# HG changeset patch
# User wenzelm
# Date 1733863384 -3600
# Node ID 972fecd8907a933bacf659c6a68d9d3604457d96
# Parent  693a95492008c77ad706de61efcfc39389298b51
fewer theories (in contrast to 05ca920cd94b);

diff -r 693a95492008 -r 972fecd8907a src/HOL/HOLCF/Domain.thy
--- 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)
 
diff -r 693a95492008 -r 972fecd8907a src/HOL/HOLCF/Domain_Aux.thy
--- 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