# Theory Domain_Aux

```(*  Title:      HOL/HOLCF/Domain_Aux.thy
Author:     Brian Huffman
*)

section ‹Domain package support›

theory Domain_Aux
imports Map_Functions Fixrec
begin

subsection ‹Continuous isomorphisms›

text ‹A locale for continuous isomorphisms›

locale iso =
fixes abs :: "'a → 'b"
fixes rep :: "'b → 'a"
assumes abs_iso [simp]: "rep⋅(abs⋅x) = x"
assumes rep_iso [simp]: "abs⋅(rep⋅y) = y"
begin

lemma swap: "iso rep abs"
by (rule iso.intro [OF rep_iso abs_iso])

lemma abs_below: "(abs⋅x ⊑ abs⋅y) = (x ⊑ y)"
proof
assume "abs⋅x ⊑ abs⋅y"
then have "rep⋅(abs⋅x) ⊑ rep⋅(abs⋅y)" by (rule monofun_cfun_arg)
then show "x ⊑ y" by simp
next
assume "x ⊑ y"
then show "abs⋅x ⊑ abs⋅y" by (rule monofun_cfun_arg)
qed

lemma rep_below: "(rep⋅x ⊑ rep⋅y) = (x ⊑ y)"
by (rule iso.abs_below [OF swap])

lemma abs_eq: "(abs⋅x = abs⋅y) = (x = y)"

lemma rep_eq: "(rep⋅x = rep⋅y) = (x = y)"
by (rule iso.abs_eq [OF swap])

lemma abs_strict: "abs⋅⊥ = ⊥"
proof -
have "⊥ ⊑ rep⋅⊥" ..
then have "abs⋅⊥ ⊑ abs⋅(rep⋅⊥)" by (rule monofun_cfun_arg)
then have "abs⋅⊥ ⊑ ⊥" by simp
then show ?thesis by (rule bottomI)
qed

lemma rep_strict: "rep⋅⊥ = ⊥"
by (rule iso.abs_strict [OF swap])

lemma abs_defin': "abs⋅x = ⊥ ⟹ x = ⊥"
proof -
have "x = rep⋅(abs⋅x)" by simp
also assume "abs⋅x = ⊥"
also note rep_strict
finally show "x = ⊥" .
qed

lemma rep_defin': "rep⋅z = ⊥ ⟹ z = ⊥"
by (rule iso.abs_defin' [OF swap])

lemma abs_defined: "z ≠ ⊥ ⟹ abs⋅z ≠ ⊥"
by (erule contrapos_nn, erule abs_defin')

lemma rep_defined: "z ≠ ⊥ ⟹ rep⋅z ≠ ⊥"
by (rule iso.abs_defined [OF iso.swap]) (rule iso_axioms)

lemma abs_bottom_iff: "(abs⋅x = ⊥) = (x = ⊥)"
by (auto elim: abs_defin' intro: abs_strict)

lemma rep_bottom_iff: "(rep⋅x = ⊥) = (x = ⊥)"
by (rule iso.abs_bottom_iff [OF iso.swap]) (rule iso_axioms)

lemma casedist_rule: "rep⋅x = ⊥ ∨ P ⟹ x = ⊥ ∨ P"

lemma compact_abs_rev: "compact (abs⋅x) ⟹ compact x"
proof (unfold compact_def)
assume "adm (λy. abs⋅x \<notsqsubseteq> y)"
with cont_Rep_cfun2
then show "adm (λy. x \<notsqsubseteq> y)" using abs_below by simp
qed

lemma compact_rep_rev: "compact (rep⋅x) ⟹ compact x"
by (rule iso.compact_abs_rev [OF iso.swap]) (rule iso_axioms)

lemma compact_abs: "compact x ⟹ compact (abs⋅x)"
by (rule compact_rep_rev) simp

lemma compact_rep: "compact x ⟹ compact (rep⋅x)"
by (rule iso.compact_abs [OF iso.swap]) (rule iso_axioms)

lemma iso_swap: "(x = abs⋅y) = (rep⋅x = y)"
proof
assume "x = abs⋅y"
then have "rep⋅x = rep⋅(abs⋅y)" by simp
then show "rep⋅x = y" by simp
next
assume "rep⋅x = y"
then have "abs⋅(rep⋅x) = abs⋅y" by simp
then show "x = abs⋅y" by simp
qed

end

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: "⋀x. rep⋅(abs⋅x) = x"
assumes rep_iso: "⋀y. abs⋅(rep⋅y) = y"
shows "deflation d ⟹ 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: "⋀n. deflation (d n)"
shows "d m⋅(d n⋅x) = d (min m n)⋅x"
proof (rule linorder_le_cases)
assume "m ≤ n"
with chain have "d m ⊑ d n" by (rule chain_mono)
then have "d m⋅(d n⋅x) = d m⋅x"
by (rule deflation_below_comp1 [OF defl defl])
moreover from ‹m ≤ n› have "min m n = m" by simp
ultimately show ?thesis by simp
next
assume "n ≤ m"
with chain have "d n ⊑ d m" by (rule chain_mono)
then have "d m⋅(d n⋅x) = d n⋅x"
by (rule deflation_below_comp2 [OF defl defl])
moreover from ‹n ≤ m› have "min m n = n" by simp
ultimately show ?thesis by simp
qed

lemma lub_ID_take_lemma:
assumes "chain t" and "(⨆n. t n) = ID"
assumes "⋀n. t n⋅x = t n⋅y" shows "x = y"
proof -
have "(⨆n. t n⋅x) = (⨆n. t n⋅y)"
using assms(3) by simp
then have "(⨆n. t n)⋅x = (⨆n. t n)⋅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 "(⨆n. t n) = ID"
shows "(⨆n. t n⋅x) = x"
using assms by (simp add: lub_distribs)

lemma lub_ID_take_induct:
assumes "chain t" and "(⨆n. t n) = ID"
assumes "adm P" and "⋀n. P (t n⋅x)" shows "P x"
proof -
from ‹chain t› have "chain (λn. t n⋅x)" by simp
from ‹adm P› this ‹⋀n. P (t n⋅x)› have "P (⨆n. t n⋅x)" by (rule admD)
with ‹chain t› ‹(⨆n. t n) = ID› show "P x" by (simp add: lub_distribs)
qed

subsection ‹Finiteness›

text ‹
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.
›

definition
decisive :: "('a::pcpo → 'a) ⇒ bool"
where
"decisive d ⟷ (∀x. d⋅x = x ∨ d⋅x = ⊥)"

lemma decisiveI: "(⋀x. d⋅x = x ∨ d⋅x = ⊥) ⟹ decisive d"
unfolding decisive_def by simp

lemma decisive_cases:
assumes "decisive d" obtains "d⋅x = x" | "d⋅x = ⊥"
using assms unfolding decisive_def by auto

lemma decisive_bottom: "decisive ⊥"
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⋅f⋅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⋅f⋅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⋅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: "(⨆n. d n) = ID"
assumes decisive: "⋀n. decisive (d n)"
shows "∃n. d n⋅x = x"
proof -
have 1: "chain (λn. d n⋅x)" using chain by simp
have 2: "(⨆n. d n⋅x) = x" using chain lub by (rule lub_ID_reach)
have "∀n. d n⋅x = x ∨ d n⋅x = ⊥"
using decisive unfolding decisive_def by simp
hence "range (λn. d n⋅x) ⊆ {x, ⊥}"
by auto
hence "finite (range (λn. d n⋅x))"
by (rule finite_subset, simp)
with 1 have "finite_chain (λn. d n⋅x)"
by (rule finite_range_imp_finch)
then have "∃n. (⨆n. d n⋅x) = d n⋅x"
unfolding finite_chain_def by (auto simp add: maxinch_is_thelub)
with 2 show "∃n. d n⋅x = x" by (auto elim: sym)
qed

lemma lub_ID_finite_take_induct:
assumes "chain d" and "(⨆n. d n) = ID" and "⋀n. decisive (d n)"
shows "(⋀n. P (d n⋅x)) ⟹ P x"
using lub_ID_finite [OF assms] by metis

text ‹Lemmas for proving nchotomy rule:›

lemma ex_one_bottom_iff:
"(∃x. P x ∧ x ≠ ⊥) = P ONE"
by simp

lemma ex_up_bottom_iff:
"(∃x. P x ∧ x ≠ ⊥) = (∃x. P (up⋅x))"
by (safe, case_tac x, auto)

lemma ex_sprod_bottom_iff:
"(∃y. P y ∧ y ≠ ⊥) =
(∃x y. (P (:x, y:) ∧ x ≠ ⊥) ∧ y ≠ ⊥)"
by (safe, case_tac y, auto)

lemma ex_sprod_up_bottom_iff:
"(∃y. P y ∧ y ≠ ⊥) =
(∃x y. P (:up⋅x, y:) ∧ y ≠ ⊥)"
by (safe, case_tac y, simp, case_tac x, auto)

lemma ex_ssum_bottom_iff:
"(∃x. P x ∧ x ≠ ⊥) =
((∃x. P (sinl⋅x) ∧ x ≠ ⊥) ∨
(∃x. P (sinr⋅x) ∧ x ≠ ⊥))"
by (safe, case_tac x, auto)

lemma exh_start: "p = ⊥ ∨ (∃x. p = x ∧ x ≠ ⊥)"
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 ‹Rules for turning nchotomy into exhaust:›

lemma exh_casedist0: "⟦R; R ⟹ P⟧ ⟹ P" (* like make_elim *)
by auto

lemma exh_casedist1: "((P ∨ Q ⟹ R) ⟹ S) ≡ (⟦P ⟹ R; Q ⟹ R⟧ ⟹ S)"
by rule auto

lemma exh_casedist2: "(∃x. P x ⟹ Q) ≡ (⋀x. P x ⟹ Q)"
by rule auto

lemma exh_casedist3: "(P ∧ Q ⟹ R) ≡ (P ⟹ Q ⟹ R)"
by rule auto

lemmas exh_casedists = exh_casedist1 exh_casedist2 exh_casedist3

text ‹Rules for proving constructor properties›

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⋅ID⋅⊥⋅(sinr⋅x) = ⊥"
"sscase⋅ID⋅⊥⋅(sinl⋅x) = x"
"sscase⋅⊥⋅ID⋅(sinl⋅x) = ⊥"
"sscase⋅⊥⋅ID⋅(sinr⋅x) = x"
"fup⋅ID⋅(up⋅x) = x"
by (cases "x = ⊥", 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 ‹ML setup›

named_theorems domain_deflation "theorems like deflation a ==> deflation (foo_map\$a)"
and domain_map_ID "theorems like foo_map\$ID = ID"

ML_file ‹Tools/Domain/domain_take_proofs.ML›
ML_file ‹Tools/cont_consts.ML›
ML_file ‹Tools/cont_proc.ML›
simproc_setup cont ("cont f") = ‹K ContProc.cont_proc›

ML_file ‹Tools/Domain/domain_constructors.ML›
ML_file ‹Tools/Domain/domain_induction.ML›

end
```