src/HOL/BNF_Constructions_on_Wellorders.thy

+(*  Title:      HOL/BNF_Constructions_on_Wellorders.thy
+    Author:     Andrei Popescu, TU Muenchen
+    Copyright   2012
+
+Constructions on wellorders (BNF).
+*)
+
+header {* Constructions on Wellorders (BNF) *}
+
+theory BNF_Constructions_on_Wellorders
+imports BNF_Wellorder_Embedding
+begin
+
+
+text {* In this section, we study basic constructions on well-orders, such as restriction to
+a set/order filter, copy via direct images, ordinal-like sum of disjoint well-orders,
+and bounded square.  We also define between well-orders
+the relations @{text "ordLeq"}, of being embedded (abbreviated @{text "\<le>o"}),
+@{text "ordLess"}, of being strictly embedded (abbreviated @{text "<o"}), and
+@{text "ordIso"}, of being isomorphic (abbreviated @{text "=o"}).  We study the
+connections between these relations, order filters, and the aforementioned constructions.
+A main result of this section is that @{text "<o"} is well-founded.  *}
+
+
+subsection {* Restriction to a set  *}
+
+
+abbreviation Restr :: "'a rel \<Rightarrow> 'a set \<Rightarrow> 'a rel"
+where "Restr r A \<equiv> r Int (A \<times> A)"
+
+
+lemma Restr_subset:
+"A \<le> B \<Longrightarrow> Restr (Restr r B) A = Restr r A"
+by blast
+
+
+lemma Restr_Field: "Restr r (Field r) = r"
+unfolding Field_def by auto
+
+
+lemma Refl_Restr: "Refl r \<Longrightarrow> Refl(Restr r A)"
+unfolding refl_on_def Field_def by auto
+
+
+lemma antisym_Restr:
+"antisym r \<Longrightarrow> antisym(Restr r A)"
+unfolding antisym_def Field_def by auto
+
+
+lemma Total_Restr:
+"Total r \<Longrightarrow> Total(Restr r A)"
+unfolding total_on_def Field_def by auto
+
+
+lemma trans_Restr:
+"trans r \<Longrightarrow> trans(Restr r A)"
+unfolding trans_def Field_def by blast
+
+
+lemma Preorder_Restr:
+"Preorder r \<Longrightarrow> Preorder(Restr r A)"
+unfolding preorder_on_def by (simp add: Refl_Restr trans_Restr)
+
+
+lemma Partial_order_Restr:
+"Partial_order r \<Longrightarrow> Partial_order(Restr r A)"
+unfolding partial_order_on_def by (simp add: Preorder_Restr antisym_Restr)
+
+
+lemma Linear_order_Restr:
+"Linear_order r \<Longrightarrow> Linear_order(Restr r A)"
+unfolding linear_order_on_def by (simp add: Partial_order_Restr Total_Restr)
+
+
+lemma Well_order_Restr:
+assumes "Well_order r"
+shows "Well_order(Restr r A)"
+proof-
+  have "Restr r A - Id \<le> r - Id" using Restr_subset by blast
+  hence "wf(Restr r A - Id)" using assms
+  using well_order_on_def wf_subset by blast
+  thus ?thesis using assms unfolding well_order_on_def
+  by (simp add: Linear_order_Restr)
+qed
+
+
+lemma Field_Restr_subset: "Field(Restr r A) \<le> A"
+by (auto simp add: Field_def)
+
+
+lemma Refl_Field_Restr:
+"Refl r \<Longrightarrow> Field(Restr r A) = (Field r) Int A"
+unfolding refl_on_def Field_def by blast
+
+
+lemma Refl_Field_Restr2:
+"\<lbrakk>Refl r; A \<le> Field r\<rbrakk> \<Longrightarrow> Field(Restr r A) = A"
+by (auto simp add: Refl_Field_Restr)
+
+
+lemma well_order_on_Restr:
+assumes WELL: "Well_order r" and SUB: "A \<le> Field r"
+shows "well_order_on A (Restr r A)"
+using assms
+using Well_order_Restr[of r A] Refl_Field_Restr2[of r A]
+     order_on_defs[of "Field r" r] by auto
+
+
+subsection {* Order filters versus restrictions and embeddings  *}
+
+
+lemma Field_Restr_ofilter:
+"\<lbrakk>Well_order r; wo_rel.ofilter r A\<rbrakk> \<Longrightarrow> Field(Restr r A) = A"
+by (auto simp add: wo_rel_def wo_rel.ofilter_def wo_rel.REFL Refl_Field_Restr2)
+
+
+lemma ofilter_Restr_under:
+assumes WELL: "Well_order r" and OF: "wo_rel.ofilter r A" and IN: "a \<in> A"
+shows "under (Restr r A) a = under r a"
+using assms wo_rel_def
+proof(auto simp add: wo_rel.ofilter_def under_def)
+  fix b assume *: "a \<in> A" and "(b,a) \<in> r"
+  hence "b \<in> under r a \<and> a \<in> Field r"
+  unfolding under_def using Field_def by fastforce
+  thus "b \<in> A" using * assms by (auto simp add: wo_rel_def wo_rel.ofilter_def)
+qed
+
+
+lemma ofilter_embed:
+assumes "Well_order r"
+shows "wo_rel.ofilter r A = (A \<le> Field r \<and> embed (Restr r A) r id)"
+proof
+  assume *: "wo_rel.ofilter r A"
+  show "A \<le> Field r \<and> embed (Restr r A) r id"
+  proof(unfold embed_def, auto)
+    fix a assume "a \<in> A" thus "a \<in> Field r" using assms *
+    by (auto simp add: wo_rel_def wo_rel.ofilter_def)
+  next
+    fix a assume "a \<in> Field (Restr r A)"
+    thus "bij_betw id (under (Restr r A) a) (under r a)" using assms *
+    by (simp add: ofilter_Restr_under Field_Restr_ofilter)
+  qed
+next
+  assume *: "A \<le> Field r \<and> embed (Restr r A) r id"
+  hence "Field(Restr r A) \<le> Field r"
+  using assms  embed_Field[of "Restr r A" r id] id_def
+        Well_order_Restr[of r] by auto
+  {fix a assume "a \<in> A"
+   hence "a \<in> Field(Restr r A)" using * assms
+   by (simp add: order_on_defs Refl_Field_Restr2)
+   hence "bij_betw id (under (Restr r A) a) (under r a)"
+   using * unfolding embed_def by auto
+   hence "under r a \<le> under (Restr r A) a"
+   unfolding bij_betw_def by auto
+   also have "\<dots> \<le> Field(Restr r A)" by (simp add: under_Field)
+   also have "\<dots> \<le> A" by (simp add: Field_Restr_subset)
+   finally have "under r a \<le> A" .
+  }
+  thus "wo_rel.ofilter r A" using assms * by (simp add: wo_rel_def wo_rel.ofilter_def)
+qed
+
+
+lemma ofilter_Restr_Int:
+assumes WELL: "Well_order r" and OFA: "wo_rel.ofilter r A"
+shows "wo_rel.ofilter (Restr r B) (A Int B)"
+proof-
+  let ?rB = "Restr r B"
+  have Well: "wo_rel r" unfolding wo_rel_def using WELL .
+  hence Refl: "Refl r" by (simp add: wo_rel.REFL)
+  hence Field: "Field ?rB = Field r Int B"
+  using Refl_Field_Restr by blast
+  have WellB: "wo_rel ?rB \<and> Well_order ?rB" using WELL
+  by (simp add: Well_order_Restr wo_rel_def)
+  (* Main proof *)
+  show ?thesis using WellB assms
+  proof(auto simp add: wo_rel.ofilter_def under_def)
+    fix a assume "a \<in> A" and *: "a \<in> B"
+    hence "a \<in> Field r" using OFA Well by (auto simp add: wo_rel.ofilter_def)
+    with * show "a \<in> Field ?rB" using Field by auto
+  next
+    fix a b assume "a \<in> A" and "(b,a) \<in> r"
+    thus "b \<in> A" using Well OFA by (auto simp add: wo_rel.ofilter_def under_def)
+  qed
+qed
+
+
+lemma ofilter_Restr_subset:
+assumes WELL: "Well_order r" and OFA: "wo_rel.ofilter r A" and SUB: "A \<le> B"
+shows "wo_rel.ofilter (Restr r B) A"
+proof-
+  have "A Int B = A" using SUB by blast
+  thus ?thesis using assms ofilter_Restr_Int[of r A B] by auto
+qed
+
+
+lemma ofilter_subset_embed:
+assumes WELL: "Well_order r" and
+        OFA: "wo_rel.ofilter r A" and OFB: "wo_rel.ofilter r B"
+shows "(A \<le> B) = (embed (Restr r A) (Restr r B) id)"
+proof-
+  let ?rA = "Restr r A"  let ?rB = "Restr r B"
+  have Well: "wo_rel r" unfolding wo_rel_def using WELL .
+  hence Refl: "Refl r" by (simp add: wo_rel.REFL)
+  hence FieldA: "Field ?rA = Field r Int A"
+  using Refl_Field_Restr by blast
+  have FieldB: "Field ?rB = Field r Int B"
+  using Refl Refl_Field_Restr by blast
+  have WellA: "wo_rel ?rA \<and> Well_order ?rA" using WELL
+  by (simp add: Well_order_Restr wo_rel_def)
+  have WellB: "wo_rel ?rB \<and> Well_order ?rB" using WELL
+  by (simp add: Well_order_Restr wo_rel_def)
+  (* Main proof *)
+  show ?thesis
+  proof
+    assume *: "A \<le> B"
+    hence "wo_rel.ofilter (Restr r B) A" using assms
+    by (simp add: ofilter_Restr_subset)
+    hence "embed (Restr ?rB A) (Restr r B) id"
+    using WellB ofilter_embed[of "?rB" A] by auto
+    thus "embed (Restr r A) (Restr r B) id"
+    using * by (simp add: Restr_subset)
+  next
+    assume *: "embed (Restr r A) (Restr r B) id"
+    {fix a assume **: "a \<in> A"
+     hence "a \<in> Field r" using Well OFA by (auto simp add: wo_rel.ofilter_def)
+     with ** FieldA have "a \<in> Field ?rA" by auto
+     hence "a \<in> Field ?rB" using * WellA embed_Field[of ?rA ?rB id] by auto
+     hence "a \<in> B" using FieldB by auto
+    }
+    thus "A \<le> B" by blast
+  qed
+qed
+
+
+lemma ofilter_subset_embedS_iso:
+assumes WELL: "Well_order r" and
+        OFA: "wo_rel.ofilter r A" and OFB: "wo_rel.ofilter r B"
+shows "((A < B) = (embedS (Restr r A) (Restr r B) id)) \<and>
+       ((A = B) = (iso (Restr r A) (Restr r B) id))"
+proof-
+  let ?rA = "Restr r A"  let ?rB = "Restr r B"
+  have Well: "wo_rel r" unfolding wo_rel_def using WELL .
+  hence Refl: "Refl r" by (simp add: wo_rel.REFL)
+  hence "Field ?rA = Field r Int A"
+  using Refl_Field_Restr by blast
+  hence FieldA: "Field ?rA = A" using OFA Well
+  by (auto simp add: wo_rel.ofilter_def)
+  have "Field ?rB = Field r Int B"
+  using Refl Refl_Field_Restr by blast
+  hence FieldB: "Field ?rB = B" using OFB Well
+  by (auto simp add: wo_rel.ofilter_def)
+  (* Main proof *)
+  show ?thesis unfolding embedS_def iso_def
+  using assms ofilter_subset_embed[of r A B]
+        FieldA FieldB bij_betw_id_iff[of A B] by auto
+qed
+
+
+lemma ofilter_subset_embedS:
+assumes WELL: "Well_order r" and
+        OFA: "wo_rel.ofilter r A" and OFB: "wo_rel.ofilter r B"
+shows "(A < B) = embedS (Restr r A) (Restr r B) id"
+using assms by (simp add: ofilter_subset_embedS_iso)
+
+
+lemma embed_implies_iso_Restr:
+assumes WELL: "Well_order r" and WELL': "Well_order r'" and
+        EMB: "embed r' r f"
+shows "iso r' (Restr r (f ` (Field r'))) f"
+proof-
+  let ?A' = "Field r'"
+  let ?r'' = "Restr r (f ` ?A')"
+  have 0: "Well_order ?r''" using WELL Well_order_Restr by blast
+  have 1: "wo_rel.ofilter r (f ` ?A')" using assms embed_Field_ofilter  by blast
+  hence "Field ?r'' = f ` (Field r')" using WELL Field_Restr_ofilter by blast
+  hence "bij_betw f ?A' (Field ?r'')"
+  using EMB embed_inj_on WELL' unfolding bij_betw_def by blast
+  moreover
+  {have "\<forall>a b. (a,b) \<in> r' \<longrightarrow> a \<in> Field r' \<and> b \<in> Field r'"
+   unfolding Field_def by auto
+   hence "compat r' ?r'' f"
+   using assms embed_iff_compat_inj_on_ofilter
+   unfolding compat_def by blast
+  }
+  ultimately show ?thesis using WELL' 0 iso_iff3 by blast
+qed
+
+
+subsection {* The strict inclusion on proper ofilters is well-founded *}
+
+
+definition ofilterIncl :: "'a rel \<Rightarrow> 'a set rel"
+where
+"ofilterIncl r \<equiv> {(A,B). wo_rel.ofilter r A \<and> A \<noteq> Field r \<and>
+                         wo_rel.ofilter r B \<and> B \<noteq> Field r \<and> A < B}"
+
+
+lemma wf_ofilterIncl:
+assumes WELL: "Well_order r"
+shows "wf(ofilterIncl r)"
+proof-
+  have Well: "wo_rel r" using WELL by (simp add: wo_rel_def)
+  hence Lo: "Linear_order r" by (simp add: wo_rel.LIN)
+  let ?h = "(\<lambda> A. wo_rel.suc r A)"
+  let ?rS = "r - Id"
+  have "wf ?rS" using WELL by (simp add: order_on_defs)
+  moreover
+  have "compat (ofilterIncl r) ?rS ?h"
+  proof(unfold compat_def ofilterIncl_def,
+        intro allI impI, simp, elim conjE)
+    fix A B
+    assume *: "wo_rel.ofilter r A" "A \<noteq> Field r" and
+           **: "wo_rel.ofilter r B" "B \<noteq> Field r" and ***: "A < B"
+    then obtain a and b where 0: "a \<in> Field r \<and> b \<in> Field r" and
+                         1: "A = underS r a \<and> B = underS r b"
+    using Well by (auto simp add: wo_rel.ofilter_underS_Field)
+    hence "a \<noteq> b" using *** by auto
+    moreover
+    have "(a,b) \<in> r" using 0 1 Lo ***
+    by (auto simp add: underS_incl_iff)
+    moreover
+    have "a = wo_rel.suc r A \<and> b = wo_rel.suc r B"
+    using Well 0 1 by (simp add: wo_rel.suc_underS)
+    ultimately
+    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"
+    by simp
+  qed
+  ultimately show "wf (ofilterIncl r)" by (simp add: compat_wf)
+qed
+
+
+
+subsection {* Ordering the well-orders by existence of embeddings *}
+
+
+text {* We define three relations between well-orders:
+\begin{itemize}
+\item @{text "ordLeq"}, of being embedded (abbreviated @{text "\<le>o"});
+\item @{text "ordLess"}, of being strictly embedded (abbreviated @{text "<o"});
+\item @{text "ordIso"}, of being isomorphic (abbreviated @{text "=o"}).
+\end{itemize}
+%
+The prefix "ord" and the index "o" in these names stand for "ordinal-like".
+These relations shall be proved to be inter-connected in a similar fashion as the trio
+@{text "\<le>"}, @{text "<"}, @{text "="} associated to a total order on a set.
+*}
+
+
+definition ordLeq :: "('a rel * 'a' rel) set"
+where
+"ordLeq = {(r,r'). Well_order r \<and> Well_order r' \<and> (\<exists>f. embed r r' f)}"
+
+
+abbreviation ordLeq2 :: "'a rel \<Rightarrow> 'a' rel \<Rightarrow> bool" (infix "<=o" 50)
+where "r <=o r' \<equiv> (r,r') \<in> ordLeq"
+
+
+abbreviation ordLeq3 :: "'a rel \<Rightarrow> 'a' rel \<Rightarrow> bool" (infix "\<le>o" 50)
+where "r \<le>o r' \<equiv> r <=o r'"
+
+
+definition ordLess :: "('a rel * 'a' rel) set"
+where
+"ordLess = {(r,r'). Well_order r \<and> Well_order r' \<and> (\<exists>f. embedS r r' f)}"
+
+abbreviation ordLess2 :: "'a rel \<Rightarrow> 'a' rel \<Rightarrow> bool" (infix "<o" 50)
+where "r <o r' \<equiv> (r,r') \<in> ordLess"
+
+
+definition ordIso :: "('a rel * 'a' rel) set"
+where
+"ordIso = {(r,r'). Well_order r \<and> Well_order r' \<and> (\<exists>f. iso r r' f)}"
+
+abbreviation ordIso2 :: "'a rel \<Rightarrow> 'a' rel \<Rightarrow> bool" (infix "=o" 50)
+where "r =o r' \<equiv> (r,r') \<in> ordIso"
+
+
+lemmas ordRels_def = ordLeq_def ordLess_def ordIso_def
+
+lemma ordLeq_Well_order_simp:
+assumes "r \<le>o r'"
+shows "Well_order r \<and> Well_order r'"
+using assms unfolding ordLeq_def by simp
+
+
+text{* Notice that the relations @{text "\<le>o"}, @{text "<o"}, @{text "=o"} connect well-orders
+on potentially {\em distinct} types. However, some of the lemmas below, including the next one,
+restrict implicitly the type of these relations to @{text "(('a rel) * ('a rel)) set"} , i.e.,
+to @{text "'a rel rel"}.  *}
+
+
+lemma ordLeq_reflexive:
+"Well_order r \<Longrightarrow> r \<le>o r"
+unfolding ordLeq_def using id_embed[of r] by blast
+
+
+lemma ordLeq_transitive[trans]:
+assumes *: "r \<le>o r'" and **: "r' \<le>o r''"
+shows "r \<le>o r''"
+proof-
+  obtain f and f'
+  where 1: "Well_order r \<and> Well_order r' \<and> Well_order r''" and
+        "embed r r' f" and "embed r' r'' f'"
+  using * ** unfolding ordLeq_def by blast
+  hence "embed r r'' (f' o f)"
+  using comp_embed[of r r' f r'' f'] by auto
+  thus "r \<le>o r''" unfolding ordLeq_def using 1 by auto
+qed
+
+
+lemma ordLeq_total:
+"\<lbrakk>Well_order r; Well_order r'\<rbrakk> \<Longrightarrow> r \<le>o r' \<or> r' \<le>o r"
+unfolding ordLeq_def using wellorders_totally_ordered by blast
+
+
+lemma ordIso_reflexive:
+"Well_order r \<Longrightarrow> r =o r"
+unfolding ordIso_def using id_iso[of r] by blast
+
+
+lemma ordIso_transitive[trans]:
+assumes *: "r =o r'" and **: "r' =o r''"
+shows "r =o r''"
+proof-
+  obtain f and f'
+  where 1: "Well_order r \<and> Well_order r' \<and> Well_order r''" and
+        "iso r r' f" and 3: "iso r' r'' f'"
+  using * ** unfolding ordIso_def by auto
+  hence "iso r r'' (f' o f)"
+  using comp_iso[of r r' f r'' f'] by auto
+  thus "r =o r''" unfolding ordIso_def using 1 by auto
+qed
+
+
+lemma ordIso_symmetric:
+assumes *: "r =o r'"
+shows "r' =o r"
+proof-
+  obtain f where 1: "Well_order r \<and> Well_order r'" and
+                 2: "embed r r' f \<and> bij_betw f (Field r) (Field r')"
+  using * by (auto simp add: ordIso_def iso_def)
+  let ?f' = "inv_into (Field r) f"
+  have "embed r' r ?f' \<and> bij_betw ?f' (Field r') (Field r)"
+  using 1 2 by (simp add: bij_betw_inv_into inv_into_Field_embed_bij_betw)
+  thus "r' =o r" unfolding ordIso_def using 1 by (auto simp add: iso_def)
+qed
+
+
+lemma ordLeq_ordLess_trans[trans]:
+assumes "r \<le>o r'" and " r' <o r''"
+shows "r <o r''"
+proof-
+  have "Well_order r \<and> Well_order r''"
+  using assms unfolding ordLeq_def ordLess_def by auto
+  thus ?thesis using assms unfolding ordLeq_def ordLess_def
+  using embed_comp_embedS by blast
+qed
+
+
+lemma ordLess_ordLeq_trans[trans]:
+assumes "r <o r'" and " r' \<le>o r''"
+shows "r <o r''"
+proof-
+  have "Well_order r \<and> Well_order r''"
+  using assms unfolding ordLeq_def ordLess_def by auto
+  thus ?thesis using assms unfolding ordLeq_def ordLess_def
+  using embedS_comp_embed by blast
+qed
+
+
+lemma ordLeq_ordIso_trans[trans]:
+assumes "r \<le>o r'" and " r' =o r''"
+shows "r \<le>o r''"
+proof-
+  have "Well_order r \<and> Well_order r''"
+  using assms unfolding ordLeq_def ordIso_def by auto
+  thus ?thesis using assms unfolding ordLeq_def ordIso_def
+  using embed_comp_iso by blast
+qed
+
+
+lemma ordIso_ordLeq_trans[trans]:
+assumes "r =o r'" and " r' \<le>o r''"
+shows "r \<le>o r''"
+proof-
+  have "Well_order r \<and> Well_order r''"
+  using assms unfolding ordLeq_def ordIso_def by auto
+  thus ?thesis using assms unfolding ordLeq_def ordIso_def
+  using iso_comp_embed by blast
+qed
+
+
+lemma ordLess_ordIso_trans[trans]:
+assumes "r <o r'" and " r' =o r''"
+shows "r <o r''"
+proof-
+  have "Well_order r \<and> Well_order r''"
+  using assms unfolding ordLess_def ordIso_def by auto
+  thus ?thesis using assms unfolding ordLess_def ordIso_def
+  using embedS_comp_iso by blast
+qed
+
+
+lemma ordIso_ordLess_trans[trans]:
+assumes "r =o r'" and " r' <o r''"
+shows "r <o r''"
+proof-
+  have "Well_order r \<and> Well_order r''"
+  using assms unfolding ordLess_def ordIso_def by auto
+  thus ?thesis using assms unfolding ordLess_def ordIso_def
+  using iso_comp_embedS by blast
+qed
+
+
+lemma ordLess_not_embed:
+assumes "r <o r'"
+shows "\<not>(\<exists>f'. embed r' r f')"
+proof-
+  obtain f where 1: "Well_order r \<and> Well_order r'" and 2: "embed r r' f" and
+                 3: " \<not> bij_betw f (Field r) (Field r')"
+  using assms unfolding ordLess_def by (auto simp add: embedS_def)
+  {fix f' assume *: "embed r' r f'"
+   hence "bij_betw f (Field r) (Field r')" using 1 2
+   by (simp add: embed_bothWays_Field_bij_betw)
+   with 3 have False by contradiction
+  }
+  thus ?thesis by blast
+qed
+
+
+lemma ordLess_Field:
+assumes OL: "r1 <o r2" and EMB: "embed r1 r2 f"
+shows "\<not> (f`(Field r1) = Field r2)"
+proof-
+  let ?A1 = "Field r1"  let ?A2 = "Field r2"
+  obtain g where
+  0: "Well_order r1 \<and> Well_order r2" and
+  1: "embed r1 r2 g \<and> \<not>(bij_betw g ?A1 ?A2)"
+  using OL unfolding ordLess_def by (auto simp add: embedS_def)
+  hence "\<forall>a \<in> ?A1. f a = g a"
+  using 0 EMB embed_unique[of r1] by auto
+  hence "\<not>(bij_betw f ?A1 ?A2)"
+  using 1 bij_betw_cong[of ?A1] by blast
+  moreover
+  have "inj_on f ?A1" using EMB 0 by (simp add: embed_inj_on)
+  ultimately show ?thesis by (simp add: bij_betw_def)
+qed
+
+
+lemma ordLess_iff:
+"r <o r' = (Well_order r \<and> Well_order r' \<and> \<not>(\<exists>f'. embed r' r f'))"
+proof
+  assume *: "r <o r'"
+  hence "\<not>(\<exists>f'. embed r' r f')" using ordLess_not_embed[of r r'] by simp
+  with * show "Well_order r \<and> Well_order r' \<and> \<not> (\<exists>f'. embed r' r f')"
+  unfolding ordLess_def by auto
+next
+  assume *: "Well_order r \<and> Well_order r' \<and> \<not> (\<exists>f'. embed r' r f')"
+  then obtain f where 1: "embed r r' f"
+  using wellorders_totally_ordered[of r r'] by blast
+  moreover
+  {assume "bij_betw f (Field r) (Field r')"
+   with * 1 have "embed r' r (inv_into (Field r) f) "
+   using inv_into_Field_embed_bij_betw[of r r' f] by auto
+   with * have False by blast
+  }
+  ultimately show "(r,r') \<in> ordLess"
+  unfolding ordLess_def using * by (fastforce simp add: embedS_def)
+qed
+
+
+lemma ordLess_irreflexive: "\<not> r <o r"
+proof
+  assume "r <o r"
+  hence "Well_order r \<and>  \<not>(\<exists>f. embed r r f)"
+  unfolding ordLess_iff ..
+  moreover have "embed r r id" using id_embed[of r] .
+  ultimately show False by blast
+qed
+
+
+lemma ordLeq_iff_ordLess_or_ordIso:
+"r \<le>o r' = (r <o r' \<or> r =o r')"
+unfolding ordRels_def embedS_defs iso_defs by blast
+
+
+lemma ordIso_iff_ordLeq:
+"(r =o r') = (r \<le>o r' \<and> r' \<le>o r)"
+proof
+  assume "r =o r'"
+  then obtain f where 1: "Well_order r \<and> Well_order r' \<and>
+                     embed r r' f \<and> bij_betw f (Field r) (Field r')"
+  unfolding ordIso_def iso_defs by auto
+  hence "embed r r' f \<and> embed r' r (inv_into (Field r) f)"
+  by (simp add: inv_into_Field_embed_bij_betw)
+  thus  "r \<le>o r' \<and> r' \<le>o r"
+  unfolding ordLeq_def using 1 by auto
+next
+  assume "r \<le>o r' \<and> r' \<le>o r"
+  then obtain f and g where 1: "Well_order r \<and> Well_order r' \<and>
+                           embed r r' f \<and> embed r' r g"
+  unfolding ordLeq_def by auto
+  hence "iso r r' f" by (auto simp add: embed_bothWays_iso)
+  thus "r =o r'" unfolding ordIso_def using 1 by auto
+qed
+
+
+lemma not_ordLess_ordLeq:
+"r <o r' \<Longrightarrow> \<not> r' \<le>o r"
+using ordLess_ordLeq_trans ordLess_irreflexive by blast
+
+
+lemma ordLess_or_ordLeq:
+assumes WELL: "Well_order r" and WELL': "Well_order r'"
+shows "r <o r' \<or> r' \<le>o r"
+proof-
+  have "r \<le>o r' \<or> r' \<le>o r"
+  using assms by (simp add: ordLeq_total)
+  moreover
+  {assume "\<not> r <o r' \<and> r \<le>o r'"
+   hence "r =o r'" using ordLeq_iff_ordLess_or_ordIso by blast
+   hence "r' \<le>o r" using ordIso_symmetric ordIso_iff_ordLeq by blast
+  }
+  ultimately show ?thesis by blast
+qed
+
+
+lemma not_ordLess_ordIso:
+"r <o r' \<Longrightarrow> \<not> r =o r'"
+using assms ordLess_ordIso_trans ordIso_symmetric ordLess_irreflexive by blast
+
+
+lemma not_ordLeq_iff_ordLess:
+assumes WELL: "Well_order r" and WELL': "Well_order r'"
+shows "(\<not> r' \<le>o r) = (r <o r')"
+using assms not_ordLess_ordLeq ordLess_or_ordLeq by blast
+
+
+lemma not_ordLess_iff_ordLeq:
+assumes WELL: "Well_order r" and WELL': "Well_order r'"
+shows "(\<not> r' <o r) = (r \<le>o r')"
+using assms not_ordLess_ordLeq ordLess_or_ordLeq by blast
+
+
+lemma ordLess_transitive[trans]:
+"\<lbrakk>r <o r'; r' <o r''\<rbrakk> \<Longrightarrow> r <o r''"
+using assms ordLess_ordLeq_trans ordLeq_iff_ordLess_or_ordIso by blast
+
+
+corollary ordLess_trans: "trans ordLess"
+unfolding trans_def using ordLess_transitive by blast
+
+
+lemmas ordIso_equivalence = ordIso_transitive ordIso_reflexive ordIso_symmetric
+
+
+lemma ordIso_imp_ordLeq:
+"r =o r' \<Longrightarrow> r \<le>o r'"
+using ordIso_iff_ordLeq by blast
+
+
+lemma ordLess_imp_ordLeq:
+"r <o r' \<Longrightarrow> r \<le>o r'"
+using ordLeq_iff_ordLess_or_ordIso by blast
+
+
+lemma ofilter_subset_ordLeq:
+assumes WELL: "Well_order r" and
+        OFA: "wo_rel.ofilter r A" and OFB: "wo_rel.ofilter r B"
+shows "(A \<le> B) = (Restr r A \<le>o Restr r B)"
+proof
+  assume "A \<le> B"
+  thus "Restr r A \<le>o Restr r B"
+  unfolding ordLeq_def using assms
+  Well_order_Restr Well_order_Restr ofilter_subset_embed by blast
+next
+  assume *: "Restr r A \<le>o Restr r B"
+  then obtain f where "embed (Restr r A) (Restr r B) f"
+  unfolding ordLeq_def by blast
+  {assume "B < A"
+   hence "Restr r B <o Restr r A"
+   unfolding ordLess_def using assms
+   Well_order_Restr Well_order_Restr ofilter_subset_embedS by blast
+   hence False using * not_ordLess_ordLeq by blast
+  }
+  thus "A \<le> B" using OFA OFB WELL
+  wo_rel_def[of r] wo_rel.ofilter_linord[of r A B] by blast
+qed
+
+
+lemma ofilter_subset_ordLess:
+assumes WELL: "Well_order r" and
+        OFA: "wo_rel.ofilter r A" and OFB: "wo_rel.ofilter r B"
+shows "(A < B) = (Restr r A <o Restr r B)"
+proof-
+  let ?rA = "Restr r A" let ?rB = "Restr r B"
+  have 1: "Well_order ?rA \<and> Well_order ?rB"
+  using WELL Well_order_Restr by blast
+  have "(A < B) = (\<not> B \<le> A)" using assms
+  wo_rel_def wo_rel.ofilter_linord[of r A B] by blast
+  also have "\<dots> = (\<not> Restr r B \<le>o Restr r A)"
+  using assms ofilter_subset_ordLeq by blast
+  also have "\<dots> = (Restr r A <o Restr r B)"
+  using 1 not_ordLeq_iff_ordLess by blast
+  finally show ?thesis .
+qed
+
+
+lemma ofilter_ordLess:
+"\<lbrakk>Well_order r; wo_rel.ofilter r A\<rbrakk> \<Longrightarrow> (A < Field r) = (Restr r A <o r)"
+by (simp add: ofilter_subset_ordLess wo_rel.Field_ofilter
+    wo_rel_def Restr_Field)
+
+
+corollary underS_Restr_ordLess:
+assumes "Well_order r" and "Field r \<noteq> {}"
+shows "Restr r (underS r a) <o r"
+proof-
+  have "underS r a < Field r" using assms
+  by (simp add: underS_Field3)
+  thus ?thesis using assms
+  by (simp add: ofilter_ordLess wo_rel.underS_ofilter wo_rel_def)
+qed
+
+
+lemma embed_ordLess_ofilterIncl:
+assumes
+  OL12: "r1 <o r2" and OL23: "r2 <o r3" and
+  EMB13: "embed r1 r3 f13" and EMB23: "embed r2 r3 f23"
+shows "(f13`(Field r1), f23`(Field r2)) \<in> (ofilterIncl r3)"
+proof-
+  have OL13: "r1 <o r3"
+  using OL12 OL23 using ordLess_transitive by auto
+  let ?A1 = "Field r1"  let ?A2 ="Field r2" let ?A3 ="Field r3"
+  obtain f12 g23 where
+  0: "Well_order r1 \<and> Well_order r2 \<and> Well_order r3" and
+  1: "embed r1 r2 f12 \<and> \<not>(bij_betw f12 ?A1 ?A2)" and
+  2: "embed r2 r3 g23 \<and> \<not>(bij_betw g23 ?A2 ?A3)"
+  using OL12 OL23 by (auto simp add: ordLess_def embedS_def)
+  hence "\<forall>a \<in> ?A2. f23 a = g23 a"
+  using EMB23 embed_unique[of r2 r3] by blast
+  hence 3: "\<not>(bij_betw f23 ?A2 ?A3)"
+  using 2 bij_betw_cong[of ?A2 f23 g23] by blast
+  (*  *)
+  have 4: "wo_rel.ofilter r2 (f12 ` ?A1) \<and> f12 ` ?A1 \<noteq> ?A2"
+  using 0 1 OL12 by (simp add: embed_Field_ofilter ordLess_Field)
+  have 5: "wo_rel.ofilter r3 (f23 ` ?A2) \<and> f23 ` ?A2 \<noteq> ?A3"
+  using 0 EMB23 OL23 by (simp add: embed_Field_ofilter ordLess_Field)
+  have 6: "wo_rel.ofilter r3 (f13 ` ?A1)  \<and> f13 ` ?A1 \<noteq> ?A3"
+  using 0 EMB13 OL13 by (simp add: embed_Field_ofilter ordLess_Field)
+  (*  *)
+  have "f12 ` ?A1 < ?A2"
+  using 0 4 by (auto simp add: wo_rel_def wo_rel.ofilter_def)
+  moreover have "inj_on f23 ?A2"
+  using EMB23 0 by (simp add: wo_rel_def embed_inj_on)
+  ultimately
+  have "f23 ` (f12 ` ?A1) < f23 ` ?A2" by (simp add: inj_on_strict_subset)
+  moreover
+  {have "embed r1 r3 (f23 o f12)"
+   using 1 EMB23 0 by (auto simp add: comp_embed)
+   hence "\<forall>a \<in> ?A1. f23(f12 a) = f13 a"
+   using EMB13 0 embed_unique[of r1 r3 "f23 o f12" f13] by auto
+   hence "f23 ` (f12 ` ?A1) = f13 ` ?A1" by force
+  }
+  ultimately
+  have "f13 ` ?A1 < f23 ` ?A2" by simp
+  (*  *)
+  with 5 6 show ?thesis
+  unfolding ofilterIncl_def by auto
+qed
+
+
+lemma ordLess_iff_ordIso_Restr:
+assumes WELL: "Well_order r" and WELL': "Well_order r'"
+shows "(r' <o r) = (\<exists>a \<in> Field r. r' =o Restr r (underS r a))"
+proof(auto)
+  fix a assume *: "a \<in> Field r" and **: "r' =o Restr r (underS r a)"
+  hence "Restr r (underS r a) <o r" using WELL underS_Restr_ordLess[of r] by blast
+  thus "r' <o r" using ** ordIso_ordLess_trans by blast
+next
+  assume "r' <o r"
+  then obtain f where 1: "Well_order r \<and> Well_order r'" and
+                      2: "embed r' r f \<and> f ` (Field r') \<noteq> Field r"
+  unfolding ordLess_def embedS_def[abs_def] bij_betw_def using embed_inj_on by blast
+  hence "wo_rel.ofilter r (f ` (Field r'))" using embed_Field_ofilter by blast
+  then obtain a where 3: "a \<in> Field r" and 4: "underS r a = f ` (Field r')"
+  using 1 2 by (auto simp add: wo_rel.ofilter_underS_Field wo_rel_def)
+  have "iso r' (Restr r (f ` (Field r'))) f"
+  using embed_implies_iso_Restr 2 assms by blast
+  moreover have "Well_order (Restr r (f ` (Field r')))"
+  using WELL Well_order_Restr by blast
+  ultimately have "r' =o Restr r (f ` (Field r'))"
+  using WELL' unfolding ordIso_def by auto
+  hence "r' =o Restr r (underS r a)" using 4 by auto
+  thus "\<exists>a \<in> Field r. r' =o Restr r (underS r a)" using 3 by auto
+qed
+
+
+lemma internalize_ordLess:
+"(r' <o r) = (\<exists>p. Field p < Field r \<and> r' =o p \<and> p <o r)"
+proof
+  assume *: "r' <o r"
+  hence 0: "Well_order r \<and> Well_order r'" unfolding ordLess_def by auto
+  with * obtain a where 1: "a \<in> Field r" and 2: "r' =o Restr r (underS r a)"
+  using ordLess_iff_ordIso_Restr by blast
+  let ?p = "Restr r (underS r a)"
+  have "wo_rel.ofilter r (underS r a)" using 0
+  by (simp add: wo_rel_def wo_rel.underS_ofilter)
+  hence "Field ?p = underS r a" using 0 Field_Restr_ofilter by blast
+  hence "Field ?p < Field r" using underS_Field2 1 by fast
+  moreover have "?p <o r" using underS_Restr_ordLess[of r a] 0 1 by blast
+  ultimately
+  show "\<exists>p. Field p < Field r \<and> r' =o p \<and> p <o r" using 2 by blast
+next
+  assume "\<exists>p. Field p < Field r \<and> r' =o p \<and> p <o r"
+  thus "r' <o r" using ordIso_ordLess_trans by blast
+qed
+
+
+lemma internalize_ordLeq:
+"(r' \<le>o r) = (\<exists>p. Field p \<le> Field r \<and> r' =o p \<and> p \<le>o r)"
+proof
+  assume *: "r' \<le>o r"
+  moreover
+  {assume "r' <o r"
+   then obtain p where "Field p < Field r \<and> r' =o p \<and> p <o r"
+   using internalize_ordLess[of r' r] by blast
+   hence "\<exists>p. Field p \<le> Field r \<and> r' =o p \<and> p \<le>o r"
+   using ordLeq_iff_ordLess_or_ordIso by blast
+  }
+  moreover
+  have "r \<le>o r" using * ordLeq_def ordLeq_reflexive by blast
+  ultimately show "\<exists>p. Field p \<le> Field r \<and> r' =o p \<and> p \<le>o r"
+  using ordLeq_iff_ordLess_or_ordIso by blast
+next
+  assume "\<exists>p. Field p \<le> Field r \<and> r' =o p \<and> p \<le>o r"
+  thus "r' \<le>o r" using ordIso_ordLeq_trans by blast
+qed
+
+
+lemma ordLeq_iff_ordLess_Restr:
+assumes WELL: "Well_order r" and WELL': "Well_order r'"
+shows "(r \<le>o r') = (\<forall>a \<in> Field r. Restr r (underS r a) <o r')"
+proof(auto)
+  assume *: "r \<le>o r'"
+  fix a assume "a \<in> Field r"
+  hence "Restr r (underS r a) <o r"
+  using WELL underS_Restr_ordLess[of r] by blast
+  thus "Restr r (underS r a) <o r'"
+  using * ordLess_ordLeq_trans by blast
+next
+  assume *: "\<forall>a \<in> Field r. Restr r (underS r a) <o r'"
+  {assume "r' <o r"
+   then obtain a where "a \<in> Field r \<and> r' =o Restr r (underS r a)"
+   using assms ordLess_iff_ordIso_Restr by blast
+   hence False using * not_ordLess_ordIso ordIso_symmetric by blast
+  }
+  thus "r \<le>o r'" using ordLess_or_ordLeq assms by blast
+qed
+
+
+lemma finite_ordLess_infinite:
+assumes WELL: "Well_order r" and WELL': "Well_order r'" and
+        FIN: "finite(Field r)" and INF: "\<not>finite(Field r')"
+shows "r <o r'"
+proof-
+  {assume "r' \<le>o r"
+   then obtain h where "inj_on h (Field r') \<and> h ` (Field r') \<le> Field r"
+   unfolding ordLeq_def using assms embed_inj_on embed_Field by blast
+   hence False using finite_imageD finite_subset FIN INF by metis
+  }
+  thus ?thesis using WELL WELL' ordLess_or_ordLeq by blast
+qed
+
+
+lemma finite_well_order_on_ordIso:
+assumes FIN: "finite A" and
+        WELL: "well_order_on A r" and WELL': "well_order_on A r'"
+shows "r =o r'"
+proof-
+  have 0: "Well_order r \<and> Well_order r' \<and> Field r = A \<and> Field r' = A"
+  using assms well_order_on_Well_order by blast
+  moreover
+  have "\<forall>r r'. well_order_on A r \<and> well_order_on A r' \<and> r \<le>o r'
+                  \<longrightarrow> r =o r'"
+  proof(clarify)
+    fix r r' assume *: "well_order_on A r" and **: "well_order_on A r'"
+    have 2: "Well_order r \<and> Well_order r' \<and> Field r = A \<and> Field r' = A"
+    using * ** well_order_on_Well_order by blast
+    assume "r \<le>o r'"
+    then obtain f where 1: "embed r r' f" and
+                        "inj_on f A \<and> f ` A \<le> A"
+    unfolding ordLeq_def using 2 embed_inj_on embed_Field by blast
+    hence "bij_betw f A A" unfolding bij_betw_def using FIN endo_inj_surj by blast
+    thus "r =o r'" unfolding ordIso_def iso_def[abs_def] using 1 2 by auto
+  qed
+  ultimately show ?thesis using assms ordLeq_total ordIso_symmetric by metis
+qed
+
+
+subsection{* @{text "<o"} is well-founded *}
+
+
+text {* Of course, it only makes sense to state that the @{text "<o"} is well-founded
+on the restricted type @{text "'a rel rel"}.  We prove this by first showing that, for any set
+of well-orders all embedded in a fixed well-order, the function mapping each well-order
+in the set to an order filter of the fixed well-order is compatible w.r.t. to @{text "<o"} versus
+{\em strict inclusion}; and we already know that strict inclusion of order filters is well-founded. *}
+
+
+definition ord_to_filter :: "'a rel \<Rightarrow> 'a rel \<Rightarrow> 'a set"
+where "ord_to_filter r0 r \<equiv> (SOME f. embed r r0 f) ` (Field r)"
+
+
+lemma ord_to_filter_compat:
+"compat (ordLess Int (ordLess^-1``{r0} \<times> ordLess^-1``{r0}))
+        (ofilterIncl r0)
+        (ord_to_filter r0)"
+proof(unfold compat_def ord_to_filter_def, clarify)
+  fix r1::"'a rel" and r2::"'a rel"
+  let ?A1 = "Field r1"  let ?A2 ="Field r2" let ?A0 ="Field r0"
+  let ?phi10 = "\<lambda> f10. embed r1 r0 f10" let ?f10 = "SOME f. ?phi10 f"
+  let ?phi20 = "\<lambda> f20. embed r2 r0 f20" let ?f20 = "SOME f. ?phi20 f"
+  assume *: "r1 <o r0" "r2 <o r0" and **: "r1 <o r2"
+  hence "(\<exists>f. ?phi10 f) \<and> (\<exists>f. ?phi20 f)"
+  by (auto simp add: ordLess_def embedS_def)
+  hence "?phi10 ?f10 \<and> ?phi20 ?f20" by (auto simp add: someI_ex)
+  thus "(?f10 ` ?A1, ?f20 ` ?A2) \<in> ofilterIncl r0"
+  using * ** by (simp add: embed_ordLess_ofilterIncl)
+qed
+
+
+theorem wf_ordLess: "wf ordLess"
+proof-
+  {fix r0 :: "('a \<times> 'a) set"
+   (* need to annotate here!*)
+   let ?ordLess = "ordLess::('d rel * 'd rel) set"
+   let ?R = "?ordLess Int (?ordLess^-1``{r0} \<times> ?ordLess^-1``{r0})"
+   {assume Case1: "Well_order r0"
+    hence "wf ?R"
+    using wf_ofilterIncl[of r0]
+          compat_wf[of ?R "ofilterIncl r0" "ord_to_filter r0"]
+          ord_to_filter_compat[of r0] by auto
+   }
+   moreover
+   {assume Case2: "\<not> Well_order r0"
+    hence "?R = {}" unfolding ordLess_def by auto
+    hence "wf ?R" using wf_empty by simp
+   }
+   ultimately have "wf ?R" by blast
+  }
+  thus ?thesis by (simp add: trans_wf_iff ordLess_trans)
+qed
+
+corollary exists_minim_Well_order:
+assumes NE: "R \<noteq> {}" and WELL: "\<forall>r \<in> R. Well_order r"
+shows "\<exists>r \<in> R. \<forall>r' \<in> R. r \<le>o r'"
+proof-
+  obtain r where "r \<in> R \<and> (\<forall>r' \<in> R. \<not> r' <o r)"
+  using NE spec[OF spec[OF subst[OF wf_eq_minimal, of "%x. x", OF wf_ordLess]], of _ R]
+    equals0I[of R] by blast
+  with not_ordLeq_iff_ordLess WELL show ?thesis by blast
+qed
+
+
+
+subsection {* Copy via direct images  *}
+
+
+text{* The direct image operator is the dual of the inverse image operator @{text "inv_image"}
+from @{text "Relation.thy"}.  It is useful for transporting a well-order between
+different types. *}
+
+
+definition dir_image :: "'a rel \<Rightarrow> ('a \<Rightarrow> 'a') \<Rightarrow> 'a' rel"
+where
+"dir_image r f = {(f a, f b)| a b. (a,b) \<in> r}"
+
+
+lemma dir_image_Field:
+"Field(dir_image r f) \<le> f ` (Field r)"
+unfolding dir_image_def Field_def by auto
+
+
+lemma dir_image_minus_Id:
+"inj_on f (Field r) \<Longrightarrow> (dir_image r f) - Id = dir_image (r - Id) f"
+unfolding inj_on_def Field_def dir_image_def by auto
+
+
+lemma Refl_dir_image:
+assumes "Refl r"
+shows "Refl(dir_image r f)"
+proof-
+  {fix a' b'
+   assume "(a',b') \<in> dir_image r f"
+   then obtain a b where 1: "a' = f a \<and> b' = f b \<and> (a,b) \<in> r"
+   unfolding dir_image_def by blast
+   hence "a \<in> Field r \<and> b \<in> Field r" using Field_def by fastforce
+   hence "(a,a) \<in> r \<and> (b,b) \<in> r" using assms by (simp add: refl_on_def)
+   with 1 have "(a',a') \<in> dir_image r f \<and> (b',b') \<in> dir_image r f"
+   unfolding dir_image_def by auto
+  }
+  thus ?thesis
+  by(unfold refl_on_def Field_def Domain_def Range_def, auto)
+qed
+
+
+lemma trans_dir_image:
+assumes TRANS: "trans r" and INJ: "inj_on f (Field r)"
+shows "trans(dir_image r f)"
+proof(unfold trans_def, auto)
+  fix a' b' c'
+  assume "(a',b') \<in> dir_image r f" "(b',c') \<in> dir_image r f"
+  then obtain a b1 b2 c where 1: "a' = f a \<and> b' = f b1 \<and> b' = f b2 \<and> c' = f c" and
+                         2: "(a,b1) \<in> r \<and> (b2,c) \<in> r"
+  unfolding dir_image_def by blast
+  hence "b1 \<in> Field r \<and> b2 \<in> Field r"
+  unfolding Field_def by auto
+  hence "b1 = b2" using 1 INJ unfolding inj_on_def by auto
+  hence "(a,c): r" using 2 TRANS unfolding trans_def by blast
+  thus "(a',c') \<in> dir_image r f"
+  unfolding dir_image_def using 1 by auto
+qed
+
+
+lemma Preorder_dir_image:
+"\<lbrakk>Preorder r; inj_on f (Field r)\<rbrakk> \<Longrightarrow> Preorder (dir_image r f)"
+by (simp add: preorder_on_def Refl_dir_image trans_dir_image)
+
+
+lemma antisym_dir_image:
+assumes AN: "antisym r" and INJ: "inj_on f (Field r)"
+shows "antisym(dir_image r f)"
+proof(unfold antisym_def, auto)
+  fix a' b'
+  assume "(a',b') \<in> dir_image r f" "(b',a') \<in> dir_image r f"
+  then obtain a1 b1 a2 b2 where 1: "a' = f a1 \<and> a' = f a2 \<and> b' = f b1 \<and> b' = f b2" and
+                           2: "(a1,b1) \<in> r \<and> (b2,a2) \<in> r " and
+                           3: "{a1,a2,b1,b2} \<le> Field r"
+  unfolding dir_image_def Field_def by blast
+  hence "a1 = a2 \<and> b1 = b2" using INJ unfolding inj_on_def by auto
+  hence "a1 = b2" using 2 AN unfolding antisym_def by auto
+  thus "a' = b'" using 1 by auto
+qed
+
+
+lemma Partial_order_dir_image:
+"\<lbrakk>Partial_order r; inj_on f (Field r)\<rbrakk> \<Longrightarrow> Partial_order (dir_image r f)"
+by (simp add: partial_order_on_def Preorder_dir_image antisym_dir_image)
+
+
+lemma Total_dir_image:
+assumes TOT: "Total r" and INJ: "inj_on f (Field r)"
+shows "Total(dir_image r f)"
+proof(unfold total_on_def, intro ballI impI)
+  fix a' b'
+  assume "a' \<in> Field (dir_image r f)" "b' \<in> Field (dir_image r f)"
+  then obtain a and b where 1: "a \<in> Field r \<and> b \<in> Field r \<and> f a = a' \<and> f b = b'"
+  using dir_image_Field[of r f] by blast
+  moreover assume "a' \<noteq> b'"
+  ultimately have "a \<noteq> b" using INJ unfolding inj_on_def by auto
+  hence "(a,b) \<in> r \<or> (b,a) \<in> r" using 1 TOT unfolding total_on_def by auto
+  thus "(a',b') \<in> dir_image r f \<or> (b',a') \<in> dir_image r f"
+  using 1 unfolding dir_image_def by auto
+qed
+
+
+lemma Linear_order_dir_image:
+"\<lbrakk>Linear_order r; inj_on f (Field r)\<rbrakk> \<Longrightarrow> Linear_order (dir_image r f)"
+by (simp add: linear_order_on_def Partial_order_dir_image Total_dir_image)
+
+
+lemma wf_dir_image:
+assumes WF: "wf r" and INJ: "inj_on f (Field r)"
+shows "wf(dir_image r f)"
+proof(unfold wf_eq_minimal2, intro allI impI, elim conjE)
+  fix A'::"'b set"
+  assume SUB: "A' \<le> Field(dir_image r f)" and NE: "A' \<noteq> {}"
+  obtain A where A_def: "A = {a \<in> Field r. f a \<in> A'}" by blast
+  have "A \<noteq> {} \<and> A \<le> Field r"
+  using A_def dir_image_Field[of r f] SUB NE by blast
+  then obtain a where 1: "a \<in> A \<and> (\<forall>b \<in> A. (b,a) \<notin> r)"
+  using WF unfolding wf_eq_minimal2 by metis
+  have "\<forall>b' \<in> A'. (b',f a) \<notin> dir_image r f"
+  proof(clarify)
+    fix b' assume *: "b' \<in> A'" and **: "(b',f a) \<in> dir_image r f"
+    obtain b1 a1 where 2: "b' = f b1 \<and> f a = f a1" and
+                       3: "(b1,a1) \<in> r \<and> {a1,b1} \<le> Field r"
+    using ** unfolding dir_image_def Field_def by blast
+    hence "a = a1" using 1 A_def INJ unfolding inj_on_def by auto
+    hence "b1 \<in> A \<and> (b1,a) \<in> r" using 2 3 A_def * by auto
+    with 1 show False by auto
+  qed
+  thus "\<exists>a'\<in>A'. \<forall>b'\<in>A'. (b', a') \<notin> dir_image r f"
+  using A_def 1 by blast
+qed
+
+
+lemma Well_order_dir_image:
+"\<lbrakk>Well_order r; inj_on f (Field r)\<rbrakk> \<Longrightarrow> Well_order (dir_image r f)"
+using assms unfolding well_order_on_def
+using Linear_order_dir_image[of r f] wf_dir_image[of "r - Id" f]
+  dir_image_minus_Id[of f r]
+  subset_inj_on[of f "Field r" "Field(r - Id)"]
+  mono_Field[of "r - Id" r] by auto
+
+
+lemma dir_image_Field2:
+"Refl r \<Longrightarrow> Field(dir_image r f) = f ` (Field r)"
+unfolding Field_def dir_image_def refl_on_def Domain_def Range_def by blast
+
+
+lemma dir_image_bij_betw:
+"\<lbrakk>Well_order r; inj_on f (Field r)\<rbrakk> \<Longrightarrow> bij_betw f (Field r) (Field (dir_image r f))"
+unfolding bij_betw_def
+by (simp add: dir_image_Field2 order_on_defs)
+
+
+lemma dir_image_compat:
+"compat r (dir_image r f) f"
+unfolding compat_def dir_image_def by auto
+
+
+lemma dir_image_iso:
+"\<lbrakk>Well_order r; inj_on f (Field r)\<rbrakk>  \<Longrightarrow> iso r (dir_image r f) f"
+using iso_iff3 dir_image_compat dir_image_bij_betw Well_order_dir_image by blast
+
+
+lemma dir_image_ordIso:
+"\<lbrakk>Well_order r; inj_on f (Field r)\<rbrakk>  \<Longrightarrow> r =o dir_image r f"
+unfolding ordIso_def using dir_image_iso Well_order_dir_image by blast
+
+
+lemma Well_order_iso_copy:
+assumes WELL: "well_order_on A r" and BIJ: "bij_betw f A A'"
+shows "\<exists>r'. well_order_on A' r' \<and> r =o r'"
+proof-
+   let ?r' = "dir_image r f"
+   have 1: "A = Field r \<and> Well_order r"
+   using WELL well_order_on_Well_order by blast
+   hence 2: "iso r ?r' f"
+   using dir_image_iso using BIJ unfolding bij_betw_def by auto
+   hence "f ` (Field r) = Field ?r'" using 1 iso_iff[of r ?r'] by blast
+   hence "Field ?r' = A'"
+   using 1 BIJ unfolding bij_betw_def by auto
+   moreover have "Well_order ?r'"
+   using 1 Well_order_dir_image BIJ unfolding bij_betw_def by blast
+   ultimately show ?thesis unfolding ordIso_def using 1 2 by blast
+qed
+
+
+
+subsection {* Bounded square  *}
+
+
+text{* This construction essentially defines, for an order relation @{text "r"}, a lexicographic
+order @{text "bsqr r"} on @{text "(Field r) \<times> (Field r)"}, applying the
+following criteria (in this order):
+\begin{itemize}
+\item compare the maximums;
+\item compare the first components;
+\item compare the second components.
+\end{itemize}
+%
+The only application of this construction that we are aware of is
+at proving that the square of an infinite set has the same cardinal
+as that set. The essential property required there (and which is ensured by this
+construction) is that any proper order filter of the product order is included in a rectangle, i.e.,
+in a product of proper filters on the original relation (assumed to be a well-order). *}
+
+
+definition bsqr :: "'a rel => ('a * 'a)rel"
+where
+"bsqr r = {((a1,a2),(b1,b2)).
+           {a1,a2,b1,b2} \<le> Field r \<and>
+           (a1 = b1 \<and> a2 = b2 \<or>
+            (wo_rel.max2 r a1 a2, wo_rel.max2 r b1 b2) \<in> r - Id \<or>
+            wo_rel.max2 r a1 a2 = wo_rel.max2 r b1 b2 \<and> (a1,b1) \<in> r - Id \<or>
+            wo_rel.max2 r a1 a2 = wo_rel.max2 r b1 b2 \<and> a1 = b1  \<and> (a2,b2) \<in> r - Id
+           )}"
+
+
+lemma Field_bsqr:
+"Field (bsqr r) = Field r \<times> Field r"
+proof
+  show "Field (bsqr r) \<le> Field r \<times> Field r"
+  proof-
+    {fix a1 a2 assume "(a1,a2) \<in> Field (bsqr r)"
+     moreover
+     have "\<And> b1 b2. ((a1,a2),(b1,b2)) \<in> bsqr r \<or> ((b1,b2),(a1,a2)) \<in> bsqr r \<Longrightarrow>
+                      a1 \<in> Field r \<and> a2 \<in> Field r" unfolding bsqr_def by auto
+     ultimately have "a1 \<in> Field r \<and> a2 \<in> Field r" unfolding Field_def by auto
+    }
+    thus ?thesis unfolding Field_def by force
+  qed
+next
+  show "Field r \<times> Field r \<le> Field (bsqr r)"
+  proof(auto)
+    fix a1 a2 assume "a1 \<in> Field r" and "a2 \<in> Field r"
+    hence "((a1,a2),(a1,a2)) \<in> bsqr r" unfolding bsqr_def by blast
+    thus "(a1,a2) \<in> Field (bsqr r)" unfolding Field_def by auto
+  qed
+qed
+
+
+lemma bsqr_Refl: "Refl(bsqr r)"
+by(unfold refl_on_def Field_bsqr, auto simp add: bsqr_def)
+
+
+lemma bsqr_Trans:
+assumes "Well_order r"
+shows "trans (bsqr r)"
+proof(unfold trans_def, auto)
+  (* Preliminary facts *)
+  have Well: "wo_rel r" using assms wo_rel_def by auto
+  hence Trans: "trans r" using wo_rel.TRANS by auto
+  have Anti: "antisym r" using wo_rel.ANTISYM Well by auto
+  hence TransS: "trans(r - Id)" using Trans by (simp add: trans_diff_Id)
+  (* Main proof *)
+  fix a1 a2 b1 b2 c1 c2
+  assume *: "((a1,a2),(b1,b2)) \<in> bsqr r" and **: "((b1,b2),(c1,c2)) \<in> bsqr r"
+  hence 0: "{a1,a2,b1,b2,c1,c2} \<le> Field r" unfolding bsqr_def by auto
+  have 1: "a1 = b1 \<and> a2 = b2 \<or> (wo_rel.max2 r a1 a2, wo_rel.max2 r b1 b2) \<in> r - Id \<or>
+           wo_rel.max2 r a1 a2 = wo_rel.max2 r b1 b2 \<and> (a1,b1) \<in> r - Id \<or>
+           wo_rel.max2 r a1 a2 = wo_rel.max2 r b1 b2 \<and> a1 = b1 \<and> (a2,b2) \<in> r - Id"
+  using * unfolding bsqr_def by auto
+  have 2: "b1 = c1 \<and> b2 = c2 \<or> (wo_rel.max2 r b1 b2, wo_rel.max2 r c1 c2) \<in> r - Id \<or>
+           wo_rel.max2 r b1 b2 = wo_rel.max2 r c1 c2 \<and> (b1,c1) \<in> r - Id \<or>
+           wo_rel.max2 r b1 b2 = wo_rel.max2 r c1 c2 \<and> b1 = c1 \<and> (b2,c2) \<in> r - Id"
+  using ** unfolding bsqr_def by auto
+  show "((a1,a2),(c1,c2)) \<in> bsqr r"
+  proof-
+    {assume Case1: "a1 = b1 \<and> a2 = b2"
+     hence ?thesis using ** by simp
+    }
+    moreover
+    {assume Case2: "(wo_rel.max2 r a1 a2, wo_rel.max2 r b1 b2) \<in> r - Id"
+     {assume Case21: "b1 = c1 \<and> b2 = c2"
+      hence ?thesis using * by simp
+     }
+     moreover
+     {assume Case22: "(wo_rel.max2 r b1 b2, wo_rel.max2 r c1 c2) \<in> r - Id"
+      hence "(wo_rel.max2 r a1 a2, wo_rel.max2 r c1 c2) \<in> r - Id"
+      using Case2 TransS trans_def[of "r - Id"] by blast
+      hence ?thesis using 0 unfolding bsqr_def by auto
+     }
+     moreover
+     {assume Case23_4: "wo_rel.max2 r b1 b2 = wo_rel.max2 r c1 c2"
+      hence ?thesis using Case2 0 unfolding bsqr_def by auto
+     }
+     ultimately have ?thesis using 0 2 by auto
+    }
+    moreover
+    {assume Case3: "wo_rel.max2 r a1 a2 = wo_rel.max2 r b1 b2 \<and> (a1,b1) \<in> r - Id"
+     {assume Case31: "b1 = c1 \<and> b2 = c2"
+      hence ?thesis using * by simp
+     }
+     moreover
+     {assume Case32: "(wo_rel.max2 r b1 b2, wo_rel.max2 r c1 c2) \<in> r - Id"
+      hence ?thesis using Case3 0 unfolding bsqr_def by auto
+     }
+     moreover
+     {assume Case33: "wo_rel.max2 r b1 b2 = wo_rel.max2 r c1 c2 \<and> (b1,c1) \<in> r - Id"
+      hence "(a1,c1) \<in> r - Id"
+      using Case3 TransS trans_def[of "r - Id"] by blast
+      hence ?thesis using Case3 Case33 0 unfolding bsqr_def by auto
+     }
+     moreover
+     {assume Case33: "wo_rel.max2 r b1 b2 = wo_rel.max2 r c1 c2 \<and> b1 = c1"
+      hence ?thesis using Case3 0 unfolding bsqr_def by auto
+     }
+     ultimately have ?thesis using 0 2 by auto
+    }
+    moreover
+    {assume Case4: "wo_rel.max2 r a1 a2 = wo_rel.max2 r b1 b2 \<and> a1 = b1 \<and> (a2,b2) \<in> r - Id"
+     {assume Case41: "b1 = c1 \<and> b2 = c2"
+      hence ?thesis using * by simp
+     }
+     moreover
+     {assume Case42: "(wo_rel.max2 r b1 b2, wo_rel.max2 r c1 c2) \<in> r - Id"
+      hence ?thesis using Case4 0 unfolding bsqr_def by force
+     }
+     moreover
+     {assume Case43: "wo_rel.max2 r b1 b2 = wo_rel.max2 r c1 c2 \<and> (b1,c1) \<in> r - Id"
+      hence ?thesis using Case4 0 unfolding bsqr_def by auto
+     }
+     moreover
+     {assume Case44: "wo_rel.max2 r b1 b2 = wo_rel.max2 r c1 c2 \<and> b1 = c1 \<and> (b2,c2) \<in> r - Id"
+      hence "(a2,c2) \<in> r - Id"
+      using Case4 TransS trans_def[of "r - Id"] by blast
+      hence ?thesis using Case4 Case44 0 unfolding bsqr_def by auto
+     }
+     ultimately have ?thesis using 0 2 by auto
+    }
+    ultimately show ?thesis using 0 1 by auto
+  qed
+qed
+
+
+lemma bsqr_antisym:
+assumes "Well_order r"
+shows "antisym (bsqr r)"
+proof(unfold antisym_def, clarify)
+  (* Preliminary facts *)
+  have Well: "wo_rel r" using assms wo_rel_def by auto
+  hence Trans: "trans r" using wo_rel.TRANS by auto
+  have Anti: "antisym r" using wo_rel.ANTISYM Well by auto
+  hence TransS: "trans(r - Id)" using Trans by (simp add: trans_diff_Id)
+  hence IrrS: "\<forall>a b. \<not>((a,b) \<in> r - Id \<and> (b,a) \<in> r - Id)"
+  using Anti trans_def[of "r - Id"] antisym_def[of "r - Id"] by blast
+  (* Main proof *)
+  fix a1 a2 b1 b2
+  assume *: "((a1,a2),(b1,b2)) \<in> bsqr r" and **: "((b1,b2),(a1,a2)) \<in> bsqr r"
+  hence 0: "{a1,a2,b1,b2} \<le> Field r" unfolding bsqr_def by auto
+  have 1: "a1 = b1 \<and> a2 = b2 \<or> (wo_rel.max2 r a1 a2, wo_rel.max2 r b1 b2) \<in> r - Id \<or>
+           wo_rel.max2 r a1 a2 = wo_rel.max2 r b1 b2 \<and> (a1,b1) \<in> r - Id \<or>
+           wo_rel.max2 r a1 a2 = wo_rel.max2 r b1 b2 \<and> a1 = b1 \<and> (a2,b2) \<in> r - Id"
+  using * unfolding bsqr_def by auto
+  have 2: "b1 = a1 \<and> b2 = a2 \<or> (wo_rel.max2 r b1 b2, wo_rel.max2 r a1 a2) \<in> r - Id \<or>
+           wo_rel.max2 r b1 b2 = wo_rel.max2 r a1 a2 \<and> (b1,a1) \<in> r - Id \<or>
+           wo_rel.max2 r b1 b2 = wo_rel.max2 r a1 a2 \<and> b1 = a1 \<and> (b2,a2) \<in> r - Id"
+  using ** unfolding bsqr_def by auto
+  show "a1 = b1 \<and> a2 = b2"
+  proof-
+    {assume Case1: "(wo_rel.max2 r a1 a2, wo_rel.max2 r b1 b2) \<in> r - Id"
+     {assume Case11: "(wo_rel.max2 r b1 b2, wo_rel.max2 r a1 a2) \<in> r - Id"
+      hence False using Case1 IrrS by blast
+     }
+     moreover
+     {assume Case12_3: "wo_rel.max2 r b1 b2 = wo_rel.max2 r a1 a2"
+      hence False using Case1 by auto
+     }
+     ultimately have ?thesis using 0 2 by auto
+    }
+    moreover
+    {assume Case2: "wo_rel.max2 r a1 a2 = wo_rel.max2 r b1 b2 \<and> (a1,b1) \<in> r - Id"
+     {assume Case21: "(wo_rel.max2 r b1 b2, wo_rel.max2 r a1 a2) \<in> r - Id"
+       hence False using Case2 by auto
+     }
+     moreover
+     {assume Case22: "(b1,a1) \<in> r - Id"
+      hence False using Case2 IrrS by blast
+     }
+     moreover
+     {assume Case23: "b1 = a1"
+      hence False using Case2 by auto
+     }
+     ultimately have ?thesis using 0 2 by auto
+    }
+    moreover
+    {assume Case3: "wo_rel.max2 r a1 a2 = wo_rel.max2 r b1 b2 \<and> a1 = b1 \<and> (a2,b2) \<in> r - Id"
+     moreover
+     {assume Case31: "(wo_rel.max2 r b1 b2, wo_rel.max2 r a1 a2) \<in> r - Id"
+      hence False using Case3 by auto
+     }
+     moreover
+     {assume Case32: "(b1,a1) \<in> r - Id"
+      hence False using Case3 by auto
+     }
+     moreover
+     {assume Case33: "(b2,a2) \<in> r - Id"
+      hence False using Case3 IrrS by blast
+     }
+     ultimately have ?thesis using 0 2 by auto
+    }
+    ultimately show ?thesis using 0 1 by blast
+  qed
+qed
+
+
+lemma bsqr_Total:
+assumes "Well_order r"
+shows "Total(bsqr r)"
+proof-
+  (* Preliminary facts *)
+  have Well: "wo_rel r" using assms wo_rel_def by auto
+  hence Total: "\<forall>a \<in> Field r. \<forall>b \<in> Field r. (a,b) \<in> r \<or> (b,a) \<in> r"
+  using wo_rel.TOTALS by auto
+  (* Main proof *)
+  {fix a1 a2 b1 b2 assume "{(a1,a2), (b1,b2)} \<le> Field(bsqr r)"
+   hence 0: "a1 \<in> Field r \<and> a2 \<in> Field r \<and> b1 \<in> Field r \<and> b2 \<in> Field r"
+   using Field_bsqr by blast
+   have "((a1,a2) = (b1,b2) \<or> ((a1,a2),(b1,b2)) \<in> bsqr r \<or> ((b1,b2),(a1,a2)) \<in> bsqr r)"
+   proof(rule wo_rel.cases_Total[of r a1 a2], clarsimp simp add: Well, simp add: 0)
+       (* Why didn't clarsimp simp add: Well 0 do the same job? *)
+     assume Case1: "(a1,a2) \<in> r"
+     hence 1: "wo_rel.max2 r a1 a2 = a2"
+     using Well 0 by (simp add: wo_rel.max2_equals2)
+     show ?thesis
+     proof(rule wo_rel.cases_Total[of r b1 b2], clarsimp simp add: Well, simp add: 0)
+       assume Case11: "(b1,b2) \<in> r"
+       hence 2: "wo_rel.max2 r b1 b2 = b2"
+       using Well 0 by (simp add: wo_rel.max2_equals2)
+       show ?thesis
+       proof(rule wo_rel.cases_Total3[of r a2 b2], clarsimp simp add: Well, simp add: 0)
+         assume Case111: "(a2,b2) \<in> r - Id \<or> (b2,a2) \<in> r - Id"
+         thus ?thesis using 0 1 2 unfolding bsqr_def by auto
+       next
+         assume Case112: "a2 = b2"
+         show ?thesis
+         proof(rule wo_rel.cases_Total3[of r a1 b1], clarsimp simp add: Well, simp add: 0)
+           assume Case1121: "(a1,b1) \<in> r - Id \<or> (b1,a1) \<in> r - Id"
+           thus ?thesis using 0 1 2 Case112 unfolding bsqr_def by auto
+         next
+           assume Case1122: "a1 = b1"
+           thus ?thesis using Case112 by auto
+         qed
+       qed
+     next
+       assume Case12: "(b2,b1) \<in> r"
+       hence 3: "wo_rel.max2 r b1 b2 = b1" using Well 0 by (simp add: wo_rel.max2_equals1)
+       show ?thesis
+       proof(rule wo_rel.cases_Total3[of r a2 b1], clarsimp simp add: Well, simp add: 0)
+         assume Case121: "(a2,b1) \<in> r - Id \<or> (b1,a2) \<in> r - Id"
+         thus ?thesis using 0 1 3 unfolding bsqr_def by auto
+       next
+         assume Case122: "a2 = b1"
+         show ?thesis
+         proof(rule wo_rel.cases_Total3[of r a1 b1], clarsimp simp add: Well, simp add: 0)
+           assume Case1221: "(a1,b1) \<in> r - Id \<or> (b1,a1) \<in> r - Id"
+           thus ?thesis using 0 1 3 Case122 unfolding bsqr_def by auto
+         next
+           assume Case1222: "a1 = b1"
+           show ?thesis
+           proof(rule wo_rel.cases_Total3[of r a2 b2], clarsimp simp add: Well, simp add: 0)
+             assume Case12221: "(a2,b2) \<in> r - Id \<or> (b2,a2) \<in> r - Id"
+             thus ?thesis using 0 1 3 Case122 Case1222 unfolding bsqr_def by auto
+           next
+             assume Case12222: "a2 = b2"
+             thus ?thesis using Case122 Case1222 by auto
+           qed
+         qed
+       qed
+     qed
+   next
+     assume Case2: "(a2,a1) \<in> r"
+     hence 1: "wo_rel.max2 r a1 a2 = a1" using Well 0 by (simp add: wo_rel.max2_equals1)
+     show ?thesis
+     proof(rule wo_rel.cases_Total[of r b1 b2], clarsimp simp add: Well, simp add: 0)
+       assume Case21: "(b1,b2) \<in> r"
+       hence 2: "wo_rel.max2 r b1 b2 = b2" using Well 0 by (simp add: wo_rel.max2_equals2)
+       show ?thesis
+       proof(rule wo_rel.cases_Total3[of r a1 b2], clarsimp simp add: Well, simp add: 0)
+         assume Case211: "(a1,b2) \<in> r - Id \<or> (b2,a1) \<in> r - Id"
+         thus ?thesis using 0 1 2 unfolding bsqr_def by auto
+       next
+         assume Case212: "a1 = b2"
+         show ?thesis
+         proof(rule wo_rel.cases_Total3[of r a1 b1], clarsimp simp add: Well, simp add: 0)
+           assume Case2121: "(a1,b1) \<in> r - Id \<or> (b1,a1) \<in> r - Id"
+           thus ?thesis using 0 1 2 Case212 unfolding bsqr_def by auto
+         next
+           assume Case2122: "a1 = b1"
+           show ?thesis
+           proof(rule wo_rel.cases_Total3[of r a2 b2], clarsimp simp add: Well, simp add: 0)
+             assume Case21221: "(a2,b2) \<in> r - Id \<or> (b2,a2) \<in> r - Id"
+             thus ?thesis using 0 1 2 Case212 Case2122 unfolding bsqr_def by auto
+           next
+             assume Case21222: "a2 = b2"
+             thus ?thesis using Case2122 Case212 by auto
+           qed
+         qed
+       qed
+     next
+       assume Case22: "(b2,b1) \<in> r"
+       hence 3: "wo_rel.max2 r b1 b2 = b1"  using Well 0 by (simp add: wo_rel.max2_equals1)
+       show ?thesis
+       proof(rule wo_rel.cases_Total3[of r a1 b1], clarsimp simp add: Well, simp add: 0)
+         assume Case221: "(a1,b1) \<in> r - Id \<or> (b1,a1) \<in> r - Id"
+         thus ?thesis using 0 1 3 unfolding bsqr_def by auto
+       next
+         assume Case222: "a1 = b1"
+         show ?thesis
+         proof(rule wo_rel.cases_Total3[of r a2 b2], clarsimp simp add: Well, simp add: 0)
+           assume Case2221: "(a2,b2) \<in> r - Id \<or> (b2,a2) \<in> r - Id"
+           thus ?thesis using 0 1 3 Case222 unfolding bsqr_def by auto
+         next
+           assume Case2222: "a2 = b2"
+           thus ?thesis using Case222 by auto
+         qed
+       qed
+     qed
+   qed
+  }
+  thus ?thesis unfolding total_on_def by fast
+qed
+
+
+lemma bsqr_Linear_order:
+assumes "Well_order r"
+shows "Linear_order(bsqr r)"
+unfolding order_on_defs
+using assms bsqr_Refl bsqr_Trans bsqr_antisym bsqr_Total by blast
+
+
+lemma bsqr_Well_order:
+assumes "Well_order r"
+shows "Well_order(bsqr r)"
+using assms
+proof(simp add: bsqr_Linear_order Linear_order_Well_order_iff, intro allI impI)
+  have 0: "\<forall>A \<le> Field r. A \<noteq> {} \<longrightarrow> (\<exists>a \<in> A. \<forall>a' \<in> A. (a,a') \<in> r)"
+  using assms well_order_on_def Linear_order_Well_order_iff by blast
+  fix D assume *: "D \<le> Field (bsqr r)" and **: "D \<noteq> {}"
+  hence 1: "D \<le> Field r \<times> Field r" unfolding Field_bsqr by simp
+  (*  *)
+  obtain M where M_def: "M = {wo_rel.max2 r a1 a2| a1 a2. (a1,a2) \<in> D}" by blast
+  have "M \<noteq> {}" using 1 M_def ** by auto
+  moreover
+  have "M \<le> Field r" unfolding M_def
+  using 1 assms wo_rel_def[of r] wo_rel.max2_among[of r] by fastforce
+  ultimately obtain m where m_min: "m \<in> M \<and> (\<forall>a \<in> M. (m,a) \<in> r)"
+  using 0 by blast
+  (*  *)
+  obtain A1 where A1_def: "A1 = {a1. \<exists>a2. (a1,a2) \<in> D \<and> wo_rel.max2 r a1 a2 = m}" by blast
+  have "A1 \<le> Field r" unfolding A1_def using 1 by auto
+  moreover have "A1 \<noteq> {}" unfolding A1_def using m_min unfolding M_def by blast
+  ultimately obtain a1 where a1_min: "a1 \<in> A1 \<and> (\<forall>a \<in> A1. (a1,a) \<in> r)"
+  using 0 by blast
+  (*  *)
+  obtain A2 where A2_def: "A2 = {a2. (a1,a2) \<in> D \<and> wo_rel.max2 r a1 a2 = m}" by blast
+  have "A2 \<le> Field r" unfolding A2_def using 1 by auto
+  moreover have "A2 \<noteq> {}" unfolding A2_def
+  using m_min a1_min unfolding A1_def M_def by blast
+  ultimately obtain a2 where a2_min: "a2 \<in> A2 \<and> (\<forall>a \<in> A2. (a2,a) \<in> r)"
+  using 0 by blast
+  (*   *)
+  have 2: "wo_rel.max2 r a1 a2 = m"
+  using a1_min a2_min unfolding A1_def A2_def by auto
+  have 3: "(a1,a2) \<in> D" using a2_min unfolding A2_def by auto
+  (*  *)
+  moreover
+  {fix b1 b2 assume ***: "(b1,b2) \<in> D"
+   hence 4: "{a1,a2,b1,b2} \<le> Field r" using 1 3 by blast
+   have 5: "(wo_rel.max2 r a1 a2, wo_rel.max2 r b1 b2) \<in> r"
+   using *** a1_min a2_min m_min unfolding A1_def A2_def M_def by auto
+   have "((a1,a2),(b1,b2)) \<in> bsqr r"
+   proof(cases "wo_rel.max2 r a1 a2 = wo_rel.max2 r b1 b2")
+     assume Case1: "wo_rel.max2 r a1 a2 \<noteq> wo_rel.max2 r b1 b2"
+     thus ?thesis unfolding bsqr_def using 4 5 by auto
+   next
+     assume Case2: "wo_rel.max2 r a1 a2 = wo_rel.max2 r b1 b2"
+     hence "b1 \<in> A1" unfolding A1_def using 2 *** by auto
+     hence 6: "(a1,b1) \<in> r" using a1_min by auto
+     show ?thesis
+     proof(cases "a1 = b1")
+       assume Case21: "a1 \<noteq> b1"
+       thus ?thesis unfolding bsqr_def using 4 Case2 6 by auto
+     next
+       assume Case22: "a1 = b1"
+       hence "b2 \<in> A2" unfolding A2_def using 2 *** Case2 by auto
+       hence 7: "(a2,b2) \<in> r" using a2_min by auto
+       thus ?thesis unfolding bsqr_def using 4 7 Case2 Case22 by auto
+     qed
+   qed
+  }
+  (*  *)
+  ultimately show "\<exists>d \<in> D. \<forall>d' \<in> D. (d,d') \<in> bsqr r" by fastforce
+qed
+
+
+lemma bsqr_max2:
+assumes WELL: "Well_order r" and LEQ: "((a1,a2),(b1,b2)) \<in> bsqr r"
+shows "(wo_rel.max2 r a1 a2, wo_rel.max2 r b1 b2) \<in> r"
+proof-
+  have "{(a1,a2),(b1,b2)} \<le> Field(bsqr r)"
+  using LEQ unfolding Field_def by auto
+  hence "{a1,a2,b1,b2} \<le> Field r" unfolding Field_bsqr by auto
+  hence "{wo_rel.max2 r a1 a2, wo_rel.max2 r b1 b2} \<le> Field r"
+  using WELL wo_rel_def[of r] wo_rel.max2_among[of r] by fastforce
+  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"
+  using LEQ unfolding bsqr_def by auto
+  ultimately show ?thesis using WELL unfolding order_on_defs refl_on_def by auto
+qed
+
+
+lemma bsqr_ofilter:
+assumes WELL: "Well_order r" and
+        OF: "wo_rel.ofilter (bsqr r) D" and SUB: "D < Field r \<times> Field r" and
+        NE: "\<not> (\<exists>a. Field r = under r a)"
+shows "\<exists>A. wo_rel.ofilter r A \<and> A < Field r \<and> D \<le> A \<times> A"
+proof-
+  let ?r' = "bsqr r"
+  have Well: "wo_rel r" using WELL wo_rel_def by blast
+  hence Trans: "trans r" using wo_rel.TRANS by blast
+  have Well': "Well_order ?r' \<and> wo_rel ?r'"
+  using WELL bsqr_Well_order wo_rel_def by blast
+  (*  *)
+  have "D < Field ?r'" unfolding Field_bsqr using SUB .
+  with OF obtain a1 and a2 where
+  "(a1,a2) \<in> Field ?r'" and 1: "D = underS ?r' (a1,a2)"
+  using Well' wo_rel.ofilter_underS_Field[of ?r' D] by auto
+  hence 2: "{a1,a2} \<le> Field r" unfolding Field_bsqr by auto
+  let ?m = "wo_rel.max2 r a1 a2"
+  have "D \<le> (under r ?m) \<times> (under r ?m)"
+  proof(unfold 1)
+    {fix b1 b2
+     let ?n = "wo_rel.max2 r b1 b2"
+     assume "(b1,b2) \<in> underS ?r' (a1,a2)"
+     hence 3: "((b1,b2),(a1,a2)) \<in> ?r'"
+     unfolding underS_def by blast
+     hence "(?n,?m) \<in> r" using WELL by (simp add: bsqr_max2)
+     moreover
+     {have "(b1,b2) \<in> Field ?r'" using 3 unfolding Field_def by auto
+      hence "{b1,b2} \<le> Field r" unfolding Field_bsqr by auto
+      hence "(b1,?n) \<in> r \<and> (b2,?n) \<in> r"
+      using Well by (simp add: wo_rel.max2_greater)
+     }
+     ultimately have "(b1,?m) \<in> r \<and> (b2,?m) \<in> r"
+     using Trans trans_def[of r] by blast
+     hence "(b1,b2) \<in> (under r ?m) \<times> (under r ?m)" unfolding under_def by simp}
+     thus "underS ?r' (a1,a2) \<le> (under r ?m) \<times> (under r ?m)" by auto
+  qed
+  moreover have "wo_rel.ofilter r (under r ?m)"
+  using Well by (simp add: wo_rel.under_ofilter)
+  moreover have "under r ?m < Field r"
+  using NE under_Field[of r ?m] by blast
+  ultimately show ?thesis by blast
+qed
+
+definition Func where
+"Func A B = {f . (\<forall> a \<in> A. f a \<in> B) \<and> (\<forall> a. a \<notin> A \<longrightarrow> f a = undefined)}"
+
+lemma Func_empty:
+"Func {} B = {\<lambda>x. undefined}"
+unfolding Func_def by auto
+
+lemma Func_elim:
+assumes "g \<in> Func A B" and "a \<in> A"
+shows "\<exists> b. b \<in> B \<and> g a = b"
+using assms unfolding Func_def by (cases "g a = undefined") auto
+
+definition curr where
+"curr A f \<equiv> \<lambda> a. if a \<in> A then \<lambda>b. f (a,b) else undefined"
+
+lemma curr_in:
+assumes f: "f \<in> Func (A <*> B) C"
+shows "curr A f \<in> Func A (Func B C)"
+using assms unfolding curr_def Func_def by auto
+
+lemma curr_inj:
+assumes "f1 \<in> Func (A <*> B) C" and "f2 \<in> Func (A <*> B) C"
+shows "curr A f1 = curr A f2 \<longleftrightarrow> f1 = f2"
+proof safe
+  assume c: "curr A f1 = curr A f2"
+  show "f1 = f2"
+  proof (rule ext, clarify)
+    fix a b show "f1 (a, b) = f2 (a, b)"
+    proof (cases "(a,b) \<in> A <*> B")
+      case False
+      thus ?thesis using assms unfolding Func_def by auto
+    next
+      case True hence a: "a \<in> A" and b: "b \<in> B" by auto
+      thus ?thesis
+      using c unfolding curr_def fun_eq_iff by(elim allE[of _ a]) simp
+    qed
+  qed
+qed
+
+lemma curr_surj:
+assumes "g \<in> Func A (Func B C)"
+shows "\<exists> f \<in> Func (A <*> B) C. curr A f = g"
+proof
+  let ?f = "\<lambda> ab. if fst ab \<in> A \<and> snd ab \<in> B then g (fst ab) (snd ab) else undefined"
+  show "curr A ?f = g"
+  proof (rule ext)
+    fix a show "curr A ?f a = g a"
+    proof (cases "a \<in> A")
+      case False
+      hence "g a = undefined" using assms unfolding Func_def by auto
+      thus ?thesis unfolding curr_def using False by simp
+    next
+      case True
+      obtain g1 where "g1 \<in> Func B C" and "g a = g1"
+      using assms using Func_elim[OF assms True] by blast
+      thus ?thesis using True unfolding Func_def curr_def by auto
+    qed
+  qed
+  show "?f \<in> Func (A <*> B) C" using assms unfolding Func_def mem_Collect_eq by auto
+qed
+
+lemma bij_betw_curr:
+"bij_betw (curr A) (Func (A <*> B) C) (Func A (Func B C))"
+unfolding bij_betw_def inj_on_def image_def
+apply (intro impI conjI ballI)
+apply (erule curr_inj[THEN iffD1], assumption+)
+apply auto
+apply (erule curr_in)
+using curr_surj by blast
+
+definition Func_map where
+"Func_map B2 f1 f2 g b2 \<equiv> if b2 \<in> B2 then f1 (g (f2 b2)) else undefined"
+
+lemma Func_map:
+assumes g: "g \<in> Func A2 A1" and f1: "f1 ` A1 \<subseteq> B1" and f2: "f2 ` B2 \<subseteq> A2"
+shows "Func_map B2 f1 f2 g \<in> Func B2 B1"
+using assms unfolding Func_def Func_map_def mem_Collect_eq by auto
+
+lemma Func_non_emp:
+assumes "B \<noteq> {}"
+shows "Func A B \<noteq> {}"
+proof-
+  obtain b where b: "b \<in> B" using assms by auto
+  hence "(\<lambda> a. if a \<in> A then b else undefined) \<in> Func A B" unfolding Func_def by auto
+  thus ?thesis by blast
+qed
+
+lemma Func_is_emp:
+"Func A B = {} \<longleftrightarrow> A \<noteq> {} \<and> B = {}" (is "?L \<longleftrightarrow> ?R")
+proof
+  assume L: ?L
+  moreover {assume "A = {}" hence False using L Func_empty by auto}
+  moreover {assume "B \<noteq> {}" hence False using L Func_non_emp by metis}
+  ultimately show ?R by blast
+next
+  assume R: ?R
+  moreover
+  {fix f assume "f \<in> Func A B"
+   moreover obtain a where "a \<in> A" using R by blast
+   ultimately obtain b where "b \<in> B" unfolding Func_def by blast
+   with R have False by blast
+  }
+  thus ?L by blast
+qed
+
+lemma Func_map_surj:
+assumes B1: "f1 ` A1 = B1" and A2: "inj_on f2 B2" "f2 ` B2 \<subseteq> A2"
+and B2A2: "B2 = {} \<Longrightarrow> A2 = {}"
+shows "Func B2 B1 = Func_map B2 f1 f2 ` Func A2 A1"
+proof(cases "B2 = {}")
+  case True
+  thus ?thesis using B2A2 by (auto simp: Func_empty Func_map_def)
+next
+  case False note B2 = False
+  show ?thesis
+  proof safe
+    fix h assume h: "h \<in> Func B2 B1"
+    def j1 \<equiv> "inv_into A1 f1"
+    have "\<forall> a2 \<in> f2 ` B2. \<exists> b2. b2 \<in> B2 \<and> f2 b2 = a2" by blast
+    then obtain k where k: "\<forall> a2 \<in> f2 ` B2. k a2 \<in> B2 \<and> f2 (k a2) = a2" by metis
+    {fix b2 assume b2: "b2 \<in> B2"
+     hence "f2 (k (f2 b2)) = f2 b2" using k A2(2) by auto
+     moreover have "k (f2 b2) \<in> B2" using b2 A2(2) k by auto
+     ultimately have "k (f2 b2) = b2" using b2 A2(1) unfolding inj_on_def by blast
+    } note kk = this
+    obtain b22 where b22: "b22 \<in> B2" using B2 by auto
+    def j2 \<equiv> "\<lambda> a2. if a2 \<in> f2 ` B2 then k a2 else b22"
+    have j2A2: "j2 ` A2 \<subseteq> B2" unfolding j2_def using k b22 by auto
+    have j2: "\<And> b2. b2 \<in> B2 \<Longrightarrow> j2 (f2 b2) = b2"
+    using kk unfolding j2_def by auto
+    def g \<equiv> "Func_map A2 j1 j2 h"
+    have "Func_map B2 f1 f2 g = h"
+    proof (rule ext)
+      fix b2 show "Func_map B2 f1 f2 g b2 = h b2"
+      proof(cases "b2 \<in> B2")
+        case True
+        show ?thesis
+        proof (cases "h b2 = undefined")
+          case True
+          hence b1: "h b2 \<in> f1 ` A1" using h `b2 \<in> B2` unfolding B1 Func_def by auto
+          show ?thesis using A2 f_inv_into_f[OF b1]
+            unfolding True g_def Func_map_def j1_def j2[OF `b2 \<in> B2`] by auto
+        qed(insert A2 True j2[OF True] h B1, unfold j1_def g_def Func_def Func_map_def,
+          auto intro: f_inv_into_f)
+      qed(insert h, unfold Func_def Func_map_def, auto)
+    qed
+    moreover have "g \<in> Func A2 A1" unfolding g_def apply(rule Func_map[OF h])
+    using inv_into_into j2A2 B1 A2 inv_into_into
+    unfolding j1_def image_def by fast+
+    ultimately show "h \<in> Func_map B2 f1 f2 ` Func A2 A1"
+    unfolding Func_map_def[abs_def] unfolding image_def by auto
+  qed(insert B1 Func_map[OF _ _ A2(2)], auto)
+qed
+
+end