# HG changeset patch
# User huffman
# Date 1257862949 28800
# Node ID ea9becc59636306a8e45e2dd301a78d6e4ec73f6
# Parent  54f98d22516398f664a8e7f2e4c3cb3a42e22294
theory of representable cpos

diff -r 54f98d225163 -r ea9becc59636 src/HOLCF/HOLCF.thy
--- a/src/HOLCF/HOLCF.thy	Mon Nov 09 15:51:32 2009 -0800
+++ b/src/HOLCF/HOLCF.thy	Tue Nov 10 06:22:29 2009 -0800
@@ -7,6 +7,7 @@
 theory HOLCF
 imports
   Domain ConvexPD Algebraic Universal Sum_Cpo Main
+  Representable
 uses
   "holcf_logic.ML"
   "Tools/adm_tac.ML"
diff -r 54f98d225163 -r ea9becc59636 src/HOLCF/Representable.thy
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOLCF/Representable.thy	Tue Nov 10 06:22:29 2009 -0800
@@ -0,0 +1,991 @@
+header {* Representable Types *}
+
+theory Representable
+imports Algebraic Universal Ssum Sprod One ConvexPD
+begin
+
+subsection {* Class of representable types *}
+
+text "Overloaded embedding and projection functions between
+      a representable type and the universal domain."
+
+class rep = bifinite +
+  fixes emb :: "'a::pcpo \<rightarrow> udom"
+  fixes prj :: "udom \<rightarrow> 'a::pcpo"
+  assumes ep_pair_emb_prj: "ep_pair emb prj"
+
+interpretation rep!:
+  pcpo_ep_pair
+    "emb :: 'a::rep \<rightarrow> udom"
+    "prj :: udom \<rightarrow> 'a::rep"
+  unfolding pcpo_ep_pair_def
+  by (rule ep_pair_emb_prj)
+
+lemmas emb_inverse = rep.e_inverse
+lemmas emb_prj_below = rep.e_p_below
+lemmas emb_eq_iff = rep.e_eq_iff
+lemmas emb_strict = rep.e_strict
+lemmas prj_strict = rep.p_strict
+
+
+subsection {* Making @{term rep} the default class *}
+
+text {*
+  From now on, free type variables are assumed to be in class
+  @{term rep}, unless specified otherwise.
+*}
+
+defaultsort rep
+
+subsection {* Representations of types *}
+
+text "A TypeRep is an algebraic deflation over the universe of values."
+
+types TypeRep = "udom alg_defl"
+translations "TypeRep" \<leftharpoondown> (type) "udom alg_defl"
+
+definition
+  Rep_of :: "'a::rep itself \<Rightarrow> TypeRep"
+where
+  "Rep_of TYPE('a::rep) =
+    (\<Squnion>i. alg_defl_principal (Abs_fin_defl
+      (emb oo (approx i :: 'a \<rightarrow> 'a) oo prj)))"
+
+syntax "_REP" :: "type \<Rightarrow> TypeRep"  ("(1REP/(1'(_')))")
+translations "REP(t)" \<rightleftharpoons> "CONST Rep_of TYPE(t)"
+
+lemma cast_REP:
+  "cast\<cdot>REP('a::rep) = (emb::'a \<rightarrow> udom) oo (prj::udom \<rightarrow> 'a)"
+proof -
+  let ?a = "\<lambda>i. emb oo approx i oo (prj::udom \<rightarrow> 'a)"
+  have a: "\<And>i. finite_deflation (?a i)"
+    apply (rule rep.finite_deflation_e_d_p)
+    apply (rule finite_deflation_approx)
+    done
+  show ?thesis
+    unfolding Rep_of_def
+    apply (subst contlub_cfun_arg)
+    apply (rule chainI)
+    apply (rule alg_defl.principal_mono)
+    apply (rule Abs_fin_defl_mono [OF a a])
+    apply (rule chainE, simp)
+    apply (subst cast_alg_defl_principal)
+    apply (simp add: Abs_fin_defl_inverse a)
+    apply (simp add: expand_cfun_eq lub_distribs)
+    done
+qed
+
+lemma emb_prj: "emb\<cdot>((prj\<cdot>x)::'a::rep) = cast\<cdot>REP('a)\<cdot>x"
+by (simp add: cast_REP)
+
+lemma in_REP_iff:
+  "x ::: REP('a::rep) \<longleftrightarrow> emb\<cdot>((prj\<cdot>x)::'a) = x"
+by (simp add: in_deflation_def cast_REP)
+
+lemma prj_inverse:
+  "x ::: REP('a::rep) \<Longrightarrow> emb\<cdot>((prj\<cdot>x)::'a) = x"
+by (simp only: in_REP_iff)
+
+lemma emb_in_REP [simp]:
+  "emb\<cdot>(x::'a::rep) ::: REP('a)"
+by (simp add: in_REP_iff)
+
+subsection {* Coerce operator *}
+
+definition coerce :: "'a \<rightarrow> 'b"
+where "coerce = prj oo emb"
+
+lemma beta_coerce: "coerce\<cdot>x = prj\<cdot>(emb\<cdot>x)"
+by (simp add: coerce_def)
+
+lemma prj_emb: "prj\<cdot>(emb\<cdot>x) = coerce\<cdot>x"
+by (simp add: coerce_def)
+
+lemma coerce_strict [simp]: "coerce\<cdot>\<bottom> = \<bottom>"
+by (simp add: coerce_def)
+
+lemma coerce_eq_ID [simp]: "(coerce :: 'a \<rightarrow> 'a) = ID"
+by (rule ext_cfun, simp add: beta_coerce)
+
+lemma emb_coerce:
+  "REP('a) \<sqsubseteq> REP('b)
+   \<Longrightarrow> emb\<cdot>((coerce::'a \<rightarrow> 'b)\<cdot>x) = emb\<cdot>x"
+ apply (simp add: beta_coerce)
+ apply (rule prj_inverse)
+ apply (erule subdeflationD)
+ apply (rule emb_in_REP)
+done
+
+lemma coerce_prj:
+  "REP('a) \<sqsubseteq> REP('b)
+   \<Longrightarrow> (coerce::'b \<rightarrow> 'a)\<cdot>(prj\<cdot>x) = prj\<cdot>x"
+ apply (simp add: coerce_def)
+ apply (rule emb_eq_iff [THEN iffD1])
+ apply (simp only: emb_prj)
+ apply (rule deflation_below_comp1)
+   apply (rule deflation_cast)
+  apply (rule deflation_cast)
+ apply (erule monofun_cfun_arg)
+done
+
+lemma coerce_coerce [simp]:
+  "REP('a) \<sqsubseteq> REP('b)
+   \<Longrightarrow> coerce\<cdot>((coerce::'a \<rightarrow> 'b)\<cdot>x) = coerce\<cdot>x"
+by (simp add: beta_coerce prj_inverse subdeflationD)
+
+lemma coerce_inverse:
+  "emb\<cdot>(x::'a) ::: REP('b) \<Longrightarrow> coerce\<cdot>(coerce\<cdot>x :: 'b) = x"
+by (simp only: beta_coerce prj_inverse emb_inverse)
+
+lemma coerce_type:
+  "REP('a) \<sqsubseteq> REP('b)
+   \<Longrightarrow> emb\<cdot>((coerce::'a \<rightarrow> 'b)\<cdot>x) ::: REP('a)"
+by (simp add: beta_coerce prj_inverse subdeflationD)
+
+lemma ep_pair_coerce:
+  "REP('a) \<sqsubseteq> REP('b)
+   \<Longrightarrow> ep_pair (coerce::'a \<rightarrow> 'b) (coerce::'b \<rightarrow> 'a)"
+ apply (rule ep_pair.intro)
+  apply simp
+ apply (simp only: beta_coerce)
+ apply (rule below_trans)
+  apply (rule monofun_cfun_arg)
+  apply (rule emb_prj_below)
+ apply simp
+done
+
+subsection {* Proving a subtype is representable *}
+
+text {*
+  Temporarily relax type constraints for @{term "approx"},
+  @{term emb}, and @{term prj}.
+*}
+
+setup {* Sign.add_const_constraint
+  (@{const_name "approx"}, SOME @{typ "nat \<Rightarrow> 'a::cpo \<rightarrow> 'a"}) *}
+
+setup {* Sign.add_const_constraint
+  (@{const_name emb}, SOME @{typ "'a::pcpo \<rightarrow> udom"}) *}
+
+setup {* Sign.add_const_constraint
+  (@{const_name prj}, SOME @{typ "udom \<rightarrow> 'a::pcpo"}) *}
+
+lemma typedef_rep_class:
+  fixes Rep :: "'a::pcpo \<Rightarrow> udom"
+  fixes Abs :: "udom \<Rightarrow> 'a::pcpo"
+  fixes t :: TypeRep
+  assumes type: "type_definition Rep Abs {x. x ::: t}"
+  assumes below: "op \<sqsubseteq> \<equiv> \<lambda>x y. Rep x \<sqsubseteq> Rep y"
+  assumes emb: "emb = (\<Lambda> x. Rep x)"
+  assumes prj: "prj = (\<Lambda> x. Abs (cast\<cdot>t\<cdot>x))"
+  assumes approx:
+    "(approx :: nat \<Rightarrow> 'a \<rightarrow> 'a) = (\<lambda>i. prj oo cast\<cdot>(approx i\<cdot>t) oo emb)"
+  shows "OFCLASS('a, rep_class)"
+proof
+  have adm: "adm (\<lambda>x. x \<in> {x. x ::: t})"
+    by (simp add: adm_in_deflation)
+  have emb_beta: "\<And>x. emb\<cdot>x = Rep x"
+    unfolding emb
+    apply (rule beta_cfun)
+    apply (rule typedef_cont_Rep [OF type below adm])
+    done
+  have prj_beta: "\<And>y. prj\<cdot>y = Abs (cast\<cdot>t\<cdot>y)"
+    unfolding prj
+    apply (rule beta_cfun)
+    apply (rule typedef_cont_Abs [OF type below adm])
+    apply simp_all
+    done
+  have emb_in_deflation: "\<And>x::'a. emb\<cdot>x ::: t"
+    using type_definition.Rep [OF type]
+    by (simp add: emb_beta)
+  have prj_emb: "\<And>x::'a. prj\<cdot>(emb\<cdot>x) = x"
+    unfolding prj_beta
+    apply (simp add: cast_fixed [OF emb_in_deflation])
+    apply (simp add: emb_beta type_definition.Rep_inverse [OF type])
+    done
+  have emb_prj: "\<And>y. emb\<cdot>(prj\<cdot>y :: 'a) = cast\<cdot>t\<cdot>y"
+    unfolding prj_beta emb_beta
+    by (simp add: type_definition.Abs_inverse [OF type])
+  show "ep_pair (emb :: 'a \<rightarrow> udom) prj"
+    apply default
+    apply (simp add: prj_emb)
+    apply (simp add: emb_prj cast.below)
+    done
+  show "chain (approx :: nat \<Rightarrow> 'a \<rightarrow> 'a)"
+    unfolding approx by simp
+  show "\<And>x::'a. (\<Squnion>i. approx i\<cdot>x) = x"
+    unfolding approx
+    apply (simp add: lub_distribs)
+    apply (subst cast_fixed [OF emb_in_deflation])
+    apply (rule prj_emb)
+    done
+  have cast_cast_approx:
+    "\<And>i x. cast\<cdot>t\<cdot>(cast\<cdot>(approx i\<cdot>t)\<cdot>x) = cast\<cdot>(approx i\<cdot>t)\<cdot>x"
+    apply (rule cast_fixed)
+    apply (rule subdeflationD)
+    apply (rule approx.below)
+    apply (rule cast_in_deflation)
+    done
+  show "\<And>(i::nat) (x::'a). approx i\<cdot>(approx i\<cdot>x) = approx i\<cdot>x"
+    unfolding approx
+    apply simp
+    apply (simp add: emb_prj)
+    apply (simp add: cast_cast_approx)
+    done
+  show "\<And>i::nat. finite {x::'a. approx i\<cdot>x = x}"
+    apply (rule_tac B="(\<lambda>x. prj\<cdot>x) ` {x. cast\<cdot>(approx i\<cdot>t)\<cdot>x = x}"
+           in finite_subset)
+    apply (clarsimp simp add: approx)
+    apply (drule_tac f="\<lambda>x. emb\<cdot>x" in arg_cong)
+    apply (rule image_eqI)
+    apply (rule prj_emb [symmetric])
+    apply (simp add: emb_prj)
+    apply (simp add: cast_cast_approx)
+    apply (rule finite_imageI)
+    apply (simp add: cast_approx_fixed_iff)
+    apply (simp add: Collect_conj_eq)
+    apply (simp add: finite_fixes_approx)
+    done
+qed
+
+text {* Restore original typing constraints *}
+
+setup {* Sign.add_const_constraint
+  (@{const_name "approx"}, SOME @{typ "nat \<Rightarrow> 'a::profinite \<rightarrow> 'a"}) *}
+
+setup {* Sign.add_const_constraint
+  (@{const_name emb}, SOME @{typ "'a::rep \<rightarrow> udom"}) *}
+
+setup {* Sign.add_const_constraint
+  (@{const_name prj}, SOME @{typ "udom \<rightarrow> 'a::rep"}) *}
+
+lemma typedef_REP:
+  fixes Rep :: "'a::rep \<Rightarrow> udom"
+  fixes Abs :: "udom \<Rightarrow> 'a::rep"
+  fixes t :: TypeRep
+  assumes type: "type_definition Rep Abs {x. x ::: t}"
+  assumes below: "op \<sqsubseteq> \<equiv> \<lambda>x y. Rep x \<sqsubseteq> Rep y"
+  assumes emb: "emb = (\<Lambda> x. Rep x)"
+  assumes prj: "prj = (\<Lambda> x. Abs (cast\<cdot>t\<cdot>x))"
+  shows "REP('a) = t"
+proof -
+  have adm: "adm (\<lambda>x. x \<in> {x. x ::: t})"
+    by (simp add: adm_in_deflation)
+  have emb_beta: "\<And>x. emb\<cdot>x = Rep x"
+    unfolding emb
+    apply (rule beta_cfun)
+    apply (rule typedef_cont_Rep [OF type below adm])
+    done
+  have prj_beta: "\<And>y. prj\<cdot>y = Abs (cast\<cdot>t\<cdot>y)"
+    unfolding prj
+    apply (rule beta_cfun)
+    apply (rule typedef_cont_Abs [OF type below adm])
+    apply simp_all
+    done
+  have emb_in_deflation: "\<And>x::'a. emb\<cdot>x ::: t"
+    using type_definition.Rep [OF type]
+    by (simp add: emb_beta)
+  have prj_emb: "\<And>x::'a. prj\<cdot>(emb\<cdot>x) = x"
+    unfolding prj_beta
+    apply (simp add: cast_fixed [OF emb_in_deflation])
+    apply (simp add: emb_beta type_definition.Rep_inverse [OF type])
+    done
+  have emb_prj: "\<And>y. emb\<cdot>(prj\<cdot>y :: 'a) = cast\<cdot>t\<cdot>y"
+    unfolding prj_beta emb_beta
+    by (simp add: type_definition.Abs_inverse [OF type])
+  show "REP('a) = t"
+    apply (rule cast_eq_imp_eq, rule ext_cfun)
+    apply (simp add: cast_REP emb_prj)
+    done
+qed
+
+
+subsection {* Instances of class @{text rep} *}
+
+subsubsection {* Universal Domain *}
+
+text "The Universal Domain itself is trivially representable."
+
+instantiation udom :: rep
+begin
+
+definition emb_udom_def [simp]: "emb = (ID :: udom \<rightarrow> udom)"
+definition prj_udom_def [simp]: "prj = (ID :: udom \<rightarrow> udom)"
+
+instance
+ apply (intro_classes)
+ apply (simp_all add: ep_pair.intro)
+done
+
+end
+
+subsubsection {* Lifted types *}
+
+instantiation lift :: (countable) rep
+begin
+
+definition emb_lift_def:
+  "emb = udom_emb oo (FLIFT x. Def (to_nat x))"
+
+definition prj_lift_def:
+  "prj = (FLIFT n. if (\<exists>x::'a::countable. n = to_nat x)
+                    then Def (THE x::'a. n = to_nat x) else \<bottom>) oo udom_prj"
+
+instance
+ apply (intro_classes, unfold emb_lift_def prj_lift_def)
+ apply (rule ep_pair_comp [OF _ ep_pair_udom])
+ apply (rule ep_pair.intro)
+  apply (case_tac x, simp, simp)
+ apply (case_tac y, simp, clarsimp)
+done
+
+end
+
+subsubsection {* Representable type constructors *}
+
+text "Functions between representable types are representable."
+
+instantiation "->" :: (rep, rep) rep
+begin
+
+definition emb_cfun_def: "emb = udom_emb oo cfun_map\<cdot>prj\<cdot>emb"
+definition prj_cfun_def: "prj = cfun_map\<cdot>emb\<cdot>prj oo udom_prj"
+
+instance
+ apply (intro_classes, unfold emb_cfun_def prj_cfun_def)
+ apply (intro ep_pair_comp ep_pair_cfun_map ep_pair_emb_prj ep_pair_udom)
+done
+
+end
+
+text "Strict products of representable types are representable."
+
+instantiation "**" :: (rep, rep) rep
+begin
+
+definition emb_sprod_def: "emb = udom_emb oo sprod_map\<cdot>emb\<cdot>emb"
+definition prj_sprod_def: "prj = sprod_map\<cdot>prj\<cdot>prj oo udom_prj"
+
+instance
+ apply (intro_classes, unfold emb_sprod_def prj_sprod_def)
+ apply (intro ep_pair_comp ep_pair_sprod_map ep_pair_emb_prj ep_pair_udom)
+done
+
+end
+
+text "Strict sums of representable types are representable."
+
+instantiation "++" :: (rep, rep) rep
+begin
+
+definition emb_ssum_def: "emb = udom_emb oo ssum_map\<cdot>emb\<cdot>emb"
+definition prj_ssum_def: "prj = ssum_map\<cdot>prj\<cdot>prj oo udom_prj"
+
+instance
+ apply (intro_classes, unfold emb_ssum_def prj_ssum_def)
+ apply (intro ep_pair_comp ep_pair_ssum_map ep_pair_emb_prj ep_pair_udom)
+done
+
+end
+
+text "Up of a representable type is representable."
+
+instantiation "u" :: (rep) rep
+begin
+
+definition emb_u_def: "emb = udom_emb oo u_map\<cdot>emb"
+definition prj_u_def: "prj = u_map\<cdot>prj oo udom_prj"
+
+instance
+ apply (intro_classes, unfold emb_u_def prj_u_def)
+ apply (intro ep_pair_comp ep_pair_u_map ep_pair_emb_prj ep_pair_udom)
+done
+
+end
+
+text "Cartesian products of representable types are representable."
+
+instantiation "*" :: (rep, rep) rep
+begin
+
+definition emb_cprod_def: "emb = udom_emb oo cprod_map\<cdot>emb\<cdot>emb"
+definition prj_cprod_def: "prj = cprod_map\<cdot>prj\<cdot>prj oo udom_prj"
+
+instance
+ apply (intro_classes, unfold emb_cprod_def prj_cprod_def)
+ apply (intro ep_pair_comp ep_pair_cprod_map ep_pair_emb_prj ep_pair_udom)
+done
+
+end
+
+text "Upper powerdomain of a representable type is representable."
+
+instantiation upper_pd :: (rep) rep
+begin
+
+definition emb_upper_pd_def: "emb = udom_emb oo upper_map\<cdot>emb"
+definition prj_upper_pd_def: "prj = upper_map\<cdot>prj oo udom_prj"
+
+instance
+ apply (intro_classes, unfold emb_upper_pd_def prj_upper_pd_def)
+ apply (intro ep_pair_comp ep_pair_upper_map ep_pair_emb_prj ep_pair_udom)
+done
+
+end
+
+text "Lower powerdomain of a representable type is representable."
+
+instantiation lower_pd :: (rep) rep
+begin
+
+definition emb_lower_pd_def: "emb = udom_emb oo lower_map\<cdot>emb"
+definition prj_lower_pd_def: "prj = lower_map\<cdot>prj oo udom_prj"
+
+instance
+ apply (intro_classes, unfold emb_lower_pd_def prj_lower_pd_def)
+ apply (intro ep_pair_comp ep_pair_lower_map ep_pair_emb_prj ep_pair_udom)
+done
+
+end
+
+text "Convex powerdomain of a representable type is representable."
+
+instantiation convex_pd :: (rep) rep
+begin
+
+definition emb_convex_pd_def: "emb = udom_emb oo convex_map\<cdot>emb"
+definition prj_convex_pd_def: "prj = convex_map\<cdot>prj oo udom_prj"
+
+instance
+ apply (intro_classes, unfold emb_convex_pd_def prj_convex_pd_def)
+ apply (intro ep_pair_comp ep_pair_convex_map ep_pair_emb_prj ep_pair_udom)
+done
+
+end
+
+subsection {* Finite deflation lemmas *}
+
+text "TODO: move these lemmas somewhere else"
+
+lemma finite_compact_range_imp_finite_range:
+  fixes d :: "'a::profinite \<rightarrow> 'b::cpo"
+  assumes "finite ((\<lambda>x. d\<cdot>x) ` {x. compact x})"
+  shows "finite (range (\<lambda>x. d\<cdot>x))"
+proof (rule finite_subset [OF _ prems])
+  {
+    fix x :: 'a
+    have "range (\<lambda>i. d\<cdot>(approx i\<cdot>x)) \<subseteq> (\<lambda>x. d\<cdot>x) ` {x. compact x}"
+      by auto
+    hence "finite (range (\<lambda>i. d\<cdot>(approx i\<cdot>x)))"
+      using prems by (rule finite_subset)
+    hence "finite_chain (\<lambda>i. d\<cdot>(approx i\<cdot>x))"
+      by (simp add: finite_range_imp_finch)
+    hence "\<exists>i. (\<Squnion>i. d\<cdot>(approx i\<cdot>x)) = d\<cdot>(approx i\<cdot>x)"
+      by (simp add: finite_chain_def maxinch_is_thelub)
+    hence "\<exists>i. d\<cdot>x = d\<cdot>(approx i\<cdot>x)"
+      by (simp add: lub_distribs)
+    hence "d\<cdot>x \<in> (\<lambda>x. d\<cdot>x) ` {x. compact x}"
+      by auto
+  }
+  thus "range (\<lambda>x. d\<cdot>x) \<subseteq> (\<lambda>x. d\<cdot>x) ` {x. compact x}"
+    by clarsimp
+qed
+
+lemma finite_deflation_upper_map:
+  assumes "finite_deflation d" shows "finite_deflation (upper_map\<cdot>d)"
+proof (intro finite_deflation.intro finite_deflation_axioms.intro)
+  interpret d: finite_deflation d by fact
+  have "deflation d" by fact
+  thus "deflation (upper_map\<cdot>d)" by (rule deflation_upper_map)
+  have "finite (range (\<lambda>x. d\<cdot>x))" by (rule d.finite_range)
+  hence "finite (Rep_compact_basis -` range (\<lambda>x. d\<cdot>x))"
+    by (rule finite_vimageI, simp add: inj_on_def Rep_compact_basis_inject)
+  hence "finite (Pow (Rep_compact_basis -` range (\<lambda>x. d\<cdot>x)))" by simp
+  hence "finite (Rep_pd_basis -` (Pow (Rep_compact_basis -` range (\<lambda>x. d\<cdot>x))))"
+    by (rule finite_vimageI, simp add: inj_on_def Rep_pd_basis_inject)
+  hence "finite (upper_principal ` Rep_pd_basis -` (Pow (Rep_compact_basis -` range (\<lambda>x. d\<cdot>x))))" by simp
+  hence "finite ((\<lambda>xs. upper_map\<cdot>d\<cdot>xs) ` range upper_principal)"
+    apply (rule finite_subset [COMP swap_prems_rl])
+    apply (clarsimp, rename_tac t)
+    apply (induct_tac t rule: pd_basis_induct)
+    apply (simp only: upper_unit_Rep_compact_basis [symmetric] upper_map_unit)
+    apply (subgoal_tac "\<exists>b. d\<cdot>(Rep_compact_basis a) = Rep_compact_basis b")
+    apply clarsimp
+    apply (rule imageI)
+    apply (rule vimageI2)
+    apply (simp add: Rep_PDUnit)
+    apply (rule image_eqI)
+    apply (erule sym)
+    apply simp
+    apply (rule exI)
+    apply (rule Abs_compact_basis_inverse [symmetric])
+    apply (simp add: d.compact)
+    apply (simp only: upper_plus_principal [symmetric] upper_map_plus)
+    apply clarsimp
+    apply (rule imageI)
+    apply (rule vimageI2)
+    apply (simp add: Rep_PDPlus)
+    done
+  moreover have "{xs::'a upper_pd. compact xs} = range upper_principal"
+    by (auto dest: upper_pd.compact_imp_principal)
+  ultimately have "finite ((\<lambda>xs. upper_map\<cdot>d\<cdot>xs) ` {xs::'a upper_pd. compact xs})"
+    by simp
+  hence "finite (range (\<lambda>xs. upper_map\<cdot>d\<cdot>xs))"
+    by (rule finite_compact_range_imp_finite_range)
+  thus "finite {xs. upper_map\<cdot>d\<cdot>xs = xs}"
+    by (rule finite_range_imp_finite_fixes)
+qed
+
+lemma finite_deflation_lower_map:
+  assumes "finite_deflation d" shows "finite_deflation (lower_map\<cdot>d)"
+proof (intro finite_deflation.intro finite_deflation_axioms.intro)
+  interpret d: finite_deflation d by fact
+  have "deflation d" by fact
+  thus "deflation (lower_map\<cdot>d)" by (rule deflation_lower_map)
+  have "finite (range (\<lambda>x. d\<cdot>x))" by (rule d.finite_range)
+  hence "finite (Rep_compact_basis -` range (\<lambda>x. d\<cdot>x))"
+    by (rule finite_vimageI, simp add: inj_on_def Rep_compact_basis_inject)
+  hence "finite (Pow (Rep_compact_basis -` range (\<lambda>x. d\<cdot>x)))" by simp
+  hence "finite (Rep_pd_basis -` (Pow (Rep_compact_basis -` range (\<lambda>x. d\<cdot>x))))"
+    by (rule finite_vimageI, simp add: inj_on_def Rep_pd_basis_inject)
+  hence "finite (lower_principal ` Rep_pd_basis -` (Pow (Rep_compact_basis -` range (\<lambda>x. d\<cdot>x))))" by simp
+  hence "finite ((\<lambda>xs. lower_map\<cdot>d\<cdot>xs) ` range lower_principal)"
+    apply (rule finite_subset [COMP swap_prems_rl])
+    apply (clarsimp, rename_tac t)
+    apply (induct_tac t rule: pd_basis_induct)
+    apply (simp only: lower_unit_Rep_compact_basis [symmetric] lower_map_unit)
+    apply (subgoal_tac "\<exists>b. d\<cdot>(Rep_compact_basis a) = Rep_compact_basis b")
+    apply clarsimp
+    apply (rule imageI)
+    apply (rule vimageI2)
+    apply (simp add: Rep_PDUnit)
+    apply (rule image_eqI)
+    apply (erule sym)
+    apply simp
+    apply (rule exI)
+    apply (rule Abs_compact_basis_inverse [symmetric])
+    apply (simp add: d.compact)
+    apply (simp only: lower_plus_principal [symmetric] lower_map_plus)
+    apply clarsimp
+    apply (rule imageI)
+    apply (rule vimageI2)
+    apply (simp add: Rep_PDPlus)
+    done
+  moreover have "{xs::'a lower_pd. compact xs} = range lower_principal"
+    by (auto dest: lower_pd.compact_imp_principal)
+  ultimately have "finite ((\<lambda>xs. lower_map\<cdot>d\<cdot>xs) ` {xs::'a lower_pd. compact xs})"
+    by simp
+  hence "finite (range (\<lambda>xs. lower_map\<cdot>d\<cdot>xs))"
+    by (rule finite_compact_range_imp_finite_range)
+  thus "finite {xs. lower_map\<cdot>d\<cdot>xs = xs}"
+    by (rule finite_range_imp_finite_fixes)
+qed
+
+lemma finite_deflation_convex_map:
+  assumes "finite_deflation d" shows "finite_deflation (convex_map\<cdot>d)"
+proof (intro finite_deflation.intro finite_deflation_axioms.intro)
+  interpret d: finite_deflation d by fact
+  have "deflation d" by fact
+  thus "deflation (convex_map\<cdot>d)" by (rule deflation_convex_map)
+  have "finite (range (\<lambda>x. d\<cdot>x))" by (rule d.finite_range)
+  hence "finite (Rep_compact_basis -` range (\<lambda>x. d\<cdot>x))"
+    by (rule finite_vimageI, simp add: inj_on_def Rep_compact_basis_inject)
+  hence "finite (Pow (Rep_compact_basis -` range (\<lambda>x. d\<cdot>x)))" by simp
+  hence "finite (Rep_pd_basis -` (Pow (Rep_compact_basis -` range (\<lambda>x. d\<cdot>x))))"
+    by (rule finite_vimageI, simp add: inj_on_def Rep_pd_basis_inject)
+  hence "finite (convex_principal ` Rep_pd_basis -` (Pow (Rep_compact_basis -` range (\<lambda>x. d\<cdot>x))))" by simp
+  hence "finite ((\<lambda>xs. convex_map\<cdot>d\<cdot>xs) ` range convex_principal)"
+    apply (rule finite_subset [COMP swap_prems_rl])
+    apply (clarsimp, rename_tac t)
+    apply (induct_tac t rule: pd_basis_induct)
+    apply (simp only: convex_unit_Rep_compact_basis [symmetric] convex_map_unit)
+    apply (subgoal_tac "\<exists>b. d\<cdot>(Rep_compact_basis a) = Rep_compact_basis b")
+    apply clarsimp
+    apply (rule imageI)
+    apply (rule vimageI2)
+    apply (simp add: Rep_PDUnit)
+    apply (rule image_eqI)
+    apply (erule sym)
+    apply simp
+    apply (rule exI)
+    apply (rule Abs_compact_basis_inverse [symmetric])
+    apply (simp add: d.compact)
+    apply (simp only: convex_plus_principal [symmetric] convex_map_plus)
+    apply clarsimp
+    apply (rule imageI)
+    apply (rule vimageI2)
+    apply (simp add: Rep_PDPlus)
+    done
+  moreover have "{xs::'a convex_pd. compact xs} = range convex_principal"
+    by (auto dest: convex_pd.compact_imp_principal)
+  ultimately have "finite ((\<lambda>xs. convex_map\<cdot>d\<cdot>xs) ` {xs::'a convex_pd. compact xs})"
+    by simp
+  hence "finite (range (\<lambda>xs. convex_map\<cdot>d\<cdot>xs))"
+    by (rule finite_compact_range_imp_finite_range)
+  thus "finite {xs. convex_map\<cdot>d\<cdot>xs = xs}"
+    by (rule finite_range_imp_finite_fixes)
+qed
+
+subsection {* Type combinators *}
+
+definition
+  TypeRep_fun1 ::
+    "((udom \<rightarrow> udom) \<rightarrow> ('a \<rightarrow> 'a))
+      \<Rightarrow> (TypeRep \<rightarrow> TypeRep)"
+where
+  "TypeRep_fun1 f =
+    alg_defl.basis_fun (\<lambda>a.
+      alg_defl_principal (
+        Abs_fin_defl (udom_emb oo f\<cdot>(Rep_fin_defl a) oo udom_prj)))"
+
+definition
+  TypeRep_fun2 ::
+    "((udom \<rightarrow> udom) \<rightarrow> (udom \<rightarrow> udom) \<rightarrow> ('a \<rightarrow> 'a))
+      \<Rightarrow> (TypeRep \<rightarrow> TypeRep \<rightarrow> TypeRep)"
+where
+  "TypeRep_fun2 f =
+    alg_defl.basis_fun (\<lambda>a.
+      alg_defl.basis_fun (\<lambda>b.
+        alg_defl_principal (
+          Abs_fin_defl (udom_emb oo
+            f\<cdot>(Rep_fin_defl a)\<cdot>(Rep_fin_defl b) oo udom_prj))))"
+
+definition "one_typ = REP(one)"
+definition "tr_typ = REP(tr)"
+definition "cfun_typ = TypeRep_fun2 cfun_map"
+definition "ssum_typ = TypeRep_fun2 ssum_map"
+definition "sprod_typ = TypeRep_fun2 sprod_map"
+definition "cprod_typ = TypeRep_fun2 cprod_map"
+definition "u_typ = TypeRep_fun1 u_map"
+definition "upper_typ = TypeRep_fun1 upper_map"
+definition "lower_typ = TypeRep_fun1 lower_map"
+definition "convex_typ = TypeRep_fun1 convex_map"
+
+lemma Rep_fin_defl_mono: "a \<sqsubseteq> b \<Longrightarrow> Rep_fin_defl a \<sqsubseteq> Rep_fin_defl b"
+unfolding below_fin_defl_def .
+
+lemma cast_TypeRep_fun1:
+  assumes f: "\<And>a. finite_deflation a \<Longrightarrow> finite_deflation (f\<cdot>a)"
+  shows "cast\<cdot>(TypeRep_fun1 f\<cdot>A) = udom_emb oo f\<cdot>(cast\<cdot>A) oo udom_prj"
+proof -
+  have 1: "\<And>a. finite_deflation (udom_emb oo f\<cdot>(Rep_fin_defl a) oo udom_prj)"
+    apply (rule ep_pair.finite_deflation_e_d_p [OF ep_pair_udom])
+    apply (rule f, rule finite_deflation_Rep_fin_defl)
+    done
+  show ?thesis
+    by (induct A rule: alg_defl.principal_induct, simp)
+       (simp only: TypeRep_fun1_def
+                   alg_defl.basis_fun_principal
+                   alg_defl.basis_fun_mono
+                   alg_defl.principal_mono
+                   Abs_fin_defl_mono [OF 1 1]
+                   monofun_cfun below_refl
+                   Rep_fin_defl_mono
+                   cast_alg_defl_principal
+                   Abs_fin_defl_inverse [unfolded mem_Collect_eq, OF 1])
+qed
+
+lemma cast_TypeRep_fun2:
+  assumes f: "\<And>a b. finite_deflation a \<Longrightarrow> finite_deflation b \<Longrightarrow>
+                finite_deflation (f\<cdot>a\<cdot>b)"
+  shows "cast\<cdot>(TypeRep_fun2 f\<cdot>A\<cdot>B) = udom_emb oo f\<cdot>(cast\<cdot>A)\<cdot>(cast\<cdot>B) oo udom_prj"
+proof -
+  have 1: "\<And>a b. finite_deflation
+           (udom_emb oo f\<cdot>(Rep_fin_defl a)\<cdot>(Rep_fin_defl b) oo udom_prj)"
+    apply (rule ep_pair.finite_deflation_e_d_p [OF ep_pair_udom])
+    apply (rule f, (rule finite_deflation_Rep_fin_defl)+)
+    done
+  show ?thesis
+    by (induct A B rule: alg_defl.principal_induct2, simp, simp)
+       (simp only: TypeRep_fun2_def
+                   alg_defl.basis_fun_principal
+                   alg_defl.basis_fun_mono
+                   alg_defl.principal_mono
+                   Abs_fin_defl_mono [OF 1 1]
+                   monofun_cfun below_refl
+                   Rep_fin_defl_mono
+                   cast_alg_defl_principal
+                   Abs_fin_defl_inverse [unfolded mem_Collect_eq, OF 1])
+qed
+
+lemma cast_cfun_typ:
+  "cast\<cdot>(cfun_typ\<cdot>A\<cdot>B) = udom_emb oo cfun_map\<cdot>(cast\<cdot>A)\<cdot>(cast\<cdot>B) oo udom_prj"
+unfolding cfun_typ_def
+apply (rule cast_TypeRep_fun2)
+apply (erule (1) finite_deflation_cfun_map)
+done
+
+lemma cast_ssum_typ:
+  "cast\<cdot>(ssum_typ\<cdot>A\<cdot>B) = udom_emb oo ssum_map\<cdot>(cast\<cdot>A)\<cdot>(cast\<cdot>B) oo udom_prj"
+unfolding ssum_typ_def
+apply (rule cast_TypeRep_fun2)
+apply (erule (1) finite_deflation_ssum_map)
+done
+
+lemma cast_sprod_typ:
+  "cast\<cdot>(sprod_typ\<cdot>A\<cdot>B) = udom_emb oo sprod_map\<cdot>(cast\<cdot>A)\<cdot>(cast\<cdot>B) oo udom_prj"
+unfolding sprod_typ_def
+apply (rule cast_TypeRep_fun2)
+apply (erule (1) finite_deflation_sprod_map)
+done
+
+lemma cast_cprod_typ:
+  "cast\<cdot>(cprod_typ\<cdot>A\<cdot>B) = udom_emb oo cprod_map\<cdot>(cast\<cdot>A)\<cdot>(cast\<cdot>B) oo udom_prj"
+unfolding cprod_typ_def
+apply (rule cast_TypeRep_fun2)
+apply (erule (1) finite_deflation_cprod_map)
+done
+
+lemma cast_u_typ:
+  "cast\<cdot>(u_typ\<cdot>A) = udom_emb oo u_map\<cdot>(cast\<cdot>A) oo udom_prj"
+unfolding u_typ_def
+apply (rule cast_TypeRep_fun1)
+apply (erule finite_deflation_u_map)
+done
+
+lemma cast_upper_typ:
+  "cast\<cdot>(upper_typ\<cdot>A) = udom_emb oo upper_map\<cdot>(cast\<cdot>A) oo udom_prj"
+unfolding upper_typ_def
+apply (rule cast_TypeRep_fun1)
+apply (erule finite_deflation_upper_map)
+done
+
+lemma cast_lower_typ:
+  "cast\<cdot>(lower_typ\<cdot>A) = udom_emb oo lower_map\<cdot>(cast\<cdot>A) oo udom_prj"
+unfolding lower_typ_def
+apply (rule cast_TypeRep_fun1)
+apply (erule finite_deflation_lower_map)
+done
+
+lemma cast_convex_typ:
+  "cast\<cdot>(convex_typ\<cdot>A) = udom_emb oo convex_map\<cdot>(cast\<cdot>A) oo udom_prj"
+unfolding convex_typ_def
+apply (rule cast_TypeRep_fun1)
+apply (erule finite_deflation_convex_map)
+done
+
+text {* REP of type constructor = type combinator *}
+
+lemma REP_one: "REP(one) = one_typ"
+by (simp only: one_typ_def)
+
+lemma REP_tr: "REP(tr) = tr_typ"
+by (simp only: tr_typ_def)
+
+lemma REP_cfun: "REP('a \<rightarrow> 'b) = cfun_typ\<cdot>REP('a)\<cdot>REP('b)"
+apply (rule cast_eq_imp_eq, rule ext_cfun)
+apply (simp add: cast_REP cast_cfun_typ)
+apply (simp add: cfun_map_def)
+apply (simp only: prj_cfun_def emb_cfun_def)
+apply (simp add: expand_cfun_eq ep_pair.e_eq_iff [OF ep_pair_udom])
+done
+
+
+lemma REP_ssum: "REP('a \<oplus> 'b) = ssum_typ\<cdot>REP('a)\<cdot>REP('b)"
+apply (rule cast_eq_imp_eq, rule ext_cfun)
+apply (simp add: cast_REP cast_ssum_typ)
+apply (simp add: prj_ssum_def)
+apply (simp add: emb_ssum_def)
+apply (simp add: ssum_map_map cfcomp1)
+done
+
+lemma REP_sprod: "REP('a \<otimes> 'b) = sprod_typ\<cdot>REP('a)\<cdot>REP('b)"
+apply (rule cast_eq_imp_eq, rule ext_cfun)
+apply (simp add: cast_REP cast_sprod_typ)
+apply (simp add: prj_sprod_def)
+apply (simp add: emb_sprod_def)
+apply (simp add: sprod_map_map cfcomp1)
+done
+
+lemma REP_cprod: "REP('a \<times> 'b) = cprod_typ\<cdot>REP('a)\<cdot>REP('b)"
+apply (rule cast_eq_imp_eq, rule ext_cfun)
+apply (simp add: cast_REP cast_cprod_typ)
+apply (simp add: prj_cprod_def)
+apply (simp add: emb_cprod_def)
+apply (simp add: cprod_map_map cfcomp1)
+done
+
+lemma REP_up: "REP('a u) = u_typ\<cdot>REP('a)"
+apply (rule cast_eq_imp_eq, rule ext_cfun)
+apply (simp add: cast_REP cast_u_typ)
+apply (simp add: prj_u_def)
+apply (simp add: emb_u_def)
+apply (simp add: u_map_map cfcomp1)
+done
+
+lemma REP_upper: "REP('a upper_pd) = upper_typ\<cdot>REP('a)"
+apply (rule cast_eq_imp_eq, rule ext_cfun)
+apply (simp add: cast_REP cast_upper_typ)
+apply (simp add: prj_upper_pd_def)
+apply (simp add: emb_upper_pd_def)
+apply (simp add: upper_map_map cfcomp1)
+done
+
+lemma REP_lower: "REP('a lower_pd) = lower_typ\<cdot>REP('a)"
+apply (rule cast_eq_imp_eq, rule ext_cfun)
+apply (simp add: cast_REP cast_lower_typ)
+apply (simp add: prj_lower_pd_def)
+apply (simp add: emb_lower_pd_def)
+apply (simp add: lower_map_map cfcomp1)
+done
+
+lemma REP_convex: "REP('a convex_pd) = convex_typ\<cdot>REP('a)"
+apply (rule cast_eq_imp_eq, rule ext_cfun)
+apply (simp add: cast_REP cast_convex_typ)
+apply (simp add: prj_convex_pd_def)
+apply (simp add: emb_convex_pd_def)
+apply (simp add: convex_map_map cfcomp1)
+done
+
+lemmas REP_simps =
+  REP_one
+  REP_tr
+  REP_cfun
+  REP_ssum
+  REP_sprod
+  REP_cprod
+  REP_up
+  REP_upper
+  REP_lower
+  REP_convex
+
+subsection {* Isomorphic deflations *}
+
+definition
+  isodefl :: "('a::rep \<rightarrow> 'a) \<Rightarrow> udom alg_defl \<Rightarrow> bool"
+where
+  "isodefl d t \<longleftrightarrow> cast\<cdot>t = emb oo d oo prj"
+
+lemma isodeflI: "(\<And>x. cast\<cdot>t\<cdot>x = emb\<cdot>(d\<cdot>(prj\<cdot>x))) \<Longrightarrow> isodefl d t"
+unfolding isodefl_def by (simp add: ext_cfun)
+
+lemma cast_isodefl: "isodefl d t \<Longrightarrow> cast\<cdot>t = (\<Lambda> x. emb\<cdot>(d\<cdot>(prj\<cdot>x)))"
+unfolding isodefl_def by (simp add: ext_cfun)
+
+lemma isodefl_strict: "isodefl d t \<Longrightarrow> d\<cdot>\<bottom> = \<bottom>"
+unfolding isodefl_def
+by (drule cfun_fun_cong [where x="\<bottom>"], simp)
+
+lemma isodefl_imp_deflation:
+  fixes d :: "'a::rep \<rightarrow> 'a"
+  assumes "isodefl d t" shows "deflation d"
+proof
+  note prems [unfolded isodefl_def, simp]
+  fix x :: 'a
+  show "d\<cdot>(d\<cdot>x) = d\<cdot>x"
+    using cast.idem [of t "emb\<cdot>x"] by simp
+  show "d\<cdot>x \<sqsubseteq> x"
+    using cast.below [of t "emb\<cdot>x"] by simp
+qed
+
+lemma isodefl_ID_REP: "isodefl (ID :: 'a \<rightarrow> 'a) REP('a)"
+unfolding isodefl_def by (simp add: cast_REP)
+
+lemma isodefl_REP_imp_ID: "isodefl (d :: 'a \<rightarrow> 'a) REP('a) \<Longrightarrow> d = ID"
+unfolding isodefl_def
+apply (simp add: cast_REP)
+apply (simp add: expand_cfun_eq)
+apply (rule allI)
+apply (drule_tac x="emb\<cdot>x" in spec)
+apply simp
+done
+
+lemma isodefl_bottom: "isodefl \<bottom> \<bottom>"
+unfolding isodefl_def by (simp add: expand_cfun_eq)
+
+lemma adm_isodefl:
+  "cont f \<Longrightarrow> cont g \<Longrightarrow> adm (\<lambda>x. isodefl (f x) (g x))"
+unfolding isodefl_def by simp
+
+lemma isodefl_lub:
+  assumes "chain d" and "chain t"
+  assumes "\<And>i. isodefl (d i) (t i)"
+  shows "isodefl (\<Squnion>i. d i) (\<Squnion>i. t i)"
+using prems unfolding isodefl_def
+by (simp add: contlub_cfun_arg contlub_cfun_fun)
+
+lemma isodefl_fix:
+  assumes "\<And>d t. isodefl d t \<Longrightarrow> isodefl (f\<cdot>d) (g\<cdot>t)"
+  shows "isodefl (fix\<cdot>f) (fix\<cdot>g)"
+unfolding fix_def2
+apply (rule isodefl_lub, simp, simp)
+apply (induct_tac i)
+apply (simp add: isodefl_bottom)
+apply (simp add: prems)
+done
+
+lemma isodefl_coerce:
+  fixes d :: "'a \<rightarrow> 'a"
+  assumes REP: "REP('b) = REP('a)"
+  shows "isodefl d t \<Longrightarrow> isodefl (coerce oo d oo coerce :: 'b \<rightarrow> 'b) t"
+unfolding isodefl_def
+apply (simp add: expand_cfun_eq)
+apply (simp add: emb_coerce coerce_prj REP)
+done
+
+lemma isodefl_cfun:
+  "isodefl d1 t1 \<Longrightarrow> isodefl d2 t2 \<Longrightarrow>
+    isodefl (cfun_map\<cdot>d1\<cdot>d2) (cfun_typ\<cdot>t1\<cdot>t2)"
+apply (rule isodeflI)
+apply (simp add: cast_cfun_typ cast_isodefl)
+apply (simp add: emb_cfun_def prj_cfun_def)
+apply (simp add: cfun_map_map cfcomp1)
+done
+
+lemma isodefl_ssum:
+  "isodefl d1 t1 \<Longrightarrow> isodefl d2 t2 \<Longrightarrow>
+    isodefl (ssum_map\<cdot>d1\<cdot>d2) (ssum_typ\<cdot>t1\<cdot>t2)"
+apply (rule isodeflI)
+apply (simp add: cast_ssum_typ cast_isodefl)
+apply (simp add: emb_ssum_def prj_ssum_def)
+apply (simp add: ssum_map_map isodefl_strict)
+done
+
+lemma isodefl_sprod:
+  "isodefl d1 t1 \<Longrightarrow> isodefl d2 t2 \<Longrightarrow>
+    isodefl (sprod_map\<cdot>d1\<cdot>d2) (sprod_typ\<cdot>t1\<cdot>t2)"
+apply (rule isodeflI)
+apply (simp add: cast_sprod_typ cast_isodefl)
+apply (simp add: emb_sprod_def prj_sprod_def)
+apply (simp add: sprod_map_map isodefl_strict)
+done
+
+lemma isodefl_u:
+  "isodefl d t \<Longrightarrow> isodefl (u_map\<cdot>d) (u_typ\<cdot>t)"
+apply (rule isodeflI)
+apply (simp add: cast_u_typ cast_isodefl)
+apply (simp add: emb_u_def prj_u_def)
+apply (simp add: u_map_map)
+done
+
+lemma isodefl_one: "isodefl (ID :: one \<rightarrow> one) one_typ"
+unfolding one_typ_def by (rule isodefl_ID_REP)
+
+lemma isodefl_tr: "isodefl (ID :: tr \<rightarrow> tr) tr_typ"
+unfolding tr_typ_def by (rule isodefl_ID_REP)
+
+lemma isodefl_upper:
+  "isodefl d t \<Longrightarrow> isodefl (upper_map\<cdot>d) (upper_typ\<cdot>t)"
+apply (rule isodeflI)
+apply (simp add: cast_upper_typ cast_isodefl)
+apply (simp add: emb_upper_pd_def prj_upper_pd_def)
+apply (simp add: upper_map_map)
+done
+
+lemma isodefl_lower:
+  "isodefl d t \<Longrightarrow> isodefl (lower_map\<cdot>d) (lower_typ\<cdot>t)"
+apply (rule isodeflI)
+apply (simp add: cast_lower_typ cast_isodefl)
+apply (simp add: emb_lower_pd_def prj_lower_pd_def)
+apply (simp add: lower_map_map)
+done
+
+lemma isodefl_convex:
+  "isodefl d t \<Longrightarrow> isodefl (convex_map\<cdot>d) (convex_typ\<cdot>t)"
+apply (rule isodeflI)
+apply (simp add: cast_convex_typ cast_isodefl)
+apply (simp add: emb_convex_pd_def prj_convex_pd_def)
+apply (simp add: convex_map_map)
+done
+
+end