [Pow_def] "!!A B. A : Pow(B) ==> A<=B" ``` clasohm@923 ` 464` ``` (fn _=> [ (etac CollectD 1) ]); ``` clasohm@923 ` 465` clasohm@923 ` 466` ```val Pow_bottom = empty_subsetI RS PowI; (* {}: Pow(B) *) ``` clasohm@923 ` 467` ```val Pow_top = subset_refl RS PowI; (* A : Pow(A) *) ```