(* 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 set_mp) 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) { fix R assume "R ∈ Chains I" then 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 have "⋃R ∈ ?K ∧ (∀r∈R. (r,⋃R) ∈ I)" using mono_Chains [OF I_init] and ‹R ∈ Chains I› by (simp (no_asm) add: I_def del: Field_Union) (metis Chains_wo) } then have 1: "∀R∈Chains I. ∃u∈Field I. ∀r∈R. (r, u) ∈ I" 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 1: "Field ?r = A" using wo univ by (fastforce simp: Field_def order_on_defs refl_on_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