Theory Universal

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

section ‹A universal bifinite domain›

theory Universal
imports Bifinite Completion "HOL-Library.Nat_Bijection"
begin

no_notation binomial  (infix ‹choose› 64)

subsection ‹Basis for universal domain›

subsubsection ‹Basis datatype›

type_synonym ubasis = nat

definition
  node :: "nat ⇒ ubasis ⇒ ubasis set ⇒ ubasis"
where
  "node i a S = Suc (prod_encode (i, prod_encode (a, set_encode S)))"

lemma node_not_0 [simp]: "node i a S ≠ 0"
unfolding node_def by simp

lemma node_gt_0 [simp]: "0 < node i a S"
unfolding node_def by simp

lemma node_inject [simp]:
  "⟦finite S; finite T⟧
    ⟹ node i a S = node j b T ⟷ i = j ∧ a = b ∧ S = T"
unfolding node_def by (simp add: prod_encode_eq set_encode_eq)

lemma node_gt0: "i < node i a S"
unfolding node_def less_Suc_eq_le
by (rule le_prod_encode_1)

lemma node_gt1: "a < node i a S"
unfolding node_def less_Suc_eq_le
by (rule order_trans [OF le_prod_encode_1 le_prod_encode_2])

lemma nat_less_power2: "n < 2^n"
  by (fact less_exp)

lemma node_gt2: "⟦finite S; b ∈ S⟧ ⟹ b < node i a S"
unfolding node_def less_Suc_eq_le set_encode_def
apply (rule order_trans [OF _ le_prod_encode_2])
apply (rule order_trans [OF _ le_prod_encode_2])
apply (rule order_trans [where y="sum ((^) 2) {b}"])
apply (simp add: nat_less_power2 [THEN order_less_imp_le])
apply (erule sum_mono2, simp, simp)
done

lemma eq_prod_encode_pairI:
  "⟦fst (prod_decode x) = a; snd (prod_decode x) = b⟧ ⟹ x = prod_encode (a, b)"
  by auto

lemma node_cases:
  assumes 1: "x = 0 ⟹ P"
  assumes 2: "⋀i a S. ⟦finite S; x = node i a S⟧ ⟹ P"
  shows "P"
 apply (cases x)
  apply (erule 1)
 apply (rule 2)
  apply (rule finite_set_decode)
 apply (simp add: node_def)
 apply (rule eq_prod_encode_pairI [OF refl])
 apply (rule eq_prod_encode_pairI [OF refl refl])
done

lemma node_induct:
  assumes 1: "P 0"
  assumes 2: "⋀i a S. ⟦P a; finite S; ∀b∈S. P b⟧ ⟹ P (node i a S)"
  shows "P x"
 apply (induct x rule: nat_less_induct)
 apply (case_tac n rule: node_cases)
  apply (simp add: 1)
 apply (simp add: 2 node_gt1 node_gt2)
done

subsubsection ‹Basis ordering›

inductive
  ubasis_le :: "nat ⇒ nat ⇒ bool"
where
  ubasis_le_refl: "ubasis_le a a"
| ubasis_le_trans:
    "⟦ubasis_le a b; ubasis_le b c⟧ ⟹ ubasis_le a c"
| ubasis_le_lower:
    "finite S ⟹ ubasis_le a (node i a S)"
| ubasis_le_upper:
    "⟦finite S; b ∈ S; ubasis_le a b⟧ ⟹ ubasis_le (node i a S) b"

lemma ubasis_le_minimal: "ubasis_le 0 x"
apply (induct x rule: node_induct)
apply (rule ubasis_le_refl)
apply (erule ubasis_le_trans)
apply (erule ubasis_le_lower)
done

interpretation udom: preorder ubasis_le
apply standard
apply (rule ubasis_le_refl)
apply (erule (1) ubasis_le_trans)
done

subsubsection ‹Generic take function›

function
  ubasis_until :: "(ubasis ⇒ bool) ⇒ ubasis ⇒ ubasis"
where
  "ubasis_until P 0 = 0"
| "finite S ⟹ ubasis_until P (node i a S) =
    (if P (node i a S) then node i a S else ubasis_until P a)"
   apply clarify
   apply (rule_tac x=b in node_cases)
    apply simp_all
done

termination ubasis_until
apply (relation "measure snd")
apply (rule wf_measure)
apply (simp add: node_gt1)
done

lemma ubasis_until: "P 0 ⟹ P (ubasis_until P x)"
by (induct x rule: node_induct) simp_all

lemma ubasis_until': "0 < ubasis_until P x ⟹ P (ubasis_until P x)"
by (induct x rule: node_induct) auto

lemma ubasis_until_same: "P x ⟹ ubasis_until P x = x"
by (induct x rule: node_induct) simp_all

lemma ubasis_until_idem:
  "P 0 ⟹ ubasis_until P (ubasis_until P x) = ubasis_until P x"
by (rule ubasis_until_same [OF ubasis_until])

lemma ubasis_until_0:
  "∀x. x ≠ 0 ⟶ ¬ P x ⟹ ubasis_until P x = 0"
by (induct x rule: node_induct) simp_all

lemma ubasis_until_less: "ubasis_le (ubasis_until P x) x"
apply (induct x rule: node_induct)
apply (simp add: ubasis_le_refl)
  by (metis ubasis_le.simps ubasis_until.simps(2))

lemma ubasis_until_chain:
  assumes PQ: "⋀x. P x ⟹ Q x"
  shows "ubasis_le (ubasis_until P x) (ubasis_until Q x)"
apply (induct x rule: node_induct)
apply (simp add: ubasis_le_refl)
  by (metis assms ubasis_until.simps(2) ubasis_until_less)

lemma ubasis_until_mono:
  assumes "⋀i a S b. ⟦finite S; P (node i a S); b ∈ S; ubasis_le a b⟧ ⟹ P b"
  shows "ubasis_le a b ⟹ ubasis_le (ubasis_until P a) (ubasis_until P b)"
proof (induct set: ubasis_le)
  case (ubasis_le_refl a) show ?case by (rule ubasis_le.ubasis_le_refl)
next
  case (ubasis_le_trans a b c) thus ?case by - (rule ubasis_le.ubasis_le_trans)
next
  case (ubasis_le_lower S a i) thus ?case
    by (metis ubasis_le.simps ubasis_until.simps(2) ubasis_until_less)
next
  case (ubasis_le_upper S b a i) thus ?case
    by (metis assms ubasis_le.simps ubasis_until.simps(2) ubasis_until_same)
qed

lemma finite_range_ubasis_until:
  "finite {x. P x} ⟹ finite (range (ubasis_until P))"
apply (rule finite_subset [where B="insert 0 {x. P x}"])
apply (clarsimp simp add: ubasis_until')
apply simp
done


subsection ‹Defining the universal domain by ideal completion›

typedef udom = "{S. udom.ideal S}"
by (rule udom.ex_ideal)

instantiation udom :: below
begin

definition
  "x ⊑ y ⟷ Rep_udom x ⊆ Rep_udom y"

instance ..
end

instance udom :: po
using type_definition_udom below_udom_def
by (rule udom.typedef_ideal_po)

instance udom :: cpo
using type_definition_udom below_udom_def
by (rule udom.typedef_ideal_cpo)

definition
  udom_principal :: "nat ⇒ udom" where
  "udom_principal t = Abs_udom {u. ubasis_le u t}"

lemma ubasis_countable: "∃f::ubasis ⇒ nat. inj f"
by (rule exI, rule inj_on_id)

interpretation udom:
  ideal_completion ubasis_le udom_principal Rep_udom
using type_definition_udom below_udom_def
using udom_principal_def ubasis_countable
by (rule udom.typedef_ideal_completion)

text ‹Universal domain is pointed›

lemma udom_minimal: "udom_principal 0 ⊑ x"
apply (induct x rule: udom.principal_induct)
apply (simp, simp add: ubasis_le_minimal)
done

instance udom :: pcpo
by intro_classes (fast intro: udom_minimal)

lemma inst_udom_pcpo: "⊥ = udom_principal 0"
by (rule udom_minimal [THEN bottomI, symmetric])


subsection ‹Compact bases of domains›

typedef 'a compact_basis = "{x::'a::pcpo. compact x}"
by auto

lemma Rep_compact_basis' [simp]: "compact (Rep_compact_basis a)"
by (rule Rep_compact_basis [unfolded mem_Collect_eq])

lemma Abs_compact_basis_inverse' [simp]:
   "compact x ⟹ Rep_compact_basis (Abs_compact_basis x) = x"
by (rule Abs_compact_basis_inverse [unfolded mem_Collect_eq])

instantiation compact_basis :: (pcpo) below
begin

definition
  compact_le_def:
    "(⊑) ≡ (λx y. Rep_compact_basis x ⊑ Rep_compact_basis y)"

instance ..
end

instance compact_basis :: (pcpo) po
using type_definition_compact_basis compact_le_def
by (rule typedef_po)

definition
  approximants :: "'a ⇒ 'a compact_basis set" where
  "approximants = (λx. {a. Rep_compact_basis a ⊑ x})"

definition
  compact_bot :: "'a::pcpo compact_basis" where
  "compact_bot = Abs_compact_basis ⊥"

lemma Rep_compact_bot [simp]: "Rep_compact_basis compact_bot = ⊥"
unfolding compact_bot_def by simp

lemma compact_bot_minimal [simp]: "compact_bot ⊑ a"
unfolding compact_le_def Rep_compact_bot by simp


subsection ‹Universality of \emph{udom}›

text ‹We use a locale to parameterize the construction over a chain
of approx functions on the type to be embedded.›

locale bifinite_approx_chain =
  approx_chain approx for approx :: "nat ⇒ 'a::bifinite → 'a"
begin

subsubsection ‹Choosing a maximal element from a finite set›

lemma finite_has_maximal:
  fixes A :: "'a compact_basis set"
  shows "⟦finite A; A ≠ {}⟧ ⟹ ∃x∈A. ∀y∈A. x ⊑ y ⟶ x = y"
proof (induct rule: finite_ne_induct)
  case (singleton x)
    show ?case by simp
next
  case (insert a A)
  from ‹∃x∈A. ∀y∈A. x ⊑ y ⟶ x = y›
  obtain x where x: "x ∈ A"
           and x_eq: "⋀y. ⟦y ∈ A; x ⊑ y⟧ ⟹ x = y" by fast
  show ?case
  proof (intro bexI ballI impI)
    fix y
    assume "y ∈ insert a A" and "(if x ⊑ a then a else x) ⊑ y"
    thus "(if x ⊑ a then a else x) = y"
      apply auto
      apply (frule (1) below_trans)
      apply (frule (1) x_eq)
      apply (rule below_antisym, assumption)
      apply simp
      apply (erule (1) x_eq)
      done
  next
    show "(if x ⊑ a then a else x) ∈ insert a A"
      by (simp add: x)
  qed
qed

definition
  choose :: "'a compact_basis set ⇒ 'a compact_basis"
where
  "choose A = (SOME x. x ∈ {x∈A. ∀y∈A. x ⊑ y ⟶ x = y})"

lemma choose_lemma:
  "⟦finite A; A ≠ {}⟧ ⟹ choose A ∈ {x∈A. ∀y∈A. x ⊑ y ⟶ x = y}"
unfolding choose_def
apply (rule someI_ex)
apply (frule (1) finite_has_maximal, fast)
done

lemma maximal_choose:
  "⟦finite A; y ∈ A; choose A ⊑ y⟧ ⟹ choose A = y"
apply (cases "A = {}", simp)
apply (frule (1) choose_lemma, simp)
done

lemma choose_in: "⟦finite A; A ≠ {}⟧ ⟹ choose A ∈ A"
by (frule (1) choose_lemma, simp)

function
  choose_pos :: "'a compact_basis set ⇒ 'a compact_basis ⇒ nat"
where
  "choose_pos A x =
    (if finite A ∧ x ∈ A ∧ x ≠ choose A
      then Suc (choose_pos (A - {choose A}) x) else 0)"
by auto

termination choose_pos
apply (relation "measure (card ∘ fst)", simp)
apply clarsimp
apply (rule card_Diff1_less)
apply assumption
apply (erule choose_in)
apply clarsimp
done

declare choose_pos.simps [simp del]

lemma choose_pos_choose: "finite A ⟹ choose_pos A (choose A) = 0"
by (simp add: choose_pos.simps)

lemma inj_on_choose_pos [OF refl]:
  "⟦card A = n; finite A⟧ ⟹ inj_on (choose_pos A) A"
 apply (induct n arbitrary: A)
  apply simp
 apply (case_tac "A = {}", simp)
 apply (frule (1) choose_in)
 apply (rule inj_onI)
 apply (drule_tac x="A - {choose A}" in meta_spec, simp)
 apply (simp add: choose_pos.simps)
 apply (simp split: if_split_asm)
 apply (erule (1) inj_onD, simp, simp)
done

lemma choose_pos_bounded [OF refl]:
  "⟦card A = n; finite A; x ∈ A⟧ ⟹ choose_pos A x < n"
apply (induct n arbitrary: A)
apply simp
 apply (case_tac "A = {}", simp)
 apply (frule (1) choose_in)
apply (subst choose_pos.simps)
apply simp
done

lemma choose_pos_lessD:
  "⟦choose_pos A x < choose_pos A y; finite A; x ∈ A; y ∈ A⟧ ⟹ x \<notsqsubseteq> y"
 apply (induct A x arbitrary: y rule: choose_pos.induct)
 apply simp
 apply (case_tac "x = choose A")
  apply simp
  apply (rule notI)
  apply (frule (2) maximal_choose)
  apply simp
 apply (case_tac "y = choose A")
  apply (simp add: choose_pos_choose)
 apply (drule_tac x=y in meta_spec)
 apply simp
 apply (erule meta_mp)
 apply (simp add: choose_pos.simps)
done

subsubsection ‹Compact basis take function›

primrec
  cb_take :: "nat ⇒ 'a compact_basis ⇒ 'a compact_basis" where
  "cb_take 0 = (λx. compact_bot)"
| "cb_take (Suc n) = (λa. Abs_compact_basis (approx n⋅(Rep_compact_basis a)))"

declare cb_take.simps [simp del]

lemma cb_take_zero [simp]: "cb_take 0 a = compact_bot"
by (simp only: cb_take.simps)

lemma Rep_cb_take:
  "Rep_compact_basis (cb_take (Suc n) a) = approx n⋅(Rep_compact_basis a)"
by (simp add: cb_take.simps(2))

lemmas approx_Rep_compact_basis = Rep_cb_take [symmetric]

lemma cb_take_covers: "∃n. cb_take n x = x"
apply (subgoal_tac "∃n. cb_take (Suc n) x = x", fast)
apply (simp add: Rep_compact_basis_inject [symmetric])
apply (simp add: Rep_cb_take)
apply (rule compact_eq_approx)
apply (rule Rep_compact_basis')
done

lemma cb_take_less: "cb_take n x ⊑ x"
unfolding compact_le_def
by (cases n, simp, simp add: Rep_cb_take approx_below)

lemma cb_take_idem: "cb_take n (cb_take n x) = cb_take n x"
unfolding Rep_compact_basis_inject [symmetric]
by (cases n, simp, simp add: Rep_cb_take approx_idem)

lemma cb_take_mono: "x ⊑ y ⟹ cb_take n x ⊑ cb_take n y"
unfolding compact_le_def
by (cases n, simp, simp add: Rep_cb_take monofun_cfun_arg)

lemma cb_take_chain_le: "m ≤ n ⟹ cb_take m x ⊑ cb_take n x"
unfolding compact_le_def
apply (cases m, simp, cases n, simp)
apply (simp add: Rep_cb_take, rule chain_mono, simp, simp)
done

lemma finite_range_cb_take: "finite (range (cb_take n))"
apply (cases n)
apply (subgoal_tac "range (cb_take 0) = {compact_bot}", simp, force)
apply (rule finite_imageD [where f="Rep_compact_basis"])
apply (rule finite_subset [where B="range (λx. approx (n - 1)⋅x)"])
apply (clarsimp simp add: Rep_cb_take)
apply (rule finite_range_approx)
apply (rule inj_onI, simp add: Rep_compact_basis_inject)
done

subsubsection ‹Rank of basis elements›

definition
  rank :: "'a compact_basis ⇒ nat"
where
  "rank x = (LEAST n. cb_take n x = x)"

lemma compact_approx_rank: "cb_take (rank x) x = x"
unfolding rank_def
apply (rule LeastI_ex)
apply (rule cb_take_covers)
done

lemma rank_leD: "rank x ≤ n ⟹ cb_take n x = x"
apply (rule below_antisym [OF cb_take_less])
apply (subst compact_approx_rank [symmetric])
apply (erule cb_take_chain_le)
done

lemma rank_leI: "cb_take n x = x ⟹ rank x ≤ n"
unfolding rank_def by (rule Least_le)

lemma rank_le_iff: "rank x ≤ n ⟷ cb_take n x = x"
by (rule iffI [OF rank_leD rank_leI])

lemma rank_compact_bot [simp]: "rank compact_bot = 0"
using rank_leI [of 0 compact_bot] by simp

lemma rank_eq_0_iff [simp]: "rank x = 0 ⟷ x = compact_bot"
using rank_le_iff [of x 0] by auto

definition
  rank_le :: "'a compact_basis ⇒ 'a compact_basis set"
where
  "rank_le x = {y. rank y ≤ rank x}"

definition
  rank_lt :: "'a compact_basis ⇒ 'a compact_basis set"
where
  "rank_lt x = {y. rank y < rank x}"

definition
  rank_eq :: "'a compact_basis ⇒ 'a compact_basis set"
where
  "rank_eq x = {y. rank y = rank x}"

lemma rank_eq_cong: "rank x = rank y ⟹ rank_eq x = rank_eq y"
unfolding rank_eq_def by simp

lemma rank_lt_cong: "rank x = rank y ⟹ rank_lt x = rank_lt y"
unfolding rank_lt_def by simp

lemma rank_eq_subset: "rank_eq x ⊆ rank_le x"
unfolding rank_eq_def rank_le_def by auto

lemma rank_lt_subset: "rank_lt x ⊆ rank_le x"
unfolding rank_lt_def rank_le_def by auto

lemma finite_rank_le: "finite (rank_le x)"
unfolding rank_le_def
apply (rule finite_subset [where B="range (cb_take (rank x))"])
apply clarify
apply (rule range_eqI)
apply (erule rank_leD [symmetric])
apply (rule finite_range_cb_take)
done

lemma finite_rank_eq: "finite (rank_eq x)"
by (rule finite_subset [OF rank_eq_subset finite_rank_le])

lemma finite_rank_lt: "finite (rank_lt x)"
by (rule finite_subset [OF rank_lt_subset finite_rank_le])

lemma rank_lt_Int_rank_eq: "rank_lt x ∩ rank_eq x = {}"
unfolding rank_lt_def rank_eq_def rank_le_def by auto

lemma rank_lt_Un_rank_eq: "rank_lt x ∪ rank_eq x = rank_le x"
unfolding rank_lt_def rank_eq_def rank_le_def by auto

subsubsection ‹Sequencing basis elements›

definition
  place :: "'a compact_basis ⇒ nat"
where
  "place x = card (rank_lt x) + choose_pos (rank_eq x) x"

lemma place_bounded: "place x < card (rank_le x)"
unfolding place_def
 apply (rule ord_less_eq_trans)
  apply (rule add_strict_left_mono)
  apply (rule choose_pos_bounded)
   apply (rule finite_rank_eq)
  apply (simp add: rank_eq_def)
 apply (subst card_Un_disjoint [symmetric])
    apply (rule finite_rank_lt)
   apply (rule finite_rank_eq)
  apply (rule rank_lt_Int_rank_eq)
 apply (simp add: rank_lt_Un_rank_eq)
done

lemma place_ge: "card (rank_lt x) ≤ place x"
unfolding place_def by simp

lemma place_rank_mono:
  fixes x y :: "'a compact_basis"
  shows "rank x < rank y ⟹ place x < place y"
apply (rule less_le_trans [OF place_bounded])
apply (rule order_trans [OF _ place_ge])
apply (rule card_mono)
apply (rule finite_rank_lt)
apply (simp add: rank_le_def rank_lt_def subset_eq)
done

lemma place_eqD: "place x = place y ⟹ x = y"
 apply (rule linorder_cases [where x="rank x" and y="rank y"])
   apply (drule place_rank_mono, simp)
  apply (simp add: place_def)
  apply (rule inj_on_choose_pos [where A="rank_eq x", THEN inj_onD])
     apply (rule finite_rank_eq)
    apply (simp cong: rank_lt_cong rank_eq_cong)
   apply (simp add: rank_eq_def)
  apply (simp add: rank_eq_def)
 apply (drule place_rank_mono, simp)
done

lemma inj_place: "inj place"
by (rule inj_onI, erule place_eqD)

subsubsection ‹Embedding and projection on basis elements›

definition
  sub :: "'a compact_basis ⇒ 'a compact_basis"
where
  "sub x = (case rank x of 0 ⇒ compact_bot | Suc k ⇒ cb_take k x)"

lemma rank_sub_less: "x ≠ compact_bot ⟹ rank (sub x) < rank x"
unfolding sub_def
apply (cases "rank x", simp)
apply (simp add: less_Suc_eq_le)
apply (rule rank_leI)
apply (rule cb_take_idem)
done

lemma place_sub_less: "x ≠ compact_bot ⟹ place (sub x) < place x"
apply (rule place_rank_mono)
apply (erule rank_sub_less)
done

lemma sub_below: "sub x ⊑ x"
unfolding sub_def by (cases "rank x", simp_all add: cb_take_less)

lemma rank_less_imp_below_sub: "⟦x ⊑ y; rank x < rank y⟧ ⟹ x ⊑ sub y"
unfolding sub_def
apply (cases "rank y", simp)
apply (simp add: less_Suc_eq_le)
apply (subgoal_tac "cb_take nat x ⊑ cb_take nat y")
apply (simp add: rank_leD)
apply (erule cb_take_mono)
done

function basis_emb :: "'a compact_basis ⇒ ubasis"
  where "basis_emb x = (if x = compact_bot then 0 else
    node (place x) (basis_emb (sub x))
      (basis_emb ` {y. place y < place x ∧ x ⊑ y}))"
  by simp_all

termination basis_emb
  by (relation "measure place") (simp_all add: place_sub_less)

declare basis_emb.simps [simp del]

lemma basis_emb_compact_bot [simp]:
  "basis_emb compact_bot = 0"
  using basis_emb.simps [of compact_bot] by simp

lemma basis_emb_rec:
  "basis_emb x = node (place x) (basis_emb (sub x)) (basis_emb ` {y. place y < place x ∧ x ⊑ y})"
  if "x ≠ compact_bot"
  using that basis_emb.simps [of x] by simp

lemma basis_emb_eq_0_iff [simp]:
  "basis_emb x = 0 ⟷ x = compact_bot"
  by (cases "x = compact_bot") (simp_all add: basis_emb_rec)

lemma fin1: "finite {y. place y < place x ∧ x ⊑ y}"
apply (subst Collect_conj_eq)
apply (rule finite_Int)
apply (rule disjI1)
apply (subgoal_tac "finite (place -` {n. n < place x})", simp)
apply (rule finite_vimageI [OF _ inj_place])
apply (simp add: lessThan_def [symmetric])
done

lemma fin2: "finite (basis_emb ` {y. place y < place x ∧ x ⊑ y})"
by (rule finite_imageI [OF fin1])

lemma rank_place_mono:
  "⟦place x < place y; x ⊑ y⟧ ⟹ rank x < rank y"
apply (rule linorder_cases, assumption)
apply (simp add: place_def cong: rank_lt_cong rank_eq_cong)
apply (drule choose_pos_lessD)
apply (rule finite_rank_eq)
apply (simp add: rank_eq_def)
apply (simp add: rank_eq_def)
apply simp
apply (drule place_rank_mono, simp)
done

lemma basis_emb_mono:
  "x ⊑ y ⟹ ubasis_le (basis_emb x) (basis_emb y)"
proof (induct "max (place x) (place y)" arbitrary: x y rule: less_induct)
  case less
  show ?case proof (rule linorder_cases)
    assume "place x < place y"
    then have "rank x < rank y"
      using ‹x ⊑ y› by (rule rank_place_mono)
    with ‹place x < place y› show ?case
      apply (case_tac "y = compact_bot", simp)
      apply (simp add: basis_emb.simps [of y])
      apply (rule ubasis_le_trans [OF _ ubasis_le_lower [OF fin2]])
      apply (rule less)
       apply (simp add: less_max_iff_disj)
       apply (erule place_sub_less)
      apply (erule rank_less_imp_below_sub [OF ‹x ⊑ y›])
      done
  next
    assume "place x = place y"
    hence "x = y" by (rule place_eqD)
    thus ?case by (simp add: ubasis_le_refl)
  next
    assume "place x > place y"
    with ‹x ⊑ y› show ?case
      apply (case_tac "x = compact_bot", simp add: ubasis_le_minimal)
      apply (simp add: basis_emb.simps [of x])
      apply (rule ubasis_le_upper [OF fin2], simp)
      apply (rule less)
       apply (simp add: less_max_iff_disj)
       apply (erule place_sub_less)
      apply (erule rev_below_trans)
      apply (rule sub_below)
      done
  qed
qed

lemma inj_basis_emb: "inj basis_emb"
proof (rule injI)
  fix x y
  assume "basis_emb x = basis_emb y"
  then show "x = y"
    by (cases "x = compact_bot ∨ y = compact_bot") (auto simp add: basis_emb_rec fin2 place_eqD)
qed

definition
  basis_prj :: "ubasis ⇒ 'a compact_basis"
where
  "basis_prj x = inv basis_emb
    (ubasis_until (λx. x ∈ range (basis_emb :: 'a compact_basis ⇒ ubasis)) x)"

lemma basis_prj_basis_emb: "⋀x. basis_prj (basis_emb x) = x"
unfolding basis_prj_def
 apply (subst ubasis_until_same)
  apply (rule rangeI)
 apply (rule inv_f_f)
 apply (rule inj_basis_emb)
done

lemma basis_prj_node:
  "⟦finite S; node i a S ∉ range (basis_emb :: 'a compact_basis ⇒ nat)⟧
    ⟹ basis_prj (node i a S) = (basis_prj a :: 'a compact_basis)"
unfolding basis_prj_def by simp

lemma basis_prj_0: "basis_prj 0 = compact_bot"
apply (subst basis_emb_compact_bot [symmetric])
apply (rule basis_prj_basis_emb)
done

lemma node_eq_basis_emb_iff:
  "finite S ⟹ node i a S = basis_emb x ⟷
    x ≠ compact_bot ∧ i = place x ∧ a = basis_emb (sub x) ∧
        S = basis_emb ` {y. place y < place x ∧ x ⊑ y}"
apply (cases "x = compact_bot", simp)
apply (simp add: basis_emb.simps [of x])
apply (simp add: fin2)
done

lemma basis_prj_mono: "ubasis_le a b ⟹ basis_prj a ⊑ basis_prj b"
proof (induct a b rule: ubasis_le.induct)
  case (ubasis_le_refl a) show ?case by (rule below_refl)
next
  case (ubasis_le_trans a b c) thus ?case by - (rule below_trans)
next
  case (ubasis_le_lower S a i) thus ?case
    apply (cases "node i a S ∈ range (basis_emb :: 'a compact_basis ⇒ nat)")
     apply (erule rangeE, rename_tac x)
     apply (simp add: basis_prj_basis_emb)
     apply (simp add: node_eq_basis_emb_iff)
     apply (simp add: basis_prj_basis_emb)
     apply (rule sub_below)
    apply (simp add: basis_prj_node)
    done
next
  case (ubasis_le_upper S b a i) thus ?case
    apply (cases "node i a S ∈ range (basis_emb :: 'a compact_basis ⇒ nat)")
     apply (erule rangeE, rename_tac x)
     apply (simp add: basis_prj_basis_emb)
     apply (clarsimp simp add: node_eq_basis_emb_iff)
     apply (simp add: basis_prj_basis_emb)
    apply (simp add: basis_prj_node)
    done
qed

lemma basis_emb_prj_less: "ubasis_le (basis_emb (basis_prj x)) x"
unfolding basis_prj_def
 apply (subst f_inv_into_f [where f=basis_emb])
  apply (rule ubasis_until)
  apply (rule range_eqI [where x=compact_bot])
  apply simp
 apply (rule ubasis_until_less)
done

lemma ideal_completion:
  "ideal_completion below Rep_compact_basis (approximants :: 'a ⇒ _)"
proof
  fix w :: "'a"
  show "below.ideal (approximants w)"
  proof (rule below.idealI)
    have "Abs_compact_basis (approx 0⋅w) ∈ approximants w"
      by (simp add: approximants_def approx_below)
    thus "∃x. x ∈ approximants w" ..
  next
    fix x y :: "'a compact_basis"
    assume x: "x ∈ approximants w" and y: "y ∈ approximants w"
    obtain i where i: "approx i⋅(Rep_compact_basis x) = Rep_compact_basis x"
      using compact_eq_approx Rep_compact_basis' by fast
    obtain j where j: "approx j⋅(Rep_compact_basis y) = Rep_compact_basis y"
      using compact_eq_approx Rep_compact_basis' by fast
    let ?z = "Abs_compact_basis (approx (max i j)⋅w)"
    have "?z ∈ approximants w"
      by (simp add: approximants_def approx_below)
    moreover from x y have "x ⊑ ?z ∧ y ⊑ ?z"
      by (simp add: approximants_def compact_le_def)
         (metis i j monofun_cfun chain_mono chain_approx max.cobounded1 max.cobounded2)
    ultimately show "∃z ∈ approximants w. x ⊑ z ∧ y ⊑ z" ..
  next
    fix x y :: "'a compact_basis"
    assume "x ⊑ y" "y ∈ approximants w" thus "x ∈ approximants w"
      unfolding approximants_def compact_le_def
      by (auto elim: below_trans)
  qed
next
  fix Y :: "nat ⇒ 'a"
  assume "chain Y"
  thus "approximants (⨆i. Y i) = (⋃i. approximants (Y i))"
    unfolding approximants_def
    by (auto simp add: compact_below_lub_iff)
next
  fix a :: "'a compact_basis"
  show "approximants (Rep_compact_basis a) = {b. b ⊑ a}"
    unfolding approximants_def compact_le_def ..
next
  fix x y :: "'a"
  assume "approximants x ⊆ approximants y"
  hence "∀z. compact z ⟶ z ⊑ x ⟶ z ⊑ y"
    by (simp add: approximants_def subset_eq)
       (metis Abs_compact_basis_inverse')
  hence "(⨆i. approx i⋅x) ⊑ y"
    by (simp add: lub_below approx_below)
  thus "x ⊑ y"
    by (simp add: lub_distribs)
next
  show "∃f::'a compact_basis ⇒ nat. inj f"
    by (rule exI, rule inj_place)
qed

end

interpretation compact_basis:
  ideal_completion below Rep_compact_basis
    "approximants :: 'a::bifinite ⇒ 'a compact_basis set"
proof -
  obtain a :: "nat ⇒ 'a → 'a" where "approx_chain a"
    using bifinite ..
  hence "bifinite_approx_chain a"
    unfolding bifinite_approx_chain_def .
  thus "ideal_completion below Rep_compact_basis (approximants :: 'a ⇒ _)"
    by (rule bifinite_approx_chain.ideal_completion)
qed

subsubsection ‹EP-pair from any bifinite domain into \emph{udom}›

context bifinite_approx_chain begin

definition
  udom_emb :: "'a → udom"
where
  "udom_emb = compact_basis.extension (λx. udom_principal (basis_emb x))"

definition
  udom_prj :: "udom → 'a"
where
  "udom_prj = udom.extension (λx. Rep_compact_basis (basis_prj x))"

lemma udom_emb_principal:
  "udom_emb⋅(Rep_compact_basis x) = udom_principal (basis_emb x)"
unfolding udom_emb_def
apply (rule compact_basis.extension_principal)
apply (rule udom.principal_mono)
apply (erule basis_emb_mono)
done

lemma udom_prj_principal:
  "udom_prj⋅(udom_principal x) = Rep_compact_basis (basis_prj x)"
unfolding udom_prj_def
apply (rule udom.extension_principal)
apply (rule compact_basis.principal_mono)
apply (erule basis_prj_mono)
done

lemma ep_pair_udom: "ep_pair udom_emb udom_prj"
 apply standard
  apply (rule compact_basis.principal_induct, simp)
  apply (simp add: udom_emb_principal udom_prj_principal)
  apply (simp add: basis_prj_basis_emb)
 apply (rule udom.principal_induct, simp)
 apply (simp add: udom_emb_principal udom_prj_principal)
 apply (rule basis_emb_prj_less)
done

end

abbreviation "udom_emb ≡ bifinite_approx_chain.udom_emb"
abbreviation "udom_prj ≡ bifinite_approx_chain.udom_prj"

lemmas ep_pair_udom =
  bifinite_approx_chain.ep_pair_udom [unfolded bifinite_approx_chain_def]

subsection ‹Chain of approx functions for type \emph{udom}›

definition
  udom_approx :: "nat ⇒ udom → udom"
where
  "udom_approx i =
    udom.extension (λx. udom_principal (ubasis_until (λy. y ≤ i) x))"

lemma udom_approx_mono:
  "ubasis_le a b ⟹
    udom_principal (ubasis_until (λy. y ≤ i) a) ⊑
    udom_principal (ubasis_until (λy. y ≤ i) b)"
apply (rule udom.principal_mono)
apply (rule ubasis_until_mono)
apply (frule (2) order_less_le_trans [OF node_gt2])
apply (erule order_less_imp_le)
apply assumption
done

lemma adm_mem_finite: "⟦cont f; finite S⟧ ⟹ adm (λx. f x ∈ S)"
by (erule adm_subst, induct set: finite, simp_all)

lemma udom_approx_principal:
  "udom_approx i⋅(udom_principal x) =
    udom_principal (ubasis_until (λy. y ≤ i) x)"
unfolding udom_approx_def
apply (rule udom.extension_principal)
apply (erule udom_approx_mono)
done

lemma finite_deflation_udom_approx: "finite_deflation (udom_approx i)"
proof
  fix x show "udom_approx i⋅(udom_approx i⋅x) = udom_approx i⋅x"
    by (induct x rule: udom.principal_induct, simp)
       (simp add: udom_approx_principal ubasis_until_idem)
next
  fix x show "udom_approx i⋅x ⊑ x"
    by (induct x rule: udom.principal_induct, simp)
       (simp add: udom_approx_principal ubasis_until_less)
next
  have *: "finite (range (λx. udom_principal (ubasis_until (λy. y ≤ i) x)))"
    apply (subst range_composition [where f=udom_principal])
    apply (simp add: finite_range_ubasis_until)
    done
  show "finite {x. udom_approx i⋅x = x}"
    apply (rule finite_range_imp_finite_fixes)
    apply (rule rev_finite_subset [OF *])
    apply (clarsimp, rename_tac x)
    apply (induct_tac x rule: udom.principal_induct)
    apply (simp add: adm_mem_finite *)
    apply (simp add: udom_approx_principal)
    done
qed

interpretation udom_approx: finite_deflation "udom_approx i"
by (rule finite_deflation_udom_approx)

lemma chain_udom_approx [simp]: "chain (λi. udom_approx i)"
unfolding udom_approx_def
apply (rule chainI)
apply (rule udom.extension_mono)
apply (erule udom_approx_mono)
apply (erule udom_approx_mono)
apply (rule udom.principal_mono)
apply (rule ubasis_until_chain, simp)
done

lemma lub_udom_approx [simp]: "(⨆i. udom_approx i) = ID"
apply (rule cfun_eqI, simp add: contlub_cfun_fun)
apply (rule below_antisym)
apply (rule lub_below)
apply (simp)
apply (rule udom_approx.below)
apply (rule_tac x=x in udom.principal_induct)
apply (simp add: lub_distribs)
apply (rule_tac i=a in below_lub)
apply simp
apply (simp add: udom_approx_principal)
apply (simp add: ubasis_until_same ubasis_le_refl)
done

lemma udom_approx [simp]: "approx_chain udom_approx"
proof
  show "chain (λi. udom_approx i)"
    by (rule chain_udom_approx)
  show "(⨆i. udom_approx i) = ID"
    by (rule lub_udom_approx)
qed

instance udom :: bifinite
  by standard (fast intro: udom_approx)

hide_const (open) node

notation binomial  (infixl ‹choose› 65)

end