src/HOL/Set.ML
 author paulson Thu Jan 09 10:23:39 1997 +0100 (1997-01-09) changeset 2499 0bc87b063447 parent 2031 03a843f0f447 child 2608 450c9b682a92 permissions -rw-r--r--
Tidying of proofs. New theorems are enterred immediately into the
relevant clasets or simpsets.
 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` paulson@1985 ` 6` ```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` paulson@2499 ` 13` ```AddIffs [mem_Collect_eq]; ``` paulson@2499 ` 14` paulson@2499 ` 15` ```goal Set.thy "!!a. P(a) ==> a : {x.P(x)}"; ``` paulson@2499 ` 16` ```by (Asm_simp_tac 1); ``` clasohm@923 ` 17` ```qed "CollectI"; ``` clasohm@923 ` 18` paulson@2499 ` 19` ```val prems = goal Set.thy "!!a. a : {x.P(x)} ==> P(a)"; ``` paulson@2499 ` 20` ```by (Asm_full_simp_tac 1); ``` clasohm@923 ` 21` ```qed "CollectD"; ``` clasohm@923 ` 22` clasohm@923 ` 23` ```val [prem] = goal Set.thy "[| !!x. (x:A) = (x:B) |] ==> A = B"; ``` clasohm@923 ` 24` ```by (rtac (prem RS ext RS arg_cong RS box_equals) 1); ``` clasohm@923 ` 25` ```by (rtac Collect_mem_eq 1); ``` clasohm@923 ` 26` ```by (rtac Collect_mem_eq 1); ``` clasohm@923 ` 27` ```qed "set_ext"; ``` clasohm@923 ` 28` clasohm@923 ` 29` ```val [prem] = goal Set.thy "[| !!x. P(x)=Q(x) |] ==> {x. P(x)} = {x. Q(x)}"; ``` clasohm@923 ` 30` ```by (rtac (prem RS ext RS arg_cong) 1); ``` clasohm@923 ` 31` ```qed "Collect_cong"; ``` clasohm@923 ` 32` clasohm@923 ` 33` ```val CollectE = make_elim CollectD; ``` clasohm@923 ` 34` paulson@2499 ` 35` ```AddSIs [CollectI]; ``` paulson@2499 ` 36` ```AddSEs [CollectE]; ``` paulson@2499 ` 37` paulson@2499 ` 38` nipkow@1548 ` 39` ```section "Bounded quantifiers"; ``` clasohm@923 ` 40` clasohm@923 ` 41` ```val prems = goalw Set.thy [Ball_def] ``` clasohm@923 ` 42` ``` "[| !!x. x:A ==> P(x) |] ==> ! x:A. P(x)"; ``` clasohm@923 ` 43` ```by (REPEAT (ares_tac (prems @ [allI,impI]) 1)); ``` clasohm@923 ` 44` ```qed "ballI"; ``` clasohm@923 ` 45` clasohm@923 ` 46` ```val [major,minor] = goalw Set.thy [Ball_def] ``` clasohm@923 ` 47` ``` "[| ! x:A. P(x); x:A |] ==> P(x)"; ``` clasohm@923 ` 48` ```by (rtac (minor RS (major RS spec RS mp)) 1); ``` clasohm@923 ` 49` ```qed "bspec"; ``` clasohm@923 ` 50` clasohm@923 ` 51` ```val major::prems = goalw Set.thy [Ball_def] ``` clasohm@923 ` 52` ``` "[| ! x:A. P(x); P(x) ==> Q; x~:A ==> Q |] ==> Q"; ``` clasohm@923 ` 53` ```by (rtac (major RS spec RS impCE) 1); ``` clasohm@923 ` 54` ```by (REPEAT (eresolve_tac prems 1)); ``` clasohm@923 ` 55` ```qed "ballE"; ``` clasohm@923 ` 56` clasohm@923 ` 57` ```(*Takes assumptions ! x:A.P(x) and a:A; creates assumption P(a)*) ``` clasohm@923 ` 58` ```fun ball_tac i = etac ballE i THEN contr_tac (i+1); ``` clasohm@923 ` 59` paulson@2499 ` 60` ```AddSIs [ballI]; ``` paulson@2499 ` 61` ```AddEs [ballE]; ``` paulson@2499 ` 62` clasohm@923 ` 63` ```val prems = goalw Set.thy [Bex_def] ``` clasohm@923 ` 64` ``` "[| P(x); x:A |] ==> ? x:A. P(x)"; ``` clasohm@923 ` 65` ```by (REPEAT (ares_tac (prems @ [exI,conjI]) 1)); ``` clasohm@923 ` 66` ```qed "bexI"; ``` clasohm@923 ` 67` clasohm@923 ` 68` ```qed_goal "bexCI" Set.thy ``` clasohm@923 ` 69` ``` "[| ! x:A. ~P(x) ==> P(a); a:A |] ==> ? x:A.P(x)" ``` clasohm@923 ` 70` ``` (fn prems=> ``` clasohm@923 ` 71` ``` [ (rtac classical 1), ``` clasohm@923 ` 72` ``` (REPEAT (ares_tac (prems@[bexI,ballI,notI,notE]) 1)) ]); ``` clasohm@923 ` 73` clasohm@923 ` 74` ```val major::prems = goalw Set.thy [Bex_def] ``` clasohm@923 ` 75` ``` "[| ? x:A. P(x); !!x. [| x:A; P(x) |] ==> Q |] ==> Q"; ``` clasohm@923 ` 76` ```by (rtac (major RS exE) 1); ``` clasohm@923 ` 77` ```by (REPEAT (eresolve_tac (prems @ [asm_rl,conjE]) 1)); ``` clasohm@923 ` 78` ```qed "bexE"; ``` clasohm@923 ` 79` paulson@2499 ` 80` ```AddIs [bexI]; ``` paulson@2499 ` 81` ```AddSEs [bexE]; ``` paulson@2499 ` 82` clasohm@923 ` 83` ```(*Trival rewrite rule; (! x:A.P)=P holds only if A is nonempty!*) ``` paulson@1882 ` 84` ```goalw Set.thy [Ball_def] "(! x:A. True) = True"; ``` paulson@1882 ` 85` ```by (Simp_tac 1); ``` paulson@1816 ` 86` ```qed "ball_True"; ``` paulson@1816 ` 87` paulson@1882 ` 88` ```(*Dual form for existentials*) ``` paulson@1882 ` 89` ```goalw Set.thy [Bex_def] "(? x:A. False) = False"; ``` paulson@1882 ` 90` ```by (Simp_tac 1); ``` paulson@1882 ` 91` ```qed "bex_False"; ``` paulson@1882 ` 92` paulson@1882 ` 93` ```Addsimps [ball_True, bex_False]; ``` clasohm@923 ` 94` clasohm@923 ` 95` ```(** Congruence rules **) ``` clasohm@923 ` 96` clasohm@923 ` 97` ```val prems = goal Set.thy ``` clasohm@923 ` 98` ``` "[| A=B; !!x. x:B ==> P(x) = Q(x) |] ==> \ ``` clasohm@923 ` 99` ```\ (! x:A. P(x)) = (! x:B. Q(x))"; ``` clasohm@923 ` 100` ```by (resolve_tac (prems RL [ssubst]) 1); ``` clasohm@923 ` 101` ```by (REPEAT (ares_tac [ballI,iffI] 1 ``` clasohm@923 ` 102` ``` ORELSE eresolve_tac ([make_elim bspec, mp] @ (prems RL [iffE])) 1)); ``` clasohm@923 ` 103` ```qed "ball_cong"; ``` clasohm@923 ` 104` clasohm@923 ` 105` ```val prems = goal Set.thy ``` clasohm@923 ` 106` ``` "[| A=B; !!x. x:B ==> P(x) = Q(x) |] ==> \ ``` clasohm@923 ` 107` ```\ (? x:A. P(x)) = (? x:B. Q(x))"; ``` clasohm@923 ` 108` ```by (resolve_tac (prems RL [ssubst]) 1); ``` clasohm@923 ` 109` ```by (REPEAT (etac bexE 1 ``` clasohm@923 ` 110` ``` ORELSE ares_tac ([bexI,iffI] @ (prems RL [iffD1,iffD2])) 1)); ``` clasohm@923 ` 111` ```qed "bex_cong"; ``` clasohm@923 ` 112` nipkow@1548 ` 113` ```section "Subsets"; ``` clasohm@923 ` 114` clasohm@923 ` 115` ```val prems = goalw Set.thy [subset_def] "(!!x.x:A ==> x:B) ==> A <= B"; ``` clasohm@923 ` 116` ```by (REPEAT (ares_tac (prems @ [ballI]) 1)); ``` clasohm@923 ` 117` ```qed "subsetI"; ``` clasohm@923 ` 118` clasohm@923 ` 119` ```(*Rule in Modus Ponens style*) ``` clasohm@923 ` 120` ```val major::prems = goalw Set.thy [subset_def] "[| A <= B; c:A |] ==> c:B"; ``` clasohm@923 ` 121` ```by (rtac (major RS bspec) 1); ``` clasohm@923 ` 122` ```by (resolve_tac prems 1); ``` clasohm@923 ` 123` ```qed "subsetD"; ``` clasohm@923 ` 124` clasohm@923 ` 125` ```(*The same, with reversed premises for use with etac -- cf rev_mp*) ``` clasohm@923 ` 126` ```qed_goal "rev_subsetD" Set.thy "[| c:A; A <= B |] ==> c:B" ``` clasohm@923 ` 127` ``` (fn prems=> [ (REPEAT (resolve_tac (prems@[subsetD]) 1)) ]); ``` clasohm@923 ` 128` paulson@1920 ` 129` ```(*Converts A<=B to x:A ==> x:B*) ``` paulson@1920 ` 130` ```fun impOfSubs th = th RSN (2, rev_subsetD); ``` paulson@1920 ` 131` paulson@1841 ` 132` ```qed_goal "contra_subsetD" Set.thy "!!c. [| A <= B; c ~: B |] ==> c ~: A" ``` paulson@1841 ` 133` ``` (fn prems=> [ (REPEAT (eresolve_tac [asm_rl, contrapos, subsetD] 1)) ]); ``` paulson@1841 ` 134` paulson@1841 ` 135` ```qed_goal "rev_contra_subsetD" Set.thy "!!c. [| c ~: B; A <= B |] ==> c ~: A" ``` paulson@1841 ` 136` ``` (fn prems=> [ (REPEAT (eresolve_tac [asm_rl, contrapos, subsetD] 1)) ]); ``` paulson@1841 ` 137` clasohm@923 ` 138` ```(*Classical elimination rule*) ``` clasohm@923 ` 139` ```val major::prems = goalw Set.thy [subset_def] ``` clasohm@923 ` 140` ``` "[| A <= B; c~:A ==> P; c:B ==> P |] ==> P"; ``` clasohm@923 ` 141` ```by (rtac (major RS ballE) 1); ``` clasohm@923 ` 142` ```by (REPEAT (eresolve_tac prems 1)); ``` clasohm@923 ` 143` ```qed "subsetCE"; ``` clasohm@923 ` 144` clasohm@923 ` 145` ```(*Takes assumptions A<=B; c:A and creates the assumption c:B *) ``` clasohm@923 ` 146` ```fun set_mp_tac i = etac subsetCE i THEN mp_tac i; ``` clasohm@923 ` 147` paulson@2499 ` 148` ```AddSIs [subsetI]; ``` paulson@2499 ` 149` ```AddEs [subsetD, subsetCE]; ``` clasohm@923 ` 150` paulson@2499 ` 151` ```qed_goal "subset_refl" Set.thy "A <= (A::'a set)" ``` paulson@2499 ` 152` ``` (fn _=> [Fast_tac 1]); ``` paulson@2499 ` 153` paulson@2499 ` 154` ```val prems = goal Set.thy "!!B. [| A<=B; B<=C |] ==> A<=(C::'a set)"; ``` paulson@2499 ` 155` ```by (Fast_tac 1); ``` clasohm@923 ` 156` ```qed "subset_trans"; ``` clasohm@923 ` 157` clasohm@923 ` 158` nipkow@1548 ` 159` ```section "Equality"; ``` clasohm@923 ` 160` clasohm@923 ` 161` ```(*Anti-symmetry of the subset relation*) ``` clasohm@923 ` 162` ```val prems = goal Set.thy "[| A <= B; B <= A |] ==> A = (B::'a set)"; ``` clasohm@923 ` 163` ```by (rtac (iffI RS set_ext) 1); ``` clasohm@923 ` 164` ```by (REPEAT (ares_tac (prems RL [subsetD]) 1)); ``` clasohm@923 ` 165` ```qed "subset_antisym"; ``` clasohm@923 ` 166` ```val equalityI = subset_antisym; ``` clasohm@923 ` 167` berghofe@1762 ` 168` ```AddSIs [equalityI]; ``` berghofe@1762 ` 169` clasohm@923 ` 170` ```(* Equality rules from ZF set theory -- are they appropriate here? *) ``` clasohm@923 ` 171` ```val prems = goal Set.thy "A = B ==> A<=(B::'a set)"; ``` clasohm@923 ` 172` ```by (resolve_tac (prems RL [subst]) 1); ``` clasohm@923 ` 173` ```by (rtac subset_refl 1); ``` clasohm@923 ` 174` ```qed "equalityD1"; ``` clasohm@923 ` 175` clasohm@923 ` 176` ```val prems = goal Set.thy "A = B ==> B<=(A::'a set)"; ``` clasohm@923 ` 177` ```by (resolve_tac (prems RL [subst]) 1); ``` clasohm@923 ` 178` ```by (rtac subset_refl 1); ``` clasohm@923 ` 179` ```qed "equalityD2"; ``` clasohm@923 ` 180` clasohm@923 ` 181` ```val prems = goal Set.thy ``` clasohm@923 ` 182` ``` "[| A = B; [| A<=B; B<=(A::'a set) |] ==> P |] ==> P"; ``` clasohm@923 ` 183` ```by (resolve_tac prems 1); ``` clasohm@923 ` 184` ```by (REPEAT (resolve_tac (prems RL [equalityD1,equalityD2]) 1)); ``` clasohm@923 ` 185` ```qed "equalityE"; ``` clasohm@923 ` 186` clasohm@923 ` 187` ```val major::prems = goal Set.thy ``` clasohm@923 ` 188` ``` "[| A = B; [| c:A; c:B |] ==> P; [| c~:A; c~:B |] ==> P |] ==> P"; ``` clasohm@923 ` 189` ```by (rtac (major RS equalityE) 1); ``` clasohm@923 ` 190` ```by (REPEAT (contr_tac 1 ORELSE eresolve_tac ([asm_rl,subsetCE]@prems) 1)); ``` clasohm@923 ` 191` ```qed "equalityCE"; ``` clasohm@923 ` 192` clasohm@923 ` 193` ```(*Lemma for creating induction formulae -- for "pattern matching" on p ``` clasohm@923 ` 194` ``` To make the induction hypotheses usable, apply "spec" or "bspec" to ``` clasohm@923 ` 195` ``` put universal quantifiers over the free variables in p. *) ``` clasohm@923 ` 196` ```val prems = goal Set.thy ``` clasohm@923 ` 197` ``` "[| p:A; !!z. z:A ==> p=z --> R |] ==> R"; ``` clasohm@923 ` 198` ```by (rtac mp 1); ``` clasohm@923 ` 199` ```by (REPEAT (resolve_tac (refl::prems) 1)); ``` clasohm@923 ` 200` ```qed "setup_induction"; ``` clasohm@923 ` 201` clasohm@923 ` 202` nipkow@1548 ` 203` ```section "Set complement -- Compl"; ``` clasohm@923 ` 204` paulson@2499 ` 205` ```qed_goalw "Compl_iff" Set.thy [Compl_def] "(c : Compl(A)) = (c~:A)" ``` paulson@2499 ` 206` ``` (fn _ => [ (Fast_tac 1) ]); ``` paulson@2499 ` 207` paulson@2499 ` 208` ```Addsimps [Compl_iff]; ``` paulson@2499 ` 209` clasohm@923 ` 210` ```val prems = goalw Set.thy [Compl_def] ``` clasohm@923 ` 211` ``` "[| c:A ==> False |] ==> c : Compl(A)"; ``` clasohm@923 ` 212` ```by (REPEAT (ares_tac (prems @ [CollectI,notI]) 1)); ``` clasohm@923 ` 213` ```qed "ComplI"; ``` clasohm@923 ` 214` clasohm@923 ` 215` ```(*This form, with negated conclusion, works well with the Classical prover. ``` clasohm@923 ` 216` ``` Negated assumptions behave like formulae on the right side of the notional ``` clasohm@923 ` 217` ``` turnstile...*) ``` clasohm@923 ` 218` ```val major::prems = goalw Set.thy [Compl_def] ``` paulson@2499 ` 219` ``` "c : Compl(A) ==> c~:A"; ``` clasohm@923 ` 220` ```by (rtac (major RS CollectD) 1); ``` clasohm@923 ` 221` ```qed "ComplD"; ``` clasohm@923 ` 222` clasohm@923 ` 223` ```val ComplE = make_elim ComplD; ``` clasohm@923 ` 224` paulson@2499 ` 225` ```AddSIs [ComplI]; ``` paulson@2499 ` 226` ```AddSEs [ComplE]; ``` paulson@1640 ` 227` clasohm@923 ` 228` nipkow@1548 ` 229` ```section "Binary union -- Un"; ``` clasohm@923 ` 230` paulson@2499 ` 231` ```qed_goalw "Un_iff" Set.thy [Un_def] "(c : A Un B) = (c:A | c:B)" ``` paulson@2499 ` 232` ``` (fn _ => [ Fast_tac 1 ]); ``` paulson@2499 ` 233` paulson@2499 ` 234` ```Addsimps [Un_iff]; ``` paulson@2499 ` 235` paulson@2499 ` 236` ```goal Set.thy "!!c. c:A ==> c : A Un B"; ``` paulson@2499 ` 237` ```by (Asm_simp_tac 1); ``` clasohm@923 ` 238` ```qed "UnI1"; ``` clasohm@923 ` 239` paulson@2499 ` 240` ```goal Set.thy "!!c. c:B ==> c : A Un B"; ``` paulson@2499 ` 241` ```by (Asm_simp_tac 1); ``` clasohm@923 ` 242` ```qed "UnI2"; ``` clasohm@923 ` 243` clasohm@923 ` 244` ```(*Classical introduction rule: no commitment to A vs B*) ``` clasohm@923 ` 245` ```qed_goal "UnCI" Set.thy "(c~:B ==> c:A) ==> c : A Un B" ``` clasohm@923 ` 246` ``` (fn prems=> ``` paulson@2499 ` 247` ``` [ (Simp_tac 1), ``` paulson@2499 ` 248` ``` (REPEAT (ares_tac (prems@[disjCI]) 1)) ]); ``` clasohm@923 ` 249` clasohm@923 ` 250` ```val major::prems = goalw Set.thy [Un_def] ``` clasohm@923 ` 251` ``` "[| c : A Un B; c:A ==> P; c:B ==> P |] ==> P"; ``` clasohm@923 ` 252` ```by (rtac (major RS CollectD RS disjE) 1); ``` clasohm@923 ` 253` ```by (REPEAT (eresolve_tac prems 1)); ``` clasohm@923 ` 254` ```qed "UnE"; ``` clasohm@923 ` 255` paulson@2499 ` 256` ```AddSIs [UnCI]; ``` paulson@2499 ` 257` ```AddSEs [UnE]; ``` paulson@1640 ` 258` clasohm@923 ` 259` nipkow@1548 ` 260` ```section "Binary intersection -- Int"; ``` clasohm@923 ` 261` paulson@2499 ` 262` ```qed_goalw "Int_iff" Set.thy [Int_def] "(c : A Int B) = (c:A & c:B)" ``` paulson@2499 ` 263` ``` (fn _ => [ (Fast_tac 1) ]); ``` paulson@2499 ` 264` paulson@2499 ` 265` ```Addsimps [Int_iff]; ``` paulson@2499 ` 266` paulson@2499 ` 267` ```goal Set.thy "!!c. [| c:A; c:B |] ==> c : A Int B"; ``` paulson@2499 ` 268` ```by (Asm_simp_tac 1); ``` clasohm@923 ` 269` ```qed "IntI"; ``` clasohm@923 ` 270` paulson@2499 ` 271` ```goal Set.thy "!!c. c : A Int B ==> c:A"; ``` paulson@2499 ` 272` ```by (Asm_full_simp_tac 1); ``` clasohm@923 ` 273` ```qed "IntD1"; ``` clasohm@923 ` 274` paulson@2499 ` 275` ```goal Set.thy "!!c. c : A Int B ==> c:B"; ``` paulson@2499 ` 276` ```by (Asm_full_simp_tac 1); ``` clasohm@923 ` 277` ```qed "IntD2"; ``` clasohm@923 ` 278` clasohm@923 ` 279` ```val [major,minor] = goal Set.thy ``` clasohm@923 ` 280` ``` "[| c : A Int B; [| c:A; c:B |] ==> P |] ==> P"; ``` clasohm@923 ` 281` ```by (rtac minor 1); ``` clasohm@923 ` 282` ```by (rtac (major RS IntD1) 1); ``` clasohm@923 ` 283` ```by (rtac (major RS IntD2) 1); ``` clasohm@923 ` 284` ```qed "IntE"; ``` clasohm@923 ` 285` paulson@2499 ` 286` ```AddSIs [IntI]; ``` paulson@2499 ` 287` ```AddSEs [IntE]; ``` clasohm@923 ` 288` nipkow@1548 ` 289` ```section "Set difference"; ``` clasohm@923 ` 290` paulson@2499 ` 291` ```qed_goalw "Diff_iff" Set.thy [set_diff_def] "(c : A-B) = (c:A & c~:B)" ``` paulson@2499 ` 292` ``` (fn _ => [ (Fast_tac 1) ]); ``` clasohm@923 ` 293` paulson@2499 ` 294` ```Addsimps [Diff_iff]; ``` paulson@2499 ` 295` paulson@2499 ` 296` ```qed_goal "DiffI" Set.thy "!!c. [| c : A; c ~: B |] ==> c : A - B" ``` paulson@2499 ` 297` ``` (fn _=> [ Asm_simp_tac 1 ]); ``` clasohm@923 ` 298` paulson@2499 ` 299` ```qed_goal "DiffD1" Set.thy "!!c. c : A - B ==> c : A" ``` paulson@2499 ` 300` ``` (fn _=> [ (Asm_full_simp_tac 1) ]); ``` clasohm@923 ` 301` paulson@2499 ` 302` ```qed_goal "DiffD2" Set.thy "!!c. [| c : A - B; c : B |] ==> P" ``` paulson@2499 ` 303` ``` (fn _=> [ (Asm_full_simp_tac 1) ]); ``` paulson@2499 ` 304` paulson@2499 ` 305` ```qed_goal "DiffE" Set.thy "[| c : A - B; [| c:A; c~:B |] ==> P |] ==> P" ``` clasohm@923 ` 306` ``` (fn prems=> ``` clasohm@923 ` 307` ``` [ (resolve_tac prems 1), ``` clasohm@923 ` 308` ``` (REPEAT (ares_tac (prems RL [DiffD1, DiffD2 RS notI]) 1)) ]); ``` clasohm@923 ` 309` paulson@2499 ` 310` ```AddSIs [DiffI]; ``` paulson@2499 ` 311` ```AddSEs [DiffE]; ``` clasohm@923 ` 312` nipkow@1548 ` 313` ```section "The empty set -- {}"; ``` clasohm@923 ` 314` paulson@2499 ` 315` ```qed_goalw "empty_iff" Set.thy [empty_def] "(c : {}) = False" ``` paulson@2499 ` 316` ``` (fn _ => [ (Fast_tac 1) ]); ``` paulson@2499 ` 317` paulson@2499 ` 318` ```Addsimps [empty_iff]; ``` paulson@2499 ` 319` paulson@2499 ` 320` ```qed_goal "emptyE" Set.thy "!!a. a:{} ==> P" ``` paulson@2499 ` 321` ``` (fn _ => [Full_simp_tac 1]); ``` paulson@2499 ` 322` paulson@2499 ` 323` ```AddSEs [emptyE]; ``` clasohm@923 ` 324` clasohm@923 ` 325` ```qed_goal "empty_subsetI" Set.thy "{} <= A" ``` paulson@2499 ` 326` ``` (fn _ => [ (Fast_tac 1) ]); ``` clasohm@923 ` 327` clasohm@923 ` 328` ```qed_goal "equals0I" Set.thy "[| !!y. y:A ==> False |] ==> A={}" ``` paulson@2499 ` 329` ``` (fn [prem]=> ``` paulson@2499 ` 330` ``` [ (fast_tac (!claset addIs [prem RS FalseE]) 1) ]); ``` clasohm@923 ` 331` paulson@2499 ` 332` ```qed_goal "equals0D" Set.thy "!!a. [| A={}; a:A |] ==> P" ``` paulson@2499 ` 333` ``` (fn _ => [ (Fast_tac 1) ]); ``` paulson@1640 ` 334` paulson@1816 ` 335` ```goal Set.thy "Ball {} P = True"; ``` paulson@1816 ` 336` ```by (simp_tac (HOL_ss addsimps [mem_Collect_eq, Ball_def, empty_def]) 1); ``` paulson@1816 ` 337` ```qed "ball_empty"; ``` paulson@1816 ` 338` paulson@1816 ` 339` ```goal Set.thy "Bex {} P = False"; ``` paulson@1816 ` 340` ```by (simp_tac (HOL_ss addsimps [mem_Collect_eq, Bex_def, empty_def]) 1); ``` paulson@1816 ` 341` ```qed "bex_empty"; ``` paulson@1816 ` 342` ```Addsimps [ball_empty, bex_empty]; ``` paulson@1816 ` 343` clasohm@923 ` 344` nipkow@1548 ` 345` ```section "Augmenting a set -- insert"; ``` clasohm@923 ` 346` paulson@2499 ` 347` ```qed_goalw "insert_iff" Set.thy [insert_def] "a : insert b A = (a=b | a:A)" ``` paulson@2499 ` 348` ``` (fn _ => [Fast_tac 1]); ``` paulson@2499 ` 349` paulson@2499 ` 350` ```Addsimps [insert_iff]; ``` clasohm@923 ` 351` paulson@2499 ` 352` ```qed_goal "insertI1" Set.thy "a : insert a B" ``` paulson@2499 ` 353` ``` (fn _ => [Simp_tac 1]); ``` paulson@2499 ` 354` paulson@2499 ` 355` ```qed_goal "insertI2" Set.thy "!!a. a : B ==> a : insert b B" ``` paulson@2499 ` 356` ``` (fn _=> [Asm_simp_tac 1]); ``` clasohm@923 ` 357` clasohm@923 ` 358` ```qed_goalw "insertE" Set.thy [insert_def] ``` clasohm@923 ` 359` ``` "[| a : insert b A; a=b ==> P; a:A ==> P |] ==> P" ``` clasohm@923 ` 360` ``` (fn major::prems=> ``` clasohm@923 ` 361` ``` [ (rtac (major RS UnE) 1), ``` clasohm@923 ` 362` ``` (REPEAT (eresolve_tac (prems @ [CollectE]) 1)) ]); ``` clasohm@923 ` 363` clasohm@923 ` 364` ```(*Classical introduction rule*) ``` clasohm@923 ` 365` ```qed_goal "insertCI" Set.thy "(a~:B ==> a=b) ==> a: insert b B" ``` paulson@2499 ` 366` ``` (fn prems=> ``` paulson@2499 ` 367` ``` [ (Simp_tac 1), ``` paulson@2499 ` 368` ``` (REPEAT (ares_tac (prems@[disjCI]) 1)) ]); ``` paulson@2499 ` 369` paulson@2499 ` 370` ```AddSIs [insertCI]; ``` paulson@2499 ` 371` ```AddSEs [insertE]; ``` clasohm@923 ` 372` nipkow@1548 ` 373` ```section "Singletons, using insert"; ``` clasohm@923 ` 374` clasohm@923 ` 375` ```qed_goal "singletonI" Set.thy "a : {a}" ``` clasohm@923 ` 376` ``` (fn _=> [ (rtac insertI1 1) ]); ``` clasohm@923 ` 377` paulson@2499 ` 378` ```goal Set.thy "!!a. b : {a} ==> b=a"; ``` paulson@2499 ` 379` ```by (Fast_tac 1); ``` clasohm@923 ` 380` ```qed "singletonD"; ``` clasohm@923 ` 381` oheimb@1776 ` 382` ```bind_thm ("singletonE", make_elim singletonD); ``` oheimb@1776 ` 383` paulson@2499 ` 384` ```qed_goal "singleton_iff" thy "(b : {a}) = (b=a)" ``` paulson@2499 ` 385` ```(fn _ => [Fast_tac 1]); ``` clasohm@923 ` 386` paulson@2499 ` 387` ```goal Set.thy "!!a b. {a}={b} ==> a=b"; ``` paulson@2499 ` 388` ```by (fast_tac (!claset addEs [equalityE]) 1); ``` clasohm@923 ` 389` ```qed "singleton_inject"; ``` clasohm@923 ` 390` paulson@2499 ` 391` ```AddSDs [singleton_inject]; ``` paulson@2499 ` 392` nipkow@1531 ` 393` nipkow@1548 ` 394` ```section "The universal set -- UNIV"; ``` nipkow@1531 ` 395` paulson@1882 ` 396` ```qed_goal "UNIV_I" Set.thy "x : UNIV" ``` paulson@1882 ` 397` ``` (fn _ => [rtac ComplI 1, etac emptyE 1]); ``` paulson@1882 ` 398` nipkow@1531 ` 399` ```qed_goal "subset_UNIV" Set.thy "A <= UNIV" ``` paulson@1882 ` 400` ``` (fn _ => [rtac subsetI 1, rtac UNIV_I 1]); ``` nipkow@1531 ` 401` nipkow@1531 ` 402` nipkow@1548 ` 403` ```section "Unions of families -- UNION x:A. B(x) is Union(B``A)"; ``` clasohm@923 ` 404` paulson@2499 ` 405` ```goalw Set.thy [UNION_def] "(b: (UN x:A. B(x))) = (EX x:A. b: B(x))"; ``` paulson@2499 ` 406` ```by (Fast_tac 1); ``` paulson@2499 ` 407` ```qed "UN_iff"; ``` paulson@2499 ` 408` paulson@2499 ` 409` ```Addsimps [UN_iff]; ``` paulson@2499 ` 410` clasohm@923 ` 411` ```(*The order of the premises presupposes that A is rigid; b may be flexible*) ``` paulson@2499 ` 412` ```goal Set.thy "!!b. [| a:A; b: B(a) |] ==> b: (UN x:A. B(x))"; ``` paulson@2499 ` 413` ```by (Auto_tac()); ``` clasohm@923 ` 414` ```qed "UN_I"; ``` clasohm@923 ` 415` clasohm@923 ` 416` ```val major::prems = goalw Set.thy [UNION_def] ``` clasohm@923 ` 417` ``` "[| b : (UN x:A. B(x)); !!x.[| x:A; b: B(x) |] ==> R |] ==> R"; ``` clasohm@923 ` 418` ```by (rtac (major RS CollectD RS bexE) 1); ``` clasohm@923 ` 419` ```by (REPEAT (ares_tac prems 1)); ``` clasohm@923 ` 420` ```qed "UN_E"; ``` clasohm@923 ` 421` paulson@2499 ` 422` ```AddIs [UN_I]; ``` paulson@2499 ` 423` ```AddSEs [UN_E]; ``` paulson@2499 ` 424` clasohm@923 ` 425` ```val prems = goal Set.thy ``` clasohm@923 ` 426` ``` "[| A=B; !!x. x:B ==> C(x) = D(x) |] ==> \ ``` clasohm@923 ` 427` ```\ (UN x:A. C(x)) = (UN x:B. D(x))"; ``` clasohm@923 ` 428` ```by (REPEAT (etac UN_E 1 ``` clasohm@923 ` 429` ``` ORELSE ares_tac ([UN_I,equalityI,subsetI] @ ``` clasohm@1465 ` 430` ``` (prems RL [equalityD1,equalityD2] RL [subsetD])) 1)); ``` clasohm@923 ` 431` ```qed "UN_cong"; ``` clasohm@923 ` 432` clasohm@923 ` 433` nipkow@1548 ` 434` ```section "Intersections of families -- INTER x:A. B(x) is Inter(B``A)"; ``` clasohm@923 ` 435` paulson@2499 ` 436` ```goalw Set.thy [INTER_def] "(b: (INT x:A. B(x))) = (ALL x:A. b: B(x))"; ``` paulson@2499 ` 437` ```by (Auto_tac()); ``` paulson@2499 ` 438` ```qed "INT_iff"; ``` paulson@2499 ` 439` paulson@2499 ` 440` ```Addsimps [INT_iff]; ``` paulson@2499 ` 441` clasohm@923 ` 442` ```val prems = goalw Set.thy [INTER_def] ``` clasohm@923 ` 443` ``` "(!!x. x:A ==> b: B(x)) ==> b : (INT x:A. B(x))"; ``` clasohm@923 ` 444` ```by (REPEAT (ares_tac ([CollectI,ballI] @ prems) 1)); ``` clasohm@923 ` 445` ```qed "INT_I"; ``` clasohm@923 ` 446` paulson@2499 ` 447` ```goal Set.thy "!!b. [| b : (INT x:A. B(x)); a:A |] ==> b: B(a)"; ``` paulson@2499 ` 448` ```by (Auto_tac()); ``` clasohm@923 ` 449` ```qed "INT_D"; ``` clasohm@923 ` 450` clasohm@923 ` 451` ```(*"Classical" elimination -- by the Excluded Middle on a:A *) ``` clasohm@923 ` 452` ```val major::prems = goalw Set.thy [INTER_def] ``` clasohm@923 ` 453` ``` "[| b : (INT x:A. B(x)); b: B(a) ==> R; a~:A ==> R |] ==> R"; ``` clasohm@923 ` 454` ```by (rtac (major RS CollectD RS ballE) 1); ``` clasohm@923 ` 455` ```by (REPEAT (eresolve_tac prems 1)); ``` clasohm@923 ` 456` ```qed "INT_E"; ``` clasohm@923 ` 457` paulson@2499 ` 458` ```AddSIs [INT_I]; ``` paulson@2499 ` 459` ```AddEs [INT_D, INT_E]; ``` paulson@2499 ` 460` clasohm@923 ` 461` ```val prems = goal Set.thy ``` clasohm@923 ` 462` ``` "[| A=B; !!x. x:B ==> C(x) = D(x) |] ==> \ ``` clasohm@923 ` 463` ```\ (INT x:A. C(x)) = (INT x:B. D(x))"; ``` clasohm@923 ` 464` ```by (REPEAT_FIRST (resolve_tac [INT_I,equalityI,subsetI])); ``` clasohm@923 ` 465` ```by (REPEAT (dtac INT_D 1 ``` clasohm@923 ` 466` ``` ORELSE ares_tac (prems RL [equalityD1,equalityD2] RL [subsetD]) 1)); ``` clasohm@923 ` 467` ```qed "INT_cong"; ``` clasohm@923 ` 468` clasohm@923 ` 469` nipkow@1548 ` 470` ```section "Unions over a type; UNION1(B) = Union(range(B))"; ``` clasohm@923 ` 471` paulson@2499 ` 472` ```goalw Set.thy [UNION1_def] "(b: (UN x. B(x))) = (EX x. b: B(x))"; ``` paulson@2499 ` 473` ```by (Simp_tac 1); ``` paulson@2499 ` 474` ```by (Fast_tac 1); ``` paulson@2499 ` 475` ```qed "UN1_iff"; ``` paulson@2499 ` 476` paulson@2499 ` 477` ```Addsimps [UN1_iff]; ``` paulson@2499 ` 478` clasohm@923 ` 479` ```(*The order of the premises presupposes that A is rigid; b may be flexible*) ``` paulson@2499 ` 480` ```goal Set.thy "!!b. b: B(x) ==> b: (UN x. B(x))"; ``` paulson@2499 ` 481` ```by (Auto_tac()); ``` clasohm@923 ` 482` ```qed "UN1_I"; ``` clasohm@923 ` 483` clasohm@923 ` 484` ```val major::prems = goalw Set.thy [UNION1_def] ``` clasohm@923 ` 485` ``` "[| b : (UN x. B(x)); !!x. b: B(x) ==> R |] ==> R"; ``` clasohm@923 ` 486` ```by (rtac (major RS UN_E) 1); ``` clasohm@923 ` 487` ```by (REPEAT (ares_tac prems 1)); ``` clasohm@923 ` 488` ```qed "UN1_E"; ``` clasohm@923 ` 489` paulson@2499 ` 490` ```AddIs [UN1_I]; ``` paulson@2499 ` 491` ```AddSEs [UN1_E]; ``` paulson@2499 ` 492` clasohm@923 ` 493` nipkow@1548 ` 494` ```section "Intersections over a type; INTER1(B) = Inter(range(B))"; ``` clasohm@923 ` 495` paulson@2499 ` 496` ```goalw Set.thy [INTER1_def] "(b: (INT x. B(x))) = (ALL x. b: B(x))"; ``` paulson@2499 ` 497` ```by (Simp_tac 1); ``` paulson@2499 ` 498` ```by (Fast_tac 1); ``` paulson@2499 ` 499` ```qed "INT1_iff"; ``` paulson@2499 ` 500` paulson@2499 ` 501` ```Addsimps [INT1_iff]; ``` paulson@2499 ` 502` clasohm@923 ` 503` ```val prems = goalw Set.thy [INTER1_def] ``` clasohm@923 ` 504` ``` "(!!x. b: B(x)) ==> b : (INT x. B(x))"; ``` clasohm@923 ` 505` ```by (REPEAT (ares_tac (INT_I::prems) 1)); ``` clasohm@923 ` 506` ```qed "INT1_I"; ``` clasohm@923 ` 507` paulson@2499 ` 508` ```goal Set.thy "!!b. b : (INT x. B(x)) ==> b: B(a)"; ``` paulson@2499 ` 509` ```by (Asm_full_simp_tac 1); ``` clasohm@923 ` 510` ```qed "INT1_D"; ``` clasohm@923 ` 511` paulson@2499 ` 512` ```AddSIs [INT1_I]; ``` paulson@2499 ` 513` ```AddDs [INT1_D]; ``` paulson@2499 ` 514` paulson@2499 ` 515` nipkow@1548 ` 516` ```section "Union"; ``` clasohm@923 ` 517` paulson@2499 ` 518` ```goalw Set.thy [Union_def] "(A : Union(C)) = (EX X:C. A:X)"; ``` paulson@2499 ` 519` ```by (Fast_tac 1); ``` paulson@2499 ` 520` ```qed "Union_iff"; ``` paulson@2499 ` 521` paulson@2499 ` 522` ```Addsimps [Union_iff]; ``` paulson@2499 ` 523` clasohm@923 ` 524` ```(*The order of the premises presupposes that C is rigid; A may be flexible*) ``` paulson@2499 ` 525` ```goal Set.thy "!!X. [| X:C; A:X |] ==> A : Union(C)"; ``` paulson@2499 ` 526` ```by (Auto_tac()); ``` clasohm@923 ` 527` ```qed "UnionI"; ``` clasohm@923 ` 528` clasohm@923 ` 529` ```val major::prems = goalw Set.thy [Union_def] ``` clasohm@923 ` 530` ``` "[| A : Union(C); !!X.[| A:X; X:C |] ==> R |] ==> R"; ``` clasohm@923 ` 531` ```by (rtac (major RS UN_E) 1); ``` clasohm@923 ` 532` ```by (REPEAT (ares_tac prems 1)); ``` clasohm@923 ` 533` ```qed "UnionE"; ``` clasohm@923 ` 534` paulson@2499 ` 535` ```AddIs [UnionI]; ``` paulson@2499 ` 536` ```AddSEs [UnionE]; ``` paulson@2499 ` 537` paulson@2499 ` 538` nipkow@1548 ` 539` ```section "Inter"; ``` clasohm@923 ` 540` paulson@2499 ` 541` ```goalw Set.thy [Inter_def] "(A : Inter(C)) = (ALL X:C. A:X)"; ``` paulson@2499 ` 542` ```by (Fast_tac 1); ``` paulson@2499 ` 543` ```qed "Inter_iff"; ``` paulson@2499 ` 544` paulson@2499 ` 545` ```Addsimps [Inter_iff]; ``` paulson@2499 ` 546` clasohm@923 ` 547` ```val prems = goalw Set.thy [Inter_def] ``` clasohm@923 ` 548` ``` "[| !!X. X:C ==> A:X |] ==> A : Inter(C)"; ``` clasohm@923 ` 549` ```by (REPEAT (ares_tac ([INT_I] @ prems) 1)); ``` clasohm@923 ` 550` ```qed "InterI"; ``` clasohm@923 ` 551` clasohm@923 ` 552` ```(*A "destruct" rule -- every X in C contains A as an element, but ``` clasohm@923 ` 553` ``` A:X can hold when X:C does not! This rule is analogous to "spec". *) ``` paulson@2499 ` 554` ```goal Set.thy "!!X. [| A : Inter(C); X:C |] ==> A:X"; ``` paulson@2499 ` 555` ```by (Auto_tac()); ``` clasohm@923 ` 556` ```qed "InterD"; ``` clasohm@923 ` 557` clasohm@923 ` 558` ```(*"Classical" elimination rule -- does not require proving X:C *) ``` clasohm@923 ` 559` ```val major::prems = goalw Set.thy [Inter_def] ``` clasohm@923 ` 560` ``` "[| A : Inter(C); A:X ==> R; X~:C ==> R |] ==> R"; ``` clasohm@923 ` 561` ```by (rtac (major RS INT_E) 1); ``` clasohm@923 ` 562` ```by (REPEAT (eresolve_tac prems 1)); ``` clasohm@923 ` 563` ```qed "InterE"; ``` clasohm@923 ` 564` paulson@2499 ` 565` ```AddSIs [InterI]; ``` paulson@2499 ` 566` ```AddEs [InterD, InterE]; ``` paulson@2499 ` 567` paulson@2499 ` 568` nipkow@1548 ` 569` ```section "The Powerset operator -- Pow"; ``` clasohm@923 ` 570` paulson@2499 ` 571` ```qed_goalw "Pow_iff" Set.thy [Pow_def] "(A : Pow(B)) = (A <= B)" ``` paulson@2499 ` 572` ``` (fn _ => [ (Asm_simp_tac 1) ]); ``` paulson@2499 ` 573` paulson@2499 ` 574` ```AddIffs [Pow_iff]; ``` paulson@2499 ` 575` clasohm@923 ` 576` ```qed_goalw "PowI" Set.thy [Pow_def] "!!A B. A <= B ==> A : Pow(B)" ``` clasohm@923 ` 577` ``` (fn _ => [ (etac CollectI 1) ]); ``` clasohm@923 ` 578` clasohm@923 ` 579` ```qed_goalw "PowD" Set.thy [Pow_def] "!!A B. A : Pow(B) ==> A<=B" ``` clasohm@923 ` 580` ``` (fn _=> [ (etac CollectD 1) ]); ``` clasohm@923 ` 581` clasohm@923 ` 582` ```val Pow_bottom = empty_subsetI RS PowI; (* {}: Pow(B) *) ``` clasohm@923 ` 583` ```val Pow_top = subset_refl RS PowI; (* A : Pow(A) *) ``` oheimb@1776 ` 584` oheimb@1776 ` 585` oheimb@1776 ` 586` oheimb@1776 ` 587` ```(*** Set reasoning tools ***) ``` oheimb@1776 ` 588` oheimb@1776 ` 589` paulson@2499 ` 590` ```(*Each of these has ALREADY been added to !simpset above.*) ``` paulson@2024 ` 591` ```val mem_simps = [insert_iff, empty_iff, Un_iff, Int_iff, Compl_iff, Diff_iff, ``` paulson@2499 ` 592` ``` mem_Collect_eq, ``` paulson@2499 ` 593` ``` UN_iff, UN1_iff, Union_iff, ``` paulson@2499 ` 594` ``` INT_iff, INT1_iff, Inter_iff]; ``` oheimb@1776 ` 595` paulson@1937 ` 596` ```(*Not for Addsimps -- it can cause goals to blow up!*) ``` paulson@1937 ` 597` ```goal Set.thy "(a : (if Q then x else y)) = ((Q --> a:x) & (~Q --> a : y))"; ``` paulson@1937 ` 598` ```by (simp_tac (!simpset setloop split_tac [expand_if]) 1); ``` paulson@1937 ` 599` ```qed "mem_if"; ``` paulson@1937 ` 600` oheimb@1776 ` 601` ```val mksimps_pairs = ("Ball",[bspec]) :: mksimps_pairs; ``` oheimb@1776 ` 602` paulson@2499 ` 603` ```simpset := !simpset addcongs [ball_cong,bex_cong] ``` oheimb@1776 ` 604` ``` setmksimps (mksimps mksimps_pairs); ```