src/HOL/BNF_Wellorder_Constructions.thy
author wenzelm
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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 total_on_imp_Total_Restr: "total_on A r \<Longrightarrow> Total (Restr r A)"
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  by (auto simp: Field_def total_on_def)
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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)"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   120
proof
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   121
  assume *: "wo_rel.ofilter r A"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   122
  show "A \<le> Field r \<and> embed (Restr r A) r id"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   123
  proof(unfold embed_def, auto)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   124
    fix a assume "a \<in> A" thus "a \<in> Field r" using assms *
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   125
    by (auto simp add: wo_rel_def wo_rel.ofilter_def)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   126
  next
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   127
    fix a assume "a \<in> Field (Restr r A)"
55023
38db7814481d get rid of 'rel' locale, to facilitate inclusion of 'Order_Relation_More_FP' into 'Order_Relation'
blanchet
parents: 54980
diff changeset
   128
    thus "bij_betw id (under (Restr r A) a) (under r a)" using assms *
48975
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   129
    by (simp add: ofilter_Restr_under Field_Restr_ofilter)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   130
  qed
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   131
next
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   132
  assume *: "A \<le> Field r \<and> embed (Restr r A) r id"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   133
  hence "Field(Restr r A) \<le> Field r"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   134
  using assms  embed_Field[of "Restr r A" r id] id_def
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   135
        Well_order_Restr[of r] by auto
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   136
  {fix a assume "a \<in> A"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   137
   hence "a \<in> Field(Restr r A)" using * assms
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   138
   by (simp add: order_on_defs Refl_Field_Restr2)
55023
38db7814481d get rid of 'rel' locale, to facilitate inclusion of 'Order_Relation_More_FP' into 'Order_Relation'
blanchet
parents: 54980
diff changeset
   139
   hence "bij_betw id (under (Restr r A) a) (under r a)"
48975
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   140
   using * unfolding embed_def by auto
55023
38db7814481d get rid of 'rel' locale, to facilitate inclusion of 'Order_Relation_More_FP' into 'Order_Relation'
blanchet
parents: 54980
diff changeset
   141
   hence "under r a \<le> under (Restr r A) a"
48975
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   142
   unfolding bij_betw_def by auto
55023
38db7814481d get rid of 'rel' locale, to facilitate inclusion of 'Order_Relation_More_FP' into 'Order_Relation'
blanchet
parents: 54980
diff changeset
   143
   also have "\<dots> \<le> Field(Restr r A)" by (simp add: under_Field)
48975
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   144
   also have "\<dots> \<le> A" by (simp add: Field_Restr_subset)
55023
38db7814481d get rid of 'rel' locale, to facilitate inclusion of 'Order_Relation_More_FP' into 'Order_Relation'
blanchet
parents: 54980
diff changeset
   145
   finally have "under r a \<le> A" .
48975
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   146
  }
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   147
  thus "wo_rel.ofilter r A" using assms * by (simp add: wo_rel_def wo_rel.ofilter_def)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   148
qed
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   149
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   150
lemma ofilter_Restr_Int:
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   151
assumes WELL: "Well_order r" and OFA: "wo_rel.ofilter r A"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   152
shows "wo_rel.ofilter (Restr r B) (A Int B)"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   153
proof-
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   154
  let ?rB = "Restr r B"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   155
  have Well: "wo_rel r" unfolding wo_rel_def using WELL .
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   156
  hence Refl: "Refl r" by (simp add: wo_rel.REFL)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   157
  hence Field: "Field ?rB = Field r Int B"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   158
  using Refl_Field_Restr by blast
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   159
  have WellB: "wo_rel ?rB \<and> Well_order ?rB" using WELL
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   160
  by (simp add: Well_order_Restr wo_rel_def)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   161
  (* Main proof *)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   162
  show ?thesis using WellB assms
55023
38db7814481d get rid of 'rel' locale, to facilitate inclusion of 'Order_Relation_More_FP' into 'Order_Relation'
blanchet
parents: 54980
diff changeset
   163
  proof(auto simp add: wo_rel.ofilter_def under_def)
48975
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   164
    fix a assume "a \<in> A" and *: "a \<in> B"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   165
    hence "a \<in> Field r" using OFA Well by (auto simp add: wo_rel.ofilter_def)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   166
    with * show "a \<in> Field ?rB" using Field by auto
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   167
  next
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   168
    fix a b assume "a \<in> A" and "(b,a) \<in> r"
55023
38db7814481d get rid of 'rel' locale, to facilitate inclusion of 'Order_Relation_More_FP' into 'Order_Relation'
blanchet
parents: 54980
diff changeset
   169
    thus "b \<in> A" using Well OFA by (auto simp add: wo_rel.ofilter_def under_def)
48975
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   170
  qed
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   171
qed
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   172
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   173
lemma ofilter_Restr_subset:
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   174
assumes WELL: "Well_order r" and OFA: "wo_rel.ofilter r A" and SUB: "A \<le> B"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   175
shows "wo_rel.ofilter (Restr r B) A"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   176
proof-
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   177
  have "A Int B = A" using SUB by blast
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   178
  thus ?thesis using assms ofilter_Restr_Int[of r A B] by auto
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   179
qed
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   180
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   181
lemma ofilter_subset_embed:
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   182
assumes WELL: "Well_order r" and
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   183
        OFA: "wo_rel.ofilter r A" and OFB: "wo_rel.ofilter r B"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   184
shows "(A \<le> B) = (embed (Restr r A) (Restr r B) id)"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   185
proof-
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   186
  let ?rA = "Restr r A"  let ?rB = "Restr r B"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   187
  have Well: "wo_rel r" unfolding wo_rel_def using WELL .
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   188
  hence Refl: "Refl r" by (simp add: wo_rel.REFL)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   189
  hence FieldA: "Field ?rA = Field r Int A"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   190
  using Refl_Field_Restr by blast
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   191
  have FieldB: "Field ?rB = Field r Int B"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   192
  using Refl Refl_Field_Restr by blast
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   193
  have WellA: "wo_rel ?rA \<and> Well_order ?rA" using WELL
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   194
  by (simp add: Well_order_Restr wo_rel_def)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   195
  have WellB: "wo_rel ?rB \<and> Well_order ?rB" using WELL
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   196
  by (simp add: Well_order_Restr wo_rel_def)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   197
  (* Main proof *)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   198
  show ?thesis
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   199
  proof
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   200
    assume *: "A \<le> B"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   201
    hence "wo_rel.ofilter (Restr r B) A" using assms
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   202
    by (simp add: ofilter_Restr_subset)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   203
    hence "embed (Restr ?rB A) (Restr r B) id"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   204
    using WellB ofilter_embed[of "?rB" A] by auto
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   205
    thus "embed (Restr r A) (Restr r B) id"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   206
    using * by (simp add: Restr_subset)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   207
  next
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   208
    assume *: "embed (Restr r A) (Restr r B) id"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   209
    {fix a assume **: "a \<in> A"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   210
     hence "a \<in> Field r" using Well OFA by (auto simp add: wo_rel.ofilter_def)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   211
     with ** FieldA have "a \<in> Field ?rA" by auto
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   212
     hence "a \<in> Field ?rB" using * WellA embed_Field[of ?rA ?rB id] by auto
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   213
     hence "a \<in> B" using FieldB by auto
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   214
    }
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   215
    thus "A \<le> B" by blast
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   216
  qed
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   217
qed
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   218
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   219
lemma ofilter_subset_embedS_iso:
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   220
assumes WELL: "Well_order r" and
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   221
        OFA: "wo_rel.ofilter r A" and OFB: "wo_rel.ofilter r B"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   222
shows "((A < B) = (embedS (Restr r A) (Restr r B) id)) \<and>
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   223
       ((A = B) = (iso (Restr r A) (Restr r B) id))"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   224
proof-
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   225
  let ?rA = "Restr r A"  let ?rB = "Restr r B"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   226
  have Well: "wo_rel r" unfolding wo_rel_def using WELL .
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   227
  hence Refl: "Refl r" by (simp add: wo_rel.REFL)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   228
  hence "Field ?rA = Field r Int A"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   229
  using Refl_Field_Restr by blast
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   230
  hence FieldA: "Field ?rA = A" using OFA Well
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   231
  by (auto simp add: wo_rel.ofilter_def)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   232
  have "Field ?rB = Field r Int B"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   233
  using Refl Refl_Field_Restr by blast
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   234
  hence FieldB: "Field ?rB = B" using OFB Well
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   235
  by (auto simp add: wo_rel.ofilter_def)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   236
  (* Main proof *)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   237
  show ?thesis unfolding embedS_def iso_def
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   238
  using assms ofilter_subset_embed[of r A B]
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   239
        FieldA FieldB bij_betw_id_iff[of A B] by auto
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   240
qed
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   241
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   242
lemma ofilter_subset_embedS:
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   243
assumes WELL: "Well_order r" and
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   244
        OFA: "wo_rel.ofilter r A" and OFB: "wo_rel.ofilter r B"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   245
shows "(A < B) = embedS (Restr r A) (Restr r B) id"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   246
using assms by (simp add: ofilter_subset_embedS_iso)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   247
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   248
lemma embed_implies_iso_Restr:
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   249
assumes WELL: "Well_order r" and WELL': "Well_order r'" and
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   250
        EMB: "embed r' r f"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   251
shows "iso r' (Restr r (f ` (Field r'))) f"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   252
proof-
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   253
  let ?A' = "Field r'"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   254
  let ?r'' = "Restr r (f ` ?A')"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   255
  have 0: "Well_order ?r''" using WELL Well_order_Restr by blast
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   256
  have 1: "wo_rel.ofilter r (f ` ?A')" using assms embed_Field_ofilter  by blast
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   257
  hence "Field ?r'' = f ` (Field r')" using WELL Field_Restr_ofilter by blast
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   258
  hence "bij_betw f ?A' (Field ?r'')"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   259
  using EMB embed_inj_on WELL' unfolding bij_betw_def by blast
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   260
  moreover
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   261
  {have "\<forall>a b. (a,b) \<in> r' \<longrightarrow> a \<in> Field r' \<and> b \<in> Field r'"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   262
   unfolding Field_def by auto
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   263
   hence "compat r' ?r'' f"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   264
   using assms embed_iff_compat_inj_on_ofilter
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   265
   unfolding compat_def by blast
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   266
  }
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   267
  ultimately show ?thesis using WELL' 0 iso_iff3 by blast
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   268
qed
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   269
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   270
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 58889
diff changeset
   271
subsection \<open>The strict inclusion on proper ofilters is well-founded\<close>
48975
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   272
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   273
definition ofilterIncl :: "'a rel \<Rightarrow> 'a set rel"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   274
where
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   275
"ofilterIncl r \<equiv> {(A,B). wo_rel.ofilter r A \<and> A \<noteq> Field r \<and>
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   276
                         wo_rel.ofilter r B \<and> B \<noteq> Field r \<and> A < B}"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   277
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   278
lemma wf_ofilterIncl:
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   279
assumes WELL: "Well_order r"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   280
shows "wf(ofilterIncl r)"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   281
proof-
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   282
  have Well: "wo_rel r" using WELL by (simp add: wo_rel_def)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   283
  hence Lo: "Linear_order r" by (simp add: wo_rel.LIN)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   284
  let ?h = "(\<lambda> A. wo_rel.suc r A)"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   285
  let ?rS = "r - Id"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   286
  have "wf ?rS" using WELL by (simp add: order_on_defs)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   287
  moreover
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   288
  have "compat (ofilterIncl r) ?rS ?h"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   289
  proof(unfold compat_def ofilterIncl_def,
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   290
        intro allI impI, simp, elim conjE)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   291
    fix A B
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   292
    assume *: "wo_rel.ofilter r A" "A \<noteq> Field r" and
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   293
           **: "wo_rel.ofilter r B" "B \<noteq> Field r" and ***: "A < B"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   294
    then obtain a and b where 0: "a \<in> Field r \<and> b \<in> Field r" and
55023
38db7814481d get rid of 'rel' locale, to facilitate inclusion of 'Order_Relation_More_FP' into 'Order_Relation'
blanchet
parents: 54980
diff changeset
   295
                         1: "A = underS r a \<and> B = underS r b"
48975
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   296
    using Well by (auto simp add: wo_rel.ofilter_underS_Field)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   297
    hence "a \<noteq> b" using *** by auto
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   298
    moreover
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   299
    have "(a,b) \<in> r" using 0 1 Lo ***
55023
38db7814481d get rid of 'rel' locale, to facilitate inclusion of 'Order_Relation_More_FP' into 'Order_Relation'
blanchet
parents: 54980
diff changeset
   300
    by (auto simp add: underS_incl_iff)
48975
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   301
    moreover
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   302
    have "a = wo_rel.suc r A \<and> b = wo_rel.suc r B"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   303
    using Well 0 1 by (simp add: wo_rel.suc_underS)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   304
    ultimately
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   305
    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"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   306
    by simp
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   307
  qed
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   308
  ultimately show "wf (ofilterIncl r)" by (simp add: compat_wf)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   309
qed
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   310
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   311
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 58889
diff changeset
   312
subsection \<open>Ordering the well-orders by existence of embeddings\<close>
48975
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   313
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 58889
diff changeset
   314
text \<open>We define three relations between well-orders:
48975
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   315
\begin{itemize}
61799
4cf66f21b764 isabelle update_cartouches -c -t;
wenzelm
parents: 60758
diff changeset
   316
\item \<open>ordLeq\<close>, of being embedded (abbreviated \<open>\<le>o\<close>);
4cf66f21b764 isabelle update_cartouches -c -t;
wenzelm
parents: 60758
diff changeset
   317
\item \<open>ordLess\<close>, of being strictly embedded (abbreviated \<open><o\<close>);
4cf66f21b764 isabelle update_cartouches -c -t;
wenzelm
parents: 60758
diff changeset
   318
\item \<open>ordIso\<close>, of being isomorphic (abbreviated \<open>=o\<close>).
48975
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   319
\end{itemize}
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   320
%
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   321
The prefix "ord" and the index "o" in these names stand for "ordinal-like".
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   322
These relations shall be proved to be inter-connected in a similar fashion as the trio
61799
4cf66f21b764 isabelle update_cartouches -c -t;
wenzelm
parents: 60758
diff changeset
   323
\<open>\<le>\<close>, \<open><\<close>, \<open>=\<close> associated to a total order on a set.
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 58889
diff changeset
   324
\<close>
48975
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   325
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   326
definition ordLeq :: "('a rel * 'a' rel) set"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   327
where
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   328
"ordLeq = {(r,r'). Well_order r \<and> Well_order r' \<and> (\<exists>f. embed r r' f)}"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   329
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   330
abbreviation ordLeq2 :: "'a rel \<Rightarrow> 'a' rel \<Rightarrow> bool" (infix "<=o" 50)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   331
where "r <=o r' \<equiv> (r,r') \<in> ordLeq"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   332
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   333
abbreviation ordLeq3 :: "'a rel \<Rightarrow> 'a' rel \<Rightarrow> bool" (infix "\<le>o" 50)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   334
where "r \<le>o r' \<equiv> r <=o r'"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   335
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   336
definition ordLess :: "('a rel * 'a' rel) set"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   337
where
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   338
"ordLess = {(r,r'). Well_order r \<and> Well_order r' \<and> (\<exists>f. embedS r r' f)}"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   339
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   340
abbreviation ordLess2 :: "'a rel \<Rightarrow> 'a' rel \<Rightarrow> bool" (infix "<o" 50)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   341
where "r <o r' \<equiv> (r,r') \<in> ordLess"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   342
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   343
definition ordIso :: "('a rel * 'a' rel) set"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   344
where
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   345
"ordIso = {(r,r'). Well_order r \<and> Well_order r' \<and> (\<exists>f. iso r r' f)}"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   346
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   347
abbreviation ordIso2 :: "'a rel \<Rightarrow> 'a' rel \<Rightarrow> bool" (infix "=o" 50)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   348
where "r =o r' \<equiv> (r,r') \<in> ordIso"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   349
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   350
lemmas ordRels_def = ordLeq_def ordLess_def ordIso_def
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   351
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   352
lemma ordLeq_Well_order_simp:
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   353
assumes "r \<le>o r'"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   354
shows "Well_order r \<and> Well_order r'"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   355
using assms unfolding ordLeq_def by simp
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   356
61799
4cf66f21b764 isabelle update_cartouches -c -t;
wenzelm
parents: 60758
diff changeset
   357
text\<open>Notice that the relations \<open>\<le>o\<close>, \<open><o\<close>, \<open>=o\<close> connect well-orders
48975
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   358
on potentially {\em distinct} types. However, some of the lemmas below, including the next one,
61799
4cf66f21b764 isabelle update_cartouches -c -t;
wenzelm
parents: 60758
diff changeset
   359
restrict implicitly the type of these relations to \<open>(('a rel) * ('a rel)) set\<close> , i.e.,
4cf66f21b764 isabelle update_cartouches -c -t;
wenzelm
parents: 60758
diff changeset
   360
to \<open>'a rel rel\<close>.\<close>
48975
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   361
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   362
lemma ordLeq_reflexive:
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   363
"Well_order r \<Longrightarrow> r \<le>o r"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   364
unfolding ordLeq_def using id_embed[of r] by blast
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   365
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   366
lemma ordLeq_transitive[trans]:
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   367
assumes *: "r \<le>o r'" and **: "r' \<le>o r''"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   368
shows "r \<le>o r''"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   369
proof-
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   370
  obtain f and f'
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   371
  where 1: "Well_order r \<and> Well_order r' \<and> Well_order r''" and
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   372
        "embed r r' f" and "embed r' r'' f'"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   373
  using * ** unfolding ordLeq_def by blast
67091
1393c2340eec more symbols;
wenzelm
parents: 63561
diff changeset
   374
  hence "embed r r'' (f' \<circ> f)"
48975
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   375
  using comp_embed[of r r' f r'' f'] by auto
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   376
  thus "r \<le>o r''" unfolding ordLeq_def using 1 by auto
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   377
qed
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   378
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   379
lemma ordLeq_total:
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   380
"\<lbrakk>Well_order r; Well_order r'\<rbrakk> \<Longrightarrow> r \<le>o r' \<or> r' \<le>o r"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   381
unfolding ordLeq_def using wellorders_totally_ordered by blast
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   382
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   383
lemma ordIso_reflexive:
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   384
"Well_order r \<Longrightarrow> r =o r"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   385
unfolding ordIso_def using id_iso[of r] by blast
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   386
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   387
lemma ordIso_transitive[trans]:
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   388
assumes *: "r =o r'" and **: "r' =o r''"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   389
shows "r =o r''"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   390
proof-
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   391
  obtain f and f'
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   392
  where 1: "Well_order r \<and> Well_order r' \<and> Well_order r''" and
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   393
        "iso r r' f" and 3: "iso r' r'' f'"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   394
  using * ** unfolding ordIso_def by auto
67091
1393c2340eec more symbols;
wenzelm
parents: 63561
diff changeset
   395
  hence "iso r r'' (f' \<circ> f)"
48975
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   396
  using comp_iso[of r r' f r'' f'] by auto
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   397
  thus "r =o r''" unfolding ordIso_def using 1 by auto
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   398
qed
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   399
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   400
lemma ordIso_symmetric:
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   401
assumes *: "r =o r'"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   402
shows "r' =o r"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   403
proof-
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   404
  obtain f where 1: "Well_order r \<and> Well_order r'" and
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   405
                 2: "embed r r' f \<and> bij_betw f (Field r) (Field r')"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   406
  using * by (auto simp add: ordIso_def iso_def)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   407
  let ?f' = "inv_into (Field r) f"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   408
  have "embed r' r ?f' \<and> bij_betw ?f' (Field r') (Field r)"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   409
  using 1 2 by (simp add: bij_betw_inv_into inv_into_Field_embed_bij_betw)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   410
  thus "r' =o r" unfolding ordIso_def using 1 by (auto simp add: iso_def)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   411
qed
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   412
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   413
lemma ordLeq_ordLess_trans[trans]:
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   414
assumes "r \<le>o r'" and " r' <o r''"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   415
shows "r <o r''"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   416
proof-
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   417
  have "Well_order r \<and> Well_order r''"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   418
  using assms unfolding ordLeq_def ordLess_def by auto
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   419
  thus ?thesis using assms unfolding ordLeq_def ordLess_def
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   420
  using embed_comp_embedS by blast
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   421
qed
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   422
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   423
lemma ordLess_ordLeq_trans[trans]:
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   424
assumes "r <o r'" and " r' \<le>o r''"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   425
shows "r <o r''"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   426
proof-
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   427
  have "Well_order r \<and> Well_order r''"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   428
  using assms unfolding ordLeq_def ordLess_def by auto
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   429
  thus ?thesis using assms unfolding ordLeq_def ordLess_def
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   430
  using embedS_comp_embed by blast
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   431
qed
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   432
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   433
lemma ordLeq_ordIso_trans[trans]:
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   434
assumes "r \<le>o r'" and " r' =o r''"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   435
shows "r \<le>o r''"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   436
proof-
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   437
  have "Well_order r \<and> Well_order r''"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   438
  using assms unfolding ordLeq_def ordIso_def by auto
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   439
  thus ?thesis using assms unfolding ordLeq_def ordIso_def
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   440
  using embed_comp_iso by blast
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   441
qed
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   442
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   443
lemma ordIso_ordLeq_trans[trans]:
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   444
assumes "r =o r'" and " r' \<le>o r''"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   445
shows "r \<le>o r''"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   446
proof-
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   447
  have "Well_order r \<and> Well_order r''"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   448
  using assms unfolding ordLeq_def ordIso_def by auto
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   449
  thus ?thesis using assms unfolding ordLeq_def ordIso_def
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   450
  using iso_comp_embed by blast
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   451
qed
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   452
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   453
lemma ordLess_ordIso_trans[trans]:
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   454
assumes "r <o r'" and " r' =o r''"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   455
shows "r <o r''"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   456
proof-
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   457
  have "Well_order r \<and> Well_order r''"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   458
  using assms unfolding ordLess_def ordIso_def by auto
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   459
  thus ?thesis using assms unfolding ordLess_def ordIso_def
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   460
  using embedS_comp_iso by blast
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   461
qed
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   462
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   463
lemma ordIso_ordLess_trans[trans]:
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   464
assumes "r =o r'" and " r' <o r''"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   465
shows "r <o r''"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   466
proof-
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   467
  have "Well_order r \<and> Well_order r''"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   468
  using assms unfolding ordLess_def ordIso_def by auto
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   469
  thus ?thesis using assms unfolding ordLess_def ordIso_def
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   470
  using iso_comp_embedS by blast
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   471
qed
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   472
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   473
lemma ordLess_not_embed:
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   474
assumes "r <o r'"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   475
shows "\<not>(\<exists>f'. embed r' r f')"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   476
proof-
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   477
  obtain f where 1: "Well_order r \<and> Well_order r'" and 2: "embed r r' f" and
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   478
                 3: " \<not> bij_betw f (Field r) (Field r')"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   479
  using assms unfolding ordLess_def by (auto simp add: embedS_def)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   480
  {fix f' assume *: "embed r' r f'"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   481
   hence "bij_betw f (Field r) (Field r')" using 1 2
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   482
   by (simp add: embed_bothWays_Field_bij_betw)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   483
   with 3 have False by contradiction
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   484
  }
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   485
  thus ?thesis by blast
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   486
qed
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   487
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   488
lemma ordLess_Field:
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   489
assumes OL: "r1 <o r2" and EMB: "embed r1 r2 f"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   490
shows "\<not> (f`(Field r1) = Field r2)"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   491
proof-
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   492
  let ?A1 = "Field r1"  let ?A2 = "Field r2"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   493
  obtain g where
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   494
  0: "Well_order r1 \<and> Well_order r2" and
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   495
  1: "embed r1 r2 g \<and> \<not>(bij_betw g ?A1 ?A2)"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   496
  using OL unfolding ordLess_def by (auto simp add: embedS_def)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   497
  hence "\<forall>a \<in> ?A1. f a = g a"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   498
  using 0 EMB embed_unique[of r1] by auto
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   499
  hence "\<not>(bij_betw f ?A1 ?A2)"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   500
  using 1 bij_betw_cong[of ?A1] by blast
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   501
  moreover
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   502
  have "inj_on f ?A1" using EMB 0 by (simp add: embed_inj_on)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   503
  ultimately show ?thesis by (simp add: bij_betw_def)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   504
qed
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   505
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   506
lemma ordLess_iff:
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   507
"r <o r' = (Well_order r \<and> Well_order r' \<and> \<not>(\<exists>f'. embed r' r f'))"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   508
proof
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   509
  assume *: "r <o r'"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   510
  hence "\<not>(\<exists>f'. embed r' r f')" using ordLess_not_embed[of r r'] by simp
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   511
  with * show "Well_order r \<and> Well_order r' \<and> \<not> (\<exists>f'. embed r' r f')"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   512
  unfolding ordLess_def by auto
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   513
next
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   514
  assume *: "Well_order r \<and> Well_order r' \<and> \<not> (\<exists>f'. embed r' r f')"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   515
  then obtain f where 1: "embed r r' f"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   516
  using wellorders_totally_ordered[of r r'] by blast
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   517
  moreover
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   518
  {assume "bij_betw f (Field r) (Field r')"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   519
   with * 1 have "embed r' r (inv_into (Field r) f) "
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   520
   using inv_into_Field_embed_bij_betw[of r r' f] by auto
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   521
   with * have False by blast
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   522
  }
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   523
  ultimately show "(r,r') \<in> ordLess"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   524
  unfolding ordLess_def using * by (fastforce simp add: embedS_def)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   525
qed
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   526
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   527
lemma ordLess_irreflexive: "\<not> r <o r"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   528
proof
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   529
  assume "r <o r"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   530
  hence "Well_order r \<and>  \<not>(\<exists>f. embed r r f)"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   531
  unfolding ordLess_iff ..
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   532
  moreover have "embed r r id" using id_embed[of r] .
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   533
  ultimately show False by blast
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   534
qed
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   535
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   536
lemma ordLeq_iff_ordLess_or_ordIso:
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   537
"r \<le>o r' = (r <o r' \<or> r =o r')"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   538
unfolding ordRels_def embedS_defs iso_defs by blast
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   539
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   540
lemma ordIso_iff_ordLeq:
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   541
"(r =o r') = (r \<le>o r' \<and> r' \<le>o r)"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   542
proof
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   543
  assume "r =o r'"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   544
  then obtain f where 1: "Well_order r \<and> Well_order r' \<and>
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   545
                     embed r r' f \<and> bij_betw f (Field r) (Field r')"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   546
  unfolding ordIso_def iso_defs by auto
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   547
  hence "embed r r' f \<and> embed r' r (inv_into (Field r) f)"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   548
  by (simp add: inv_into_Field_embed_bij_betw)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   549
  thus  "r \<le>o r' \<and> r' \<le>o r"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   550
  unfolding ordLeq_def using 1 by auto
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   551
next
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   552
  assume "r \<le>o r' \<and> r' \<le>o r"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   553
  then obtain f and g where 1: "Well_order r \<and> Well_order r' \<and>
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   554
                           embed r r' f \<and> embed r' r g"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   555
  unfolding ordLeq_def by auto
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   556
  hence "iso r r' f" by (auto simp add: embed_bothWays_iso)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   557
  thus "r =o r'" unfolding ordIso_def using 1 by auto
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   558
qed
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   559
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   560
lemma not_ordLess_ordLeq:
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   561
"r <o r' \<Longrightarrow> \<not> r' \<le>o r"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   562
using ordLess_ordLeq_trans ordLess_irreflexive by blast
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   563
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   564
lemma ordLess_or_ordLeq:
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   565
assumes WELL: "Well_order r" and WELL': "Well_order r'"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   566
shows "r <o r' \<or> r' \<le>o r"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   567
proof-
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   568
  have "r \<le>o r' \<or> r' \<le>o r"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   569
  using assms by (simp add: ordLeq_total)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   570
  moreover
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   571
  {assume "\<not> r <o r' \<and> r \<le>o r'"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   572
   hence "r =o r'" using ordLeq_iff_ordLess_or_ordIso by blast
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   573
   hence "r' \<le>o r" using ordIso_symmetric ordIso_iff_ordLeq by blast
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   574
  }
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   575
  ultimately show ?thesis by blast
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   576
qed
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   577
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   578
lemma not_ordLess_ordIso:
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   579
"r <o r' \<Longrightarrow> \<not> r =o r'"
63092
a949b2a5f51d eliminated use of empty "assms";
wenzelm
parents: 63040
diff changeset
   580
using ordLess_ordIso_trans ordIso_symmetric ordLess_irreflexive by blast
48975
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   581
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   582
lemma not_ordLeq_iff_ordLess:
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   583
assumes WELL: "Well_order r" and WELL': "Well_order r'"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   584
shows "(\<not> r' \<le>o r) = (r <o r')"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   585
using assms not_ordLess_ordLeq ordLess_or_ordLeq by blast
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   586
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   587
lemma not_ordLess_iff_ordLeq:
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   588
assumes WELL: "Well_order r" and WELL': "Well_order r'"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   589
shows "(\<not> r' <o r) = (r \<le>o r')"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   590
using assms not_ordLess_ordLeq ordLess_or_ordLeq by blast
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   591
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   592
lemma ordLess_transitive[trans]:
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   593
"\<lbrakk>r <o r'; r' <o r''\<rbrakk> \<Longrightarrow> r <o r''"
63092
a949b2a5f51d eliminated use of empty "assms";
wenzelm
parents: 63040
diff changeset
   594
using ordLess_ordLeq_trans ordLeq_iff_ordLess_or_ordIso by blast
48975
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   595
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   596
corollary ordLess_trans: "trans ordLess"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   597
unfolding trans_def using ordLess_transitive by blast
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   598
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   599
lemmas ordIso_equivalence = ordIso_transitive ordIso_reflexive ordIso_symmetric
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   600
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   601
lemma ordIso_imp_ordLeq:
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   602
"r =o r' \<Longrightarrow> r \<le>o r'"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   603
using ordIso_iff_ordLeq by blast
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   604
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   605
lemma ordLess_imp_ordLeq:
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   606
"r <o r' \<Longrightarrow> r \<le>o r'"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   607
using ordLeq_iff_ordLess_or_ordIso by blast
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   608
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   609
lemma ofilter_subset_ordLeq:
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   610
assumes WELL: "Well_order r" and
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   611
        OFA: "wo_rel.ofilter r A" and OFB: "wo_rel.ofilter r B"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   612
shows "(A \<le> B) = (Restr r A \<le>o Restr r B)"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   613
proof
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   614
  assume "A \<le> B"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   615
  thus "Restr r A \<le>o Restr r B"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   616
  unfolding ordLeq_def using assms
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   617
  Well_order_Restr Well_order_Restr ofilter_subset_embed by blast
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   618
next
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   619
  assume *: "Restr r A \<le>o Restr r B"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   620
  then obtain f where "embed (Restr r A) (Restr r B) f"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   621
  unfolding ordLeq_def by blast
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   622
  {assume "B < A"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   623
   hence "Restr r B <o Restr r A"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   624
   unfolding ordLess_def using assms
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   625
   Well_order_Restr Well_order_Restr ofilter_subset_embedS by blast
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   626
   hence False using * not_ordLess_ordLeq by blast
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   627
  }
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   628
  thus "A \<le> B" using OFA OFB WELL
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   629
  wo_rel_def[of r] wo_rel.ofilter_linord[of r A B] by blast
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   630
qed
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   631
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   632
lemma ofilter_subset_ordLess:
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   633
assumes WELL: "Well_order r" and
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   634
        OFA: "wo_rel.ofilter r A" and OFB: "wo_rel.ofilter r B"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   635
shows "(A < B) = (Restr r A <o Restr r B)"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   636
proof-
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   637
  let ?rA = "Restr r A" let ?rB = "Restr r B"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   638
  have 1: "Well_order ?rA \<and> Well_order ?rB"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   639
  using WELL Well_order_Restr by blast
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   640
  have "(A < B) = (\<not> B \<le> A)" using assms
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   641
  wo_rel_def wo_rel.ofilter_linord[of r A B] by blast
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   642
  also have "\<dots> = (\<not> Restr r B \<le>o Restr r A)"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   643
  using assms ofilter_subset_ordLeq by blast
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   644
  also have "\<dots> = (Restr r A <o Restr r B)"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   645
  using 1 not_ordLeq_iff_ordLess by blast
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   646
  finally show ?thesis .
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   647
qed
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   648
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   649
lemma ofilter_ordLess:
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   650
"\<lbrakk>Well_order r; wo_rel.ofilter r A\<rbrakk> \<Longrightarrow> (A < Field r) = (Restr r A <o r)"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   651
by (simp add: ofilter_subset_ordLess wo_rel.Field_ofilter
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   652
    wo_rel_def Restr_Field)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   653
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   654
corollary underS_Restr_ordLess:
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   655
assumes "Well_order r" and "Field r \<noteq> {}"
55023
38db7814481d get rid of 'rel' locale, to facilitate inclusion of 'Order_Relation_More_FP' into 'Order_Relation'
blanchet
parents: 54980
diff changeset
   656
shows "Restr r (underS r a) <o r"
48975
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   657
proof-
55023
38db7814481d get rid of 'rel' locale, to facilitate inclusion of 'Order_Relation_More_FP' into 'Order_Relation'
blanchet
parents: 54980
diff changeset
   658
  have "underS r a < Field r" using assms
38db7814481d get rid of 'rel' locale, to facilitate inclusion of 'Order_Relation_More_FP' into 'Order_Relation'
blanchet
parents: 54980
diff changeset
   659
  by (simp add: underS_Field3)
48975
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   660
  thus ?thesis using assms
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   661
  by (simp add: ofilter_ordLess wo_rel.underS_ofilter wo_rel_def)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   662
qed
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   663
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   664
lemma embed_ordLess_ofilterIncl:
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   665
assumes
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   666
  OL12: "r1 <o r2" and OL23: "r2 <o r3" and
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   667
  EMB13: "embed r1 r3 f13" and EMB23: "embed r2 r3 f23"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   668
shows "(f13`(Field r1), f23`(Field r2)) \<in> (ofilterIncl r3)"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   669
proof-
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   670
  have OL13: "r1 <o r3"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   671
  using OL12 OL23 using ordLess_transitive by auto
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   672
  let ?A1 = "Field r1"  let ?A2 ="Field r2" let ?A3 ="Field r3"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   673
  obtain f12 g23 where
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   674
  0: "Well_order r1 \<and> Well_order r2 \<and> Well_order r3" and
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   675
  1: "embed r1 r2 f12 \<and> \<not>(bij_betw f12 ?A1 ?A2)" and
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   676
  2: "embed r2 r3 g23 \<and> \<not>(bij_betw g23 ?A2 ?A3)"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   677
  using OL12 OL23 by (auto simp add: ordLess_def embedS_def)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   678
  hence "\<forall>a \<in> ?A2. f23 a = g23 a"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   679
  using EMB23 embed_unique[of r2 r3] by blast
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   680
  hence 3: "\<not>(bij_betw f23 ?A2 ?A3)"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   681
  using 2 bij_betw_cong[of ?A2 f23 g23] by blast
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   682
  (*  *)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   683
  have 4: "wo_rel.ofilter r2 (f12 ` ?A1) \<and> f12 ` ?A1 \<noteq> ?A2"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   684
  using 0 1 OL12 by (simp add: embed_Field_ofilter ordLess_Field)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   685
  have 5: "wo_rel.ofilter r3 (f23 ` ?A2) \<and> f23 ` ?A2 \<noteq> ?A3"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   686
  using 0 EMB23 OL23 by (simp add: embed_Field_ofilter ordLess_Field)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   687
  have 6: "wo_rel.ofilter r3 (f13 ` ?A1)  \<and> f13 ` ?A1 \<noteq> ?A3"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   688
  using 0 EMB13 OL13 by (simp add: embed_Field_ofilter ordLess_Field)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   689
  (*  *)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   690
  have "f12 ` ?A1 < ?A2"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   691
  using 0 4 by (auto simp add: wo_rel_def wo_rel.ofilter_def)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   692
  moreover have "inj_on f23 ?A2"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   693
  using EMB23 0 by (simp add: wo_rel_def embed_inj_on)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   694
  ultimately
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   695
  have "f23 ` (f12 ` ?A1) < f23 ` ?A2" by (simp add: inj_on_strict_subset)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   696
  moreover
67091
1393c2340eec more symbols;
wenzelm
parents: 63561
diff changeset
   697
  {have "embed r1 r3 (f23 \<circ> f12)"
48975
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   698
   using 1 EMB23 0 by (auto simp add: comp_embed)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   699
   hence "\<forall>a \<in> ?A1. f23(f12 a) = f13 a"
67091
1393c2340eec more symbols;
wenzelm
parents: 63561
diff changeset
   700
   using EMB13 0 embed_unique[of r1 r3 "f23 \<circ> f12" f13] by auto
48975
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   701
   hence "f23 ` (f12 ` ?A1) = f13 ` ?A1" by force
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   702
  }
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   703
  ultimately
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   704
  have "f13 ` ?A1 < f23 ` ?A2" by simp
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   705
  (*  *)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   706
  with 5 6 show ?thesis
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   707
  unfolding ofilterIncl_def by auto
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   708
qed
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   709
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   710
lemma ordLess_iff_ordIso_Restr:
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   711
assumes WELL: "Well_order r" and WELL': "Well_order r'"
55023
38db7814481d get rid of 'rel' locale, to facilitate inclusion of 'Order_Relation_More_FP' into 'Order_Relation'
blanchet
parents: 54980
diff changeset
   712
shows "(r' <o r) = (\<exists>a \<in> Field r. r' =o Restr r (underS r a))"
48975
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   713
proof(auto)
55023
38db7814481d get rid of 'rel' locale, to facilitate inclusion of 'Order_Relation_More_FP' into 'Order_Relation'
blanchet
parents: 54980
diff changeset
   714
  fix a assume *: "a \<in> Field r" and **: "r' =o Restr r (underS r a)"
38db7814481d get rid of 'rel' locale, to facilitate inclusion of 'Order_Relation_More_FP' into 'Order_Relation'
blanchet
parents: 54980
diff changeset
   715
  hence "Restr r (underS r a) <o r" using WELL underS_Restr_ordLess[of r] by blast
48975
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   716
  thus "r' <o r" using ** ordIso_ordLess_trans by blast
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   717
next
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   718
  assume "r' <o r"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   719
  then obtain f where 1: "Well_order r \<and> Well_order r'" and
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   720
                      2: "embed r' r f \<and> f ` (Field r') \<noteq> Field r"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   721
  unfolding ordLess_def embedS_def[abs_def] bij_betw_def using embed_inj_on by blast
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   722
  hence "wo_rel.ofilter r (f ` (Field r'))" using embed_Field_ofilter by blast
55023
38db7814481d get rid of 'rel' locale, to facilitate inclusion of 'Order_Relation_More_FP' into 'Order_Relation'
blanchet
parents: 54980
diff changeset
   723
  then obtain a where 3: "a \<in> Field r" and 4: "underS r a = f ` (Field r')"
48975
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   724
  using 1 2 by (auto simp add: wo_rel.ofilter_underS_Field wo_rel_def)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   725
  have "iso r' (Restr r (f ` (Field r'))) f"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   726
  using embed_implies_iso_Restr 2 assms by blast
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   727
  moreover have "Well_order (Restr r (f ` (Field r')))"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   728
  using WELL Well_order_Restr by blast
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   729
  ultimately have "r' =o Restr r (f ` (Field r'))"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   730
  using WELL' unfolding ordIso_def by auto
55023
38db7814481d get rid of 'rel' locale, to facilitate inclusion of 'Order_Relation_More_FP' into 'Order_Relation'
blanchet
parents: 54980
diff changeset
   731
  hence "r' =o Restr r (underS r a)" using 4 by auto
38db7814481d get rid of 'rel' locale, to facilitate inclusion of 'Order_Relation_More_FP' into 'Order_Relation'
blanchet
parents: 54980
diff changeset
   732
  thus "\<exists>a \<in> Field r. r' =o Restr r (underS r a)" using 3 by auto
48975
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   733
qed
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   734
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   735
lemma internalize_ordLess:
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   736
"(r' <o r) = (\<exists>p. Field p < Field r \<and> r' =o p \<and> p <o r)"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   737
proof
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   738
  assume *: "r' <o r"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   739
  hence 0: "Well_order r \<and> Well_order r'" unfolding ordLess_def by auto
55023
38db7814481d get rid of 'rel' locale, to facilitate inclusion of 'Order_Relation_More_FP' into 'Order_Relation'
blanchet
parents: 54980
diff changeset
   740
  with * obtain a where 1: "a \<in> Field r" and 2: "r' =o Restr r (underS r a)"
48975
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   741
  using ordLess_iff_ordIso_Restr by blast
55023
38db7814481d get rid of 'rel' locale, to facilitate inclusion of 'Order_Relation_More_FP' into 'Order_Relation'
blanchet
parents: 54980
diff changeset
   742
  let ?p = "Restr r (underS r a)"
38db7814481d get rid of 'rel' locale, to facilitate inclusion of 'Order_Relation_More_FP' into 'Order_Relation'
blanchet
parents: 54980
diff changeset
   743
  have "wo_rel.ofilter r (underS r a)" using 0
48975
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   744
  by (simp add: wo_rel_def wo_rel.underS_ofilter)
55023
38db7814481d get rid of 'rel' locale, to facilitate inclusion of 'Order_Relation_More_FP' into 'Order_Relation'
blanchet
parents: 54980
diff changeset
   745
  hence "Field ?p = underS r a" using 0 Field_Restr_ofilter by blast
38db7814481d get rid of 'rel' locale, to facilitate inclusion of 'Order_Relation_More_FP' into 'Order_Relation'
blanchet
parents: 54980
diff changeset
   746
  hence "Field ?p < Field r" using underS_Field2 1 by fast
48975
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   747
  moreover have "?p <o r" using underS_Restr_ordLess[of r a] 0 1 by blast
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   748
  ultimately
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   749
  show "\<exists>p. Field p < Field r \<and> r' =o p \<and> p <o r" using 2 by blast
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   750
next
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   751
  assume "\<exists>p. Field p < Field r \<and> r' =o p \<and> p <o r"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   752
  thus "r' <o r" using ordIso_ordLess_trans by blast
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   753
qed
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   754
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   755
lemma internalize_ordLeq:
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   756
"(r' \<le>o r) = (\<exists>p. Field p \<le> Field r \<and> r' =o p \<and> p \<le>o r)"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   757
proof
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   758
  assume *: "r' \<le>o r"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   759
  moreover
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   760
  {assume "r' <o r"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   761
   then obtain p where "Field p < Field r \<and> r' =o p \<and> p <o r"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   762
   using internalize_ordLess[of r' r] by blast
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   763
   hence "\<exists>p. Field p \<le> Field r \<and> r' =o p \<and> p \<le>o r"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   764
   using ordLeq_iff_ordLess_or_ordIso by blast
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   765
  }
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   766
  moreover
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   767
  have "r \<le>o r" using * ordLeq_def ordLeq_reflexive by blast
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   768
  ultimately show "\<exists>p. Field p \<le> Field r \<and> r' =o p \<and> p \<le>o r"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   769
  using ordLeq_iff_ordLess_or_ordIso by blast
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   770
next
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   771
  assume "\<exists>p. Field p \<le> Field r \<and> r' =o p \<and> p \<le>o r"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   772
  thus "r' \<le>o r" using ordIso_ordLeq_trans by blast
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   773
qed
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   774
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   775
lemma ordLeq_iff_ordLess_Restr:
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   776
assumes WELL: "Well_order r" and WELL': "Well_order r'"
55023
38db7814481d get rid of 'rel' locale, to facilitate inclusion of 'Order_Relation_More_FP' into 'Order_Relation'
blanchet
parents: 54980
diff changeset
   777
shows "(r \<le>o r') = (\<forall>a \<in> Field r. Restr r (underS r a) <o r')"
48975
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   778
proof(auto)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   779
  assume *: "r \<le>o r'"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   780
  fix a assume "a \<in> Field r"
55023
38db7814481d get rid of 'rel' locale, to facilitate inclusion of 'Order_Relation_More_FP' into 'Order_Relation'
blanchet
parents: 54980
diff changeset
   781
  hence "Restr r (underS r a) <o r"
48975
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   782
  using WELL underS_Restr_ordLess[of r] by blast
55023
38db7814481d get rid of 'rel' locale, to facilitate inclusion of 'Order_Relation_More_FP' into 'Order_Relation'
blanchet
parents: 54980
diff changeset
   783
  thus "Restr r (underS r a) <o r'"
48975
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   784
  using * ordLess_ordLeq_trans by blast
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   785
next
55023
38db7814481d get rid of 'rel' locale, to facilitate inclusion of 'Order_Relation_More_FP' into 'Order_Relation'
blanchet
parents: 54980
diff changeset
   786
  assume *: "\<forall>a \<in> Field r. Restr r (underS r a) <o r'"
48975
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   787
  {assume "r' <o r"
55023
38db7814481d get rid of 'rel' locale, to facilitate inclusion of 'Order_Relation_More_FP' into 'Order_Relation'
blanchet
parents: 54980
diff changeset
   788
   then obtain a where "a \<in> Field r \<and> r' =o Restr r (underS r a)"
48975
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   789
   using assms ordLess_iff_ordIso_Restr by blast
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   790
   hence False using * not_ordLess_ordIso ordIso_symmetric by blast
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   791
  }
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   792
  thus "r \<le>o r'" using ordLess_or_ordLeq assms by blast
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   793
qed
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   794
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   795
lemma finite_ordLess_infinite:
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   796
assumes WELL: "Well_order r" and WELL': "Well_order r'" and
54578
9387251b6a46 eliminated dependence of BNF on Infinite_Set by moving 3 theorems from the latter to Main
traytel
parents: 54482
diff changeset
   797
        FIN: "finite(Field r)" and INF: "\<not>finite(Field r')"
48975
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   798
shows "r <o r'"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   799
proof-
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   800
  {assume "r' \<le>o r"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   801
   then obtain h where "inj_on h (Field r') \<and> h ` (Field r') \<le> Field r"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   802
   unfolding ordLeq_def using assms embed_inj_on embed_Field by blast
55811
aa1acc25126b load Metis a little later
traytel
parents: 55603
diff changeset
   803
   hence False using finite_imageD finite_subset FIN INF by blast
48975
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   804
  }
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   805
  thus ?thesis using WELL WELL' ordLess_or_ordLeq by blast
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   806
qed
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   807
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   808
lemma finite_well_order_on_ordIso:
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   809
assumes FIN: "finite A" and
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   810
        WELL: "well_order_on A r" and WELL': "well_order_on A r'"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   811
shows "r =o r'"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   812
proof-
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   813
  have 0: "Well_order r \<and> Well_order r' \<and> Field r = A \<and> Field r' = A"
55023
38db7814481d get rid of 'rel' locale, to facilitate inclusion of 'Order_Relation_More_FP' into 'Order_Relation'
blanchet
parents: 54980
diff changeset
   814
  using assms well_order_on_Well_order by blast
48975
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   815
  moreover
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   816
  have "\<forall>r r'. well_order_on A r \<and> well_order_on A r' \<and> r \<le>o r'
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   817
                  \<longrightarrow> r =o r'"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   818
  proof(clarify)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   819
    fix r r' assume *: "well_order_on A r" and **: "well_order_on A r'"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   820
    have 2: "Well_order r \<and> Well_order r' \<and> Field r = A \<and> Field r' = A"
55023
38db7814481d get rid of 'rel' locale, to facilitate inclusion of 'Order_Relation_More_FP' into 'Order_Relation'
blanchet
parents: 54980
diff changeset
   821
    using * ** well_order_on_Well_order by blast
48975
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   822
    assume "r \<le>o r'"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   823
    then obtain f where 1: "embed r r' f" and
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   824
                        "inj_on f A \<and> f ` A \<le> A"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   825
    unfolding ordLeq_def using 2 embed_inj_on embed_Field by blast
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   826
    hence "bij_betw f A A" unfolding bij_betw_def using FIN endo_inj_surj by blast
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   827
    thus "r =o r'" unfolding ordIso_def iso_def[abs_def] using 1 2 by auto
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   828
  qed
55811
aa1acc25126b load Metis a little later
traytel
parents: 55603
diff changeset
   829
  ultimately show ?thesis using assms ordLeq_total ordIso_symmetric by blast
48975
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   830
qed
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   831
61799
4cf66f21b764 isabelle update_cartouches -c -t;
wenzelm
parents: 60758
diff changeset
   832
subsection\<open>\<open><o\<close> is well-founded\<close>
48975
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   833
61799
4cf66f21b764 isabelle update_cartouches -c -t;
wenzelm
parents: 60758
diff changeset
   834
text \<open>Of course, it only makes sense to state that the \<open><o\<close> is well-founded
4cf66f21b764 isabelle update_cartouches -c -t;
wenzelm
parents: 60758
diff changeset
   835
on the restricted type \<open>'a rel rel\<close>.  We prove this by first showing that, for any set
48975
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   836
of well-orders all embedded in a fixed well-order, the function mapping each well-order
61799
4cf66f21b764 isabelle update_cartouches -c -t;
wenzelm
parents: 60758
diff changeset
   837
in the set to an order filter of the fixed well-order is compatible w.r.t. to \<open><o\<close> versus
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 58889
diff changeset
   838
{\em strict inclusion}; and we already know that strict inclusion of order filters is well-founded.\<close>
48975
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   839
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   840
definition ord_to_filter :: "'a rel \<Rightarrow> 'a rel \<Rightarrow> 'a set"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   841
where "ord_to_filter r0 r \<equiv> (SOME f. embed r r0 f) ` (Field r)"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   842
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   843
lemma ord_to_filter_compat:
67613
ce654b0e6d69 more symbols;
wenzelm
parents: 67091
diff changeset
   844
"compat (ordLess Int (ordLess\<inverse>``{r0} \<times> ordLess\<inverse>``{r0}))
48975
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   845
        (ofilterIncl r0)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   846
        (ord_to_filter r0)"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   847
proof(unfold compat_def ord_to_filter_def, clarify)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   848
  fix r1::"'a rel" and r2::"'a rel"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   849
  let ?A1 = "Field r1"  let ?A2 ="Field r2" let ?A0 ="Field r0"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   850
  let ?phi10 = "\<lambda> f10. embed r1 r0 f10" let ?f10 = "SOME f. ?phi10 f"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   851
  let ?phi20 = "\<lambda> f20. embed r2 r0 f20" let ?f20 = "SOME f. ?phi20 f"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   852
  assume *: "r1 <o r0" "r2 <o r0" and **: "r1 <o r2"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   853
  hence "(\<exists>f. ?phi10 f) \<and> (\<exists>f. ?phi20 f)"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   854
  by (auto simp add: ordLess_def embedS_def)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   855
  hence "?phi10 ?f10 \<and> ?phi20 ?f20" by (auto simp add: someI_ex)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   856
  thus "(?f10 ` ?A1, ?f20 ` ?A2) \<in> ofilterIncl r0"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   857
  using * ** by (simp add: embed_ordLess_ofilterIncl)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   858
qed
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   859
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   860
theorem wf_ordLess: "wf ordLess"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   861
proof-
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   862
  {fix r0 :: "('a \<times> 'a) set"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   863
   (* need to annotate here!*)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   864
   let ?ordLess = "ordLess::('d rel * 'd rel) set"
67613
ce654b0e6d69 more symbols;
wenzelm
parents: 67091
diff changeset
   865
   let ?R = "?ordLess Int (?ordLess\<inverse>``{r0} \<times> ?ordLess\<inverse>``{r0})"
48975
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   866
   {assume Case1: "Well_order r0"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   867
    hence "wf ?R"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   868
    using wf_ofilterIncl[of r0]
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   869
          compat_wf[of ?R "ofilterIncl r0" "ord_to_filter r0"]
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   870
          ord_to_filter_compat[of r0] by auto
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   871
   }
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   872
   moreover
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   873
   {assume Case2: "\<not> Well_order r0"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   874
    hence "?R = {}" unfolding ordLess_def by auto
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   875
    hence "wf ?R" using wf_empty by simp
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   876
   }
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   877
   ultimately have "wf ?R" by blast
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   878
  }
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   879
  thus ?thesis by (simp add: trans_wf_iff ordLess_trans)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   880
qed
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   881
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   882
corollary exists_minim_Well_order:
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   883
assumes NE: "R \<noteq> {}" and WELL: "\<forall>r \<in> R. Well_order r"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   884
shows "\<exists>r \<in> R. \<forall>r' \<in> R. r \<le>o r'"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   885
proof-
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   886
  obtain r where "r \<in> R \<and> (\<forall>r' \<in> R. \<not> r' <o r)"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   887
  using NE spec[OF spec[OF subst[OF wf_eq_minimal, of "%x. x", OF wf_ordLess]], of _ R]
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   888
    equals0I[of R] by blast
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   889
  with not_ordLeq_iff_ordLess WELL show ?thesis by blast
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   890
qed
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   891
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   892
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 58889
diff changeset
   893
subsection \<open>Copy via direct images\<close>
48975
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   894
61799
4cf66f21b764 isabelle update_cartouches -c -t;
wenzelm
parents: 60758
diff changeset
   895
text\<open>The direct image operator is the dual of the inverse image operator \<open>inv_image\<close>
4cf66f21b764 isabelle update_cartouches -c -t;
wenzelm
parents: 60758
diff changeset
   896
from \<open>Relation.thy\<close>.  It is useful for transporting a well-order between
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 58889
diff changeset
   897
different types.\<close>
48975
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   898
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   899
definition dir_image :: "'a rel \<Rightarrow> ('a \<Rightarrow> 'a') \<Rightarrow> 'a' rel"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   900
where
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   901
"dir_image r f = {(f a, f b)| a b. (a,b) \<in> r}"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   902
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   903
lemma dir_image_Field:
56191
159b0c88b4a4 tuned proofs; removed duplicated facts
traytel
parents: 56077
diff changeset
   904
"Field(dir_image r f) = f ` (Field r)"
159b0c88b4a4 tuned proofs; removed duplicated facts
traytel
parents: 56077
diff changeset
   905
unfolding dir_image_def Field_def Range_def Domain_def by fast
48975
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   906
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   907
lemma dir_image_minus_Id:
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   908
"inj_on f (Field r) \<Longrightarrow> (dir_image r f) - Id = dir_image (r - Id) f"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   909
unfolding inj_on_def Field_def dir_image_def by auto
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   910
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   911
lemma Refl_dir_image:
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   912
assumes "Refl r"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   913
shows "Refl(dir_image r f)"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   914
proof-
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   915
  {fix a' b'
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   916
   assume "(a',b') \<in> dir_image r f"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   917
   then obtain a b where 1: "a' = f a \<and> b' = f b \<and> (a,b) \<in> r"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   918
   unfolding dir_image_def by blast
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   919
   hence "a \<in> Field r \<and> b \<in> Field r" using Field_def by fastforce
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   920
   hence "(a,a) \<in> r \<and> (b,b) \<in> r" using assms by (simp add: refl_on_def)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   921
   with 1 have "(a',a') \<in> dir_image r f \<and> (b',b') \<in> dir_image r f"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   922
   unfolding dir_image_def by auto
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   923
  }
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   924
  thus ?thesis
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   925
  by(unfold refl_on_def Field_def Domain_def Range_def, auto)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   926
qed
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   927
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   928
lemma trans_dir_image:
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   929
assumes TRANS: "trans r" and INJ: "inj_on f (Field r)"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   930
shows "trans(dir_image r f)"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   931
proof(unfold trans_def, auto)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   932
  fix a' b' c'
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   933
  assume "(a',b') \<in> dir_image r f" "(b',c') \<in> dir_image r f"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   934
  then obtain a b1 b2 c where 1: "a' = f a \<and> b' = f b1 \<and> b' = f b2 \<and> c' = f c" and
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   935
                         2: "(a,b1) \<in> r \<and> (b2,c) \<in> r"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   936
  unfolding dir_image_def by blast
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   937
  hence "b1 \<in> Field r \<and> b2 \<in> Field r"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   938
  unfolding Field_def by auto
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   939
  hence "b1 = b2" using 1 INJ unfolding inj_on_def by auto
67613
ce654b0e6d69 more symbols;
wenzelm
parents: 67091
diff changeset
   940
  hence "(a,c) \<in> r" using 2 TRANS unfolding trans_def by blast
48975
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   941
  thus "(a',c') \<in> dir_image r f"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   942
  unfolding dir_image_def using 1 by auto
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   943
qed
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   944
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   945
lemma Preorder_dir_image:
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   946
"\<lbrakk>Preorder r; inj_on f (Field r)\<rbrakk> \<Longrightarrow> Preorder (dir_image r f)"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   947
by (simp add: preorder_on_def Refl_dir_image trans_dir_image)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   948
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   949
lemma antisym_dir_image:
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   950
assumes AN: "antisym r" and INJ: "inj_on f (Field r)"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   951
shows "antisym(dir_image r f)"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   952
proof(unfold antisym_def, auto)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   953
  fix a' b'
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   954
  assume "(a',b') \<in> dir_image r f" "(b',a') \<in> dir_image r f"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   955
  then obtain a1 b1 a2 b2 where 1: "a' = f a1 \<and> a' = f a2 \<and> b' = f b1 \<and> b' = f b2" and
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   956
                           2: "(a1,b1) \<in> r \<and> (b2,a2) \<in> r " and
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   957
                           3: "{a1,a2,b1,b2} \<le> Field r"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   958
  unfolding dir_image_def Field_def by blast
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   959
  hence "a1 = a2 \<and> b1 = b2" using INJ unfolding inj_on_def by auto
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   960
  hence "a1 = b2" using 2 AN unfolding antisym_def by auto
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   961
  thus "a' = b'" using 1 by auto
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   962
qed
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   963
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   964
lemma Partial_order_dir_image:
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   965
"\<lbrakk>Partial_order r; inj_on f (Field r)\<rbrakk> \<Longrightarrow> Partial_order (dir_image r f)"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   966
by (simp add: partial_order_on_def Preorder_dir_image antisym_dir_image)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   967
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   968
lemma Total_dir_image:
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   969
assumes TOT: "Total r" and INJ: "inj_on f (Field r)"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   970
shows "Total(dir_image r f)"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   971
proof(unfold total_on_def, intro ballI impI)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   972
  fix a' b'
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   973
  assume "a' \<in> Field (dir_image r f)" "b' \<in> Field (dir_image r f)"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   974
  then obtain a and b where 1: "a \<in> Field r \<and> b \<in> Field r \<and> f a = a' \<and> f b = b'"
56191
159b0c88b4a4 tuned proofs; removed duplicated facts
traytel
parents: 56077
diff changeset
   975
    unfolding dir_image_Field[of r f] by blast
48975
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   976
  moreover assume "a' \<noteq> b'"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   977
  ultimately have "a \<noteq> b" using INJ unfolding inj_on_def by auto
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   978
  hence "(a,b) \<in> r \<or> (b,a) \<in> r" using 1 TOT unfolding total_on_def by auto
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   979
  thus "(a',b') \<in> dir_image r f \<or> (b',a') \<in> dir_image r f"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   980
  using 1 unfolding dir_image_def by auto
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   981
qed
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   982
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   983
lemma Linear_order_dir_image:
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   984
"\<lbrakk>Linear_order r; inj_on f (Field r)\<rbrakk> \<Longrightarrow> Linear_order (dir_image r f)"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   985
by (simp add: linear_order_on_def Partial_order_dir_image Total_dir_image)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   986
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   987
lemma wf_dir_image:
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   988
assumes WF: "wf r" and INJ: "inj_on f (Field r)"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   989
shows "wf(dir_image r f)"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   990
proof(unfold wf_eq_minimal2, intro allI impI, elim conjE)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   991
  fix A'::"'b set"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   992
  assume SUB: "A' \<le> Field(dir_image r f)" and NE: "A' \<noteq> {}"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   993
  obtain A where A_def: "A = {a \<in> Field r. f a \<in> A'}" by blast
56191
159b0c88b4a4 tuned proofs; removed duplicated facts
traytel
parents: 56077
diff changeset
   994
  have "A \<noteq> {} \<and> A \<le> Field r" using A_def SUB NE by (auto simp: dir_image_Field)
48975
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   995
  then obtain a where 1: "a \<in> A \<and> (\<forall>b \<in> A. (b,a) \<notin> r)"
56191
159b0c88b4a4 tuned proofs; removed duplicated facts
traytel
parents: 56077
diff changeset
   996
  using spec[OF WF[unfolded wf_eq_minimal2], of A] by blast
48975
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   997
  have "\<forall>b' \<in> A'. (b',f a) \<notin> dir_image r f"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   998
  proof(clarify)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   999
    fix b' assume *: "b' \<in> A'" and **: "(b',f a) \<in> dir_image r f"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1000
    obtain b1 a1 where 2: "b' = f b1 \<and> f a = f a1" and
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1001
                       3: "(b1,a1) \<in> r \<and> {a1,b1} \<le> Field r"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1002
    using ** unfolding dir_image_def Field_def by blast
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1003
    hence "a = a1" using 1 A_def INJ unfolding inj_on_def by auto
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1004
    hence "b1 \<in> A \<and> (b1,a) \<in> r" using 2 3 A_def * by auto
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1005
    with 1 show False by auto
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1006
  qed
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1007
  thus "\<exists>a'\<in>A'. \<forall>b'\<in>A'. (b', a') \<notin> dir_image r f"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1008
  using A_def 1 by blast
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1009
qed
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1010
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1011
lemma Well_order_dir_image:
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1012
"\<lbrakk>Well_order r; inj_on f (Field r)\<rbrakk> \<Longrightarrow> Well_order (dir_image r f)"
63092
a949b2a5f51d eliminated use of empty "assms";
wenzelm
parents: 63040
diff changeset
  1013
unfolding well_order_on_def
48975
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1014
using Linear_order_dir_image[of r f] wf_dir_image[of "r - Id" f]
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1015
  dir_image_minus_Id[of f r]
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1016
  subset_inj_on[of f "Field r" "Field(r - Id)"]
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1017
  mono_Field[of "r - Id" r] by auto
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1018
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1019
lemma dir_image_bij_betw:
56191
159b0c88b4a4 tuned proofs; removed duplicated facts
traytel
parents: 56077
diff changeset
  1020
"\<lbrakk>inj_on f (Field r)\<rbrakk> \<Longrightarrow> bij_betw f (Field r) (Field (dir_image r f))"
159b0c88b4a4 tuned proofs; removed duplicated facts
traytel
parents: 56077
diff changeset
  1021
unfolding bij_betw_def by (simp add: dir_image_Field order_on_defs)
48975
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1022
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1023
lemma dir_image_compat:
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1024
"compat r (dir_image r f) f"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1025
unfolding compat_def dir_image_def by auto
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1026
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1027
lemma dir_image_iso:
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1028
"\<lbrakk>Well_order r; inj_on f (Field r)\<rbrakk>  \<Longrightarrow> iso r (dir_image r f) f"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1029
using iso_iff3 dir_image_compat dir_image_bij_betw Well_order_dir_image by blast
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1030
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1031
lemma dir_image_ordIso:
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1032
"\<lbrakk>Well_order r; inj_on f (Field r)\<rbrakk>  \<Longrightarrow> r =o dir_image r f"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1033
unfolding ordIso_def using dir_image_iso Well_order_dir_image by blast
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1034
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1035
lemma Well_order_iso_copy:
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1036
assumes WELL: "well_order_on A r" and BIJ: "bij_betw f A A'"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1037
shows "\<exists>r'. well_order_on A' r' \<and> r =o r'"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1038
proof-
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1039
   let ?r' = "dir_image r f"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1040
   have 1: "A = Field r \<and> Well_order r"
55023
38db7814481d get rid of 'rel' locale, to facilitate inclusion of 'Order_Relation_More_FP' into 'Order_Relation'
blanchet
parents: 54980
diff changeset
  1041
   using WELL well_order_on_Well_order by blast
48975
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1042
   hence 2: "iso r ?r' f"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1043
   using dir_image_iso using BIJ unfolding bij_betw_def by auto
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1044
   hence "f ` (Field r) = Field ?r'" using 1 iso_iff[of r ?r'] by blast
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1045
   hence "Field ?r' = A'"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1046
   using 1 BIJ unfolding bij_betw_def by auto
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1047
   moreover have "Well_order ?r'"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1048
   using 1 Well_order_dir_image BIJ unfolding bij_betw_def by blast
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1049
   ultimately show ?thesis unfolding ordIso_def using 1 2 by blast
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1050
qed
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1051
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1052
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 58889
diff changeset
  1053
subsection \<open>Bounded square\<close>
48975
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1054
61799
4cf66f21b764 isabelle update_cartouches -c -t;
wenzelm
parents: 60758
diff changeset
  1055
text\<open>This construction essentially defines, for an order relation \<open>r\<close>, a lexicographic
4cf66f21b764 isabelle update_cartouches -c -t;
wenzelm
parents: 60758
diff changeset
  1056
order \<open>bsqr r\<close> on \<open>(Field r) \<times> (Field r)\<close>, applying the
48975
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1057
following criteria (in this order):
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1058
\begin{itemize}
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1059
\item compare the maximums;
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1060
\item compare the first components;
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1061
\item compare the second components.
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1062
\end{itemize}
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1063
%
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1064
The only application of this construction that we are aware of is
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1065
at proving that the square of an infinite set has the same cardinal
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1066
as that set. The essential property required there (and which is ensured by this
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1067
construction) is that any proper order filter of the product order is included in a rectangle, i.e.,
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 58889
diff changeset
  1068
in a product of proper filters on the original relation (assumed to be a well-order).\<close>
48975
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1069
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1070
definition bsqr :: "'a rel => ('a * 'a)rel"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1071
where
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1072
"bsqr r = {((a1,a2),(b1,b2)).
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1073
           {a1,a2,b1,b2} \<le> Field r \<and>
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1074
           (a1 = b1 \<and> a2 = b2 \<or>
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1075
            (wo_rel.max2 r a1 a2, wo_rel.max2 r b1 b2) \<in> r - Id \<or>
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1076
            wo_rel.max2 r a1 a2 = wo_rel.max2 r b1 b2 \<and> (a1,b1) \<in> r - Id \<or>
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1077
            wo_rel.max2 r a1 a2 = wo_rel.max2 r b1 b2 \<and> a1 = b1  \<and> (a2,b2) \<in> r - Id
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1078
           )}"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1079
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1080
lemma Field_bsqr:
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1081
"Field (bsqr r) = Field r \<times> Field r"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1082
proof
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1083
  show "Field (bsqr r) \<le> Field r \<times> Field r"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1084
  proof-
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1085
    {fix a1 a2 assume "(a1,a2) \<in> Field (bsqr r)"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1086
     moreover
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1087
     have "\<And> b1 b2. ((a1,a2),(b1,b2)) \<in> bsqr r \<or> ((b1,b2),(a1,a2)) \<in> bsqr r \<Longrightarrow>
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1088
                      a1 \<in> Field r \<and> a2 \<in> Field r" unfolding bsqr_def by auto
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1089
     ultimately have "a1 \<in> Field r \<and> a2 \<in> Field r" unfolding Field_def by auto
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1090
    }
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1091
    thus ?thesis unfolding Field_def by force
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1092
  qed
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1093
next
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1094
  show "Field r \<times> Field r \<le> Field (bsqr r)"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1095
  proof(auto)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1096
    fix a1 a2 assume "a1 \<in> Field r" and "a2 \<in> Field r"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1097
    hence "((a1,a2),(a1,a2)) \<in> bsqr r" unfolding bsqr_def by blast
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1098
    thus "(a1,a2) \<in> Field (bsqr r)" unfolding Field_def by auto
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1099
  qed
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1100
qed
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1101
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1102
lemma bsqr_Refl: "Refl(bsqr r)"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1103
by(unfold refl_on_def Field_bsqr, auto simp add: bsqr_def)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1104
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1105
lemma bsqr_Trans:
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1106
assumes "Well_order r"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1107
shows "trans (bsqr r)"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1108
proof(unfold trans_def, auto)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1109
  (* Preliminary facts *)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1110
  have Well: "wo_rel r" using assms wo_rel_def by auto
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1111
  hence Trans: "trans r" using wo_rel.TRANS by auto
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1112
  have Anti: "antisym r" using wo_rel.ANTISYM Well by auto
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1113
  hence TransS: "trans(r - Id)" using Trans by (simp add: trans_diff_Id)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1114
  (* Main proof *)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1115
  fix a1 a2 b1 b2 c1 c2
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1116
  assume *: "((a1,a2),(b1,b2)) \<in> bsqr r" and **: "((b1,b2),(c1,c2)) \<in> bsqr r"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1117
  hence 0: "{a1,a2,b1,b2,c1,c2} \<le> Field r" unfolding bsqr_def by auto
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1118
  have 1: "a1 = b1 \<and> a2 = b2 \<or> (wo_rel.max2 r a1 a2, wo_rel.max2 r b1 b2) \<in> r - Id \<or>
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1119
           wo_rel.max2 r a1 a2 = wo_rel.max2 r b1 b2 \<and> (a1,b1) \<in> r - Id \<or>
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1120
           wo_rel.max2 r a1 a2 = wo_rel.max2 r b1 b2 \<and> a1 = b1 \<and> (a2,b2) \<in> r - Id"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1121
  using * unfolding bsqr_def by auto
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1122
  have 2: "b1 = c1 \<and> b2 = c2 \<or> (wo_rel.max2 r b1 b2, wo_rel.max2 r c1 c2) \<in> r - Id \<or>
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1123
           wo_rel.max2 r b1 b2 = wo_rel.max2 r c1 c2 \<and> (b1,c1) \<in> r - Id \<or>
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1124
           wo_rel.max2 r b1 b2 = wo_rel.max2 r c1 c2 \<and> b1 = c1 \<and> (b2,c2) \<in> r - Id"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1125
  using ** unfolding bsqr_def by auto
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1126
  show "((a1,a2),(c1,c2)) \<in> bsqr r"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1127
  proof-
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1128
    {assume Case1: "a1 = b1 \<and> a2 = b2"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1129
     hence ?thesis using ** by simp
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1130
    }
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1131
    moreover
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1132
    {assume Case2: "(wo_rel.max2 r a1 a2, wo_rel.max2 r b1 b2) \<in> r - Id"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1133
     {assume Case21: "b1 = c1 \<and> b2 = c2"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1134
      hence ?thesis using * by simp
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1135
     }
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1136
     moreover
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1137
     {assume Case22: "(wo_rel.max2 r b1 b2, wo_rel.max2 r c1 c2) \<in> r - Id"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1138
      hence "(wo_rel.max2 r a1 a2, wo_rel.max2 r c1 c2) \<in> r - Id"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1139
      using Case2 TransS trans_def[of "r - Id"] by blast
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1140
      hence ?thesis using 0 unfolding bsqr_def by auto
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1141
     }
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1142
     moreover
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1143
     {assume Case23_4: "wo_rel.max2 r b1 b2 = wo_rel.max2 r c1 c2"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1144
      hence ?thesis using Case2 0 unfolding bsqr_def by auto
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1145
     }
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1146
     ultimately have ?thesis using 0 2 by auto
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1147
    }
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1148
    moreover
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1149
    {assume Case3: "wo_rel.max2 r a1 a2 = wo_rel.max2 r b1 b2 \<and> (a1,b1) \<in> r - Id"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1150
     {assume Case31: "b1 = c1 \<and> b2 = c2"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1151
      hence ?thesis using * by simp
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1152
     }
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1153
     moreover
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1154
     {assume Case32: "(wo_rel.max2 r b1 b2, wo_rel.max2 r c1 c2) \<in> r - Id"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1155
      hence ?thesis using Case3 0 unfolding bsqr_def by auto
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1156
     }
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1157
     moreover
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1158
     {assume Case33: "wo_rel.max2 r b1 b2 = wo_rel.max2 r c1 c2 \<and> (b1,c1) \<in> r - Id"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1159
      hence "(a1,c1) \<in> r - Id"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1160
      using Case3 TransS trans_def[of "r - Id"] by blast
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1161
      hence ?thesis using Case3 Case33 0 unfolding bsqr_def by auto
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1162
     }
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1163
     moreover
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1164
     {assume Case33: "wo_rel.max2 r b1 b2 = wo_rel.max2 r c1 c2 \<and> b1 = c1"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1165
      hence ?thesis using Case3 0 unfolding bsqr_def by auto
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1166
     }
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1167
     ultimately have ?thesis using 0 2 by auto
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1168
    }
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1169
    moreover
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1170
    {assume Case4: "wo_rel.max2 r a1 a2 = wo_rel.max2 r b1 b2 \<and> a1 = b1 \<and> (a2,b2) \<in> r - Id"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1171
     {assume Case41: "b1 = c1 \<and> b2 = c2"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1172
      hence ?thesis using * by simp
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1173
     }
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1174
     moreover
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1175
     {assume Case42: "(wo_rel.max2 r b1 b2, wo_rel.max2 r c1 c2) \<in> r - Id"
51764
67f05cb13e08 optimized proofs
traytel
parents: 49310
diff changeset
  1176
      hence ?thesis using Case4 0 unfolding bsqr_def by force
48975
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1177
     }
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1178
     moreover
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1179
     {assume Case43: "wo_rel.max2 r b1 b2 = wo_rel.max2 r c1 c2 \<and> (b1,c1) \<in> r - Id"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1180
      hence ?thesis using Case4 0 unfolding bsqr_def by auto
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1181
     }
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1182
     moreover
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1183
     {assume Case44: "wo_rel.max2 r b1 b2 = wo_rel.max2 r c1 c2 \<and> b1 = c1 \<and> (b2,c2) \<in> r - Id"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1184
      hence "(a2,c2) \<in> r - Id"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1185
      using Case4 TransS trans_def[of "r - Id"] by blast
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1186
      hence ?thesis using Case4 Case44 0 unfolding bsqr_def by auto
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1187
     }
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1188
     ultimately have ?thesis using 0 2 by auto
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1189
    }
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1190
    ultimately show ?thesis using 0 1 by auto
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1191
  qed
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1192
qed
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1193
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1194
lemma bsqr_antisym:
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1195
assumes "Well_order r"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1196
shows "antisym (bsqr r)"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1197
proof(unfold antisym_def, clarify)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1198
  (* Preliminary facts *)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1199
  have Well: "wo_rel r" using assms wo_rel_def by auto
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1200
  hence Trans: "trans r" using wo_rel.TRANS by auto
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1201
  have Anti: "antisym r" using wo_rel.ANTISYM Well by auto
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1202
  hence TransS: "trans(r - Id)" using Trans by (simp add: trans_diff_Id)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1203
  hence IrrS: "\<forall>a b. \<not>((a,b) \<in> r - Id \<and> (b,a) \<in> r - Id)"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1204
  using Anti trans_def[of "r - Id"] antisym_def[of "r - Id"] by blast
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1205
  (* Main proof *)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1206
  fix a1 a2 b1 b2
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1207
  assume *: "((a1,a2),(b1,b2)) \<in> bsqr r" and **: "((b1,b2),(a1,a2)) \<in> bsqr r"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1208
  hence 0: "{a1,a2,b1,b2} \<le> Field r" unfolding bsqr_def by auto
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1209
  have 1: "a1 = b1 \<and> a2 = b2 \<or> (wo_rel.max2 r a1 a2, wo_rel.max2 r b1 b2) \<in> r - Id \<or>
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1210
           wo_rel.max2 r a1 a2 = wo_rel.max2 r b1 b2 \<and> (a1,b1) \<in> r - Id \<or>
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1211
           wo_rel.max2 r a1 a2 = wo_rel.max2 r b1 b2 \<and> a1 = b1 \<and> (a2,b2) \<in> r - Id"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1212
  using * unfolding bsqr_def by auto
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1213
  have 2: "b1 = a1 \<and> b2 = a2 \<or> (wo_rel.max2 r b1 b2, wo_rel.max2 r a1 a2) \<in> r - Id \<or>
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1214
           wo_rel.max2 r b1 b2 = wo_rel.max2 r a1 a2 \<and> (b1,a1) \<in> r - Id \<or>
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1215
           wo_rel.max2 r b1 b2 = wo_rel.max2 r a1 a2 \<and> b1 = a1 \<and> (b2,a2) \<in> r - Id"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1216
  using ** unfolding bsqr_def by auto
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1217
  show "a1 = b1 \<and> a2 = b2"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1218
  proof-
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1219
    {assume Case1: "(wo_rel.max2 r a1 a2, wo_rel.max2 r b1 b2) \<in> r - Id"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1220
     {assume Case11: "(wo_rel.max2 r b1 b2, wo_rel.max2 r a1 a2) \<in> r - Id"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1221
      hence False using Case1 IrrS by blast
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1222
     }
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1223
     moreover
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1224
     {assume Case12_3: "wo_rel.max2 r b1 b2 = wo_rel.max2 r a1 a2"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1225
      hence False using Case1 by auto
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1226
     }
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1227
     ultimately have ?thesis using 0 2 by auto
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1228
    }
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1229
    moreover
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1230
    {assume Case2: "wo_rel.max2 r a1 a2 = wo_rel.max2 r b1 b2 \<and> (a1,b1) \<in> r - Id"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1231
     {assume Case21: "(wo_rel.max2 r b1 b2, wo_rel.max2 r a1 a2) \<in> r - Id"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1232
       hence False using Case2 by auto
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1233
     }
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1234
     moreover
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1235
     {assume Case22: "(b1,a1) \<in> r - Id"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1236
      hence False using Case2 IrrS by blast
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1237
     }
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1238
     moreover
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1239
     {assume Case23: "b1 = a1"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1240
      hence False using Case2 by auto
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1241
     }
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1242
     ultimately have ?thesis using 0 2 by auto
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1243
    }
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1244
    moreover
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1245
    {assume Case3: "wo_rel.max2 r a1 a2 = wo_rel.max2 r b1 b2 \<and> a1 = b1 \<and> (a2,b2) \<in> r - Id"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1246
     moreover
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1247
     {assume Case31: "(wo_rel.max2 r b1 b2, wo_rel.max2 r a1 a2) \<in> r - Id"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1248
      hence False using Case3 by auto
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1249
     }
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1250
     moreover
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1251
     {assume Case32: "(b1,a1) \<in> r - Id"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1252
      hence False using Case3 by auto
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1253
     }
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1254
     moreover
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1255
     {assume Case33: "(b2,a2) \<in> r - Id"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1256
      hence False using Case3 IrrS by blast
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1257
     }
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1258
     ultimately have ?thesis using 0 2 by auto
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1259
    }
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1260
    ultimately show ?thesis using 0 1 by blast
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1261
  qed
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1262
qed
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1263
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1264
lemma bsqr_Total:
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1265
assumes "Well_order r"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1266
shows "Total(bsqr r)"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1267
proof-
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1268
  (* Preliminary facts *)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1269
  have Well: "wo_rel r" using assms wo_rel_def by auto
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1270
  hence Total: "\<forall>a \<in> Field r. \<forall>b \<in> Field r. (a,b) \<in> r \<or> (b,a) \<in> r"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1271
  using wo_rel.TOTALS by auto
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1272
  (* Main proof *)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1273
  {fix a1 a2 b1 b2 assume "{(a1,a2), (b1,b2)} \<le> Field(bsqr r)"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1274
   hence 0: "a1 \<in> Field r \<and> a2 \<in> Field r \<and> b1 \<in> Field r \<and> b2 \<in> Field r"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1275
   using Field_bsqr by blast
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1276
   have "((a1,a2) = (b1,b2) \<or> ((a1,a2),(b1,b2)) \<in> bsqr r \<or> ((b1,b2),(a1,a2)) \<in> bsqr r)"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1277
   proof(rule wo_rel.cases_Total[of r a1 a2], clarsimp simp add: Well, simp add: 0)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1278
       (* Why didn't clarsimp simp add: Well 0 do the same job? *)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1279
     assume Case1: "(a1,a2) \<in> r"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1280
     hence 1: "wo_rel.max2 r a1 a2 = a2"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1281
     using Well 0 by (simp add: wo_rel.max2_equals2)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1282
     show ?thesis
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1283
     proof(rule wo_rel.cases_Total[of r b1 b2], clarsimp simp add: Well, simp add: 0)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1284
       assume Case11: "(b1,b2) \<in> r"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1285
       hence 2: "wo_rel.max2 r b1 b2 = b2"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1286
       using Well 0 by (simp add: wo_rel.max2_equals2)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1287
       show ?thesis
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1288
       proof(rule wo_rel.cases_Total3[of r a2 b2], clarsimp simp add: Well, simp add: 0)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1289
         assume Case111: "(a2,b2) \<in> r - Id \<or> (b2,a2) \<in> r - Id"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1290
         thus ?thesis using 0 1 2 unfolding bsqr_def by auto
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1291
       next
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1292
         assume Case112: "a2 = b2"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1293
         show ?thesis
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1294
         proof(rule wo_rel.cases_Total3[of r a1 b1], clarsimp simp add: Well, simp add: 0)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1295
           assume Case1121: "(a1,b1) \<in> r - Id \<or> (b1,a1) \<in> r - Id"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1296
           thus ?thesis using 0 1 2 Case112 unfolding bsqr_def by auto
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1297
         next
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1298
           assume Case1122: "a1 = b1"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1299
           thus ?thesis using Case112 by auto
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1300
         qed
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1301
       qed
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1302
     next
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1303
       assume Case12: "(b2,b1) \<in> r"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1304
       hence 3: "wo_rel.max2 r b1 b2 = b1" using Well 0 by (simp add: wo_rel.max2_equals1)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1305
       show ?thesis
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1306
       proof(rule wo_rel.cases_Total3[of r a2 b1], clarsimp simp add: Well, simp add: 0)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1307
         assume Case121: "(a2,b1) \<in> r - Id \<or> (b1,a2) \<in> r - Id"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1308
         thus ?thesis using 0 1 3 unfolding bsqr_def by auto
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1309
       next
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1310
         assume Case122: "a2 = b1"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1311
         show ?thesis
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1312
         proof(rule wo_rel.cases_Total3[of r a1 b1], clarsimp simp add: Well, simp add: 0)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1313
           assume Case1221: "(a1,b1) \<in> r - Id \<or> (b1,a1) \<in> r - Id"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1314
           thus ?thesis using 0 1 3 Case122 unfolding bsqr_def by auto
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1315
         next
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1316
           assume Case1222: "a1 = b1"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1317
           show ?thesis
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1318
           proof(rule wo_rel.cases_Total3[of r a2 b2], clarsimp simp add: Well, simp add: 0)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1319
             assume Case12221: "(a2,b2) \<in> r - Id \<or> (b2,a2) \<in> r - Id"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1320
             thus ?thesis using 0 1 3 Case122 Case1222 unfolding bsqr_def by auto
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1321
           next
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1322
             assume Case12222: "a2 = b2"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1323
             thus ?thesis using Case122 Case1222 by auto
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1324
           qed
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1325
         qed
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1326
       qed
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1327
     qed
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1328
   next
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1329
     assume Case2: "(a2,a1) \<in> r"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1330
     hence 1: "wo_rel.max2 r a1 a2 = a1" using Well 0 by (simp add: wo_rel.max2_equals1)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1331
     show ?thesis
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1332
     proof(rule wo_rel.cases_Total[of r b1 b2], clarsimp simp add: Well, simp add: 0)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1333
       assume Case21: "(b1,b2) \<in> r"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1334
       hence 2: "wo_rel.max2 r b1 b2 = b2" using Well 0 by (simp add: wo_rel.max2_equals2)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1335
       show ?thesis
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1336
       proof(rule wo_rel.cases_Total3[of r a1 b2], clarsimp simp add: Well, simp add: 0)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1337
         assume Case211: "(a1,b2) \<in> r - Id \<or> (b2,a1) \<in> r - Id"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1338
         thus ?thesis using 0 1 2 unfolding bsqr_def by auto
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1339
       next
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1340
         assume Case212: "a1 = b2"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1341
         show ?thesis
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1342
         proof(rule wo_rel.cases_Total3[of r a1 b1], clarsimp simp add: Well, simp add: 0)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1343
           assume Case2121: "(a1,b1) \<in> r - Id \<or> (b1,a1) \<in> r - Id"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1344
           thus ?thesis using 0 1 2 Case212 unfolding bsqr_def by auto
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1345
         next
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1346
           assume Case2122: "a1 = b1"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1347
           show ?thesis
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1348
           proof(rule wo_rel.cases_Total3[of r a2 b2], clarsimp simp add: Well, simp add: 0)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1349
             assume Case21221: "(a2,b2) \<in> r - Id \<or> (b2,a2) \<in> r - Id"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1350
             thus ?thesis using 0 1 2 Case212 Case2122 unfolding bsqr_def by auto
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1351
           next
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1352
             assume Case21222: "a2 = b2"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1353
             thus ?thesis using Case2122 Case212 by auto
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1354
           qed
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1355
         qed
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1356
       qed
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1357
     next
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1358
       assume Case22: "(b2,b1) \<in> r"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1359
       hence 3: "wo_rel.max2 r b1 b2 = b1"  using Well 0 by (simp add: wo_rel.max2_equals1)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1360
       show ?thesis
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1361
       proof(rule wo_rel.cases_Total3[of r a1 b1], clarsimp simp add: Well, simp add: 0)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1362
         assume Case221: "(a1,b1) \<in> r - Id \<or> (b1,a1) \<in> r - Id"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1363
         thus ?thesis using 0 1 3 unfolding bsqr_def by auto
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1364
       next
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1365
         assume Case222: "a1 = b1"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1366
         show ?thesis
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1367
         proof(rule wo_rel.cases_Total3[of r a2 b2], clarsimp simp add: Well, simp add: 0)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1368
           assume Case2221: "(a2,b2) \<in> r - Id \<or> (b2,a2) \<in> r - Id"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1369
           thus ?thesis using 0 1 3 Case222 unfolding bsqr_def by auto
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1370
         next
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1371
           assume Case2222: "a2 = b2"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1372
           thus ?thesis using Case222 by auto
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1373
         qed
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1374
       qed
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1375
     qed
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1376
   qed
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1377
  }
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1378
  thus ?thesis unfolding total_on_def by fast
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1379
qed
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1380
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1381
lemma bsqr_Linear_order:
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1382
assumes "Well_order r"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1383
shows "Linear_order(bsqr r)"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1384
unfolding order_on_defs
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1385
using assms bsqr_Refl bsqr_Trans bsqr_antisym bsqr_Total by blast
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1386
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1387
lemma bsqr_Well_order:
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1388
assumes "Well_order r"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1389
shows "Well_order(bsqr r)"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1390
using assms
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1391
proof(simp add: bsqr_Linear_order Linear_order_Well_order_iff, intro allI impI)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1392
  have 0: "\<forall>A \<le> Field r. A \<noteq> {} \<longrightarrow> (\<exists>a \<in> A. \<forall>a' \<in> A. (a,a') \<in> r)"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1393
  using assms well_order_on_def Linear_order_Well_order_iff by blast
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1394
  fix D assume *: "D \<le> Field (bsqr r)" and **: "D \<noteq> {}"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1395
  hence 1: "D \<le> Field r \<times> Field r" unfolding Field_bsqr by simp
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1396
  (*  *)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1397
  obtain M where M_def: "M = {wo_rel.max2 r a1 a2| a1 a2. (a1,a2) \<in> D}" by blast
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1398
  have "M \<noteq> {}" using 1 M_def ** by auto
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1399
  moreover
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1400
  have "M \<le> Field r" unfolding M_def
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1401
  using 1 assms wo_rel_def[of r] wo_rel.max2_among[of r] by fastforce
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1402
  ultimately obtain m where m_min: "m \<in> M \<and> (\<forall>a \<in> M. (m,a) \<in> r)"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1403
  using 0 by blast
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1404
  (*  *)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1405
  obtain A1 where A1_def: "A1 = {a1. \<exists>a2. (a1,a2) \<in> D \<and> wo_rel.max2 r a1 a2 = m}" by blast
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1406
  have "A1 \<le> Field r" unfolding A1_def using 1 by auto
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1407
  moreover have "A1 \<noteq> {}" unfolding A1_def using m_min unfolding M_def by blast
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1408
  ultimately obtain a1 where a1_min: "a1 \<in> A1 \<and> (\<forall>a \<in> A1. (a1,a) \<in> r)"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1409
  using 0 by blast
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1410
  (*  *)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1411
  obtain A2 where A2_def: "A2 = {a2. (a1,a2) \<in> D \<and> wo_rel.max2 r a1 a2 = m}" by blast
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1412
  have "A2 \<le> Field r" unfolding A2_def using 1 by auto
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1413
  moreover have "A2 \<noteq> {}" unfolding A2_def
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1414
  using m_min a1_min unfolding A1_def M_def by blast
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1415
  ultimately obtain a2 where a2_min: "a2 \<in> A2 \<and> (\<forall>a \<in> A2. (a2,a) \<in> r)"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1416
  using 0 by blast
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1417
  (*   *)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1418
  have 2: "wo_rel.max2 r a1 a2 = m"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1419
  using a1_min a2_min unfolding A1_def A2_def by auto
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1420
  have 3: "(a1,a2) \<in> D" using a2_min unfolding A2_def by auto
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1421
  (*  *)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1422
  moreover
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1423
  {fix b1 b2 assume ***: "(b1,b2) \<in> D"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1424
   hence 4: "{a1,a2,b1,b2} \<le> Field r" using 1 3 by blast
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1425
   have 5: "(wo_rel.max2 r a1 a2, wo_rel.max2 r b1 b2) \<in> r"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1426
   using *** a1_min a2_min m_min unfolding A1_def A2_def M_def by auto
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1427
   have "((a1,a2),(b1,b2)) \<in> bsqr r"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1428
   proof(cases "wo_rel.max2 r a1 a2 = wo_rel.max2 r b1 b2")
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1429
     assume Case1: "wo_rel.max2 r a1 a2 \<noteq> wo_rel.max2 r b1 b2"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1430
     thus ?thesis unfolding bsqr_def using 4 5 by auto
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1431
   next
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1432
     assume Case2: "wo_rel.max2 r a1 a2 = wo_rel.max2 r b1 b2"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1433
     hence "b1 \<in> A1" unfolding A1_def using 2 *** by auto
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1434
     hence 6: "(a1,b1) \<in> r" using a1_min by auto
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1435
     show ?thesis
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1436
     proof(cases "a1 = b1")
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1437
       assume Case21: "a1 \<noteq> b1"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1438
       thus ?thesis unfolding bsqr_def using 4 Case2 6 by auto
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1439
     next
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1440
       assume Case22: "a1 = b1"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1441
       hence "b2 \<in> A2" unfolding A2_def using 2 *** Case2 by auto
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1442
       hence 7: "(a2,b2) \<in> r" using a2_min by auto
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1443
       thus ?thesis unfolding bsqr_def using 4 7 Case2 Case22 by auto
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1444
     qed
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1445
   qed
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1446
  }
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1447
  (*  *)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1448
  ultimately show "\<exists>d \<in> D. \<forall>d' \<in> D. (d,d') \<in> bsqr r" by fastforce
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1449
qed
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1450
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1451
lemma bsqr_max2:
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1452
assumes WELL: "Well_order r" and LEQ: "((a1,a2),(b1,b2)) \<in> bsqr r"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1453
shows "(wo_rel.max2 r a1 a2, wo_rel.max2 r b1 b2) \<in> r"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1454
proof-
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1455
  have "{(a1,a2),(b1,b2)} \<le> Field(bsqr r)"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1456
  using LEQ unfolding Field_def by auto
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1457
  hence "{a1,a2,b1,b2} \<le> Field r" unfolding Field_bsqr by auto
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1458
  hence "{wo_rel.max2 r a1 a2, wo_rel.max2 r b1 b2} \<le> Field r"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1459
  using WELL wo_rel_def[of r] wo_rel.max2_among[of r] by fastforce
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1460
  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"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1461
  using LEQ unfolding bsqr_def by auto
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1462
  ultimately show ?thesis using WELL unfolding order_on_defs refl_on_def by auto
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1463
qed
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1464
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1465
lemma bsqr_ofilter:
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1466
assumes WELL: "Well_order r" and
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1467
        OF: "wo_rel.ofilter (bsqr r) D" and SUB: "D < Field r \<times> Field r" and
55023
38db7814481d get rid of 'rel' locale, to facilitate inclusion of 'Order_Relation_More_FP' into 'Order_Relation'
blanchet
parents: 54980
diff changeset
  1468
        NE: "\<not> (\<exists>a. Field r = under r a)"
48975
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1469
shows "\<exists>A. wo_rel.ofilter r A \<and> A < Field r \<and> D \<le> A \<times> A"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1470
proof-
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1471
  let ?r' = "bsqr r"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1472
  have Well: "wo_rel r" using WELL wo_rel_def by blast
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1473
  hence Trans: "trans r" using wo_rel.TRANS by blast
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1474
  have Well': "Well_order ?r' \<and> wo_rel ?r'"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1475
  using WELL bsqr_Well_order wo_rel_def by blast
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1476
  (*  *)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1477
  have "D < Field ?r'" unfolding Field_bsqr using SUB .
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1478
  with OF obtain a1 and a2 where
55023
38db7814481d get rid of 'rel' locale, to facilitate inclusion of 'Order_Relation_More_FP' into 'Order_Relation'
blanchet
parents: 54980
diff changeset
  1479
  "(a1,a2) \<in> Field ?r'" and 1: "D = underS ?r' (a1,a2)"
48975
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1480
  using Well' wo_rel.ofilter_underS_Field[of ?r' D] by auto
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1481
  hence 2: "{a1,a2} \<le> Field r" unfolding Field_bsqr by auto
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1482
  let ?m = "wo_rel.max2 r a1 a2"
55023
38db7814481d get rid of 'rel' locale, to facilitate inclusion of 'Order_Relation_More_FP' into 'Order_Relation'
blanchet
parents: 54980
diff changeset
  1483
  have "D \<le> (under r ?m) \<times> (under r ?m)"
48975
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1484
  proof(unfold 1)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1485
    {fix b1 b2
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1486
     let ?n = "wo_rel.max2 r b1 b2"
55023
38db7814481d get rid of 'rel' locale, to facilitate inclusion of 'Order_Relation_More_FP' into 'Order_Relation'
blanchet
parents: 54980
diff changeset
  1487
     assume "(b1,b2) \<in> underS ?r' (a1,a2)"
48975
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1488
     hence 3: "((b1,b2),(a1,a2)) \<in> ?r'"
55023
38db7814481d get rid of 'rel' locale, to facilitate inclusion of 'Order_Relation_More_FP' into 'Order_Relation'
blanchet
parents: 54980
diff changeset
  1489
     unfolding underS_def by blast
48975
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1490
     hence "(?n,?m) \<in> r" using WELL by (simp add: bsqr_max2)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1491
     moreover
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1492
     {have "(b1,b2) \<in> Field ?r'" using 3 unfolding Field_def by auto
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1493
      hence "{b1,b2} \<le> Field r" unfolding Field_bsqr by auto
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1494
      hence "(b1,?n) \<in> r \<and> (b2,?n) \<in> r"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1495
      using Well by (simp add: wo_rel.max2_greater)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1496
     }
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1497
     ultimately have "(b1,?m) \<in> r \<and> (b2,?m) \<in> r"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1498
     using Trans trans_def[of r] by blast
55023
38db7814481d get rid of 'rel' locale, to facilitate inclusion of 'Order_Relation_More_FP' into 'Order_Relation'
blanchet
parents: 54980
diff changeset
  1499
     hence "(b1,b2) \<in> (under r ?m) \<times> (under r ?m)" unfolding under_def by simp}
38db7814481d get rid of 'rel' locale, to facilitate inclusion of 'Order_Relation_More_FP' into 'Order_Relation'
blanchet
parents: 54980
diff changeset
  1500
     thus "underS ?r' (a1,a2) \<le> (under r ?m) \<times> (under r ?m)" by auto
48975
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1501
  qed
55023
38db7814481d get rid of 'rel' locale, to facilitate inclusion of 'Order_Relation_More_FP' into 'Order_Relation'
blanchet
parents: 54980
diff changeset
  1502
  moreover have "wo_rel.ofilter r (under r ?m)"
48975
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1503
  using Well by (simp add: wo_rel.under_ofilter)
55023
38db7814481d get rid of 'rel' locale, to facilitate inclusion of 'Order_Relation_More_FP' into 'Order_Relation'
blanchet
parents: 54980
diff changeset
  1504
  moreover have "under r ?m < Field r"
38db7814481d get rid of 'rel' locale, to facilitate inclusion of 'Order_Relation_More_FP' into 'Order_Relation'
blanchet
parents: 54980
diff changeset
  1505
  using NE under_Field[of r ?m] by blast
48975
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1506
  ultimately show ?thesis by blast
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1507
qed
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1508
54980
7e0573a490ee basic ordinal arithmetic and cardinals library extension (not relevant for BNFs)
traytel
parents: 54578
diff changeset
  1509
definition Func where
7e0573a490ee basic ordinal arithmetic and cardinals library extension (not relevant for BNFs)
traytel
parents: 54578
diff changeset
  1510
"Func A B = {f . (\<forall> a \<in> A. f a \<in> B) \<and> (\<forall> a. a \<notin> A \<longrightarrow> f a = undefined)}"
7e0573a490ee basic ordinal arithmetic and cardinals library extension (not relevant for BNFs)
traytel
parents: 54578
diff changeset
  1511
7e0573a490ee basic ordinal arithmetic and cardinals library extension (not relevant for BNFs)
traytel
parents: 54578
diff changeset
  1512
lemma Func_empty:
7e0573a490ee basic ordinal arithmetic and cardinals library extension (not relevant for BNFs)
traytel
parents: 54578
diff changeset
  1513
"Func {} B = {\<lambda>x. undefined}"
7e0573a490ee basic ordinal arithmetic and cardinals library extension (not relevant for BNFs)
traytel
parents: 54578
diff changeset
  1514
unfolding Func_def by auto
7e0573a490ee basic ordinal arithmetic and cardinals library extension (not relevant for BNFs)
traytel
parents: 54578
diff changeset
  1515
7e0573a490ee basic ordinal arithmetic and cardinals library extension (not relevant for BNFs)
traytel
parents: 54578
diff changeset
  1516
lemma Func_elim:
7e0573a490ee basic ordinal arithmetic and cardinals library extension (not relevant for BNFs)
traytel
parents: 54578
diff changeset
  1517
assumes "g \<in> Func A B" and "a \<in> A"
7e0573a490ee basic ordinal arithmetic and cardinals library extension (not relevant for BNFs)
traytel
parents: 54578
diff changeset
  1518
shows "\<exists> b. b \<in> B \<and> g a = b"
7e0573a490ee basic ordinal arithmetic and cardinals library extension (not relevant for BNFs)
traytel
parents: 54578
diff changeset
  1519
using assms unfolding Func_def by (cases "g a = undefined") auto
7e0573a490ee basic ordinal arithmetic and cardinals library extension (not relevant for BNFs)
traytel
parents: 54578
diff changeset
  1520
7e0573a490ee basic ordinal arithmetic and cardinals library extension (not relevant for BNFs)
traytel
parents: 54578
diff changeset
  1521
definition curr where
7e0573a490ee basic ordinal arithmetic and cardinals library extension (not relevant for BNFs)
traytel
parents: 54578
diff changeset
  1522
"curr A f \<equiv> \<lambda> a. if a \<in> A then \<lambda>b. f (a,b) else undefined"
7e0573a490ee basic ordinal arithmetic and cardinals library extension (not relevant for BNFs)
traytel
parents: 54578
diff changeset
  1523
7e0573a490ee basic ordinal arithmetic and cardinals library extension (not relevant for BNFs)
traytel
parents: 54578
diff changeset
  1524
lemma curr_in:
61943
7fba644ed827 discontinued ASCII replacement syntax <*>;
wenzelm
parents: 61799
diff changeset
  1525
assumes f: "f \<in> Func (A \<times> B) C"
54980
7e0573a490ee basic ordinal arithmetic and cardinals library extension (not relevant for BNFs)
traytel
parents: 54578
diff changeset
  1526
shows "curr A f \<in> Func A (Func B C)"
7e0573a490ee basic ordinal arithmetic and cardinals library extension (not relevant for BNFs)
traytel
parents: 54578
diff changeset
  1527
using assms unfolding curr_def Func_def by auto
7e0573a490ee basic ordinal arithmetic and cardinals library extension (not relevant for BNFs)
traytel
parents: 54578
diff changeset
  1528
7e0573a490ee basic ordinal arithmetic and cardinals library extension (not relevant for BNFs)
traytel
parents: 54578
diff changeset
  1529
lemma curr_inj:
61943
7fba644ed827 discontinued ASCII replacement syntax <*>;
wenzelm
parents: 61799
diff changeset
  1530
assumes "f1 \<in> Func (A \<times> B) C" and "f2 \<in> Func (A \<times> B) C"
54980
7e0573a490ee basic ordinal arithmetic and cardinals library extension (not relevant for BNFs)
traytel
parents: 54578
diff changeset
  1531
shows "curr A f1 = curr A f2 \<longleftrightarrow> f1 = f2"
7e0573a490ee basic ordinal arithmetic and cardinals library extension (not relevant for BNFs)
traytel
parents: 54578
diff changeset
  1532
proof safe
7e0573a490ee basic ordinal arithmetic and cardinals library extension (not relevant for BNFs)
traytel
parents: 54578
diff changeset
  1533
  assume c: "curr A f1 = curr A f2"
7e0573a490ee basic ordinal arithmetic and cardinals library extension (not relevant for BNFs)
traytel
parents: 54578
diff changeset
  1534
  show "f1 = f2"
7e0573a490ee basic ordinal arithmetic and cardinals library extension (not relevant for BNFs)
traytel
parents: 54578
diff changeset
  1535
  proof (rule ext, clarify)
7e0573a490ee basic ordinal arithmetic and cardinals library extension (not relevant for BNFs)
traytel
parents: 54578
diff changeset
  1536
    fix a b show "f1 (a, b) = f2 (a, b)"
61943
7fba644ed827 discontinued ASCII replacement syntax <*>;
wenzelm
parents: 61799
diff changeset
  1537
    proof (cases "(a,b) \<in> A \<times> B")
54980
7e0573a490ee basic ordinal arithmetic and cardinals library extension (not relevant for BNFs)
traytel
parents: 54578
diff changeset
  1538
      case False
7e0573a490ee basic ordinal arithmetic and cardinals library extension (not relevant for BNFs)
traytel
parents: 54578
diff changeset
  1539
      thus ?thesis using assms unfolding Func_def by auto
7e0573a490ee basic ordinal arithmetic and cardinals library extension (not relevant for BNFs)
traytel
parents: 54578
diff changeset
  1540
    next
7e0573a490ee basic ordinal arithmetic and cardinals library extension (not relevant for BNFs)
traytel
parents: 54578
diff changeset
  1541
      case True hence a: "a \<in> A" and b: "b \<in> B" by auto
7e0573a490ee basic ordinal arithmetic and cardinals library extension (not relevant for BNFs)
traytel
parents: 54578
diff changeset
  1542
      thus ?thesis
7e0573a490ee basic ordinal arithmetic and cardinals library extension (not relevant for BNFs)
traytel
parents: 54578
diff changeset
  1543
      using c unfolding curr_def fun_eq_iff by(elim allE[of _ a]) simp
7e0573a490ee basic ordinal arithmetic and cardinals library extension (not relevant for BNFs)
traytel
parents: 54578
diff changeset
  1544
    qed
7e0573a490ee basic ordinal arithmetic and cardinals library extension (not relevant for BNFs)
traytel
parents: 54578
diff changeset
  1545
  qed
7e0573a490ee basic ordinal arithmetic and cardinals library extension (not relevant for BNFs)
traytel
parents: 54578
diff changeset
  1546
qed
7e0573a490ee basic ordinal arithmetic and cardinals library extension (not relevant for BNFs)
traytel
parents: 54578
diff changeset
  1547
7e0573a490ee basic ordinal arithmetic and cardinals library extension (not relevant for BNFs)
traytel
parents: 54578
diff changeset
  1548
lemma curr_surj:
7e0573a490ee basic ordinal arithmetic and cardinals library extension (not relevant for BNFs)
traytel
parents: 54578
diff changeset
  1549
assumes "g \<in> Func A (Func B C)"
61943
7fba644ed827 discontinued ASCII replacement syntax <*>;
wenzelm
parents: 61799
diff changeset
  1550
shows "\<exists> f \<in> Func (A \<times> B) C. curr A f = g"
54980
7e0573a490ee basic ordinal arithmetic and cardinals library extension (not relevant for BNFs)
traytel
parents: 54578
diff changeset
  1551
proof
7e0573a490ee basic ordinal arithmetic and cardinals library extension (not relevant for BNFs)
traytel
parents: 54578
diff changeset
  1552
  let ?f = "\<lambda> ab. if fst ab \<in> A \<and> snd ab \<in> B then g (fst ab) (snd ab) else undefined"
7e0573a490ee basic ordinal arithmetic and cardinals library extension (not relevant for BNFs)
traytel
parents: 54578
diff changeset
  1553
  show "curr A ?f = g"
7e0573a490ee basic ordinal arithmetic and cardinals library extension (not relevant for BNFs)
traytel
parents: 54578
diff changeset
  1554
  proof (rule ext)
7e0573a490ee basic ordinal arithmetic and cardinals library extension (not relevant for BNFs)
traytel
parents: 54578
diff changeset
  1555
    fix a show "curr A ?f a = g a"
7e0573a490ee basic ordinal arithmetic and cardinals library extension (not relevant for BNFs)
traytel
parents: 54578
diff changeset
  1556
    proof (cases "a \<in> A")
7e0573a490ee basic ordinal arithmetic and cardinals library extension (not relevant for BNFs)
traytel
parents: 54578
diff changeset
  1557
      case False
7e0573a490ee basic ordinal arithmetic and cardinals library extension (not relevant for BNFs)
traytel
parents: 54578
diff changeset
  1558
      hence "g a = undefined" using assms unfolding Func_def by auto
7e0573a490ee basic ordinal arithmetic and cardinals library extension (not relevant for BNFs)
traytel
parents: 54578
diff changeset
  1559
      thus ?thesis unfolding curr_def using False by simp
7e0573a490ee basic ordinal arithmetic and cardinals library extension (not relevant for BNFs)
traytel
parents: 54578
diff changeset
  1560
    next
7e0573a490ee basic ordinal arithmetic and cardinals library extension (not relevant for BNFs)
traytel
parents: 54578
diff changeset
  1561
      case True
7e0573a490ee basic ordinal arithmetic and cardinals library extension (not relevant for BNFs)
traytel
parents: 54578
diff changeset
  1562
      obtain g1 where "g1 \<in> Func B C" and "g a = g1"
7e0573a490ee basic ordinal arithmetic and cardinals library extension (not relevant for BNFs)
traytel
parents: 54578
diff changeset
  1563
      using assms using Func_elim[OF assms True] by blast
7e0573a490ee basic ordinal arithmetic and cardinals library extension (not relevant for BNFs)
traytel
parents: 54578
diff changeset
  1564
      thus ?thesis using True unfolding Func_def curr_def by auto
7e0573a490ee basic ordinal arithmetic and cardinals library extension (not relevant for BNFs)
traytel
parents: 54578
diff changeset
  1565
    qed
7e0573a490ee basic ordinal arithmetic and cardinals library extension (not relevant for BNFs)
traytel
parents: 54578
diff changeset
  1566
  qed
61943
7fba644ed827 discontinued ASCII replacement syntax <*>;
wenzelm
parents: 61799
diff changeset
  1567
  show "?f \<in> Func (A \<times> B) C" using assms unfolding Func_def mem_Collect_eq by auto
54980
7e0573a490ee basic ordinal arithmetic and cardinals library extension (not relevant for BNFs)
traytel
parents: 54578
diff changeset
  1568
qed
7e0573a490ee basic ordinal arithmetic and cardinals library extension (not relevant for BNFs)
traytel
parents: 54578
diff changeset
  1569
7e0573a490ee basic ordinal arithmetic and cardinals library extension (not relevant for BNFs)
traytel
parents: 54578
diff changeset
  1570
lemma bij_betw_curr:
61943
7fba644ed827 discontinued ASCII replacement syntax <*>;
wenzelm
parents: 61799
diff changeset
  1571
"bij_betw (curr A) (Func (A \<times> B) C) (Func A (Func B C))"
54980
7e0573a490ee basic ordinal arithmetic and cardinals library extension (not relevant for BNFs)
traytel
parents: 54578
diff changeset
  1572
unfolding bij_betw_def inj_on_def image_def
7e0573a490ee basic ordinal arithmetic and cardinals library extension (not relevant for BNFs)
traytel
parents: 54578
diff changeset
  1573
apply (intro impI conjI ballI)
7e0573a490ee basic ordinal arithmetic and cardinals library extension (not relevant for BNFs)
traytel
parents: 54578
diff changeset
  1574
apply (erule curr_inj[THEN iffD1], assumption+)
7e0573a490ee basic ordinal arithmetic and cardinals library extension (not relevant for BNFs)
traytel
parents: 54578
diff changeset
  1575
apply auto
7e0573a490ee basic ordinal arithmetic and cardinals library extension (not relevant for BNFs)
traytel
parents: 54578
diff changeset
  1576
apply (erule curr_in)
7e0573a490ee basic ordinal arithmetic and cardinals library extension (not relevant for BNFs)
traytel
parents: 54578
diff changeset
  1577
using curr_surj by blast
7e0573a490ee basic ordinal arithmetic and cardinals library extension (not relevant for BNFs)
traytel
parents: 54578
diff changeset
  1578
7e0573a490ee basic ordinal arithmetic and cardinals library extension (not relevant for BNFs)
traytel
parents: 54578
diff changeset
  1579
definition Func_map where
7e0573a490ee basic ordinal arithmetic and cardinals library extension (not relevant for BNFs)
traytel
parents: 54578
diff changeset
  1580
"Func_map B2 f1 f2 g b2 \<equiv> if b2 \<in> B2 then f1 (g (f2 b2)) else undefined"
7e0573a490ee basic ordinal arithmetic and cardinals library extension (not relevant for BNFs)
traytel
parents: 54578
diff changeset
  1581
7e0573a490ee basic ordinal arithmetic and cardinals library extension (not relevant for BNFs)
traytel
parents: 54578
diff changeset
  1582
lemma Func_map:
7e0573a490ee basic ordinal arithmetic and cardinals library extension (not relevant for BNFs)
traytel
parents: 54578
diff changeset
  1583
assumes g: "g \<in> Func A2 A1" and f1: "f1 ` A1 \<subseteq> B1" and f2: "f2 ` B2 \<subseteq> A2"
7e0573a490ee basic ordinal arithmetic and cardinals library extension (not relevant for BNFs)
traytel
parents: 54578
diff changeset
  1584
shows "Func_map B2 f1 f2 g \<in> Func B2 B1"
7e0573a490ee basic ordinal arithmetic and cardinals library extension (not relevant for BNFs)
traytel
parents: 54578
diff changeset
  1585
using assms unfolding Func_def Func_map_def mem_Collect_eq by auto
7e0573a490ee basic ordinal arithmetic and cardinals library extension (not relevant for BNFs)
traytel
parents: 54578
diff changeset
  1586
7e0573a490ee basic ordinal arithmetic and cardinals library extension (not relevant for BNFs)
traytel
parents: 54578
diff changeset
  1587
lemma Func_non_emp:
7e0573a490ee basic ordinal arithmetic and cardinals library extension (not relevant for BNFs)
traytel
parents: 54578
diff changeset
  1588
assumes "B \<noteq> {}"
7e0573a490ee basic ordinal arithmetic and cardinals library extension (not relevant for BNFs)
traytel
parents: 54578
diff changeset
  1589
shows "Func A B \<noteq> {}"
7e0573a490ee basic ordinal arithmetic and cardinals library extension (not relevant for BNFs)
traytel
parents: 54578
diff changeset
  1590
proof-
7e0573a490ee basic ordinal arithmetic and cardinals library extension (not relevant for BNFs)
traytel
parents: 54578
diff changeset
  1591
  obtain b where b: "b \<in> B" using assms by auto
7e0573a490ee basic ordinal arithmetic and cardinals library extension (not relevant for BNFs)
traytel
parents: 54578
diff changeset
  1592
  hence "(\<lambda> a. if a \<in> A then b else undefined) \<in> Func A B" unfolding Func_def by auto
7e0573a490ee basic ordinal arithmetic and cardinals library extension (not relevant for BNFs)
traytel
parents: 54578
diff changeset
  1593
  thus ?thesis by blast
7e0573a490ee basic ordinal arithmetic and cardinals library extension (not relevant for BNFs)
traytel
parents: 54578
diff changeset
  1594
qed
7e0573a490ee basic ordinal arithmetic and cardinals library extension (not relevant for BNFs)
traytel
parents: 54578
diff changeset
  1595
7e0573a490ee basic ordinal arithmetic and cardinals library extension (not relevant for BNFs)
traytel
parents: 54578
diff changeset
  1596
lemma Func_is_emp:
7e0573a490ee basic ordinal arithmetic and cardinals library extension (not relevant for BNFs)
traytel
parents: 54578
diff changeset
  1597
"Func A B = {} \<longleftrightarrow> A \<noteq> {} \<and> B = {}" (is "?L \<longleftrightarrow> ?R")
7e0573a490ee basic ordinal arithmetic and cardinals library extension (not relevant for BNFs)
traytel
parents: 54578
diff changeset
  1598
proof
7e0573a490ee basic ordinal arithmetic and cardinals library extension (not relevant for BNFs)
traytel
parents: 54578
diff changeset
  1599
  assume L: ?L
7e0573a490ee basic ordinal arithmetic and cardinals library extension (not relevant for BNFs)
traytel
parents: 54578
diff changeset
  1600
  moreover {assume "A = {}" hence False using L Func_empty by auto}
55811
aa1acc25126b load Metis a little later
traytel
parents: 55603
diff changeset
  1601
  moreover {assume "B \<noteq> {}" hence False using L Func_non_emp[of B A] by simp }
54980
7e0573a490ee basic ordinal arithmetic and cardinals library extension (not relevant for BNFs)
traytel
parents: 54578
diff changeset
  1602
  ultimately show ?R by blast
7e0573a490ee basic ordinal arithmetic and cardinals library extension (not relevant for BNFs)
traytel
parents: 54578
diff changeset
  1603
next
7e0573a490ee basic ordinal arithmetic and cardinals library extension (not relevant for BNFs)
traytel
parents: 54578
diff changeset
  1604
  assume R: ?R
7e0573a490ee basic ordinal arithmetic and cardinals library extension (not relevant for BNFs)
traytel
parents: 54578
diff changeset
  1605
  moreover
7e0573a490ee basic ordinal arithmetic and cardinals library extension (not relevant for BNFs)
traytel
parents: 54578
diff changeset
  1606
  {fix f assume "f \<in> Func A B"
7e0573a490ee basic ordinal arithmetic and cardinals library extension (not relevant for BNFs)
traytel
parents: 54578
diff changeset
  1607
   moreover obtain a where "a \<in> A" using R by blast
7e0573a490ee basic ordinal arithmetic and cardinals library extension (not relevant for BNFs)
traytel
parents: 54578
diff changeset
  1608
   ultimately obtain b where "b \<in> B" unfolding Func_def by blast
7e0573a490ee basic ordinal arithmetic and cardinals library extension (not relevant for BNFs)
traytel
parents: 54578
diff changeset
  1609
   with R have False by blast
7e0573a490ee basic ordinal arithmetic and cardinals library extension (not relevant for BNFs)
traytel
parents: 54578
diff changeset
  1610
  }
7e0573a490ee basic ordinal arithmetic and cardinals library extension (not relevant for BNFs)
traytel
parents: 54578
diff changeset
  1611
  thus ?L by blast
7e0573a490ee basic ordinal arithmetic and cardinals library extension (not relevant for BNFs)
traytel
parents: 54578
diff changeset
  1612
qed
7e0573a490ee basic ordinal arithmetic and cardinals library extension (not relevant for BNFs)
traytel
parents: 54578
diff changeset
  1613
7e0573a490ee basic ordinal arithmetic and cardinals library extension (not relevant for BNFs)
traytel
parents: 54578
diff changeset
  1614
lemma Func_map_surj:
7e0573a490ee basic ordinal arithmetic and cardinals library extension (not relevant for BNFs)
traytel
parents: 54578
diff changeset
  1615
assumes B1: "f1 ` A1 = B1" and A2: "inj_on f2 B2" "f2 ` B2 \<subseteq> A2"
7e0573a490ee basic ordinal arithmetic and cardinals library extension (not relevant for BNFs)
traytel
parents: 54578
diff changeset
  1616
and B2A2: "B2 = {} \<Longrightarrow> A2 = {}"
7e0573a490ee basic ordinal arithmetic and cardinals library extension (not relevant for BNFs)
traytel
parents: 54578
diff changeset
  1617
shows "Func B2 B1 = Func_map B2 f1 f2 ` Func A2 A1"
7e0573a490ee basic ordinal arithmetic and cardinals library extension (not relevant for BNFs)
traytel
parents: 54578
diff changeset
  1618
proof(cases "B2 = {}")
7e0573a490ee basic ordinal arithmetic and cardinals library extension (not relevant for BNFs)
traytel
parents: 54578
diff changeset
  1619
  case True
7e0573a490ee basic ordinal arithmetic and cardinals library extension (not relevant for BNFs)
traytel
parents: 54578
diff changeset
  1620
  thus ?thesis using B2A2 by (auto simp: Func_empty Func_map_def)
7e0573a490ee basic ordinal arithmetic and cardinals library extension (not relevant for BNFs)
traytel
parents: 54578
diff changeset
  1621
next
7e0573a490ee basic ordinal arithmetic and cardinals library extension (not relevant for BNFs)
traytel
parents: 54578
diff changeset
  1622
  case False note B2 = False
7e0573a490ee basic ordinal arithmetic and cardinals library extension (not relevant for BNFs)
traytel
parents: 54578
diff changeset
  1623
  show ?thesis
7e0573a490ee basic ordinal arithmetic and cardinals library extension (not relevant for BNFs)
traytel
parents: 54578
diff changeset
  1624
  proof safe
7e0573a490ee basic ordinal arithmetic and cardinals library extension (not relevant for BNFs)
traytel
parents: 54578
diff changeset
  1625
    fix h assume h: "h \<in> Func B2 B1"
63040
eb4ddd18d635 eliminated old 'def';
wenzelm
parents: 61943
diff changeset
  1626
    define j1 where "j1 = inv_into A1 f1"
54980
7e0573a490ee basic ordinal arithmetic and cardinals library extension (not relevant for BNFs)
traytel
parents: 54578
diff changeset
  1627
    have "\<forall> a2 \<in> f2 ` B2. \<exists> b2. b2 \<in> B2 \<and> f2 b2 = a2" by blast
55811
aa1acc25126b load Metis a little later
traytel
parents: 55603
diff changeset
  1628
    then obtain k where k: "\<forall> a2 \<in> f2 ` B2. k a2 \<in> B2 \<and> f2 (k a2) = a2"
aa1acc25126b load Metis a little later
traytel
parents: 55603
diff changeset
  1629
      by atomize_elim (rule bchoice)
54980
7e0573a490ee basic ordinal arithmetic and cardinals library extension (not relevant for BNFs)
traytel
parents: 54578
diff changeset
  1630
    {fix b2 assume b2: "b2 \<in> B2"
7e0573a490ee basic ordinal arithmetic and cardinals library extension (not relevant for BNFs)
traytel
parents: 54578
diff changeset
  1631
     hence "f2 (k (f2 b2)) = f2 b2" using k A2(2) by auto
7e0573a490ee basic ordinal arithmetic and cardinals library extension (not relevant for BNFs)
traytel
parents: 54578
diff changeset
  1632
     moreover have "k (f2 b2) \<in> B2" using b2 A2(2) k by auto
7e0573a490ee basic ordinal arithmetic and cardinals library extension (not relevant for BNFs)
traytel
parents: 54578
diff changeset
  1633
     ultimately have "k (f2 b2) = b2" using b2 A2(1) unfolding inj_on_def by blast
7e0573a490ee basic ordinal arithmetic and cardinals library extension (not relevant for BNFs)
traytel
parents: 54578
diff changeset
  1634
    } note kk = this
7e0573a490ee basic ordinal arithmetic and cardinals library extension (not relevant for BNFs)
traytel
parents: 54578
diff changeset
  1635
    obtain b22 where b22: "b22 \<in> B2" using B2 by auto
63040
eb4ddd18d635 eliminated old 'def';
wenzelm
parents: 61943
diff changeset
  1636
    define j2 where [abs_def]: "j2 a2 = (if a2 \<in> f2 ` B2 then k a2 else b22)" for a2
54980
7e0573a490ee basic ordinal arithmetic and cardinals library extension (not relevant for BNFs)
traytel
parents: 54578
diff changeset
  1637
    have j2A2: "j2 ` A2 \<subseteq> B2" unfolding j2_def using k b22 by auto
7e0573a490ee basic ordinal arithmetic and cardinals library extension (not relevant for BNFs)
traytel
parents: 54578
diff changeset
  1638
    have j2: "\<And> b2. b2 \<in> B2 \<Longrightarrow> j2 (f2 b2) = b2"
7e0573a490ee basic ordinal arithmetic and cardinals library extension (not relevant for BNFs)
traytel
parents: 54578
diff changeset
  1639
    using kk unfolding j2_def by auto
63040
eb4ddd18d635 eliminated old 'def';
wenzelm
parents: 61943
diff changeset
  1640
    define g where "g = Func_map A2 j1 j2 h"
54980
7e0573a490ee basic ordinal arithmetic and cardinals library extension (not relevant for BNFs)
traytel
parents: 54578
diff changeset
  1641
    have "Func_map B2 f1 f2 g = h"
7e0573a490ee basic ordinal arithmetic and cardinals library extension (not relevant for BNFs)
traytel
parents: 54578
diff changeset
  1642
    proof (rule ext)
7e0573a490ee basic ordinal arithmetic and cardinals library extension (not relevant for BNFs)
traytel
parents: 54578
diff changeset
  1643
      fix b2 show "Func_map B2 f1 f2 g b2 = h b2"
7e0573a490ee basic ordinal arithmetic and cardinals library extension (not relevant for BNFs)
traytel
parents: 54578
diff changeset
  1644
      proof(cases "b2 \<in> B2")
7e0573a490ee basic ordinal arithmetic and cardinals library extension (not relevant for BNFs)
traytel
parents: 54578
diff changeset
  1645
        case True
7e0573a490ee basic ordinal arithmetic and cardinals library extension (not relevant for BNFs)
traytel
parents: 54578
diff changeset
  1646
        show ?thesis
7e0573a490ee basic ordinal arithmetic and cardinals library extension (not relevant for BNFs)
traytel
parents: 54578
diff changeset
  1647
        proof (cases "h b2 = undefined")
7e0573a490ee basic ordinal arithmetic and cardinals library extension (not relevant for BNFs)
traytel
parents: 54578
diff changeset
  1648
          case True
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 58889
diff changeset
  1649
          hence b1: "h b2 \<in> f1 ` A1" using h \<open>b2 \<in> B2\<close> unfolding B1 Func_def by auto
54980
7e0573a490ee basic ordinal arithmetic and cardinals library extension (not relevant for BNFs)
traytel
parents: 54578
diff changeset
  1650
          show ?thesis using A2 f_inv_into_f[OF b1]
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 58889
diff changeset
  1651
            unfolding True g_def Func_map_def j1_def j2[OF \<open>b2 \<in> B2\<close>] by auto
54980
7e0573a490ee basic ordinal arithmetic and cardinals library extension (not relevant for BNFs)
traytel
parents: 54578
diff changeset
  1652
        qed(insert A2 True j2[OF True] h B1, unfold j1_def g_def Func_def Func_map_def,
7e0573a490ee basic ordinal arithmetic and cardinals library extension (not relevant for BNFs)
traytel
parents: 54578
diff changeset
  1653
          auto intro: f_inv_into_f)
7e0573a490ee basic ordinal arithmetic and cardinals library extension (not relevant for BNFs)
traytel
parents: 54578
diff changeset
  1654
      qed(insert h, unfold Func_def Func_map_def, auto)
7e0573a490ee basic ordinal arithmetic and cardinals library extension (not relevant for BNFs)
traytel
parents: 54578
diff changeset
  1655
    qed
7e0573a490ee basic ordinal arithmetic and cardinals library extension (not relevant for BNFs)
traytel
parents: 54578
diff changeset
  1656
    moreover have "g \<in> Func A2 A1" unfolding g_def apply(rule Func_map[OF h])
56077
d397030fb27e tuned proofs
haftmann
parents: 55811
diff changeset
  1657
    using j2A2 B1 A2 unfolding j1_def by (fast intro: inv_into_into)+
54980
7e0573a490ee basic ordinal arithmetic and cardinals library extension (not relevant for BNFs)
traytel
parents: 54578
diff changeset
  1658
    ultimately show "h \<in> Func_map B2 f1 f2 ` Func A2 A1"
56077
d397030fb27e tuned proofs
haftmann
parents: 55811
diff changeset
  1659
    unfolding Func_map_def[abs_def] by auto
54980
7e0573a490ee basic ordinal arithmetic and cardinals library extension (not relevant for BNFs)
traytel
parents: 54578
diff changeset
  1660
  qed(insert B1 Func_map[OF _ _ A2(2)], auto)
7e0573a490ee basic ordinal arithmetic and cardinals library extension (not relevant for BNFs)
traytel
parents: 54578
diff changeset
  1661
qed
48975
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1662
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1663
end