# 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 = (xA. yA. (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
(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:
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  (rR. (r,R)  I)" if "R  Chains I" for R
proof -
from that have Ris:  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 "rR. Refl r" and "rR. trans r" and "rR. antisym r" and
"rR. Total r" and "rR. 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 rR. Refl r  unfolding refl_on_def by fastforce
moreover have "trans (R)"
by (rule chain_subset_trans_Union [OF subch rR. trans r])
moreover have "antisym (R)"
by (rule chain_subset_antisym_Union [OF subch rR. antisym r])
moreover have "Total (R)"
by (rule chain_subset_Total_Union [OF subch rR. Total r])
moreover have "wf ((R) - Id)"
proof -
have "(R) - Id = {r - Id | r. r  R}" by blast
with rR. 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 "rR. r initial_segment_of R" using Ris
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: "uField I. rR. (r, u)  I" if "RChains 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 termx as topmost element to termm.›
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