src/HOL/BNF_Wellorder_Constructions.thy
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Mon Jun 05 15:59:41 2017 +0200 (2017-06-05)
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(*  Title:      HOL/BNF_Wellorder_Constructions.thy
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    Author:     Andrei Popescu, TU Muenchen
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    Copyright   2012
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Constructions on wellorders as needed by bounded natural functors.
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*)
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section \<open>Constructions on Wellorders as Needed by Bounded Natural Functors\<close>
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theory BNF_Wellorder_Constructions
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imports BNF_Wellorder_Embedding
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begin
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text \<open>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 \<open>ordLeq\<close>, of being embedded (abbreviated \<open>\<le>o\<close>),
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\<open>ordLess\<close>, of being strictly embedded (abbreviated \<open><o\<close>), and
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\<open>ordIso\<close>, of being isomorphic (abbreviated \<open>=o\<close>).  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 \<open><o\<close> is well-founded.\<close>
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subsection \<open>Restriction to a set\<close>
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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 linear_order_on_Restr:
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  "linear_order_on A r \<Longrightarrow> linear_order_on (A \<inter> above r x) (Restr r (above r x))"
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by(simp add: order_on_defs refl_on_def trans_def antisym_def total_on_def)(safe; blast)
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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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unfolding refl_on_def Field_def by blast
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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 \<open>Order filters versus restrictions and embeddings\<close>
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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 "under (Restr r A) a = under r a"
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using assms wo_rel_def
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proof(auto simp add: wo_rel.ofilter_def under_def)
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  fix b assume *: "a \<in> A" and "(b,a) \<in> r"
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  hence "b \<in> under r a \<and> a \<in> Field r"
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  unfolding 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 (under (Restr r A) a) (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 (under (Restr r A) a) (under r a)"
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   using * unfolding embed_def by auto
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   hence "under r a \<le> 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: under_Field)
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   also have "\<dots> \<le> A" by (simp add: Field_Restr_subset)
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   finally have "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 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 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 \<open>The strict inclusion on proper ofilters is well-founded\<close>
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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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  proof(unfold compat_def ofilterIncl_def,
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        intro allI impI, simp, elim conjE)
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    fix A B
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    assume *: "wo_rel.ofilter r A" "A \<noteq> Field r" and
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           **: "wo_rel.ofilter r B" "B \<noteq> Field r" and ***: "A < B"
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    then obtain a and b where 0: "a \<in> Field r \<and> b \<in> Field r" and
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   292
                         1: "A = underS r a \<and> B = underS r b"
blanchet@48975
   293
    using Well by (auto simp add: wo_rel.ofilter_underS_Field)
blanchet@48975
   294
    hence "a \<noteq> b" using *** by auto
blanchet@48975
   295
    moreover
blanchet@48975
   296
    have "(a,b) \<in> r" using 0 1 Lo ***
blanchet@55023
   297
    by (auto simp add: underS_incl_iff)
blanchet@48975
   298
    moreover
blanchet@48975
   299
    have "a = wo_rel.suc r A \<and> b = wo_rel.suc r B"
blanchet@48975
   300
    using Well 0 1 by (simp add: wo_rel.suc_underS)
blanchet@48975
   301
    ultimately
blanchet@48975
   302
    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
   303
    by simp
blanchet@48975
   304
  qed
blanchet@48975
   305
  ultimately show "wf (ofilterIncl r)" by (simp add: compat_wf)
blanchet@48975
   306
qed
blanchet@48975
   307
blanchet@48975
   308
wenzelm@60758
   309
subsection \<open>Ordering the well-orders by existence of embeddings\<close>
blanchet@48975
   310
wenzelm@60758
   311
text \<open>We define three relations between well-orders:
blanchet@48975
   312
\begin{itemize}
wenzelm@61799
   313
\item \<open>ordLeq\<close>, of being embedded (abbreviated \<open>\<le>o\<close>);
wenzelm@61799
   314
\item \<open>ordLess\<close>, of being strictly embedded (abbreviated \<open><o\<close>);
wenzelm@61799
   315
\item \<open>ordIso\<close>, of being isomorphic (abbreviated \<open>=o\<close>).
blanchet@48975
   316
\end{itemize}
blanchet@48975
   317
%
blanchet@48975
   318
The prefix "ord" and the index "o" in these names stand for "ordinal-like".
blanchet@48975
   319
These relations shall be proved to be inter-connected in a similar fashion as the trio
wenzelm@61799
   320
\<open>\<le>\<close>, \<open><\<close>, \<open>=\<close> associated to a total order on a set.
wenzelm@60758
   321
\<close>
blanchet@48975
   322
blanchet@48975
   323
definition ordLeq :: "('a rel * 'a' rel) set"
blanchet@48975
   324
where
blanchet@48975
   325
"ordLeq = {(r,r'). Well_order r \<and> Well_order r' \<and> (\<exists>f. embed r r' f)}"
blanchet@48975
   326
blanchet@48975
   327
abbreviation ordLeq2 :: "'a rel \<Rightarrow> 'a' rel \<Rightarrow> bool" (infix "<=o" 50)
blanchet@48975
   328
where "r <=o r' \<equiv> (r,r') \<in> ordLeq"
blanchet@48975
   329
blanchet@48975
   330
abbreviation ordLeq3 :: "'a rel \<Rightarrow> 'a' rel \<Rightarrow> bool" (infix "\<le>o" 50)
blanchet@48975
   331
where "r \<le>o r' \<equiv> r <=o r'"
blanchet@48975
   332
blanchet@48975
   333
definition ordLess :: "('a rel * 'a' rel) set"
blanchet@48975
   334
where
blanchet@48975
   335
"ordLess = {(r,r'). Well_order r \<and> Well_order r' \<and> (\<exists>f. embedS r r' f)}"
blanchet@48975
   336
blanchet@48975
   337
abbreviation ordLess2 :: "'a rel \<Rightarrow> 'a' rel \<Rightarrow> bool" (infix "<o" 50)
blanchet@48975
   338
where "r <o r' \<equiv> (r,r') \<in> ordLess"
blanchet@48975
   339
blanchet@48975
   340
definition ordIso :: "('a rel * 'a' rel) set"
blanchet@48975
   341
where
blanchet@48975
   342
"ordIso = {(r,r'). Well_order r \<and> Well_order r' \<and> (\<exists>f. iso r r' f)}"
blanchet@48975
   343
blanchet@48975
   344
abbreviation ordIso2 :: "'a rel \<Rightarrow> 'a' rel \<Rightarrow> bool" (infix "=o" 50)
blanchet@48975
   345
where "r =o r' \<equiv> (r,r') \<in> ordIso"
blanchet@48975
   346
blanchet@48975
   347
lemmas ordRels_def = ordLeq_def ordLess_def ordIso_def
blanchet@48975
   348
blanchet@48975
   349
lemma ordLeq_Well_order_simp:
blanchet@48975
   350
assumes "r \<le>o r'"
blanchet@48975
   351
shows "Well_order r \<and> Well_order r'"
blanchet@48975
   352
using assms unfolding ordLeq_def by simp
blanchet@48975
   353
wenzelm@61799
   354
text\<open>Notice that the relations \<open>\<le>o\<close>, \<open><o\<close>, \<open>=o\<close> connect well-orders
blanchet@48975
   355
on potentially {\em distinct} types. However, some of the lemmas below, including the next one,
wenzelm@61799
   356
restrict implicitly the type of these relations to \<open>(('a rel) * ('a rel)) set\<close> , i.e.,
wenzelm@61799
   357
to \<open>'a rel rel\<close>.\<close>
blanchet@48975
   358
blanchet@48975
   359
lemma ordLeq_reflexive:
blanchet@48975
   360
"Well_order r \<Longrightarrow> r \<le>o r"
blanchet@48975
   361
unfolding ordLeq_def using id_embed[of r] by blast
blanchet@48975
   362
blanchet@48975
   363
lemma ordLeq_transitive[trans]:
blanchet@48975
   364
assumes *: "r \<le>o r'" and **: "r' \<le>o r''"
blanchet@48975
   365
shows "r \<le>o r''"
blanchet@48975
   366
proof-
blanchet@48975
   367
  obtain f and f'
blanchet@48975
   368
  where 1: "Well_order r \<and> Well_order r' \<and> Well_order r''" and
blanchet@48975
   369
        "embed r r' f" and "embed r' r'' f'"
blanchet@48975
   370
  using * ** unfolding ordLeq_def by blast
blanchet@48975
   371
  hence "embed r r'' (f' o f)"
blanchet@48975
   372
  using comp_embed[of r r' f r'' f'] by auto
blanchet@48975
   373
  thus "r \<le>o r''" unfolding ordLeq_def using 1 by auto
blanchet@48975
   374
qed
blanchet@48975
   375
blanchet@48975
   376
lemma ordLeq_total:
blanchet@48975
   377
"\<lbrakk>Well_order r; Well_order r'\<rbrakk> \<Longrightarrow> r \<le>o r' \<or> r' \<le>o r"
blanchet@48975
   378
unfolding ordLeq_def using wellorders_totally_ordered by blast
blanchet@48975
   379
blanchet@48975
   380
lemma ordIso_reflexive:
blanchet@48975
   381
"Well_order r \<Longrightarrow> r =o r"
blanchet@48975
   382
unfolding ordIso_def using id_iso[of r] by blast
blanchet@48975
   383
blanchet@48975
   384
lemma ordIso_transitive[trans]:
blanchet@48975
   385
assumes *: "r =o r'" and **: "r' =o r''"
blanchet@48975
   386
shows "r =o r''"
blanchet@48975
   387
proof-
blanchet@48975
   388
  obtain f and f'
blanchet@48975
   389
  where 1: "Well_order r \<and> Well_order r' \<and> Well_order r''" and
blanchet@48975
   390
        "iso r r' f" and 3: "iso r' r'' f'"
blanchet@48975
   391
  using * ** unfolding ordIso_def by auto
blanchet@48975
   392
  hence "iso r r'' (f' o f)"
blanchet@48975
   393
  using comp_iso[of r r' f r'' f'] by auto
blanchet@48975
   394
  thus "r =o r''" unfolding ordIso_def using 1 by auto
blanchet@48975
   395
qed
blanchet@48975
   396
blanchet@48975
   397
lemma ordIso_symmetric:
blanchet@48975
   398
assumes *: "r =o r'"
blanchet@48975
   399
shows "r' =o r"
blanchet@48975
   400
proof-
blanchet@48975
   401
  obtain f where 1: "Well_order r \<and> Well_order r'" and
blanchet@48975
   402
                 2: "embed r r' f \<and> bij_betw f (Field r) (Field r')"
blanchet@48975
   403
  using * by (auto simp add: ordIso_def iso_def)
blanchet@48975
   404
  let ?f' = "inv_into (Field r) f"
blanchet@48975
   405
  have "embed r' r ?f' \<and> bij_betw ?f' (Field r') (Field r)"
blanchet@48975
   406
  using 1 2 by (simp add: bij_betw_inv_into inv_into_Field_embed_bij_betw)
blanchet@48975
   407
  thus "r' =o r" unfolding ordIso_def using 1 by (auto simp add: iso_def)
blanchet@48975
   408
qed
blanchet@48975
   409
blanchet@48975
   410
lemma ordLeq_ordLess_trans[trans]:
blanchet@48975
   411
assumes "r \<le>o r'" and " r' <o r''"
blanchet@48975
   412
shows "r <o r''"
blanchet@48975
   413
proof-
blanchet@48975
   414
  have "Well_order r \<and> Well_order r''"
blanchet@48975
   415
  using assms unfolding ordLeq_def ordLess_def by auto
blanchet@48975
   416
  thus ?thesis using assms unfolding ordLeq_def ordLess_def
blanchet@48975
   417
  using embed_comp_embedS by blast
blanchet@48975
   418
qed
blanchet@48975
   419
blanchet@48975
   420
lemma ordLess_ordLeq_trans[trans]:
blanchet@48975
   421
assumes "r <o r'" and " r' \<le>o r''"
blanchet@48975
   422
shows "r <o r''"
blanchet@48975
   423
proof-
blanchet@48975
   424
  have "Well_order r \<and> Well_order r''"
blanchet@48975
   425
  using assms unfolding ordLeq_def ordLess_def by auto
blanchet@48975
   426
  thus ?thesis using assms unfolding ordLeq_def ordLess_def
blanchet@48975
   427
  using embedS_comp_embed by blast
blanchet@48975
   428
qed
blanchet@48975
   429
blanchet@48975
   430
lemma ordLeq_ordIso_trans[trans]:
blanchet@48975
   431
assumes "r \<le>o r'" and " r' =o r''"
blanchet@48975
   432
shows "r \<le>o r''"
blanchet@48975
   433
proof-
blanchet@48975
   434
  have "Well_order r \<and> Well_order r''"
blanchet@48975
   435
  using assms unfolding ordLeq_def ordIso_def by auto
blanchet@48975
   436
  thus ?thesis using assms unfolding ordLeq_def ordIso_def
blanchet@48975
   437
  using embed_comp_iso by blast
blanchet@48975
   438
qed
blanchet@48975
   439
blanchet@48975
   440
lemma ordIso_ordLeq_trans[trans]:
blanchet@48975
   441
assumes "r =o r'" and " r' \<le>o r''"
blanchet@48975
   442
shows "r \<le>o r''"
blanchet@48975
   443
proof-
blanchet@48975
   444
  have "Well_order r \<and> Well_order r''"
blanchet@48975
   445
  using assms unfolding ordLeq_def ordIso_def by auto
blanchet@48975
   446
  thus ?thesis using assms unfolding ordLeq_def ordIso_def
blanchet@48975
   447
  using iso_comp_embed by blast
blanchet@48975
   448
qed
blanchet@48975
   449
blanchet@48975
   450
lemma ordLess_ordIso_trans[trans]:
blanchet@48975
   451
assumes "r <o r'" and " r' =o r''"
blanchet@48975
   452
shows "r <o r''"
blanchet@48975
   453
proof-
blanchet@48975
   454
  have "Well_order r \<and> Well_order r''"
blanchet@48975
   455
  using assms unfolding ordLess_def ordIso_def by auto
blanchet@48975
   456
  thus ?thesis using assms unfolding ordLess_def ordIso_def
blanchet@48975
   457
  using embedS_comp_iso by blast
blanchet@48975
   458
qed
blanchet@48975
   459
blanchet@48975
   460
lemma ordIso_ordLess_trans[trans]:
blanchet@48975
   461
assumes "r =o r'" and " r' <o r''"
blanchet@48975
   462
shows "r <o r''"
blanchet@48975
   463
proof-
blanchet@48975
   464
  have "Well_order r \<and> Well_order r''"
blanchet@48975
   465
  using assms unfolding ordLess_def ordIso_def by auto
blanchet@48975
   466
  thus ?thesis using assms unfolding ordLess_def ordIso_def
blanchet@48975
   467
  using iso_comp_embedS by blast
blanchet@48975
   468
qed
blanchet@48975
   469
blanchet@48975
   470
lemma ordLess_not_embed:
blanchet@48975
   471
assumes "r <o r'"
blanchet@48975
   472
shows "\<not>(\<exists>f'. embed r' r f')"
blanchet@48975
   473
proof-
blanchet@48975
   474
  obtain f where 1: "Well_order r \<and> Well_order r'" and 2: "embed r r' f" and
blanchet@48975
   475
                 3: " \<not> bij_betw f (Field r) (Field r')"
blanchet@48975
   476
  using assms unfolding ordLess_def by (auto simp add: embedS_def)
blanchet@48975
   477
  {fix f' assume *: "embed r' r f'"
blanchet@48975
   478
   hence "bij_betw f (Field r) (Field r')" using 1 2
blanchet@48975
   479
   by (simp add: embed_bothWays_Field_bij_betw)
blanchet@48975
   480
   with 3 have False by contradiction
blanchet@48975
   481
  }
blanchet@48975
   482
  thus ?thesis by blast
blanchet@48975
   483
qed
blanchet@48975
   484
blanchet@48975
   485
lemma ordLess_Field:
blanchet@48975
   486
assumes OL: "r1 <o r2" and EMB: "embed r1 r2 f"
blanchet@48975
   487
shows "\<not> (f`(Field r1) = Field r2)"
blanchet@48975
   488
proof-
blanchet@48975
   489
  let ?A1 = "Field r1"  let ?A2 = "Field r2"
blanchet@48975
   490
  obtain g where
blanchet@48975
   491
  0: "Well_order r1 \<and> Well_order r2" and
blanchet@48975
   492
  1: "embed r1 r2 g \<and> \<not>(bij_betw g ?A1 ?A2)"
blanchet@48975
   493
  using OL unfolding ordLess_def by (auto simp add: embedS_def)
blanchet@48975
   494
  hence "\<forall>a \<in> ?A1. f a = g a"
blanchet@48975
   495
  using 0 EMB embed_unique[of r1] by auto
blanchet@48975
   496
  hence "\<not>(bij_betw f ?A1 ?A2)"
blanchet@48975
   497
  using 1 bij_betw_cong[of ?A1] by blast
blanchet@48975
   498
  moreover
blanchet@48975
   499
  have "inj_on f ?A1" using EMB 0 by (simp add: embed_inj_on)
blanchet@48975
   500
  ultimately show ?thesis by (simp add: bij_betw_def)
blanchet@48975
   501
qed
blanchet@48975
   502
blanchet@48975
   503
lemma ordLess_iff:
blanchet@48975
   504
"r <o r' = (Well_order r \<and> Well_order r' \<and> \<not>(\<exists>f'. embed r' r f'))"
blanchet@48975
   505
proof
blanchet@48975
   506
  assume *: "r <o r'"
blanchet@48975
   507
  hence "\<not>(\<exists>f'. embed r' r f')" using ordLess_not_embed[of r r'] by simp
blanchet@48975
   508
  with * show "Well_order r \<and> Well_order r' \<and> \<not> (\<exists>f'. embed r' r f')"
blanchet@48975
   509
  unfolding ordLess_def by auto
blanchet@48975
   510
next
blanchet@48975
   511
  assume *: "Well_order r \<and> Well_order r' \<and> \<not> (\<exists>f'. embed r' r f')"
blanchet@48975
   512
  then obtain f where 1: "embed r r' f"
blanchet@48975
   513
  using wellorders_totally_ordered[of r r'] by blast
blanchet@48975
   514
  moreover
blanchet@48975
   515
  {assume "bij_betw f (Field r) (Field r')"
blanchet@48975
   516
   with * 1 have "embed r' r (inv_into (Field r) f) "
blanchet@48975
   517
   using inv_into_Field_embed_bij_betw[of r r' f] by auto
blanchet@48975
   518
   with * have False by blast
blanchet@48975
   519
  }
blanchet@48975
   520
  ultimately show "(r,r') \<in> ordLess"
blanchet@48975
   521
  unfolding ordLess_def using * by (fastforce simp add: embedS_def)
blanchet@48975
   522
qed
blanchet@48975
   523
blanchet@48975
   524
lemma ordLess_irreflexive: "\<not> r <o r"
blanchet@48975
   525
proof
blanchet@48975
   526
  assume "r <o r"
blanchet@48975
   527
  hence "Well_order r \<and>  \<not>(\<exists>f. embed r r f)"
blanchet@48975
   528
  unfolding ordLess_iff ..
blanchet@48975
   529
  moreover have "embed r r id" using id_embed[of r] .
blanchet@48975
   530
  ultimately show False by blast
blanchet@48975
   531
qed
blanchet@48975
   532
blanchet@48975
   533
lemma ordLeq_iff_ordLess_or_ordIso:
blanchet@48975
   534
"r \<le>o r' = (r <o r' \<or> r =o r')"
blanchet@48975
   535
unfolding ordRels_def embedS_defs iso_defs by blast
blanchet@48975
   536
blanchet@48975
   537
lemma ordIso_iff_ordLeq:
blanchet@48975
   538
"(r =o r') = (r \<le>o r' \<and> r' \<le>o r)"
blanchet@48975
   539
proof
blanchet@48975
   540
  assume "r =o r'"
blanchet@48975
   541
  then obtain f where 1: "Well_order r \<and> Well_order r' \<and>
blanchet@48975
   542
                     embed r r' f \<and> bij_betw f (Field r) (Field r')"
blanchet@48975
   543
  unfolding ordIso_def iso_defs by auto
blanchet@48975
   544
  hence "embed r r' f \<and> embed r' r (inv_into (Field r) f)"
blanchet@48975
   545
  by (simp add: inv_into_Field_embed_bij_betw)
blanchet@48975
   546
  thus  "r \<le>o r' \<and> r' \<le>o r"
blanchet@48975
   547
  unfolding ordLeq_def using 1 by auto
blanchet@48975
   548
next
blanchet@48975
   549
  assume "r \<le>o r' \<and> r' \<le>o r"
blanchet@48975
   550
  then obtain f and g where 1: "Well_order r \<and> Well_order r' \<and>
blanchet@48975
   551
                           embed r r' f \<and> embed r' r g"
blanchet@48975
   552
  unfolding ordLeq_def by auto
blanchet@48975
   553
  hence "iso r r' f" by (auto simp add: embed_bothWays_iso)
blanchet@48975
   554
  thus "r =o r'" unfolding ordIso_def using 1 by auto
blanchet@48975
   555
qed
blanchet@48975
   556
blanchet@48975
   557
lemma not_ordLess_ordLeq:
blanchet@48975
   558
"r <o r' \<Longrightarrow> \<not> r' \<le>o r"
blanchet@48975
   559
using ordLess_ordLeq_trans ordLess_irreflexive by blast
blanchet@48975
   560
blanchet@48975
   561
lemma ordLess_or_ordLeq:
blanchet@48975
   562
assumes WELL: "Well_order r" and WELL': "Well_order r'"
blanchet@48975
   563
shows "r <o r' \<or> r' \<le>o r"
blanchet@48975
   564
proof-
blanchet@48975
   565
  have "r \<le>o r' \<or> r' \<le>o r"
blanchet@48975
   566
  using assms by (simp add: ordLeq_total)
blanchet@48975
   567
  moreover
blanchet@48975
   568
  {assume "\<not> r <o r' \<and> r \<le>o r'"
blanchet@48975
   569
   hence "r =o r'" using ordLeq_iff_ordLess_or_ordIso by blast
blanchet@48975
   570
   hence "r' \<le>o r" using ordIso_symmetric ordIso_iff_ordLeq by blast
blanchet@48975
   571
  }
blanchet@48975
   572
  ultimately show ?thesis by blast
blanchet@48975
   573
qed
blanchet@48975
   574
blanchet@48975
   575
lemma not_ordLess_ordIso:
blanchet@48975
   576
"r <o r' \<Longrightarrow> \<not> r =o r'"
wenzelm@63092
   577
using ordLess_ordIso_trans ordIso_symmetric ordLess_irreflexive by blast
blanchet@48975
   578
blanchet@48975
   579
lemma not_ordLeq_iff_ordLess:
blanchet@48975
   580
assumes WELL: "Well_order r" and WELL': "Well_order r'"
blanchet@48975
   581
shows "(\<not> r' \<le>o r) = (r <o r')"
blanchet@48975
   582
using assms not_ordLess_ordLeq ordLess_or_ordLeq by blast
blanchet@48975
   583
blanchet@48975
   584
lemma not_ordLess_iff_ordLeq:
blanchet@48975
   585
assumes WELL: "Well_order r" and WELL': "Well_order r'"
blanchet@48975
   586
shows "(\<not> r' <o r) = (r \<le>o r')"
blanchet@48975
   587
using assms not_ordLess_ordLeq ordLess_or_ordLeq by blast
blanchet@48975
   588
blanchet@48975
   589
lemma ordLess_transitive[trans]:
blanchet@48975
   590
"\<lbrakk>r <o r'; r' <o r''\<rbrakk> \<Longrightarrow> r <o r''"
wenzelm@63092
   591
using ordLess_ordLeq_trans ordLeq_iff_ordLess_or_ordIso by blast
blanchet@48975
   592
blanchet@48975
   593
corollary ordLess_trans: "trans ordLess"
blanchet@48975
   594
unfolding trans_def using ordLess_transitive by blast
blanchet@48975
   595
blanchet@48975
   596
lemmas ordIso_equivalence = ordIso_transitive ordIso_reflexive ordIso_symmetric
blanchet@48975
   597
blanchet@48975
   598
lemma ordIso_imp_ordLeq:
blanchet@48975
   599
"r =o r' \<Longrightarrow> r \<le>o r'"
blanchet@48975
   600
using ordIso_iff_ordLeq by blast
blanchet@48975
   601
blanchet@48975
   602
lemma ordLess_imp_ordLeq:
blanchet@48975
   603
"r <o r' \<Longrightarrow> r \<le>o r'"
blanchet@48975
   604
using ordLeq_iff_ordLess_or_ordIso by blast
blanchet@48975
   605
blanchet@48975
   606
lemma ofilter_subset_ordLeq:
blanchet@48975
   607
assumes WELL: "Well_order r" and
blanchet@48975
   608
        OFA: "wo_rel.ofilter r A" and OFB: "wo_rel.ofilter r B"
blanchet@48975
   609
shows "(A \<le> B) = (Restr r A \<le>o Restr r B)"
blanchet@48975
   610
proof
blanchet@48975
   611
  assume "A \<le> B"
blanchet@48975
   612
  thus "Restr r A \<le>o Restr r B"
blanchet@48975
   613
  unfolding ordLeq_def using assms
blanchet@48975
   614
  Well_order_Restr Well_order_Restr ofilter_subset_embed by blast
blanchet@48975
   615
next
blanchet@48975
   616
  assume *: "Restr r A \<le>o Restr r B"
blanchet@48975
   617
  then obtain f where "embed (Restr r A) (Restr r B) f"
blanchet@48975
   618
  unfolding ordLeq_def by blast
blanchet@48975
   619
  {assume "B < A"
blanchet@48975
   620
   hence "Restr r B <o Restr r A"
blanchet@48975
   621
   unfolding ordLess_def using assms
blanchet@48975
   622
   Well_order_Restr Well_order_Restr ofilter_subset_embedS by blast
blanchet@48975
   623
   hence False using * not_ordLess_ordLeq by blast
blanchet@48975
   624
  }
blanchet@48975
   625
  thus "A \<le> B" using OFA OFB WELL
blanchet@48975
   626
  wo_rel_def[of r] wo_rel.ofilter_linord[of r A B] by blast
blanchet@48975
   627
qed
blanchet@48975
   628
blanchet@48975
   629
lemma ofilter_subset_ordLess:
blanchet@48975
   630
assumes WELL: "Well_order r" and
blanchet@48975
   631
        OFA: "wo_rel.ofilter r A" and OFB: "wo_rel.ofilter r B"
blanchet@48975
   632
shows "(A < B) = (Restr r A <o Restr r B)"
blanchet@48975
   633
proof-
blanchet@48975
   634
  let ?rA = "Restr r A" let ?rB = "Restr r B"
blanchet@48975
   635
  have 1: "Well_order ?rA \<and> Well_order ?rB"
blanchet@48975
   636
  using WELL Well_order_Restr by blast
blanchet@48975
   637
  have "(A < B) = (\<not> B \<le> A)" using assms
blanchet@48975
   638
  wo_rel_def wo_rel.ofilter_linord[of r A B] by blast
blanchet@48975
   639
  also have "\<dots> = (\<not> Restr r B \<le>o Restr r A)"
blanchet@48975
   640
  using assms ofilter_subset_ordLeq by blast
blanchet@48975
   641
  also have "\<dots> = (Restr r A <o Restr r B)"
blanchet@48975
   642
  using 1 not_ordLeq_iff_ordLess by blast
blanchet@48975
   643
  finally show ?thesis .
blanchet@48975
   644
qed
blanchet@48975
   645
blanchet@48975
   646
lemma ofilter_ordLess:
blanchet@48975
   647
"\<lbrakk>Well_order r; wo_rel.ofilter r A\<rbrakk> \<Longrightarrow> (A < Field r) = (Restr r A <o r)"
blanchet@48975
   648
by (simp add: ofilter_subset_ordLess wo_rel.Field_ofilter
blanchet@48975
   649
    wo_rel_def Restr_Field)
blanchet@48975
   650
blanchet@48975
   651
corollary underS_Restr_ordLess:
blanchet@48975
   652
assumes "Well_order r" and "Field r \<noteq> {}"
blanchet@55023
   653
shows "Restr r (underS r a) <o r"
blanchet@48975
   654
proof-
blanchet@55023
   655
  have "underS r a < Field r" using assms
blanchet@55023
   656
  by (simp add: underS_Field3)
blanchet@48975
   657
  thus ?thesis using assms
blanchet@48975
   658
  by (simp add: ofilter_ordLess wo_rel.underS_ofilter wo_rel_def)
blanchet@48975
   659
qed
blanchet@48975
   660
blanchet@48975
   661
lemma embed_ordLess_ofilterIncl:
blanchet@48975
   662
assumes
blanchet@48975
   663
  OL12: "r1 <o r2" and OL23: "r2 <o r3" and
blanchet@48975
   664
  EMB13: "embed r1 r3 f13" and EMB23: "embed r2 r3 f23"
blanchet@48975
   665
shows "(f13`(Field r1), f23`(Field r2)) \<in> (ofilterIncl r3)"
blanchet@48975
   666
proof-
blanchet@48975
   667
  have OL13: "r1 <o r3"
blanchet@48975
   668
  using OL12 OL23 using ordLess_transitive by auto
blanchet@48975
   669
  let ?A1 = "Field r1"  let ?A2 ="Field r2" let ?A3 ="Field r3"
blanchet@48975
   670
  obtain f12 g23 where
blanchet@48975
   671
  0: "Well_order r1 \<and> Well_order r2 \<and> Well_order r3" and
blanchet@48975
   672
  1: "embed r1 r2 f12 \<and> \<not>(bij_betw f12 ?A1 ?A2)" and
blanchet@48975
   673
  2: "embed r2 r3 g23 \<and> \<not>(bij_betw g23 ?A2 ?A3)"
blanchet@48975
   674
  using OL12 OL23 by (auto simp add: ordLess_def embedS_def)
blanchet@48975
   675
  hence "\<forall>a \<in> ?A2. f23 a = g23 a"
blanchet@48975
   676
  using EMB23 embed_unique[of r2 r3] by blast
blanchet@48975
   677
  hence 3: "\<not>(bij_betw f23 ?A2 ?A3)"
blanchet@48975
   678
  using 2 bij_betw_cong[of ?A2 f23 g23] by blast
blanchet@48975
   679
  (*  *)
blanchet@48975
   680
  have 4: "wo_rel.ofilter r2 (f12 ` ?A1) \<and> f12 ` ?A1 \<noteq> ?A2"
blanchet@48975
   681
  using 0 1 OL12 by (simp add: embed_Field_ofilter ordLess_Field)
blanchet@48975
   682
  have 5: "wo_rel.ofilter r3 (f23 ` ?A2) \<and> f23 ` ?A2 \<noteq> ?A3"
blanchet@48975
   683
  using 0 EMB23 OL23 by (simp add: embed_Field_ofilter ordLess_Field)
blanchet@48975
   684
  have 6: "wo_rel.ofilter r3 (f13 ` ?A1)  \<and> f13 ` ?A1 \<noteq> ?A3"
blanchet@48975
   685
  using 0 EMB13 OL13 by (simp add: embed_Field_ofilter ordLess_Field)
blanchet@48975
   686
  (*  *)
blanchet@48975
   687
  have "f12 ` ?A1 < ?A2"
blanchet@48975
   688
  using 0 4 by (auto simp add: wo_rel_def wo_rel.ofilter_def)
blanchet@48975
   689
  moreover have "inj_on f23 ?A2"
blanchet@48975
   690
  using EMB23 0 by (simp add: wo_rel_def embed_inj_on)
blanchet@48975
   691
  ultimately
blanchet@48975
   692
  have "f23 ` (f12 ` ?A1) < f23 ` ?A2" by (simp add: inj_on_strict_subset)
blanchet@48975
   693
  moreover
blanchet@48975
   694
  {have "embed r1 r3 (f23 o f12)"
blanchet@48975
   695
   using 1 EMB23 0 by (auto simp add: comp_embed)
blanchet@48975
   696
   hence "\<forall>a \<in> ?A1. f23(f12 a) = f13 a"
blanchet@48975
   697
   using EMB13 0 embed_unique[of r1 r3 "f23 o f12" f13] by auto
blanchet@48975
   698
   hence "f23 ` (f12 ` ?A1) = f13 ` ?A1" by force
blanchet@48975
   699
  }
blanchet@48975
   700
  ultimately
blanchet@48975
   701
  have "f13 ` ?A1 < f23 ` ?A2" by simp
blanchet@48975
   702
  (*  *)
blanchet@48975
   703
  with 5 6 show ?thesis
blanchet@48975
   704
  unfolding ofilterIncl_def by auto
blanchet@48975
   705
qed
blanchet@48975
   706
blanchet@48975
   707
lemma ordLess_iff_ordIso_Restr:
blanchet@48975
   708
assumes WELL: "Well_order r" and WELL': "Well_order r'"
blanchet@55023
   709
shows "(r' <o r) = (\<exists>a \<in> Field r. r' =o Restr r (underS r a))"
blanchet@48975
   710
proof(auto)
blanchet@55023
   711
  fix a assume *: "a \<in> Field r" and **: "r' =o Restr r (underS r a)"
blanchet@55023
   712
  hence "Restr r (underS r a) <o r" using WELL underS_Restr_ordLess[of r] by blast
blanchet@48975
   713
  thus "r' <o r" using ** ordIso_ordLess_trans by blast
blanchet@48975
   714
next
blanchet@48975
   715
  assume "r' <o r"
blanchet@48975
   716
  then obtain f where 1: "Well_order r \<and> Well_order r'" and
blanchet@48975
   717
                      2: "embed r' r f \<and> f ` (Field r') \<noteq> Field r"
blanchet@48975
   718
  unfolding ordLess_def embedS_def[abs_def] bij_betw_def using embed_inj_on by blast
blanchet@48975
   719
  hence "wo_rel.ofilter r (f ` (Field r'))" using embed_Field_ofilter by blast
blanchet@55023
   720
  then obtain a where 3: "a \<in> Field r" and 4: "underS r a = f ` (Field r')"
blanchet@48975
   721
  using 1 2 by (auto simp add: wo_rel.ofilter_underS_Field wo_rel_def)
blanchet@48975
   722
  have "iso r' (Restr r (f ` (Field r'))) f"
blanchet@48975
   723
  using embed_implies_iso_Restr 2 assms by blast
blanchet@48975
   724
  moreover have "Well_order (Restr r (f ` (Field r')))"
blanchet@48975
   725
  using WELL Well_order_Restr by blast
blanchet@48975
   726
  ultimately have "r' =o Restr r (f ` (Field r'))"
blanchet@48975
   727
  using WELL' unfolding ordIso_def by auto
blanchet@55023
   728
  hence "r' =o Restr r (underS r a)" using 4 by auto
blanchet@55023
   729
  thus "\<exists>a \<in> Field r. r' =o Restr r (underS r a)" using 3 by auto
blanchet@48975
   730
qed
blanchet@48975
   731
blanchet@48975
   732
lemma internalize_ordLess:
blanchet@48975
   733
"(r' <o r) = (\<exists>p. Field p < Field r \<and> r' =o p \<and> p <o r)"
blanchet@48975
   734
proof
blanchet@48975
   735
  assume *: "r' <o r"
blanchet@48975
   736
  hence 0: "Well_order r \<and> Well_order r'" unfolding ordLess_def by auto
blanchet@55023
   737
  with * obtain a where 1: "a \<in> Field r" and 2: "r' =o Restr r (underS r a)"
blanchet@48975
   738
  using ordLess_iff_ordIso_Restr by blast
blanchet@55023
   739
  let ?p = "Restr r (underS r a)"
blanchet@55023
   740
  have "wo_rel.ofilter r (underS r a)" using 0
blanchet@48975
   741
  by (simp add: wo_rel_def wo_rel.underS_ofilter)
blanchet@55023
   742
  hence "Field ?p = underS r a" using 0 Field_Restr_ofilter by blast
blanchet@55023
   743
  hence "Field ?p < Field r" using underS_Field2 1 by fast
blanchet@48975
   744
  moreover have "?p <o r" using underS_Restr_ordLess[of r a] 0 1 by blast
blanchet@48975
   745
  ultimately
blanchet@48975
   746
  show "\<exists>p. Field p < Field r \<and> r' =o p \<and> p <o r" using 2 by blast
blanchet@48975
   747
next
blanchet@48975
   748
  assume "\<exists>p. Field p < Field r \<and> r' =o p \<and> p <o r"
blanchet@48975
   749
  thus "r' <o r" using ordIso_ordLess_trans by blast
blanchet@48975
   750
qed
blanchet@48975
   751
blanchet@48975
   752
lemma internalize_ordLeq:
blanchet@48975
   753
"(r' \<le>o r) = (\<exists>p. Field p \<le> Field r \<and> r' =o p \<and> p \<le>o r)"
blanchet@48975
   754
proof
blanchet@48975
   755
  assume *: "r' \<le>o r"
blanchet@48975
   756
  moreover
blanchet@48975
   757
  {assume "r' <o r"
blanchet@48975
   758
   then obtain p where "Field p < Field r \<and> r' =o p \<and> p <o r"
blanchet@48975
   759
   using internalize_ordLess[of r' r] by blast
blanchet@48975
   760
   hence "\<exists>p. Field p \<le> Field r \<and> r' =o p \<and> p \<le>o r"
blanchet@48975
   761
   using ordLeq_iff_ordLess_or_ordIso by blast
blanchet@48975
   762
  }
blanchet@48975
   763
  moreover
blanchet@48975
   764
  have "r \<le>o r" using * ordLeq_def ordLeq_reflexive by blast
blanchet@48975
   765
  ultimately show "\<exists>p. Field p \<le> Field r \<and> r' =o p \<and> p \<le>o r"
blanchet@48975
   766
  using ordLeq_iff_ordLess_or_ordIso by blast
blanchet@48975
   767
next
blanchet@48975
   768
  assume "\<exists>p. Field p \<le> Field r \<and> r' =o p \<and> p \<le>o r"
blanchet@48975
   769
  thus "r' \<le>o r" using ordIso_ordLeq_trans by blast
blanchet@48975
   770
qed
blanchet@48975
   771
blanchet@48975
   772
lemma ordLeq_iff_ordLess_Restr:
blanchet@48975
   773
assumes WELL: "Well_order r" and WELL': "Well_order r'"
blanchet@55023
   774
shows "(r \<le>o r') = (\<forall>a \<in> Field r. Restr r (underS r a) <o r')"
blanchet@48975
   775
proof(auto)
blanchet@48975
   776
  assume *: "r \<le>o r'"
blanchet@48975
   777
  fix a assume "a \<in> Field r"
blanchet@55023
   778
  hence "Restr r (underS r a) <o r"
blanchet@48975
   779
  using WELL underS_Restr_ordLess[of r] by blast
blanchet@55023
   780
  thus "Restr r (underS r a) <o r'"
blanchet@48975
   781
  using * ordLess_ordLeq_trans by blast
blanchet@48975
   782
next
blanchet@55023
   783
  assume *: "\<forall>a \<in> Field r. Restr r (underS r a) <o r'"
blanchet@48975
   784
  {assume "r' <o r"
blanchet@55023
   785
   then obtain a where "a \<in> Field r \<and> r' =o Restr r (underS r a)"
blanchet@48975
   786
   using assms ordLess_iff_ordIso_Restr by blast
blanchet@48975
   787
   hence False using * not_ordLess_ordIso ordIso_symmetric by blast
blanchet@48975
   788
  }
blanchet@48975
   789
  thus "r \<le>o r'" using ordLess_or_ordLeq assms by blast
blanchet@48975
   790
qed
blanchet@48975
   791
blanchet@48975
   792
lemma finite_ordLess_infinite:
blanchet@48975
   793
assumes WELL: "Well_order r" and WELL': "Well_order r'" and
traytel@54578
   794
        FIN: "finite(Field r)" and INF: "\<not>finite(Field r')"
blanchet@48975
   795
shows "r <o r'"
blanchet@48975
   796
proof-
blanchet@48975
   797
  {assume "r' \<le>o r"
blanchet@48975
   798
   then obtain h where "inj_on h (Field r') \<and> h ` (Field r') \<le> Field r"
blanchet@48975
   799
   unfolding ordLeq_def using assms embed_inj_on embed_Field by blast
traytel@55811
   800
   hence False using finite_imageD finite_subset FIN INF by blast
blanchet@48975
   801
  }
blanchet@48975
   802
  thus ?thesis using WELL WELL' ordLess_or_ordLeq by blast
blanchet@48975
   803
qed
blanchet@48975
   804
blanchet@48975
   805
lemma finite_well_order_on_ordIso:
blanchet@48975
   806
assumes FIN: "finite A" and
blanchet@48975
   807
        WELL: "well_order_on A r" and WELL': "well_order_on A r'"
blanchet@48975
   808
shows "r =o r'"
blanchet@48975
   809
proof-
blanchet@48975
   810
  have 0: "Well_order r \<and> Well_order r' \<and> Field r = A \<and> Field r' = A"
blanchet@55023
   811
  using assms well_order_on_Well_order by blast
blanchet@48975
   812
  moreover
blanchet@48975
   813
  have "\<forall>r r'. well_order_on A r \<and> well_order_on A r' \<and> r \<le>o r'
blanchet@48975
   814
                  \<longrightarrow> r =o r'"
blanchet@48975
   815
  proof(clarify)
blanchet@48975
   816
    fix r r' assume *: "well_order_on A r" and **: "well_order_on A r'"
blanchet@48975
   817
    have 2: "Well_order r \<and> Well_order r' \<and> Field r = A \<and> Field r' = A"
blanchet@55023
   818
    using * ** well_order_on_Well_order by blast
blanchet@48975
   819
    assume "r \<le>o r'"
blanchet@48975
   820
    then obtain f where 1: "embed r r' f" and
blanchet@48975
   821
                        "inj_on f A \<and> f ` A \<le> A"
blanchet@48975
   822
    unfolding ordLeq_def using 2 embed_inj_on embed_Field by blast
blanchet@48975
   823
    hence "bij_betw f A A" unfolding bij_betw_def using FIN endo_inj_surj by blast
blanchet@48975
   824
    thus "r =o r'" unfolding ordIso_def iso_def[abs_def] using 1 2 by auto
blanchet@48975
   825
  qed
traytel@55811
   826
  ultimately show ?thesis using assms ordLeq_total ordIso_symmetric by blast
blanchet@48975
   827
qed
blanchet@48975
   828
wenzelm@61799
   829
subsection\<open>\<open><o\<close> is well-founded\<close>
blanchet@48975
   830
wenzelm@61799
   831
text \<open>Of course, it only makes sense to state that the \<open><o\<close> is well-founded
wenzelm@61799
   832
on the restricted type \<open>'a rel rel\<close>.  We prove this by first showing that, for any set
blanchet@48975
   833
of well-orders all embedded in a fixed well-order, the function mapping each well-order
wenzelm@61799
   834
in the set to an order filter of the fixed well-order is compatible w.r.t. to \<open><o\<close> versus
wenzelm@60758
   835
{\em strict inclusion}; and we already know that strict inclusion of order filters is well-founded.\<close>
blanchet@48975
   836
blanchet@48975
   837
definition ord_to_filter :: "'a rel \<Rightarrow> 'a rel \<Rightarrow> 'a set"
blanchet@48975
   838
where "ord_to_filter r0 r \<equiv> (SOME f. embed r r0 f) ` (Field r)"
blanchet@48975
   839
blanchet@48975
   840
lemma ord_to_filter_compat:
blanchet@48975
   841
"compat (ordLess Int (ordLess^-1``{r0} \<times> ordLess^-1``{r0}))
blanchet@48975
   842
        (ofilterIncl r0)
blanchet@48975
   843
        (ord_to_filter r0)"
blanchet@48975
   844
proof(unfold compat_def ord_to_filter_def, clarify)
blanchet@48975
   845
  fix r1::"'a rel" and r2::"'a rel"
blanchet@48975
   846
  let ?A1 = "Field r1"  let ?A2 ="Field r2" let ?A0 ="Field r0"
blanchet@48975
   847
  let ?phi10 = "\<lambda> f10. embed r1 r0 f10" let ?f10 = "SOME f. ?phi10 f"
blanchet@48975
   848
  let ?phi20 = "\<lambda> f20. embed r2 r0 f20" let ?f20 = "SOME f. ?phi20 f"
blanchet@48975
   849
  assume *: "r1 <o r0" "r2 <o r0" and **: "r1 <o r2"
blanchet@48975
   850
  hence "(\<exists>f. ?phi10 f) \<and> (\<exists>f. ?phi20 f)"
blanchet@48975
   851
  by (auto simp add: ordLess_def embedS_def)
blanchet@48975
   852
  hence "?phi10 ?f10 \<and> ?phi20 ?f20" by (auto simp add: someI_ex)
blanchet@48975
   853
  thus "(?f10 ` ?A1, ?f20 ` ?A2) \<in> ofilterIncl r0"
blanchet@48975
   854
  using * ** by (simp add: embed_ordLess_ofilterIncl)
blanchet@48975
   855
qed
blanchet@48975
   856
blanchet@48975
   857
theorem wf_ordLess: "wf ordLess"
blanchet@48975
   858
proof-
blanchet@48975
   859
  {fix r0 :: "('a \<times> 'a) set"
blanchet@48975
   860
   (* need to annotate here!*)
blanchet@48975
   861
   let ?ordLess = "ordLess::('d rel * 'd rel) set"
blanchet@48975
   862
   let ?R = "?ordLess Int (?ordLess^-1``{r0} \<times> ?ordLess^-1``{r0})"
blanchet@48975
   863
   {assume Case1: "Well_order r0"
blanchet@48975
   864
    hence "wf ?R"
blanchet@48975
   865
    using wf_ofilterIncl[of r0]
blanchet@48975
   866
          compat_wf[of ?R "ofilterIncl r0" "ord_to_filter r0"]
blanchet@48975
   867
          ord_to_filter_compat[of r0] by auto
blanchet@48975
   868
   }
blanchet@48975
   869
   moreover
blanchet@48975
   870
   {assume Case2: "\<not> Well_order r0"
blanchet@48975
   871
    hence "?R = {}" unfolding ordLess_def by auto
blanchet@48975
   872
    hence "wf ?R" using wf_empty by simp
blanchet@48975
   873
   }
blanchet@48975
   874
   ultimately have "wf ?R" by blast
blanchet@48975
   875
  }
blanchet@48975
   876
  thus ?thesis by (simp add: trans_wf_iff ordLess_trans)
blanchet@48975
   877
qed
blanchet@48975
   878
blanchet@48975
   879
corollary exists_minim_Well_order:
blanchet@48975
   880
assumes NE: "R \<noteq> {}" and WELL: "\<forall>r \<in> R. Well_order r"
blanchet@48975
   881
shows "\<exists>r \<in> R. \<forall>r' \<in> R. r \<le>o r'"
blanchet@48975
   882
proof-
blanchet@48975
   883
  obtain r where "r \<in> R \<and> (\<forall>r' \<in> R. \<not> r' <o r)"
blanchet@48975
   884
  using NE spec[OF spec[OF subst[OF wf_eq_minimal, of "%x. x", OF wf_ordLess]], of _ R]
blanchet@48975
   885
    equals0I[of R] by blast
blanchet@48975
   886
  with not_ordLeq_iff_ordLess WELL show ?thesis by blast
blanchet@48975
   887
qed
blanchet@48975
   888
blanchet@48975
   889
wenzelm@60758
   890
subsection \<open>Copy via direct images\<close>
blanchet@48975
   891
wenzelm@61799
   892
text\<open>The direct image operator is the dual of the inverse image operator \<open>inv_image\<close>
wenzelm@61799
   893
from \<open>Relation.thy\<close>.  It is useful for transporting a well-order between
wenzelm@60758
   894
different types.\<close>
blanchet@48975
   895
blanchet@48975
   896
definition dir_image :: "'a rel \<Rightarrow> ('a \<Rightarrow> 'a') \<Rightarrow> 'a' rel"
blanchet@48975
   897
where
blanchet@48975
   898
"dir_image r f = {(f a, f b)| a b. (a,b) \<in> r}"
blanchet@48975
   899
blanchet@48975
   900
lemma dir_image_Field:
traytel@56191
   901
"Field(dir_image r f) = f ` (Field r)"
traytel@56191
   902
unfolding dir_image_def Field_def Range_def Domain_def by fast
blanchet@48975
   903
blanchet@48975
   904
lemma dir_image_minus_Id:
blanchet@48975
   905
"inj_on f (Field r) \<Longrightarrow> (dir_image r f) - Id = dir_image (r - Id) f"
blanchet@48975
   906
unfolding inj_on_def Field_def dir_image_def by auto
blanchet@48975
   907
blanchet@48975
   908
lemma Refl_dir_image:
blanchet@48975
   909
assumes "Refl r"
blanchet@48975
   910
shows "Refl(dir_image r f)"
blanchet@48975
   911
proof-
blanchet@48975
   912
  {fix a' b'
blanchet@48975
   913
   assume "(a',b') \<in> dir_image r f"
blanchet@48975
   914
   then obtain a b where 1: "a' = f a \<and> b' = f b \<and> (a,b) \<in> r"
blanchet@48975
   915
   unfolding dir_image_def by blast
blanchet@48975
   916
   hence "a \<in> Field r \<and> b \<in> Field r" using Field_def by fastforce
blanchet@48975
   917
   hence "(a,a) \<in> r \<and> (b,b) \<in> r" using assms by (simp add: refl_on_def)
blanchet@48975
   918
   with 1 have "(a',a') \<in> dir_image r f \<and> (b',b') \<in> dir_image r f"
blanchet@48975
   919
   unfolding dir_image_def by auto
blanchet@48975
   920
  }
blanchet@48975
   921
  thus ?thesis
blanchet@48975
   922
  by(unfold refl_on_def Field_def Domain_def Range_def, auto)
blanchet@48975
   923
qed
blanchet@48975
   924
blanchet@48975
   925
lemma trans_dir_image:
blanchet@48975
   926
assumes TRANS: "trans r" and INJ: "inj_on f (Field r)"
blanchet@48975
   927
shows "trans(dir_image r f)"
blanchet@48975
   928
proof(unfold trans_def, auto)
blanchet@48975
   929
  fix a' b' c'
blanchet@48975
   930
  assume "(a',b') \<in> dir_image r f" "(b',c') \<in> dir_image r f"
blanchet@48975
   931
  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
   932
                         2: "(a,b1) \<in> r \<and> (b2,c) \<in> r"
blanchet@48975
   933
  unfolding dir_image_def by blast
blanchet@48975
   934
  hence "b1 \<in> Field r \<and> b2 \<in> Field r"
blanchet@48975
   935
  unfolding Field_def by auto
blanchet@48975
   936
  hence "b1 = b2" using 1 INJ unfolding inj_on_def by auto
blanchet@48975
   937
  hence "(a,c): r" using 2 TRANS unfolding trans_def by blast
blanchet@48975
   938
  thus "(a',c') \<in> dir_image r f"
blanchet@48975
   939
  unfolding dir_image_def using 1 by auto
blanchet@48975
   940
qed
blanchet@48975
   941
blanchet@48975
   942
lemma Preorder_dir_image:
blanchet@48975
   943
"\<lbrakk>Preorder r; inj_on f (Field r)\<rbrakk> \<Longrightarrow> Preorder (dir_image r f)"
blanchet@48975
   944
by (simp add: preorder_on_def Refl_dir_image trans_dir_image)
blanchet@48975
   945
blanchet@48975
   946
lemma antisym_dir_image:
blanchet@48975
   947
assumes AN: "antisym r" and INJ: "inj_on f (Field r)"
blanchet@48975
   948
shows "antisym(dir_image r f)"
blanchet@48975
   949
proof(unfold antisym_def, auto)
blanchet@48975
   950
  fix a' b'
blanchet@48975
   951
  assume "(a',b') \<in> dir_image r f" "(b',a') \<in> dir_image r f"
blanchet@48975
   952
  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
   953
                           2: "(a1,b1) \<in> r \<and> (b2,a2) \<in> r " and
blanchet@48975
   954
                           3: "{a1,a2,b1,b2} \<le> Field r"
blanchet@48975
   955
  unfolding dir_image_def Field_def by blast
blanchet@48975
   956
  hence "a1 = a2 \<and> b1 = b2" using INJ unfolding inj_on_def by auto
blanchet@48975
   957
  hence "a1 = b2" using 2 AN unfolding antisym_def by auto
blanchet@48975
   958
  thus "a' = b'" using 1 by auto
blanchet@48975
   959
qed
blanchet@48975
   960
blanchet@48975
   961
lemma Partial_order_dir_image:
blanchet@48975
   962
"\<lbrakk>Partial_order r; inj_on f (Field r)\<rbrakk> \<Longrightarrow> Partial_order (dir_image r f)"
blanchet@48975
   963
by (simp add: partial_order_on_def Preorder_dir_image antisym_dir_image)
blanchet@48975
   964
blanchet@48975
   965
lemma Total_dir_image:
blanchet@48975
   966
assumes TOT: "Total r" and INJ: "inj_on f (Field r)"
blanchet@48975
   967
shows "Total(dir_image r f)"
blanchet@48975
   968
proof(unfold total_on_def, intro ballI impI)
blanchet@48975
   969
  fix a' b'
blanchet@48975
   970
  assume "a' \<in> Field (dir_image r f)" "b' \<in> Field (dir_image r f)"
blanchet@48975
   971
  then obtain a and b where 1: "a \<in> Field r \<and> b \<in> Field r \<and> f a = a' \<and> f b = b'"
traytel@56191
   972
    unfolding dir_image_Field[of r f] by blast
blanchet@48975
   973
  moreover assume "a' \<noteq> b'"
blanchet@48975
   974
  ultimately have "a \<noteq> b" using INJ unfolding inj_on_def by auto
blanchet@48975
   975
  hence "(a,b) \<in> r \<or> (b,a) \<in> r" using 1 TOT unfolding total_on_def by auto
blanchet@48975
   976
  thus "(a',b') \<in> dir_image r f \<or> (b',a') \<in> dir_image r f"
blanchet@48975
   977
  using 1 unfolding dir_image_def by auto
blanchet@48975
   978
qed
blanchet@48975
   979
blanchet@48975
   980
lemma Linear_order_dir_image:
blanchet@48975
   981
"\<lbrakk>Linear_order r; inj_on f (Field r)\<rbrakk> \<Longrightarrow> Linear_order (dir_image r f)"
blanchet@48975
   982
by (simp add: linear_order_on_def Partial_order_dir_image Total_dir_image)
blanchet@48975
   983
blanchet@48975
   984
lemma wf_dir_image:
blanchet@48975
   985
assumes WF: "wf r" and INJ: "inj_on f (Field r)"
blanchet@48975
   986
shows "wf(dir_image r f)"
blanchet@48975
   987
proof(unfold wf_eq_minimal2, intro allI impI, elim conjE)
blanchet@48975
   988
  fix A'::"'b set"
blanchet@48975
   989
  assume SUB: "A' \<le> Field(dir_image r f)" and NE: "A' \<noteq> {}"
blanchet@48975
   990
  obtain A where A_def: "A = {a \<in> Field r. f a \<in> A'}" by blast
traytel@56191
   991
  have "A \<noteq> {} \<and> A \<le> Field r" using A_def SUB NE by (auto simp: dir_image_Field)
blanchet@48975
   992
  then obtain a where 1: "a \<in> A \<and> (\<forall>b \<in> A. (b,a) \<notin> r)"
traytel@56191
   993
  using spec[OF WF[unfolded wf_eq_minimal2], of A] by blast
blanchet@48975
   994
  have "\<forall>b' \<in> A'. (b',f a) \<notin> dir_image r f"
blanchet@48975
   995
  proof(clarify)
blanchet@48975
   996
    fix b' assume *: "b' \<in> A'" and **: "(b',f a) \<in> dir_image r f"
blanchet@48975
   997
    obtain b1 a1 where 2: "b' = f b1 \<and> f a = f a1" and
blanchet@48975
   998
                       3: "(b1,a1) \<in> r \<and> {a1,b1} \<le> Field r"
blanchet@48975
   999
    using ** unfolding dir_image_def Field_def by blast
blanchet@48975
  1000
    hence "a = a1" using 1 A_def INJ unfolding inj_on_def by auto
blanchet@48975
  1001
    hence "b1 \<in> A \<and> (b1,a) \<in> r" using 2 3 A_def * by auto
blanchet@48975
  1002
    with 1 show False by auto
blanchet@48975
  1003
  qed
blanchet@48975
  1004
  thus "\<exists>a'\<in>A'. \<forall>b'\<in>A'. (b', a') \<notin> dir_image r f"
blanchet@48975
  1005
  using A_def 1 by blast
blanchet@48975
  1006
qed
blanchet@48975
  1007
blanchet@48975
  1008
lemma Well_order_dir_image:
blanchet@48975
  1009
"\<lbrakk>Well_order r; inj_on f (Field r)\<rbrakk> \<Longrightarrow> Well_order (dir_image r f)"
wenzelm@63092
  1010
unfolding well_order_on_def
blanchet@48975
  1011
using Linear_order_dir_image[of r f] wf_dir_image[of "r - Id" f]
blanchet@48975
  1012
  dir_image_minus_Id[of f r]
blanchet@48975
  1013
  subset_inj_on[of f "Field r" "Field(r - Id)"]
blanchet@48975
  1014
  mono_Field[of "r - Id" r] by auto
blanchet@48975
  1015
blanchet@48975
  1016
lemma dir_image_bij_betw:
traytel@56191
  1017
"\<lbrakk>inj_on f (Field r)\<rbrakk> \<Longrightarrow> bij_betw f (Field r) (Field (dir_image r f))"
traytel@56191
  1018
unfolding bij_betw_def by (simp add: dir_image_Field order_on_defs)
blanchet@48975
  1019
blanchet@48975
  1020
lemma dir_image_compat:
blanchet@48975
  1021
"compat r (dir_image r f) f"
blanchet@48975
  1022
unfolding compat_def dir_image_def by auto
blanchet@48975
  1023
blanchet@48975
  1024
lemma dir_image_iso:
blanchet@48975
  1025
"\<lbrakk>Well_order r; inj_on f (Field r)\<rbrakk>  \<Longrightarrow> iso r (dir_image r f) f"
blanchet@48975
  1026
using iso_iff3 dir_image_compat dir_image_bij_betw Well_order_dir_image by blast
blanchet@48975
  1027
blanchet@48975
  1028
lemma dir_image_ordIso:
blanchet@48975
  1029
"\<lbrakk>Well_order r; inj_on f (Field r)\<rbrakk>  \<Longrightarrow> r =o dir_image r f"
blanchet@48975
  1030
unfolding ordIso_def using dir_image_iso Well_order_dir_image by blast
blanchet@48975
  1031
blanchet@48975
  1032
lemma Well_order_iso_copy:
blanchet@48975
  1033
assumes WELL: "well_order_on A r" and BIJ: "bij_betw f A A'"
blanchet@48975
  1034
shows "\<exists>r'. well_order_on A' r' \<and> r =o r'"
blanchet@48975
  1035
proof-
blanchet@48975
  1036
   let ?r' = "dir_image r f"
blanchet@48975
  1037
   have 1: "A = Field r \<and> Well_order r"
blanchet@55023
  1038
   using WELL well_order_on_Well_order by blast
blanchet@48975
  1039
   hence 2: "iso r ?r' f"
blanchet@48975
  1040
   using dir_image_iso using BIJ unfolding bij_betw_def by auto
blanchet@48975
  1041
   hence "f ` (Field r) = Field ?r'" using 1 iso_iff[of r ?r'] by blast
blanchet@48975
  1042
   hence "Field ?r' = A'"
blanchet@48975
  1043
   using 1 BIJ unfolding bij_betw_def by auto
blanchet@48975
  1044
   moreover have "Well_order ?r'"
blanchet@48975
  1045
   using 1 Well_order_dir_image BIJ unfolding bij_betw_def by blast
blanchet@48975
  1046
   ultimately show ?thesis unfolding ordIso_def using 1 2 by blast
blanchet@48975
  1047
qed
blanchet@48975
  1048
blanchet@48975
  1049
wenzelm@60758
  1050
subsection \<open>Bounded square\<close>
blanchet@48975
  1051
wenzelm@61799
  1052
text\<open>This construction essentially defines, for an order relation \<open>r\<close>, a lexicographic
wenzelm@61799
  1053
order \<open>bsqr r\<close> on \<open>(Field r) \<times> (Field r)\<close>, applying the
blanchet@48975
  1054
following criteria (in this order):
blanchet@48975
  1055
\begin{itemize}
blanchet@48975
  1056
\item compare the maximums;
blanchet@48975
  1057
\item compare the first components;
blanchet@48975
  1058
\item compare the second components.
blanchet@48975
  1059
\end{itemize}
blanchet@48975
  1060
%
blanchet@48975
  1061
The only application of this construction that we are aware of is
blanchet@48975
  1062
at proving that the square of an infinite set has the same cardinal
blanchet@48975
  1063
as that set. The essential property required there (and which is ensured by this
blanchet@48975
  1064
construction) is that any proper order filter of the product order is included in a rectangle, i.e.,
wenzelm@60758
  1065
in a product of proper filters on the original relation (assumed to be a well-order).\<close>
blanchet@48975
  1066
blanchet@48975
  1067
definition bsqr :: "'a rel => ('a * 'a)rel"
blanchet@48975
  1068
where
blanchet@48975
  1069
"bsqr r = {((a1,a2),(b1,b2)).
blanchet@48975
  1070
           {a1,a2,b1,b2} \<le> Field r \<and>
blanchet@48975
  1071
           (a1 = b1 \<and> a2 = b2 \<or>
blanchet@48975
  1072
            (wo_rel.max2 r a1 a2, wo_rel.max2 r b1 b2) \<in> r - Id \<or>
blanchet@48975
  1073
            wo_rel.max2 r a1 a2 = wo_rel.max2 r b1 b2 \<and> (a1,b1) \<in> r - Id \<or>
blanchet@48975
  1074
            wo_rel.max2 r a1 a2 = wo_rel.max2 r b1 b2 \<and> a1 = b1  \<and> (a2,b2) \<in> r - Id
blanchet@48975
  1075
           )}"
blanchet@48975
  1076
blanchet@48975
  1077
lemma Field_bsqr:
blanchet@48975
  1078
"Field (bsqr r) = Field r \<times> Field r"
blanchet@48975
  1079
proof
blanchet@48975
  1080
  show "Field (bsqr r) \<le> Field r \<times> Field r"
blanchet@48975
  1081
  proof-
blanchet@48975
  1082
    {fix a1 a2 assume "(a1,a2) \<in> Field (bsqr r)"
blanchet@48975
  1083
     moreover
blanchet@48975
  1084
     have "\<And> b1 b2. ((a1,a2),(b1,b2)) \<in> bsqr r \<or> ((b1,b2),(a1,a2)) \<in> bsqr r \<Longrightarrow>
blanchet@48975
  1085
                      a1 \<in> Field r \<and> a2 \<in> Field r" unfolding bsqr_def by auto
blanchet@48975
  1086
     ultimately have "a1 \<in> Field r \<and> a2 \<in> Field r" unfolding Field_def by auto
blanchet@48975
  1087
    }
blanchet@48975
  1088
    thus ?thesis unfolding Field_def by force
blanchet@48975
  1089
  qed
blanchet@48975
  1090
next
blanchet@48975
  1091
  show "Field r \<times> Field r \<le> Field (bsqr r)"
blanchet@48975
  1092
  proof(auto)
blanchet@48975
  1093
    fix a1 a2 assume "a1 \<in> Field r" and "a2 \<in> Field r"
blanchet@48975
  1094
    hence "((a1,a2),(a1,a2)) \<in> bsqr r" unfolding bsqr_def by blast
blanchet@48975
  1095
    thus "(a1,a2) \<in> Field (bsqr r)" unfolding Field_def by auto
blanchet@48975
  1096
  qed
blanchet@48975
  1097
qed
blanchet@48975
  1098
blanchet@48975
  1099
lemma bsqr_Refl: "Refl(bsqr r)"
blanchet@48975
  1100
by(unfold refl_on_def Field_bsqr, auto simp add: bsqr_def)
blanchet@48975
  1101
blanchet@48975
  1102
lemma bsqr_Trans:
blanchet@48975
  1103
assumes "Well_order r"
blanchet@48975
  1104
shows "trans (bsqr r)"
blanchet@48975
  1105
proof(unfold trans_def, auto)
blanchet@48975
  1106
  (* Preliminary facts *)
blanchet@48975
  1107
  have Well: "wo_rel r" using assms wo_rel_def by auto
blanchet@48975
  1108
  hence Trans: "trans r" using wo_rel.TRANS by auto
blanchet@48975
  1109
  have Anti: "antisym r" using wo_rel.ANTISYM Well by auto
blanchet@48975
  1110
  hence TransS: "trans(r - Id)" using Trans by (simp add: trans_diff_Id)
blanchet@48975
  1111
  (* Main proof *)
blanchet@48975
  1112
  fix a1 a2 b1 b2 c1 c2
blanchet@48975
  1113
  assume *: "((a1,a2),(b1,b2)) \<in> bsqr r" and **: "((b1,b2),(c1,c2)) \<in> bsqr r"
blanchet@48975
  1114
  hence 0: "{a1,a2,b1,b2,c1,c2} \<le> Field r" unfolding bsqr_def by auto
blanchet@48975
  1115
  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
  1116
           wo_rel.max2 r a1 a2 = wo_rel.max2 r b1 b2 \<and> (a1,b1) \<in> r - Id \<or>
blanchet@48975
  1117
           wo_rel.max2 r a1 a2 = wo_rel.max2 r b1 b2 \<and> a1 = b1 \<and> (a2,b2) \<in> r - Id"
blanchet@48975
  1118
  using * unfolding bsqr_def by auto
blanchet@48975
  1119
  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
  1120
           wo_rel.max2 r b1 b2 = wo_rel.max2 r c1 c2 \<and> (b1,c1) \<in> r - Id \<or>
blanchet@48975
  1121
           wo_rel.max2 r b1 b2 = wo_rel.max2 r c1 c2 \<and> b1 = c1 \<and> (b2,c2) \<in> r - Id"
blanchet@48975
  1122
  using ** unfolding bsqr_def by auto
blanchet@48975
  1123
  show "((a1,a2),(c1,c2)) \<in> bsqr r"
blanchet@48975
  1124
  proof-
blanchet@48975
  1125
    {assume Case1: "a1 = b1 \<and> a2 = b2"
blanchet@48975
  1126
     hence ?thesis using ** by simp
blanchet@48975
  1127
    }
blanchet@48975
  1128
    moreover
blanchet@48975
  1129
    {assume Case2: "(wo_rel.max2 r a1 a2, wo_rel.max2 r b1 b2) \<in> r - Id"
blanchet@48975
  1130
     {assume Case21: "b1 = c1 \<and> b2 = c2"
blanchet@48975
  1131
      hence ?thesis using * by simp
blanchet@48975
  1132
     }
blanchet@48975
  1133
     moreover
blanchet@48975
  1134
     {assume Case22: "(wo_rel.max2 r b1 b2, wo_rel.max2 r c1 c2) \<in> r - Id"
blanchet@48975
  1135
      hence "(wo_rel.max2 r a1 a2, wo_rel.max2 r c1 c2) \<in> r - Id"
blanchet@48975
  1136
      using Case2 TransS trans_def[of "r - Id"] by blast
blanchet@48975
  1137
      hence ?thesis using 0 unfolding bsqr_def by auto
blanchet@48975
  1138
     }
blanchet@48975
  1139
     moreover
blanchet@48975
  1140
     {assume Case23_4: "wo_rel.max2 r b1 b2 = wo_rel.max2 r c1 c2"
blanchet@48975
  1141
      hence ?thesis using Case2 0 unfolding bsqr_def by auto
blanchet@48975
  1142
     }
blanchet@48975
  1143
     ultimately have ?thesis using 0 2 by auto
blanchet@48975
  1144
    }
blanchet@48975
  1145
    moreover
blanchet@48975
  1146
    {assume Case3: "wo_rel.max2 r a1 a2 = wo_rel.max2 r b1 b2 \<and> (a1,b1) \<in> r - Id"
blanchet@48975
  1147
     {assume Case31: "b1 = c1 \<and> b2 = c2"
blanchet@48975
  1148
      hence ?thesis using * by simp
blanchet@48975
  1149
     }
blanchet@48975
  1150
     moreover
blanchet@48975
  1151
     {assume Case32: "(wo_rel.max2 r b1 b2, wo_rel.max2 r c1 c2) \<in> r - Id"
blanchet@48975
  1152
      hence ?thesis using Case3 0 unfolding bsqr_def by auto
blanchet@48975
  1153
     }
blanchet@48975
  1154
     moreover
blanchet@48975
  1155
     {assume Case33: "wo_rel.max2 r b1 b2 = wo_rel.max2 r c1 c2 \<and> (b1,c1) \<in> r - Id"
blanchet@48975
  1156
      hence "(a1,c1) \<in> r - Id"
blanchet@48975
  1157
      using Case3 TransS trans_def[of "r - Id"] by blast
blanchet@48975
  1158
      hence ?thesis using Case3 Case33 0 unfolding bsqr_def by auto
blanchet@48975
  1159
     }
blanchet@48975
  1160
     moreover
blanchet@48975
  1161
     {assume Case33: "wo_rel.max2 r b1 b2 = wo_rel.max2 r c1 c2 \<and> b1 = c1"
blanchet@48975
  1162
      hence ?thesis using Case3 0 unfolding bsqr_def by auto
blanchet@48975
  1163
     }
blanchet@48975
  1164
     ultimately have ?thesis using 0 2 by auto
blanchet@48975
  1165
    }
blanchet@48975
  1166
    moreover
blanchet@48975
  1167
    {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
  1168
     {assume Case41: "b1 = c1 \<and> b2 = c2"
blanchet@48975
  1169
      hence ?thesis using * by simp
blanchet@48975
  1170
     }
blanchet@48975
  1171
     moreover
blanchet@48975
  1172
     {assume Case42: "(wo_rel.max2 r b1 b2, wo_rel.max2 r c1 c2) \<in> r - Id"
traytel@51764
  1173
      hence ?thesis using Case4 0 unfolding bsqr_def by force
blanchet@48975
  1174
     }
blanchet@48975
  1175
     moreover
blanchet@48975
  1176
     {assume Case43: "wo_rel.max2 r b1 b2 = wo_rel.max2 r c1 c2 \<and> (b1,c1) \<in> r - Id"
blanchet@48975
  1177
      hence ?thesis using Case4 0 unfolding bsqr_def by auto
blanchet@48975
  1178
     }
blanchet@48975
  1179
     moreover
blanchet@48975
  1180
     {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
  1181
      hence "(a2,c2) \<in> r - Id"
blanchet@48975
  1182
      using Case4 TransS trans_def[of "r - Id"] by blast
blanchet@48975
  1183
      hence ?thesis using Case4 Case44 0 unfolding bsqr_def by auto
blanchet@48975
  1184
     }
blanchet@48975
  1185
     ultimately have ?thesis using 0 2 by auto
blanchet@48975
  1186
    }
blanchet@48975
  1187
    ultimately show ?thesis using 0 1 by auto
blanchet@48975
  1188
  qed
blanchet@48975
  1189
qed
blanchet@48975
  1190
blanchet@48975
  1191
lemma bsqr_antisym:
blanchet@48975
  1192
assumes "Well_order r"
blanchet@48975
  1193
shows "antisym (bsqr r)"
blanchet@48975
  1194
proof(unfold antisym_def, clarify)
blanchet@48975
  1195
  (* Preliminary facts *)
blanchet@48975
  1196
  have Well: "wo_rel r" using assms wo_rel_def by auto
blanchet@48975
  1197
  hence Trans: "trans r" using wo_rel.TRANS by auto
blanchet@48975
  1198
  have Anti: "antisym r" using wo_rel.ANTISYM Well by auto
blanchet@48975
  1199
  hence TransS: "trans(r - Id)" using Trans by (simp add: trans_diff_Id)
blanchet@48975
  1200
  hence IrrS: "\<forall>a b. \<not>((a,b) \<in> r - Id \<and> (b,a) \<in> r - Id)"
blanchet@48975
  1201
  using Anti trans_def[of "r - Id"] antisym_def[of "r - Id"] by blast
blanchet@48975
  1202
  (* Main proof *)
blanchet@48975
  1203
  fix a1 a2 b1 b2
blanchet@48975
  1204
  assume *: "((a1,a2),(b1,b2)) \<in> bsqr r" and **: "((b1,b2),(a1,a2)) \<in> bsqr r"
blanchet@48975
  1205
  hence 0: "{a1,a2,b1,b2} \<le> Field r" unfolding bsqr_def by auto
blanchet@48975
  1206
  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
  1207
           wo_rel.max2 r a1 a2 = wo_rel.max2 r b1 b2 \<and> (a1,b1) \<in> r - Id \<or>
blanchet@48975
  1208
           wo_rel.max2 r a1 a2 = wo_rel.max2 r b1 b2 \<and> a1 = b1 \<and> (a2,b2) \<in> r - Id"
blanchet@48975
  1209
  using * unfolding bsqr_def by auto
blanchet@48975
  1210
  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
  1211
           wo_rel.max2 r b1 b2 = wo_rel.max2 r a1 a2 \<and> (b1,a1) \<in> r - Id \<or>
blanchet@48975
  1212
           wo_rel.max2 r b1 b2 = wo_rel.max2 r a1 a2 \<and> b1 = a1 \<and> (b2,a2) \<in> r - Id"
blanchet@48975
  1213
  using ** unfolding bsqr_def by auto
blanchet@48975
  1214
  show "a1 = b1 \<and> a2 = b2"
blanchet@48975
  1215
  proof-
blanchet@48975
  1216
    {assume Case1: "(wo_rel.max2 r a1 a2, wo_rel.max2 r b1 b2) \<in> r - Id"
blanchet@48975
  1217
     {assume Case11: "(wo_rel.max2 r b1 b2, wo_rel.max2 r a1 a2) \<in> r - Id"
blanchet@48975
  1218
      hence False using Case1 IrrS by blast
blanchet@48975
  1219
     }
blanchet@48975
  1220
     moreover
blanchet@48975
  1221
     {assume Case12_3: "wo_rel.max2 r b1 b2 = wo_rel.max2 r a1 a2"
blanchet@48975
  1222
      hence False using Case1 by auto
blanchet@48975
  1223
     }
blanchet@48975
  1224
     ultimately have ?thesis using 0 2 by auto
blanchet@48975
  1225
    }
blanchet@48975
  1226
    moreover
blanchet@48975
  1227
    {assume Case2: "wo_rel.max2 r a1 a2 = wo_rel.max2 r b1 b2 \<and> (a1,b1) \<in> r - Id"
blanchet@48975
  1228
     {assume Case21: "(wo_rel.max2 r b1 b2, wo_rel.max2 r a1 a2) \<in> r - Id"
blanchet@48975
  1229
       hence False using Case2 by auto
blanchet@48975
  1230
     }
blanchet@48975
  1231
     moreover
blanchet@48975
  1232
     {assume Case22: "(b1,a1) \<in> r - Id"
blanchet@48975
  1233
      hence False using Case2 IrrS by blast
blanchet@48975
  1234
     }
blanchet@48975
  1235
     moreover
blanchet@48975
  1236
     {assume Case23: "b1 = a1"
blanchet@48975
  1237
      hence False using Case2 by auto
blanchet@48975
  1238
     }
blanchet@48975
  1239
     ultimately have ?thesis using 0 2 by auto
blanchet@48975
  1240
    }
blanchet@48975
  1241
    moreover
blanchet@48975
  1242
    {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
  1243
     moreover
blanchet@48975
  1244
     {assume Case31: "(wo_rel.max2 r b1 b2, wo_rel.max2 r a1 a2) \<in> r - Id"
blanchet@48975
  1245
      hence False using Case3 by auto
blanchet@48975
  1246
     }
blanchet@48975
  1247
     moreover
blanchet@48975
  1248
     {assume Case32: "(b1,a1) \<in> r - Id"
blanchet@48975
  1249
      hence False using Case3 by auto
blanchet@48975
  1250
     }
blanchet@48975
  1251
     moreover
blanchet@48975
  1252
     {assume Case33: "(b2,a2) \<in> r - Id"
blanchet@48975
  1253
      hence False using Case3 IrrS by blast
blanchet@48975
  1254
     }
blanchet@48975
  1255
     ultimately have ?thesis using 0 2 by auto
blanchet@48975
  1256
    }
blanchet@48975
  1257
    ultimately show ?thesis using 0 1 by blast
blanchet@48975
  1258
  qed
blanchet@48975
  1259
qed
blanchet@48975
  1260
blanchet@48975
  1261
lemma bsqr_Total:
blanchet@48975
  1262
assumes "Well_order r"
blanchet@48975
  1263
shows "Total(bsqr r)"
blanchet@48975
  1264
proof-
blanchet@48975
  1265
  (* Preliminary facts *)
blanchet@48975
  1266
  have Well: "wo_rel r" using assms wo_rel_def by auto
blanchet@48975
  1267
  hence Total: "\<forall>a \<in> Field r. \<forall>b \<in> Field r. (a,b) \<in> r \<or> (b,a) \<in> r"
blanchet@48975
  1268
  using wo_rel.TOTALS by auto
blanchet@48975
  1269
  (* Main proof *)
blanchet@48975
  1270
  {fix a1 a2 b1 b2 assume "{(a1,a2), (b1,b2)} \<le> Field(bsqr r)"
blanchet@48975
  1271
   hence 0: "a1 \<in> Field r \<and> a2 \<in> Field r \<and> b1 \<in> Field r \<and> b2 \<in> Field r"
blanchet@48975
  1272
   using Field_bsqr by blast
blanchet@48975
  1273
   have "((a1,a2) = (b1,b2) \<or> ((a1,a2),(b1,b2)) \<in> bsqr r \<or> ((b1,b2),(a1,a2)) \<in> bsqr r)"
blanchet@48975
  1274
   proof(rule wo_rel.cases_Total[of r a1 a2], clarsimp simp add: Well, simp add: 0)
blanchet@48975
  1275
       (* Why didn't clarsimp simp add: Well 0 do the same job? *)
blanchet@48975
  1276
     assume Case1: "(a1,a2) \<in> r"
blanchet@48975
  1277
     hence 1: "wo_rel.max2 r a1 a2 = a2"
blanchet@48975
  1278
     using Well 0 by (simp add: wo_rel.max2_equals2)
blanchet@48975
  1279
     show ?thesis
blanchet@48975
  1280
     proof(rule wo_rel.cases_Total[of r b1 b2], clarsimp simp add: Well, simp add: 0)
blanchet@48975
  1281
       assume Case11: "(b1,b2) \<in> r"
blanchet@48975
  1282
       hence 2: "wo_rel.max2 r b1 b2 = b2"
blanchet@48975
  1283
       using Well 0 by (simp add: wo_rel.max2_equals2)
blanchet@48975
  1284
       show ?thesis
blanchet@48975
  1285
       proof(rule wo_rel.cases_Total3[of r a2 b2], clarsimp simp add: Well, simp add: 0)
blanchet@48975
  1286
         assume Case111: "(a2,b2) \<in> r - Id \<or> (b2,a2) \<in> r - Id"
blanchet@48975
  1287
         thus ?thesis using 0 1 2 unfolding bsqr_def by auto
blanchet@48975
  1288
       next
blanchet@48975
  1289
         assume Case112: "a2 = b2"
blanchet@48975
  1290
         show ?thesis
blanchet@48975
  1291
         proof(rule wo_rel.cases_Total3[of r a1 b1], clarsimp simp add: Well, simp add: 0)
blanchet@48975
  1292
           assume Case1121: "(a1,b1) \<in> r - Id \<or> (b1,a1) \<in> r - Id"
blanchet@48975
  1293
           thus ?thesis using 0 1 2 Case112 unfolding bsqr_def by auto
blanchet@48975
  1294
         next
blanchet@48975
  1295
           assume Case1122: "a1 = b1"
blanchet@48975
  1296
           thus ?thesis using Case112 by auto
blanchet@48975
  1297
         qed
blanchet@48975
  1298
       qed
blanchet@48975
  1299
     next
blanchet@48975
  1300
       assume Case12: "(b2,b1) \<in> r"
blanchet@48975
  1301
       hence 3: "wo_rel.max2 r b1 b2 = b1" using Well 0 by (simp add: wo_rel.max2_equals1)
blanchet@48975
  1302
       show ?thesis
blanchet@48975
  1303
       proof(rule wo_rel.cases_Total3[of r a2 b1], clarsimp simp add: Well, simp add: 0)
blanchet@48975
  1304
         assume Case121: "(a2,b1) \<in> r - Id \<or> (b1,a2) \<in> r - Id"
blanchet@48975
  1305
         thus ?thesis using 0 1 3 unfolding bsqr_def by auto
blanchet@48975
  1306
       next
blanchet@48975
  1307
         assume Case122: "a2 = b1"
blanchet@48975
  1308
         show ?thesis
blanchet@48975
  1309
         proof(rule wo_rel.cases_Total3[of r a1 b1], clarsimp simp add: Well, simp add: 0)
blanchet@48975
  1310
           assume Case1221: "(a1,b1) \<in> r - Id \<or> (b1,a1) \<in> r - Id"
blanchet@48975
  1311
           thus ?thesis using 0 1 3 Case122 unfolding bsqr_def by auto
blanchet@48975
  1312
         next
blanchet@48975
  1313
           assume Case1222: "a1 = b1"
blanchet@48975
  1314
           show ?thesis
blanchet@48975
  1315
           proof(rule wo_rel.cases_Total3[of r a2 b2], clarsimp simp add: Well, simp add: 0)
blanchet@48975
  1316
             assume Case12221: "(a2,b2) \<in> r - Id \<or> (b2,a2) \<in> r - Id"
blanchet@48975
  1317
             thus ?thesis using 0 1 3 Case122 Case1222 unfolding bsqr_def by auto
blanchet@48975
  1318
           next
blanchet@48975
  1319
             assume Case12222: "a2 = b2"
blanchet@48975
  1320
             thus ?thesis using Case122 Case1222 by auto
blanchet@48975
  1321
           qed
blanchet@48975
  1322
         qed
blanchet@48975
  1323
       qed
blanchet@48975
  1324
     qed
blanchet@48975
  1325
   next
blanchet@48975
  1326
     assume Case2: "(a2,a1) \<in> r"
blanchet@48975
  1327
     hence 1: "wo_rel.max2 r a1 a2 = a1" using Well 0 by (simp add: wo_rel.max2_equals1)
blanchet@48975
  1328
     show ?thesis
blanchet@48975
  1329
     proof(rule wo_rel.cases_Total[of r b1 b2], clarsimp simp add: Well, simp add: 0)
blanchet@48975
  1330
       assume Case21: "(b1,b2) \<in> r"
blanchet@48975
  1331
       hence 2: "wo_rel.max2 r b1 b2 = b2" using Well 0 by (simp add: wo_rel.max2_equals2)
blanchet@48975
  1332
       show ?thesis
blanchet@48975
  1333
       proof(rule wo_rel.cases_Total3[of r a1 b2], clarsimp simp add: Well, simp add: 0)
blanchet@48975
  1334
         assume Case211: "(a1,b2) \<in> r - Id \<or> (b2,a1) \<in> r - Id"
blanchet@48975
  1335
         thus ?thesis using 0 1 2 unfolding bsqr_def by auto
blanchet@48975
  1336
       next
blanchet@48975
  1337
         assume Case212: "a1 = b2"
blanchet@48975
  1338
         show ?thesis
blanchet@48975
  1339
         proof(rule wo_rel.cases_Total3[of r a1 b1], clarsimp simp add: Well, simp add: 0)
blanchet@48975
  1340
           assume Case2121: "(a1,b1) \<in> r - Id \<or> (b1,a1) \<in> r - Id"
blanchet@48975
  1341
           thus ?thesis using 0 1 2 Case212 unfolding bsqr_def by auto
blanchet@48975
  1342
         next
blanchet@48975
  1343
           assume Case2122: "a1 = b1"
blanchet@48975
  1344
           show ?thesis
blanchet@48975
  1345
           proof(rule wo_rel.cases_Total3[of r a2 b2], clarsimp simp add: Well, simp add: 0)
blanchet@48975
  1346
             assume Case21221: "(a2,b2) \<in> r - Id \<or> (b2,a2) \<in> r - Id"
blanchet@48975
  1347
             thus ?thesis using 0 1 2 Case212 Case2122 unfolding bsqr_def by auto
blanchet@48975
  1348
           next
blanchet@48975
  1349
             assume Case21222: "a2 = b2"
blanchet@48975
  1350
             thus ?thesis using Case2122 Case212 by auto
blanchet@48975
  1351
           qed
blanchet@48975
  1352
         qed
blanchet@48975
  1353
       qed
blanchet@48975
  1354
     next
blanchet@48975
  1355
       assume Case22: "(b2,b1) \<in> r"
blanchet@48975
  1356
       hence 3: "wo_rel.max2 r b1 b2 = b1"  using Well 0 by (simp add: wo_rel.max2_equals1)
blanchet@48975
  1357
       show ?thesis
blanchet@48975
  1358
       proof(rule wo_rel.cases_Total3[of r a1 b1], clarsimp simp add: Well, simp add: 0)
blanchet@48975
  1359
         assume Case221: "(a1,b1) \<in> r - Id \<or> (b1,a1) \<in> r - Id"
blanchet@48975
  1360
         thus ?thesis using 0 1 3 unfolding bsqr_def by auto
blanchet@48975
  1361
       next
blanchet@48975
  1362
         assume Case222: "a1 = b1"
blanchet@48975
  1363
         show ?thesis
blanchet@48975
  1364
         proof(rule wo_rel.cases_Total3[of r a2 b2], clarsimp simp add: Well, simp add: 0)
blanchet@48975
  1365
           assume Case2221: "(a2,b2) \<in> r - Id \<or> (b2,a2) \<in> r - Id"
blanchet@48975
  1366
           thus ?thesis using 0 1 3 Case222 unfolding bsqr_def by auto
blanchet@48975
  1367
         next
blanchet@48975
  1368
           assume Case2222: "a2 = b2"
blanchet@48975
  1369
           thus ?thesis using Case222 by auto
blanchet@48975
  1370
         qed
blanchet@48975
  1371
       qed
blanchet@48975
  1372
     qed
blanchet@48975
  1373
   qed
blanchet@48975
  1374
  }
blanchet@48975
  1375
  thus ?thesis unfolding total_on_def by fast
blanchet@48975
  1376
qed
blanchet@48975
  1377
blanchet@48975
  1378
lemma bsqr_Linear_order:
blanchet@48975
  1379
assumes "Well_order r"
blanchet@48975
  1380
shows "Linear_order(bsqr r)"
blanchet@48975
  1381
unfolding order_on_defs
blanchet@48975
  1382
using assms bsqr_Refl bsqr_Trans bsqr_antisym bsqr_Total by blast
blanchet@48975
  1383
blanchet@48975
  1384
lemma bsqr_Well_order:
blanchet@48975
  1385
assumes "Well_order r"
blanchet@48975
  1386
shows "Well_order(bsqr r)"
blanchet@48975
  1387
using assms
blanchet@48975
  1388
proof(simp add: bsqr_Linear_order Linear_order_Well_order_iff, intro allI impI)
blanchet@48975
  1389
  have 0: "\<forall>A \<le> Field r. A \<noteq> {} \<longrightarrow> (\<exists>a \<in> A. \<forall>a' \<in> A. (a,a') \<in> r)"
blanchet@48975
  1390
  using assms well_order_on_def Linear_order_Well_order_iff by blast
blanchet@48975
  1391
  fix D assume *: "D \<le> Field (bsqr r)" and **: "D \<noteq> {}"
blanchet@48975
  1392
  hence 1: "D \<le> Field r \<times> Field r" unfolding Field_bsqr by simp
blanchet@48975
  1393
  (*  *)
blanchet@48975
  1394
  obtain M where M_def: "M = {wo_rel.max2 r a1 a2| a1 a2. (a1,a2) \<in> D}" by blast
blanchet@48975
  1395
  have "M \<noteq> {}" using 1 M_def ** by auto
blanchet@48975
  1396
  moreover
blanchet@48975
  1397
  have "M \<le> Field r" unfolding M_def
blanchet@48975
  1398
  using 1 assms wo_rel_def[of r] wo_rel.max2_among[of r] by fastforce
blanchet@48975
  1399
  ultimately obtain m where m_min: "m \<in> M \<and> (\<forall>a \<in> M. (m,a) \<in> r)"
blanchet@48975
  1400
  using 0 by blast
blanchet@48975
  1401
  (*  *)
blanchet@48975
  1402
  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
  1403
  have "A1 \<le> Field r" unfolding A1_def using 1 by auto
blanchet@48975
  1404
  moreover have "A1 \<noteq> {}" unfolding A1_def using m_min unfolding M_def by blast
blanchet@48975
  1405
  ultimately obtain a1 where a1_min: "a1 \<in> A1 \<and> (\<forall>a \<in> A1. (a1,a) \<in> r)"
blanchet@48975
  1406
  using 0 by blast
blanchet@48975
  1407
  (*  *)
blanchet@48975
  1408
  obtain A2 where A2_def: "A2 = {a2. (a1,a2) \<in> D \<and> wo_rel.max2 r a1 a2 = m}" by blast
blanchet@48975
  1409
  have "A2 \<le> Field r" unfolding A2_def using 1 by auto
blanchet@48975
  1410
  moreover have "A2 \<noteq> {}" unfolding A2_def
blanchet@48975
  1411
  using m_min a1_min unfolding A1_def M_def by blast
blanchet@48975
  1412
  ultimately obtain a2 where a2_min: "a2 \<in> A2 \<and> (\<forall>a \<in> A2. (a2,a) \<in> r)"
blanchet@48975
  1413
  using 0 by blast
blanchet@48975
  1414
  (*   *)
blanchet@48975
  1415
  have 2: "wo_rel.max2 r a1 a2 = m"
blanchet@48975
  1416
  using a1_min a2_min unfolding A1_def A2_def by auto
blanchet@48975
  1417
  have 3: "(a1,a2) \<in> D" using a2_min unfolding A2_def by auto
blanchet@48975
  1418
  (*  *)
blanchet@48975
  1419
  moreover
blanchet@48975
  1420
  {fix b1 b2 assume ***: "(b1,b2) \<in> D"
blanchet@48975
  1421
   hence 4: "{a1,a2,b1,b2} \<le> Field r" using 1 3 by blast
blanchet@48975
  1422
   have 5: "(wo_rel.max2 r a1 a2, wo_rel.max2 r b1 b2) \<in> r"
blanchet@48975
  1423
   using *** a1_min a2_min m_min unfolding A1_def A2_def M_def by auto
blanchet@48975
  1424
   have "((a1,a2),(b1,b2)) \<in> bsqr r"
blanchet@48975
  1425
   proof(cases "wo_rel.max2 r a1 a2 = wo_rel.max2 r b1 b2")
blanchet@48975
  1426
     assume Case1: "wo_rel.max2 r a1 a2 \<noteq> wo_rel.max2 r b1 b2"
blanchet@48975
  1427
     thus ?thesis unfolding bsqr_def using 4 5 by auto
blanchet@48975
  1428
   next
blanchet@48975
  1429
     assume Case2: "wo_rel.max2 r a1 a2 = wo_rel.max2 r b1 b2"
blanchet@48975
  1430
     hence "b1 \<in> A1" unfolding A1_def using 2 *** by auto
blanchet@48975
  1431
     hence 6: "(a1,b1) \<in> r" using a1_min by auto
blanchet@48975
  1432
     show ?thesis
blanchet@48975
  1433
     proof(cases "a1 = b1")
blanchet@48975
  1434
       assume Case21: "a1 \<noteq> b1"
blanchet@48975
  1435
       thus ?thesis unfolding bsqr_def using 4 Case2 6 by auto
blanchet@48975
  1436
     next
blanchet@48975
  1437
       assume Case22: "a1 = b1"
blanchet@48975
  1438
       hence "b2 \<in> A2" unfolding A2_def using 2 *** Case2 by auto
blanchet@48975
  1439
       hence 7: "(a2,b2) \<in> r" using a2_min by auto
blanchet@48975
  1440
       thus ?thesis unfolding bsqr_def using 4 7 Case2 Case22 by auto
blanchet@48975
  1441
     qed
blanchet@48975
  1442
   qed
blanchet@48975
  1443
  }
blanchet@48975
  1444
  (*  *)
blanchet@48975
  1445
  ultimately show "\<exists>d \<in> D. \<forall>d' \<in> D. (d,d') \<in> bsqr r" by fastforce
blanchet@48975
  1446
qed
blanchet@48975
  1447
blanchet@48975
  1448
lemma bsqr_max2:
blanchet@48975
  1449
assumes WELL: "Well_order r" and LEQ: "((a1,a2),(b1,b2)) \<in> bsqr r"
blanchet@48975
  1450
shows "(wo_rel.max2 r a1 a2, wo_rel.max2 r b1 b2) \<in> r"
blanchet@48975
  1451
proof-
blanchet@48975
  1452
  have "{(a1,a2),(b1,b2)} \<le> Field(bsqr r)"
blanchet@48975
  1453
  using LEQ unfolding Field_def by auto
blanchet@48975
  1454
  hence "{a1,a2,b1,b2} \<le> Field r" unfolding Field_bsqr by auto
blanchet@48975
  1455
  hence "{wo_rel.max2 r a1 a2, wo_rel.max2 r b1 b2} \<le> Field r"
blanchet@48975
  1456
  using WELL wo_rel_def[of r] wo_rel.max2_among[of r] by fastforce
blanchet@48975
  1457
  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
  1458
  using LEQ unfolding bsqr_def by auto
blanchet@48975
  1459
  ultimately show ?thesis using WELL unfolding order_on_defs refl_on_def by auto
blanchet@48975
  1460
qed
blanchet@48975
  1461
blanchet@48975
  1462
lemma bsqr_ofilter:
blanchet@48975
  1463
assumes WELL: "Well_order r" and
blanchet@48975
  1464
        OF: "wo_rel.ofilter (bsqr r) D" and SUB: "D < Field r \<times> Field r" and
blanchet@55023
  1465
        NE: "\<not> (\<exists>a. Field r = under r a)"
blanchet@48975
  1466
shows "\<exists>A. wo_rel.ofilter r A \<and> A < Field r \<and> D \<le> A \<times> A"
blanchet@48975
  1467
proof-
blanchet@48975
  1468
  let ?r' = "bsqr r"
blanchet@48975
  1469
  have Well: "wo_rel r" using WELL wo_rel_def by blast
blanchet@48975
  1470
  hence Trans: "trans r" using wo_rel.TRANS by blast
blanchet@48975
  1471
  have Well': "Well_order ?r' \<and> wo_rel ?r'"
blanchet@48975
  1472
  using WELL bsqr_Well_order wo_rel_def by blast
blanchet@48975
  1473
  (*  *)
blanchet@48975
  1474
  have "D < Field ?r'" unfolding Field_bsqr using SUB .
blanchet@48975
  1475
  with OF obtain a1 and a2 where
blanchet@55023
  1476
  "(a1,a2) \<in> Field ?r'" and 1: "D = underS ?r' (a1,a2)"
blanchet@48975
  1477
  using Well' wo_rel.ofilter_underS_Field[of ?r' D] by auto
blanchet@48975
  1478
  hence 2: "{a1,a2} \<le> Field r" unfolding Field_bsqr by auto
blanchet@48975
  1479
  let ?m = "wo_rel.max2 r a1 a2"
blanchet@55023
  1480
  have "D \<le> (under r ?m) \<times> (under r ?m)"
blanchet@48975
  1481
  proof(unfold 1)
blanchet@48975
  1482
    {fix b1 b2
blanchet@48975
  1483
     let ?n = "wo_rel.max2 r b1 b2"
blanchet@55023
  1484
     assume "(b1,b2) \<in> underS ?r' (a1,a2)"
blanchet@48975
  1485
     hence 3: "((b1,b2),(a1,a2)) \<in> ?r'"
blanchet@55023
  1486
     unfolding underS_def by blast
blanchet@48975
  1487
     hence "(?n,?m) \<in> r" using WELL by (simp add: bsqr_max2)
blanchet@48975
  1488
     moreover
blanchet@48975
  1489
     {have "(b1,b2) \<in> Field ?r'" using 3 unfolding Field_def by auto
blanchet@48975
  1490
      hence "{b1,b2} \<le> Field r" unfolding Field_bsqr by auto
blanchet@48975
  1491
      hence "(b1,?n) \<in> r \<and> (b2,?n) \<in> r"
blanchet@48975
  1492
      using Well by (simp add: wo_rel.max2_greater)
blanchet@48975
  1493
     }
blanchet@48975
  1494
     ultimately have "(b1,?m) \<in> r \<and> (b2,?m) \<in> r"
blanchet@48975
  1495
     using Trans trans_def[of r] by blast
blanchet@55023
  1496
     hence "(b1,b2) \<in> (under r ?m) \<times> (under r ?m)" unfolding under_def by simp}
blanchet@55023
  1497
     thus "underS ?r' (a1,a2) \<le> (under r ?m) \<times> (under r ?m)" by auto
blanchet@48975
  1498
  qed
blanchet@55023
  1499
  moreover have "wo_rel.ofilter r (under r ?m)"
blanchet@48975
  1500
  using Well by (simp add: wo_rel.under_ofilter)
blanchet@55023
  1501
  moreover have "under r ?m < Field r"
blanchet@55023
  1502
  using NE under_Field[of r ?m] by blast
blanchet@48975
  1503
  ultimately show ?thesis by blast
blanchet@48975
  1504
qed
blanchet@48975
  1505
traytel@54980
  1506
definition Func where
traytel@54980
  1507
"Func A B = {f . (\<forall> a \<in> A. f a \<in> B) \<and> (\<forall> a. a \<notin> A \<longrightarrow> f a = undefined)}"
traytel@54980
  1508
traytel@54980
  1509
lemma Func_empty:
traytel@54980
  1510
"Func {} B = {\<lambda>x. undefined}"
traytel@54980
  1511
unfolding Func_def by auto
traytel@54980
  1512
traytel@54980
  1513
lemma Func_elim:
traytel@54980
  1514
assumes "g \<in> Func A B" and "a \<in> A"
traytel@54980
  1515
shows "\<exists> b. b \<in> B \<and> g a = b"
traytel@54980
  1516
using assms unfolding Func_def by (cases "g a = undefined") auto
traytel@54980
  1517
traytel@54980
  1518
definition curr where
traytel@54980
  1519
"curr A f \<equiv> \<lambda> a. if a \<in> A then \<lambda>b. f (a,b) else undefined"
traytel@54980
  1520
traytel@54980
  1521
lemma curr_in:
wenzelm@61943
  1522
assumes f: "f \<in> Func (A \<times> B) C"
traytel@54980
  1523
shows "curr A f \<in> Func A (Func B C)"
traytel@54980
  1524
using assms unfolding curr_def Func_def by auto
traytel@54980
  1525
traytel@54980
  1526
lemma curr_inj:
wenzelm@61943
  1527
assumes "f1 \<in> Func (A \<times> B) C" and "f2 \<in> Func (A \<times> B) C"
traytel@54980
  1528
shows "curr A f1 = curr A f2 \<longleftrightarrow> f1 = f2"
traytel@54980
  1529
proof safe
traytel@54980
  1530
  assume c: "curr A f1 = curr A f2"
traytel@54980
  1531
  show "f1 = f2"
traytel@54980
  1532
  proof (rule ext, clarify)
traytel@54980
  1533
    fix a b show "f1 (a, b) = f2 (a, b)"
wenzelm@61943
  1534
    proof (cases "(a,b) \<in> A \<times> B")
traytel@54980
  1535
      case False
traytel@54980
  1536
      thus ?thesis using assms unfolding Func_def by auto
traytel@54980
  1537
    next
traytel@54980
  1538
      case True hence a: "a \<in> A" and b: "b \<in> B" by auto
traytel@54980
  1539
      thus ?thesis
traytel@54980
  1540
      using c unfolding curr_def fun_eq_iff by(elim allE[of _ a]) simp
traytel@54980
  1541
    qed
traytel@54980
  1542
  qed
traytel@54980
  1543
qed
traytel@54980
  1544
traytel@54980
  1545
lemma curr_surj:
traytel@54980
  1546
assumes "g \<in> Func A (Func B C)"
wenzelm@61943
  1547
shows "\<exists> f \<in> Func (A \<times> B) C. curr A f = g"
traytel@54980
  1548
proof
traytel@54980
  1549
  let ?f = "\<lambda> ab. if fst ab \<in> A \<and> snd ab \<in> B then g (fst ab) (snd ab) else undefined"
traytel@54980
  1550
  show "curr A ?f = g"
traytel@54980
  1551
  proof (rule ext)
traytel@54980
  1552
    fix a show "curr A ?f a = g a"
traytel@54980
  1553
    proof (cases "a \<in> A")
traytel@54980
  1554
      case False
traytel@54980
  1555
      hence "g a = undefined" using assms unfolding Func_def by auto
traytel@54980
  1556
      thus ?thesis unfolding curr_def using False by simp
traytel@54980
  1557
    next
traytel@54980
  1558
      case True
traytel@54980
  1559
      obtain g1 where "g1 \<in> Func B C" and "g a = g1"
traytel@54980
  1560
      using assms using Func_elim[OF assms True] by blast
traytel@54980
  1561
      thus ?thesis using True unfolding Func_def curr_def by auto
traytel@54980
  1562
    qed
traytel@54980
  1563
  qed
wenzelm@61943
  1564
  show "?f \<in> Func (A \<times> B) C" using assms unfolding Func_def mem_Collect_eq by auto
traytel@54980
  1565
qed
traytel@54980
  1566
traytel@54980
  1567
lemma bij_betw_curr:
wenzelm@61943
  1568
"bij_betw (curr A) (Func (A \<times> B) C) (Func A (Func B C))"
traytel@54980
  1569
unfolding bij_betw_def inj_on_def image_def
traytel@54980
  1570
apply (intro impI conjI ballI)
traytel@54980
  1571
apply (erule curr_inj[THEN iffD1], assumption+)
traytel@54980
  1572
apply auto
traytel@54980
  1573
apply (erule curr_in)
traytel@54980
  1574
using curr_surj by blast
traytel@54980
  1575
traytel@54980
  1576
definition Func_map where
traytel@54980
  1577
"Func_map B2 f1 f2 g b2 \<equiv> if b2 \<in> B2 then f1 (g (f2 b2)) else undefined"
traytel@54980
  1578
traytel@54980
  1579
lemma Func_map:
traytel@54980
  1580
assumes g: "g \<in> Func A2 A1" and f1: "f1 ` A1 \<subseteq> B1" and f2: "f2 ` B2 \<subseteq> A2"
traytel@54980
  1581
shows "Func_map B2 f1 f2 g \<in> Func B2 B1"
traytel@54980
  1582
using assms unfolding Func_def Func_map_def mem_Collect_eq by auto
traytel@54980
  1583
traytel@54980
  1584
lemma Func_non_emp:
traytel@54980
  1585
assumes "B \<noteq> {}"
traytel@54980
  1586
shows "Func A B \<noteq> {}"
traytel@54980
  1587
proof-
traytel@54980
  1588
  obtain b where b: "b \<in> B" using assms by auto
traytel@54980
  1589
  hence "(\<lambda> a. if a \<in> A then b else undefined) \<in> Func A B" unfolding Func_def by auto
traytel@54980
  1590
  thus ?thesis by blast
traytel@54980
  1591
qed
traytel@54980
  1592
traytel@54980
  1593
lemma Func_is_emp:
traytel@54980
  1594
"Func A B = {} \<longleftrightarrow> A \<noteq> {} \<and> B = {}" (is "?L \<longleftrightarrow> ?R")
traytel@54980
  1595
proof
traytel@54980
  1596
  assume L: ?L
traytel@54980
  1597
  moreover {assume "A = {}" hence False using L Func_empty by auto}
traytel@55811
  1598
  moreover {assume "B \<noteq> {}" hence False using L Func_non_emp[of B A] by simp }
traytel@54980
  1599
  ultimately show ?R by blast
traytel@54980
  1600
next
traytel@54980
  1601
  assume R: ?R
traytel@54980
  1602
  moreover
traytel@54980
  1603
  {fix f assume "f \<in> Func A B"
traytel@54980
  1604
   moreover obtain a where "a \<in> A" using R by blast
traytel@54980
  1605
   ultimately obtain b where "b \<in> B" unfolding Func_def by blast
traytel@54980
  1606
   with R have False by blast
traytel@54980
  1607
  }
traytel@54980
  1608
  thus ?L by blast
traytel@54980
  1609
qed
traytel@54980
  1610
traytel@54980
  1611
lemma Func_map_surj:
traytel@54980
  1612
assumes B1: "f1 ` A1 = B1" and A2: "inj_on f2 B2" "f2 ` B2 \<subseteq> A2"
traytel@54980
  1613
and B2A2: "B2 = {} \<Longrightarrow> A2 = {}"
traytel@54980
  1614
shows "Func B2 B1 = Func_map B2 f1 f2 ` Func A2 A1"
traytel@54980
  1615
proof(cases "B2 = {}")
traytel@54980
  1616
  case True
traytel@54980
  1617
  thus ?thesis using B2A2 by (auto simp: Func_empty Func_map_def)
traytel@54980
  1618
next
traytel@54980
  1619
  case False note B2 = False
traytel@54980
  1620
  show ?thesis
traytel@54980
  1621
  proof safe
traytel@54980
  1622
    fix h assume h: "h \<in> Func B2 B1"
wenzelm@63040
  1623
    define j1 where "j1 = inv_into A1 f1"
traytel@54980
  1624
    have "\<forall> a2 \<in> f2 ` B2. \<exists> b2. b2 \<in> B2 \<and> f2 b2 = a2" by blast
traytel@55811
  1625
    then obtain k where k: "\<forall> a2 \<in> f2 ` B2. k a2 \<in> B2 \<and> f2 (k a2) = a2"
traytel@55811
  1626
      by atomize_elim (rule bchoice)
traytel@54980
  1627
    {fix b2 assume b2: "b2 \<in> B2"
traytel@54980
  1628
     hence "f2 (k (f2 b2)) = f2 b2" using k A2(2) by auto
traytel@54980
  1629
     moreover have "k (f2 b2) \<in> B2" using b2 A2(2) k by auto
traytel@54980
  1630
     ultimately have "k (f2 b2) = b2" using b2 A2(1) unfolding inj_on_def by blast
traytel@54980
  1631
    } note kk = this
traytel@54980
  1632
    obtain b22 where b22: "b22 \<in> B2" using B2 by auto
wenzelm@63040
  1633
    define j2 where [abs_def]: "j2 a2 = (if a2 \<in> f2 ` B2 then k a2 else b22)" for a2
traytel@54980
  1634
    have j2A2: "j2 ` A2 \<subseteq> B2" unfolding j2_def using k b22 by auto
traytel@54980
  1635
    have j2: "\<And> b2. b2 \<in> B2 \<Longrightarrow> j2 (f2 b2) = b2"
traytel@54980
  1636
    using kk unfolding j2_def by auto
wenzelm@63040
  1637
    define g where "g = Func_map A2 j1 j2 h"
traytel@54980
  1638
    have "Func_map B2 f1 f2 g = h"
traytel@54980
  1639
    proof (rule ext)
traytel@54980
  1640
      fix b2 show "Func_map B2 f1 f2 g b2 = h b2"
traytel@54980
  1641
      proof(cases "b2 \<in> B2")
traytel@54980
  1642
        case True
traytel@54980
  1643
        show ?thesis
traytel@54980
  1644
        proof (cases "h b2 = undefined")
traytel@54980
  1645
          case True
wenzelm@60758
  1646
          hence b1: "h b2 \<in> f1 ` A1" using h \<open>b2 \<in> B2\<close> unfolding B1 Func_def by auto
traytel@54980
  1647
          show ?thesis using A2 f_inv_into_f[OF b1]
wenzelm@60758
  1648
            unfolding True g_def Func_map_def j1_def j2[OF \<open>b2 \<in> B2\<close>] by auto
traytel@54980
  1649
        qed(insert A2 True j2[OF True] h B1, unfold j1_def g_def Func_def Func_map_def,
traytel@54980
  1650
          auto intro: f_inv_into_f)
traytel@54980
  1651
      qed(insert h, unfold Func_def Func_map_def, auto)
traytel@54980
  1652
    qed
traytel@54980
  1653
    moreover have "g \<in> Func A2 A1" unfolding g_def apply(rule Func_map[OF h])
haftmann@56077
  1654
    using j2A2 B1 A2 unfolding j1_def by (fast intro: inv_into_into)+
traytel@54980
  1655
    ultimately show "h \<in> Func_map B2 f1 f2 ` Func A2 A1"
haftmann@56077
  1656
    unfolding Func_map_def[abs_def] by auto
traytel@54980
  1657
  qed(insert B1 Func_map[OF _ _ A2(2)], auto)
traytel@54980
  1658
qed
blanchet@48975
  1659
blanchet@48975
  1660
end