# Theory Order_Union

theory Order_Union
imports Main
```(*  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"]
}
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'"]
}
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'