1(* Title: HOL/HOLCF/Domain_Aux.thy 2 Author: Brian Huffman 3*) 4 5section \<open>Domain package support\<close> 6 7theory Domain_Aux 8imports Map_Functions Fixrec 9begin 10 11subsection \<open>Continuous isomorphisms\<close> 12 13text \<open>A locale for continuous isomorphisms\<close> 14 15locale iso = 16 fixes abs :: "'a \<rightarrow> 'b" 17 fixes rep :: "'b \<rightarrow> 'a" 18 assumes abs_iso [simp]: "rep\<cdot>(abs\<cdot>x) = x" 19 assumes rep_iso [simp]: "abs\<cdot>(rep\<cdot>y) = y" 20begin 21 22lemma swap: "iso rep abs" 23 by (rule iso.intro [OF rep_iso abs_iso]) 24 25lemma abs_below: "(abs\<cdot>x \<sqsubseteq> abs\<cdot>y) = (x \<sqsubseteq> y)" 26proof 27 assume "abs\<cdot>x \<sqsubseteq> abs\<cdot>y" 28 then have "rep\<cdot>(abs\<cdot>x) \<sqsubseteq> rep\<cdot>(abs\<cdot>y)" by (rule monofun_cfun_arg) 29 then show "x \<sqsubseteq> y" by simp 30next 31 assume "x \<sqsubseteq> y" 32 then show "abs\<cdot>x \<sqsubseteq> abs\<cdot>y" by (rule monofun_cfun_arg) 33qed 34 35lemma rep_below: "(rep\<cdot>x \<sqsubseteq> rep\<cdot>y) = (x \<sqsubseteq> y)" 36 by (rule iso.abs_below [OF swap]) 37 38lemma abs_eq: "(abs\<cdot>x = abs\<cdot>y) = (x = y)" 39 by (simp add: po_eq_conv abs_below) 40 41lemma rep_eq: "(rep\<cdot>x = rep\<cdot>y) = (x = y)" 42 by (rule iso.abs_eq [OF swap]) 43 44lemma abs_strict: "abs\<cdot>\<bottom> = \<bottom>" 45proof - 46 have "\<bottom> \<sqsubseteq> rep\<cdot>\<bottom>" .. 47 then have "abs\<cdot>\<bottom> \<sqsubseteq> abs\<cdot>(rep\<cdot>\<bottom>)" by (rule monofun_cfun_arg) 48 then have "abs\<cdot>\<bottom> \<sqsubseteq> \<bottom>" by simp 49 then show ?thesis by (rule bottomI) 50qed 51 52lemma rep_strict: "rep\<cdot>\<bottom> = \<bottom>" 53 by (rule iso.abs_strict [OF swap]) 54 55lemma abs_defin': "abs\<cdot>x = \<bottom> \<Longrightarrow> x = \<bottom>" 56proof - 57 have "x = rep\<cdot>(abs\<cdot>x)" by simp 58 also assume "abs\<cdot>x = \<bottom>" 59 also note rep_strict 60 finally show "x = \<bottom>" . 61qed 62 63lemma rep_defin': "rep\<cdot>z = \<bottom> \<Longrightarrow> z = \<bottom>" 64 by (rule iso.abs_defin' [OF swap]) 65 66lemma abs_defined: "z \<noteq> \<bottom> \<Longrightarrow> abs\<cdot>z \<noteq> \<bottom>" 67 by (erule contrapos_nn, erule abs_defin') 68 69lemma rep_defined: "z \<noteq> \<bottom> \<Longrightarrow> rep\<cdot>z \<noteq> \<bottom>" 70 by (rule iso.abs_defined [OF iso.swap]) (rule iso_axioms) 71 72lemma abs_bottom_iff: "(abs\<cdot>x = \<bottom>) = (x = \<bottom>)" 73 by (auto elim: abs_defin' intro: abs_strict) 74 75lemma rep_bottom_iff: "(rep\<cdot>x = \<bottom>) = (x = \<bottom>)" 76 by (rule iso.abs_bottom_iff [OF iso.swap]) (rule iso_axioms) 77 78lemma casedist_rule: "rep\<cdot>x = \<bottom> \<or> P \<Longrightarrow> x = \<bottom> \<or> P" 79 by (simp add: rep_bottom_iff) 80 81lemma compact_abs_rev: "compact (abs\<cdot>x) \<Longrightarrow> compact x" 82proof (unfold compact_def) 83 assume "adm (\<lambda>y. abs\<cdot>x \<notsqsubseteq> y)" 84 with cont_Rep_cfun2 85 have "adm (\<lambda>y. abs\<cdot>x \<notsqsubseteq> abs\<cdot>y)" by (rule adm_subst) 86 then show "adm (\<lambda>y. x \<notsqsubseteq> y)" using abs_below by simp 87qed 88 89lemma compact_rep_rev: "compact (rep\<cdot>x) \<Longrightarrow> compact x" 90 by (rule iso.compact_abs_rev [OF iso.swap]) (rule iso_axioms) 91 92lemma compact_abs: "compact x \<Longrightarrow> compact (abs\<cdot>x)" 93 by (rule compact_rep_rev) simp 94 95lemma compact_rep: "compact x \<Longrightarrow> compact (rep\<cdot>x)" 96 by (rule iso.compact_abs [OF iso.swap]) (rule iso_axioms) 97 98lemma iso_swap: "(x = abs\<cdot>y) = (rep\<cdot>x = y)" 99proof 100 assume "x = abs\<cdot>y" 101 then have "rep\<cdot>x = rep\<cdot>(abs\<cdot>y)" by simp 102 then show "rep\<cdot>x = y" by simp 103next 104 assume "rep\<cdot>x = y" 105 then have "abs\<cdot>(rep\<cdot>x) = abs\<cdot>y" by simp 106 then show "x = abs\<cdot>y" by simp 107qed 108 109end 110 111subsection \<open>Proofs about take functions\<close> 112 113text \<open> 114 This section contains lemmas that are used in a module that supports 115 the domain isomorphism package; the module contains proofs related 116 to take functions and the finiteness predicate. 117\<close> 118 119lemma deflation_abs_rep: 120 fixes abs and rep and d 121 assumes abs_iso: "\<And>x. rep\<cdot>(abs\<cdot>x) = x" 122 assumes rep_iso: "\<And>y. abs\<cdot>(rep\<cdot>y) = y" 123 shows "deflation d \<Longrightarrow> deflation (abs oo d oo rep)" 124by (rule ep_pair.deflation_e_d_p) (simp add: ep_pair.intro assms) 125 126lemma deflation_chain_min: 127 assumes chain: "chain d" 128 assumes defl: "\<And>n. deflation (d n)" 129 shows "d m\<cdot>(d n\<cdot>x) = d (min m n)\<cdot>x" 130proof (rule linorder_le_cases) 131 assume "m \<le> n" 132 with chain have "d m \<sqsubseteq> d n" by (rule chain_mono) 133 then have "d m\<cdot>(d n\<cdot>x) = d m\<cdot>x" 134 by (rule deflation_below_comp1 [OF defl defl]) 135 moreover from \<open>m \<le> n\<close> have "min m n = m" by simp 136 ultimately show ?thesis by simp 137next 138 assume "n \<le> m" 139 with chain have "d n \<sqsubseteq> d m" by (rule chain_mono) 140 then have "d m\<cdot>(d n\<cdot>x) = d n\<cdot>x" 141 by (rule deflation_below_comp2 [OF defl defl]) 142 moreover from \<open>n \<le> m\<close> have "min m n = n" by simp 143 ultimately show ?thesis by simp 144qed 145 146lemma lub_ID_take_lemma: 147 assumes "chain t" and "(\<Squnion>n. t n) = ID" 148 assumes "\<And>n. t n\<cdot>x = t n\<cdot>y" shows "x = y" 149proof - 150 have "(\<Squnion>n. t n\<cdot>x) = (\<Squnion>n. t n\<cdot>y)" 151 using assms(3) by simp 152 then have "(\<Squnion>n. t n)\<cdot>x = (\<Squnion>n. t n)\<cdot>y" 153 using assms(1) by (simp add: lub_distribs) 154 then show "x = y" 155 using assms(2) by simp 156qed 157 158lemma lub_ID_reach: 159 assumes "chain t" and "(\<Squnion>n. t n) = ID" 160 shows "(\<Squnion>n. t n\<cdot>x) = x" 161using assms by (simp add: lub_distribs) 162 163lemma lub_ID_take_induct: 164 assumes "chain t" and "(\<Squnion>n. t n) = ID" 165 assumes "adm P" and "\<And>n. P (t n\<cdot>x)" shows "P x" 166proof - 167 from \<open>chain t\<close> have "chain (\<lambda>n. t n\<cdot>x)" by simp 168 from \<open>adm P\<close> this \<open>\<And>n. P (t n\<cdot>x)\<close> have "P (\<Squnion>n. t n\<cdot>x)" by (rule admD) 169 with \<open>chain t\<close> \<open>(\<Squnion>n. t n) = ID\<close> show "P x" by (simp add: lub_distribs) 170qed 171 172subsection \<open>Finiteness\<close> 173 174text \<open> 175 Let a ``decisive'' function be a deflation that maps every input to 176 either itself or bottom. Then if a domain's take functions are all 177 decisive, then all values in the domain are finite. 178\<close> 179 180definition 181 decisive :: "('a::pcpo \<rightarrow> 'a) \<Rightarrow> bool" 182where 183 "decisive d \<longleftrightarrow> (\<forall>x. d\<cdot>x = x \<or> d\<cdot>x = \<bottom>)" 184 185lemma decisiveI: "(\<And>x. d\<cdot>x = x \<or> d\<cdot>x = \<bottom>) \<Longrightarrow> decisive d" 186 unfolding decisive_def by simp 187 188lemma decisive_cases: 189 assumes "decisive d" obtains "d\<cdot>x = x" | "d\<cdot>x = \<bottom>" 190using assms unfolding decisive_def by auto 191 192lemma decisive_bottom: "decisive \<bottom>" 193 unfolding decisive_def by simp 194 195lemma decisive_ID: "decisive ID" 196 unfolding decisive_def by simp 197 198lemma decisive_ssum_map: 199 assumes f: "decisive f" 200 assumes g: "decisive g" 201 shows "decisive (ssum_map\<cdot>f\<cdot>g)" 202 apply (rule decisiveI) 203 subgoal for s 204 apply (cases s, simp_all) 205 apply (rule_tac x=x in decisive_cases [OF f], simp_all) 206 apply (rule_tac x=y in decisive_cases [OF g], simp_all) 207 done 208 done 209 210lemma decisive_sprod_map: 211 assumes f: "decisive f" 212 assumes g: "decisive g" 213 shows "decisive (sprod_map\<cdot>f\<cdot>g)" 214 apply (rule decisiveI) 215 subgoal for s 216 apply (cases s, simp) 217 subgoal for x y 218 apply (rule decisive_cases [OF f, where x = x], simp_all) 219 apply (rule decisive_cases [OF g, where x = y], simp_all) 220 done 221 done 222 done 223 224lemma decisive_abs_rep: 225 fixes abs rep 226 assumes iso: "iso abs rep" 227 assumes d: "decisive d" 228 shows "decisive (abs oo d oo rep)" 229 apply (rule decisiveI) 230 subgoal for s 231 apply (rule decisive_cases [OF d, where x="rep\<cdot>s"]) 232 apply (simp add: iso.rep_iso [OF iso]) 233 apply (simp add: iso.abs_strict [OF iso]) 234 done 235 done 236 237lemma lub_ID_finite: 238 assumes chain: "chain d" 239 assumes lub: "(\<Squnion>n. d n) = ID" 240 assumes decisive: "\<And>n. decisive (d n)" 241 shows "\<exists>n. d n\<cdot>x = x" 242proof - 243 have 1: "chain (\<lambda>n. d n\<cdot>x)" using chain by simp 244 have 2: "(\<Squnion>n. d n\<cdot>x) = x" using chain lub by (rule lub_ID_reach) 245 have "\<forall>n. d n\<cdot>x = x \<or> d n\<cdot>x = \<bottom>" 246 using decisive unfolding decisive_def by simp 247 hence "range (\<lambda>n. d n\<cdot>x) \<subseteq> {x, \<bottom>}" 248 by auto 249 hence "finite (range (\<lambda>n. d n\<cdot>x))" 250 by (rule finite_subset, simp) 251 with 1 have "finite_chain (\<lambda>n. d n\<cdot>x)" 252 by (rule finite_range_imp_finch) 253 then have "\<exists>n. (\<Squnion>n. d n\<cdot>x) = d n\<cdot>x" 254 unfolding finite_chain_def by (auto simp add: maxinch_is_thelub) 255 with 2 show "\<exists>n. d n\<cdot>x = x" by (auto elim: sym) 256qed 257 258lemma lub_ID_finite_take_induct: 259 assumes "chain d" and "(\<Squnion>n. d n) = ID" and "\<And>n. decisive (d n)" 260 shows "(\<And>n. P (d n\<cdot>x)) \<Longrightarrow> P x" 261using lub_ID_finite [OF assms] by metis 262 263subsection \<open>Proofs about constructor functions\<close> 264 265text \<open>Lemmas for proving nchotomy rule:\<close> 266 267lemma ex_one_bottom_iff: 268 "(\<exists>x. P x \<and> x \<noteq> \<bottom>) = P ONE" 269by simp 270 271lemma ex_up_bottom_iff: 272 "(\<exists>x. P x \<and> x \<noteq> \<bottom>) = (\<exists>x. P (up\<cdot>x))" 273by (safe, case_tac x, auto) 274 275lemma ex_sprod_bottom_iff: 276 "(\<exists>y. P y \<and> y \<noteq> \<bottom>) = 277 (\<exists>x y. (P (:x, y:) \<and> x \<noteq> \<bottom>) \<and> y \<noteq> \<bottom>)" 278by (safe, case_tac y, auto) 279 280lemma ex_sprod_up_bottom_iff: 281 "(\<exists>y. P y \<and> y \<noteq> \<bottom>) = 282 (\<exists>x y. P (:up\<cdot>x, y:) \<and> y \<noteq> \<bottom>)" 283by (safe, case_tac y, simp, case_tac x, auto) 284 285lemma ex_ssum_bottom_iff: 286 "(\<exists>x. P x \<and> x \<noteq> \<bottom>) = 287 ((\<exists>x. P (sinl\<cdot>x) \<and> x \<noteq> \<bottom>) \<or> 288 (\<exists>x. P (sinr\<cdot>x) \<and> x \<noteq> \<bottom>))" 289by (safe, case_tac x, auto) 290 291lemma exh_start: "p = \<bottom> \<or> (\<exists>x. p = x \<and> x \<noteq> \<bottom>)" 292 by auto 293 294lemmas ex_bottom_iffs = 295 ex_ssum_bottom_iff 296 ex_sprod_up_bottom_iff 297 ex_sprod_bottom_iff 298 ex_up_bottom_iff 299 ex_one_bottom_iff 300 301text \<open>Rules for turning nchotomy into exhaust:\<close> 302 303lemma exh_casedist0: "\<lbrakk>R; R \<Longrightarrow> P\<rbrakk> \<Longrightarrow> P" (* like make_elim *) 304 by auto 305 306lemma exh_casedist1: "((P \<or> Q \<Longrightarrow> R) \<Longrightarrow> S) \<equiv> (\<lbrakk>P \<Longrightarrow> R; Q \<Longrightarrow> R\<rbrakk> \<Longrightarrow> S)" 307 by rule auto 308 309lemma exh_casedist2: "(\<exists>x. P x \<Longrightarrow> Q) \<equiv> (\<And>x. P x \<Longrightarrow> Q)" 310 by rule auto 311 312lemma exh_casedist3: "(P \<and> Q \<Longrightarrow> R) \<equiv> (P \<Longrightarrow> Q \<Longrightarrow> R)" 313 by rule auto 314 315lemmas exh_casedists = exh_casedist1 exh_casedist2 exh_casedist3 316 317text \<open>Rules for proving constructor properties\<close> 318 319lemmas con_strict_rules = 320 sinl_strict sinr_strict spair_strict1 spair_strict2 321 322lemmas con_bottom_iff_rules = 323 sinl_bottom_iff sinr_bottom_iff spair_bottom_iff up_defined ONE_defined 324 325lemmas con_below_iff_rules = 326 sinl_below sinr_below sinl_below_sinr sinr_below_sinl con_bottom_iff_rules 327 328lemmas con_eq_iff_rules = 329 sinl_eq sinr_eq sinl_eq_sinr sinr_eq_sinl con_bottom_iff_rules 330 331lemmas sel_strict_rules = 332 cfcomp2 sscase1 sfst_strict ssnd_strict fup1 333 334lemma sel_app_extra_rules: 335 "sscase\<cdot>ID\<cdot>\<bottom>\<cdot>(sinr\<cdot>x) = \<bottom>" 336 "sscase\<cdot>ID\<cdot>\<bottom>\<cdot>(sinl\<cdot>x) = x" 337 "sscase\<cdot>\<bottom>\<cdot>ID\<cdot>(sinl\<cdot>x) = \<bottom>" 338 "sscase\<cdot>\<bottom>\<cdot>ID\<cdot>(sinr\<cdot>x) = x" 339 "fup\<cdot>ID\<cdot>(up\<cdot>x) = x" 340by (cases "x = \<bottom>", simp, simp)+ 341 342lemmas sel_app_rules = 343 sel_strict_rules sel_app_extra_rules 344 ssnd_spair sfst_spair up_defined spair_defined 345 346lemmas sel_bottom_iff_rules = 347 cfcomp2 sfst_bottom_iff ssnd_bottom_iff 348 349lemmas take_con_rules = 350 ssum_map_sinl' ssum_map_sinr' sprod_map_spair' u_map_up 351 deflation_strict deflation_ID ID1 cfcomp2 352 353subsection \<open>ML setup\<close> 354 355named_theorems domain_deflation "theorems like deflation a ==> deflation (foo_map$a)" 356 and domain_map_ID "theorems like foo_map$ID = ID" 357 358ML_file \<open>Tools/Domain/domain_take_proofs.ML\<close> 359ML_file \<open>Tools/cont_consts.ML\<close> 360ML_file \<open>Tools/cont_proc.ML\<close> 361ML_file \<open>Tools/Domain/domain_constructors.ML\<close> 362ML_file \<open>Tools/Domain/domain_induction.ML\<close> 363 364end 365