(* 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')" proof(unfold trans_def, auto) fix x y z assume *: "(x, y) ∈ r ∪o r'" and **: "(y, z) ∈ r ∪o r'" show "(x, z) ∈ r ∪o r'" proof- {assume Case1: "(x,y) ∈ r" hence 1: "x ∈ Field r ∧ y ∈ Field r" unfolding Field_def by auto have ?thesis proof- {assume Case11: "(y,z) ∈ r" hence "(x,z) ∈ r" using Case1 TRANS trans_def[of r] by blast hence ?thesis unfolding Osum_def by auto } moreover {assume Case12: "(y,z) ∈ r'" hence "y ∈ Field r'" unfolding Field_def by auto hence False using FLD 1 by auto } moreover {assume Case13: "z ∈ Field r'" hence ?thesis using 1 unfolding Osum_def by auto } ultimately show ?thesis using ** unfolding Osum_def by blast qed } moreover {assume Case2: "(x,y) ∈ r'" hence 2: "x ∈ Field r' ∧ y ∈ Field r'" unfolding Field_def by auto have ?thesis proof- {assume Case21: "(y,z) ∈ r" hence "y ∈ Field r" unfolding Field_def by auto hence False using FLD 2 by auto } moreover {assume Case22: "(y,z) ∈ r'" hence "(x,z) ∈ r'" using Case2 TRANS' trans_def[of r'] by blast hence ?thesis unfolding Osum_def by auto } moreover {assume Case23: "y ∈ Field r" hence False using FLD 2 by auto } ultimately show ?thesis using ** unfolding Osum_def by blast qed } moreover {assume Case3: "x ∈ Field r ∧ y ∈ Field r'" have ?thesis proof- {assume Case31: "(y,z) ∈ r" hence "y ∈ Field r" unfolding Field_def by auto hence False using FLD Case3 by auto } moreover {assume Case32: "(y,z) ∈ r'" hence "z ∈ Field r'" unfolding Field_def by blast hence ?thesis unfolding Osum_def using Case3 by auto } moreover {assume Case33: "y ∈ Field r" hence False using FLD Case3 by auto } ultimately show ?thesis using ** unfolding Osum_def by blast qed } ultimately show ?thesis using * unfolding Osum_def by blast qed qed 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')" proof(unfold antisym_def, auto) fix x y assume *: "(x, y) ∈ r ∪o r'" and **: "(y, x) ∈ r ∪o r'" show "x = y" proof- {assume Case1: "(x,y) ∈ r" hence 1: "x ∈ Field r ∧ y ∈ Field r" unfolding Field_def by auto have ?thesis proof- have "(y,x) ∈ r ⟹ ?thesis" using Case1 AN antisym_def[of r] by blast moreover {assume "(y,x) ∈ r'" hence "y ∈ Field r'" unfolding Field_def by auto hence False using FLD 1 by auto } moreover have "x ∈ Field r' ⟹ False" using FLD 1 by auto ultimately show ?thesis using ** unfolding Osum_def by blast qed } moreover {assume Case2: "(x,y) ∈ r'" hence 2: "x ∈ Field r' ∧ y ∈ Field r'" unfolding Field_def by auto have ?thesis proof- {assume "(y,x) ∈ r" hence "y ∈ Field r" unfolding Field_def by auto hence False using FLD 2 by auto } moreover have "(y,x) ∈ r' ⟹ ?thesis" using Case2 AN' antisym_def[of r'] by blast moreover {assume "y ∈ Field r" hence False using FLD 2 by auto } ultimately show ?thesis using ** unfolding Osum_def by blast qed } moreover {assume Case3: "x ∈ Field r ∧ y ∈ Field r'" have ?thesis proof- {assume "(y,x) ∈ r" hence "y ∈ Field r" unfolding Field_def by auto hence False using FLD Case3 by auto } moreover {assume Case32: "(y,x) ∈ r'" hence "x ∈ Field r'" unfolding Field_def by blast hence False using FLD Case3 by auto } moreover have "¬ y ∈ Field r" using FLD Case3 by auto ultimately show ?thesis using ** unfolding Osum_def by blast qed } ultimately show ?thesis using * unfolding Osum_def by blast qed qed 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')" unfolding linear_order_on_def using Osum_Partial_order Osum_Total by blast lemma Osum_minus_Id1: assumes "r ≤ Id" shows "(r Osum r') - Id ≤ (r' - Id) ∪ (Field r × Field r')" proof- let ?Left = "(r Osum r') - Id" let ?Right = "(r' - Id) ∪ (Field r × Field r')" {fix a::'a and b assume *: "(a,b) ∉ Id" {assume "(a,b) ∈ r" with * have False using assms by auto } moreover {assume "(a,b) ∈ r'" with * have "(a,b) ∈ r' - Id" by auto } ultimately have "(a,b) ∈ ?Left ⟹ (a,b) ∈ ?Right" unfolding Osum_def by auto } thus ?thesis by auto qed lemma Osum_minus_Id2: assumes "r' ≤ Id" shows "(r Osum r') - Id ≤ (r - Id) ∪ (Field r × Field r')" proof- let ?Left = "(r Osum r') - Id" let ?Right = "(r - Id) ∪ (Field r × Field r')" {fix a::'a and b assume *: "(a,b) ∉ Id" {assume "(a,b) ∈ r'" with * have False using assms by auto } moreover {assume "(a,b) ∈ r" with * have "(a,b) ∈ r - Id" by auto } ultimately have "(a,b) ∈ ?Left ⟹ (a,b) ∈ ?Right" unfolding Osum_def by auto } thus ?thesis by auto qed 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)" proof- {fix a a' assume *: "(a,a') ∈ (r Osum r')" and **: "a ≠ a'" have "(a,a') ∈ (r - Id) Osum (r' - Id)" proof- {assume "(a,a') ∈ r ∨ (a,a') ∈ r'" with ** have ?thesis unfolding Osum_def by auto } moreover {assume "a ∈ Field r ∧ a' ∈ Field r'" hence "a ∈ Field(r - Id) ∧ a' ∈ Field (r' - Id)" using assms Total_Id_Field by blast hence ?thesis unfolding Osum_def by auto } ultimately show ?thesis using * unfolding Osum_def by fast qed } thus ?thesis by(auto simp add: Osum_def) qed 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 FLD mono_Field[of "r - Id" r] mono_Field[of "r' - Id" r'] Diff_subset[of r Id] Diff_subset[of r' Id] by blast thus ?thesis using Case1 Osum_minus_Id[of r r'] assms Osum_wf[of "r - Id" "r' - Id"] wf_subset[of "(r - Id) ∪o (r' - Id)" "(r Osum r') - Id"] by auto 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