--- a/src/HOL/HOLCF/Domain.thy Tue Dec 10 21:06:04 2024 +0100
+++ b/src/HOL/HOLCF/Domain.thy Tue Dec 10 21:43:04 2024 +0100
@@ -5,17 +5,373 @@
section \<open>Domain package\<close>
theory Domain
-imports Representable Domain_Aux
+imports Representable Map_Functions Fixrec
keywords
"lazy" "unsafe" and
"domaindef" "domain" :: thy_defn and
"domain_isomorphism" :: thy_decl
begin
-default_sort "domain"
+subsection \<open>Continuous isomorphisms\<close>
+
+text \<open>A locale for continuous isomorphisms\<close>
+
+locale iso =
+ fixes abs :: "'a \<rightarrow> 'b"
+ fixes rep :: "'b \<rightarrow> 'a"
+ assumes abs_iso [simp]: "rep\<cdot>(abs\<cdot>x) = x"
+ assumes rep_iso [simp]: "abs\<cdot>(rep\<cdot>y) = y"
+begin
+
+lemma swap: "iso rep abs"
+ by (rule iso.intro [OF rep_iso abs_iso])
+
+lemma abs_below: "(abs\<cdot>x \<sqsubseteq> abs\<cdot>y) = (x \<sqsubseteq> y)"
+proof
+ assume "abs\<cdot>x \<sqsubseteq> abs\<cdot>y"
+ then have "rep\<cdot>(abs\<cdot>x) \<sqsubseteq> rep\<cdot>(abs\<cdot>y)" by (rule monofun_cfun_arg)
+ then show "x \<sqsubseteq> y" by simp
+next
+ assume "x \<sqsubseteq> y"
+ then show "abs\<cdot>x \<sqsubseteq> abs\<cdot>y" by (rule monofun_cfun_arg)
+qed
+
+lemma rep_below: "(rep\<cdot>x \<sqsubseteq> rep\<cdot>y) = (x \<sqsubseteq> y)"
+ by (rule iso.abs_below [OF swap])
+
+lemma abs_eq: "(abs\<cdot>x = abs\<cdot>y) = (x = y)"
+ by (simp add: po_eq_conv abs_below)
+
+lemma rep_eq: "(rep\<cdot>x = rep\<cdot>y) = (x = y)"
+ by (rule iso.abs_eq [OF swap])
+
+lemma abs_strict: "abs\<cdot>\<bottom> = \<bottom>"
+proof -
+ have "\<bottom> \<sqsubseteq> rep\<cdot>\<bottom>" ..
+ then have "abs\<cdot>\<bottom> \<sqsubseteq> abs\<cdot>(rep\<cdot>\<bottom>)" by (rule monofun_cfun_arg)
+ then have "abs\<cdot>\<bottom> \<sqsubseteq> \<bottom>" by simp
+ then show ?thesis by (rule bottomI)
+qed
+
+lemma rep_strict: "rep\<cdot>\<bottom> = \<bottom>"
+ by (rule iso.abs_strict [OF swap])
+
+lemma abs_defin': "abs\<cdot>x = \<bottom> \<Longrightarrow> x = \<bottom>"
+proof -
+ have "x = rep\<cdot>(abs\<cdot>x)" by simp
+ also assume "abs\<cdot>x = \<bottom>"
+ also note rep_strict
+ finally show "x = \<bottom>" .
+qed
+
+lemma rep_defin': "rep\<cdot>z = \<bottom> \<Longrightarrow> z = \<bottom>"
+ by (rule iso.abs_defin' [OF swap])
+
+lemma abs_defined: "z \<noteq> \<bottom> \<Longrightarrow> abs\<cdot>z \<noteq> \<bottom>"
+ by (erule contrapos_nn, erule abs_defin')
+
+lemma rep_defined: "z \<noteq> \<bottom> \<Longrightarrow> rep\<cdot>z \<noteq> \<bottom>"
+ by (rule iso.abs_defined [OF iso.swap]) (rule iso_axioms)
+
+lemma abs_bottom_iff: "(abs\<cdot>x = \<bottom>) = (x = \<bottom>)"
+ by (auto elim: abs_defin' intro: abs_strict)
+
+lemma rep_bottom_iff: "(rep\<cdot>x = \<bottom>) = (x = \<bottom>)"
+ by (rule iso.abs_bottom_iff [OF iso.swap]) (rule iso_axioms)
+
+lemma casedist_rule: "rep\<cdot>x = \<bottom> \<or> P \<Longrightarrow> x = \<bottom> \<or> P"
+ by (simp add: rep_bottom_iff)
+
+lemma compact_abs_rev: "compact (abs\<cdot>x) \<Longrightarrow> compact x"
+proof (unfold compact_def)
+ assume "adm (\<lambda>y. abs\<cdot>x \<notsqsubseteq> y)"
+ with cont_Rep_cfun2
+ have "adm (\<lambda>y. abs\<cdot>x \<notsqsubseteq> abs\<cdot>y)" by (rule adm_subst)
+ then show "adm (\<lambda>y. x \<notsqsubseteq> y)" using abs_below by simp
+qed
+
+lemma compact_rep_rev: "compact (rep\<cdot>x) \<Longrightarrow> compact x"
+ by (rule iso.compact_abs_rev [OF iso.swap]) (rule iso_axioms)
+
+lemma compact_abs: "compact x \<Longrightarrow> compact (abs\<cdot>x)"
+ by (rule compact_rep_rev) simp
+
+lemma compact_rep: "compact x \<Longrightarrow> compact (rep\<cdot>x)"
+ by (rule iso.compact_abs [OF iso.swap]) (rule iso_axioms)
+
+lemma iso_swap: "(x = abs\<cdot>y) = (rep\<cdot>x = y)"
+proof
+ assume "x = abs\<cdot>y"
+ then have "rep\<cdot>x = rep\<cdot>(abs\<cdot>y)" by simp
+ then show "rep\<cdot>x = y" by simp
+next
+ assume "rep\<cdot>x = y"
+ then have "abs\<cdot>(rep\<cdot>x) = abs\<cdot>y" by simp
+ then show "x = abs\<cdot>y" by simp
+qed
+
+end
+
+subsection \<open>Proofs about take functions\<close>
+
+text \<open>
+ 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.
+\<close>
+
+lemma deflation_abs_rep:
+ fixes abs and rep and d
+ assumes abs_iso: "\<And>x. rep\<cdot>(abs\<cdot>x) = x"
+ assumes rep_iso: "\<And>y. abs\<cdot>(rep\<cdot>y) = y"
+ shows "deflation d \<Longrightarrow> 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: "\<And>n. deflation (d n)"
+ shows "d m\<cdot>(d n\<cdot>x) = d (min m n)\<cdot>x"
+proof (rule linorder_le_cases)
+ assume "m \<le> n"
+ with chain have "d m \<sqsubseteq> d n" by (rule chain_mono)
+ then have "d m\<cdot>(d n\<cdot>x) = d m\<cdot>x"
+ by (rule deflation_below_comp1 [OF defl defl])
+ moreover from \<open>m \<le> n\<close> have "min m n = m" by simp
+ ultimately show ?thesis by simp
+next
+ assume "n \<le> m"
+ with chain have "d n \<sqsubseteq> d m" by (rule chain_mono)
+ then have "d m\<cdot>(d n\<cdot>x) = d n\<cdot>x"
+ by (rule deflation_below_comp2 [OF defl defl])
+ moreover from \<open>n \<le> m\<close> 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 "\<And>n. t n\<cdot>x = t n\<cdot>y" shows "x = y"
+proof -
+ have "(\<Squnion>n. t n\<cdot>x) = (\<Squnion>n. t n\<cdot>y)"
+ using assms(3) by simp
+ then have "(\<Squnion>n. t n)\<cdot>x = (\<Squnion>n. t n)\<cdot>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\<cdot>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 "\<And>n. P (t n\<cdot>x)" shows "P x"
+proof -
+ from \<open>chain t\<close> have "chain (\<lambda>n. t n\<cdot>x)" by simp
+ 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)
+ with \<open>chain t\<close> \<open>(\<Squnion>n. t n) = ID\<close> show "P x" by (simp add: lub_distribs)
+qed
+
+subsection \<open>Finiteness\<close>
+
+text \<open>
+ 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.
+\<close>
+
+definition
+ decisive :: "('a::pcpo \<rightarrow> 'a) \<Rightarrow> bool"
+where
+ "decisive d \<longleftrightarrow> (\<forall>x. d\<cdot>x = x \<or> d\<cdot>x = \<bottom>)"
+
+lemma decisiveI: "(\<And>x. d\<cdot>x = x \<or> d\<cdot>x = \<bottom>) \<Longrightarrow> decisive d"
+ unfolding decisive_def by simp
+
+lemma decisive_cases:
+ assumes "decisive d" obtains "d\<cdot>x = x" | "d\<cdot>x = \<bottom>"
+using assms unfolding decisive_def by auto
+
+lemma decisive_bottom: "decisive \<bottom>"
+ 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\<cdot>f\<cdot>g)"
+ apply (rule decisiveI)
+ subgoal for s
+ apply (cases 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
+ done
+
+lemma decisive_sprod_map:
+ assumes f: "decisive f"
+ assumes g: "decisive g"
+ shows "decisive (sprod_map\<cdot>f\<cdot>g)"
+ apply (rule decisiveI)
+ subgoal for s
+ apply (cases s, simp)
+ subgoal for x y
+ apply (rule decisive_cases [OF f, where x = x], simp_all)
+ apply (rule decisive_cases [OF g, where x = y], simp_all)
+ done
+ done
+ 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)
+ subgoal for s
+ apply (rule decisive_cases [OF d, where x="rep\<cdot>s"])
+ apply (simp add: iso.rep_iso [OF iso])
+ apply (simp add: iso.abs_strict [OF iso])
+ done
+ done
+
+lemma lub_ID_finite:
+ assumes chain: "chain d"
+ assumes lub: "(\<Squnion>n. d n) = ID"
+ assumes decisive: "\<And>n. decisive (d n)"
+ shows "\<exists>n. d n\<cdot>x = x"
+proof -
+ have 1: "chain (\<lambda>n. d n\<cdot>x)" using chain by simp
+ have 2: "(\<Squnion>n. d n\<cdot>x) = x" using chain lub by (rule lub_ID_reach)
+ have "\<forall>n. d n\<cdot>x = x \<or> d n\<cdot>x = \<bottom>"
+ using decisive unfolding decisive_def by simp
+ hence "range (\<lambda>n. d n\<cdot>x) \<subseteq> {x, \<bottom>}"
+ by auto
+ hence "finite (range (\<lambda>n. d n\<cdot>x))"
+ by (rule finite_subset, simp)
+ with 1 have "finite_chain (\<lambda>n. d n\<cdot>x)"
+ by (rule finite_range_imp_finch)
+ then have "\<exists>n. (\<Squnion>n. d n\<cdot>x) = d n\<cdot>x"
+ unfolding finite_chain_def by (auto simp add: maxinch_is_thelub)
+ with 2 show "\<exists>n. d n\<cdot>x = x" by (auto elim: sym)
+qed
+
+lemma lub_ID_finite_take_induct:
+ assumes "chain d" and "(\<Squnion>n. d n) = ID" and "\<And>n. decisive (d n)"
+ shows "(\<And>n. P (d n\<cdot>x)) \<Longrightarrow> P x"
+using lub_ID_finite [OF assms] by metis
+
+subsection \<open>Proofs about constructor functions\<close>
+
+text \<open>Lemmas for proving nchotomy rule:\<close>
+
+lemma ex_one_bottom_iff:
+ "(\<exists>x. P x \<and> x \<noteq> \<bottom>) = P ONE"
+by simp
+
+lemma ex_up_bottom_iff:
+ "(\<exists>x. P x \<and> x \<noteq> \<bottom>) = (\<exists>x. P (up\<cdot>x))"
+by (safe, case_tac x, auto)
+
+lemma ex_sprod_bottom_iff:
+ "(\<exists>y. P y \<and> y \<noteq> \<bottom>) =
+ (\<exists>x y. (P (:x, y:) \<and> x \<noteq> \<bottom>) \<and> y \<noteq> \<bottom>)"
+by (safe, case_tac y, auto)
+
+lemma ex_sprod_up_bottom_iff:
+ "(\<exists>y. P y \<and> y \<noteq> \<bottom>) =
+ (\<exists>x y. P (:up\<cdot>x, y:) \<and> y \<noteq> \<bottom>)"
+by (safe, case_tac y, simp, case_tac x, auto)
+
+lemma ex_ssum_bottom_iff:
+ "(\<exists>x. P x \<and> x \<noteq> \<bottom>) =
+ ((\<exists>x. P (sinl\<cdot>x) \<and> x \<noteq> \<bottom>) \<or>
+ (\<exists>x. P (sinr\<cdot>x) \<and> x \<noteq> \<bottom>))"
+by (safe, case_tac x, auto)
+
+lemma exh_start: "p = \<bottom> \<or> (\<exists>x. p = x \<and> x \<noteq> \<bottom>)"
+ 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 \<open>Rules for turning nchotomy into exhaust:\<close>
+
+lemma exh_casedist0: "\<lbrakk>R; R \<Longrightarrow> P\<rbrakk> \<Longrightarrow> P" (* like make_elim *)
+ by auto
+
+lemma exh_casedist1: "((P \<or> Q \<Longrightarrow> R) \<Longrightarrow> S) \<equiv> (\<lbrakk>P \<Longrightarrow> R; Q \<Longrightarrow> R\<rbrakk> \<Longrightarrow> S)"
+ by rule auto
+
+lemma exh_casedist2: "(\<exists>x. P x \<Longrightarrow> Q) \<equiv> (\<And>x. P x \<Longrightarrow> Q)"
+ by rule auto
+
+lemma exh_casedist3: "(P \<and> Q \<Longrightarrow> R) \<equiv> (P \<Longrightarrow> Q \<Longrightarrow> R)"
+ by rule auto
+
+lemmas exh_casedists = exh_casedist1 exh_casedist2 exh_casedist3
+
+text \<open>Rules for proving constructor properties\<close>
+
+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\<cdot>ID\<cdot>\<bottom>\<cdot>(sinr\<cdot>x) = \<bottom>"
+ "sscase\<cdot>ID\<cdot>\<bottom>\<cdot>(sinl\<cdot>x) = x"
+ "sscase\<cdot>\<bottom>\<cdot>ID\<cdot>(sinl\<cdot>x) = \<bottom>"
+ "sscase\<cdot>\<bottom>\<cdot>ID\<cdot>(sinr\<cdot>x) = x"
+ "fup\<cdot>ID\<cdot>(up\<cdot>x) = x"
+by (cases "x = \<bottom>", 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 \<open>ML setup\<close>
+
+named_theorems domain_deflation "theorems like deflation a ==> deflation (foo_map$a)"
+ and domain_map_ID "theorems like foo_map$ID = ID"
+
+ML_file \<open>Tools/Domain/domain_take_proofs.ML\<close>
+ML_file \<open>Tools/cont_consts.ML\<close>
+ML_file \<open>Tools/cont_proc.ML\<close>
+simproc_setup cont ("cont f") = \<open>K ContProc.cont_proc\<close>
+
+ML_file \<open>Tools/Domain/domain_constructors.ML\<close>
+ML_file \<open>Tools/Domain/domain_induction.ML\<close>
+
subsection \<open>Representations of types\<close>
+default_sort "domain"
+
lemma emb_prj: "emb\<cdot>((prj\<cdot>x)::'a) = cast\<cdot>DEFL('a)\<cdot>x"
by (simp add: cast_DEFL)
--- a/src/HOL/HOLCF/Domain_Aux.thy Tue Dec 10 21:06:04 2024 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,366 +0,0 @@
-(* Title: HOL/HOLCF/Domain_Aux.thy
- Author: Brian Huffman
-*)
-
-section \<open>Domain package support\<close>
-
-theory Domain_Aux
-imports Map_Functions Fixrec
-begin
-
-subsection \<open>Continuous isomorphisms\<close>
-
-text \<open>A locale for continuous isomorphisms\<close>
-
-locale iso =
- fixes abs :: "'a \<rightarrow> 'b"
- fixes rep :: "'b \<rightarrow> 'a"
- assumes abs_iso [simp]: "rep\<cdot>(abs\<cdot>x) = x"
- assumes rep_iso [simp]: "abs\<cdot>(rep\<cdot>y) = y"
-begin
-
-lemma swap: "iso rep abs"
- by (rule iso.intro [OF rep_iso abs_iso])
-
-lemma abs_below: "(abs\<cdot>x \<sqsubseteq> abs\<cdot>y) = (x \<sqsubseteq> y)"
-proof
- assume "abs\<cdot>x \<sqsubseteq> abs\<cdot>y"
- then have "rep\<cdot>(abs\<cdot>x) \<sqsubseteq> rep\<cdot>(abs\<cdot>y)" by (rule monofun_cfun_arg)
- then show "x \<sqsubseteq> y" by simp
-next
- assume "x \<sqsubseteq> y"
- then show "abs\<cdot>x \<sqsubseteq> abs\<cdot>y" by (rule monofun_cfun_arg)
-qed
-
-lemma rep_below: "(rep\<cdot>x \<sqsubseteq> rep\<cdot>y) = (x \<sqsubseteq> y)"
- by (rule iso.abs_below [OF swap])
-
-lemma abs_eq: "(abs\<cdot>x = abs\<cdot>y) = (x = y)"
- by (simp add: po_eq_conv abs_below)
-
-lemma rep_eq: "(rep\<cdot>x = rep\<cdot>y) = (x = y)"
- by (rule iso.abs_eq [OF swap])
-
-lemma abs_strict: "abs\<cdot>\<bottom> = \<bottom>"
-proof -
- have "\<bottom> \<sqsubseteq> rep\<cdot>\<bottom>" ..
- then have "abs\<cdot>\<bottom> \<sqsubseteq> abs\<cdot>(rep\<cdot>\<bottom>)" by (rule monofun_cfun_arg)
- then have "abs\<cdot>\<bottom> \<sqsubseteq> \<bottom>" by simp
- then show ?thesis by (rule bottomI)
-qed
-
-lemma rep_strict: "rep\<cdot>\<bottom> = \<bottom>"
- by (rule iso.abs_strict [OF swap])
-
-lemma abs_defin': "abs\<cdot>x = \<bottom> \<Longrightarrow> x = \<bottom>"
-proof -
- have "x = rep\<cdot>(abs\<cdot>x)" by simp
- also assume "abs\<cdot>x = \<bottom>"
- also note rep_strict
- finally show "x = \<bottom>" .
-qed
-
-lemma rep_defin': "rep\<cdot>z = \<bottom> \<Longrightarrow> z = \<bottom>"
- by (rule iso.abs_defin' [OF swap])
-
-lemma abs_defined: "z \<noteq> \<bottom> \<Longrightarrow> abs\<cdot>z \<noteq> \<bottom>"
- by (erule contrapos_nn, erule abs_defin')
-
-lemma rep_defined: "z \<noteq> \<bottom> \<Longrightarrow> rep\<cdot>z \<noteq> \<bottom>"
- by (rule iso.abs_defined [OF iso.swap]) (rule iso_axioms)
-
-lemma abs_bottom_iff: "(abs\<cdot>x = \<bottom>) = (x = \<bottom>)"
- by (auto elim: abs_defin' intro: abs_strict)
-
-lemma rep_bottom_iff: "(rep\<cdot>x = \<bottom>) = (x = \<bottom>)"
- by (rule iso.abs_bottom_iff [OF iso.swap]) (rule iso_axioms)
-
-lemma casedist_rule: "rep\<cdot>x = \<bottom> \<or> P \<Longrightarrow> x = \<bottom> \<or> P"
- by (simp add: rep_bottom_iff)
-
-lemma compact_abs_rev: "compact (abs\<cdot>x) \<Longrightarrow> compact x"
-proof (unfold compact_def)
- assume "adm (\<lambda>y. abs\<cdot>x \<notsqsubseteq> y)"
- with cont_Rep_cfun2
- have "adm (\<lambda>y. abs\<cdot>x \<notsqsubseteq> abs\<cdot>y)" by (rule adm_subst)
- then show "adm (\<lambda>y. x \<notsqsubseteq> y)" using abs_below by simp
-qed
-
-lemma compact_rep_rev: "compact (rep\<cdot>x) \<Longrightarrow> compact x"
- by (rule iso.compact_abs_rev [OF iso.swap]) (rule iso_axioms)
-
-lemma compact_abs: "compact x \<Longrightarrow> compact (abs\<cdot>x)"
- by (rule compact_rep_rev) simp
-
-lemma compact_rep: "compact x \<Longrightarrow> compact (rep\<cdot>x)"
- by (rule iso.compact_abs [OF iso.swap]) (rule iso_axioms)
-
-lemma iso_swap: "(x = abs\<cdot>y) = (rep\<cdot>x = y)"
-proof
- assume "x = abs\<cdot>y"
- then have "rep\<cdot>x = rep\<cdot>(abs\<cdot>y)" by simp
- then show "rep\<cdot>x = y" by simp
-next
- assume "rep\<cdot>x = y"
- then have "abs\<cdot>(rep\<cdot>x) = abs\<cdot>y" by simp
- then show "x = abs\<cdot>y" by simp
-qed
-
-end
-
-subsection \<open>Proofs about take functions\<close>
-
-text \<open>
- 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.
-\<close>
-
-lemma deflation_abs_rep:
- fixes abs and rep and d
- assumes abs_iso: "\<And>x. rep\<cdot>(abs\<cdot>x) = x"
- assumes rep_iso: "\<And>y. abs\<cdot>(rep\<cdot>y) = y"
- shows "deflation d \<Longrightarrow> 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: "\<And>n. deflation (d n)"
- shows "d m\<cdot>(d n\<cdot>x) = d (min m n)\<cdot>x"
-proof (rule linorder_le_cases)
- assume "m \<le> n"
- with chain have "d m \<sqsubseteq> d n" by (rule chain_mono)
- then have "d m\<cdot>(d n\<cdot>x) = d m\<cdot>x"
- by (rule deflation_below_comp1 [OF defl defl])
- moreover from \<open>m \<le> n\<close> have "min m n = m" by simp
- ultimately show ?thesis by simp
-next
- assume "n \<le> m"
- with chain have "d n \<sqsubseteq> d m" by (rule chain_mono)
- then have "d m\<cdot>(d n\<cdot>x) = d n\<cdot>x"
- by (rule deflation_below_comp2 [OF defl defl])
- moreover from \<open>n \<le> m\<close> 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 "\<And>n. t n\<cdot>x = t n\<cdot>y" shows "x = y"
-proof -
- have "(\<Squnion>n. t n\<cdot>x) = (\<Squnion>n. t n\<cdot>y)"
- using assms(3) by simp
- then have "(\<Squnion>n. t n)\<cdot>x = (\<Squnion>n. t n)\<cdot>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\<cdot>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 "\<And>n. P (t n\<cdot>x)" shows "P x"
-proof -
- from \<open>chain t\<close> have "chain (\<lambda>n. t n\<cdot>x)" by simp
- 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)
- with \<open>chain t\<close> \<open>(\<Squnion>n. t n) = ID\<close> show "P x" by (simp add: lub_distribs)
-qed
-
-subsection \<open>Finiteness\<close>
-
-text \<open>
- 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.
-\<close>
-
-definition
- decisive :: "('a::pcpo \<rightarrow> 'a) \<Rightarrow> bool"
-where
- "decisive d \<longleftrightarrow> (\<forall>x. d\<cdot>x = x \<or> d\<cdot>x = \<bottom>)"
-
-lemma decisiveI: "(\<And>x. d\<cdot>x = x \<or> d\<cdot>x = \<bottom>) \<Longrightarrow> decisive d"
- unfolding decisive_def by simp
-
-lemma decisive_cases:
- assumes "decisive d" obtains "d\<cdot>x = x" | "d\<cdot>x = \<bottom>"
-using assms unfolding decisive_def by auto
-
-lemma decisive_bottom: "decisive \<bottom>"
- 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\<cdot>f\<cdot>g)"
- apply (rule decisiveI)
- subgoal for s
- apply (cases 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
- done
-
-lemma decisive_sprod_map:
- assumes f: "decisive f"
- assumes g: "decisive g"
- shows "decisive (sprod_map\<cdot>f\<cdot>g)"
- apply (rule decisiveI)
- subgoal for s
- apply (cases s, simp)
- subgoal for x y
- apply (rule decisive_cases [OF f, where x = x], simp_all)
- apply (rule decisive_cases [OF g, where x = y], simp_all)
- done
- done
- 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)
- subgoal for s
- apply (rule decisive_cases [OF d, where x="rep\<cdot>s"])
- apply (simp add: iso.rep_iso [OF iso])
- apply (simp add: iso.abs_strict [OF iso])
- done
- done
-
-lemma lub_ID_finite:
- assumes chain: "chain d"
- assumes lub: "(\<Squnion>n. d n) = ID"
- assumes decisive: "\<And>n. decisive (d n)"
- shows "\<exists>n. d n\<cdot>x = x"
-proof -
- have 1: "chain (\<lambda>n. d n\<cdot>x)" using chain by simp
- have 2: "(\<Squnion>n. d n\<cdot>x) = x" using chain lub by (rule lub_ID_reach)
- have "\<forall>n. d n\<cdot>x = x \<or> d n\<cdot>x = \<bottom>"
- using decisive unfolding decisive_def by simp
- hence "range (\<lambda>n. d n\<cdot>x) \<subseteq> {x, \<bottom>}"
- by auto
- hence "finite (range (\<lambda>n. d n\<cdot>x))"
- by (rule finite_subset, simp)
- with 1 have "finite_chain (\<lambda>n. d n\<cdot>x)"
- by (rule finite_range_imp_finch)
- then have "\<exists>n. (\<Squnion>n. d n\<cdot>x) = d n\<cdot>x"
- unfolding finite_chain_def by (auto simp add: maxinch_is_thelub)
- with 2 show "\<exists>n. d n\<cdot>x = x" by (auto elim: sym)
-qed
-
-lemma lub_ID_finite_take_induct:
- assumes "chain d" and "(\<Squnion>n. d n) = ID" and "\<And>n. decisive (d n)"
- shows "(\<And>n. P (d n\<cdot>x)) \<Longrightarrow> P x"
-using lub_ID_finite [OF assms] by metis
-
-subsection \<open>Proofs about constructor functions\<close>
-
-text \<open>Lemmas for proving nchotomy rule:\<close>
-
-lemma ex_one_bottom_iff:
- "(\<exists>x. P x \<and> x \<noteq> \<bottom>) = P ONE"
-by simp
-
-lemma ex_up_bottom_iff:
- "(\<exists>x. P x \<and> x \<noteq> \<bottom>) = (\<exists>x. P (up\<cdot>x))"
-by (safe, case_tac x, auto)
-
-lemma ex_sprod_bottom_iff:
- "(\<exists>y. P y \<and> y \<noteq> \<bottom>) =
- (\<exists>x y. (P (:x, y:) \<and> x \<noteq> \<bottom>) \<and> y \<noteq> \<bottom>)"
-by (safe, case_tac y, auto)
-
-lemma ex_sprod_up_bottom_iff:
- "(\<exists>y. P y \<and> y \<noteq> \<bottom>) =
- (\<exists>x y. P (:up\<cdot>x, y:) \<and> y \<noteq> \<bottom>)"
-by (safe, case_tac y, simp, case_tac x, auto)
-
-lemma ex_ssum_bottom_iff:
- "(\<exists>x. P x \<and> x \<noteq> \<bottom>) =
- ((\<exists>x. P (sinl\<cdot>x) \<and> x \<noteq> \<bottom>) \<or>
- (\<exists>x. P (sinr\<cdot>x) \<and> x \<noteq> \<bottom>))"
-by (safe, case_tac x, auto)
-
-lemma exh_start: "p = \<bottom> \<or> (\<exists>x. p = x \<and> x \<noteq> \<bottom>)"
- 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 \<open>Rules for turning nchotomy into exhaust:\<close>
-
-lemma exh_casedist0: "\<lbrakk>R; R \<Longrightarrow> P\<rbrakk> \<Longrightarrow> P" (* like make_elim *)
- by auto
-
-lemma exh_casedist1: "((P \<or> Q \<Longrightarrow> R) \<Longrightarrow> S) \<equiv> (\<lbrakk>P \<Longrightarrow> R; Q \<Longrightarrow> R\<rbrakk> \<Longrightarrow> S)"
- by rule auto
-
-lemma exh_casedist2: "(\<exists>x. P x \<Longrightarrow> Q) \<equiv> (\<And>x. P x \<Longrightarrow> Q)"
- by rule auto
-
-lemma exh_casedist3: "(P \<and> Q \<Longrightarrow> R) \<equiv> (P \<Longrightarrow> Q \<Longrightarrow> R)"
- by rule auto
-
-lemmas exh_casedists = exh_casedist1 exh_casedist2 exh_casedist3
-
-text \<open>Rules for proving constructor properties\<close>
-
-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\<cdot>ID\<cdot>\<bottom>\<cdot>(sinr\<cdot>x) = \<bottom>"
- "sscase\<cdot>ID\<cdot>\<bottom>\<cdot>(sinl\<cdot>x) = x"
- "sscase\<cdot>\<bottom>\<cdot>ID\<cdot>(sinl\<cdot>x) = \<bottom>"
- "sscase\<cdot>\<bottom>\<cdot>ID\<cdot>(sinr\<cdot>x) = x"
- "fup\<cdot>ID\<cdot>(up\<cdot>x) = x"
-by (cases "x = \<bottom>", 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 \<open>ML setup\<close>
-
-named_theorems domain_deflation "theorems like deflation a ==> deflation (foo_map$a)"
- and domain_map_ID "theorems like foo_map$ID = ID"
-
-ML_file \<open>Tools/Domain/domain_take_proofs.ML\<close>
-ML_file \<open>Tools/cont_consts.ML\<close>
-ML_file \<open>Tools/cont_proc.ML\<close>
-simproc_setup cont ("cont f") = \<open>K ContProc.cont_proc\<close>
-
-ML_file \<open>Tools/Domain/domain_constructors.ML\<close>
-ML_file \<open>Tools/Domain/domain_induction.ML\<close>
-
-end