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src/HOL/TLA/Intensional.thy

author | haftmann |

Fri Nov 27 08:41:10 2009 +0100 (2009-11-27) | |

changeset 33963 | 977b94b64905 |

parent 31945 | d5f186aa0bed |

child 35108 | e384e27c229f |

permissions | -rw-r--r-- |

renamed former datatype.ML to datatype_data.ML; datatype.ML provides uniform view on datatype.ML and datatype_rep_proofs.ML

1 (*

2 File: TLA/Intensional.thy

3 ID: $Id$

4 Author: Stephan Merz

5 Copyright: 1998 University of Munich

6 *)

8 header {* A framework for "intensional" (possible-world based) logics

9 on top of HOL, with lifting of constants and functions *}

11 theory Intensional

12 imports Main

13 begin

15 axclass

16 world < type

18 (** abstract syntax **)

20 types

21 ('w,'a) expr = "'w => 'a" (* intention: 'w::world, 'a::type *)

22 'w form = "('w, bool) expr"

24 consts

25 Valid :: "('w::world) form => bool"

26 const :: "'a => ('w::world, 'a) expr"

27 lift :: "['a => 'b, ('w::world, 'a) expr] => ('w,'b) expr"

28 lift2 :: "['a => 'b => 'c, ('w::world,'a) expr, ('w,'b) expr] => ('w,'c) expr"

29 lift3 :: "['a => 'b => 'c => 'd, ('w::world,'a) expr, ('w,'b) expr, ('w,'c) expr] => ('w,'d) expr"

31 (* "Rigid" quantification (logic level) *)

32 RAll :: "('a => ('w::world) form) => 'w form" (binder "Rall " 10)

33 REx :: "('a => ('w::world) form) => 'w form" (binder "Rex " 10)

34 REx1 :: "('a => ('w::world) form) => 'w form" (binder "Rex! " 10)

36 (** concrete syntax **)

38 nonterminals

39 lift

40 liftargs

42 syntax

43 "" :: "id => lift" ("_")

44 "" :: "longid => lift" ("_")

45 "" :: "var => lift" ("_")

46 "_applC" :: "[lift, cargs] => lift" ("(1_/ _)" [1000, 1000] 999)

47 "" :: "lift => lift" ("'(_')")

48 "_lambda" :: "[idts, 'a] => lift" ("(3%_./ _)" [0, 3] 3)

49 "_constrain" :: "[lift, type] => lift" ("(_::_)" [4, 0] 3)

50 "" :: "lift => liftargs" ("_")

51 "_liftargs" :: "[lift, liftargs] => liftargs" ("_,/ _")

52 "_Valid" :: "lift => bool" ("(|- _)" 5)

53 "_holdsAt" :: "['a, lift] => bool" ("(_ |= _)" [100,10] 10)

55 (* Syntax for lifted expressions outside the scope of |- or |= *)

56 "LIFT" :: "lift => 'a" ("LIFT _")

58 (* generic syntax for lifted constants and functions *)

59 "_const" :: "'a => lift" ("(#_)" [1000] 999)

60 "_lift" :: "['a, lift] => lift" ("(_<_>)" [1000] 999)

61 "_lift2" :: "['a, lift, lift] => lift" ("(_<_,/ _>)" [1000] 999)

62 "_lift3" :: "['a, lift, lift, lift] => lift" ("(_<_,/ _,/ _>)" [1000] 999)

64 (* concrete syntax for common infix functions: reuse same symbol *)

65 "_liftEqu" :: "[lift, lift] => lift" ("(_ =/ _)" [50,51] 50)

66 "_liftNeq" :: "[lift, lift] => lift" ("(_ ~=/ _)" [50,51] 50)

67 "_liftNot" :: "lift => lift" ("(~ _)" [40] 40)

68 "_liftAnd" :: "[lift, lift] => lift" ("(_ &/ _)" [36,35] 35)

69 "_liftOr" :: "[lift, lift] => lift" ("(_ |/ _)" [31,30] 30)

70 "_liftImp" :: "[lift, lift] => lift" ("(_ -->/ _)" [26,25] 25)

71 "_liftIf" :: "[lift, lift, lift] => lift" ("(if (_)/ then (_)/ else (_))" 10)

72 "_liftPlus" :: "[lift, lift] => lift" ("(_ +/ _)" [66,65] 65)

73 "_liftMinus" :: "[lift, lift] => lift" ("(_ -/ _)" [66,65] 65)

74 "_liftTimes" :: "[lift, lift] => lift" ("(_ */ _)" [71,70] 70)

75 "_liftDiv" :: "[lift, lift] => lift" ("(_ div _)" [71,70] 70)

76 "_liftMod" :: "[lift, lift] => lift" ("(_ mod _)" [71,70] 70)

77 "_liftLess" :: "[lift, lift] => lift" ("(_/ < _)" [50, 51] 50)

78 "_liftLeq" :: "[lift, lift] => lift" ("(_/ <= _)" [50, 51] 50)

79 "_liftMem" :: "[lift, lift] => lift" ("(_/ : _)" [50, 51] 50)

80 "_liftNotMem" :: "[lift, lift] => lift" ("(_/ ~: _)" [50, 51] 50)

81 "_liftFinset" :: "liftargs => lift" ("{(_)}")

82 (** TODO: syntax for lifted collection / comprehension **)

83 "_liftPair" :: "[lift,liftargs] => lift" ("(1'(_,/ _'))")

84 (* infix syntax for list operations *)

85 "_liftCons" :: "[lift, lift] => lift" ("(_ #/ _)" [65,66] 65)

86 "_liftApp" :: "[lift, lift] => lift" ("(_ @/ _)" [65,66] 65)

87 "_liftList" :: "liftargs => lift" ("[(_)]")

89 (* Rigid quantification (syntax level) *)

90 "_ARAll" :: "[idts, lift] => lift" ("(3! _./ _)" [0, 10] 10)

91 "_AREx" :: "[idts, lift] => lift" ("(3? _./ _)" [0, 10] 10)

92 "_AREx1" :: "[idts, lift] => lift" ("(3?! _./ _)" [0, 10] 10)

93 "_RAll" :: "[idts, lift] => lift" ("(3ALL _./ _)" [0, 10] 10)

94 "_REx" :: "[idts, lift] => lift" ("(3EX _./ _)" [0, 10] 10)

95 "_REx1" :: "[idts, lift] => lift" ("(3EX! _./ _)" [0, 10] 10)

97 translations

98 "_const" == "const"

99 "_lift" == "lift"

100 "_lift2" == "lift2"

101 "_lift3" == "lift3"

102 "_Valid" == "Valid"

103 "_RAll x A" == "Rall x. A"

104 "_REx x A" == "Rex x. A"

105 "_REx1 x A" == "Rex! x. A"

106 "_ARAll" => "_RAll"

107 "_AREx" => "_REx"

108 "_AREx1" => "_REx1"

110 "w |= A" => "A w"

111 "LIFT A" => "A::_=>_"

113 "_liftEqu" == "_lift2 (op =)"

114 "_liftNeq u v" == "_liftNot (_liftEqu u v)"

115 "_liftNot" == "_lift Not"

116 "_liftAnd" == "_lift2 (op &)"

117 "_liftOr" == "_lift2 (op | )"

118 "_liftImp" == "_lift2 (op -->)"

119 "_liftIf" == "_lift3 If"

120 "_liftPlus" == "_lift2 (op +)"

121 "_liftMinus" == "_lift2 (op -)"

122 "_liftTimes" == "_lift2 (op *)"

123 "_liftDiv" == "_lift2 (op div)"

124 "_liftMod" == "_lift2 (op mod)"

125 "_liftLess" == "_lift2 (op <)"

126 "_liftLeq" == "_lift2 (op <=)"

127 "_liftMem" == "_lift2 (op :)"

128 "_liftNotMem x xs" == "_liftNot (_liftMem x xs)"

129 "_liftFinset (_liftargs x xs)" == "_lift2 CONST insert x (_liftFinset xs)"

130 "_liftFinset x" == "_lift2 CONST insert x (_const {})"

131 "_liftPair x (_liftargs y z)" == "_liftPair x (_liftPair y z)"

132 "_liftPair" == "_lift2 Pair"

133 "_liftCons" == "lift2 Cons"

134 "_liftApp" == "lift2 (op @)"

135 "_liftList (_liftargs x xs)" == "_liftCons x (_liftList xs)"

136 "_liftList x" == "_liftCons x (_const [])"

140 "w |= ~A" <= "_liftNot A w"

141 "w |= A & B" <= "_liftAnd A B w"

142 "w |= A | B" <= "_liftOr A B w"

143 "w |= A --> B" <= "_liftImp A B w"

144 "w |= u = v" <= "_liftEqu u v w"

145 "w |= ALL x. A" <= "_RAll x A w"

146 "w |= EX x. A" <= "_REx x A w"

147 "w |= EX! x. A" <= "_REx1 x A w"

149 syntax (xsymbols)

150 "_Valid" :: "lift => bool" ("(\<turnstile> _)" 5)

151 "_holdsAt" :: "['a, lift] => bool" ("(_ \<Turnstile> _)" [100,10] 10)

152 "_liftNeq" :: "[lift, lift] => lift" (infixl "\<noteq>" 50)

153 "_liftNot" :: "lift => lift" ("\<not> _" [40] 40)

154 "_liftAnd" :: "[lift, lift] => lift" (infixr "\<and>" 35)

155 "_liftOr" :: "[lift, lift] => lift" (infixr "\<or>" 30)

156 "_liftImp" :: "[lift, lift] => lift" (infixr "\<longrightarrow>" 25)

157 "_RAll" :: "[idts, lift] => lift" ("(3\<forall>_./ _)" [0, 10] 10)

158 "_REx" :: "[idts, lift] => lift" ("(3\<exists>_./ _)" [0, 10] 10)

159 "_REx1" :: "[idts, lift] => lift" ("(3\<exists>!_./ _)" [0, 10] 10)

160 "_liftLeq" :: "[lift, lift] => lift" ("(_/ \<le> _)" [50, 51] 50)

161 "_liftMem" :: "[lift, lift] => lift" ("(_/ \<in> _)" [50, 51] 50)

162 "_liftNotMem" :: "[lift, lift] => lift" ("(_/ \<notin> _)" [50, 51] 50)

164 syntax (HTML output)

165 "_liftNeq" :: "[lift, lift] => lift" (infixl "\<noteq>" 50)

166 "_liftNot" :: "lift => lift" ("\<not> _" [40] 40)

167 "_liftAnd" :: "[lift, lift] => lift" (infixr "\<and>" 35)

168 "_liftOr" :: "[lift, lift] => lift" (infixr "\<or>" 30)

169 "_RAll" :: "[idts, lift] => lift" ("(3\<forall>_./ _)" [0, 10] 10)

170 "_REx" :: "[idts, lift] => lift" ("(3\<exists>_./ _)" [0, 10] 10)

171 "_REx1" :: "[idts, lift] => lift" ("(3\<exists>!_./ _)" [0, 10] 10)

172 "_liftLeq" :: "[lift, lift] => lift" ("(_/ \<le> _)" [50, 51] 50)

173 "_liftMem" :: "[lift, lift] => lift" ("(_/ \<in> _)" [50, 51] 50)

174 "_liftNotMem" :: "[lift, lift] => lift" ("(_/ \<notin> _)" [50, 51] 50)

176 axioms

177 Valid_def: "|- A == ALL w. w |= A"

179 unl_con: "LIFT #c w == c"

180 unl_lift: "lift f x w == f (x w)"

181 unl_lift2: "LIFT f<x, y> w == f (x w) (y w)"

182 unl_lift3: "LIFT f<x, y, z> w == f (x w) (y w) (z w)"

184 unl_Rall: "w |= ALL x. A x == ALL x. (w |= A x)"

185 unl_Rex: "w |= EX x. A x == EX x. (w |= A x)"

186 unl_Rex1: "w |= EX! x. A x == EX! x. (w |= A x)"

189 subsection {* Lemmas and tactics for "intensional" logics. *}

191 lemmas intensional_rews [simp] =

192 unl_con unl_lift unl_lift2 unl_lift3 unl_Rall unl_Rex unl_Rex1

194 lemma inteq_reflection: "|- x=y ==> (x==y)"

195 apply (unfold Valid_def unl_lift2)

196 apply (rule eq_reflection)

197 apply (rule ext)

198 apply (erule spec)

199 done

201 lemma intI [intro!]: "(!!w. w |= A) ==> |- A"

202 apply (unfold Valid_def)

203 apply (rule allI)

204 apply (erule meta_spec)

205 done

207 lemma intD [dest]: "|- A ==> w |= A"

208 apply (unfold Valid_def)

209 apply (erule spec)

210 done

212 (** Lift usual HOL simplifications to "intensional" level. **)

214 lemma int_simps:

215 "|- (x=x) = #True"

216 "|- (~#True) = #False" "|- (~#False) = #True" "|- (~~ P) = P"

217 "|- ((~P) = P) = #False" "|- (P = (~P)) = #False"

218 "|- (P ~= Q) = (P = (~Q))"

219 "|- (#True=P) = P" "|- (P=#True) = P"

220 "|- (#True --> P) = P" "|- (#False --> P) = #True"

221 "|- (P --> #True) = #True" "|- (P --> P) = #True"

222 "|- (P --> #False) = (~P)" "|- (P --> ~P) = (~P)"

223 "|- (P & #True) = P" "|- (#True & P) = P"

224 "|- (P & #False) = #False" "|- (#False & P) = #False"

225 "|- (P & P) = P" "|- (P & ~P) = #False" "|- (~P & P) = #False"

226 "|- (P | #True) = #True" "|- (#True | P) = #True"

227 "|- (P | #False) = P" "|- (#False | P) = P"

228 "|- (P | P) = P" "|- (P | ~P) = #True" "|- (~P | P) = #True"

229 "|- (! x. P) = P" "|- (? x. P) = P"

230 "|- (~Q --> ~P) = (P --> Q)"

231 "|- (P|Q --> R) = ((P-->R)&(Q-->R))"

232 apply (unfold Valid_def intensional_rews)

233 apply blast+

234 done

236 declare int_simps [THEN inteq_reflection, simp]

238 lemma TrueW [simp]: "|- #True"

239 by (simp add: Valid_def unl_con)

243 (* ======== Functions to "unlift" intensional implications into HOL rules ====== *)

245 ML {*

246 (* Basic unlifting introduces a parameter "w" and applies basic rewrites, e.g.

247 |- F = G becomes F w = G w

248 |- F --> G becomes F w --> G w

249 *)

251 fun int_unlift th =

252 rewrite_rule @{thms intensional_rews} (th RS @{thm intD} handle THM _ => th);

254 (* Turn |- F = G into meta-level rewrite rule F == G *)

255 fun int_rewrite th =

256 zero_var_indexes (rewrite_rule @{thms intensional_rews} (th RS @{thm inteq_reflection}))

258 (* flattening turns "-->" into "==>" and eliminates conjunctions in the

259 antecedent. For example,

261 P & Q --> (R | S --> T) becomes [| P; Q; R | S |] ==> T

263 Flattening can be useful with "intensional" lemmas (after unlifting).

264 Naive resolution with mp and conjI may run away because of higher-order

265 unification, therefore the code is a little awkward.

266 *)

267 fun flatten t =

268 let

269 (* analogous to RS, but using matching instead of resolution *)

270 fun matchres tha i thb =

271 case Seq.chop 2 (Thm.biresolution true [(false,tha)] i thb) of

272 ([th],_) => th

273 | ([],_) => raise THM("matchres: no match", i, [tha,thb])

274 | _ => raise THM("matchres: multiple unifiers", i, [tha,thb])

276 (* match tha with some premise of thb *)

277 fun matchsome tha thb =

278 let fun hmatch 0 = raise THM("matchsome: no match", 0, [tha,thb])

279 | hmatch n = matchres tha n thb handle THM _ => hmatch (n-1)

280 in hmatch (nprems_of thb) end

282 fun hflatten t =

283 case (concl_of t) of

284 Const _ $ (Const ("op -->", _) $ _ $ _) => hflatten (t RS mp)

285 | _ => (hflatten (matchsome conjI t)) handle THM _ => zero_var_indexes t

286 in

287 hflatten t

288 end

290 fun int_use th =

291 case (concl_of th) of

292 Const _ $ (Const ("Intensional.Valid", _) $ _) =>

293 (flatten (int_unlift th) handle THM _ => th)

294 | _ => th

295 *}

297 attribute_setup int_unlift = {* Scan.succeed (Thm.rule_attribute (K int_unlift)) *} ""

298 attribute_setup int_rewrite = {* Scan.succeed (Thm.rule_attribute (K int_rewrite)) *} ""

299 attribute_setup flatten = {* Scan.succeed (Thm.rule_attribute (K flatten)) *} ""

300 attribute_setup int_use = {* Scan.succeed (Thm.rule_attribute (K int_use)) *} ""

302 lemma Not_Rall: "|- (~(! x. F x)) = (? x. ~F x)"

303 by (simp add: Valid_def)

305 lemma Not_Rex: "|- (~ (? x. F x)) = (! x. ~ F x)"

306 by (simp add: Valid_def)

308 end