src/HOLCF/Representable.thy
 author huffman Fri, 05 Mar 2010 14:05:25 -0800 changeset 35596 49a02dab35ed parent 35595 1785d387627a child 35652 05ca920cd94b permissions -rw-r--r--
```
(*  Title:      HOLCF/Representable.thy
Author:     Brian Huffman
*)

theory Representable
imports Algebraic Universal Ssum Sprod One Fixrec
uses
("Tools/repdef.ML")
("Tools/Domain/domain_take_proofs.ML")
("Tools/Domain/domain_isomorphism.ML")
begin

subsection {* Preliminaries: Take proofs *}

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: "\<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 `m \<le> n` 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 `n \<le> m` have "min m n = n" by simp
ultimately show ?thesis by simp
qed

use "Tools/Domain/domain_take_proofs.ML"

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 (type) "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)
done
qed

lemma emb_prj: "emb\<cdot>((prj\<cdot>x)::'a::rep) = cast\<cdot>REP('a)\<cdot>x"

lemma in_REP_iff:
"x ::: REP('a::rep) \<longleftrightarrow> emb\<cdot>((prj\<cdot>x)::'a) = x"

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)"

subsection {* Coerce operator *}

definition coerce :: "'a \<rightarrow> 'b"
where "coerce = prj oo emb"

lemma beta_coerce: "coerce\<cdot>x = prj\<cdot>(emb\<cdot>x)"

lemma prj_emb: "prj\<cdot>(emb\<cdot>x) = coerce\<cdot>x"

lemma coerce_strict [simp]: "coerce\<cdot>\<bottom> = \<bottom>"

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 (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 (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

text {* Isomorphism lemmas used internally by the domain package: *}

lemma domain_abs_iso:
fixes abs and rep
assumes REP: "REP('b) = REP('a)"
assumes abs_def: "abs \<equiv> (coerce :: 'a \<rightarrow> 'b)"
assumes rep_def: "rep \<equiv> (coerce :: 'b \<rightarrow> 'a)"
shows "rep\<cdot>(abs\<cdot>x) = x"
unfolding abs_def rep_def by (simp add: REP)

lemma domain_rep_iso:
fixes abs and rep
assumes REP: "REP('b) = REP('a)"
assumes abs_def: "abs \<equiv> (coerce :: 'a \<rightarrow> 'b)"
assumes rep_def: "rep \<equiv> (coerce :: 'b \<rightarrow> 'a)"
shows "abs\<cdot>(rep\<cdot>x) = x"
unfolding abs_def rep_def by (simp add: REP [symmetric])

subsection {* Proving a subtype is representable *}

text {*
Temporarily relax type constraints for @{term "approx"},
@{term emb}, and @{term prj}.
*}

(@{const_name "approx"}, SOME @{typ "nat \<Rightarrow> 'a::cpo \<rightarrow> 'a"}) *}

(@{const_name emb}, SOME @{typ "'a::pcpo \<rightarrow> udom"}) *}

(@{const_name prj}, SOME @{typ "udom \<rightarrow> 'a::pcpo"}) *}

definition
repdef_approx ::
"('a::pcpo \<Rightarrow> udom) \<Rightarrow> (udom \<Rightarrow> 'a) \<Rightarrow> udom alg_defl \<Rightarrow> nat \<Rightarrow> 'a \<rightarrow> 'a"
where
"repdef_approx Rep Abs t = (\<lambda>i. \<Lambda> x. Abs (cast\<cdot>(approx i\<cdot>t)\<cdot>(Rep x)))"

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 \<equiv> (\<Lambda> x. Rep x)"
assumes prj: "prj \<equiv> (\<Lambda> x. Abs (cast\<cdot>t\<cdot>x))"
assumes approx: "(approx :: nat \<Rightarrow> 'a \<rightarrow> 'a) \<equiv> repdef_approx Rep Abs t"
shows "OFCLASS('a, rep_class)"
proof
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 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
have approx':
"approx = (\<lambda>i. \<Lambda>(x::'a). prj\<cdot>(cast\<cdot>(approx i\<cdot>t)\<cdot>(emb\<cdot>x)))"
unfolding approx repdef_approx_def
apply (subst cast_cast_approx [symmetric])
apply (simp add: prj_beta [symmetric] emb_beta [symmetric])
done
have emb_in_deflation: "\<And>x::'a. emb\<cdot>x ::: t"
using type_definition.Rep [OF type]
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
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 (subst cast_fixed [OF emb_in_deflation])
apply (rule prj_emb)
done
show "\<And>(i::nat) (x::'a). approx i\<cdot>(approx i\<cdot>x) = approx i\<cdot>x"
unfolding approx'
apply simp
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 (drule_tac f="\<lambda>x. emb\<cdot>x" in arg_cong)
apply (rule image_eqI)
apply (rule prj_emb [symmetric])
apply (rule finite_imageI)
done
qed

text {* Restore original typing constraints *}

(@{const_name "approx"}, SOME @{typ "nat \<Rightarrow> 'a::profinite \<rightarrow> 'a"}) *}

(@{const_name emb}, SOME @{typ "'a::rep \<rightarrow> udom"}) *}

(@{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 \<equiv> (\<Lambda> x. Rep x)"
assumes prj: "prj \<equiv> (\<Lambda> x. Abs (cast\<cdot>t\<cdot>x))"
shows "REP('a) = t"
proof -
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]
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)
done
qed

use "Tools/repdef.ML"

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)
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 cfun :: (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 sprod :: (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 ssum :: (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

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 "cfun_defl = TypeRep_fun2 cfun_map"
definition "ssum_defl = TypeRep_fun2 ssum_map"
definition "sprod_defl = TypeRep_fun2 sprod_map"
definition "cprod_defl = TypeRep_fun2 cprod_map"
definition "u_defl = TypeRep_fun1 u_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_defl:
"cast\<cdot>(cfun_defl\<cdot>A\<cdot>B) = udom_emb oo cfun_map\<cdot>(cast\<cdot>A)\<cdot>(cast\<cdot>B) oo udom_prj"
unfolding cfun_defl_def
apply (rule cast_TypeRep_fun2)
apply (erule (1) finite_deflation_cfun_map)
done

lemma cast_ssum_defl:
"cast\<cdot>(ssum_defl\<cdot>A\<cdot>B) = udom_emb oo ssum_map\<cdot>(cast\<cdot>A)\<cdot>(cast\<cdot>B) oo udom_prj"
unfolding ssum_defl_def
apply (rule cast_TypeRep_fun2)
apply (erule (1) finite_deflation_ssum_map)
done

lemma cast_sprod_defl:
"cast\<cdot>(sprod_defl\<cdot>A\<cdot>B) = udom_emb oo sprod_map\<cdot>(cast\<cdot>A)\<cdot>(cast\<cdot>B) oo udom_prj"
unfolding sprod_defl_def
apply (rule cast_TypeRep_fun2)
apply (erule (1) finite_deflation_sprod_map)
done

lemma cast_cprod_defl:
"cast\<cdot>(cprod_defl\<cdot>A\<cdot>B) = udom_emb oo cprod_map\<cdot>(cast\<cdot>A)\<cdot>(cast\<cdot>B) oo udom_prj"
unfolding cprod_defl_def
apply (rule cast_TypeRep_fun2)
apply (erule (1) finite_deflation_cprod_map)
done

lemma cast_u_defl:
"cast\<cdot>(u_defl\<cdot>A) = udom_emb oo u_map\<cdot>(cast\<cdot>A) oo udom_prj"
unfolding u_defl_def
apply (rule cast_TypeRep_fun1)
apply (erule finite_deflation_u_map)
done

text {* REP of type constructor = type combinator *}

lemma REP_cfun: "REP('a \<rightarrow> 'b) = cfun_defl\<cdot>REP('a)\<cdot>REP('b)"
apply (rule cast_eq_imp_eq, rule ext_cfun)
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_defl\<cdot>REP('a)\<cdot>REP('b)"
apply (rule cast_eq_imp_eq, rule ext_cfun)
done

lemma REP_sprod: "REP('a \<otimes> 'b) = sprod_defl\<cdot>REP('a)\<cdot>REP('b)"
apply (rule cast_eq_imp_eq, rule ext_cfun)
done

lemma REP_cprod: "REP('a \<times> 'b) = cprod_defl\<cdot>REP('a)\<cdot>REP('b)"
apply (rule cast_eq_imp_eq, rule ext_cfun)
done

lemma REP_up: "REP('a u) = u_defl\<cdot>REP('a)"
apply (rule cast_eq_imp_eq, rule ext_cfun)
done

lemmas REP_simps =
REP_cfun
REP_ssum
REP_sprod
REP_cprod
REP_up

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 (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)

"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

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)
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: emb_coerce coerce_prj REP)
done

lemma isodefl_abs_rep:
fixes abs and rep and d
assumes REP: "REP('b) = REP('a)"
assumes abs_def: "abs \<equiv> (coerce :: 'a \<rightarrow> 'b)"
assumes rep_def: "rep \<equiv> (coerce :: 'b \<rightarrow> 'a)"
shows "isodefl d t \<Longrightarrow> isodefl (abs oo d oo rep) t"
unfolding abs_def rep_def using REP by (rule isodefl_coerce)

lemma isodefl_cfun:
"isodefl d1 t1 \<Longrightarrow> isodefl d2 t2 \<Longrightarrow>
isodefl (cfun_map\<cdot>d1\<cdot>d2) (cfun_defl\<cdot>t1\<cdot>t2)"
apply (rule isodeflI)
done

lemma isodefl_ssum:
"isodefl d1 t1 \<Longrightarrow> isodefl d2 t2 \<Longrightarrow>
isodefl (ssum_map\<cdot>d1\<cdot>d2) (ssum_defl\<cdot>t1\<cdot>t2)"
apply (rule isodeflI)
done

lemma isodefl_sprod:
"isodefl d1 t1 \<Longrightarrow> isodefl d2 t2 \<Longrightarrow>
isodefl (sprod_map\<cdot>d1\<cdot>d2) (sprod_defl\<cdot>t1\<cdot>t2)"
apply (rule isodeflI)
done

lemma isodefl_cprod:
"isodefl d1 t1 \<Longrightarrow> isodefl d2 t2 \<Longrightarrow>
isodefl (cprod_map\<cdot>d1\<cdot>d2) (cprod_defl\<cdot>t1\<cdot>t2)"
apply (rule isodeflI)
done

lemma isodefl_u:
"isodefl d t \<Longrightarrow> isodefl (u_map\<cdot>d) (u_defl\<cdot>t)"
apply (rule isodeflI)
done

subsection {* Constructing Domain Isomorphisms *}

use "Tools/Domain/domain_isomorphism.ML"

setup {*
[(@{type_name cfun}, @{term cfun_defl}, @{const_name cfun_map}, @{thm REP_cfun},
@{thm isodefl_cfun}, @{thm cfun_map_ID}, @{thm deflation_cfun_map}),

(@{type_name ssum}, @{term ssum_defl}, @{const_name ssum_map}, @{thm REP_ssum},
@{thm isodefl_ssum}, @{thm ssum_map_ID}, @{thm deflation_ssum_map}),

(@{type_name sprod}, @{term sprod_defl}, @{const_name sprod_map}, @{thm REP_sprod},
@{thm isodefl_sprod}, @{thm sprod_map_ID}, @{thm deflation_sprod_map}),

(@{type_name "*"}, @{term cprod_defl}, @{const_name cprod_map}, @{thm REP_cprod},
@{thm isodefl_cprod}, @{thm cprod_map_ID}, @{thm deflation_cprod_map}),

(@{type_name "u"}, @{term u_defl}, @{const_name u_map}, @{thm REP_up},
@{thm isodefl_u}, @{thm u_map_ID}, @{thm deflation_u_map})]
*}

end
```