(*  Title:      HOLCF/Universal.thy

     Author:     Brian Huffman

 *)

 theory Universal

 imports CompactBasis NatIso

 begin

 subsection {* Basis datatype *}

 types ubasis = nat

 definition

   node :: "nat \<Rightarrow> ubasis \<Rightarrow> ubasis set \<Rightarrow> ubasis"

 where

   "node i a S = Suc (prod2nat (i, prod2nat (a, set2nat S)))"

 lemma node_not_0 [simp]: "node i a S \<noteq> 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]:

   "\<lbrakk>finite S; finite T\<rbrakk>

     \<Longrightarrow> node i a S = node j b T \<longleftrightarrow> i = j \<and> a = b \<and> S = T"

 unfolding node_def by simp

 lemma node_gt0: "i < node i a S"

 unfolding node_def less_Suc_eq_le

 by (rule le_prod2nat_1)

 lemma node_gt1: "a < node i a S"

 unfolding node_def less_Suc_eq_le

 by (rule order_trans [OF le_prod2nat_1 le_prod2nat_2])

 lemma nat_less_power2: "n < 2^n"

 by (induct n) simp_all

 lemma node_gt2: "\<lbrakk>finite S; b \<in> S\<rbrakk> \<Longrightarrow> b < node i a S"

 unfolding node_def less_Suc_eq_le set2nat_def

 apply (rule order_trans [OF _ le_prod2nat_2])

 apply (rule order_trans [OF _ le_prod2nat_2])

 apply (rule order_trans [where y="setsum (op ^ 2) {b}"])

 apply (simp add: nat_less_power2 [THEN order_less_imp_le])

 apply (erule setsum_mono2, simp, simp)

 done

 lemma eq_prod2nat_pairI:

   "\<lbrakk>fst (nat2prod x) = a; snd (nat2prod x) = b\<rbrakk> \<Longrightarrow> x = prod2nat (a, b)"

 by (erule subst, erule subst, simp)

 lemma node_cases:

   assumes 1: "x = 0 \<Longrightarrow> P"

   assumes 2: "\<And>i a S. \<lbrakk>finite S; x = node i a S\<rbrakk> \<Longrightarrow> P"

   shows "P"

  apply (cases x)

   apply (erule 1)

  apply (rule 2)

   apply (rule finite_nat2set)

  apply (simp add: node_def)

  apply (rule eq_prod2nat_pairI [OF refl])

  apply (rule eq_prod2nat_pairI [OF refl refl])

 done

 lemma node_induct:

   assumes 1: "P 0"

   assumes 2: "\<And>i a S. \<lbrakk>P a; finite S; \<forall>b\<in>S. P b\<rbrakk> \<Longrightarrow> 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

 subsection {* Basis ordering *}

 inductive

   ubasis_le :: "nat \<Rightarrow> nat \<Rightarrow> bool"

 where

   ubasis_le_refl: "ubasis_le a a"

 | ubasis_le_trans:

     "\<lbrakk>ubasis_le a b; ubasis_le b c\<rbrakk> \<Longrightarrow> ubasis_le a c"

 | ubasis_le_lower:

     "finite S \<Longrightarrow> ubasis_le a (node i a S)"

 | ubasis_le_upper:

     "\<lbrakk>finite S; b \<in> S; ubasis_le a b\<rbrakk> \<Longrightarrow> 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

 subsubsection {* Generic take function *}

 function

   ubasis_until :: "(ubasis \<Rightarrow> bool) \<Rightarrow> ubasis \<Rightarrow> ubasis"

 where

   "ubasis_until P 0 = 0"

 | "finite S \<Longrightarrow> 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

     apply simp

     apply fast

    apply simp

   apply simp

  apply simp

 done

 termination ubasis_until

 apply (relation "measure snd")

 apply (rule wf_measure)

 apply (simp add: node_gt1)

 done

 lemma ubasis_until: "P 0 \<Longrightarrow> P (ubasis_until P x)"

 by (induct x rule: node_induct) simp_all

 lemma ubasis_until': "0 < ubasis_until P x \<Longrightarrow> P (ubasis_until P x)"

 by (induct x rule: node_induct) auto

 lemma ubasis_until_same: "P x \<Longrightarrow> ubasis_until P x = x"

 by (induct x rule: node_induct) simp_all

 lemma ubasis_until_idem:

   "P 0 \<Longrightarrow> ubasis_until P (ubasis_until P x) = ubasis_until P x"

 by (rule ubasis_until_same [OF ubasis_until])

 lemma ubasis_until_0:

   "\<forall>x. x \<noteq> 0 \<longrightarrow> \<not> P x \<Longrightarrow> 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)

 apply (simp add: ubasis_le_refl)

 apply (rule impI)

 apply (erule ubasis_le_trans)

 apply (erule ubasis_le_lower)

 done

 lemma ubasis_until_chain:

   assumes PQ: "\<And>x. P x \<Longrightarrow> 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)

 apply (simp add: ubasis_le_refl)

 apply (simp add: PQ)

 apply clarify

 apply (rule ubasis_le_trans)

 apply (rule ubasis_until_less)

 apply (erule ubasis_le_lower)

 done

 lemma ubasis_until_mono:

   assumes "\<And>i a S b. \<lbrakk>finite S; P (node i a S); b \<in> S; ubasis_le a b\<rbrakk> \<Longrightarrow> P b"

   shows "ubasis_le a b \<Longrightarrow> 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

     apply (clarsimp simp add: ubasis_le_refl)

     apply (rule ubasis_le_trans [OF ubasis_until_less])

     apply (erule ubasis_le.ubasis_le_lower)

     done

 next

   case (ubasis_le_upper S b a i) thus ?case

     apply clarsimp

     apply (subst ubasis_until_same)

      apply (erule (3) prems)

     apply (erule (2) ubasis_le.ubasis_le_upper)

     done

 qed

 lemma finite_range_ubasis_until:

   "finite {x. P x} \<Longrightarrow> 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

 subsubsection {* Take function for @{typ ubasis} *}

 definition

   ubasis_take :: "nat \<Rightarrow> ubasis \<Rightarrow> ubasis"

 where

   "ubasis_take n = ubasis_until (\<lambda>x. x \<le> n)"

 lemma ubasis_take_le: "ubasis_take n x \<le> n"

 unfolding ubasis_take_def by (rule ubasis_until, rule le0)

 lemma ubasis_take_same: "x \<le> n \<Longrightarrow> ubasis_take n x = x"

 unfolding ubasis_take_def by (rule ubasis_until_same)

 lemma ubasis_take_idem: "ubasis_take n (ubasis_take n x) = ubasis_take n x"

 by (rule ubasis_take_same [OF ubasis_take_le])

 lemma ubasis_take_0 [simp]: "ubasis_take 0 x = 0"

 unfolding ubasis_take_def by (simp add: ubasis_until_0)

 lemma ubasis_take_less: "ubasis_le (ubasis_take n x) x"

 unfolding ubasis_take_def by (rule ubasis_until_less)

 lemma ubasis_take_chain: "ubasis_le (ubasis_take n x) (ubasis_take (Suc n) x)"

 unfolding ubasis_take_def by (rule ubasis_until_chain) simp

 lemma ubasis_take_mono:

   assumes "ubasis_le x y"

   shows "ubasis_le (ubasis_take n x) (ubasis_take n y)"

 unfolding ubasis_take_def

  apply (rule ubasis_until_mono [OF _ prems])

  apply (frule (2) order_less_le_trans [OF node_gt2])

  apply (erule order_less_imp_le)

 done

 lemma finite_range_ubasis_take: "finite (range (ubasis_take n))"

 apply (rule finite_subset [where B="{..n}"])

 apply (simp add: subset_eq ubasis_take_le)

 apply simp

 done

 lemma ubasis_take_covers: "\<exists>n. ubasis_take n x = x"

 apply (rule exI [where x=x])

 apply (simp add: ubasis_take_same)

 done

 interpretation udom: preorder ubasis_le

 apply default

 apply (rule ubasis_le_refl)

 apply (erule (1) ubasis_le_trans)

 done

 interpretation udom: basis_take ubasis_le ubasis_take

 apply default

 apply (rule ubasis_take_less)

 apply (rule ubasis_take_idem)

 apply (erule ubasis_take_mono)

 apply (rule ubasis_take_chain)

 apply (rule finite_range_ubasis_take)

 apply (rule ubasis_take_covers)

 done

 subsection {* Defining the universal domain by ideal completion *}

 typedef (open) udom = "{S. udom.ideal S}"

 by (fast intro: udom.ideal_principal)

 instantiation udom :: below

 begin

 definition

   "x \<sqsubseteq> y \<longleftrightarrow> Rep_udom x \<subseteq> Rep_udom y"

 instance ..

 end

 instance udom :: po

 by (rule udom.typedef_ideal_po

     [OF type_definition_udom below_udom_def])

 instance udom :: cpo

 by (rule udom.typedef_ideal_cpo

     [OF type_definition_udom below_udom_def])

 lemma Rep_udom_lub:

   "chain Y \<Longrightarrow> Rep_udom (\<Squnion>i. Y i) = (\<Union>i. Rep_udom (Y i))"

 by (rule udom.typedef_ideal_rep_contlub

     [OF type_definition_udom below_udom_def])

 lemma ideal_Rep_udom: "udom.ideal (Rep_udom xs)"

 by (rule Rep_udom [unfolded mem_Collect_eq])

 definition

   udom_principal :: "nat \<Rightarrow> udom" where

   "udom_principal t = Abs_udom {u. ubasis_le u t}"

 lemma Rep_udom_principal:

   "Rep_udom (udom_principal t) = {u. ubasis_le u t}"

 unfolding udom_principal_def

 by (simp add: Abs_udom_inverse udom.ideal_principal)

 interpretation udom:

   ideal_completion ubasis_le ubasis_take udom_principal Rep_udom

 apply unfold_locales

 apply (rule ideal_Rep_udom)

 apply (erule Rep_udom_lub)

 apply (rule Rep_udom_principal)

 apply (simp only: below_udom_def)

 done

 text {* Universal domain is pointed *}

 lemma udom_minimal: "udom_principal 0 \<sqsubseteq> 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: "\<bottom> = udom_principal 0"

 by (rule udom_minimal [THEN UU_I, symmetric])

 text {* Universal domain is bifinite *}

 instantiation udom :: bifinite

 begin

 definition

   approx_udom_def: "approx = udom.completion_approx"

 instance

 apply (intro_classes, unfold approx_udom_def)

 apply (rule udom.chain_completion_approx)

 apply (rule udom.lub_completion_approx)

 apply (rule udom.completion_approx_idem)

 apply (rule udom.finite_fixes_completion_approx)

 done

 end

 lemma approx_udom_principal [simp]:

   "approx n\<cdot>(udom_principal x) = udom_principal (ubasis_take n x)"

 unfolding approx_udom_def

 by (rule udom.completion_approx_principal)

 lemma approx_eq_udom_principal:

   "\<exists>a\<in>Rep_udom x. approx n\<cdot>x = udom_principal (ubasis_take n a)"

 unfolding approx_udom_def

 by (rule udom.completion_approx_eq_principal)

 subsection {* Universality of @{typ udom} *}

 defaultsort bifinite

 subsubsection {* Choosing a maximal element from a finite set *}

 lemma finite_has_maximal:

   fixes A :: "'a::po set"

   shows "\<lbrakk>finite A; A \<noteq> {}\<rbrakk> \<Longrightarrow> \<exists>x\<in>A. \<forall>y\<in>A. x \<sqsubseteq> y \<longrightarrow> x = y"

 proof (induct rule: finite_ne_induct)

   case (singleton x)

     show ?case by simp

 next

   case (insert a A)

   from `\<exists>x\<in>A. \<forall>y\<in>A. x \<sqsubseteq> y \<longrightarrow> x = y`

   obtain x where x: "x \<in> A"

            and x_eq: "\<And>y. \<lbrakk>y \<in> A; x \<sqsubseteq> y\<rbrakk> \<Longrightarrow> x = y" by fast

   show ?case

   proof (intro bexI ballI impI)

     fix y

     assume "y \<in> insert a A" and "(if x \<sqsubseteq> a then a else x) \<sqsubseteq> y"

     thus "(if x \<sqsubseteq> 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 \<sqsubseteq> a then a else x) \<in> insert a A"

       by (simp add: x)

   qed

 qed

 definition

   choose :: "'a compact_basis set \<Rightarrow> 'a compact_basis"

 where

   "choose A = (SOME x. x \<in> {x\<in>A. \<forall>y\<in>A. x \<sqsubseteq> y \<longrightarrow> x = y})"

 lemma choose_lemma:

   "\<lbrakk>finite A; A \<noteq> {}\<rbrakk> \<Longrightarrow> choose A \<in> {x\<in>A. \<forall>y\<in>A. x \<sqsubseteq> y \<longrightarrow> x = y}"

 unfolding choose_def

 apply (rule someI_ex)

 apply (frule (1) finite_has_maximal, fast)

 done

 lemma maximal_choose:

   "\<lbrakk>finite A; y \<in> A; choose A \<sqsubseteq> y\<rbrakk> \<Longrightarrow> choose A = y"

 apply (cases "A = {}", simp)

 apply (frule (1) choose_lemma, simp)

 done

 lemma choose_in: "\<lbrakk>finite A; A \<noteq> {}\<rbrakk> \<Longrightarrow> choose A \<in> A"

 by (frule (1) choose_lemma, simp)

 function

   choose_pos :: "'a compact_basis set \<Rightarrow> 'a compact_basis \<Rightarrow> nat"

 where

   "choose_pos A x =

     (if finite A \<and> x \<in> A \<and> x \<noteq> choose A

       then Suc (choose_pos (A - {choose A}) x) else 0)"

 by auto

 termination choose_pos

 apply (relation "measure (card \<circ> 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 \<Longrightarrow> choose_pos A (choose A) = 0"

 by (simp add: choose_pos.simps)

 lemma inj_on_choose_pos [OF refl]:

   "\<lbrakk>card A = n; finite A\<rbrakk> \<Longrightarrow> 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: split_if_asm)

  apply (erule (1) inj_onD, simp, simp)

 done

 lemma choose_pos_bounded [OF refl]:

   "\<lbrakk>card A = n; finite A; x \<in> A\<rbrakk> \<Longrightarrow> 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:

   "\<lbrakk>choose_pos A x < choose_pos A y; finite A; x \<in> A; y \<in> A\<rbrakk> \<Longrightarrow> \<not> x \<sqsubseteq> 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 {* Rank of basis elements *}

 primrec

   cb_take :: "nat \<Rightarrow> 'a compact_basis \<Rightarrow> 'a compact_basis"

 where

   "cb_take 0 = (\<lambda>x. compact_bot)"

 | "cb_take (Suc n) = compact_take n"

 lemma cb_take_covers: "\<exists>n. cb_take n x = x"

 apply (rule exE [OF compact_basis.take_covers [where a=x]])

 apply (rename_tac n, rule_tac x="Suc n" in exI, simp)

 done

 lemma cb_take_less: "cb_take n x \<sqsubseteq> x"

 by (cases n, simp, simp add: compact_basis.take_less)

 lemma cb_take_idem: "cb_take n (cb_take n x) = cb_take n x"

 by (cases n, simp, simp add: compact_basis.take_take)

 lemma cb_take_mono: "x \<sqsubseteq> y \<Longrightarrow> cb_take n x \<sqsubseteq> cb_take n y"

 by (cases n, simp, simp add: compact_basis.take_mono)

 lemma cb_take_chain_le: "m \<le> n \<Longrightarrow> cb_take m x \<sqsubseteq> cb_take n x"

 apply (cases m, simp)

 apply (cases n, simp)

 apply (simp add: compact_basis.take_chain_le)

 done

 lemma range_const: "range (\<lambda>x. c) = {c}"

 by auto

 lemma finite_range_cb_take: "finite (range (cb_take n))"

 apply (cases n)

 apply (simp add: range_const)

 apply (simp add: compact_basis.finite_range_take)

 done

 definition

   rank :: "'a compact_basis \<Rightarrow> 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 \<le> n \<Longrightarrow> 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 \<Longrightarrow> rank x \<le> n"

 unfolding rank_def by (rule Least_le)

 lemma rank_le_iff: "rank x \<le> n \<longleftrightarrow> 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 \<longleftrightarrow> x = compact_bot"

 using rank_le_iff [of x 0] by auto

 definition

   rank_le :: "'a compact_basis \<Rightarrow> 'a compact_basis set"

 where

   "rank_le x = {y. rank y \<le> rank x}"

 definition

   rank_lt :: "'a compact_basis \<Rightarrow> 'a compact_basis set"

 where

   "rank_lt x = {y. rank y < rank x}"

 definition

   rank_eq :: "'a compact_basis \<Rightarrow> 'a compact_basis set"

 where

   "rank_eq x = {y. rank y = rank x}"

 lemma rank_eq_cong: "rank x = rank y \<Longrightarrow> rank_eq x = rank_eq y"

 unfolding rank_eq_def by simp

 lemma rank_lt_cong: "rank x = rank y \<Longrightarrow> rank_lt x = rank_lt y"

 unfolding rank_lt_def by simp

 lemma rank_eq_subset: "rank_eq x \<subseteq> rank_le x"

 unfolding rank_eq_def rank_le_def by auto

 lemma rank_lt_subset: "rank_lt x \<subseteq> 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 \<inter> rank_eq x = {}"

 unfolding rank_lt_def rank_eq_def rank_le_def by auto

 lemma rank_lt_Un_rank_eq: "rank_lt x \<union> 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 \<Rightarrow> 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) \<le> place x"

 unfolding place_def by simp

 lemma place_rank_mono:

   fixes x y :: "'a compact_basis"

   shows "rank x < rank y \<Longrightarrow> 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 \<Longrightarrow> 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 \<Rightarrow> 'a compact_basis"

 where

   "sub x = (case rank x of 0 \<Rightarrow> compact_bot | Suc k \<Rightarrow> cb_take k x)"

 lemma rank_sub_less: "x \<noteq> compact_bot \<Longrightarrow> 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 \<noteq> compact_bot \<Longrightarrow> place (sub x) < place x"

 apply (rule place_rank_mono)

 apply (erule rank_sub_less)

 done

 lemma sub_below: "sub x \<sqsubseteq> x"

 unfolding sub_def by (cases "rank x", simp_all add: cb_take_less)

 lemma rank_less_imp_below_sub: "\<lbrakk>x \<sqsubseteq> y; rank x < rank y\<rbrakk> \<Longrightarrow> x \<sqsubseteq> sub y"

 unfolding sub_def

 apply (cases "rank y", simp)

 apply (simp add: less_Suc_eq_le)

 apply (subgoal_tac "cb_take nat x \<sqsubseteq> cb_take nat y")

 apply (simp add: rank_leD)

 apply (erule cb_take_mono)

 done

 function

   basis_emb :: "'a compact_basis \<Rightarrow> 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 \<and> x \<sqsubseteq> y}))"

 by auto

 termination basis_emb

 apply (relation "measure place", simp)

 apply (simp add: place_sub_less)

 apply simp

 done

 declare basis_emb.simps [simp del]

 lemma basis_emb_compact_bot [simp]: "basis_emb compact_bot = 0"

 by (simp add: basis_emb.simps)

 lemma fin1: "finite {y. place y < place x \<and> x \<sqsubseteq> 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 \<and> x \<sqsubseteq> y})"

 by (rule finite_imageI [OF fin1])

 lemma rank_place_mono:

   "\<lbrakk>place x < place y; x \<sqsubseteq> y\<rbrakk> \<Longrightarrow> 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 \<sqsubseteq> y \<Longrightarrow> ubasis_le (basis_emb x) (basis_emb y)"

 proof (induct n \<equiv> "max (place x) (place y)" arbitrary: x y rule: less_induct)

   case (less n)

   hence IH:

     "\<And>(a::'a compact_basis) b.

      \<lbrakk>max (place a) (place b) < max (place x) (place y); a \<sqsubseteq> b\<rbrakk>

         \<Longrightarrow> ubasis_le (basis_emb a) (basis_emb b)"

     by simp

   show ?case proof (rule linorder_cases)

     assume "place x < place y"

     then have "rank x < rank y"

       using `x \<sqsubseteq> 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 IH)

        apply (simp add: less_max_iff_disj)

        apply (erule place_sub_less)

       apply (erule rank_less_imp_below_sub [OF `x \<sqsubseteq> 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 \<sqsubseteq> 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 IH)

        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"

  apply (rule inj_onI)

  apply (case_tac "x = compact_bot")

   apply (case_tac [!] "y = compact_bot")

     apply simp

    apply (simp add: basis_emb.simps)

   apply (simp add: basis_emb.simps)

  apply (simp add: basis_emb.simps)

  apply (simp add: fin2 inj_eq [OF inj_place])

 done

 definition

   basis_prj :: "ubasis \<Rightarrow> 'a compact_basis"

 where

   "basis_prj x = inv basis_emb

     (ubasis_until (\<lambda>x. x \<in> range (basis_emb :: 'a compact_basis \<Rightarrow> ubasis)) x)"

 lemma basis_prj_basis_emb: "\<And>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:

   "\<lbrakk>finite S; node i a S \<notin> range (basis_emb :: 'a compact_basis \<Rightarrow> nat)\<rbrakk>

     \<Longrightarrow> 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 \<Longrightarrow> node i a S = basis_emb x \<longleftrightarrow>

     x \<noteq> compact_bot \<and> i = place x \<and> a = basis_emb (sub x) \<and>

         S = basis_emb ` {y. place y < place x \<and> x \<sqsubseteq> 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 \<Longrightarrow> basis_prj a \<sqsubseteq> 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 \<in> range (basis_emb :: 'a compact_basis \<Rightarrow> 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 \<in> range (basis_emb :: 'a compact_basis \<Rightarrow> 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_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

 hide (open) const

   node

   choose

   choose_pos

   place

   sub

 subsubsection {* EP-pair from any bifinite domain into @{typ udom} *}

 definition

   udom_emb :: "'a::bifinite \<rightarrow> udom"

 where

   "udom_emb = compact_basis.basis_fun (\<lambda>x. udom_principal (basis_emb x))"

 definition

   udom_prj :: "udom \<rightarrow> 'a::bifinite"

 where

   "udom_prj = udom.basis_fun (\<lambda>x. Rep_compact_basis (basis_prj x))"

 lemma udom_emb_principal:

   "udom_emb\<cdot>(Rep_compact_basis x) = udom_principal (basis_emb x)"

 unfolding udom_emb_def

 apply (rule compact_basis.basis_fun_principal)

 apply (rule udom.principal_mono)

 apply (erule basis_emb_mono)

 done

 lemma udom_prj_principal:

   "udom_prj\<cdot>(udom_principal x) = Rep_compact_basis (basis_prj x)"

 unfolding udom_prj_def

 apply (rule udom.basis_fun_principal)

 apply (rule compact_basis.principal_mono)

 apply (erule basis_prj_mono)

 done

 lemma ep_pair_udom: "ep_pair udom_emb udom_prj"

  apply default

   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