# Theory Domain_Aux

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theory Domain_Aux
imports Map_Functions Fixrec
`(*  Title:      HOL/HOLCF/Domain_Aux.thy    Author:     Brian Huffman*)header {* Domain package support *}theory Domain_Auximports Map_Functions Fixrecbeginsubsection {* Continuous isomorphisms *}text {* A locale for continuous isomorphisms *}locale iso =  fixes abs :: "'a -> 'b"  fixes rep :: "'b -> 'a"  assumes abs_iso [simp]: "rep·(abs·x) = x"  assumes rep_iso [simp]: "abs·(rep·y) = y"beginlemma swap: "iso rep abs"  by (rule iso.intro [OF rep_iso abs_iso])lemma abs_below: "(abs·x \<sqsubseteq> abs·y) = (x \<sqsubseteq> y)"proof  assume "abs·x \<sqsubseteq> abs·y"  then have "rep·(abs·x) \<sqsubseteq> rep·(abs·y)" by (rule monofun_cfun_arg)  then show "x \<sqsubseteq> y" by simpnext  assume "x \<sqsubseteq> y"  then show "abs·x \<sqsubseteq> abs·y" by (rule monofun_cfun_arg)qedlemma rep_below: "(rep·x \<sqsubseteq> rep·y) = (x \<sqsubseteq> y)"  by (rule iso.abs_below [OF swap])lemma abs_eq: "(abs·x = abs·y) = (x = y)"  by (simp add: po_eq_conv abs_below)lemma rep_eq: "(rep·x = rep·y) = (x = y)"  by (rule iso.abs_eq [OF swap])lemma abs_strict: "abs·⊥ = ⊥"proof -  have "⊥ \<sqsubseteq> rep·⊥" ..  then have "abs·⊥ \<sqsubseteq> abs·(rep·⊥)" by (rule monofun_cfun_arg)  then have "abs·⊥ \<sqsubseteq> ⊥" by simp  then show ?thesis by (rule bottomI)qedlemma rep_strict: "rep·⊥ = ⊥"  by (rule iso.abs_strict [OF swap])lemma abs_defin': "abs·x = ⊥ ==> x = ⊥"proof -  have "x = rep·(abs·x)" by simp  also assume "abs·x = ⊥"  also note rep_strict  finally show "x = ⊥" .qedlemma rep_defin': "rep·z = ⊥ ==> z = ⊥"  by (rule iso.abs_defin' [OF swap])lemma abs_defined: "z ≠ ⊥ ==> abs·z ≠ ⊥"  by (erule contrapos_nn, erule abs_defin')lemma rep_defined: "z ≠ ⊥ ==> rep·z ≠ ⊥"  by (rule iso.abs_defined [OF iso.swap]) (rule iso_axioms)lemma abs_bottom_iff: "(abs·x = ⊥) = (x = ⊥)"  by (auto elim: abs_defin' intro: abs_strict)lemma rep_bottom_iff: "(rep·x = ⊥) = (x = ⊥)"  by (rule iso.abs_bottom_iff [OF iso.swap]) (rule iso_axioms)lemma casedist_rule: "rep·x = ⊥ ∨ P ==> x = ⊥ ∨ P"  by (simp add: rep_bottom_iff)lemma compact_abs_rev: "compact (abs·x) ==> compact x"proof (unfold compact_def)  assume "adm (λy. abs·x \<notsqsubseteq> y)"  with cont_Rep_cfun2  have "adm (λy. abs·x \<notsqsubseteq> abs·y)" by (rule adm_subst)  then show "adm (λy. x \<notsqsubseteq> y)" using abs_below by simpqedlemma compact_rep_rev: "compact (rep·x) ==> compact x"  by (rule iso.compact_abs_rev [OF iso.swap]) (rule iso_axioms)lemma compact_abs: "compact x ==> compact (abs·x)"  by (rule compact_rep_rev) simplemma compact_rep: "compact x ==> compact (rep·x)"  by (rule iso.compact_abs [OF iso.swap]) (rule iso_axioms)lemma iso_swap: "(x = abs·y) = (rep·x = y)"proof  assume "x = abs·y"  then have "rep·x = rep·(abs·y)" by simp  then show "rep·x = y" by simpnext  assume "rep·x = y"  then have "abs·(rep·x) = abs·y" by simp  then show "x = abs·y" by simpqedendsubsection {* Proofs about take functions *}text {*  This section contains lemmas that are used in a module that supports  the domain isomorphism package; the module contains proofs related  to take functions and the finiteness predicate.*}lemma deflation_abs_rep:  fixes abs and rep and d  assumes abs_iso: "!!x. rep·(abs·x) = x"  assumes rep_iso: "!!y. abs·(rep·y) = y"  shows "deflation d ==> deflation (abs oo d oo rep)"by (rule ep_pair.deflation_e_d_p) (simp add: ep_pair.intro assms)lemma deflation_chain_min:  assumes chain: "chain d"  assumes defl: "!!n. deflation (d n)"  shows "d m·(d n·x) = d (min m n)·x"proof (rule linorder_le_cases)  assume "m ≤ n"  with chain have "d m \<sqsubseteq> d n" by (rule chain_mono)  then have "d m·(d n·x) = d m·x"    by (rule deflation_below_comp1 [OF defl defl])  moreover from `m ≤ n` have "min m n = m" by simp  ultimately show ?thesis by simpnext  assume "n ≤ m"  with chain have "d n \<sqsubseteq> d m" by (rule chain_mono)  then have "d m·(d n·x) = d n·x"    by (rule deflation_below_comp2 [OF defl defl])  moreover from `n ≤ m` have "min m n = n" by simp  ultimately show ?thesis by simpqedlemma lub_ID_take_lemma:  assumes "chain t" and "(\<Squnion>n. t n) = ID"  assumes "!!n. t n·x = t n·y" shows "x = y"proof -  have "(\<Squnion>n. t n·x) = (\<Squnion>n. t n·y)"    using assms(3) by simp  then have "(\<Squnion>n. t n)·x = (\<Squnion>n. t n)·y"    using assms(1) by (simp add: lub_distribs)  then show "x = y"    using assms(2) by simpqedlemma lub_ID_reach:  assumes "chain t" and "(\<Squnion>n. t n) = ID"  shows "(\<Squnion>n. t n·x) = x"using assms by (simp add: lub_distribs)lemma lub_ID_take_induct:  assumes "chain t" and "(\<Squnion>n. t n) = ID"  assumes "adm P" and "!!n. P (t n·x)" shows "P x"proof -  from `chain t` have "chain (λn. t n·x)" by simp  from `adm P` this `!!n. P (t n·x)` have "P (\<Squnion>n. t n·x)" by (rule admD)  with `chain t` `(\<Squnion>n. t n) = ID` show "P x" by (simp add: lub_distribs)qedsubsection {* Finiteness *}text {*  Let a ``decisive'' function be a deflation that maps every input to  either itself or bottom.  Then if a domain's take functions are all  decisive, then all values in the domain are finite.*}definition  decisive :: "('a::pcpo -> 'a) => bool"where  "decisive d <-> (∀x. d·x = x ∨ d·x = ⊥)"lemma decisiveI: "(!!x. d·x = x ∨ d·x = ⊥) ==> decisive d"  unfolding decisive_def by simplemma decisive_cases:  assumes "decisive d" obtains "d·x = x" | "d·x = ⊥"using assms unfolding decisive_def by autolemma decisive_bottom: "decisive ⊥"  unfolding decisive_def by simplemma decisive_ID: "decisive ID"  unfolding decisive_def by simplemma decisive_ssum_map:  assumes f: "decisive f"  assumes g: "decisive g"  shows "decisive (ssum_map·f·g)"apply (rule decisiveI, rename_tac s)apply (case_tac s, simp_all)apply (rule_tac x=x in decisive_cases [OF f], simp_all)apply (rule_tac x=y in decisive_cases [OF g], simp_all)donelemma decisive_sprod_map:  assumes f: "decisive f"  assumes g: "decisive g"  shows "decisive (sprod_map·f·g)"apply (rule decisiveI, rename_tac s)apply (case_tac s, simp_all)apply (rule_tac x=x in decisive_cases [OF f], simp_all)apply (rule_tac x=y in decisive_cases [OF g], simp_all)donelemma decisive_abs_rep:  fixes abs rep  assumes iso: "iso abs rep"  assumes d: "decisive d"  shows "decisive (abs oo d oo rep)"apply (rule decisiveI)apply (rule_tac x="rep·x" in decisive_cases [OF d])apply (simp add: iso.rep_iso [OF iso])apply (simp add: iso.abs_strict [OF iso])donelemma lub_ID_finite:  assumes chain: "chain d"  assumes lub: "(\<Squnion>n. d n) = ID"  assumes decisive: "!!n. decisive (d n)"  shows "∃n. d n·x = x"proof -  have 1: "chain (λn. d n·x)" using chain by simp  have 2: "(\<Squnion>n. d n·x) = x" using chain lub by (rule lub_ID_reach)  have "∀n. d n·x = x ∨ d n·x = ⊥"    using decisive unfolding decisive_def by simp  hence "range (λn. d n·x) ⊆ {x, ⊥}"    by auto  hence "finite (range (λn. d n·x))"    by (rule finite_subset, simp)  with 1 have "finite_chain (λn. d n·x)"    by (rule finite_range_imp_finch)  then have "∃n. (\<Squnion>n. d n·x) = d n·x"    unfolding finite_chain_def by (auto simp add: maxinch_is_thelub)  with 2 show "∃n. d n·x = x" by (auto elim: sym)qedlemma lub_ID_finite_take_induct:  assumes "chain d" and "(\<Squnion>n. d n) = ID" and "!!n. decisive (d n)"  shows "(!!n. P (d n·x)) ==> P x"using lub_ID_finite [OF assms] by metissubsection {* Proofs about constructor functions *}text {* Lemmas for proving nchotomy rule: *}lemma ex_one_bottom_iff:  "(∃x. P x ∧ x ≠ ⊥) = P ONE"by simplemma ex_up_bottom_iff:  "(∃x. P x ∧ x ≠ ⊥) = (∃x. P (up·x))"by (safe, case_tac x, auto)lemma ex_sprod_bottom_iff: "(∃y. P y ∧ y ≠ ⊥) =  (∃x y. (P (:x, y:) ∧ x ≠ ⊥) ∧ y ≠ ⊥)"by (safe, case_tac y, auto)lemma ex_sprod_up_bottom_iff: "(∃y. P y ∧ y ≠ ⊥) =  (∃x y. P (:up·x, y:) ∧ y ≠ ⊥)"by (safe, case_tac y, simp, case_tac x, auto)lemma ex_ssum_bottom_iff: "(∃x. P x ∧ x ≠ ⊥) = ((∃x. P (sinl·x) ∧ x ≠ ⊥) ∨  (∃x. P (sinr·x) ∧ x ≠ ⊥))"by (safe, case_tac x, auto)lemma exh_start: "p = ⊥ ∨ (∃x. p = x ∧ x ≠ ⊥)"  by autolemmas ex_bottom_iffs =   ex_ssum_bottom_iff   ex_sprod_up_bottom_iff   ex_sprod_bottom_iff   ex_up_bottom_iff   ex_one_bottom_ifftext {* Rules for turning nchotomy into exhaust: *}lemma exh_casedist0: "[|R; R ==> P|] ==> P" (* like make_elim *)  by autolemma exh_casedist1: "((P ∨ Q ==> R) ==> S) ≡ ([|P ==> R; Q ==> R|] ==> S)"  by rule autolemma exh_casedist2: "(∃x. P x ==> Q) ≡ (!!x. P x ==> Q)"  by rule autolemma exh_casedist3: "(P ∧ Q ==> R) ≡ (P ==> Q ==> R)"  by rule autolemmas exh_casedists = exh_casedist1 exh_casedist2 exh_casedist3text {* Rules for proving constructor properties *}lemmas con_strict_rules =  sinl_strict sinr_strict spair_strict1 spair_strict2lemmas con_bottom_iff_rules =  sinl_bottom_iff sinr_bottom_iff spair_bottom_iff up_defined ONE_definedlemmas con_below_iff_rules =  sinl_below sinr_below sinl_below_sinr sinr_below_sinl con_bottom_iff_ruleslemmas con_eq_iff_rules =  sinl_eq sinr_eq sinl_eq_sinr sinr_eq_sinl con_bottom_iff_ruleslemmas sel_strict_rules =  cfcomp2 sscase1 sfst_strict ssnd_strict fup1lemma sel_app_extra_rules:  "sscase·ID·⊥·(sinr·x) = ⊥"  "sscase·ID·⊥·(sinl·x) = x"  "sscase·⊥·ID·(sinl·x) = ⊥"  "sscase·⊥·ID·(sinr·x) = x"  "fup·ID·(up·x) = x"by (cases "x = ⊥", simp, simp)+lemmas sel_app_rules =  sel_strict_rules sel_app_extra_rules  ssnd_spair sfst_spair up_defined spair_definedlemmas sel_bottom_iff_rules =  cfcomp2 sfst_bottom_iff ssnd_bottom_ifflemmas take_con_rules =  ssum_map_sinl' ssum_map_sinr' sprod_map_spair' u_map_up  deflation_strict deflation_ID ID1 cfcomp2subsection {* ML setup *}ML_file "Tools/Domain/domain_take_proofs.ML"ML_file "Tools/cont_consts.ML"ML_file "Tools/cont_proc.ML"ML_file "Tools/Domain/domain_constructors.ML"ML_file "Tools/Domain/domain_induction.ML"setup Domain_Take_Proofs.setupend`