src/HOL/Set.ML
 author nipkow Mon Apr 27 16:45:11 1998 +0200 (1998-04-27) changeset 4830 bd73675adbed parent 4770 3e026ace28da child 5069 3ea049f7979d permissions -rw-r--r--
Added a few lemmas.
Renamed expand_const -> split_const.
 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@3469 ` 13` ```Addsimps [Collect_mem_eq]; ``` paulson@3469 ` 14` ```AddIffs [mem_Collect_eq]; ``` paulson@2499 ` 15` wenzelm@3842 ` 16` ```goal Set.thy "!!a. P(a) ==> a : {x. P(x)}"; ``` paulson@2499 ` 17` ```by (Asm_simp_tac 1); ``` clasohm@923 ` 18` ```qed "CollectI"; ``` clasohm@923 ` 19` wenzelm@3842 ` 20` ```val prems = goal Set.thy "!!a. a : {x. P(x)} ==> P(a)"; ``` paulson@2499 ` 21` ```by (Asm_full_simp_tac 1); ``` clasohm@923 ` 22` ```qed "CollectD"; ``` clasohm@923 ` 23` clasohm@923 ` 24` ```val [prem] = goal Set.thy "[| !!x. (x:A) = (x:B) |] ==> A = B"; ``` clasohm@923 ` 25` ```by (rtac (prem RS ext RS arg_cong RS box_equals) 1); ``` clasohm@923 ` 26` ```by (rtac Collect_mem_eq 1); ``` clasohm@923 ` 27` ```by (rtac Collect_mem_eq 1); ``` clasohm@923 ` 28` ```qed "set_ext"; ``` clasohm@923 ` 29` clasohm@923 ` 30` ```val [prem] = goal Set.thy "[| !!x. P(x)=Q(x) |] ==> {x. P(x)} = {x. Q(x)}"; ``` clasohm@923 ` 31` ```by (rtac (prem RS ext RS arg_cong) 1); ``` clasohm@923 ` 32` ```qed "Collect_cong"; ``` clasohm@923 ` 33` clasohm@923 ` 34` ```val CollectE = make_elim CollectD; ``` clasohm@923 ` 35` paulson@2499 ` 36` ```AddSIs [CollectI]; ``` paulson@2499 ` 37` ```AddSEs [CollectE]; ``` paulson@2499 ` 38` paulson@2499 ` 39` nipkow@1548 ` 40` ```section "Bounded quantifiers"; ``` clasohm@923 ` 41` clasohm@923 ` 42` ```val prems = goalw Set.thy [Ball_def] ``` clasohm@923 ` 43` ``` "[| !!x. x:A ==> P(x) |] ==> ! x:A. P(x)"; ``` clasohm@923 ` 44` ```by (REPEAT (ares_tac (prems @ [allI,impI]) 1)); ``` clasohm@923 ` 45` ```qed "ballI"; ``` clasohm@923 ` 46` clasohm@923 ` 47` ```val [major,minor] = goalw Set.thy [Ball_def] ``` clasohm@923 ` 48` ``` "[| ! x:A. P(x); x:A |] ==> P(x)"; ``` clasohm@923 ` 49` ```by (rtac (minor RS (major RS spec RS mp)) 1); ``` clasohm@923 ` 50` ```qed "bspec"; ``` clasohm@923 ` 51` clasohm@923 ` 52` ```val major::prems = goalw Set.thy [Ball_def] ``` clasohm@923 ` 53` ``` "[| ! x:A. P(x); P(x) ==> Q; x~:A ==> Q |] ==> Q"; ``` clasohm@923 ` 54` ```by (rtac (major RS spec RS impCE) 1); ``` clasohm@923 ` 55` ```by (REPEAT (eresolve_tac prems 1)); ``` clasohm@923 ` 56` ```qed "ballE"; ``` clasohm@923 ` 57` clasohm@923 ` 58` ```(*Takes assumptions ! x:A.P(x) and a:A; creates assumption P(a)*) ``` clasohm@923 ` 59` ```fun ball_tac i = etac ballE i THEN contr_tac (i+1); ``` clasohm@923 ` 60` paulson@2499 ` 61` ```AddSIs [ballI]; ``` paulson@2499 ` 62` ```AddEs [ballE]; ``` paulson@2499 ` 63` clasohm@923 ` 64` ```val prems = goalw Set.thy [Bex_def] ``` clasohm@923 ` 65` ``` "[| P(x); x:A |] ==> ? x:A. P(x)"; ``` clasohm@923 ` 66` ```by (REPEAT (ares_tac (prems @ [exI,conjI]) 1)); ``` clasohm@923 ` 67` ```qed "bexI"; ``` clasohm@923 ` 68` clasohm@923 ` 69` ```qed_goal "bexCI" Set.thy ``` wenzelm@3842 ` 70` ``` "[| ! x:A. ~P(x) ==> P(a); a:A |] ==> ? x:A. P(x)" ``` clasohm@923 ` 71` ``` (fn prems=> ``` clasohm@923 ` 72` ``` [ (rtac classical 1), ``` clasohm@923 ` 73` ``` (REPEAT (ares_tac (prems@[bexI,ballI,notI,notE]) 1)) ]); ``` clasohm@923 ` 74` clasohm@923 ` 75` ```val major::prems = goalw Set.thy [Bex_def] ``` clasohm@923 ` 76` ``` "[| ? x:A. P(x); !!x. [| x:A; P(x) |] ==> Q |] ==> Q"; ``` clasohm@923 ` 77` ```by (rtac (major RS exE) 1); ``` clasohm@923 ` 78` ```by (REPEAT (eresolve_tac (prems @ [asm_rl,conjE]) 1)); ``` clasohm@923 ` 79` ```qed "bexE"; ``` clasohm@923 ` 80` paulson@2499 ` 81` ```AddIs [bexI]; ``` paulson@2499 ` 82` ```AddSEs [bexE]; ``` paulson@2499 ` 83` paulson@3420 ` 84` ```(*Trival rewrite rule*) ``` wenzelm@3842 ` 85` ```goal Set.thy "(! x:A. P) = ((? x. x:A) --> P)"; ``` wenzelm@4089 ` 86` ```by (simp_tac (simpset() addsimps [Ball_def]) 1); ``` paulson@3420 ` 87` ```qed "ball_triv"; ``` paulson@1816 ` 88` paulson@1882 ` 89` ```(*Dual form for existentials*) ``` wenzelm@3842 ` 90` ```goal Set.thy "(? x:A. P) = ((? x. x:A) & P)"; ``` wenzelm@4089 ` 91` ```by (simp_tac (simpset() addsimps [Bex_def]) 1); ``` paulson@3420 ` 92` ```qed "bex_triv"; ``` paulson@1882 ` 93` paulson@3420 ` 94` ```Addsimps [ball_triv, bex_triv]; ``` clasohm@923 ` 95` clasohm@923 ` 96` ```(** Congruence rules **) ``` clasohm@923 ` 97` clasohm@923 ` 98` ```val prems = goal Set.thy ``` clasohm@923 ` 99` ``` "[| A=B; !!x. x:B ==> P(x) = Q(x) |] ==> \ ``` clasohm@923 ` 100` ```\ (! x:A. P(x)) = (! x:B. Q(x))"; ``` clasohm@923 ` 101` ```by (resolve_tac (prems RL [ssubst]) 1); ``` clasohm@923 ` 102` ```by (REPEAT (ares_tac [ballI,iffI] 1 ``` clasohm@923 ` 103` ``` ORELSE eresolve_tac ([make_elim bspec, mp] @ (prems RL [iffE])) 1)); ``` clasohm@923 ` 104` ```qed "ball_cong"; ``` clasohm@923 ` 105` clasohm@923 ` 106` ```val prems = goal Set.thy ``` clasohm@923 ` 107` ``` "[| A=B; !!x. x:B ==> P(x) = Q(x) |] ==> \ ``` clasohm@923 ` 108` ```\ (? x:A. P(x)) = (? x:B. Q(x))"; ``` clasohm@923 ` 109` ```by (resolve_tac (prems RL [ssubst]) 1); ``` clasohm@923 ` 110` ```by (REPEAT (etac bexE 1 ``` clasohm@923 ` 111` ``` ORELSE ares_tac ([bexI,iffI] @ (prems RL [iffD1,iffD2])) 1)); ``` clasohm@923 ` 112` ```qed "bex_cong"; ``` clasohm@923 ` 113` nipkow@1548 ` 114` ```section "Subsets"; ``` clasohm@923 ` 115` wenzelm@3842 ` 116` ```val prems = goalw Set.thy [subset_def] "(!!x. x:A ==> x:B) ==> A <= B"; ``` clasohm@923 ` 117` ```by (REPEAT (ares_tac (prems @ [ballI]) 1)); ``` clasohm@923 ` 118` ```qed "subsetI"; ``` clasohm@923 ` 119` paulson@4240 ` 120` ```Blast.overloaded ("op <=", domain_type); (*The <= relation is overloaded*) ``` paulson@4059 ` 121` paulson@4059 ` 122` ```(*While (:) is not, its type must be kept ``` paulson@4059 ` 123` ``` for overloading of = to work.*) ``` paulson@4240 ` 124` ```Blast.overloaded ("op :", domain_type); ``` paulson@4240 ` 125` ```seq (fn a => Blast.overloaded (a, HOLogic.dest_setT o domain_type)) ``` paulson@4059 ` 126` ``` ["Ball", "Bex"]; ``` paulson@4059 ` 127` ```(*need UNION, INTER also?*) ``` paulson@4059 ` 128` paulson@4469 ` 129` ```(*Image: retain the type of the set being expressed*) ``` paulson@4469 ` 130` ```Blast.overloaded ("op ``", domain_type o domain_type); ``` paulson@2881 ` 131` clasohm@923 ` 132` ```(*Rule in Modus Ponens style*) ``` clasohm@923 ` 133` ```val major::prems = goalw Set.thy [subset_def] "[| A <= B; c:A |] ==> c:B"; ``` clasohm@923 ` 134` ```by (rtac (major RS bspec) 1); ``` clasohm@923 ` 135` ```by (resolve_tac prems 1); ``` clasohm@923 ` 136` ```qed "subsetD"; ``` clasohm@923 ` 137` clasohm@923 ` 138` ```(*The same, with reversed premises for use with etac -- cf rev_mp*) ``` clasohm@923 ` 139` ```qed_goal "rev_subsetD" Set.thy "[| c:A; A <= B |] ==> c:B" ``` clasohm@923 ` 140` ``` (fn prems=> [ (REPEAT (resolve_tac (prems@[subsetD]) 1)) ]); ``` clasohm@923 ` 141` paulson@1920 ` 142` ```(*Converts A<=B to x:A ==> x:B*) ``` paulson@1920 ` 143` ```fun impOfSubs th = th RSN (2, rev_subsetD); ``` paulson@1920 ` 144` paulson@1841 ` 145` ```qed_goal "contra_subsetD" Set.thy "!!c. [| A <= B; c ~: B |] ==> c ~: A" ``` paulson@1841 ` 146` ``` (fn prems=> [ (REPEAT (eresolve_tac [asm_rl, contrapos, subsetD] 1)) ]); ``` paulson@1841 ` 147` paulson@1841 ` 148` ```qed_goal "rev_contra_subsetD" Set.thy "!!c. [| c ~: B; A <= B |] ==> c ~: A" ``` paulson@1841 ` 149` ``` (fn prems=> [ (REPEAT (eresolve_tac [asm_rl, contrapos, subsetD] 1)) ]); ``` paulson@1841 ` 150` clasohm@923 ` 151` ```(*Classical elimination rule*) ``` clasohm@923 ` 152` ```val major::prems = goalw Set.thy [subset_def] ``` clasohm@923 ` 153` ``` "[| A <= B; c~:A ==> P; c:B ==> P |] ==> P"; ``` clasohm@923 ` 154` ```by (rtac (major RS ballE) 1); ``` clasohm@923 ` 155` ```by (REPEAT (eresolve_tac prems 1)); ``` clasohm@923 ` 156` ```qed "subsetCE"; ``` clasohm@923 ` 157` clasohm@923 ` 158` ```(*Takes assumptions A<=B; c:A and creates the assumption c:B *) ``` clasohm@923 ` 159` ```fun set_mp_tac i = etac subsetCE i THEN mp_tac i; ``` clasohm@923 ` 160` paulson@2499 ` 161` ```AddSIs [subsetI]; ``` paulson@2499 ` 162` ```AddEs [subsetD, subsetCE]; ``` clasohm@923 ` 163` paulson@2499 ` 164` ```qed_goal "subset_refl" Set.thy "A <= (A::'a set)" ``` paulson@4059 ` 165` ``` (fn _=> [Fast_tac 1]); (*Blast_tac would try order_refl and fail*) ``` paulson@2499 ` 166` paulson@2499 ` 167` ```val prems = goal Set.thy "!!B. [| A<=B; B<=C |] ==> A<=(C::'a set)"; ``` paulson@2891 ` 168` ```by (Blast_tac 1); ``` clasohm@923 ` 169` ```qed "subset_trans"; ``` clasohm@923 ` 170` clasohm@923 ` 171` nipkow@1548 ` 172` ```section "Equality"; ``` clasohm@923 ` 173` clasohm@923 ` 174` ```(*Anti-symmetry of the subset relation*) ``` clasohm@923 ` 175` ```val prems = goal Set.thy "[| A <= B; B <= A |] ==> A = (B::'a set)"; ``` clasohm@923 ` 176` ```by (rtac (iffI RS set_ext) 1); ``` clasohm@923 ` 177` ```by (REPEAT (ares_tac (prems RL [subsetD]) 1)); ``` clasohm@923 ` 178` ```qed "subset_antisym"; ``` clasohm@923 ` 179` ```val equalityI = subset_antisym; ``` clasohm@923 ` 180` berghofe@1762 ` 181` ```AddSIs [equalityI]; ``` berghofe@1762 ` 182` clasohm@923 ` 183` ```(* Equality rules from ZF set theory -- are they appropriate here? *) ``` clasohm@923 ` 184` ```val prems = goal Set.thy "A = B ==> A<=(B::'a set)"; ``` clasohm@923 ` 185` ```by (resolve_tac (prems RL [subst]) 1); ``` clasohm@923 ` 186` ```by (rtac subset_refl 1); ``` clasohm@923 ` 187` ```qed "equalityD1"; ``` clasohm@923 ` 188` clasohm@923 ` 189` ```val prems = goal Set.thy "A = B ==> B<=(A::'a set)"; ``` clasohm@923 ` 190` ```by (resolve_tac (prems RL [subst]) 1); ``` clasohm@923 ` 191` ```by (rtac subset_refl 1); ``` clasohm@923 ` 192` ```qed "equalityD2"; ``` clasohm@923 ` 193` clasohm@923 ` 194` ```val prems = goal Set.thy ``` clasohm@923 ` 195` ``` "[| A = B; [| A<=B; B<=(A::'a set) |] ==> P |] ==> P"; ``` clasohm@923 ` 196` ```by (resolve_tac prems 1); ``` clasohm@923 ` 197` ```by (REPEAT (resolve_tac (prems RL [equalityD1,equalityD2]) 1)); ``` clasohm@923 ` 198` ```qed "equalityE"; ``` clasohm@923 ` 199` clasohm@923 ` 200` ```val major::prems = goal Set.thy ``` clasohm@923 ` 201` ``` "[| A = B; [| c:A; c:B |] ==> P; [| c~:A; c~:B |] ==> P |] ==> P"; ``` clasohm@923 ` 202` ```by (rtac (major RS equalityE) 1); ``` clasohm@923 ` 203` ```by (REPEAT (contr_tac 1 ORELSE eresolve_tac ([asm_rl,subsetCE]@prems) 1)); ``` clasohm@923 ` 204` ```qed "equalityCE"; ``` clasohm@923 ` 205` clasohm@923 ` 206` ```(*Lemma for creating induction formulae -- for "pattern matching" on p ``` clasohm@923 ` 207` ``` To make the induction hypotheses usable, apply "spec" or "bspec" to ``` clasohm@923 ` 208` ``` put universal quantifiers over the free variables in p. *) ``` clasohm@923 ` 209` ```val prems = goal Set.thy ``` clasohm@923 ` 210` ``` "[| p:A; !!z. z:A ==> p=z --> R |] ==> R"; ``` clasohm@923 ` 211` ```by (rtac mp 1); ``` clasohm@923 ` 212` ```by (REPEAT (resolve_tac (refl::prems) 1)); ``` clasohm@923 ` 213` ```qed "setup_induction"; ``` clasohm@923 ` 214` clasohm@923 ` 215` paulson@4159 ` 216` ```section "The universal set -- UNIV"; ``` paulson@4159 ` 217` paulson@4159 ` 218` ```qed_goalw "UNIV_I" Set.thy [UNIV_def] "x : UNIV" ``` paulson@4159 ` 219` ``` (fn _ => [rtac CollectI 1, rtac TrueI 1]); ``` paulson@4159 ` 220` paulson@4434 ` 221` ```Addsimps [UNIV_I]; ``` paulson@4434 ` 222` ```AddIs [UNIV_I]; (*unsafe makes it less likely to cause problems*) ``` paulson@4159 ` 223` paulson@4159 ` 224` ```qed_goal "subset_UNIV" Set.thy "A <= UNIV" ``` paulson@4159 ` 225` ``` (fn _ => [rtac subsetI 1, rtac UNIV_I 1]); ``` paulson@4159 ` 226` paulson@4159 ` 227` ```(** Eta-contracting these two rules (to remove P) causes them to be ignored ``` paulson@4159 ` 228` ``` because of their interaction with congruence rules. **) ``` paulson@4159 ` 229` paulson@4159 ` 230` ```goalw Set.thy [Ball_def] "Ball UNIV P = All P"; ``` paulson@4159 ` 231` ```by (Simp_tac 1); ``` paulson@4159 ` 232` ```qed "ball_UNIV"; ``` paulson@4159 ` 233` paulson@4159 ` 234` ```goalw Set.thy [Bex_def] "Bex UNIV P = Ex P"; ``` paulson@4159 ` 235` ```by (Simp_tac 1); ``` paulson@4159 ` 236` ```qed "bex_UNIV"; ``` paulson@4159 ` 237` ```Addsimps [ball_UNIV, bex_UNIV]; ``` paulson@4159 ` 238` paulson@4159 ` 239` paulson@2858 ` 240` ```section "The empty set -- {}"; ``` paulson@2858 ` 241` paulson@2858 ` 242` ```qed_goalw "empty_iff" Set.thy [empty_def] "(c : {}) = False" ``` paulson@2891 ` 243` ``` (fn _ => [ (Blast_tac 1) ]); ``` paulson@2858 ` 244` paulson@2858 ` 245` ```Addsimps [empty_iff]; ``` paulson@2858 ` 246` paulson@2858 ` 247` ```qed_goal "emptyE" Set.thy "!!a. a:{} ==> P" ``` paulson@2858 ` 248` ``` (fn _ => [Full_simp_tac 1]); ``` paulson@2858 ` 249` paulson@2858 ` 250` ```AddSEs [emptyE]; ``` paulson@2858 ` 251` paulson@2858 ` 252` ```qed_goal "empty_subsetI" Set.thy "{} <= A" ``` paulson@2891 ` 253` ``` (fn _ => [ (Blast_tac 1) ]); ``` paulson@2858 ` 254` paulson@2858 ` 255` ```qed_goal "equals0I" Set.thy "[| !!y. y:A ==> False |] ==> A={}" ``` paulson@2858 ` 256` ``` (fn [prem]=> ``` wenzelm@4089 ` 257` ``` [ (blast_tac (claset() addIs [prem RS FalseE]) 1) ]); ``` paulson@2858 ` 258` paulson@2858 ` 259` ```qed_goal "equals0D" Set.thy "!!a. [| A={}; a:A |] ==> P" ``` paulson@2891 ` 260` ``` (fn _ => [ (Blast_tac 1) ]); ``` paulson@2858 ` 261` paulson@4159 ` 262` ```goalw Set.thy [Ball_def] "Ball {} P = True"; ``` paulson@4159 ` 263` ```by (Simp_tac 1); ``` paulson@4159 ` 264` ```qed "ball_empty"; ``` paulson@4159 ` 265` paulson@4159 ` 266` ```goalw Set.thy [Bex_def] "Bex {} P = False"; ``` paulson@4159 ` 267` ```by (Simp_tac 1); ``` paulson@4159 ` 268` ```qed "bex_empty"; ``` paulson@4159 ` 269` ```Addsimps [ball_empty, bex_empty]; ``` paulson@4159 ` 270` paulson@4159 ` 271` ```goal thy "UNIV ~= {}"; ``` paulson@4159 ` 272` ```by (blast_tac (claset() addEs [equalityE]) 1); ``` paulson@4159 ` 273` ```qed "UNIV_not_empty"; ``` paulson@4159 ` 274` ```AddIffs [UNIV_not_empty]; ``` paulson@4159 ` 275` paulson@4159 ` 276` paulson@2858 ` 277` paulson@2858 ` 278` ```section "The Powerset operator -- Pow"; ``` paulson@2858 ` 279` paulson@2858 ` 280` ```qed_goalw "Pow_iff" Set.thy [Pow_def] "(A : Pow(B)) = (A <= B)" ``` paulson@2858 ` 281` ``` (fn _ => [ (Asm_simp_tac 1) ]); ``` paulson@2858 ` 282` paulson@2858 ` 283` ```AddIffs [Pow_iff]; ``` paulson@2858 ` 284` paulson@2858 ` 285` ```qed_goalw "PowI" Set.thy [Pow_def] "!!A B. A <= B ==> A : Pow(B)" ``` paulson@2858 ` 286` ``` (fn _ => [ (etac CollectI 1) ]); ``` paulson@2858 ` 287` paulson@2858 ` 288` ```qed_goalw "PowD" Set.thy [Pow_def] "!!A B. A : Pow(B) ==> A<=B" ``` paulson@2858 ` 289` ``` (fn _=> [ (etac CollectD 1) ]); ``` paulson@2858 ` 290` paulson@2858 ` 291` ```val Pow_bottom = empty_subsetI RS PowI; (* {}: Pow(B) *) ``` paulson@2858 ` 292` ```val Pow_top = subset_refl RS PowI; (* A : Pow(A) *) ``` paulson@2858 ` 293` paulson@2858 ` 294` nipkow@1548 ` 295` ```section "Set complement -- Compl"; ``` clasohm@923 ` 296` paulson@2499 ` 297` ```qed_goalw "Compl_iff" Set.thy [Compl_def] "(c : Compl(A)) = (c~:A)" ``` paulson@2891 ` 298` ``` (fn _ => [ (Blast_tac 1) ]); ``` paulson@2499 ` 299` paulson@2499 ` 300` ```Addsimps [Compl_iff]; ``` paulson@2499 ` 301` clasohm@923 ` 302` ```val prems = goalw Set.thy [Compl_def] ``` clasohm@923 ` 303` ``` "[| c:A ==> False |] ==> c : Compl(A)"; ``` clasohm@923 ` 304` ```by (REPEAT (ares_tac (prems @ [CollectI,notI]) 1)); ``` clasohm@923 ` 305` ```qed "ComplI"; ``` clasohm@923 ` 306` clasohm@923 ` 307` ```(*This form, with negated conclusion, works well with the Classical prover. ``` clasohm@923 ` 308` ``` Negated assumptions behave like formulae on the right side of the notional ``` clasohm@923 ` 309` ``` turnstile...*) ``` clasohm@923 ` 310` ```val major::prems = goalw Set.thy [Compl_def] ``` paulson@2499 ` 311` ``` "c : Compl(A) ==> c~:A"; ``` clasohm@923 ` 312` ```by (rtac (major RS CollectD) 1); ``` clasohm@923 ` 313` ```qed "ComplD"; ``` clasohm@923 ` 314` clasohm@923 ` 315` ```val ComplE = make_elim ComplD; ``` clasohm@923 ` 316` paulson@2499 ` 317` ```AddSIs [ComplI]; ``` paulson@2499 ` 318` ```AddSEs [ComplE]; ``` paulson@1640 ` 319` clasohm@923 ` 320` nipkow@1548 ` 321` ```section "Binary union -- Un"; ``` clasohm@923 ` 322` paulson@2499 ` 323` ```qed_goalw "Un_iff" Set.thy [Un_def] "(c : A Un B) = (c:A | c:B)" ``` paulson@2891 ` 324` ``` (fn _ => [ Blast_tac 1 ]); ``` paulson@2499 ` 325` paulson@2499 ` 326` ```Addsimps [Un_iff]; ``` paulson@2499 ` 327` paulson@2499 ` 328` ```goal Set.thy "!!c. c:A ==> c : A Un B"; ``` paulson@2499 ` 329` ```by (Asm_simp_tac 1); ``` clasohm@923 ` 330` ```qed "UnI1"; ``` clasohm@923 ` 331` paulson@2499 ` 332` ```goal Set.thy "!!c. c:B ==> c : A Un B"; ``` paulson@2499 ` 333` ```by (Asm_simp_tac 1); ``` clasohm@923 ` 334` ```qed "UnI2"; ``` clasohm@923 ` 335` clasohm@923 ` 336` ```(*Classical introduction rule: no commitment to A vs B*) ``` clasohm@923 ` 337` ```qed_goal "UnCI" Set.thy "(c~:B ==> c:A) ==> c : A Un B" ``` clasohm@923 ` 338` ``` (fn prems=> ``` paulson@2499 ` 339` ``` [ (Simp_tac 1), ``` paulson@2499 ` 340` ``` (REPEAT (ares_tac (prems@[disjCI]) 1)) ]); ``` clasohm@923 ` 341` clasohm@923 ` 342` ```val major::prems = goalw Set.thy [Un_def] ``` clasohm@923 ` 343` ``` "[| c : A Un B; c:A ==> P; c:B ==> P |] ==> P"; ``` clasohm@923 ` 344` ```by (rtac (major RS CollectD RS disjE) 1); ``` clasohm@923 ` 345` ```by (REPEAT (eresolve_tac prems 1)); ``` clasohm@923 ` 346` ```qed "UnE"; ``` clasohm@923 ` 347` paulson@2499 ` 348` ```AddSIs [UnCI]; ``` paulson@2499 ` 349` ```AddSEs [UnE]; ``` paulson@1640 ` 350` clasohm@923 ` 351` nipkow@1548 ` 352` ```section "Binary intersection -- Int"; ``` clasohm@923 ` 353` paulson@2499 ` 354` ```qed_goalw "Int_iff" Set.thy [Int_def] "(c : A Int B) = (c:A & c:B)" ``` paulson@2891 ` 355` ``` (fn _ => [ (Blast_tac 1) ]); ``` paulson@2499 ` 356` paulson@2499 ` 357` ```Addsimps [Int_iff]; ``` paulson@2499 ` 358` paulson@2499 ` 359` ```goal Set.thy "!!c. [| c:A; c:B |] ==> c : A Int B"; ``` paulson@2499 ` 360` ```by (Asm_simp_tac 1); ``` clasohm@923 ` 361` ```qed "IntI"; ``` clasohm@923 ` 362` paulson@2499 ` 363` ```goal Set.thy "!!c. c : A Int B ==> c:A"; ``` paulson@2499 ` 364` ```by (Asm_full_simp_tac 1); ``` clasohm@923 ` 365` ```qed "IntD1"; ``` clasohm@923 ` 366` paulson@2499 ` 367` ```goal Set.thy "!!c. c : A Int B ==> c:B"; ``` paulson@2499 ` 368` ```by (Asm_full_simp_tac 1); ``` clasohm@923 ` 369` ```qed "IntD2"; ``` clasohm@923 ` 370` clasohm@923 ` 371` ```val [major,minor] = goal Set.thy ``` clasohm@923 ` 372` ``` "[| c : A Int B; [| c:A; c:B |] ==> P |] ==> P"; ``` clasohm@923 ` 373` ```by (rtac minor 1); ``` clasohm@923 ` 374` ```by (rtac (major RS IntD1) 1); ``` clasohm@923 ` 375` ```by (rtac (major RS IntD2) 1); ``` clasohm@923 ` 376` ```qed "IntE"; ``` clasohm@923 ` 377` paulson@2499 ` 378` ```AddSIs [IntI]; ``` paulson@2499 ` 379` ```AddSEs [IntE]; ``` clasohm@923 ` 380` nipkow@1548 ` 381` ```section "Set difference"; ``` clasohm@923 ` 382` paulson@2499 ` 383` ```qed_goalw "Diff_iff" Set.thy [set_diff_def] "(c : A-B) = (c:A & c~:B)" ``` paulson@2891 ` 384` ``` (fn _ => [ (Blast_tac 1) ]); ``` clasohm@923 ` 385` paulson@2499 ` 386` ```Addsimps [Diff_iff]; ``` paulson@2499 ` 387` paulson@2499 ` 388` ```qed_goal "DiffI" Set.thy "!!c. [| c : A; c ~: B |] ==> c : A - B" ``` paulson@2499 ` 389` ``` (fn _=> [ Asm_simp_tac 1 ]); ``` clasohm@923 ` 390` paulson@2499 ` 391` ```qed_goal "DiffD1" Set.thy "!!c. c : A - B ==> c : A" ``` paulson@2499 ` 392` ``` (fn _=> [ (Asm_full_simp_tac 1) ]); ``` clasohm@923 ` 393` paulson@2499 ` 394` ```qed_goal "DiffD2" Set.thy "!!c. [| c : A - B; c : B |] ==> P" ``` paulson@2499 ` 395` ``` (fn _=> [ (Asm_full_simp_tac 1) ]); ``` paulson@2499 ` 396` paulson@2499 ` 397` ```qed_goal "DiffE" Set.thy "[| c : A - B; [| c:A; c~:B |] ==> P |] ==> P" ``` clasohm@923 ` 398` ``` (fn prems=> ``` clasohm@923 ` 399` ``` [ (resolve_tac prems 1), ``` clasohm@923 ` 400` ``` (REPEAT (ares_tac (prems RL [DiffD1, DiffD2 RS notI]) 1)) ]); ``` clasohm@923 ` 401` paulson@2499 ` 402` ```AddSIs [DiffI]; ``` paulson@2499 ` 403` ```AddSEs [DiffE]; ``` clasohm@923 ` 404` clasohm@923 ` 405` nipkow@1548 ` 406` ```section "Augmenting a set -- insert"; ``` clasohm@923 ` 407` paulson@2499 ` 408` ```qed_goalw "insert_iff" Set.thy [insert_def] "a : insert b A = (a=b | a:A)" ``` paulson@2891 ` 409` ``` (fn _ => [Blast_tac 1]); ``` paulson@2499 ` 410` paulson@2499 ` 411` ```Addsimps [insert_iff]; ``` clasohm@923 ` 412` paulson@2499 ` 413` ```qed_goal "insertI1" Set.thy "a : insert a B" ``` paulson@2499 ` 414` ``` (fn _ => [Simp_tac 1]); ``` paulson@2499 ` 415` paulson@2499 ` 416` ```qed_goal "insertI2" Set.thy "!!a. a : B ==> a : insert b B" ``` paulson@2499 ` 417` ``` (fn _=> [Asm_simp_tac 1]); ``` clasohm@923 ` 418` clasohm@923 ` 419` ```qed_goalw "insertE" Set.thy [insert_def] ``` clasohm@923 ` 420` ``` "[| a : insert b A; a=b ==> P; a:A ==> P |] ==> P" ``` clasohm@923 ` 421` ``` (fn major::prems=> ``` clasohm@923 ` 422` ``` [ (rtac (major RS UnE) 1), ``` clasohm@923 ` 423` ``` (REPEAT (eresolve_tac (prems @ [CollectE]) 1)) ]); ``` clasohm@923 ` 424` clasohm@923 ` 425` ```(*Classical introduction rule*) ``` clasohm@923 ` 426` ```qed_goal "insertCI" Set.thy "(a~:B ==> a=b) ==> a: insert b B" ``` paulson@2499 ` 427` ``` (fn prems=> ``` paulson@2499 ` 428` ``` [ (Simp_tac 1), ``` paulson@2499 ` 429` ``` (REPEAT (ares_tac (prems@[disjCI]) 1)) ]); ``` paulson@2499 ` 430` paulson@2499 ` 431` ```AddSIs [insertCI]; ``` paulson@2499 ` 432` ```AddSEs [insertE]; ``` clasohm@923 ` 433` nipkow@1548 ` 434` ```section "Singletons, using insert"; ``` clasohm@923 ` 435` clasohm@923 ` 436` ```qed_goal "singletonI" Set.thy "a : {a}" ``` clasohm@923 ` 437` ``` (fn _=> [ (rtac insertI1 1) ]); ``` clasohm@923 ` 438` paulson@2499 ` 439` ```goal Set.thy "!!a. b : {a} ==> b=a"; ``` paulson@2891 ` 440` ```by (Blast_tac 1); ``` clasohm@923 ` 441` ```qed "singletonD"; ``` clasohm@923 ` 442` oheimb@1776 ` 443` ```bind_thm ("singletonE", make_elim singletonD); ``` oheimb@1776 ` 444` paulson@2499 ` 445` ```qed_goal "singleton_iff" thy "(b : {a}) = (b=a)" ``` paulson@2891 ` 446` ```(fn _ => [Blast_tac 1]); ``` clasohm@923 ` 447` paulson@2499 ` 448` ```goal Set.thy "!!a b. {a}={b} ==> a=b"; ``` wenzelm@4089 ` 449` ```by (blast_tac (claset() addEs [equalityE]) 1); ``` clasohm@923 ` 450` ```qed "singleton_inject"; ``` clasohm@923 ` 451` paulson@2858 ` 452` ```(*Redundant? But unlike insertCI, it proves the subgoal immediately!*) ``` paulson@2858 ` 453` ```AddSIs [singletonI]; ``` paulson@2499 ` 454` ```AddSDs [singleton_inject]; ``` paulson@3718 ` 455` ```AddSEs [singletonE]; ``` paulson@2499 ` 456` wenzelm@3842 ` 457` ```goal Set.thy "{x. x=a} = {a}"; ``` wenzelm@4423 ` 458` ```by (Blast_tac 1); ``` nipkow@3582 ` 459` ```qed "singleton_conv"; ``` nipkow@3582 ` 460` ```Addsimps [singleton_conv]; ``` nipkow@1531 ` 461` nipkow@1531 ` 462` nipkow@1548 ` 463` ```section "Unions of families -- UNION x:A. B(x) is Union(B``A)"; ``` clasohm@923 ` 464` paulson@2499 ` 465` ```goalw Set.thy [UNION_def] "(b: (UN x:A. B(x))) = (EX x:A. b: B(x))"; ``` paulson@2891 ` 466` ```by (Blast_tac 1); ``` paulson@2499 ` 467` ```qed "UN_iff"; ``` paulson@2499 ` 468` paulson@2499 ` 469` ```Addsimps [UN_iff]; ``` paulson@2499 ` 470` clasohm@923 ` 471` ```(*The order of the premises presupposes that A is rigid; b may be flexible*) ``` paulson@2499 ` 472` ```goal Set.thy "!!b. [| a:A; b: B(a) |] ==> b: (UN x:A. B(x))"; ``` paulson@4477 ` 473` ```by Auto_tac; ``` clasohm@923 ` 474` ```qed "UN_I"; ``` clasohm@923 ` 475` clasohm@923 ` 476` ```val major::prems = goalw Set.thy [UNION_def] ``` clasohm@923 ` 477` ``` "[| b : (UN x:A. B(x)); !!x.[| x:A; b: B(x) |] ==> R |] ==> R"; ``` clasohm@923 ` 478` ```by (rtac (major RS CollectD RS bexE) 1); ``` clasohm@923 ` 479` ```by (REPEAT (ares_tac prems 1)); ``` clasohm@923 ` 480` ```qed "UN_E"; ``` clasohm@923 ` 481` paulson@2499 ` 482` ```AddIs [UN_I]; ``` paulson@2499 ` 483` ```AddSEs [UN_E]; ``` paulson@2499 ` 484` clasohm@923 ` 485` ```val prems = goal Set.thy ``` clasohm@923 ` 486` ``` "[| A=B; !!x. x:B ==> C(x) = D(x) |] ==> \ ``` clasohm@923 ` 487` ```\ (UN x:A. C(x)) = (UN x:B. D(x))"; ``` clasohm@923 ` 488` ```by (REPEAT (etac UN_E 1 ``` clasohm@923 ` 489` ``` ORELSE ares_tac ([UN_I,equalityI,subsetI] @ ``` clasohm@1465 ` 490` ``` (prems RL [equalityD1,equalityD2] RL [subsetD])) 1)); ``` clasohm@923 ` 491` ```qed "UN_cong"; ``` clasohm@923 ` 492` clasohm@923 ` 493` nipkow@1548 ` 494` ```section "Intersections of families -- INTER x:A. B(x) is Inter(B``A)"; ``` clasohm@923 ` 495` paulson@2499 ` 496` ```goalw Set.thy [INTER_def] "(b: (INT x:A. B(x))) = (ALL x:A. b: B(x))"; ``` paulson@4477 ` 497` ```by Auto_tac; ``` paulson@2499 ` 498` ```qed "INT_iff"; ``` paulson@2499 ` 499` paulson@2499 ` 500` ```Addsimps [INT_iff]; ``` paulson@2499 ` 501` clasohm@923 ` 502` ```val prems = goalw Set.thy [INTER_def] ``` clasohm@923 ` 503` ``` "(!!x. x:A ==> b: B(x)) ==> b : (INT x:A. B(x))"; ``` clasohm@923 ` 504` ```by (REPEAT (ares_tac ([CollectI,ballI] @ prems) 1)); ``` clasohm@923 ` 505` ```qed "INT_I"; ``` clasohm@923 ` 506` paulson@2499 ` 507` ```goal Set.thy "!!b. [| b : (INT x:A. B(x)); a:A |] ==> b: B(a)"; ``` paulson@4477 ` 508` ```by Auto_tac; ``` clasohm@923 ` 509` ```qed "INT_D"; ``` clasohm@923 ` 510` clasohm@923 ` 511` ```(*"Classical" elimination -- by the Excluded Middle on a:A *) ``` clasohm@923 ` 512` ```val major::prems = goalw Set.thy [INTER_def] ``` clasohm@923 ` 513` ``` "[| b : (INT x:A. B(x)); b: B(a) ==> R; a~:A ==> R |] ==> R"; ``` clasohm@923 ` 514` ```by (rtac (major RS CollectD RS ballE) 1); ``` clasohm@923 ` 515` ```by (REPEAT (eresolve_tac prems 1)); ``` clasohm@923 ` 516` ```qed "INT_E"; ``` clasohm@923 ` 517` paulson@2499 ` 518` ```AddSIs [INT_I]; ``` paulson@2499 ` 519` ```AddEs [INT_D, INT_E]; ``` paulson@2499 ` 520` clasohm@923 ` 521` ```val prems = goal Set.thy ``` clasohm@923 ` 522` ``` "[| A=B; !!x. x:B ==> C(x) = D(x) |] ==> \ ``` clasohm@923 ` 523` ```\ (INT x:A. C(x)) = (INT x:B. D(x))"; ``` clasohm@923 ` 524` ```by (REPEAT_FIRST (resolve_tac [INT_I,equalityI,subsetI])); ``` clasohm@923 ` 525` ```by (REPEAT (dtac INT_D 1 ``` clasohm@923 ` 526` ``` ORELSE ares_tac (prems RL [equalityD1,equalityD2] RL [subsetD]) 1)); ``` clasohm@923 ` 527` ```qed "INT_cong"; ``` clasohm@923 ` 528` clasohm@923 ` 529` nipkow@1548 ` 530` ```section "Union"; ``` clasohm@923 ` 531` paulson@2499 ` 532` ```goalw Set.thy [Union_def] "(A : Union(C)) = (EX X:C. A:X)"; ``` paulson@2891 ` 533` ```by (Blast_tac 1); ``` paulson@2499 ` 534` ```qed "Union_iff"; ``` paulson@2499 ` 535` paulson@2499 ` 536` ```Addsimps [Union_iff]; ``` paulson@2499 ` 537` clasohm@923 ` 538` ```(*The order of the premises presupposes that C is rigid; A may be flexible*) ``` paulson@2499 ` 539` ```goal Set.thy "!!X. [| X:C; A:X |] ==> A : Union(C)"; ``` paulson@4477 ` 540` ```by Auto_tac; ``` clasohm@923 ` 541` ```qed "UnionI"; ``` clasohm@923 ` 542` clasohm@923 ` 543` ```val major::prems = goalw Set.thy [Union_def] ``` clasohm@923 ` 544` ``` "[| A : Union(C); !!X.[| A:X; X:C |] ==> R |] ==> R"; ``` clasohm@923 ` 545` ```by (rtac (major RS UN_E) 1); ``` clasohm@923 ` 546` ```by (REPEAT (ares_tac prems 1)); ``` clasohm@923 ` 547` ```qed "UnionE"; ``` clasohm@923 ` 548` paulson@2499 ` 549` ```AddIs [UnionI]; ``` paulson@2499 ` 550` ```AddSEs [UnionE]; ``` paulson@2499 ` 551` paulson@2499 ` 552` nipkow@1548 ` 553` ```section "Inter"; ``` clasohm@923 ` 554` paulson@2499 ` 555` ```goalw Set.thy [Inter_def] "(A : Inter(C)) = (ALL X:C. A:X)"; ``` paulson@2891 ` 556` ```by (Blast_tac 1); ``` paulson@2499 ` 557` ```qed "Inter_iff"; ``` paulson@2499 ` 558` paulson@2499 ` 559` ```Addsimps [Inter_iff]; ``` paulson@2499 ` 560` clasohm@923 ` 561` ```val prems = goalw Set.thy [Inter_def] ``` clasohm@923 ` 562` ``` "[| !!X. X:C ==> A:X |] ==> A : Inter(C)"; ``` clasohm@923 ` 563` ```by (REPEAT (ares_tac ([INT_I] @ prems) 1)); ``` clasohm@923 ` 564` ```qed "InterI"; ``` clasohm@923 ` 565` clasohm@923 ` 566` ```(*A "destruct" rule -- every X in C contains A as an element, but ``` clasohm@923 ` 567` ``` A:X can hold when X:C does not! This rule is analogous to "spec". *) ``` paulson@2499 ` 568` ```goal Set.thy "!!X. [| A : Inter(C); X:C |] ==> A:X"; ``` paulson@4477 ` 569` ```by Auto_tac; ``` clasohm@923 ` 570` ```qed "InterD"; ``` clasohm@923 ` 571` clasohm@923 ` 572` ```(*"Classical" elimination rule -- does not require proving X:C *) ``` clasohm@923 ` 573` ```val major::prems = goalw Set.thy [Inter_def] ``` paulson@2721 ` 574` ``` "[| A : Inter(C); X~:C ==> R; A:X ==> R |] ==> R"; ``` clasohm@923 ` 575` ```by (rtac (major RS INT_E) 1); ``` clasohm@923 ` 576` ```by (REPEAT (eresolve_tac prems 1)); ``` clasohm@923 ` 577` ```qed "InterE"; ``` clasohm@923 ` 578` paulson@2499 ` 579` ```AddSIs [InterI]; ``` paulson@2499 ` 580` ```AddEs [InterD, InterE]; ``` paulson@2499 ` 581` paulson@2499 ` 582` nipkow@2912 ` 583` ```(*** Image of a set under a function ***) ``` nipkow@2912 ` 584` nipkow@2912 ` 585` ```(*Frequently b does not have the syntactic form of f(x).*) ``` nipkow@2912 ` 586` ```val prems = goalw thy [image_def] "[| b=f(x); x:A |] ==> b : f``A"; ``` nipkow@2912 ` 587` ```by (REPEAT (resolve_tac (prems @ [CollectI,bexI,prem]) 1)); ``` nipkow@2912 ` 588` ```qed "image_eqI"; ``` nipkow@3909 ` 589` ```Addsimps [image_eqI]; ``` nipkow@2912 ` 590` nipkow@2912 ` 591` ```bind_thm ("imageI", refl RS image_eqI); ``` nipkow@2912 ` 592` nipkow@2912 ` 593` ```(*The eta-expansion gives variable-name preservation.*) ``` nipkow@2912 ` 594` ```val major::prems = goalw thy [image_def] ``` wenzelm@3842 ` 595` ``` "[| b : (%x. f(x))``A; !!x.[| b=f(x); x:A |] ==> P |] ==> P"; ``` nipkow@2912 ` 596` ```by (rtac (major RS CollectD RS bexE) 1); ``` nipkow@2912 ` 597` ```by (REPEAT (ares_tac prems 1)); ``` nipkow@2912 ` 598` ```qed "imageE"; ``` nipkow@2912 ` 599` nipkow@2912 ` 600` ```AddIs [image_eqI]; ``` nipkow@2912 ` 601` ```AddSEs [imageE]; ``` nipkow@2912 ` 602` nipkow@2912 ` 603` ```goalw thy [o_def] "(f o g)``r = f``(g``r)"; ``` paulson@2935 ` 604` ```by (Blast_tac 1); ``` nipkow@2912 ` 605` ```qed "image_compose"; ``` nipkow@2912 ` 606` nipkow@2912 ` 607` ```goal thy "f``(A Un B) = f``A Un f``B"; ``` paulson@2935 ` 608` ```by (Blast_tac 1); ``` nipkow@2912 ` 609` ```qed "image_Un"; ``` nipkow@2912 ` 610` paulson@4510 ` 611` ```goal thy "(z : f``A) = (EX x:A. z = f x)"; ``` paulson@3960 ` 612` ```by (Blast_tac 1); ``` paulson@3960 ` 613` ```qed "image_iff"; ``` paulson@3960 ` 614` paulson@4523 ` 615` ```(*This rewrite rule would confuse users if made default.*) ``` paulson@4523 ` 616` ```goal thy "(f``A <= B) = (ALL x:A. f(x): B)"; ``` paulson@4523 ` 617` ```by (Blast_tac 1); ``` paulson@4523 ` 618` ```qed "image_subset_iff"; ``` paulson@4523 ` 619` paulson@4523 ` 620` ```(*Replaces the three steps subsetI, imageE, hyp_subst_tac, but breaks too ``` paulson@4523 ` 621` ``` many existing proofs.*) ``` paulson@4510 ` 622` ```val prems = goal thy "(!!x. x:A ==> f(x) : B) ==> f``A <= B"; ``` paulson@4510 ` 623` ```by (blast_tac (claset() addIs prems) 1); ``` paulson@4510 ` 624` ```qed "image_subsetI"; ``` paulson@4510 ` 625` nipkow@2912 ` 626` nipkow@2912 ` 627` ```(*** Range of a function -- just a translation for image! ***) ``` nipkow@2912 ` 628` nipkow@2912 ` 629` ```goal thy "!!b. b=f(x) ==> b : range(f)"; ``` nipkow@2912 ` 630` ```by (EVERY1 [etac image_eqI, rtac UNIV_I]); ``` nipkow@2912 ` 631` ```bind_thm ("range_eqI", UNIV_I RSN (2,image_eqI)); ``` nipkow@2912 ` 632` nipkow@2912 ` 633` ```bind_thm ("rangeI", UNIV_I RS imageI); ``` nipkow@2912 ` 634` nipkow@2912 ` 635` ```val [major,minor] = goal thy ``` wenzelm@3842 ` 636` ``` "[| b : range(%x. f(x)); !!x. b=f(x) ==> P |] ==> P"; ``` nipkow@2912 ` 637` ```by (rtac (major RS imageE) 1); ``` nipkow@2912 ` 638` ```by (etac minor 1); ``` nipkow@2912 ` 639` ```qed "rangeE"; ``` nipkow@2912 ` 640` oheimb@1776 ` 641` oheimb@1776 ` 642` ```(*** Set reasoning tools ***) ``` oheimb@1776 ` 643` oheimb@1776 ` 644` paulson@3912 ` 645` ```(** Rewrite rules for boolean case-splitting: faster than ``` nipkow@4830 ` 646` ``` addsplits[split_if] ``` paulson@3912 ` 647` ```**) ``` paulson@3912 ` 648` nipkow@4830 ` 649` ```bind_thm ("split_if_eq1", read_instantiate [("P", "%x. x = ?b")] split_if); ``` nipkow@4830 ` 650` ```bind_thm ("split_if_eq2", read_instantiate [("P", "%x. ?a = x")] split_if); ``` paulson@3912 ` 651` nipkow@4830 ` 652` ```bind_thm ("split_if_mem1", ``` nipkow@4830 ` 653` ``` read_instantiate_sg (sign_of Set.thy) [("P", "%x. x : ?b")] split_if); ``` nipkow@4830 ` 654` ```bind_thm ("split_if_mem2", ``` nipkow@4830 ` 655` ``` read_instantiate_sg (sign_of Set.thy) [("P", "%x. ?a : x")] split_if); ``` paulson@3912 ` 656` nipkow@4830 ` 657` ```val split_ifs = [if_bool_eq_conj, split_if_eq1, split_if_eq2, ``` nipkow@4830 ` 658` ``` split_if_mem1, split_if_mem2]; ``` paulson@3912 ` 659` paulson@3912 ` 660` wenzelm@4089 ` 661` ```(*Each of these has ALREADY been added to simpset() above.*) ``` paulson@2024 ` 662` ```val mem_simps = [insert_iff, empty_iff, Un_iff, Int_iff, Compl_iff, Diff_iff, ``` paulson@4159 ` 663` ``` mem_Collect_eq, UN_iff, Union_iff, INT_iff, Inter_iff]; ``` oheimb@1776 ` 664` paulson@1937 ` 665` ```(*Not for Addsimps -- it can cause goals to blow up!*) ``` paulson@1937 ` 666` ```goal Set.thy "(a : (if Q then x else y)) = ((Q --> a:x) & (~Q --> a : y))"; ``` nipkow@4686 ` 667` ```by (Simp_tac 1); ``` paulson@1937 ` 668` ```qed "mem_if"; ``` paulson@1937 ` 669` oheimb@1776 ` 670` ```val mksimps_pairs = ("Ball",[bspec]) :: mksimps_pairs; ``` oheimb@1776 ` 671` wenzelm@4089 ` 672` ```simpset_ref() := simpset() addcongs [ball_cong,bex_cong] ``` oheimb@1776 ` 673` ``` setmksimps (mksimps mksimps_pairs); ``` nipkow@3222 ` 674` nipkow@3222 ` 675` ```Addsimps[subset_UNIV, empty_subsetI, subset_refl]; ``` nipkow@3222 ` 676` nipkow@3222 ` 677` nipkow@3222 ` 678` ```(*** < ***) ``` nipkow@3222 ` 679` nipkow@3222 ` 680` ```goalw Set.thy [psubset_def] "!!A::'a set. [| A <= B; A ~= B |] ==> A (x ~: A) & A<=B | x:A & A-{x}