# Theory Wellorder_Extension

(*  Title:      HOL/Cardinals/Wellorder_Extension.thy
Author:     Christian Sternagel, JAIST
*)

section ‹Extending Well-founded Relations to Wellorders›

theory Wellorder_Extension
imports Main Order_Union
begin

subsection ‹Extending Well-founded Relations to Wellorders›

text ‹A \emph{downset} (also lower set, decreasing set, initial segment, or
downward closed set) is closed w.r.t.\ smaller elements.›
definition downset_on where
"downset_on A r = (∀x y. (x, y) ∈ r ∧ y ∈ A ⟶ x ∈ A)"

(*
text {*Connection to order filters of the @{theory Cardinals} theory.*}
lemma (in wo_rel) ofilter_downset_on_conv:
"ofilter A ⟷ downset_on A r ∧ A ⊆ Field r"
by (auto simp: downset_on_def ofilter_def under_def)
*)

lemma downset_onI:
"(⋀x y. (x, y) ∈ r ⟹ y ∈ A ⟹ x ∈ A) ⟹ downset_on A r"
by (auto simp: downset_on_def)

lemma downset_onD:
"downset_on A r ⟹ (x, y) ∈ r ⟹ y ∈ A ⟹ x ∈ A"
unfolding downset_on_def by blast

text ‹Extensions of relations w.r.t.\ a given set.›
definition extension_on where
"extension_on A r s = (∀x∈A. ∀y∈A. (x, y) ∈ s ⟶ (x, y) ∈ r)"

lemma extension_onI:
"(⋀x y. ⟦x ∈ A; y ∈ A; (x, y) ∈ s⟧ ⟹ (x, y) ∈ r) ⟹ extension_on A r s"
by (auto simp: extension_on_def)

lemma extension_onD:
"extension_on A r s ⟹ x ∈ A ⟹ y ∈ A ⟹ (x, y) ∈ s ⟹ (x, y) ∈ r"
by (auto simp: extension_on_def)

lemma downset_on_Union:
assumes "⋀r. r ∈ R ⟹ downset_on (Field r) p"
shows "downset_on (Field (⋃R)) p"
using assms by (auto intro: downset_onI dest: downset_onD)

lemma chain_subset_extension_on_Union:
assumes "chain⇩⊆ R" and "⋀r. r ∈ R ⟹ extension_on (Field r) r p"
shows "extension_on (Field (⋃R)) (⋃R) p"
using assms
by (simp add: chain_subset_def extension_on_def)
(metis (no_types) mono_Field subsetD)

lemma downset_on_empty [simp]: "downset_on {} p"
by (auto simp: downset_on_def)

lemma extension_on_empty [simp]: "extension_on {} p q"
by (auto simp: extension_on_def)

text ‹Every well-founded relation can be extended to a wellorder.›
theorem well_order_extension:
assumes "wf p"
shows "∃w. p ⊆ w ∧ Well_order w"
proof -
let ?K = "{r. Well_order r ∧ downset_on (Field r) p ∧ extension_on (Field r) r p}"
define I where "I = init_seg_of ∩ ?K × ?K"
have I_init: "I ⊆ init_seg_of" by (simp add: I_def)
then have subch: "⋀R. R ∈ Chains I ⟹ chain⇩⊆ R"
by (auto simp: init_seg_of_def chain_subset_def Chains_def)
have Chains_wo: "⋀R r. R ∈ Chains I ⟹ r ∈ R ⟹
Well_order r ∧ downset_on (Field r) p ∧ extension_on (Field r) r p"
by (simp add: Chains_def I_def) blast
have FI: "Field I = ?K" by (auto simp: I_def init_seg_of_def Field_def)
then have 0: "Partial_order I"
by (auto simp: partial_order_on_def preorder_on_def antisym_def antisym_init_seg_of refl_on_def
trans_def I_def elim: trans_init_seg_of)
have "⋃R ∈ ?K ∧ (∀r∈R. (r,⋃R) ∈ I)" if "R ∈ Chains I" for R
proof -
from that have Ris: "R ∈ Chains init_seg_of" using mono_Chains [OF I_init] by blast
have subch: "chain⇩⊆ R" using ‹R ∈ Chains I› I_init
by (auto simp: init_seg_of_def chain_subset_def Chains_def)
have "∀r∈R. Refl r" and "∀r∈R. trans r" and "∀r∈R. antisym r" and
"∀r∈R. Total r" and "∀r∈R. wf (r - Id)" and
"⋀r. r ∈ R ⟹ downset_on (Field r) p" and
"⋀r. r ∈ R ⟹ extension_on (Field r) r p"
using Chains_wo [OF ‹R ∈ Chains I›] by (simp_all add: order_on_defs)
have "Refl (⋃R)" using ‹∀r∈R. Refl r›  unfolding refl_on_def by fastforce
moreover have "trans (⋃R)"
by (rule chain_subset_trans_Union [OF subch ‹∀r∈R. trans r›])
moreover have "antisym (⋃R)"
by (rule chain_subset_antisym_Union [OF subch ‹∀r∈R. antisym r›])
moreover have "Total (⋃R)"
by (rule chain_subset_Total_Union [OF subch ‹∀r∈R. Total r›])
moreover have "wf ((⋃R) - Id)"
proof -
have "(⋃R) - Id = ⋃{r - Id | r. r ∈ R}" by blast
with ‹∀r∈R. wf (r - Id)› wf_Union_wf_init_segs [OF Chains_inits_DiffI [OF Ris]]
show ?thesis by fastforce
qed
ultimately have "Well_order (⋃R)" by (simp add: order_on_defs)
moreover have "∀r∈R. r initial_segment_of ⋃R" using Ris
by (simp add: Chains_init_seg_of_Union)
moreover have "downset_on (Field (⋃R)) p"
by (rule downset_on_Union [OF ‹⋀r. r ∈ R ⟹ downset_on (Field r) p›])
moreover have "extension_on (Field (⋃R)) (⋃R) p"
by (rule chain_subset_extension_on_Union [OF subch ‹⋀r. r ∈ R ⟹ extension_on (Field r) r p›])
ultimately show ?thesis
using mono_Chains [OF I_init] and ‹R ∈ Chains I›
by (simp (no_asm) add: I_def del: Field_Union) (metis Chains_wo)
qed
then have 1: "∃u∈Field I. ∀r∈R. (r, u) ∈ I" if "R∈Chains I" for R
using that by (subst FI) blast
txt ‹Zorn's Lemma yields a maximal wellorder m.›
from Zorns_po_lemma [OF 0 1] obtain m :: "('a × 'a) set"
where "Well_order m" and "downset_on (Field m) p" and "extension_on (Field m) m p" and
max: "∀r. Well_order r ∧ downset_on (Field r) p ∧ extension_on (Field r) r p ∧
(m, r) ∈ I ⟶ r = m"
by (auto simp: FI)
have "Field p ⊆ Field m"
proof (rule ccontr)
let ?Q = "Field p - Field m"
assume "¬ (Field p ⊆ Field m)"
with assms [unfolded wf_eq_minimal, THEN spec, of ?Q]
obtain x where "x ∈ Field p" and "x ∉ Field m" and
min: "∀y. (y, x) ∈ p ⟶ y ∉ ?Q" by blast
txt ‹Add \<^term>‹x› as topmost element to \<^term>‹m›.›
let ?s = "{(y, x) | y. y ∈ Field m}"
let ?m = "insert (x, x) m ∪ ?s"
have Fm: "Field ?m = insert x (Field m)" by (auto simp: Field_def)
have "Refl m" and "trans m" and "antisym m" and "Total m" and "wf (m - Id)"
using ‹Well_order m› by (simp_all add: order_on_defs)
txt ‹We show that the extension is a wellorder.›
have "Refl ?m" using ‹Refl m› Fm by (auto simp: refl_on_def)
moreover have "trans ?m" using ‹trans m› ‹x ∉ Field m›
unfolding trans_def Field_def Domain_unfold Domain_converse [symmetric] by blast
moreover have "antisym ?m" using ‹antisym m› ‹x ∉ Field m›
unfolding antisym_def Field_def Domain_unfold Domain_converse [symmetric] by blast
moreover have "Total ?m" using ‹Total m› Fm by (auto simp: Relation.total_on_def)
moreover have "wf (?m - Id)"
proof -
have "wf ?s" using ‹x ∉ Field m›
by (simp add: wf_eq_minimal Field_def Domain_unfold Domain_converse [symmetric]) metis
thus ?thesis using ‹wf (m - Id)› ‹x ∉ Field m›
wf_subset [OF ‹wf ?s› Diff_subset]
by (fastforce intro!: wf_Un simp add: Un_Diff Field_def)
qed
ultimately have "Well_order ?m" by (simp add: order_on_defs)
moreover have "extension_on (Field ?m) ?m p"
using ‹extension_on (Field m) m p› ‹downset_on (Field m) p›
by (subst Fm) (auto simp: extension_on_def dest: downset_onD)
moreover have "downset_on (Field ?m) p"
apply (subst Fm)
using ‹downset_on (Field m) p› and min
unfolding downset_on_def Field_def by blast
moreover have "(m, ?m) ∈ I"
using ‹Well_order m› and ‹Well_order ?m› and
‹downset_on (Field m) p› and ‹downset_on (Field ?m) p› and
‹extension_on (Field m) m p› and ‹extension_on (Field ?m) ?m p› and
‹Refl m› and ‹x ∉ Field m›
by (auto simp: I_def init_seg_of_def refl_on_def)
ultimately
― ‹This contradicts maximality of m:›
show False using max and ‹x ∉ Field m› unfolding Field_def by blast
qed
have "p ⊆ m"
using ‹Field p ⊆ Field m› and ‹extension_on (Field m) m p›
unfolding Field_def extension_on_def by auto fast
with ‹Well_order m› show ?thesis by blast
qed

text ‹Every well-founded relation can be extended to a total wellorder.›
corollary total_well_order_extension:
assumes "wf p"
shows "∃w. p ⊆ w ∧ Well_order w ∧ Field w = UNIV"
proof -
from well_order_extension [OF assms] obtain w
where "p ⊆ w" and wo: "Well_order w" by blast
let ?A = "UNIV - Field w"
from well_order_on [of ?A] obtain w' where wo': "well_order_on ?A w'" ..
have [simp]: "Field w' = ?A" using well_order_on_Well_order [OF wo'] by simp
have *: "Field w ∩ Field w' = {}" by simp
let ?w = "w ∪o w'"
have "p ⊆ ?w" using ‹p ⊆ w› by (auto simp: Osum_def)
moreover have "Well_order ?w" using Osum_Well_order [OF * wo] and wo' by simp
moreover have "Field ?w = UNIV" by (simp add: Field_Osum)
ultimately show ?thesis by blast
qed

corollary well_order_on_extension:
assumes "wf p" and "Field p ⊆ A"
shows "∃w. p ⊆ w ∧ well_order_on A w"
proof -
from total_well_order_extension [OF ‹wf p›] obtain r
where "p ⊆ r" and wo: "Well_order r" and univ: "Field r = UNIV" by blast
let ?r = "{(x, y). x ∈ A ∧ y ∈ A ∧ (x, y) ∈ r}"
from ‹p ⊆ r› have "p ⊆ ?r" using ‹Field p ⊆ A› by (auto simp: Field_def)
have "Refl r" "trans r" "antisym r" "Total r" "wf (r - Id)"
using ‹Well_order r› by (simp_all add: order_on_defs)
have "refl_on A ?r" using ‹Refl r› by (auto simp: refl_on_def univ)
moreover have "trans ?r" using ‹trans r›
unfolding trans_def by blast
moreover have "antisym ?r" using ‹antisym r›
unfolding antisym_def by blast
moreover have "total_on A ?r" using ‹Total r› by (simp add: total_on_def univ)
moreover have "wf (?r - Id)" by (rule wf_subset [OF ‹wf(r - Id)›]) blast
ultimately have "well_order_on A ?r" by (simp add: order_on_defs)
with ‹p ⊆ ?r› show ?thesis by blast
qed

end