# Theory Algebraic

theory Algebraic
imports Universal
```(*  Title:      HOL/HOLCF/Algebraic.thy
Author:     Brian Huffman
*)

section ‹Algebraic deflations›

theory Algebraic
imports Universal Map_Functions
begin

default_sort bifinite

subsection ‹Type constructor for finite deflations›

typedef 'a fin_defl = "{d::'a → 'a. finite_deflation d}"
by (fast intro: finite_deflation_bottom)

instantiation fin_defl :: (bifinite) below
begin

definition below_fin_defl_def:
"below ≡ λx y. Rep_fin_defl x ⊑ Rep_fin_defl y"

instance ..
end

instance fin_defl :: (bifinite) po
using type_definition_fin_defl below_fin_defl_def
by (rule typedef_po)

lemma finite_deflation_Rep_fin_defl: "finite_deflation (Rep_fin_defl d)"
using Rep_fin_defl by simp

lemma deflation_Rep_fin_defl: "deflation (Rep_fin_defl d)"
using finite_deflation_Rep_fin_defl
by (rule finite_deflation_imp_deflation)

interpretation Rep_fin_defl: finite_deflation "Rep_fin_defl d"
by (rule finite_deflation_Rep_fin_defl)

lemma fin_defl_belowI:
"(⋀x. Rep_fin_defl a⋅x = x ⟹ Rep_fin_defl b⋅x = x) ⟹ a ⊑ b"
unfolding below_fin_defl_def
by (rule Rep_fin_defl.belowI)

lemma fin_defl_belowD:
"⟦a ⊑ b; Rep_fin_defl a⋅x = x⟧ ⟹ Rep_fin_defl b⋅x = x"
unfolding below_fin_defl_def
by (rule Rep_fin_defl.belowD)

lemma fin_defl_eqI:
"(⋀x. Rep_fin_defl a⋅x = x ⟷ Rep_fin_defl b⋅x = x) ⟹ a = b"
apply (rule below_antisym)
apply (rule fin_defl_belowI, simp)
apply (rule fin_defl_belowI, simp)
done

lemma Rep_fin_defl_mono: "a ⊑ b ⟹ Rep_fin_defl a ⊑ Rep_fin_defl b"
unfolding below_fin_defl_def .

lemma Abs_fin_defl_mono:
"⟦finite_deflation a; finite_deflation b; a ⊑ b⟧
⟹ Abs_fin_defl a ⊑ Abs_fin_defl b"
unfolding below_fin_defl_def
by (simp add: Abs_fin_defl_inverse)

lemma (in finite_deflation) compact_belowI:
assumes "⋀x. compact x ⟹ d⋅x = x ⟹ f⋅x = x" shows "d ⊑ f"
by (rule belowI, rule assms, erule subst, rule compact)

lemma compact_Rep_fin_defl [simp]: "compact (Rep_fin_defl a)"
using finite_deflation_Rep_fin_defl
by (rule finite_deflation_imp_compact)

subsection ‹Defining algebraic deflations by ideal completion›

typedef 'a defl = "{S::'a fin_defl set. below.ideal S}"
by (rule below.ex_ideal)

instantiation defl :: (bifinite) below
begin

definition
"x ⊑ y ⟷ Rep_defl x ⊆ Rep_defl y"

instance ..
end

instance defl :: (bifinite) po
using type_definition_defl below_defl_def
by (rule below.typedef_ideal_po)

instance defl :: (bifinite) cpo
using type_definition_defl below_defl_def
by (rule below.typedef_ideal_cpo)

definition
defl_principal :: "'a fin_defl ⇒ 'a defl" where
"defl_principal t = Abs_defl {u. u ⊑ t}"

lemma fin_defl_countable: "∃f::'a fin_defl ⇒ nat. inj f"
proof -
obtain f :: "'a compact_basis ⇒ nat" where inj_f: "inj f"
using compact_basis.countable ..
have *: "⋀d. finite (f ` Rep_compact_basis -` {x. Rep_fin_defl d⋅x = x})"
apply (rule finite_imageI)
apply (rule finite_vimageI)
apply (rule Rep_fin_defl.finite_fixes)
apply (simp add: inj_on_def Rep_compact_basis_inject)
done
have range_eq: "range Rep_compact_basis = {x. compact x}"
using type_definition_compact_basis by (rule type_definition.Rep_range)
have "inj (λd. set_encode
(f ` Rep_compact_basis -` {x. Rep_fin_defl d⋅x = x}))"
apply (rule inj_onI)
apply (simp only: set_encode_eq *)
apply (simp only: inj_image_eq_iff inj_f)
apply (drule_tac f="image Rep_compact_basis" in arg_cong)
apply (simp del: vimage_Collect_eq add: range_eq set_eq_iff)
apply (rule Rep_fin_defl_inject [THEN iffD1])
apply (rule below_antisym)
apply (rule Rep_fin_defl.compact_belowI, rename_tac z)
apply (drule_tac x=z in spec, simp)
apply (rule Rep_fin_defl.compact_belowI, rename_tac z)
apply (drule_tac x=z in spec, simp)
done
thus ?thesis by - (rule exI)
qed

interpretation defl: ideal_completion below defl_principal Rep_defl
using type_definition_defl below_defl_def
using defl_principal_def fin_defl_countable
by (rule below.typedef_ideal_completion)

text ‹Algebraic deflations are pointed›

lemma defl_minimal: "defl_principal (Abs_fin_defl ⊥) ⊑ x"
apply (induct x rule: defl.principal_induct, simp)
apply (rule defl.principal_mono)
apply (simp add: below_fin_defl_def)
apply (simp add: Abs_fin_defl_inverse finite_deflation_bottom)
done

instance defl :: (bifinite) pcpo
by intro_classes (fast intro: defl_minimal)

lemma inst_defl_pcpo: "⊥ = defl_principal (Abs_fin_defl ⊥)"
by (rule defl_minimal [THEN bottomI, symmetric])

subsection ‹Applying algebraic deflations›

definition
cast :: "'a defl → 'a → 'a"
where
"cast = defl.extension Rep_fin_defl"

lemma cast_defl_principal:
"cast⋅(defl_principal a) = Rep_fin_defl a"
unfolding cast_def
apply (rule defl.extension_principal)
apply (simp only: below_fin_defl_def)
done

lemma deflation_cast: "deflation (cast⋅d)"
apply (induct d rule: defl.principal_induct)
apply (rule adm_subst [OF _ adm_deflation], simp)
apply (simp add: cast_defl_principal)
apply (rule finite_deflation_imp_deflation)
apply (rule finite_deflation_Rep_fin_defl)
done

lemma finite_deflation_cast:
"compact d ⟹ finite_deflation (cast⋅d)"
apply (drule defl.compact_imp_principal, clarify)
apply (simp add: cast_defl_principal)
apply (rule finite_deflation_Rep_fin_defl)
done

interpretation cast: deflation "cast⋅d"
by (rule deflation_cast)

declare cast.idem [simp]

lemma compact_cast [simp]: "compact d ⟹ compact (cast⋅d)"
apply (rule finite_deflation_imp_compact)
apply (erule finite_deflation_cast)
done

lemma cast_below_cast: "cast⋅A ⊑ cast⋅B ⟷ A ⊑ B"
apply (induct A rule: defl.principal_induct, simp)
apply (induct B rule: defl.principal_induct, simp)
apply (simp add: cast_defl_principal below_fin_defl_def)
done

lemma compact_cast_iff: "compact (cast⋅d) ⟷ compact d"
apply (rule iffI)
apply (simp only: compact_def cast_below_cast [symmetric])
apply (erule adm_subst [OF cont_Rep_cfun2])
apply (erule compact_cast)
done

lemma cast_below_imp_below: "cast⋅A ⊑ cast⋅B ⟹ A ⊑ B"
by (simp only: cast_below_cast)

lemma cast_eq_imp_eq: "cast⋅A = cast⋅B ⟹ A = B"
by (simp add: below_antisym cast_below_imp_below)

lemma cast_strict1 [simp]: "cast⋅⊥ = ⊥"
apply (subst inst_defl_pcpo)
apply (subst cast_defl_principal)
apply (rule Abs_fin_defl_inverse)
apply (simp add: finite_deflation_bottom)
done

lemma cast_strict2 [simp]: "cast⋅A⋅⊥ = ⊥"
by (rule cast.below [THEN bottomI])

subsection ‹Deflation combinators›

definition
"defl_fun1 e p f =
defl.extension (λa.
defl_principal (Abs_fin_defl
(e oo f⋅(Rep_fin_defl a) oo p)))"

definition
"defl_fun2 e p f =
defl.extension (λa.
defl.extension (λb.
defl_principal (Abs_fin_defl
(e oo f⋅(Rep_fin_defl a)⋅(Rep_fin_defl b) oo p))))"

lemma cast_defl_fun1:
assumes ep: "ep_pair e p"
assumes f: "⋀a. finite_deflation a ⟹ finite_deflation (f⋅a)"
shows "cast⋅(defl_fun1 e p f⋅A) = e oo f⋅(cast⋅A) oo p"
proof -
have 1: "⋀a. finite_deflation (e oo f⋅(Rep_fin_defl a) oo p)"
apply (rule ep_pair.finite_deflation_e_d_p [OF ep])
apply (rule f, rule finite_deflation_Rep_fin_defl)
done
show ?thesis
by (induct A rule: defl.principal_induct, simp)
(simp only: defl_fun1_def
defl.extension_principal
defl.extension_mono
defl.principal_mono
Abs_fin_defl_mono [OF 1 1]
monofun_cfun below_refl
Rep_fin_defl_mono
cast_defl_principal
Abs_fin_defl_inverse [unfolded mem_Collect_eq, OF 1])
qed

lemma cast_defl_fun2:
assumes ep: "ep_pair e p"
assumes f: "⋀a b. finite_deflation a ⟹ finite_deflation b ⟹
finite_deflation (f⋅a⋅b)"
shows "cast⋅(defl_fun2 e p f⋅A⋅B) = e oo f⋅(cast⋅A)⋅(cast⋅B) oo p"
proof -
have 1: "⋀a b. finite_deflation
(e oo f⋅(Rep_fin_defl a)⋅(Rep_fin_defl b) oo p)"
apply (rule ep_pair.finite_deflation_e_d_p [OF ep])
apply (rule f, (rule finite_deflation_Rep_fin_defl)+)
done
show ?thesis
apply (induct A rule: defl.principal_induct, simp)
apply (induct B rule: defl.principal_induct, simp)
by (simp only: defl_fun2_def
defl.extension_principal
defl.extension_mono
defl.principal_mono
Abs_fin_defl_mono [OF 1 1]
monofun_cfun below_refl
Rep_fin_defl_mono
cast_defl_principal
Abs_fin_defl_inverse [unfolded mem_Collect_eq, OF 1])
qed

end
```