src/HOL/Ordinals_and_Cardinals/Constructions_on_Wellorders_Base.thy
author blanchet
Tue Aug 28 17:16:00 2012 +0200 (2012-08-28)
changeset 48975 7f79f94a432c
permissions -rw-r--r--
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
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(*  Title:      HOL/Ordinals_and_Cardinals/Constructions_on_Wellorders_Base.thy
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    Author:     Andrei Popescu, TU Muenchen
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    Copyright   2012
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Constructions on wellorders (base).
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*)
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header {* Constructions on Wellorders (Base) *}
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theory Constructions_on_Wellorders_Base
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imports Wellorder_Embedding_Base
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begin
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text {* In this section, we study basic constructions on well-orders, such as restriction to
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a set/order filter, copy via direct images, ordinal-like sum of disjoint well-orders,
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and bounded square.  We also define between well-orders
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the relations @{text "ordLeq"}, of being embedded (abbreviated @{text "\<le>o"}),
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@{text "ordLess"}, of being strictly embedded (abbreviated @{text "<o"}), and
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@{text "ordIso"}, of being isomorphic (abbreviated @{text "=o"}).  We study the
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connections between these relations, order filters, and the aforementioned constructions.
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A main result of this section is that @{text "<o"} is well-founded.  *}
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subsection {* Restriction to a set  *}
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abbreviation Restr :: "'a rel \<Rightarrow> 'a set \<Rightarrow> 'a rel"
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where "Restr r A \<equiv> r Int (A \<times> A)"
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lemma Restr_subset:
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"A \<le> B \<Longrightarrow> Restr (Restr r B) A = Restr r A"
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by blast
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lemma Restr_Field: "Restr r (Field r) = r"
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unfolding Field_def by auto
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lemma Refl_Restr: "Refl r \<Longrightarrow> Refl(Restr r A)"
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unfolding refl_on_def Field_def by auto
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lemma antisym_Restr:
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"antisym r \<Longrightarrow> antisym(Restr r A)"
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unfolding antisym_def Field_def by auto
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lemma Total_Restr:
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"Total r \<Longrightarrow> Total(Restr r A)"
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unfolding total_on_def Field_def by auto
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lemma trans_Restr:
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"trans r \<Longrightarrow> trans(Restr r A)"
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unfolding trans_def Field_def by blast
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lemma Preorder_Restr:
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"Preorder r \<Longrightarrow> Preorder(Restr r A)"
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unfolding preorder_on_def by (simp add: Refl_Restr trans_Restr)
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lemma Partial_order_Restr:
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"Partial_order r \<Longrightarrow> Partial_order(Restr r A)"
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unfolding partial_order_on_def by (simp add: Preorder_Restr antisym_Restr)
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lemma Linear_order_Restr:
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"Linear_order r \<Longrightarrow> Linear_order(Restr r A)"
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unfolding linear_order_on_def by (simp add: Partial_order_Restr Total_Restr)
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lemma Well_order_Restr:
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assumes "Well_order r"
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shows "Well_order(Restr r A)"
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proof-
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  have "Restr r A - Id \<le> r - Id" using Restr_subset by blast
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  hence "wf(Restr r A - Id)" using assms
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  using well_order_on_def wf_subset by blast
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  thus ?thesis using assms unfolding well_order_on_def
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  by (simp add: Linear_order_Restr)
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qed
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lemma Field_Restr_subset: "Field(Restr r A) \<le> A"
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by (auto simp add: Field_def)
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lemma Refl_Field_Restr:
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"Refl r \<Longrightarrow> Field(Restr r A) = (Field r) Int A"
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by (auto simp add: refl_on_def Field_def)
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lemma Refl_Field_Restr2:
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"\<lbrakk>Refl r; A \<le> Field r\<rbrakk> \<Longrightarrow> Field(Restr r A) = A"
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by (auto simp add: Refl_Field_Restr)
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lemma well_order_on_Restr:
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assumes WELL: "Well_order r" and SUB: "A \<le> Field r"
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shows "well_order_on A (Restr r A)"
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using assms
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using Well_order_Restr[of r A] Refl_Field_Restr2[of r A]
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     order_on_defs[of "Field r" r] by auto
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subsection {* Order filters versus restrictions and embeddings  *}
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lemma Field_Restr_ofilter:
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"\<lbrakk>Well_order r; wo_rel.ofilter r A\<rbrakk> \<Longrightarrow> Field(Restr r A) = A"
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by (auto simp add: wo_rel_def wo_rel.ofilter_def wo_rel.REFL Refl_Field_Restr2)
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lemma ofilter_Restr_under:
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assumes WELL: "Well_order r" and OF: "wo_rel.ofilter r A" and IN: "a \<in> A"
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shows "rel.under (Restr r A) a = rel.under r a"
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using assms wo_rel_def
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proof(auto simp add: wo_rel.ofilter_def rel.under_def)
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  fix b assume *: "a \<in> A" and "(b,a) \<in> r"
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  hence "b \<in> rel.under r a \<and> a \<in> Field r"
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  unfolding rel.under_def using Field_def by fastforce
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  thus "b \<in> A" using * assms by (auto simp add: wo_rel_def wo_rel.ofilter_def)
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qed
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lemma ofilter_embed:
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assumes "Well_order r"
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shows "wo_rel.ofilter r A = (A \<le> Field r \<and> embed (Restr r A) r id)"
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proof
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  assume *: "wo_rel.ofilter r A"
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  show "A \<le> Field r \<and> embed (Restr r A) r id"
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  proof(unfold embed_def, auto)
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    fix a assume "a \<in> A" thus "a \<in> Field r" using assms *
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    by (auto simp add: wo_rel_def wo_rel.ofilter_def)
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  next
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    fix a assume "a \<in> Field (Restr r A)"
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    thus "bij_betw id (rel.under (Restr r A) a) (rel.under r a)" using assms *
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    by (simp add: ofilter_Restr_under Field_Restr_ofilter)
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  qed
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next
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  assume *: "A \<le> Field r \<and> embed (Restr r A) r id"
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  hence "Field(Restr r A) \<le> Field r"
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  using assms  embed_Field[of "Restr r A" r id] id_def
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        Well_order_Restr[of r] by auto
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  {fix a assume "a \<in> A"
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   hence "a \<in> Field(Restr r A)" using * assms
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   by (simp add: order_on_defs Refl_Field_Restr2)
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   hence "bij_betw id (rel.under (Restr r A) a) (rel.under r a)"
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   using * unfolding embed_def by auto
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   hence "rel.under r a \<le> rel.under (Restr r A) a"
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   unfolding bij_betw_def by auto
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   also have "\<dots> \<le> Field(Restr r A)" by (simp add: rel.under_Field)
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   also have "\<dots> \<le> A" by (simp add: Field_Restr_subset)
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   finally have "rel.under r a \<le> A" .
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  }
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  thus "wo_rel.ofilter r A" using assms * by (simp add: wo_rel_def wo_rel.ofilter_def)
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qed
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lemma ofilter_Restr_Int:
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assumes WELL: "Well_order r" and OFA: "wo_rel.ofilter r A"
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shows "wo_rel.ofilter (Restr r B) (A Int B)"
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proof-
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  let ?rB = "Restr r B"
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  have Well: "wo_rel r" unfolding wo_rel_def using WELL .
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  hence Refl: "Refl r" by (simp add: wo_rel.REFL)
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  hence Field: "Field ?rB = Field r Int B"
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  using Refl_Field_Restr by blast
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  have WellB: "wo_rel ?rB \<and> Well_order ?rB" using WELL
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  by (simp add: Well_order_Restr wo_rel_def)
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  (* Main proof *)
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  show ?thesis using WellB assms
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  proof(auto simp add: wo_rel.ofilter_def rel.under_def)
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    fix a assume "a \<in> A" and *: "a \<in> B"
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    hence "a \<in> Field r" using OFA Well by (auto simp add: wo_rel.ofilter_def)
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    with * show "a \<in> Field ?rB" using Field by auto
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  next
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    fix a b assume "a \<in> A" and "(b,a) \<in> r"
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    thus "b \<in> A" using Well OFA by (auto simp add: wo_rel.ofilter_def rel.under_def)
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  qed
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qed
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lemma ofilter_Restr_subset:
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assumes WELL: "Well_order r" and OFA: "wo_rel.ofilter r A" and SUB: "A \<le> B"
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shows "wo_rel.ofilter (Restr r B) A"
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proof-
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  have "A Int B = A" using SUB by blast
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  thus ?thesis using assms ofilter_Restr_Int[of r A B] by auto
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qed
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lemma ofilter_subset_embed:
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assumes WELL: "Well_order r" and
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        OFA: "wo_rel.ofilter r A" and OFB: "wo_rel.ofilter r B"
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shows "(A \<le> B) = (embed (Restr r A) (Restr r B) id)"
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proof-
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  let ?rA = "Restr r A"  let ?rB = "Restr r B"
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  have Well: "wo_rel r" unfolding wo_rel_def using WELL .
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  hence Refl: "Refl r" by (simp add: wo_rel.REFL)
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  hence FieldA: "Field ?rA = Field r Int A"
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  using Refl_Field_Restr by blast
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  have FieldB: "Field ?rB = Field r Int B"
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  using Refl Refl_Field_Restr by blast
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  have WellA: "wo_rel ?rA \<and> Well_order ?rA" using WELL
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  by (simp add: Well_order_Restr wo_rel_def)
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  have WellB: "wo_rel ?rB \<and> Well_order ?rB" using WELL
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  by (simp add: Well_order_Restr wo_rel_def)
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  (* Main proof *)
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  show ?thesis
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  proof
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    assume *: "A \<le> B"
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    hence "wo_rel.ofilter (Restr r B) A" using assms
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    by (simp add: ofilter_Restr_subset)
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    hence "embed (Restr ?rB A) (Restr r B) id"
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    using WellB ofilter_embed[of "?rB" A] by auto
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    thus "embed (Restr r A) (Restr r B) id"
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    using * by (simp add: Restr_subset)
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  next
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    assume *: "embed (Restr r A) (Restr r B) id"
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    {fix a assume **: "a \<in> A"
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     hence "a \<in> Field r" using Well OFA by (auto simp add: wo_rel.ofilter_def)
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     with ** FieldA have "a \<in> Field ?rA" by auto
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     hence "a \<in> Field ?rB" using * WellA embed_Field[of ?rA ?rB id] by auto
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     hence "a \<in> B" using FieldB by auto
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    }
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    thus "A \<le> B" by blast
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  qed
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qed
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lemma ofilter_subset_embedS_iso:
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assumes WELL: "Well_order r" and
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        OFA: "wo_rel.ofilter r A" and OFB: "wo_rel.ofilter r B"
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shows "((A < B) = (embedS (Restr r A) (Restr r B) id)) \<and>
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       ((A = B) = (iso (Restr r A) (Restr r B) id))"
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proof-
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  let ?rA = "Restr r A"  let ?rB = "Restr r B"
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  have Well: "wo_rel r" unfolding wo_rel_def using WELL .
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  hence Refl: "Refl r" by (simp add: wo_rel.REFL)
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  hence "Field ?rA = Field r Int A"
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  using Refl_Field_Restr by blast
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  hence FieldA: "Field ?rA = A" using OFA Well
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  by (auto simp add: wo_rel.ofilter_def)
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  have "Field ?rB = Field r Int B"
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  using Refl Refl_Field_Restr by blast
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  hence FieldB: "Field ?rB = B" using OFB Well
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  by (auto simp add: wo_rel.ofilter_def)
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  (* Main proof *)
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  show ?thesis unfolding embedS_def iso_def
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  using assms ofilter_subset_embed[of r A B]
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        FieldA FieldB bij_betw_id_iff[of A B] by auto
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qed
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lemma ofilter_subset_embedS:
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assumes WELL: "Well_order r" and
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        OFA: "wo_rel.ofilter r A" and OFB: "wo_rel.ofilter r B"
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shows "(A < B) = embedS (Restr r A) (Restr r B) id"
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using assms by (simp add: ofilter_subset_embedS_iso)
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lemma embed_implies_iso_Restr:
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assumes WELL: "Well_order r" and WELL': "Well_order r'" and
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        EMB: "embed r' r f"
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shows "iso r' (Restr r (f ` (Field r'))) f"
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proof-
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  let ?A' = "Field r'"
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  let ?r'' = "Restr r (f ` ?A')"
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  have 0: "Well_order ?r''" using WELL Well_order_Restr by blast
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  have 1: "wo_rel.ofilter r (f ` ?A')" using assms embed_Field_ofilter  by blast
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  hence "Field ?r'' = f ` (Field r')" using WELL Field_Restr_ofilter by blast
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  hence "bij_betw f ?A' (Field ?r'')"
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  using EMB embed_inj_on WELL' unfolding bij_betw_def by blast
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  moreover
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  {have "\<forall>a b. (a,b) \<in> r' \<longrightarrow> a \<in> Field r' \<and> b \<in> Field r'"
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   unfolding Field_def by auto
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   hence "compat r' ?r'' f"
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   using assms embed_iff_compat_inj_on_ofilter
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   unfolding compat_def by blast
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  }
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  ultimately show ?thesis using WELL' 0 iso_iff3 by blast
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qed
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subsection {* The strict inclusion on proper ofilters is well-founded *}
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definition ofilterIncl :: "'a rel \<Rightarrow> 'a set rel"
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where
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"ofilterIncl r \<equiv> {(A,B). wo_rel.ofilter r A \<and> A \<noteq> Field r \<and>
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                         wo_rel.ofilter r B \<and> B \<noteq> Field r \<and> A < B}"
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lemma wf_ofilterIncl:
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assumes WELL: "Well_order r"
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shows "wf(ofilterIncl r)"
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proof-
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  have Well: "wo_rel r" using WELL by (simp add: wo_rel_def)
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  hence Lo: "Linear_order r" by (simp add: wo_rel.LIN)
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  let ?h = "(\<lambda> A. wo_rel.suc r A)"
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  let ?rS = "r - Id"
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  have "wf ?rS" using WELL by (simp add: order_on_defs)
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  moreover
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  have "compat (ofilterIncl r) ?rS ?h"
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   309
  proof(unfold compat_def ofilterIncl_def,
blanchet@48975
   310
        intro allI impI, simp, elim conjE)
blanchet@48975
   311
    fix A B
blanchet@48975
   312
    assume *: "wo_rel.ofilter r A" "A \<noteq> Field r" and
blanchet@48975
   313
           **: "wo_rel.ofilter r B" "B \<noteq> Field r" and ***: "A < B"
blanchet@48975
   314
    then obtain a and b where 0: "a \<in> Field r \<and> b \<in> Field r" and
blanchet@48975
   315
                         1: "A = rel.underS r a \<and> B = rel.underS r b"
blanchet@48975
   316
    using Well by (auto simp add: wo_rel.ofilter_underS_Field)
blanchet@48975
   317
    hence "a \<noteq> b" using *** by auto
blanchet@48975
   318
    moreover
blanchet@48975
   319
    have "(a,b) \<in> r" using 0 1 Lo ***
blanchet@48975
   320
    by (auto simp add: rel.underS_incl_iff)
blanchet@48975
   321
    moreover
blanchet@48975
   322
    have "a = wo_rel.suc r A \<and> b = wo_rel.suc r B"
blanchet@48975
   323
    using Well 0 1 by (simp add: wo_rel.suc_underS)
blanchet@48975
   324
    ultimately
blanchet@48975
   325
    show "(wo_rel.suc r A, wo_rel.suc r B) \<in> r \<and> wo_rel.suc r A \<noteq> wo_rel.suc r B"
blanchet@48975
   326
    by simp
blanchet@48975
   327
  qed
blanchet@48975
   328
  ultimately show "wf (ofilterIncl r)" by (simp add: compat_wf)
blanchet@48975
   329
qed
blanchet@48975
   330
blanchet@48975
   331
blanchet@48975
   332
blanchet@48975
   333
subsection {* Ordering the  well-orders by existence of embeddings *}
blanchet@48975
   334
blanchet@48975
   335
blanchet@48975
   336
text {* We define three relations between well-orders:
blanchet@48975
   337
\begin{itemize}
blanchet@48975
   338
\item @{text "ordLeq"}, of being embedded (abbreviated @{text "\<le>o"});
blanchet@48975
   339
\item @{text "ordLess"}, of being strictly embedded (abbreviated @{text "<o"});
blanchet@48975
   340
\item @{text "ordIso"}, of being isomorphic (abbreviated @{text "=o"}).
blanchet@48975
   341
\end{itemize}
blanchet@48975
   342
%
blanchet@48975
   343
The prefix "ord" and the index "o" in these names stand for "ordinal-like".
blanchet@48975
   344
These relations shall be proved to be inter-connected in a similar fashion as the trio
blanchet@48975
   345
@{text "\<le>"}, @{text "<"}, @{text "="} associated to a total order on a set.
blanchet@48975
   346
*}
blanchet@48975
   347
blanchet@48975
   348
blanchet@48975
   349
definition ordLeq :: "('a rel * 'a' rel) set"
blanchet@48975
   350
where
blanchet@48975
   351
"ordLeq = {(r,r'). Well_order r \<and> Well_order r' \<and> (\<exists>f. embed r r' f)}"
blanchet@48975
   352
blanchet@48975
   353
blanchet@48975
   354
abbreviation ordLeq2 :: "'a rel \<Rightarrow> 'a' rel \<Rightarrow> bool" (infix "<=o" 50)
blanchet@48975
   355
where "r <=o r' \<equiv> (r,r') \<in> ordLeq"
blanchet@48975
   356
blanchet@48975
   357
blanchet@48975
   358
abbreviation ordLeq3 :: "'a rel \<Rightarrow> 'a' rel \<Rightarrow> bool" (infix "\<le>o" 50)
blanchet@48975
   359
where "r \<le>o r' \<equiv> r <=o r'"
blanchet@48975
   360
blanchet@48975
   361
blanchet@48975
   362
definition ordLess :: "('a rel * 'a' rel) set"
blanchet@48975
   363
where
blanchet@48975
   364
"ordLess = {(r,r'). Well_order r \<and> Well_order r' \<and> (\<exists>f. embedS r r' f)}"
blanchet@48975
   365
blanchet@48975
   366
abbreviation ordLess2 :: "'a rel \<Rightarrow> 'a' rel \<Rightarrow> bool" (infix "<o" 50)
blanchet@48975
   367
where "r <o r' \<equiv> (r,r') \<in> ordLess"
blanchet@48975
   368
blanchet@48975
   369
blanchet@48975
   370
definition ordIso :: "('a rel * 'a' rel) set"
blanchet@48975
   371
where
blanchet@48975
   372
"ordIso = {(r,r'). Well_order r \<and> Well_order r' \<and> (\<exists>f. iso r r' f)}"
blanchet@48975
   373
blanchet@48975
   374
abbreviation ordIso2 :: "'a rel \<Rightarrow> 'a' rel \<Rightarrow> bool" (infix "=o" 50)
blanchet@48975
   375
where "r =o r' \<equiv> (r,r') \<in> ordIso"
blanchet@48975
   376
blanchet@48975
   377
blanchet@48975
   378
lemmas ordRels_def = ordLeq_def ordLess_def ordIso_def
blanchet@48975
   379
blanchet@48975
   380
lemma ordLeq_Well_order_simp:
blanchet@48975
   381
assumes "r \<le>o r'"
blanchet@48975
   382
shows "Well_order r \<and> Well_order r'"
blanchet@48975
   383
using assms unfolding ordLeq_def by simp
blanchet@48975
   384
blanchet@48975
   385
blanchet@48975
   386
lemma ordLess_Well_order_simp:
blanchet@48975
   387
assumes "r <o r'"
blanchet@48975
   388
shows "Well_order r \<and> Well_order r'"
blanchet@48975
   389
using assms unfolding ordLess_def by simp
blanchet@48975
   390
blanchet@48975
   391
blanchet@48975
   392
lemma ordIso_Well_order_simp:
blanchet@48975
   393
assumes "r =o r'"
blanchet@48975
   394
shows "Well_order r \<and> Well_order r'"
blanchet@48975
   395
using assms unfolding ordIso_def by simp
blanchet@48975
   396
blanchet@48975
   397
blanchet@48975
   398
text{* Notice that the relations @{text "\<le>o"}, @{text "<o"}, @{text "=o"} connect well-orders
blanchet@48975
   399
on potentially {\em distinct} types. However, some of the lemmas below, including the next one,
blanchet@48975
   400
restrict implicitly the type of these relations to @{text "(('a rel) * ('a rel)) set"} , i.e.,
blanchet@48975
   401
to @{text "'a rel rel"}.  *}
blanchet@48975
   402
blanchet@48975
   403
blanchet@48975
   404
lemma ordLeq_reflexive:
blanchet@48975
   405
"Well_order r \<Longrightarrow> r \<le>o r"
blanchet@48975
   406
unfolding ordLeq_def using id_embed[of r] by blast
blanchet@48975
   407
blanchet@48975
   408
blanchet@48975
   409
lemma ordLeq_transitive[trans]:
blanchet@48975
   410
assumes *: "r \<le>o r'" and **: "r' \<le>o r''"
blanchet@48975
   411
shows "r \<le>o r''"
blanchet@48975
   412
proof-
blanchet@48975
   413
  obtain f and f'
blanchet@48975
   414
  where 1: "Well_order r \<and> Well_order r' \<and> Well_order r''" and
blanchet@48975
   415
        "embed r r' f" and "embed r' r'' f'"
blanchet@48975
   416
  using * ** unfolding ordLeq_def by blast
blanchet@48975
   417
  hence "embed r r'' (f' o f)"
blanchet@48975
   418
  using comp_embed[of r r' f r'' f'] by auto
blanchet@48975
   419
  thus "r \<le>o r''" unfolding ordLeq_def using 1 by auto
blanchet@48975
   420
qed
blanchet@48975
   421
blanchet@48975
   422
blanchet@48975
   423
lemma ordLeq_total:
blanchet@48975
   424
"\<lbrakk>Well_order r; Well_order r'\<rbrakk> \<Longrightarrow> r \<le>o r' \<or> r' \<le>o r"
blanchet@48975
   425
unfolding ordLeq_def using wellorders_totally_ordered by blast
blanchet@48975
   426
blanchet@48975
   427
blanchet@48975
   428
lemma ordIso_reflexive:
blanchet@48975
   429
"Well_order r \<Longrightarrow> r =o r"
blanchet@48975
   430
unfolding ordIso_def using id_iso[of r] by blast
blanchet@48975
   431
blanchet@48975
   432
blanchet@48975
   433
lemma ordIso_transitive[trans]:
blanchet@48975
   434
assumes *: "r =o r'" and **: "r' =o r''"
blanchet@48975
   435
shows "r =o r''"
blanchet@48975
   436
proof-
blanchet@48975
   437
  obtain f and f'
blanchet@48975
   438
  where 1: "Well_order r \<and> Well_order r' \<and> Well_order r''" and
blanchet@48975
   439
        "iso r r' f" and 3: "iso r' r'' f'"
blanchet@48975
   440
  using * ** unfolding ordIso_def by auto
blanchet@48975
   441
  hence "iso r r'' (f' o f)"
blanchet@48975
   442
  using comp_iso[of r r' f r'' f'] by auto
blanchet@48975
   443
  thus "r =o r''" unfolding ordIso_def using 1 by auto
blanchet@48975
   444
qed
blanchet@48975
   445
blanchet@48975
   446
blanchet@48975
   447
lemma ordIso_symmetric:
blanchet@48975
   448
assumes *: "r =o r'"
blanchet@48975
   449
shows "r' =o r"
blanchet@48975
   450
proof-
blanchet@48975
   451
  obtain f where 1: "Well_order r \<and> Well_order r'" and
blanchet@48975
   452
                 2: "embed r r' f \<and> bij_betw f (Field r) (Field r')"
blanchet@48975
   453
  using * by (auto simp add: ordIso_def iso_def)
blanchet@48975
   454
  let ?f' = "inv_into (Field r) f"
blanchet@48975
   455
  have "embed r' r ?f' \<and> bij_betw ?f' (Field r') (Field r)"
blanchet@48975
   456
  using 1 2 by (simp add: bij_betw_inv_into inv_into_Field_embed_bij_betw)
blanchet@48975
   457
  thus "r' =o r" unfolding ordIso_def using 1 by (auto simp add: iso_def)
blanchet@48975
   458
qed
blanchet@48975
   459
blanchet@48975
   460
blanchet@48975
   461
lemma ordLeq_ordLess_trans[trans]:
blanchet@48975
   462
assumes "r \<le>o r'" and " r' <o r''"
blanchet@48975
   463
shows "r <o r''"
blanchet@48975
   464
proof-
blanchet@48975
   465
  have "Well_order r \<and> Well_order r''"
blanchet@48975
   466
  using assms unfolding ordLeq_def ordLess_def by auto
blanchet@48975
   467
  thus ?thesis using assms unfolding ordLeq_def ordLess_def
blanchet@48975
   468
  using embed_comp_embedS by blast
blanchet@48975
   469
qed
blanchet@48975
   470
blanchet@48975
   471
blanchet@48975
   472
lemma ordLess_ordLeq_trans[trans]:
blanchet@48975
   473
assumes "r <o r'" and " r' \<le>o r''"
blanchet@48975
   474
shows "r <o r''"
blanchet@48975
   475
proof-
blanchet@48975
   476
  have "Well_order r \<and> Well_order r''"
blanchet@48975
   477
  using assms unfolding ordLeq_def ordLess_def by auto
blanchet@48975
   478
  thus ?thesis using assms unfolding ordLeq_def ordLess_def
blanchet@48975
   479
  using embedS_comp_embed by blast
blanchet@48975
   480
qed
blanchet@48975
   481
blanchet@48975
   482
blanchet@48975
   483
lemma ordLeq_ordIso_trans[trans]:
blanchet@48975
   484
assumes "r \<le>o r'" and " r' =o r''"
blanchet@48975
   485
shows "r \<le>o r''"
blanchet@48975
   486
proof-
blanchet@48975
   487
  have "Well_order r \<and> Well_order r''"
blanchet@48975
   488
  using assms unfolding ordLeq_def ordIso_def by auto
blanchet@48975
   489
  thus ?thesis using assms unfolding ordLeq_def ordIso_def
blanchet@48975
   490
  using embed_comp_iso by blast
blanchet@48975
   491
qed
blanchet@48975
   492
blanchet@48975
   493
blanchet@48975
   494
lemma ordIso_ordLeq_trans[trans]:
blanchet@48975
   495
assumes "r =o r'" and " r' \<le>o r''"
blanchet@48975
   496
shows "r \<le>o r''"
blanchet@48975
   497
proof-
blanchet@48975
   498
  have "Well_order r \<and> Well_order r''"
blanchet@48975
   499
  using assms unfolding ordLeq_def ordIso_def by auto
blanchet@48975
   500
  thus ?thesis using assms unfolding ordLeq_def ordIso_def
blanchet@48975
   501
  using iso_comp_embed by blast
blanchet@48975
   502
qed
blanchet@48975
   503
blanchet@48975
   504
blanchet@48975
   505
lemma ordLess_ordIso_trans[trans]:
blanchet@48975
   506
assumes "r <o r'" and " r' =o r''"
blanchet@48975
   507
shows "r <o r''"
blanchet@48975
   508
proof-
blanchet@48975
   509
  have "Well_order r \<and> Well_order r''"
blanchet@48975
   510
  using assms unfolding ordLess_def ordIso_def by auto
blanchet@48975
   511
  thus ?thesis using assms unfolding ordLess_def ordIso_def
blanchet@48975
   512
  using embedS_comp_iso by blast
blanchet@48975
   513
qed
blanchet@48975
   514
blanchet@48975
   515
blanchet@48975
   516
lemma ordIso_ordLess_trans[trans]:
blanchet@48975
   517
assumes "r =o r'" and " r' <o r''"
blanchet@48975
   518
shows "r <o r''"
blanchet@48975
   519
proof-
blanchet@48975
   520
  have "Well_order r \<and> Well_order r''"
blanchet@48975
   521
  using assms unfolding ordLess_def ordIso_def by auto
blanchet@48975
   522
  thus ?thesis using assms unfolding ordLess_def ordIso_def
blanchet@48975
   523
  using iso_comp_embedS by blast
blanchet@48975
   524
qed
blanchet@48975
   525
blanchet@48975
   526
blanchet@48975
   527
lemma ordLess_not_embed:
blanchet@48975
   528
assumes "r <o r'"
blanchet@48975
   529
shows "\<not>(\<exists>f'. embed r' r f')"
blanchet@48975
   530
proof-
blanchet@48975
   531
  obtain f where 1: "Well_order r \<and> Well_order r'" and 2: "embed r r' f" and
blanchet@48975
   532
                 3: " \<not> bij_betw f (Field r) (Field r')"
blanchet@48975
   533
  using assms unfolding ordLess_def by (auto simp add: embedS_def)
blanchet@48975
   534
  {fix f' assume *: "embed r' r f'"
blanchet@48975
   535
   hence "bij_betw f (Field r) (Field r')" using 1 2
blanchet@48975
   536
   by (simp add: embed_bothWays_Field_bij_betw)
blanchet@48975
   537
   with 3 have False by contradiction
blanchet@48975
   538
  }
blanchet@48975
   539
  thus ?thesis by blast
blanchet@48975
   540
qed
blanchet@48975
   541
blanchet@48975
   542
blanchet@48975
   543
lemma ordLess_Field:
blanchet@48975
   544
assumes OL: "r1 <o r2" and EMB: "embed r1 r2 f"
blanchet@48975
   545
shows "\<not> (f`(Field r1) = Field r2)"
blanchet@48975
   546
proof-
blanchet@48975
   547
  let ?A1 = "Field r1"  let ?A2 = "Field r2"
blanchet@48975
   548
  obtain g where
blanchet@48975
   549
  0: "Well_order r1 \<and> Well_order r2" and
blanchet@48975
   550
  1: "embed r1 r2 g \<and> \<not>(bij_betw g ?A1 ?A2)"
blanchet@48975
   551
  using OL unfolding ordLess_def by (auto simp add: embedS_def)
blanchet@48975
   552
  hence "\<forall>a \<in> ?A1. f a = g a"
blanchet@48975
   553
  using 0 EMB embed_unique[of r1] by auto
blanchet@48975
   554
  hence "\<not>(bij_betw f ?A1 ?A2)"
blanchet@48975
   555
  using 1 bij_betw_cong[of ?A1] by blast
blanchet@48975
   556
  moreover
blanchet@48975
   557
  have "inj_on f ?A1" using EMB 0 by (simp add: embed_inj_on)
blanchet@48975
   558
  ultimately show ?thesis by (simp add: bij_betw_def)
blanchet@48975
   559
qed
blanchet@48975
   560
blanchet@48975
   561
blanchet@48975
   562
lemma ordLess_iff:
blanchet@48975
   563
"r <o r' = (Well_order r \<and> Well_order r' \<and> \<not>(\<exists>f'. embed r' r f'))"
blanchet@48975
   564
proof
blanchet@48975
   565
  assume *: "r <o r'"
blanchet@48975
   566
  hence "\<not>(\<exists>f'. embed r' r f')" using ordLess_not_embed[of r r'] by simp
blanchet@48975
   567
  with * show "Well_order r \<and> Well_order r' \<and> \<not> (\<exists>f'. embed r' r f')"
blanchet@48975
   568
  unfolding ordLess_def by auto
blanchet@48975
   569
next
blanchet@48975
   570
  assume *: "Well_order r \<and> Well_order r' \<and> \<not> (\<exists>f'. embed r' r f')"
blanchet@48975
   571
  then obtain f where 1: "embed r r' f"
blanchet@48975
   572
  using wellorders_totally_ordered[of r r'] by blast
blanchet@48975
   573
  moreover
blanchet@48975
   574
  {assume "bij_betw f (Field r) (Field r')"
blanchet@48975
   575
   with * 1 have "embed r' r (inv_into (Field r) f) "
blanchet@48975
   576
   using inv_into_Field_embed_bij_betw[of r r' f] by auto
blanchet@48975
   577
   with * have False by blast
blanchet@48975
   578
  }
blanchet@48975
   579
  ultimately show "(r,r') \<in> ordLess"
blanchet@48975
   580
  unfolding ordLess_def using * by (fastforce simp add: embedS_def)
blanchet@48975
   581
qed
blanchet@48975
   582
blanchet@48975
   583
blanchet@48975
   584
lemma ordLess_irreflexive: "\<not> r <o r"
blanchet@48975
   585
proof
blanchet@48975
   586
  assume "r <o r"
blanchet@48975
   587
  hence "Well_order r \<and>  \<not>(\<exists>f. embed r r f)"
blanchet@48975
   588
  unfolding ordLess_iff ..
blanchet@48975
   589
  moreover have "embed r r id" using id_embed[of r] .
blanchet@48975
   590
  ultimately show False by blast
blanchet@48975
   591
qed
blanchet@48975
   592
blanchet@48975
   593
blanchet@48975
   594
lemma ordLeq_iff_ordLess_or_ordIso:
blanchet@48975
   595
"r \<le>o r' = (r <o r' \<or> r =o r')"
blanchet@48975
   596
unfolding ordRels_def embedS_defs iso_defs by blast
blanchet@48975
   597
blanchet@48975
   598
blanchet@48975
   599
lemma ordIso_iff_ordLeq:
blanchet@48975
   600
"(r =o r') = (r \<le>o r' \<and> r' \<le>o r)"
blanchet@48975
   601
proof
blanchet@48975
   602
  assume "r =o r'"
blanchet@48975
   603
  then obtain f where 1: "Well_order r \<and> Well_order r' \<and>
blanchet@48975
   604
                     embed r r' f \<and> bij_betw f (Field r) (Field r')"
blanchet@48975
   605
  unfolding ordIso_def iso_defs by auto
blanchet@48975
   606
  hence "embed r r' f \<and> embed r' r (inv_into (Field r) f)"
blanchet@48975
   607
  by (simp add: inv_into_Field_embed_bij_betw)
blanchet@48975
   608
  thus  "r \<le>o r' \<and> r' \<le>o r"
blanchet@48975
   609
  unfolding ordLeq_def using 1 by auto
blanchet@48975
   610
next
blanchet@48975
   611
  assume "r \<le>o r' \<and> r' \<le>o r"
blanchet@48975
   612
  then obtain f and g where 1: "Well_order r \<and> Well_order r' \<and>
blanchet@48975
   613
                           embed r r' f \<and> embed r' r g"
blanchet@48975
   614
  unfolding ordLeq_def by auto
blanchet@48975
   615
  hence "iso r r' f" by (auto simp add: embed_bothWays_iso)
blanchet@48975
   616
  thus "r =o r'" unfolding ordIso_def using 1 by auto
blanchet@48975
   617
qed
blanchet@48975
   618
blanchet@48975
   619
blanchet@48975
   620
lemma not_ordLess_ordLeq:
blanchet@48975
   621
"r <o r' \<Longrightarrow> \<not> r' \<le>o r"
blanchet@48975
   622
using ordLess_ordLeq_trans ordLess_irreflexive by blast
blanchet@48975
   623
blanchet@48975
   624
blanchet@48975
   625
lemma ordLess_or_ordLeq:
blanchet@48975
   626
assumes WELL: "Well_order r" and WELL': "Well_order r'"
blanchet@48975
   627
shows "r <o r' \<or> r' \<le>o r"
blanchet@48975
   628
proof-
blanchet@48975
   629
  have "r \<le>o r' \<or> r' \<le>o r"
blanchet@48975
   630
  using assms by (simp add: ordLeq_total)
blanchet@48975
   631
  moreover
blanchet@48975
   632
  {assume "\<not> r <o r' \<and> r \<le>o r'"
blanchet@48975
   633
   hence "r =o r'" using ordLeq_iff_ordLess_or_ordIso by blast
blanchet@48975
   634
   hence "r' \<le>o r" using ordIso_symmetric ordIso_iff_ordLeq by blast
blanchet@48975
   635
  }
blanchet@48975
   636
  ultimately show ?thesis by blast
blanchet@48975
   637
qed
blanchet@48975
   638
blanchet@48975
   639
blanchet@48975
   640
lemma not_ordLess_ordIso:
blanchet@48975
   641
"r <o r' \<Longrightarrow> \<not> r =o r'"
blanchet@48975
   642
using assms ordLess_ordIso_trans ordIso_symmetric ordLess_irreflexive by blast
blanchet@48975
   643
blanchet@48975
   644
blanchet@48975
   645
lemma not_ordLeq_iff_ordLess:
blanchet@48975
   646
assumes WELL: "Well_order r" and WELL': "Well_order r'"
blanchet@48975
   647
shows "(\<not> r' \<le>o r) = (r <o r')"
blanchet@48975
   648
using assms not_ordLess_ordLeq ordLess_or_ordLeq by blast
blanchet@48975
   649
blanchet@48975
   650
blanchet@48975
   651
lemma not_ordLess_iff_ordLeq:
blanchet@48975
   652
assumes WELL: "Well_order r" and WELL': "Well_order r'"
blanchet@48975
   653
shows "(\<not> r' <o r) = (r \<le>o r')"
blanchet@48975
   654
using assms not_ordLess_ordLeq ordLess_or_ordLeq by blast
blanchet@48975
   655
blanchet@48975
   656
blanchet@48975
   657
lemma ordLess_transitive[trans]:
blanchet@48975
   658
"\<lbrakk>r <o r'; r' <o r''\<rbrakk> \<Longrightarrow> r <o r''"
blanchet@48975
   659
using assms ordLess_ordLeq_trans ordLeq_iff_ordLess_or_ordIso by blast
blanchet@48975
   660
blanchet@48975
   661
blanchet@48975
   662
corollary ordLess_trans: "trans ordLess"
blanchet@48975
   663
unfolding trans_def using ordLess_transitive by blast
blanchet@48975
   664
blanchet@48975
   665
blanchet@48975
   666
lemmas ordIso_equivalence = ordIso_transitive ordIso_reflexive ordIso_symmetric
blanchet@48975
   667
blanchet@48975
   668
blanchet@48975
   669
lemma ordIso_imp_ordLeq:
blanchet@48975
   670
"r =o r' \<Longrightarrow> r \<le>o r'"
blanchet@48975
   671
using ordIso_iff_ordLeq by blast
blanchet@48975
   672
blanchet@48975
   673
blanchet@48975
   674
lemma ordLess_imp_ordLeq:
blanchet@48975
   675
"r <o r' \<Longrightarrow> r \<le>o r'"
blanchet@48975
   676
using ordLeq_iff_ordLess_or_ordIso by blast
blanchet@48975
   677
blanchet@48975
   678
blanchet@48975
   679
lemma ofilter_subset_ordLeq:
blanchet@48975
   680
assumes WELL: "Well_order r" and
blanchet@48975
   681
        OFA: "wo_rel.ofilter r A" and OFB: "wo_rel.ofilter r B"
blanchet@48975
   682
shows "(A \<le> B) = (Restr r A \<le>o Restr r B)"
blanchet@48975
   683
proof
blanchet@48975
   684
  assume "A \<le> B"
blanchet@48975
   685
  thus "Restr r A \<le>o Restr r B"
blanchet@48975
   686
  unfolding ordLeq_def using assms
blanchet@48975
   687
  Well_order_Restr Well_order_Restr ofilter_subset_embed by blast
blanchet@48975
   688
next
blanchet@48975
   689
  assume *: "Restr r A \<le>o Restr r B"
blanchet@48975
   690
  then obtain f where "embed (Restr r A) (Restr r B) f"
blanchet@48975
   691
  unfolding ordLeq_def by blast
blanchet@48975
   692
  {assume "B < A"
blanchet@48975
   693
   hence "Restr r B <o Restr r A"
blanchet@48975
   694
   unfolding ordLess_def using assms
blanchet@48975
   695
   Well_order_Restr Well_order_Restr ofilter_subset_embedS by blast
blanchet@48975
   696
   hence False using * not_ordLess_ordLeq by blast
blanchet@48975
   697
  }
blanchet@48975
   698
  thus "A \<le> B" using OFA OFB WELL
blanchet@48975
   699
  wo_rel_def[of r] wo_rel.ofilter_linord[of r A B] by blast
blanchet@48975
   700
qed
blanchet@48975
   701
blanchet@48975
   702
blanchet@48975
   703
lemma ofilter_subset_ordLess:
blanchet@48975
   704
assumes WELL: "Well_order r" and
blanchet@48975
   705
        OFA: "wo_rel.ofilter r A" and OFB: "wo_rel.ofilter r B"
blanchet@48975
   706
shows "(A < B) = (Restr r A <o Restr r B)"
blanchet@48975
   707
proof-
blanchet@48975
   708
  let ?rA = "Restr r A" let ?rB = "Restr r B"
blanchet@48975
   709
  have 1: "Well_order ?rA \<and> Well_order ?rB"
blanchet@48975
   710
  using WELL Well_order_Restr by blast
blanchet@48975
   711
  have "(A < B) = (\<not> B \<le> A)" using assms
blanchet@48975
   712
  wo_rel_def wo_rel.ofilter_linord[of r A B] by blast
blanchet@48975
   713
  also have "\<dots> = (\<not> Restr r B \<le>o Restr r A)"
blanchet@48975
   714
  using assms ofilter_subset_ordLeq by blast
blanchet@48975
   715
  also have "\<dots> = (Restr r A <o Restr r B)"
blanchet@48975
   716
  using 1 not_ordLeq_iff_ordLess by blast
blanchet@48975
   717
  finally show ?thesis .
blanchet@48975
   718
qed
blanchet@48975
   719
blanchet@48975
   720
blanchet@48975
   721
lemma ofilter_ordLess:
blanchet@48975
   722
"\<lbrakk>Well_order r; wo_rel.ofilter r A\<rbrakk> \<Longrightarrow> (A < Field r) = (Restr r A <o r)"
blanchet@48975
   723
by (simp add: ofilter_subset_ordLess wo_rel.Field_ofilter
blanchet@48975
   724
    wo_rel_def Restr_Field)
blanchet@48975
   725
blanchet@48975
   726
blanchet@48975
   727
corollary underS_Restr_ordLess:
blanchet@48975
   728
assumes "Well_order r" and "Field r \<noteq> {}"
blanchet@48975
   729
shows "Restr r (rel.underS r a) <o r"
blanchet@48975
   730
proof-
blanchet@48975
   731
  have "rel.underS r a < Field r" using assms
blanchet@48975
   732
  by (simp add: rel.underS_Field3)
blanchet@48975
   733
  thus ?thesis using assms
blanchet@48975
   734
  by (simp add: ofilter_ordLess wo_rel.underS_ofilter wo_rel_def)
blanchet@48975
   735
qed
blanchet@48975
   736
blanchet@48975
   737
blanchet@48975
   738
lemma embed_ordLess_ofilterIncl:
blanchet@48975
   739
assumes
blanchet@48975
   740
  OL12: "r1 <o r2" and OL23: "r2 <o r3" and
blanchet@48975
   741
  EMB13: "embed r1 r3 f13" and EMB23: "embed r2 r3 f23"
blanchet@48975
   742
shows "(f13`(Field r1), f23`(Field r2)) \<in> (ofilterIncl r3)"
blanchet@48975
   743
proof-
blanchet@48975
   744
  have OL13: "r1 <o r3"
blanchet@48975
   745
  using OL12 OL23 using ordLess_transitive by auto
blanchet@48975
   746
  let ?A1 = "Field r1"  let ?A2 ="Field r2" let ?A3 ="Field r3"
blanchet@48975
   747
  obtain f12 g23 where
blanchet@48975
   748
  0: "Well_order r1 \<and> Well_order r2 \<and> Well_order r3" and
blanchet@48975
   749
  1: "embed r1 r2 f12 \<and> \<not>(bij_betw f12 ?A1 ?A2)" and
blanchet@48975
   750
  2: "embed r2 r3 g23 \<and> \<not>(bij_betw g23 ?A2 ?A3)"
blanchet@48975
   751
  using OL12 OL23 by (auto simp add: ordLess_def embedS_def)
blanchet@48975
   752
  hence "\<forall>a \<in> ?A2. f23 a = g23 a"
blanchet@48975
   753
  using EMB23 embed_unique[of r2 r3] by blast
blanchet@48975
   754
  hence 3: "\<not>(bij_betw f23 ?A2 ?A3)"
blanchet@48975
   755
  using 2 bij_betw_cong[of ?A2 f23 g23] by blast
blanchet@48975
   756
  (*  *)
blanchet@48975
   757
  have 4: "wo_rel.ofilter r2 (f12 ` ?A1) \<and> f12 ` ?A1 \<noteq> ?A2"
blanchet@48975
   758
  using 0 1 OL12 by (simp add: embed_Field_ofilter ordLess_Field)
blanchet@48975
   759
  have 5: "wo_rel.ofilter r3 (f23 ` ?A2) \<and> f23 ` ?A2 \<noteq> ?A3"
blanchet@48975
   760
  using 0 EMB23 OL23 by (simp add: embed_Field_ofilter ordLess_Field)
blanchet@48975
   761
  have 6: "wo_rel.ofilter r3 (f13 ` ?A1)  \<and> f13 ` ?A1 \<noteq> ?A3"
blanchet@48975
   762
  using 0 EMB13 OL13 by (simp add: embed_Field_ofilter ordLess_Field)
blanchet@48975
   763
  (*  *)
blanchet@48975
   764
  have "f12 ` ?A1 < ?A2"
blanchet@48975
   765
  using 0 4 by (auto simp add: wo_rel_def wo_rel.ofilter_def)
blanchet@48975
   766
  moreover have "inj_on f23 ?A2"
blanchet@48975
   767
  using EMB23 0 by (simp add: wo_rel_def embed_inj_on)
blanchet@48975
   768
  ultimately
blanchet@48975
   769
  have "f23 ` (f12 ` ?A1) < f23 ` ?A2" by (simp add: inj_on_strict_subset)
blanchet@48975
   770
  moreover
blanchet@48975
   771
  {have "embed r1 r3 (f23 o f12)"
blanchet@48975
   772
   using 1 EMB23 0 by (auto simp add: comp_embed)
blanchet@48975
   773
   hence "\<forall>a \<in> ?A1. f23(f12 a) = f13 a"
blanchet@48975
   774
   using EMB13 0 embed_unique[of r1 r3 "f23 o f12" f13] by auto
blanchet@48975
   775
   hence "f23 ` (f12 ` ?A1) = f13 ` ?A1" by force
blanchet@48975
   776
  }
blanchet@48975
   777
  ultimately
blanchet@48975
   778
  have "f13 ` ?A1 < f23 ` ?A2" by simp
blanchet@48975
   779
  (*  *)
blanchet@48975
   780
  with 5 6 show ?thesis
blanchet@48975
   781
  unfolding ofilterIncl_def by auto
blanchet@48975
   782
qed
blanchet@48975
   783
blanchet@48975
   784
blanchet@48975
   785
lemma ordLess_iff_ordIso_Restr:
blanchet@48975
   786
assumes WELL: "Well_order r" and WELL': "Well_order r'"
blanchet@48975
   787
shows "(r' <o r) = (\<exists>a \<in> Field r. r' =o Restr r (rel.underS r a))"
blanchet@48975
   788
proof(auto)
blanchet@48975
   789
  fix a assume *: "a \<in> Field r" and **: "r' =o Restr r (rel.underS r a)"
blanchet@48975
   790
  hence "Restr r (rel.underS r a) <o r" using WELL underS_Restr_ordLess[of r] by blast
blanchet@48975
   791
  thus "r' <o r" using ** ordIso_ordLess_trans by blast
blanchet@48975
   792
next
blanchet@48975
   793
  assume "r' <o r"
blanchet@48975
   794
  then obtain f where 1: "Well_order r \<and> Well_order r'" and
blanchet@48975
   795
                      2: "embed r' r f \<and> f ` (Field r') \<noteq> Field r"
blanchet@48975
   796
  unfolding ordLess_def embedS_def[abs_def] bij_betw_def using embed_inj_on by blast
blanchet@48975
   797
  hence "wo_rel.ofilter r (f ` (Field r'))" using embed_Field_ofilter by blast
blanchet@48975
   798
  then obtain a where 3: "a \<in> Field r" and 4: "rel.underS r a = f ` (Field r')"
blanchet@48975
   799
  using 1 2 by (auto simp add: wo_rel.ofilter_underS_Field wo_rel_def)
blanchet@48975
   800
  have "iso r' (Restr r (f ` (Field r'))) f"
blanchet@48975
   801
  using embed_implies_iso_Restr 2 assms by blast
blanchet@48975
   802
  moreover have "Well_order (Restr r (f ` (Field r')))"
blanchet@48975
   803
  using WELL Well_order_Restr by blast
blanchet@48975
   804
  ultimately have "r' =o Restr r (f ` (Field r'))"
blanchet@48975
   805
  using WELL' unfolding ordIso_def by auto
blanchet@48975
   806
  hence "r' =o Restr r (rel.underS r a)" using 4 by auto
blanchet@48975
   807
  thus "\<exists>a \<in> Field r. r' =o Restr r (rel.underS r a)" using 3 by auto
blanchet@48975
   808
qed
blanchet@48975
   809
blanchet@48975
   810
blanchet@48975
   811
lemma internalize_ordLess:
blanchet@48975
   812
"(r' <o r) = (\<exists>p. Field p < Field r \<and> r' =o p \<and> p <o r)"
blanchet@48975
   813
proof
blanchet@48975
   814
  assume *: "r' <o r"
blanchet@48975
   815
  hence 0: "Well_order r \<and> Well_order r'" unfolding ordLess_def by auto
blanchet@48975
   816
  with * obtain a where 1: "a \<in> Field r" and 2: "r' =o Restr r (rel.underS r a)"
blanchet@48975
   817
  using ordLess_iff_ordIso_Restr by blast
blanchet@48975
   818
  let ?p = "Restr r (rel.underS r a)"
blanchet@48975
   819
  have "wo_rel.ofilter r (rel.underS r a)" using 0
blanchet@48975
   820
  by (simp add: wo_rel_def wo_rel.underS_ofilter)
blanchet@48975
   821
  hence "Field ?p = rel.underS r a" using 0 Field_Restr_ofilter by blast
blanchet@48975
   822
  hence "Field ?p < Field r" using rel.underS_Field2 1 by fastforce
blanchet@48975
   823
  moreover have "?p <o r" using underS_Restr_ordLess[of r a] 0 1 by blast
blanchet@48975
   824
  ultimately
blanchet@48975
   825
  show "\<exists>p. Field p < Field r \<and> r' =o p \<and> p <o r" using 2 by blast
blanchet@48975
   826
next
blanchet@48975
   827
  assume "\<exists>p. Field p < Field r \<and> r' =o p \<and> p <o r"
blanchet@48975
   828
  thus "r' <o r" using ordIso_ordLess_trans by blast
blanchet@48975
   829
qed
blanchet@48975
   830
blanchet@48975
   831
blanchet@48975
   832
lemma internalize_ordLeq:
blanchet@48975
   833
"(r' \<le>o r) = (\<exists>p. Field p \<le> Field r \<and> r' =o p \<and> p \<le>o r)"
blanchet@48975
   834
proof
blanchet@48975
   835
  assume *: "r' \<le>o r"
blanchet@48975
   836
  moreover
blanchet@48975
   837
  {assume "r' <o r"
blanchet@48975
   838
   then obtain p where "Field p < Field r \<and> r' =o p \<and> p <o r"
blanchet@48975
   839
   using internalize_ordLess[of r' r] by blast
blanchet@48975
   840
   hence "\<exists>p. Field p \<le> Field r \<and> r' =o p \<and> p \<le>o r"
blanchet@48975
   841
   using ordLeq_iff_ordLess_or_ordIso by blast
blanchet@48975
   842
  }
blanchet@48975
   843
  moreover
blanchet@48975
   844
  have "r \<le>o r" using * ordLeq_def ordLeq_reflexive by blast
blanchet@48975
   845
  ultimately show "\<exists>p. Field p \<le> Field r \<and> r' =o p \<and> p \<le>o r"
blanchet@48975
   846
  using ordLeq_iff_ordLess_or_ordIso by blast
blanchet@48975
   847
next
blanchet@48975
   848
  assume "\<exists>p. Field p \<le> Field r \<and> r' =o p \<and> p \<le>o r"
blanchet@48975
   849
  thus "r' \<le>o r" using ordIso_ordLeq_trans by blast
blanchet@48975
   850
qed
blanchet@48975
   851
blanchet@48975
   852
blanchet@48975
   853
lemma ordLeq_iff_ordLess_Restr:
blanchet@48975
   854
assumes WELL: "Well_order r" and WELL': "Well_order r'"
blanchet@48975
   855
shows "(r \<le>o r') = (\<forall>a \<in> Field r. Restr r (rel.underS r a) <o r')"
blanchet@48975
   856
proof(auto)
blanchet@48975
   857
  assume *: "r \<le>o r'"
blanchet@48975
   858
  fix a assume "a \<in> Field r"
blanchet@48975
   859
  hence "Restr r (rel.underS r a) <o r"
blanchet@48975
   860
  using WELL underS_Restr_ordLess[of r] by blast
blanchet@48975
   861
  thus "Restr r (rel.underS r a) <o r'"
blanchet@48975
   862
  using * ordLess_ordLeq_trans by blast
blanchet@48975
   863
next
blanchet@48975
   864
  assume *: "\<forall>a \<in> Field r. Restr r (rel.underS r a) <o r'"
blanchet@48975
   865
  {assume "r' <o r"
blanchet@48975
   866
   then obtain a where "a \<in> Field r \<and> r' =o Restr r (rel.underS r a)"
blanchet@48975
   867
   using assms ordLess_iff_ordIso_Restr by blast
blanchet@48975
   868
   hence False using * not_ordLess_ordIso ordIso_symmetric by blast
blanchet@48975
   869
  }
blanchet@48975
   870
  thus "r \<le>o r'" using ordLess_or_ordLeq assms by blast
blanchet@48975
   871
qed
blanchet@48975
   872
blanchet@48975
   873
blanchet@48975
   874
lemma finite_ordLess_infinite:
blanchet@48975
   875
assumes WELL: "Well_order r" and WELL': "Well_order r'" and
blanchet@48975
   876
        FIN: "finite(Field r)" and INF: "infinite(Field r')"
blanchet@48975
   877
shows "r <o r'"
blanchet@48975
   878
proof-
blanchet@48975
   879
  {assume "r' \<le>o r"
blanchet@48975
   880
   then obtain h where "inj_on h (Field r') \<and> h ` (Field r') \<le> Field r"
blanchet@48975
   881
   unfolding ordLeq_def using assms embed_inj_on embed_Field by blast
blanchet@48975
   882
   hence False using finite_imageD finite_subset FIN INF by blast
blanchet@48975
   883
  }
blanchet@48975
   884
  thus ?thesis using WELL WELL' ordLess_or_ordLeq by blast
blanchet@48975
   885
qed
blanchet@48975
   886
blanchet@48975
   887
blanchet@48975
   888
lemma finite_well_order_on_ordIso:
blanchet@48975
   889
assumes FIN: "finite A" and
blanchet@48975
   890
        WELL: "well_order_on A r" and WELL': "well_order_on A r'"
blanchet@48975
   891
shows "r =o r'"
blanchet@48975
   892
proof-
blanchet@48975
   893
  have 0: "Well_order r \<and> Well_order r' \<and> Field r = A \<and> Field r' = A"
blanchet@48975
   894
  using assms rel.well_order_on_Well_order by blast
blanchet@48975
   895
  moreover
blanchet@48975
   896
  have "\<forall>r r'. well_order_on A r \<and> well_order_on A r' \<and> r \<le>o r'
blanchet@48975
   897
                  \<longrightarrow> r =o r'"
blanchet@48975
   898
  proof(clarify)
blanchet@48975
   899
    fix r r' assume *: "well_order_on A r" and **: "well_order_on A r'"
blanchet@48975
   900
    have 2: "Well_order r \<and> Well_order r' \<and> Field r = A \<and> Field r' = A"
blanchet@48975
   901
    using * ** rel.well_order_on_Well_order by blast
blanchet@48975
   902
    assume "r \<le>o r'"
blanchet@48975
   903
    then obtain f where 1: "embed r r' f" and
blanchet@48975
   904
                        "inj_on f A \<and> f ` A \<le> A"
blanchet@48975
   905
    unfolding ordLeq_def using 2 embed_inj_on embed_Field by blast
blanchet@48975
   906
    hence "bij_betw f A A" unfolding bij_betw_def using FIN endo_inj_surj by blast
blanchet@48975
   907
    thus "r =o r'" unfolding ordIso_def iso_def[abs_def] using 1 2 by auto
blanchet@48975
   908
  qed
blanchet@48975
   909
  ultimately show ?thesis using assms ordLeq_total ordIso_symmetric by blast
blanchet@48975
   910
qed
blanchet@48975
   911
blanchet@48975
   912
blanchet@48975
   913
subsection{* @{text "<o"} is well-founded *}
blanchet@48975
   914
blanchet@48975
   915
blanchet@48975
   916
text {* Of course, it only makes sense to state that the @{text "<o"} is well-founded
blanchet@48975
   917
on the restricted type @{text "'a rel rel"}.  We prove this by first showing that, for any set
blanchet@48975
   918
of well-orders all embedded in a fixed well-order, the function mapping each well-order
blanchet@48975
   919
in the set to an order filter of the fixed well-order is compatible w.r.t. to @{text "<o"} versus
blanchet@48975
   920
{\em strict inclusion}; and we already know that strict inclusion of order filters is well-founded. *}
blanchet@48975
   921
blanchet@48975
   922
blanchet@48975
   923
definition ord_to_filter :: "'a rel \<Rightarrow> 'a rel \<Rightarrow> 'a set"
blanchet@48975
   924
where "ord_to_filter r0 r \<equiv> (SOME f. embed r r0 f) ` (Field r)"
blanchet@48975
   925
blanchet@48975
   926
blanchet@48975
   927
lemma ord_to_filter_compat:
blanchet@48975
   928
"compat (ordLess Int (ordLess^-1``{r0} \<times> ordLess^-1``{r0}))
blanchet@48975
   929
        (ofilterIncl r0)
blanchet@48975
   930
        (ord_to_filter r0)"
blanchet@48975
   931
proof(unfold compat_def ord_to_filter_def, clarify)
blanchet@48975
   932
  fix r1::"'a rel" and r2::"'a rel"
blanchet@48975
   933
  let ?A1 = "Field r1"  let ?A2 ="Field r2" let ?A0 ="Field r0"
blanchet@48975
   934
  let ?phi10 = "\<lambda> f10. embed r1 r0 f10" let ?f10 = "SOME f. ?phi10 f"
blanchet@48975
   935
  let ?phi20 = "\<lambda> f20. embed r2 r0 f20" let ?f20 = "SOME f. ?phi20 f"
blanchet@48975
   936
  assume *: "r1 <o r0" "r2 <o r0" and **: "r1 <o r2"
blanchet@48975
   937
  hence "(\<exists>f. ?phi10 f) \<and> (\<exists>f. ?phi20 f)"
blanchet@48975
   938
  by (auto simp add: ordLess_def embedS_def)
blanchet@48975
   939
  hence "?phi10 ?f10 \<and> ?phi20 ?f20" by (auto simp add: someI_ex)
blanchet@48975
   940
  thus "(?f10 ` ?A1, ?f20 ` ?A2) \<in> ofilterIncl r0"
blanchet@48975
   941
  using * ** by (simp add: embed_ordLess_ofilterIncl)
blanchet@48975
   942
qed
blanchet@48975
   943
blanchet@48975
   944
blanchet@48975
   945
theorem wf_ordLess: "wf ordLess"
blanchet@48975
   946
proof-
blanchet@48975
   947
  {fix r0 :: "('a \<times> 'a) set"
blanchet@48975
   948
   (* need to annotate here!*)
blanchet@48975
   949
   let ?ordLess = "ordLess::('d rel * 'd rel) set"
blanchet@48975
   950
   let ?R = "?ordLess Int (?ordLess^-1``{r0} \<times> ?ordLess^-1``{r0})"
blanchet@48975
   951
   {assume Case1: "Well_order r0"
blanchet@48975
   952
    hence "wf ?R"
blanchet@48975
   953
    using wf_ofilterIncl[of r0]
blanchet@48975
   954
          compat_wf[of ?R "ofilterIncl r0" "ord_to_filter r0"]
blanchet@48975
   955
          ord_to_filter_compat[of r0] by auto
blanchet@48975
   956
   }
blanchet@48975
   957
   moreover
blanchet@48975
   958
   {assume Case2: "\<not> Well_order r0"
blanchet@48975
   959
    hence "?R = {}" unfolding ordLess_def by auto
blanchet@48975
   960
    hence "wf ?R" using wf_empty by simp
blanchet@48975
   961
   }
blanchet@48975
   962
   ultimately have "wf ?R" by blast
blanchet@48975
   963
  }
blanchet@48975
   964
  thus ?thesis by (simp add: trans_wf_iff ordLess_trans)
blanchet@48975
   965
qed
blanchet@48975
   966
blanchet@48975
   967
corollary exists_minim_Well_order:
blanchet@48975
   968
assumes NE: "R \<noteq> {}" and WELL: "\<forall>r \<in> R. Well_order r"
blanchet@48975
   969
shows "\<exists>r \<in> R. \<forall>r' \<in> R. r \<le>o r'"
blanchet@48975
   970
proof-
blanchet@48975
   971
  obtain r where "r \<in> R \<and> (\<forall>r' \<in> R. \<not> r' <o r)"
blanchet@48975
   972
  using NE spec[OF spec[OF subst[OF wf_eq_minimal, of "%x. x", OF wf_ordLess]], of _ R]
blanchet@48975
   973
    equals0I[of R] by blast
blanchet@48975
   974
  with not_ordLeq_iff_ordLess WELL show ?thesis by blast
blanchet@48975
   975
qed
blanchet@48975
   976
blanchet@48975
   977
blanchet@48975
   978
blanchet@48975
   979
subsection {* Copy via direct images  *}
blanchet@48975
   980
blanchet@48975
   981
blanchet@48975
   982
text{* The direct image operator is the dual of the inverse image operator @{text "inv_image"}
blanchet@48975
   983
from @{text "Relation.thy"}.  It is useful for transporting a well-order between
blanchet@48975
   984
different types. *}
blanchet@48975
   985
blanchet@48975
   986
blanchet@48975
   987
definition dir_image :: "'a rel \<Rightarrow> ('a \<Rightarrow> 'a') \<Rightarrow> 'a' rel"
blanchet@48975
   988
where
blanchet@48975
   989
"dir_image r f = {(f a, f b)| a b. (a,b) \<in> r}"
blanchet@48975
   990
blanchet@48975
   991
blanchet@48975
   992
lemma dir_image_Field:
blanchet@48975
   993
"Field(dir_image r f) \<le> f ` (Field r)"
blanchet@48975
   994
unfolding dir_image_def Field_def by auto
blanchet@48975
   995
blanchet@48975
   996
blanchet@48975
   997
lemma dir_image_minus_Id:
blanchet@48975
   998
"inj_on f (Field r) \<Longrightarrow> (dir_image r f) - Id = dir_image (r - Id) f"
blanchet@48975
   999
unfolding inj_on_def Field_def dir_image_def by auto
blanchet@48975
  1000
blanchet@48975
  1001
blanchet@48975
  1002
lemma Refl_dir_image:
blanchet@48975
  1003
assumes "Refl r"
blanchet@48975
  1004
shows "Refl(dir_image r f)"
blanchet@48975
  1005
proof-
blanchet@48975
  1006
  {fix a' b'
blanchet@48975
  1007
   assume "(a',b') \<in> dir_image r f"
blanchet@48975
  1008
   then obtain a b where 1: "a' = f a \<and> b' = f b \<and> (a,b) \<in> r"
blanchet@48975
  1009
   unfolding dir_image_def by blast
blanchet@48975
  1010
   hence "a \<in> Field r \<and> b \<in> Field r" using Field_def by fastforce
blanchet@48975
  1011
   hence "(a,a) \<in> r \<and> (b,b) \<in> r" using assms by (simp add: refl_on_def)
blanchet@48975
  1012
   with 1 have "(a',a') \<in> dir_image r f \<and> (b',b') \<in> dir_image r f"
blanchet@48975
  1013
   unfolding dir_image_def by auto
blanchet@48975
  1014
  }
blanchet@48975
  1015
  thus ?thesis
blanchet@48975
  1016
  by(unfold refl_on_def Field_def Domain_def Range_def, auto)
blanchet@48975
  1017
qed
blanchet@48975
  1018
blanchet@48975
  1019
blanchet@48975
  1020
lemma trans_dir_image:
blanchet@48975
  1021
assumes TRANS: "trans r" and INJ: "inj_on f (Field r)"
blanchet@48975
  1022
shows "trans(dir_image r f)"
blanchet@48975
  1023
proof(unfold trans_def, auto)
blanchet@48975
  1024
  fix a' b' c'
blanchet@48975
  1025
  assume "(a',b') \<in> dir_image r f" "(b',c') \<in> dir_image r f"
blanchet@48975
  1026
  then obtain a b1 b2 c where 1: "a' = f a \<and> b' = f b1 \<and> b' = f b2 \<and> c' = f c" and
blanchet@48975
  1027
                         2: "(a,b1) \<in> r \<and> (b2,c) \<in> r"
blanchet@48975
  1028
  unfolding dir_image_def by blast
blanchet@48975
  1029
  hence "b1 \<in> Field r \<and> b2 \<in> Field r"
blanchet@48975
  1030
  unfolding Field_def by auto
blanchet@48975
  1031
  hence "b1 = b2" using 1 INJ unfolding inj_on_def by auto
blanchet@48975
  1032
  hence "(a,c): r" using 2 TRANS unfolding trans_def by blast
blanchet@48975
  1033
  thus "(a',c') \<in> dir_image r f"
blanchet@48975
  1034
  unfolding dir_image_def using 1 by auto
blanchet@48975
  1035
qed
blanchet@48975
  1036
blanchet@48975
  1037
blanchet@48975
  1038
lemma Preorder_dir_image:
blanchet@48975
  1039
"\<lbrakk>Preorder r; inj_on f (Field r)\<rbrakk> \<Longrightarrow> Preorder (dir_image r f)"
blanchet@48975
  1040
by (simp add: preorder_on_def Refl_dir_image trans_dir_image)
blanchet@48975
  1041
blanchet@48975
  1042
blanchet@48975
  1043
lemma antisym_dir_image:
blanchet@48975
  1044
assumes AN: "antisym r" and INJ: "inj_on f (Field r)"
blanchet@48975
  1045
shows "antisym(dir_image r f)"
blanchet@48975
  1046
proof(unfold antisym_def, auto)
blanchet@48975
  1047
  fix a' b'
blanchet@48975
  1048
  assume "(a',b') \<in> dir_image r f" "(b',a') \<in> dir_image r f"
blanchet@48975
  1049
  then obtain a1 b1 a2 b2 where 1: "a' = f a1 \<and> a' = f a2 \<and> b' = f b1 \<and> b' = f b2" and
blanchet@48975
  1050
                           2: "(a1,b1) \<in> r \<and> (b2,a2) \<in> r " and
blanchet@48975
  1051
                           3: "{a1,a2,b1,b2} \<le> Field r"
blanchet@48975
  1052
  unfolding dir_image_def Field_def by blast
blanchet@48975
  1053
  hence "a1 = a2 \<and> b1 = b2" using INJ unfolding inj_on_def by auto
blanchet@48975
  1054
  hence "a1 = b2" using 2 AN unfolding antisym_def by auto
blanchet@48975
  1055
  thus "a' = b'" using 1 by auto
blanchet@48975
  1056
qed
blanchet@48975
  1057
blanchet@48975
  1058
blanchet@48975
  1059
lemma Partial_order_dir_image:
blanchet@48975
  1060
"\<lbrakk>Partial_order r; inj_on f (Field r)\<rbrakk> \<Longrightarrow> Partial_order (dir_image r f)"
blanchet@48975
  1061
by (simp add: partial_order_on_def Preorder_dir_image antisym_dir_image)
blanchet@48975
  1062
blanchet@48975
  1063
blanchet@48975
  1064
lemma Total_dir_image:
blanchet@48975
  1065
assumes TOT: "Total r" and INJ: "inj_on f (Field r)"
blanchet@48975
  1066
shows "Total(dir_image r f)"
blanchet@48975
  1067
proof(unfold total_on_def, intro ballI impI)
blanchet@48975
  1068
  fix a' b'
blanchet@48975
  1069
  assume "a' \<in> Field (dir_image r f)" "b' \<in> Field (dir_image r f)"
blanchet@48975
  1070
  then obtain a and b where 1: "a \<in> Field r \<and> b \<in> Field r \<and> f a = a' \<and> f b = b'"
blanchet@48975
  1071
  using dir_image_Field[of r f] by blast
blanchet@48975
  1072
  moreover assume "a' \<noteq> b'"
blanchet@48975
  1073
  ultimately have "a \<noteq> b" using INJ unfolding inj_on_def by auto
blanchet@48975
  1074
  hence "(a,b) \<in> r \<or> (b,a) \<in> r" using 1 TOT unfolding total_on_def by auto
blanchet@48975
  1075
  thus "(a',b') \<in> dir_image r f \<or> (b',a') \<in> dir_image r f"
blanchet@48975
  1076
  using 1 unfolding dir_image_def by auto
blanchet@48975
  1077
qed
blanchet@48975
  1078
blanchet@48975
  1079
blanchet@48975
  1080
lemma Linear_order_dir_image:
blanchet@48975
  1081
"\<lbrakk>Linear_order r; inj_on f (Field r)\<rbrakk> \<Longrightarrow> Linear_order (dir_image r f)"
blanchet@48975
  1082
by (simp add: linear_order_on_def Partial_order_dir_image Total_dir_image)
blanchet@48975
  1083
blanchet@48975
  1084
blanchet@48975
  1085
lemma wf_dir_image:
blanchet@48975
  1086
assumes WF: "wf r" and INJ: "inj_on f (Field r)"
blanchet@48975
  1087
shows "wf(dir_image r f)"
blanchet@48975
  1088
proof(unfold wf_eq_minimal2, intro allI impI, elim conjE)
blanchet@48975
  1089
  fix A'::"'b set"
blanchet@48975
  1090
  assume SUB: "A' \<le> Field(dir_image r f)" and NE: "A' \<noteq> {}"
blanchet@48975
  1091
  obtain A where A_def: "A = {a \<in> Field r. f a \<in> A'}" by blast
blanchet@48975
  1092
  have "A \<noteq> {} \<and> A \<le> Field r"
blanchet@48975
  1093
  using A_def dir_image_Field[of r f] SUB NE by blast
blanchet@48975
  1094
  then obtain a where 1: "a \<in> A \<and> (\<forall>b \<in> A. (b,a) \<notin> r)"
blanchet@48975
  1095
  using WF unfolding wf_eq_minimal2 by blast
blanchet@48975
  1096
  have "\<forall>b' \<in> A'. (b',f a) \<notin> dir_image r f"
blanchet@48975
  1097
  proof(clarify)
blanchet@48975
  1098
    fix b' assume *: "b' \<in> A'" and **: "(b',f a) \<in> dir_image r f"
blanchet@48975
  1099
    obtain b1 a1 where 2: "b' = f b1 \<and> f a = f a1" and
blanchet@48975
  1100
                       3: "(b1,a1) \<in> r \<and> {a1,b1} \<le> Field r"
blanchet@48975
  1101
    using ** unfolding dir_image_def Field_def by blast
blanchet@48975
  1102
    hence "a = a1" using 1 A_def INJ unfolding inj_on_def by auto
blanchet@48975
  1103
    hence "b1 \<in> A \<and> (b1,a) \<in> r" using 2 3 A_def * by auto
blanchet@48975
  1104
    with 1 show False by auto
blanchet@48975
  1105
  qed
blanchet@48975
  1106
  thus "\<exists>a'\<in>A'. \<forall>b'\<in>A'. (b', a') \<notin> dir_image r f"
blanchet@48975
  1107
  using A_def 1 by blast
blanchet@48975
  1108
qed
blanchet@48975
  1109
blanchet@48975
  1110
blanchet@48975
  1111
lemma Well_order_dir_image:
blanchet@48975
  1112
"\<lbrakk>Well_order r; inj_on f (Field r)\<rbrakk> \<Longrightarrow> Well_order (dir_image r f)"
blanchet@48975
  1113
using assms unfolding well_order_on_def
blanchet@48975
  1114
using Linear_order_dir_image[of r f] wf_dir_image[of "r - Id" f]
blanchet@48975
  1115
  dir_image_minus_Id[of f r]
blanchet@48975
  1116
  subset_inj_on[of f "Field r" "Field(r - Id)"]
blanchet@48975
  1117
  mono_Field[of "r - Id" r] by auto
blanchet@48975
  1118
blanchet@48975
  1119
blanchet@48975
  1120
lemma dir_image_Field2:
blanchet@48975
  1121
"Refl r \<Longrightarrow> Field(dir_image r f) = f ` (Field r)"
blanchet@48975
  1122
unfolding Field_def dir_image_def refl_on_def Domain_def Range_def by blast
blanchet@48975
  1123
blanchet@48975
  1124
blanchet@48975
  1125
lemma dir_image_bij_betw:
blanchet@48975
  1126
"\<lbrakk>Well_order r; inj_on f (Field r)\<rbrakk> \<Longrightarrow> bij_betw f (Field r) (Field (dir_image r f))"
blanchet@48975
  1127
unfolding bij_betw_def
blanchet@48975
  1128
by (simp add: dir_image_Field2 order_on_defs)
blanchet@48975
  1129
blanchet@48975
  1130
blanchet@48975
  1131
lemma dir_image_compat:
blanchet@48975
  1132
"compat r (dir_image r f) f"
blanchet@48975
  1133
unfolding compat_def dir_image_def by auto
blanchet@48975
  1134
blanchet@48975
  1135
blanchet@48975
  1136
lemma dir_image_iso:
blanchet@48975
  1137
"\<lbrakk>Well_order r; inj_on f (Field r)\<rbrakk>  \<Longrightarrow> iso r (dir_image r f) f"
blanchet@48975
  1138
using iso_iff3 dir_image_compat dir_image_bij_betw Well_order_dir_image by blast
blanchet@48975
  1139
blanchet@48975
  1140
blanchet@48975
  1141
lemma dir_image_ordIso:
blanchet@48975
  1142
"\<lbrakk>Well_order r; inj_on f (Field r)\<rbrakk>  \<Longrightarrow> r =o dir_image r f"
blanchet@48975
  1143
unfolding ordIso_def using dir_image_iso Well_order_dir_image by blast
blanchet@48975
  1144
blanchet@48975
  1145
blanchet@48975
  1146
lemma Well_order_iso_copy:
blanchet@48975
  1147
assumes WELL: "well_order_on A r" and BIJ: "bij_betw f A A'"
blanchet@48975
  1148
shows "\<exists>r'. well_order_on A' r' \<and> r =o r'"
blanchet@48975
  1149
proof-
blanchet@48975
  1150
   let ?r' = "dir_image r f"
blanchet@48975
  1151
   have 1: "A = Field r \<and> Well_order r"
blanchet@48975
  1152
   using WELL rel.well_order_on_Well_order by blast
blanchet@48975
  1153
   hence 2: "iso r ?r' f"
blanchet@48975
  1154
   using dir_image_iso using BIJ unfolding bij_betw_def by auto
blanchet@48975
  1155
   hence "f ` (Field r) = Field ?r'" using 1 iso_iff[of r ?r'] by blast
blanchet@48975
  1156
   hence "Field ?r' = A'"
blanchet@48975
  1157
   using 1 BIJ unfolding bij_betw_def by auto
blanchet@48975
  1158
   moreover have "Well_order ?r'"
blanchet@48975
  1159
   using 1 Well_order_dir_image BIJ unfolding bij_betw_def by blast
blanchet@48975
  1160
   ultimately show ?thesis unfolding ordIso_def using 1 2 by blast
blanchet@48975
  1161
qed
blanchet@48975
  1162
blanchet@48975
  1163
blanchet@48975
  1164
blanchet@48975
  1165
subsection {* Bounded square  *}
blanchet@48975
  1166
blanchet@48975
  1167
blanchet@48975
  1168
text{* This construction essentially defines, for an order relation @{text "r"}, a lexicographic
blanchet@48975
  1169
order @{text "bsqr r"} on @{text "(Field r) \<times> (Field r)"}, applying the
blanchet@48975
  1170
following criteria (in this order):
blanchet@48975
  1171
\begin{itemize}
blanchet@48975
  1172
\item compare the maximums;
blanchet@48975
  1173
\item compare the first components;
blanchet@48975
  1174
\item compare the second components.
blanchet@48975
  1175
\end{itemize}
blanchet@48975
  1176
%
blanchet@48975
  1177
The only application of this construction that we are aware of is
blanchet@48975
  1178
at proving that the square of an infinite set has the same cardinal
blanchet@48975
  1179
as that set. The essential property required there (and which is ensured by this
blanchet@48975
  1180
construction) is that any proper order filter of the product order is included in a rectangle, i.e.,
blanchet@48975
  1181
in a product of proper filters on the original relation (assumed to be a well-order). *}
blanchet@48975
  1182
blanchet@48975
  1183
blanchet@48975
  1184
definition bsqr :: "'a rel => ('a * 'a)rel"
blanchet@48975
  1185
where
blanchet@48975
  1186
"bsqr r = {((a1,a2),(b1,b2)).
blanchet@48975
  1187
           {a1,a2,b1,b2} \<le> Field r \<and>
blanchet@48975
  1188
           (a1 = b1 \<and> a2 = b2 \<or>
blanchet@48975
  1189
            (wo_rel.max2 r a1 a2, wo_rel.max2 r b1 b2) \<in> r - Id \<or>
blanchet@48975
  1190
            wo_rel.max2 r a1 a2 = wo_rel.max2 r b1 b2 \<and> (a1,b1) \<in> r - Id \<or>
blanchet@48975
  1191
            wo_rel.max2 r a1 a2 = wo_rel.max2 r b1 b2 \<and> a1 = b1  \<and> (a2,b2) \<in> r - Id
blanchet@48975
  1192
           )}"
blanchet@48975
  1193
blanchet@48975
  1194
blanchet@48975
  1195
lemma Field_bsqr:
blanchet@48975
  1196
"Field (bsqr r) = Field r \<times> Field r"
blanchet@48975
  1197
proof
blanchet@48975
  1198
  show "Field (bsqr r) \<le> Field r \<times> Field r"
blanchet@48975
  1199
  proof-
blanchet@48975
  1200
    {fix a1 a2 assume "(a1,a2) \<in> Field (bsqr r)"
blanchet@48975
  1201
     moreover
blanchet@48975
  1202
     have "\<And> b1 b2. ((a1,a2),(b1,b2)) \<in> bsqr r \<or> ((b1,b2),(a1,a2)) \<in> bsqr r \<Longrightarrow>
blanchet@48975
  1203
                      a1 \<in> Field r \<and> a2 \<in> Field r" unfolding bsqr_def by auto
blanchet@48975
  1204
     ultimately have "a1 \<in> Field r \<and> a2 \<in> Field r" unfolding Field_def by auto
blanchet@48975
  1205
    }
blanchet@48975
  1206
    thus ?thesis unfolding Field_def by force
blanchet@48975
  1207
  qed
blanchet@48975
  1208
next
blanchet@48975
  1209
  show "Field r \<times> Field r \<le> Field (bsqr r)"
blanchet@48975
  1210
  proof(auto)
blanchet@48975
  1211
    fix a1 a2 assume "a1 \<in> Field r" and "a2 \<in> Field r"
blanchet@48975
  1212
    hence "((a1,a2),(a1,a2)) \<in> bsqr r" unfolding bsqr_def by blast
blanchet@48975
  1213
    thus "(a1,a2) \<in> Field (bsqr r)" unfolding Field_def by auto
blanchet@48975
  1214
  qed
blanchet@48975
  1215
qed
blanchet@48975
  1216
blanchet@48975
  1217
blanchet@48975
  1218
lemma bsqr_Refl: "Refl(bsqr r)"
blanchet@48975
  1219
by(unfold refl_on_def Field_bsqr, auto simp add: bsqr_def)
blanchet@48975
  1220
blanchet@48975
  1221
blanchet@48975
  1222
lemma bsqr_Trans:
blanchet@48975
  1223
assumes "Well_order r"
blanchet@48975
  1224
shows "trans (bsqr r)"
blanchet@48975
  1225
proof(unfold trans_def, auto)
blanchet@48975
  1226
  (* Preliminary facts *)
blanchet@48975
  1227
  have Well: "wo_rel r" using assms wo_rel_def by auto
blanchet@48975
  1228
  hence Trans: "trans r" using wo_rel.TRANS by auto
blanchet@48975
  1229
  have Anti: "antisym r" using wo_rel.ANTISYM Well by auto
blanchet@48975
  1230
  hence TransS: "trans(r - Id)" using Trans by (simp add: trans_diff_Id)
blanchet@48975
  1231
  (* Main proof *)
blanchet@48975
  1232
  fix a1 a2 b1 b2 c1 c2
blanchet@48975
  1233
  assume *: "((a1,a2),(b1,b2)) \<in> bsqr r" and **: "((b1,b2),(c1,c2)) \<in> bsqr r"
blanchet@48975
  1234
  hence 0: "{a1,a2,b1,b2,c1,c2} \<le> Field r" unfolding bsqr_def by auto
blanchet@48975
  1235
  have 1: "a1 = b1 \<and> a2 = b2 \<or> (wo_rel.max2 r a1 a2, wo_rel.max2 r b1 b2) \<in> r - Id \<or>
blanchet@48975
  1236
           wo_rel.max2 r a1 a2 = wo_rel.max2 r b1 b2 \<and> (a1,b1) \<in> r - Id \<or>
blanchet@48975
  1237
           wo_rel.max2 r a1 a2 = wo_rel.max2 r b1 b2 \<and> a1 = b1 \<and> (a2,b2) \<in> r - Id"
blanchet@48975
  1238
  using * unfolding bsqr_def by auto
blanchet@48975
  1239
  have 2: "b1 = c1 \<and> b2 = c2 \<or> (wo_rel.max2 r b1 b2, wo_rel.max2 r c1 c2) \<in> r - Id \<or>
blanchet@48975
  1240
           wo_rel.max2 r b1 b2 = wo_rel.max2 r c1 c2 \<and> (b1,c1) \<in> r - Id \<or>
blanchet@48975
  1241
           wo_rel.max2 r b1 b2 = wo_rel.max2 r c1 c2 \<and> b1 = c1 \<and> (b2,c2) \<in> r - Id"
blanchet@48975
  1242
  using ** unfolding bsqr_def by auto
blanchet@48975
  1243
  show "((a1,a2),(c1,c2)) \<in> bsqr r"
blanchet@48975
  1244
  proof-
blanchet@48975
  1245
    {assume Case1: "a1 = b1 \<and> a2 = b2"
blanchet@48975
  1246
     hence ?thesis using ** by simp
blanchet@48975
  1247
    }
blanchet@48975
  1248
    moreover
blanchet@48975
  1249
    {assume Case2: "(wo_rel.max2 r a1 a2, wo_rel.max2 r b1 b2) \<in> r - Id"
blanchet@48975
  1250
     {assume Case21: "b1 = c1 \<and> b2 = c2"
blanchet@48975
  1251
      hence ?thesis using * by simp
blanchet@48975
  1252
     }
blanchet@48975
  1253
     moreover
blanchet@48975
  1254
     {assume Case22: "(wo_rel.max2 r b1 b2, wo_rel.max2 r c1 c2) \<in> r - Id"
blanchet@48975
  1255
      hence "(wo_rel.max2 r a1 a2, wo_rel.max2 r c1 c2) \<in> r - Id"
blanchet@48975
  1256
      using Case2 TransS trans_def[of "r - Id"] by blast
blanchet@48975
  1257
      hence ?thesis using 0 unfolding bsqr_def by auto
blanchet@48975
  1258
     }
blanchet@48975
  1259
     moreover
blanchet@48975
  1260
     {assume Case23_4: "wo_rel.max2 r b1 b2 = wo_rel.max2 r c1 c2"
blanchet@48975
  1261
      hence ?thesis using Case2 0 unfolding bsqr_def by auto
blanchet@48975
  1262
     }
blanchet@48975
  1263
     ultimately have ?thesis using 0 2 by auto
blanchet@48975
  1264
    }
blanchet@48975
  1265
    moreover
blanchet@48975
  1266
    {assume Case3: "wo_rel.max2 r a1 a2 = wo_rel.max2 r b1 b2 \<and> (a1,b1) \<in> r - Id"
blanchet@48975
  1267
     {assume Case31: "b1 = c1 \<and> b2 = c2"
blanchet@48975
  1268
      hence ?thesis using * by simp
blanchet@48975
  1269
     }
blanchet@48975
  1270
     moreover
blanchet@48975
  1271
     {assume Case32: "(wo_rel.max2 r b1 b2, wo_rel.max2 r c1 c2) \<in> r - Id"
blanchet@48975
  1272
      hence ?thesis using Case3 0 unfolding bsqr_def by auto
blanchet@48975
  1273
     }
blanchet@48975
  1274
     moreover
blanchet@48975
  1275
     {assume Case33: "wo_rel.max2 r b1 b2 = wo_rel.max2 r c1 c2 \<and> (b1,c1) \<in> r - Id"
blanchet@48975
  1276
      hence "(a1,c1) \<in> r - Id"
blanchet@48975
  1277
      using Case3 TransS trans_def[of "r - Id"] by blast
blanchet@48975
  1278
      hence ?thesis using Case3 Case33 0 unfolding bsqr_def by auto
blanchet@48975
  1279
     }
blanchet@48975
  1280
     moreover
blanchet@48975
  1281
     {assume Case33: "wo_rel.max2 r b1 b2 = wo_rel.max2 r c1 c2 \<and> b1 = c1"
blanchet@48975
  1282
      hence ?thesis using Case3 0 unfolding bsqr_def by auto
blanchet@48975
  1283
     }
blanchet@48975
  1284
     ultimately have ?thesis using 0 2 by auto
blanchet@48975
  1285
    }
blanchet@48975
  1286
    moreover
blanchet@48975
  1287
    {assume Case4: "wo_rel.max2 r a1 a2 = wo_rel.max2 r b1 b2 \<and> a1 = b1 \<and> (a2,b2) \<in> r - Id"
blanchet@48975
  1288
     {assume Case41: "b1 = c1 \<and> b2 = c2"
blanchet@48975
  1289
      hence ?thesis using * by simp
blanchet@48975
  1290
     }
blanchet@48975
  1291
     moreover
blanchet@48975
  1292
     {assume Case42: "(wo_rel.max2 r b1 b2, wo_rel.max2 r c1 c2) \<in> r - Id"
blanchet@48975
  1293
      hence ?thesis using Case4 0 unfolding bsqr_def by auto
blanchet@48975
  1294
     }
blanchet@48975
  1295
     moreover
blanchet@48975
  1296
     {assume Case43: "wo_rel.max2 r b1 b2 = wo_rel.max2 r c1 c2 \<and> (b1,c1) \<in> r - Id"
blanchet@48975
  1297
      hence ?thesis using Case4 0 unfolding bsqr_def by auto
blanchet@48975
  1298
     }
blanchet@48975
  1299
     moreover
blanchet@48975
  1300
     {assume Case44: "wo_rel.max2 r b1 b2 = wo_rel.max2 r c1 c2 \<and> b1 = c1 \<and> (b2,c2) \<in> r - Id"
blanchet@48975
  1301
      hence "(a2,c2) \<in> r - Id"
blanchet@48975
  1302
      using Case4 TransS trans_def[of "r - Id"] by blast
blanchet@48975
  1303
      hence ?thesis using Case4 Case44 0 unfolding bsqr_def by auto
blanchet@48975
  1304
     }
blanchet@48975
  1305
     ultimately have ?thesis using 0 2 by auto
blanchet@48975
  1306
    }
blanchet@48975
  1307
    ultimately show ?thesis using 0 1 by auto
blanchet@48975
  1308
  qed
blanchet@48975
  1309
qed
blanchet@48975
  1310
blanchet@48975
  1311
blanchet@48975
  1312
lemma bsqr_antisym:
blanchet@48975
  1313
assumes "Well_order r"
blanchet@48975
  1314
shows "antisym (bsqr r)"
blanchet@48975
  1315
proof(unfold antisym_def, clarify)
blanchet@48975
  1316
  (* Preliminary facts *)
blanchet@48975
  1317
  have Well: "wo_rel r" using assms wo_rel_def by auto
blanchet@48975
  1318
  hence Trans: "trans r" using wo_rel.TRANS by auto
blanchet@48975
  1319
  have Anti: "antisym r" using wo_rel.ANTISYM Well by auto
blanchet@48975
  1320
  hence TransS: "trans(r - Id)" using Trans by (simp add: trans_diff_Id)
blanchet@48975
  1321
  hence IrrS: "\<forall>a b. \<not>((a,b) \<in> r - Id \<and> (b,a) \<in> r - Id)"
blanchet@48975
  1322
  using Anti trans_def[of "r - Id"] antisym_def[of "r - Id"] by blast
blanchet@48975
  1323
  (* Main proof *)
blanchet@48975
  1324
  fix a1 a2 b1 b2
blanchet@48975
  1325
  assume *: "((a1,a2),(b1,b2)) \<in> bsqr r" and **: "((b1,b2),(a1,a2)) \<in> bsqr r"
blanchet@48975
  1326
  hence 0: "{a1,a2,b1,b2} \<le> Field r" unfolding bsqr_def by auto
blanchet@48975
  1327
  have 1: "a1 = b1 \<and> a2 = b2 \<or> (wo_rel.max2 r a1 a2, wo_rel.max2 r b1 b2) \<in> r - Id \<or>
blanchet@48975
  1328
           wo_rel.max2 r a1 a2 = wo_rel.max2 r b1 b2 \<and> (a1,b1) \<in> r - Id \<or>
blanchet@48975
  1329
           wo_rel.max2 r a1 a2 = wo_rel.max2 r b1 b2 \<and> a1 = b1 \<and> (a2,b2) \<in> r - Id"
blanchet@48975
  1330
  using * unfolding bsqr_def by auto
blanchet@48975
  1331
  have 2: "b1 = a1 \<and> b2 = a2 \<or> (wo_rel.max2 r b1 b2, wo_rel.max2 r a1 a2) \<in> r - Id \<or>
blanchet@48975
  1332
           wo_rel.max2 r b1 b2 = wo_rel.max2 r a1 a2 \<and> (b1,a1) \<in> r - Id \<or>
blanchet@48975
  1333
           wo_rel.max2 r b1 b2 = wo_rel.max2 r a1 a2 \<and> b1 = a1 \<and> (b2,a2) \<in> r - Id"
blanchet@48975
  1334
  using ** unfolding bsqr_def by auto
blanchet@48975
  1335
  show "a1 = b1 \<and> a2 = b2"
blanchet@48975
  1336
  proof-
blanchet@48975
  1337
    {assume Case1: "(wo_rel.max2 r a1 a2, wo_rel.max2 r b1 b2) \<in> r - Id"
blanchet@48975
  1338
     {assume Case11: "(wo_rel.max2 r b1 b2, wo_rel.max2 r a1 a2) \<in> r - Id"
blanchet@48975
  1339
      hence False using Case1 IrrS by blast
blanchet@48975
  1340
     }
blanchet@48975
  1341
     moreover
blanchet@48975
  1342
     {assume Case12_3: "wo_rel.max2 r b1 b2 = wo_rel.max2 r a1 a2"
blanchet@48975
  1343
      hence False using Case1 by auto
blanchet@48975
  1344
     }
blanchet@48975
  1345
     ultimately have ?thesis using 0 2 by auto
blanchet@48975
  1346
    }
blanchet@48975
  1347
    moreover
blanchet@48975
  1348
    {assume Case2: "wo_rel.max2 r a1 a2 = wo_rel.max2 r b1 b2 \<and> (a1,b1) \<in> r - Id"
blanchet@48975
  1349
     {assume Case21: "(wo_rel.max2 r b1 b2, wo_rel.max2 r a1 a2) \<in> r - Id"
blanchet@48975
  1350
       hence False using Case2 by auto
blanchet@48975
  1351
     }
blanchet@48975
  1352
     moreover
blanchet@48975
  1353
     {assume Case22: "(b1,a1) \<in> r - Id"
blanchet@48975
  1354
      hence False using Case2 IrrS by blast
blanchet@48975
  1355
     }
blanchet@48975
  1356
     moreover
blanchet@48975
  1357
     {assume Case23: "b1 = a1"
blanchet@48975
  1358
      hence False using Case2 by auto
blanchet@48975
  1359
     }
blanchet@48975
  1360
     ultimately have ?thesis using 0 2 by auto
blanchet@48975
  1361
    }
blanchet@48975
  1362
    moreover
blanchet@48975
  1363
    {assume Case3: "wo_rel.max2 r a1 a2 = wo_rel.max2 r b1 b2 \<and> a1 = b1 \<and> (a2,b2) \<in> r - Id"
blanchet@48975
  1364
     moreover
blanchet@48975
  1365
     {assume Case31: "(wo_rel.max2 r b1 b2, wo_rel.max2 r a1 a2) \<in> r - Id"
blanchet@48975
  1366
      hence False using Case3 by auto
blanchet@48975
  1367
     }
blanchet@48975
  1368
     moreover
blanchet@48975
  1369
     {assume Case32: "(b1,a1) \<in> r - Id"
blanchet@48975
  1370
      hence False using Case3 by auto
blanchet@48975
  1371
     }
blanchet@48975
  1372
     moreover
blanchet@48975
  1373
     {assume Case33: "(b2,a2) \<in> r - Id"
blanchet@48975
  1374
      hence False using Case3 IrrS by blast
blanchet@48975
  1375
     }
blanchet@48975
  1376
     ultimately have ?thesis using 0 2 by auto
blanchet@48975
  1377
    }
blanchet@48975
  1378
    ultimately show ?thesis using 0 1 by blast
blanchet@48975
  1379
  qed
blanchet@48975
  1380
qed
blanchet@48975
  1381
blanchet@48975
  1382
blanchet@48975
  1383
lemma bsqr_Total:
blanchet@48975
  1384
assumes "Well_order r"
blanchet@48975
  1385
shows "Total(bsqr r)"
blanchet@48975
  1386
proof-
blanchet@48975
  1387
  (* Preliminary facts *)
blanchet@48975
  1388
  have Well: "wo_rel r" using assms wo_rel_def by auto
blanchet@48975
  1389
  hence Total: "\<forall>a \<in> Field r. \<forall>b \<in> Field r. (a,b) \<in> r \<or> (b,a) \<in> r"
blanchet@48975
  1390
  using wo_rel.TOTALS by auto
blanchet@48975
  1391
  (* Main proof *)
blanchet@48975
  1392
  {fix a1 a2 b1 b2 assume "{(a1,a2), (b1,b2)} \<le> Field(bsqr r)"
blanchet@48975
  1393
   hence 0: "a1 \<in> Field r \<and> a2 \<in> Field r \<and> b1 \<in> Field r \<and> b2 \<in> Field r"
blanchet@48975
  1394
   using Field_bsqr by blast
blanchet@48975
  1395
   have "((a1,a2) = (b1,b2) \<or> ((a1,a2),(b1,b2)) \<in> bsqr r \<or> ((b1,b2),(a1,a2)) \<in> bsqr r)"
blanchet@48975
  1396
   proof(rule wo_rel.cases_Total[of r a1 a2], clarsimp simp add: Well, simp add: 0)
blanchet@48975
  1397
       (* Why didn't clarsimp simp add: Well 0 do the same job? *)
blanchet@48975
  1398
     assume Case1: "(a1,a2) \<in> r"
blanchet@48975
  1399
     hence 1: "wo_rel.max2 r a1 a2 = a2"
blanchet@48975
  1400
     using Well 0 by (simp add: wo_rel.max2_equals2)
blanchet@48975
  1401
     show ?thesis
blanchet@48975
  1402
     proof(rule wo_rel.cases_Total[of r b1 b2], clarsimp simp add: Well, simp add: 0)
blanchet@48975
  1403
       assume Case11: "(b1,b2) \<in> r"
blanchet@48975
  1404
       hence 2: "wo_rel.max2 r b1 b2 = b2"
blanchet@48975
  1405
       using Well 0 by (simp add: wo_rel.max2_equals2)
blanchet@48975
  1406
       show ?thesis
blanchet@48975
  1407
       proof(rule wo_rel.cases_Total3[of r a2 b2], clarsimp simp add: Well, simp add: 0)
blanchet@48975
  1408
         assume Case111: "(a2,b2) \<in> r - Id \<or> (b2,a2) \<in> r - Id"
blanchet@48975
  1409
         thus ?thesis using 0 1 2 unfolding bsqr_def by auto
blanchet@48975
  1410
       next
blanchet@48975
  1411
         assume Case112: "a2 = b2"
blanchet@48975
  1412
         show ?thesis
blanchet@48975
  1413
         proof(rule wo_rel.cases_Total3[of r a1 b1], clarsimp simp add: Well, simp add: 0)
blanchet@48975
  1414
           assume Case1121: "(a1,b1) \<in> r - Id \<or> (b1,a1) \<in> r - Id"
blanchet@48975
  1415
           thus ?thesis using 0 1 2 Case112 unfolding bsqr_def by auto
blanchet@48975
  1416
         next
blanchet@48975
  1417
           assume Case1122: "a1 = b1"
blanchet@48975
  1418
           thus ?thesis using Case112 by auto
blanchet@48975
  1419
         qed
blanchet@48975
  1420
       qed
blanchet@48975
  1421
     next
blanchet@48975
  1422
       assume Case12: "(b2,b1) \<in> r"
blanchet@48975
  1423
       hence 3: "wo_rel.max2 r b1 b2 = b1" using Well 0 by (simp add: wo_rel.max2_equals1)
blanchet@48975
  1424
       show ?thesis
blanchet@48975
  1425
       proof(rule wo_rel.cases_Total3[of r a2 b1], clarsimp simp add: Well, simp add: 0)
blanchet@48975
  1426
         assume Case121: "(a2,b1) \<in> r - Id \<or> (b1,a2) \<in> r - Id"
blanchet@48975
  1427
         thus ?thesis using 0 1 3 unfolding bsqr_def by auto
blanchet@48975
  1428
       next
blanchet@48975
  1429
         assume Case122: "a2 = b1"
blanchet@48975
  1430
         show ?thesis
blanchet@48975
  1431
         proof(rule wo_rel.cases_Total3[of r a1 b1], clarsimp simp add: Well, simp add: 0)
blanchet@48975
  1432
           assume Case1221: "(a1,b1) \<in> r - Id \<or> (b1,a1) \<in> r - Id"
blanchet@48975
  1433
           thus ?thesis using 0 1 3 Case122 unfolding bsqr_def by auto
blanchet@48975
  1434
         next
blanchet@48975
  1435
           assume Case1222: "a1 = b1"
blanchet@48975
  1436
           show ?thesis
blanchet@48975
  1437
           proof(rule wo_rel.cases_Total3[of r a2 b2], clarsimp simp add: Well, simp add: 0)
blanchet@48975
  1438
             assume Case12221: "(a2,b2) \<in> r - Id \<or> (b2,a2) \<in> r - Id"
blanchet@48975
  1439
             thus ?thesis using 0 1 3 Case122 Case1222 unfolding bsqr_def by auto
blanchet@48975
  1440
           next
blanchet@48975
  1441
             assume Case12222: "a2 = b2"
blanchet@48975
  1442
             thus ?thesis using Case122 Case1222 by auto
blanchet@48975
  1443
           qed
blanchet@48975
  1444
         qed
blanchet@48975
  1445
       qed
blanchet@48975
  1446
     qed
blanchet@48975
  1447
   next
blanchet@48975
  1448
     assume Case2: "(a2,a1) \<in> r"
blanchet@48975
  1449
     hence 1: "wo_rel.max2 r a1 a2 = a1" using Well 0 by (simp add: wo_rel.max2_equals1)
blanchet@48975
  1450
     show ?thesis
blanchet@48975
  1451
     proof(rule wo_rel.cases_Total[of r b1 b2], clarsimp simp add: Well, simp add: 0)
blanchet@48975
  1452
       assume Case21: "(b1,b2) \<in> r"
blanchet@48975
  1453
       hence 2: "wo_rel.max2 r b1 b2 = b2" using Well 0 by (simp add: wo_rel.max2_equals2)
blanchet@48975
  1454
       show ?thesis
blanchet@48975
  1455
       proof(rule wo_rel.cases_Total3[of r a1 b2], clarsimp simp add: Well, simp add: 0)
blanchet@48975
  1456
         assume Case211: "(a1,b2) \<in> r - Id \<or> (b2,a1) \<in> r - Id"
blanchet@48975
  1457
         thus ?thesis using 0 1 2 unfolding bsqr_def by auto
blanchet@48975
  1458
       next
blanchet@48975
  1459
         assume Case212: "a1 = b2"
blanchet@48975
  1460
         show ?thesis
blanchet@48975
  1461
         proof(rule wo_rel.cases_Total3[of r a1 b1], clarsimp simp add: Well, simp add: 0)
blanchet@48975
  1462
           assume Case2121: "(a1,b1) \<in> r - Id \<or> (b1,a1) \<in> r - Id"
blanchet@48975
  1463
           thus ?thesis using 0 1 2 Case212 unfolding bsqr_def by auto
blanchet@48975
  1464
         next
blanchet@48975
  1465
           assume Case2122: "a1 = b1"
blanchet@48975
  1466
           show ?thesis
blanchet@48975
  1467
           proof(rule wo_rel.cases_Total3[of r a2 b2], clarsimp simp add: Well, simp add: 0)
blanchet@48975
  1468
             assume Case21221: "(a2,b2) \<in> r - Id \<or> (b2,a2) \<in> r - Id"
blanchet@48975
  1469
             thus ?thesis using 0 1 2 Case212 Case2122 unfolding bsqr_def by auto
blanchet@48975
  1470
           next
blanchet@48975
  1471
             assume Case21222: "a2 = b2"
blanchet@48975
  1472
             thus ?thesis using Case2122 Case212 by auto
blanchet@48975
  1473
           qed
blanchet@48975
  1474
         qed
blanchet@48975
  1475
       qed
blanchet@48975
  1476
     next
blanchet@48975
  1477
       assume Case22: "(b2,b1) \<in> r"
blanchet@48975
  1478
       hence 3: "wo_rel.max2 r b1 b2 = b1"  using Well 0 by (simp add: wo_rel.max2_equals1)
blanchet@48975
  1479
       show ?thesis
blanchet@48975
  1480
       proof(rule wo_rel.cases_Total3[of r a1 b1], clarsimp simp add: Well, simp add: 0)
blanchet@48975
  1481
         assume Case221: "(a1,b1) \<in> r - Id \<or> (b1,a1) \<in> r - Id"
blanchet@48975
  1482
         thus ?thesis using 0 1 3 unfolding bsqr_def by auto
blanchet@48975
  1483
       next
blanchet@48975
  1484
         assume Case222: "a1 = b1"
blanchet@48975
  1485
         show ?thesis
blanchet@48975
  1486
         proof(rule wo_rel.cases_Total3[of r a2 b2], clarsimp simp add: Well, simp add: 0)
blanchet@48975
  1487
           assume Case2221: "(a2,b2) \<in> r - Id \<or> (b2,a2) \<in> r - Id"
blanchet@48975
  1488
           thus ?thesis using 0 1 3 Case222 unfolding bsqr_def by auto
blanchet@48975
  1489
         next
blanchet@48975
  1490
           assume Case2222: "a2 = b2"
blanchet@48975
  1491
           thus ?thesis using Case222 by auto
blanchet@48975
  1492
         qed
blanchet@48975
  1493
       qed
blanchet@48975
  1494
     qed
blanchet@48975
  1495
   qed
blanchet@48975
  1496
  }
blanchet@48975
  1497
  thus ?thesis unfolding total_on_def by fast
blanchet@48975
  1498
qed
blanchet@48975
  1499
blanchet@48975
  1500
blanchet@48975
  1501
lemma bsqr_Linear_order:
blanchet@48975
  1502
assumes "Well_order r"
blanchet@48975
  1503
shows "Linear_order(bsqr r)"
blanchet@48975
  1504
unfolding order_on_defs
blanchet@48975
  1505
using assms bsqr_Refl bsqr_Trans bsqr_antisym bsqr_Total by blast
blanchet@48975
  1506
blanchet@48975
  1507
blanchet@48975
  1508
lemma bsqr_Well_order:
blanchet@48975
  1509
assumes "Well_order r"
blanchet@48975
  1510
shows "Well_order(bsqr r)"
blanchet@48975
  1511
using assms
blanchet@48975
  1512
proof(simp add: bsqr_Linear_order Linear_order_Well_order_iff, intro allI impI)
blanchet@48975
  1513
  have 0: "\<forall>A \<le> Field r. A \<noteq> {} \<longrightarrow> (\<exists>a \<in> A. \<forall>a' \<in> A. (a,a') \<in> r)"
blanchet@48975
  1514
  using assms well_order_on_def Linear_order_Well_order_iff by blast
blanchet@48975
  1515
  fix D assume *: "D \<le> Field (bsqr r)" and **: "D \<noteq> {}"
blanchet@48975
  1516
  hence 1: "D \<le> Field r \<times> Field r" unfolding Field_bsqr by simp
blanchet@48975
  1517
  (*  *)
blanchet@48975
  1518
  obtain M where M_def: "M = {wo_rel.max2 r a1 a2| a1 a2. (a1,a2) \<in> D}" by blast
blanchet@48975
  1519
  have "M \<noteq> {}" using 1 M_def ** by auto
blanchet@48975
  1520
  moreover
blanchet@48975
  1521
  have "M \<le> Field r" unfolding M_def
blanchet@48975
  1522
  using 1 assms wo_rel_def[of r] wo_rel.max2_among[of r] by fastforce
blanchet@48975
  1523
  ultimately obtain m where m_min: "m \<in> M \<and> (\<forall>a \<in> M. (m,a) \<in> r)"
blanchet@48975
  1524
  using 0 by blast
blanchet@48975
  1525
  (*  *)
blanchet@48975
  1526
  obtain A1 where A1_def: "A1 = {a1. \<exists>a2. (a1,a2) \<in> D \<and> wo_rel.max2 r a1 a2 = m}" by blast
blanchet@48975
  1527
  have "A1 \<le> Field r" unfolding A1_def using 1 by auto
blanchet@48975
  1528
  moreover have "A1 \<noteq> {}" unfolding A1_def using m_min unfolding M_def by blast
blanchet@48975
  1529
  ultimately obtain a1 where a1_min: "a1 \<in> A1 \<and> (\<forall>a \<in> A1. (a1,a) \<in> r)"
blanchet@48975
  1530
  using 0 by blast
blanchet@48975
  1531
  (*  *)
blanchet@48975
  1532
  obtain A2 where A2_def: "A2 = {a2. (a1,a2) \<in> D \<and> wo_rel.max2 r a1 a2 = m}" by blast
blanchet@48975
  1533
  have "A2 \<le> Field r" unfolding A2_def using 1 by auto
blanchet@48975
  1534
  moreover have "A2 \<noteq> {}" unfolding A2_def
blanchet@48975
  1535
  using m_min a1_min unfolding A1_def M_def by blast
blanchet@48975
  1536
  ultimately obtain a2 where a2_min: "a2 \<in> A2 \<and> (\<forall>a \<in> A2. (a2,a) \<in> r)"
blanchet@48975
  1537
  using 0 by blast
blanchet@48975
  1538
  (*   *)
blanchet@48975
  1539
  have 2: "wo_rel.max2 r a1 a2 = m"
blanchet@48975
  1540
  using a1_min a2_min unfolding A1_def A2_def by auto
blanchet@48975
  1541
  have 3: "(a1,a2) \<in> D" using a2_min unfolding A2_def by auto
blanchet@48975
  1542
  (*  *)
blanchet@48975
  1543
  moreover
blanchet@48975
  1544
  {fix b1 b2 assume ***: "(b1,b2) \<in> D"
blanchet@48975
  1545
   hence 4: "{a1,a2,b1,b2} \<le> Field r" using 1 3 by blast
blanchet@48975
  1546
   have 5: "(wo_rel.max2 r a1 a2, wo_rel.max2 r b1 b2) \<in> r"
blanchet@48975
  1547
   using *** a1_min a2_min m_min unfolding A1_def A2_def M_def by auto
blanchet@48975
  1548
   have "((a1,a2),(b1,b2)) \<in> bsqr r"
blanchet@48975
  1549
   proof(cases "wo_rel.max2 r a1 a2 = wo_rel.max2 r b1 b2")
blanchet@48975
  1550
     assume Case1: "wo_rel.max2 r a1 a2 \<noteq> wo_rel.max2 r b1 b2"
blanchet@48975
  1551
     thus ?thesis unfolding bsqr_def using 4 5 by auto
blanchet@48975
  1552
   next
blanchet@48975
  1553
     assume Case2: "wo_rel.max2 r a1 a2 = wo_rel.max2 r b1 b2"
blanchet@48975
  1554
     hence "b1 \<in> A1" unfolding A1_def using 2 *** by auto
blanchet@48975
  1555
     hence 6: "(a1,b1) \<in> r" using a1_min by auto
blanchet@48975
  1556
     show ?thesis
blanchet@48975
  1557
     proof(cases "a1 = b1")
blanchet@48975
  1558
       assume Case21: "a1 \<noteq> b1"
blanchet@48975
  1559
       thus ?thesis unfolding bsqr_def using 4 Case2 6 by auto
blanchet@48975
  1560
     next
blanchet@48975
  1561
       assume Case22: "a1 = b1"
blanchet@48975
  1562
       hence "b2 \<in> A2" unfolding A2_def using 2 *** Case2 by auto
blanchet@48975
  1563
       hence 7: "(a2,b2) \<in> r" using a2_min by auto
blanchet@48975
  1564
       thus ?thesis unfolding bsqr_def using 4 7 Case2 Case22 by auto
blanchet@48975
  1565
     qed
blanchet@48975
  1566
   qed
blanchet@48975
  1567
  }
blanchet@48975
  1568
  (*  *)
blanchet@48975
  1569
  ultimately show "\<exists>d \<in> D. \<forall>d' \<in> D. (d,d') \<in> bsqr r" by fastforce
blanchet@48975
  1570
qed
blanchet@48975
  1571
blanchet@48975
  1572
blanchet@48975
  1573
lemma bsqr_max2:
blanchet@48975
  1574
assumes WELL: "Well_order r" and LEQ: "((a1,a2),(b1,b2)) \<in> bsqr r"
blanchet@48975
  1575
shows "(wo_rel.max2 r a1 a2, wo_rel.max2 r b1 b2) \<in> r"
blanchet@48975
  1576
proof-
blanchet@48975
  1577
  have "{(a1,a2),(b1,b2)} \<le> Field(bsqr r)"
blanchet@48975
  1578
  using LEQ unfolding Field_def by auto
blanchet@48975
  1579
  hence "{a1,a2,b1,b2} \<le> Field r" unfolding Field_bsqr by auto
blanchet@48975
  1580
  hence "{wo_rel.max2 r a1 a2, wo_rel.max2 r b1 b2} \<le> Field r"
blanchet@48975
  1581
  using WELL wo_rel_def[of r] wo_rel.max2_among[of r] by fastforce
blanchet@48975
  1582
  moreover have "(wo_rel.max2 r a1 a2, wo_rel.max2 r b1 b2) \<in> r \<or> wo_rel.max2 r a1 a2 = wo_rel.max2 r b1 b2"
blanchet@48975
  1583
  using LEQ unfolding bsqr_def by auto
blanchet@48975
  1584
  ultimately show ?thesis using WELL unfolding order_on_defs refl_on_def by auto
blanchet@48975
  1585
qed
blanchet@48975
  1586
blanchet@48975
  1587
blanchet@48975
  1588
lemma bsqr_ofilter:
blanchet@48975
  1589
assumes WELL: "Well_order r" and
blanchet@48975
  1590
        OF: "wo_rel.ofilter (bsqr r) D" and SUB: "D < Field r \<times> Field r" and
blanchet@48975
  1591
        NE: "\<not> (\<exists>a. Field r = rel.under r a)"
blanchet@48975
  1592
shows "\<exists>A. wo_rel.ofilter r A \<and> A < Field r \<and> D \<le> A \<times> A"
blanchet@48975
  1593
proof-
blanchet@48975
  1594
  let ?r' = "bsqr r"
blanchet@48975
  1595
  have Well: "wo_rel r" using WELL wo_rel_def by blast
blanchet@48975
  1596
  hence Trans: "trans r" using wo_rel.TRANS by blast
blanchet@48975
  1597
  have Well': "Well_order ?r' \<and> wo_rel ?r'"
blanchet@48975
  1598
  using WELL bsqr_Well_order wo_rel_def by blast
blanchet@48975
  1599
  (*  *)
blanchet@48975
  1600
  have "D < Field ?r'" unfolding Field_bsqr using SUB .
blanchet@48975
  1601
  with OF obtain a1 and a2 where
blanchet@48975
  1602
  "(a1,a2) \<in> Field ?r'" and 1: "D = rel.underS ?r' (a1,a2)"
blanchet@48975
  1603
  using Well' wo_rel.ofilter_underS_Field[of ?r' D] by auto
blanchet@48975
  1604
  hence 2: "{a1,a2} \<le> Field r" unfolding Field_bsqr by auto
blanchet@48975
  1605
  let ?m = "wo_rel.max2 r a1 a2"
blanchet@48975
  1606
  have "D \<le> (rel.under r ?m) \<times> (rel.under r ?m)"
blanchet@48975
  1607
  proof(unfold 1)
blanchet@48975
  1608
    {fix b1 b2
blanchet@48975
  1609
     let ?n = "wo_rel.max2 r b1 b2"
blanchet@48975
  1610
     assume "(b1,b2) \<in> rel.underS ?r' (a1,a2)"
blanchet@48975
  1611
     hence 3: "((b1,b2),(a1,a2)) \<in> ?r'"
blanchet@48975
  1612
     unfolding rel.underS_def by blast
blanchet@48975
  1613
     hence "(?n,?m) \<in> r" using WELL by (simp add: bsqr_max2)
blanchet@48975
  1614
     moreover
blanchet@48975
  1615
     {have "(b1,b2) \<in> Field ?r'" using 3 unfolding Field_def by auto
blanchet@48975
  1616
      hence "{b1,b2} \<le> Field r" unfolding Field_bsqr by auto
blanchet@48975
  1617
      hence "(b1,?n) \<in> r \<and> (b2,?n) \<in> r"
blanchet@48975
  1618
      using Well by (simp add: wo_rel.max2_greater)
blanchet@48975
  1619
     }
blanchet@48975
  1620
     ultimately have "(b1,?m) \<in> r \<and> (b2,?m) \<in> r"
blanchet@48975
  1621
     using Trans trans_def[of r] by blast
blanchet@48975
  1622
     hence "(b1,b2) \<in> (rel.under r ?m) \<times> (rel.under r ?m)" unfolding rel.under_def by simp}
blanchet@48975
  1623
     thus "rel.underS ?r' (a1,a2) \<le> (rel.under r ?m) \<times> (rel.under r ?m)" by auto
blanchet@48975
  1624
  qed
blanchet@48975
  1625
  moreover have "wo_rel.ofilter r (rel.under r ?m)"
blanchet@48975
  1626
  using Well by (simp add: wo_rel.under_ofilter)
blanchet@48975
  1627
  moreover have "rel.under r ?m < Field r"
blanchet@48975
  1628
  using NE rel.under_Field[of r ?m] by blast
blanchet@48975
  1629
  ultimately show ?thesis by blast
blanchet@48975
  1630
qed
blanchet@48975
  1631
blanchet@48975
  1632
blanchet@48975
  1633
end