(* Title: HOL/HOLCF/Domain.thy
Author: Brian Huffman
*)
section \<open>Domain package\<close>
theory Domain
imports Representable Map_Functions Fixrec
keywords
"lazy" "unsafe" and
"domaindef" "domain" :: thy_defn and
"domain_isomorphism" :: thy_decl
begin
subsection \<open>Continuous isomorphisms\<close>
text \<open>A locale for continuous isomorphisms\<close>
locale iso =
fixes abs :: "'a::pcpo \<rightarrow> 'b::pcpo"
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>
lemma emb_prj: "emb\<cdot>((prj\<cdot>x)::'a::domain) = cast\<cdot>DEFL('a)\<cdot>x"
by (simp add: cast_DEFL)
lemma emb_prj_emb:
fixes x :: "'a::domain"
assumes "DEFL('a) \<sqsubseteq> DEFL('b)"
shows "emb\<cdot>(prj\<cdot>(emb\<cdot>x) :: 'b::domain) = emb\<cdot>x"
unfolding emb_prj
apply (rule cast.belowD)
apply (rule monofun_cfun_arg [OF assms])
apply (simp add: cast_DEFL)
done
lemma prj_emb_prj:
assumes "DEFL('a::domain) \<sqsubseteq> DEFL('b::domain)"
shows "prj\<cdot>(emb\<cdot>(prj\<cdot>x :: 'b)) = (prj\<cdot>x :: 'a)"
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 (rule monofun_cfun_arg [OF assms])
done
text \<open>Isomorphism lemmas used internally by the domain package:\<close>
lemma domain_abs_iso:
fixes abs and rep
assumes DEFL: "DEFL('b::domain) = DEFL('a::domain)"
assumes abs_def: "(abs :: 'a \<rightarrow> 'b) \<equiv> prj oo emb"
assumes rep_def: "(rep :: 'b \<rightarrow> 'a) \<equiv> prj oo emb"
shows "rep\<cdot>(abs\<cdot>x) = x"
unfolding abs_def rep_def
by (simp add: emb_prj_emb DEFL)
lemma domain_rep_iso:
fixes abs and rep
assumes DEFL: "DEFL('b::domain) = DEFL('a::domain)"
assumes abs_def: "(abs :: 'a \<rightarrow> 'b) \<equiv> prj oo emb"
assumes rep_def: "(rep :: 'b \<rightarrow> 'a) \<equiv> prj oo emb"
shows "abs\<cdot>(rep\<cdot>x) = x"
unfolding abs_def rep_def
by (simp add: emb_prj_emb DEFL)
subsection \<open>Deflations as sets\<close>
definition defl_set :: "'a::bifinite defl \<Rightarrow> 'a set"
where "defl_set A = {x. cast\<cdot>A\<cdot>x = x}"
lemma adm_defl_set: "adm (\<lambda>x. x \<in> defl_set A)"
unfolding defl_set_def by simp
lemma defl_set_bottom: "\<bottom> \<in> defl_set A"
unfolding defl_set_def by simp
lemma defl_set_cast [simp]: "cast\<cdot>A\<cdot>x \<in> defl_set A"
unfolding defl_set_def by simp
lemma defl_set_subset_iff: "defl_set A \<subseteq> defl_set B \<longleftrightarrow> A \<sqsubseteq> B"
apply (simp add: defl_set_def subset_eq cast_below_cast [symmetric])
apply (auto simp add: cast.belowI cast.belowD)
done
subsection \<open>Proving a subtype is representable\<close>
text \<open>Temporarily relax type constraints.\<close>
setup \<open>
fold Sign.add_const_constraint
[ (\<^const_name>\<open>defl\<close>, SOME \<^typ>\<open>'a::pcpo itself \<Rightarrow> udom defl\<close>)
, (\<^const_name>\<open>emb\<close>, SOME \<^typ>\<open>'a::pcpo \<rightarrow> udom\<close>)
, (\<^const_name>\<open>prj\<close>, SOME \<^typ>\<open>udom \<rightarrow> 'a::pcpo\<close>)
, (\<^const_name>\<open>liftdefl\<close>, SOME \<^typ>\<open>'a::pcpo itself \<Rightarrow> udom u defl\<close>)
, (\<^const_name>\<open>liftemb\<close>, SOME \<^typ>\<open>'a::pcpo u \<rightarrow> udom u\<close>)
, (\<^const_name>\<open>liftprj\<close>, SOME \<^typ>\<open>udom u \<rightarrow> 'a::pcpo u\<close>) ]
\<close>
lemma typedef_domain_class:
fixes Rep :: "'a::pcpo \<Rightarrow> udom"
fixes Abs :: "udom \<Rightarrow> 'a::pcpo"
fixes t :: "udom defl"
assumes type: "type_definition Rep Abs (defl_set t)"
assumes below: "(\<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 defl: "defl \<equiv> (\<lambda> a::'a itself. t)"
assumes liftemb: "(liftemb :: 'a u \<rightarrow> udom u) \<equiv> u_map\<cdot>emb"
assumes liftprj: "(liftprj :: udom u \<rightarrow> 'a u) \<equiv> u_map\<cdot>prj"
assumes liftdefl: "(liftdefl :: 'a itself \<Rightarrow> _) \<equiv> (\<lambda>t. liftdefl_of\<cdot>DEFL('a))"
shows "OFCLASS('a, domain_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_defl_set cont_id])
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_defl_set])
apply simp_all
done
have prj_emb: "\<And>x::'a. prj\<cdot>(emb\<cdot>x) = x"
using type_definition.Rep [OF type]
unfolding prj_beta emb_beta defl_set_def
by (simp add: type_definition.Rep_inverse [OF type])
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 standard
apply (simp add: prj_emb)
apply (simp add: emb_prj cast.below)
done
show "cast\<cdot>DEFL('a) = emb oo (prj :: udom \<rightarrow> 'a)"
by (rule cfun_eqI, simp add: defl emb_prj)
qed (simp_all only: liftemb liftprj liftdefl)
lemma typedef_DEFL:
assumes "defl \<equiv> (\<lambda>a::'a::pcpo itself. t)"
shows "DEFL('a::pcpo) = t"
unfolding assms ..
text \<open>Restore original typing constraints.\<close>
setup \<open>
fold Sign.add_const_constraint
[(\<^const_name>\<open>defl\<close>, SOME \<^typ>\<open>'a::domain itself \<Rightarrow> udom defl\<close>),
(\<^const_name>\<open>emb\<close>, SOME \<^typ>\<open>'a::domain \<rightarrow> udom\<close>),
(\<^const_name>\<open>prj\<close>, SOME \<^typ>\<open>udom \<rightarrow> 'a::domain\<close>),
(\<^const_name>\<open>liftdefl\<close>, SOME \<^typ>\<open>'a::predomain itself \<Rightarrow> udom u defl\<close>),
(\<^const_name>\<open>liftemb\<close>, SOME \<^typ>\<open>'a::predomain u \<rightarrow> udom u\<close>),
(\<^const_name>\<open>liftprj\<close>, SOME \<^typ>\<open>udom u \<rightarrow> 'a::predomain u\<close>)]
\<close>
ML_file \<open>Tools/domaindef.ML\<close>
subsection \<open>Isomorphic deflations\<close>
definition isodefl :: "('a::domain \<rightarrow> 'a) \<Rightarrow> udom defl \<Rightarrow> bool"
where "isodefl d t \<longleftrightarrow> cast\<cdot>t = emb oo d oo prj"
definition isodefl' :: "('a::predomain \<rightarrow> 'a) \<Rightarrow> udom u defl \<Rightarrow> bool"
where "isodefl' d t \<longleftrightarrow> cast\<cdot>t = liftemb oo u_map\<cdot>d oo liftprj"
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: cfun_eqI)
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: cfun_eqI)
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::domain \<rightarrow> 'a"
assumes "isodefl d t" shows "deflation d"
proof
note assms [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_DEFL: "isodefl (ID :: 'a \<rightarrow> 'a) DEFL('a::domain)"
unfolding isodefl_def by (simp add: cast_DEFL)
lemma isodefl_LIFTDEFL:
"isodefl' (ID :: 'a \<rightarrow> 'a) LIFTDEFL('a::predomain)"
unfolding isodefl'_def by (simp add: cast_liftdefl u_map_ID)
lemma isodefl_DEFL_imp_ID: "isodefl (d :: 'a \<rightarrow> 'a) DEFL('a::domain) \<Longrightarrow> d = ID"
unfolding isodefl_def
apply (simp add: cast_DEFL)
apply (simp add: cfun_eq_iff)
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: cfun_eq_iff)
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 assms 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: assms)
done
lemma isodefl_abs_rep:
fixes abs and rep and d
assumes DEFL: "DEFL('b::domain) = DEFL('a::domain)"
assumes abs_def: "(abs :: 'a \<rightarrow> 'b) \<equiv> prj oo emb"
assumes rep_def: "(rep :: 'b \<rightarrow> 'a) \<equiv> prj oo emb"
shows "isodefl d t \<Longrightarrow> isodefl (abs oo d oo rep) t"
unfolding isodefl_def
by (simp add: cfun_eq_iff assms prj_emb_prj emb_prj_emb)
lemma isodefl'_liftdefl_of: "isodefl d t \<Longrightarrow> isodefl' d (liftdefl_of\<cdot>t)"
unfolding isodefl_def isodefl'_def
by (simp add: cast_liftdefl_of u_map_oo liftemb_eq liftprj_eq)
lemma isodefl_sfun:
"isodefl d1 t1 \<Longrightarrow> isodefl d2 t2 \<Longrightarrow>
isodefl (sfun_map\<cdot>d1\<cdot>d2) (sfun_defl\<cdot>t1\<cdot>t2)"
apply (rule isodeflI)
apply (simp add: cast_sfun_defl cast_isodefl)
apply (simp add: emb_sfun_def prj_sfun_def)
apply (simp add: sfun_map_map isodefl_strict)
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)
apply (simp add: cast_ssum_defl 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_defl\<cdot>t1\<cdot>t2)"
apply (rule isodeflI)
apply (simp add: cast_sprod_defl cast_isodefl)
apply (simp add: emb_sprod_def prj_sprod_def)
apply (simp add: sprod_map_map isodefl_strict)
done
lemma isodefl_prod:
"isodefl d1 t1 \<Longrightarrow> isodefl d2 t2 \<Longrightarrow>
isodefl (prod_map\<cdot>d1\<cdot>d2) (prod_defl\<cdot>t1\<cdot>t2)"
apply (rule isodeflI)
apply (simp add: cast_prod_defl cast_isodefl)
apply (simp add: emb_prod_def prj_prod_def)
apply (simp add: prod_map_map cfcomp1)
done
lemma isodefl_u:
"isodefl d t \<Longrightarrow> isodefl (u_map\<cdot>d) (u_defl\<cdot>t)"
apply (rule isodeflI)
apply (simp add: cast_u_defl cast_isodefl)
apply (simp add: emb_u_def prj_u_def liftemb_eq liftprj_eq u_map_map)
done
lemma isodefl_u_liftdefl:
"isodefl' d t \<Longrightarrow> isodefl (u_map\<cdot>d) (u_liftdefl\<cdot>t)"
apply (rule isodeflI)
apply (simp add: cast_u_liftdefl isodefl'_def)
apply (simp add: emb_u_def prj_u_def liftemb_eq liftprj_eq)
done
lemma encode_prod_u_map:
"encode_prod_u\<cdot>(u_map\<cdot>(prod_map\<cdot>f\<cdot>g)\<cdot>(decode_prod_u\<cdot>x))
= sprod_map\<cdot>(u_map\<cdot>f)\<cdot>(u_map\<cdot>g)\<cdot>x"
unfolding encode_prod_u_def decode_prod_u_def
apply (case_tac x, simp, rename_tac a b)
apply (case_tac a, simp, case_tac b, simp, simp)
done
lemma isodefl_prod_u:
assumes "isodefl' d1 t1" and "isodefl' d2 t2"
shows "isodefl' (prod_map\<cdot>d1\<cdot>d2) (prod_liftdefl\<cdot>t1\<cdot>t2)"
using assms unfolding isodefl'_def
unfolding liftemb_prod_def liftprj_prod_def
by (simp add: cast_prod_liftdefl cfcomp1 encode_prod_u_map sprod_map_map)
lemma encode_cfun_map:
"encode_cfun\<cdot>(cfun_map\<cdot>f\<cdot>g\<cdot>(decode_cfun\<cdot>x))
= sfun_map\<cdot>(u_map\<cdot>f)\<cdot>g\<cdot>x"
unfolding encode_cfun_def decode_cfun_def
apply (simp add: sfun_eq_iff cfun_map_def sfun_map_def)
apply (rule cfun_eqI, rename_tac y, case_tac y, simp_all)
done
lemma isodefl_cfun:
assumes "isodefl (u_map\<cdot>d1) t1" and "isodefl d2 t2"
shows "isodefl (cfun_map\<cdot>d1\<cdot>d2) (sfun_defl\<cdot>t1\<cdot>t2)"
using isodefl_sfun [OF assms] unfolding isodefl_def
by (simp add: emb_cfun_def prj_cfun_def cfcomp1 encode_cfun_map)
subsection \<open>Setting up the domain package\<close>
named_theorems domain_defl_simps "theorems like DEFL('a t) = t_defl$DEFL('a)"
and domain_isodefl "theorems like isodefl d t ==> isodefl (foo_map$d) (foo_defl$t)"
ML_file \<open>Tools/Domain/domain_isomorphism.ML\<close>
ML_file \<open>Tools/Domain/domain_axioms.ML\<close>
ML_file \<open>Tools/Domain/domain.ML\<close>
lemmas [domain_defl_simps] =
DEFL_cfun DEFL_sfun DEFL_ssum DEFL_sprod DEFL_prod DEFL_u
liftdefl_eq LIFTDEFL_prod u_liftdefl_liftdefl_of
lemmas [domain_map_ID] =
cfun_map_ID sfun_map_ID ssum_map_ID sprod_map_ID prod_map_ID u_map_ID
lemmas [domain_isodefl] =
isodefl_u isodefl_sfun isodefl_ssum isodefl_sprod
isodefl_cfun isodefl_prod isodefl_prod_u isodefl'_liftdefl_of
isodefl_u_liftdefl
lemmas [domain_deflation] =
deflation_cfun_map deflation_sfun_map deflation_ssum_map
deflation_sprod_map deflation_prod_map deflation_u_map
setup \<open>
fold Domain_Take_Proofs.add_rec_type
[(\<^type_name>\<open>cfun\<close>, [true, true]),
(\<^type_name>\<open>sfun\<close>, [true, true]),
(\<^type_name>\<open>ssum\<close>, [true, true]),
(\<^type_name>\<open>sprod\<close>, [true, true]),
(\<^type_name>\<open>prod\<close>, [true, true]),
(\<^type_name>\<open>u\<close>, [true])]
\<close>
end