(* Title: HOL/HOLCF/Domain_Aux.thy Author: Brian Huffman *) header {* Domain package support *} theory Domain_Aux imports Map_Functions Fixrec begin subsection {* 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" begin lemma 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 simp next assume "x \<sqsubseteq> y" then show "abs·x \<sqsubseteq> abs·y" by (rule monofun_cfun_arg) qed lemma 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) qed lemma 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 = ⊥" . qed lemma 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 simp qed lemma 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) simp lemma 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 simp next assume "rep·x = y" then have "abs·(rep·x) = abs·y" by simp then show "x = abs·y" by simp qed end subsection {* 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 simp next 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 simp qed lemma 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 simp qed lemma 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) qed subsection {* 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 simp lemma decisive_cases: assumes "decisive d" obtains "d·x = x" | "d·x = ⊥" using assms unfolding decisive_def by auto lemma decisive_bottom: "decisive ⊥" unfolding decisive_def by simp lemma decisive_ID: "decisive ID" unfolding decisive_def by simp lemma 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) done lemma 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) done lemma 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]) done lemma 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) qed lemma 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 metis subsection {* Proofs about constructor functions *} text {* Lemmas for proving nchotomy rule: *} lemma ex_one_bottom_iff: "(∃x. P x ∧ x ≠ ⊥) = P ONE" by simp lemma 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 auto lemmas ex_bottom_iffs = ex_ssum_bottom_iff ex_sprod_up_bottom_iff ex_sprod_bottom_iff ex_up_bottom_iff ex_one_bottom_iff text {* Rules for turning nchotomy into exhaust: *} lemma exh_casedist0: "[|R; R ==> P|] ==> P" (* like make_elim *) by auto lemma exh_casedist1: "((P ∨ Q ==> R) ==> S) ≡ ([|P ==> R; Q ==> R|] ==> S)" by rule auto lemma exh_casedist2: "(∃x. P x ==> Q) ≡ (!!x. P x ==> Q)" by rule auto lemma exh_casedist3: "(P ∧ Q ==> R) ≡ (P ==> Q ==> R)" by rule auto lemmas exh_casedists = exh_casedist1 exh_casedist2 exh_casedist3 text {* Rules for proving constructor properties *} lemmas con_strict_rules = sinl_strict sinr_strict spair_strict1 spair_strict2 lemmas con_bottom_iff_rules = sinl_bottom_iff sinr_bottom_iff spair_bottom_iff up_defined ONE_defined lemmas con_below_iff_rules = sinl_below sinr_below sinl_below_sinr sinr_below_sinl con_bottom_iff_rules lemmas con_eq_iff_rules = sinl_eq sinr_eq sinl_eq_sinr sinr_eq_sinl con_bottom_iff_rules lemmas sel_strict_rules = cfcomp2 sscase1 sfst_strict ssnd_strict fup1 lemma 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_defined lemmas sel_bottom_iff_rules = cfcomp2 sfst_bottom_iff ssnd_bottom_iff lemmas take_con_rules = ssum_map_sinl' ssum_map_sinr' sprod_map_spair' u_map_up deflation_strict deflation_ID ID1 cfcomp2 subsection {* 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" end