(* Title: HOL/Cardinals/Order_Union.thy Author: Andrei Popescu, TU Muenchen The ordinal-like sum of two orders with disjoint fields *) section ‹Order Union› theory Order_Union imports Main begin definition Osum :: "'a rel ⇒ 'a rel ⇒ 'a rel" (infix "Osum" 60) where "r Osum r' = r ∪ r' ∪ {(a, a'). a ∈ Field r ∧ a' ∈ Field r'}" notation Osum (infix "∪o" 60) lemma Field_Osum: "Field (r ∪o r') = Field r ∪ Field r'" unfolding Osum_def Field_def by blast lemma Osum_wf: assumes FLD: "Field r Int Field r' = {}" and WF: "wf r" and WF': "wf r'" shows "wf (r Osum r')" unfolding wf_eq_minimal2 unfolding Field_Osum proof(intro allI impI, elim conjE) fix A assume *: "A ⊆ Field r ∪ Field r'" and **: "A ≠ {}" obtain B where B_def: "B = A Int Field r" by blast show "∃a∈A. ∀a'∈A. (a', a) ∉ r ∪o r'" proof(cases "B = {}") assume Case1: "B ≠ {}" hence "B ≠ {} ∧ B ≤ Field r" using B_def by auto then obtain a where 1: "a ∈ B" and 2: "∀a1 ∈ B. (a1,a) ∉ r" using WF unfolding wf_eq_minimal2 by blast hence 3: "a ∈ Field r ∧ a ∉ Field r'" using B_def FLD by auto (* *) have "∀a1 ∈ A. (a1,a) ∉ r Osum r'" proof(intro ballI) fix a1 assume **: "a1 ∈ A" {assume Case11: "a1 ∈ Field r" hence "(a1,a) ∉ r" using B_def ** 2 by auto moreover have "(a1,a) ∉ r'" using 3 by (auto simp add: Field_def) ultimately have "(a1,a) ∉ r Osum r'" using 3 unfolding Osum_def by auto } moreover {assume Case12: "a1 ∉ Field r" hence "(a1,a) ∉ r" unfolding Field_def by auto moreover have "(a1,a) ∉ r'" using 3 unfolding Field_def by auto ultimately have "(a1,a) ∉ r Osum r'" using 3 unfolding Osum_def by auto } ultimately show "(a1,a) ∉ r Osum r'" by blast qed thus ?thesis using 1 B_def by auto next assume Case2: "B = {}" hence 1: "A ≠ {} ∧ A ≤ Field r'" using * ** B_def by auto then obtain a' where 2: "a' ∈ A" and 3: "∀a1' ∈ A. (a1',a') ∉ r'" using WF' unfolding wf_eq_minimal2 by blast hence 4: "a' ∈ Field r' ∧ a' ∉ Field r" using 1 FLD by blast (* *) have "∀a1' ∈ A. (a1',a') ∉ r Osum r'" proof(unfold Osum_def, auto simp add: 3) fix a1' assume "(a1', a') ∈ r" thus False using 4 unfolding Field_def by blast next fix a1' assume "a1' ∈ A" and "a1' ∈ Field r" thus False using Case2 B_def by auto qed thus ?thesis using 2 by blast qed qed lemma Osum_Refl: assumes FLD: "Field r Int Field r' = {}" and REFL: "Refl r" and REFL': "Refl r'" shows "Refl (r Osum r')" using assms unfolding refl_on_def Field_Osum unfolding Osum_def by blast lemma Osum_trans: assumes FLD: "Field r Int Field r' = {}" and TRANS: "trans r" and TRANS': "trans r'" shows "trans (r Osum r')" using assms unfolding Osum_def trans_def disjoint_iff Field_iff by blast lemma Osum_Preorder: "⟦Field r Int Field r' = {}; Preorder r; Preorder r'⟧ ⟹ Preorder (r Osum r')" unfolding preorder_on_def using Osum_Refl Osum_trans by blast lemma Osum_antisym: assumes FLD: "Field r Int Field r' = {}" and AN: "antisym r" and AN': "antisym r'" shows "antisym (r Osum r')" using assms by (auto simp: disjoint_iff antisym_def Osum_def Field_def) lemma Osum_Partial_order: "⟦Field r Int Field r' = {}; Partial_order r; Partial_order r'⟧ ⟹ Partial_order (r Osum r')" unfolding partial_order_on_def using Osum_Preorder Osum_antisym by blast lemma Osum_Total: assumes FLD: "Field r Int Field r' = {}" and TOT: "Total r" and TOT': "Total r'" shows "Total (r Osum r')" using assms unfolding total_on_def Field_Osum unfolding Osum_def by blast lemma Osum_Linear_order: "⟦Field r Int Field r' = {}; Linear_order r; Linear_order r'⟧ ⟹ Linear_order (r Osum r')" by (simp add: Osum_Partial_order Osum_Total linear_order_on_def) lemma Osum_minus_Id1: assumes "r ≤ Id" shows "(r Osum r') - Id ≤ (r' - Id) ∪ (Field r × Field r')" using assms by (force simp: Osum_def) lemma Osum_minus_Id2: assumes "r' ≤ Id" shows "(r Osum r') - Id ≤ (r - Id) ∪ (Field r × Field r')" using assms by (force simp: Osum_def) lemma Osum_minus_Id: assumes TOT: "Total r" and TOT': "Total r'" and NID: "¬ (r ≤ Id)" and NID': "¬ (r' ≤ Id)" shows "(r Osum r') - Id ≤ (r - Id) Osum (r' - Id)" using assms Total_Id_Field by (force simp: Osum_def) lemma wf_Int_Times: assumes "A Int B = {}" shows "wf(A × B)" unfolding wf_def using assms by blast lemma Osum_wf_Id: assumes TOT: "Total r" and TOT': "Total r'" and FLD: "Field r Int Field r' = {}" and WF: "wf(r - Id)" and WF': "wf(r' - Id)" shows "wf ((r Osum r') - Id)" proof(cases "r ≤ Id ∨ r' ≤ Id") assume Case1: "¬(r ≤ Id ∨ r' ≤ Id)" have "Field(r - Id) Int Field(r' - Id) = {}" using Case1 FLD TOT TOT' Total_Id_Field by blast thus ?thesis by (meson Case1 Osum_minus_Id Osum_wf TOT TOT' WF WF' wf_subset) next have 1: "wf(Field r × Field r')" using FLD by (auto simp add: wf_Int_Times) assume Case2: "r ≤ Id ∨ r' ≤ Id" moreover {assume Case21: "r ≤ Id" hence "(r Osum r') - Id ≤ (r' - Id) ∪ (Field r × Field r')" using Osum_minus_Id1[of r r'] by simp moreover {have "Domain(Field r × Field r') Int Range(r' - Id) = {}" using FLD unfolding Field_def by blast hence "wf((r' - Id) ∪ (Field r × Field r'))" using 1 WF' wf_Un[of "Field r × Field r'" "r' - Id"] by (auto simp add: Un_commute) } ultimately have ?thesis using wf_subset by blast } moreover {assume Case22: "r' ≤ Id" hence "(r Osum r') - Id ≤ (r - Id) ∪ (Field r × Field r')" using Osum_minus_Id2[of r' r] by simp moreover {have "Range(Field r × Field r') Int Domain(r - Id) = {}" using FLD unfolding Field_def by blast hence "wf((r - Id) ∪ (Field r × Field r'))" using 1 WF wf_Un[of "r - Id" "Field r × Field r'"] by (auto simp add: Un_commute) } ultimately have ?thesis using wf_subset by blast } ultimately show ?thesis by blast qed lemma Osum_Well_order: assumes FLD: "Field r Int Field r' = {}" and WELL: "Well_order r" and WELL': "Well_order r'" shows "Well_order (r Osum r')" proof- have "Total r ∧ Total r'" using WELL WELL' by (auto simp add: order_on_defs) thus ?thesis using assms unfolding well_order_on_def using Osum_Linear_order Osum_wf_Id by blast qed end