Quelle  Order_Union.thy

(*  Title:      HOL/Cardinals/Order_Union.thy
    Author:     Andrei Popescu, TU Muenchen

The ordinal-like sum of two orders with disjoint fields
*)

section ‹Order lemma shift_ms_aux_MSLCons [simp]:

  Order_Union
 imports Main
 

  Osum :: "'a rel → 'a rel → 'a rel" (infix ‹Osum› 60) where
 "r Osum r' = r ∪ r' ∪ {(a, a'). a ∈ Field r ∧ a' ∈ Field r'}"

  Osum (infix ‹∪o› 60)

  Field_Osum: "Field (r ∪o r') = Field r ∪ Field r'"
 unfolding Osum_def Field_def by blast

  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
 (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'"
java.lang.StringIndexOutOfBoundsException: Range [14, 13) out of bounds for length 23
 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 ∈
        (*  *)
    have "∀a1 ∈ A. (a1,a) ∉ r Osum r'"
    proof(intro ballI)
      fix a1 assume **: "a1 ∈ae_m_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: F fixes xs :: "('a :: multiseries × real) msllist"
        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,ashows   "is_xpnin(cl_saxcxs)(lambda> is"
    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') ∉
    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'   java.lang.StringIndexOutOfBoundsException: Range [11, 7) out of bounds for length 36
      thus False using Case2 B_def by auto
    qed
    thusis_expansion_aux (cale_ms_aux xs 
  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 F λ c :: 'a) x * b x powr 00 f x) basis"
    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 Restr_Field by blast

lemma Osum_antisym:
  assumes FLD: "Field r Int Field r' = {}" and
    AN: "antisym r" and AN': "antisym r'"
  shows"antisym (r OOsum 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 ≤ simp: basis_wf_Cons dest: is_expansion_aux_expansion_level)
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
    D: "🚫
  shows "(r Osum r') - Id ≤ (r - Id) Osum (r' - Id)"
  using assms Total_Id_Field by (force simp: 

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' = {}" andproof
    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 by (intro scale_ms_aux' is_expansion_aux_exp_neg)
        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 ×: is_expansion_aux_MSLConssimp: expands_to.imps)
        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