(* Title: HOL/HOLCF/Universal.thy
Author: Brian Huffman
*)
section \<open>A universal bifinite domain\<close>
theory Universal
imports Bifinite Completion "HOL-Library.Nat_Bijection"
begin
no_notation binomial (infixl "choose" 65)
subsection \<open>Basis for universal domain\<close>
subsubsection \<open>Basis datatype\<close>
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\
\<Longrightarrow> node i a S = node j b T \<longleftrightarrow> i = j \<and> a = b \<and> 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 (induct n) simp_all
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 (erule subst, erule subst, simp)
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 \<open>Basis ordering\<close>
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 \<open>Generic take function\<close>
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)
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: "\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)
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 "\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
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) assms)
apply (erule (2) ubasis_le.ubasis_le_upper)
done
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 \<open>Defining the universal domain by ideal completion\<close>
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 \<open>Universal domain is pointed\<close>
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 \<open>Compact bases of domains\<close>
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:
"(\) \ (\x 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 = (\x. {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 \<open>Universality of \emph{udom}\<close>
text \<open>We use a locale to parameterize the construction over a chain
of approx functions on the type to be embedded.\<close>
locale bifinite_approx_chain =
approx_chain approx for approx :: "nat \ 'a::bifinite \ 'a"
begin
subsubsection \<open>Choosing a maximal element from a finite set\<close>
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 \<open>\<exists>x\<in>A. \<forall>y\<in>A. x \<sqsubseteq> y \<longrightarrow> x = y\<close>
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 \<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 \ 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 \ 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 \<open>Compact basis take function\<close>
primrec
cb_take :: "nat \ 'a compact_basis \ 'a compact_basis" where
"cb_take 0 = (\x. compact_bot)"
| "cb_take (Suc n) = (\a. 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 (\x. 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 \<open>Rank of basis elements\<close>
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 \<open>Sequencing basis elements\<close>
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 \<open>Embedding and projection on basis elements\<close>
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 \<and> x \<sqsubseteq> 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 \<open>x \<sqsubseteq> y\<close> by (rule rank_place_mono)
with \<open>place x < place y\<close> 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 \<open>x \<sqsubseteq> y\<close>])
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 \<open>x \<sqsubseteq> y\<close> 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 (\<lambda>x. x \<in> range (basis_emb :: 'a compact_basis \<Rightarrow> 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)\
\<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 \ node i a S = basis_emb x \
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 \ 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 \<open>EP-pair from any bifinite domain into \emph{udom}\<close>
context bifinite_approx_chain begin
definition
udom_emb :: "'a \ udom"
where
"udom_emb = compact_basis.extension (\x. udom_principal (basis_emb x))"
definition
udom_prj :: "udom \ 'a"
where
"udom_prj = udom.extension (\x. 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 \<open>Chain of approx functions for type \emph{udom}\<close>
definition
udom_approx :: "nat \ udom \ udom"
where
"udom_approx i =
udom.extension (\<lambda>x. udom_principal (ubasis_until (\<lambda>y. y \<le> i) x))"
lemma udom_approx_mono:
"ubasis_le a b \
udom_principal (ubasis_until (\<lambda>y. y \<le> i) a) \<sqsubseteq>
udom_principal (ubasis_until (\<lambda>y. y \<le> 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 (\x. 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 (\<lambda>y. y \<le> 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 (\x. udom_principal (ubasis_until (\y. 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 (\i. 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 (\i. 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
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