src/HOL/Library/Bourbaki_Witt_Fixpoint.thy
author Andreas Lochbihler
Tue, 12 Jan 2016 15:23:54 +0100
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(*  Title:      HOL/Library/Bourbaki_Witt_Fixpoint.thy
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    Author:     Andreas Lochbihler, ETH Zurich
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  Follows G. Smolka, S. Schäfer and C. Doczkal: Transfinite Constructions in
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  Classical Type Theory. ITP 2015
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*)
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section \<open>The Bourbaki-Witt tower construction for transfinite iteration\<close>
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theory Bourbaki_Witt_Fixpoint imports Main begin
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lemma ChainsI [intro?]:
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  "(\<And>a b. \<lbrakk> a \<in> Y; b \<in> Y \<rbrakk> \<Longrightarrow> (a, b) \<in> r \<or> (b, a) \<in> r) \<Longrightarrow> Y \<in> Chains r"
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unfolding Chains_def by blast
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lemma in_Chains_subset: "\<lbrakk> M \<in> Chains r; M' \<subseteq> M \<rbrakk> \<Longrightarrow> M' \<in> Chains r"
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by(auto simp add: Chains_def)
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lemma FieldI1: "(i, j) \<in> R \<Longrightarrow> i \<in> Field R"
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  unfolding Field_def by auto
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lemma Chains_FieldD: "\<lbrakk> M \<in> Chains r; x \<in> M \<rbrakk> \<Longrightarrow> x \<in> Field r"
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by(auto simp add: Chains_def intro: FieldI1 FieldI2)
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lemma partial_order_on_trans:
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  "\<lbrakk> partial_order_on A r; (x, y) \<in> r; (y, z) \<in> r \<rbrakk> \<Longrightarrow> (x, z) \<in> r"
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by(auto simp add: order_on_defs dest: transD)
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locale bourbaki_witt_fixpoint =
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  fixes lub :: "'a set \<Rightarrow> 'a"
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  and leq :: "('a \<times> 'a) set"
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  and f :: "'a \<Rightarrow> 'a"
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  assumes po: "Partial_order leq"
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  and lub_least: "\<lbrakk> M \<in> Chains leq; M \<noteq> {}; \<And>x. x \<in> M \<Longrightarrow> (x, z) \<in> leq \<rbrakk> \<Longrightarrow> (lub M, z) \<in> leq"
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  and lub_upper: "\<lbrakk> M \<in> Chains leq; x \<in> M \<rbrakk> \<Longrightarrow> (x, lub M) \<in> leq"
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  and lub_in_Field: "\<lbrakk> M \<in> Chains leq; M \<noteq> {} \<rbrakk> \<Longrightarrow> lub M \<in> Field leq"
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  and increasing: "\<And>x. x \<in> Field leq \<Longrightarrow> (x, f x) \<in> leq"
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begin
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lemma leq_trans: "\<lbrakk> (x, y) \<in> leq; (y, z) \<in> leq \<rbrakk> \<Longrightarrow> (x, z) \<in> leq"
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by(rule partial_order_on_trans[OF po])
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lemma leq_refl: "x \<in> Field leq \<Longrightarrow> (x, x) \<in> leq"
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using po by(simp add: order_on_defs refl_on_def)
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lemma leq_antisym: "\<lbrakk> (x, y) \<in> leq; (y, x) \<in> leq \<rbrakk> \<Longrightarrow> x = y"
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using po by(simp add: order_on_defs antisym_def)
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inductive_set iterates_above :: "'a \<Rightarrow> 'a set"
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  for a
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where
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  base: "a \<in> iterates_above a"
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| step: "x \<in> iterates_above a \<Longrightarrow> f x \<in> iterates_above a"
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| Sup: "\<lbrakk> M \<in> Chains leq; M \<noteq> {}; \<And>x. x \<in> M \<Longrightarrow> x \<in> iterates_above a \<rbrakk> \<Longrightarrow> lub M \<in> iterates_above a"
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definition fixp_above :: "'a \<Rightarrow> 'a"
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where "fixp_above a = lub (iterates_above a)"
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context 
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  notes leq_refl [intro!, simp]
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  and base [intro]
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  and step [intro]
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  and Sup [intro]
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  and leq_trans [trans]
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begin
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lemma iterates_above_le_f: "\<lbrakk> x \<in> iterates_above a; a \<in> Field leq \<rbrakk> \<Longrightarrow> (x, f x) \<in> leq"
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by(induction x rule: iterates_above.induct)(blast intro: increasing FieldI2 lub_in_Field)+
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lemma iterates_above_Field: "\<lbrakk> x \<in> iterates_above a; a \<in> Field leq \<rbrakk> \<Longrightarrow> x \<in> Field leq"
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by(drule (1) iterates_above_le_f)(rule FieldI1)
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lemma iterates_above_ge:
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  assumes y: "y \<in> iterates_above a"
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  and a: "a \<in> Field leq"
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  shows "(a, y) \<in> leq"
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using y by(induction)(auto intro: a increasing iterates_above_le_f leq_trans leq_trans[OF _ lub_upper])
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lemma iterates_above_lub:
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  assumes M: "M \<in> Chains leq"
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  and nempty: "M \<noteq> {}"
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  and upper: "\<And>y. y \<in> M \<Longrightarrow> \<exists>z \<in> M. (y, z) \<in> leq \<and> z \<in> iterates_above a"
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  shows "lub M \<in> iterates_above a"
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proof -
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  let ?M = "M \<inter> iterates_above a"
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  from M have M': "?M \<in> Chains leq" by(rule in_Chains_subset)simp
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  have "?M \<noteq> {}" using nempty by(auto dest: upper)
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  with M' have "lub ?M \<in> iterates_above a" by(rule Sup) blast
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  also have "lub ?M = lub M" using nempty
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    by(intro leq_antisym)(blast intro!: lub_least[OF M] lub_least[OF M'] intro: lub_upper[OF M'] lub_upper[OF M] leq_trans dest: upper)+
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  finally show ?thesis .
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qed
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lemma iterates_above_successor:
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  assumes y: "y \<in> iterates_above a"
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  and a: "a \<in> Field leq"
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  shows "y = a \<or> y \<in> iterates_above (f a)"
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using y
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proof induction
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  case base thus ?case by simp
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next
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  case (step x) thus ?case by auto
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next
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  case (Sup M)
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  show ?case
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  proof(cases "\<exists>x. M \<subseteq> {x}")
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    case True
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    with \<open>M \<noteq> {}\<close> obtain y where M: "M = {y}" by auto
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    have "lub M = y"
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      by(rule leq_antisym)(auto intro!: lub_upper Sup lub_least ChainsI simp add: a M Sup.hyps(3)[of y, THEN iterates_above_Field] dest: iterates_above_Field)
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    with Sup.IH[of y] M show ?thesis by simp
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  next
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    case False
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    from Sup(1-2) have "lub M \<in> iterates_above (f a)"
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    proof(rule iterates_above_lub)
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      fix y
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      assume y: "y \<in> M"
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      from Sup.IH[OF this] show "\<exists>z\<in>M. (y, z) \<in> leq \<and> z \<in> iterates_above (f a)"
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      proof
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        assume "y = a"
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        from y False obtain z where z: "z \<in> M" and neq: "y \<noteq> z" by (metis insertI1 subsetI)
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        with Sup.IH[OF z] \<open>y = a\<close> Sup.hyps(3)[OF z]
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        show ?thesis by(auto dest: iterates_above_ge intro: a)
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      next
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        assume "y \<in> iterates_above (f a)"
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        moreover with increasing[OF a] have "y \<in> Field leq"
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          by(auto dest!: iterates_above_Field intro: FieldI2)
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        ultimately show ?thesis using y by(auto)
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      qed
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    qed
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    thus ?thesis by simp
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diff changeset
   132
  qed
507b39df1a57 add formalisation of Bourbaki-Witt fixpoint theorem
Andreas Lochbihler
parents:
diff changeset
   133
qed
507b39df1a57 add formalisation of Bourbaki-Witt fixpoint theorem
Andreas Lochbihler
parents:
diff changeset
   134
507b39df1a57 add formalisation of Bourbaki-Witt fixpoint theorem
Andreas Lochbihler
parents:
diff changeset
   135
lemma iterates_above_Sup_aux:
507b39df1a57 add formalisation of Bourbaki-Witt fixpoint theorem
Andreas Lochbihler
parents:
diff changeset
   136
  assumes M: "M \<in> Chains leq" "M \<noteq> {}"
507b39df1a57 add formalisation of Bourbaki-Witt fixpoint theorem
Andreas Lochbihler
parents:
diff changeset
   137
  and M': "M' \<in> Chains leq" "M' \<noteq> {}"
507b39df1a57 add formalisation of Bourbaki-Witt fixpoint theorem
Andreas Lochbihler
parents:
diff changeset
   138
  and comp: "\<And>x. x \<in> M \<Longrightarrow> x \<in> iterates_above (lub M') \<or> lub M' \<in> iterates_above x"
507b39df1a57 add formalisation of Bourbaki-Witt fixpoint theorem
Andreas Lochbihler
parents:
diff changeset
   139
  shows "(lub M, lub M') \<in> leq \<or> lub M \<in> iterates_above (lub M')"
507b39df1a57 add formalisation of Bourbaki-Witt fixpoint theorem
Andreas Lochbihler
parents:
diff changeset
   140
proof(cases "\<exists>x \<in> M. x \<in> iterates_above (lub M')")
507b39df1a57 add formalisation of Bourbaki-Witt fixpoint theorem
Andreas Lochbihler
parents:
diff changeset
   141
  case True
507b39df1a57 add formalisation of Bourbaki-Witt fixpoint theorem
Andreas Lochbihler
parents:
diff changeset
   142
  then obtain x where x: "x \<in> M" "x \<in> iterates_above (lub M')" by blast
507b39df1a57 add formalisation of Bourbaki-Witt fixpoint theorem
Andreas Lochbihler
parents:
diff changeset
   143
  have lub_M': "lub M' \<in> Field leq" using M' by(rule lub_in_Field)
507b39df1a57 add formalisation of Bourbaki-Witt fixpoint theorem
Andreas Lochbihler
parents:
diff changeset
   144
  have "lub M \<in> iterates_above (lub M')" using M
507b39df1a57 add formalisation of Bourbaki-Witt fixpoint theorem
Andreas Lochbihler
parents:
diff changeset
   145
  proof(rule iterates_above_lub)
507b39df1a57 add formalisation of Bourbaki-Witt fixpoint theorem
Andreas Lochbihler
parents:
diff changeset
   146
    fix y
507b39df1a57 add formalisation of Bourbaki-Witt fixpoint theorem
Andreas Lochbihler
parents:
diff changeset
   147
    assume y: "y \<in> M"
507b39df1a57 add formalisation of Bourbaki-Witt fixpoint theorem
Andreas Lochbihler
parents:
diff changeset
   148
    from comp[OF y] show "\<exists>z\<in>M. (y, z) \<in> leq \<and> z \<in> iterates_above (lub M')"
507b39df1a57 add formalisation of Bourbaki-Witt fixpoint theorem
Andreas Lochbihler
parents:
diff changeset
   149
    proof
507b39df1a57 add formalisation of Bourbaki-Witt fixpoint theorem
Andreas Lochbihler
parents:
diff changeset
   150
      assume "y \<in> iterates_above (lub M')"
507b39df1a57 add formalisation of Bourbaki-Witt fixpoint theorem
Andreas Lochbihler
parents:
diff changeset
   151
      from this iterates_above_Field[OF this] y lub_M' show ?thesis by blast
507b39df1a57 add formalisation of Bourbaki-Witt fixpoint theorem
Andreas Lochbihler
parents:
diff changeset
   152
    next
507b39df1a57 add formalisation of Bourbaki-Witt fixpoint theorem
Andreas Lochbihler
parents:
diff changeset
   153
      assume "lub M' \<in> iterates_above y"
507b39df1a57 add formalisation of Bourbaki-Witt fixpoint theorem
Andreas Lochbihler
parents:
diff changeset
   154
      hence "(y, lub M') \<in> leq" using Chains_FieldD[OF M(1) y] by(rule iterates_above_ge)
507b39df1a57 add formalisation of Bourbaki-Witt fixpoint theorem
Andreas Lochbihler
parents:
diff changeset
   155
      also have "(lub M', x) \<in> leq" using x(2) lub_M' by(rule iterates_above_ge)
507b39df1a57 add formalisation of Bourbaki-Witt fixpoint theorem
Andreas Lochbihler
parents:
diff changeset
   156
      finally show ?thesis using x by blast
507b39df1a57 add formalisation of Bourbaki-Witt fixpoint theorem
Andreas Lochbihler
parents:
diff changeset
   157
    qed
507b39df1a57 add formalisation of Bourbaki-Witt fixpoint theorem
Andreas Lochbihler
parents:
diff changeset
   158
  qed
507b39df1a57 add formalisation of Bourbaki-Witt fixpoint theorem
Andreas Lochbihler
parents:
diff changeset
   159
  thus ?thesis ..
507b39df1a57 add formalisation of Bourbaki-Witt fixpoint theorem
Andreas Lochbihler
parents:
diff changeset
   160
next
507b39df1a57 add formalisation of Bourbaki-Witt fixpoint theorem
Andreas Lochbihler
parents:
diff changeset
   161
  case False
507b39df1a57 add formalisation of Bourbaki-Witt fixpoint theorem
Andreas Lochbihler
parents:
diff changeset
   162
  have "(lub M, lub M') \<in> leq" using M
507b39df1a57 add formalisation of Bourbaki-Witt fixpoint theorem
Andreas Lochbihler
parents:
diff changeset
   163
  proof(rule lub_least)
507b39df1a57 add formalisation of Bourbaki-Witt fixpoint theorem
Andreas Lochbihler
parents:
diff changeset
   164
    fix x
507b39df1a57 add formalisation of Bourbaki-Witt fixpoint theorem
Andreas Lochbihler
parents:
diff changeset
   165
    assume x: "x \<in> M"
507b39df1a57 add formalisation of Bourbaki-Witt fixpoint theorem
Andreas Lochbihler
parents:
diff changeset
   166
    from comp[OF x] x False have "lub M' \<in> iterates_above x" by auto
507b39df1a57 add formalisation of Bourbaki-Witt fixpoint theorem
Andreas Lochbihler
parents:
diff changeset
   167
    moreover from M(1) x have "x \<in> Field leq" by(rule Chains_FieldD)
507b39df1a57 add formalisation of Bourbaki-Witt fixpoint theorem
Andreas Lochbihler
parents:
diff changeset
   168
    ultimately show "(x, lub M') \<in> leq" by(rule iterates_above_ge)
507b39df1a57 add formalisation of Bourbaki-Witt fixpoint theorem
Andreas Lochbihler
parents:
diff changeset
   169
  qed
507b39df1a57 add formalisation of Bourbaki-Witt fixpoint theorem
Andreas Lochbihler
parents:
diff changeset
   170
  thus ?thesis ..
507b39df1a57 add formalisation of Bourbaki-Witt fixpoint theorem
Andreas Lochbihler
parents:
diff changeset
   171
qed
507b39df1a57 add formalisation of Bourbaki-Witt fixpoint theorem
Andreas Lochbihler
parents:
diff changeset
   172
507b39df1a57 add formalisation of Bourbaki-Witt fixpoint theorem
Andreas Lochbihler
parents:
diff changeset
   173
lemma iterates_above_triangle:
507b39df1a57 add formalisation of Bourbaki-Witt fixpoint theorem
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parents:
diff changeset
   174
  assumes x: "x \<in> iterates_above a"
507b39df1a57 add formalisation of Bourbaki-Witt fixpoint theorem
Andreas Lochbihler
parents:
diff changeset
   175
  and y: "y \<in> iterates_above a"
507b39df1a57 add formalisation of Bourbaki-Witt fixpoint theorem
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parents:
diff changeset
   176
  and a: "a \<in> Field leq"
507b39df1a57 add formalisation of Bourbaki-Witt fixpoint theorem
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parents:
diff changeset
   177
  shows "x \<in> iterates_above y \<or> y \<in> iterates_above x"
507b39df1a57 add formalisation of Bourbaki-Witt fixpoint theorem
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parents:
diff changeset
   178
using x y
507b39df1a57 add formalisation of Bourbaki-Witt fixpoint theorem
Andreas Lochbihler
parents:
diff changeset
   179
proof(induction arbitrary: y)
507b39df1a57 add formalisation of Bourbaki-Witt fixpoint theorem
Andreas Lochbihler
parents:
diff changeset
   180
  case base then show ?case by simp
507b39df1a57 add formalisation of Bourbaki-Witt fixpoint theorem
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parents:
diff changeset
   181
next
507b39df1a57 add formalisation of Bourbaki-Witt fixpoint theorem
Andreas Lochbihler
parents:
diff changeset
   182
  case (step x) thus ?case using a
507b39df1a57 add formalisation of Bourbaki-Witt fixpoint theorem
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parents:
diff changeset
   183
    by(auto dest: iterates_above_successor intro: iterates_above_Field)
507b39df1a57 add formalisation of Bourbaki-Witt fixpoint theorem
Andreas Lochbihler
parents:
diff changeset
   184
next
507b39df1a57 add formalisation of Bourbaki-Witt fixpoint theorem
Andreas Lochbihler
parents:
diff changeset
   185
  case x: (Sup M)
507b39df1a57 add formalisation of Bourbaki-Witt fixpoint theorem
Andreas Lochbihler
parents:
diff changeset
   186
  hence lub: "lub M \<in> iterates_above a" by blast
507b39df1a57 add formalisation of Bourbaki-Witt fixpoint theorem
Andreas Lochbihler
parents:
diff changeset
   187
  from \<open>y \<in> iterates_above a\<close> show ?case
507b39df1a57 add formalisation of Bourbaki-Witt fixpoint theorem
Andreas Lochbihler
parents:
diff changeset
   188
  proof(induction)
507b39df1a57 add formalisation of Bourbaki-Witt fixpoint theorem
Andreas Lochbihler
parents:
diff changeset
   189
    case base show ?case using lub by simp
507b39df1a57 add formalisation of Bourbaki-Witt fixpoint theorem
Andreas Lochbihler
parents:
diff changeset
   190
  next
507b39df1a57 add formalisation of Bourbaki-Witt fixpoint theorem
Andreas Lochbihler
parents:
diff changeset
   191
    case (step y) thus ?case using a
507b39df1a57 add formalisation of Bourbaki-Witt fixpoint theorem
Andreas Lochbihler
parents:
diff changeset
   192
      by(auto dest: iterates_above_successor intro: iterates_above_Field)
507b39df1a57 add formalisation of Bourbaki-Witt fixpoint theorem
Andreas Lochbihler
parents:
diff changeset
   193
  next
507b39df1a57 add formalisation of Bourbaki-Witt fixpoint theorem
Andreas Lochbihler
parents:
diff changeset
   194
    case y: (Sup M')
507b39df1a57 add formalisation of Bourbaki-Witt fixpoint theorem
Andreas Lochbihler
parents:
diff changeset
   195
    hence lub': "lub M' \<in> iterates_above a" by blast
507b39df1a57 add formalisation of Bourbaki-Witt fixpoint theorem
Andreas Lochbihler
parents:
diff changeset
   196
    have *: "x \<in> iterates_above (lub M') \<or> lub M' \<in> iterates_above x" if "x \<in> M" for x
507b39df1a57 add formalisation of Bourbaki-Witt fixpoint theorem
Andreas Lochbihler
parents:
diff changeset
   197
      using that lub' by(rule x.IH)
507b39df1a57 add formalisation of Bourbaki-Witt fixpoint theorem
Andreas Lochbihler
parents:
diff changeset
   198
    with x(1-2) y(1-2) have "(lub M, lub M') \<in> leq \<or> lub M \<in> iterates_above (lub M')"
507b39df1a57 add formalisation of Bourbaki-Witt fixpoint theorem
Andreas Lochbihler
parents:
diff changeset
   199
      by(rule iterates_above_Sup_aux)
507b39df1a57 add formalisation of Bourbaki-Witt fixpoint theorem
Andreas Lochbihler
parents:
diff changeset
   200
    moreover from y(1-2) x(1-2) have "(lub M', lub M) \<in> leq \<or> lub M' \<in> iterates_above (lub M)"
507b39df1a57 add formalisation of Bourbaki-Witt fixpoint theorem
Andreas Lochbihler
parents:
diff changeset
   201
      by(rule iterates_above_Sup_aux)(blast dest: y.IH)
507b39df1a57 add formalisation of Bourbaki-Witt fixpoint theorem
Andreas Lochbihler
parents:
diff changeset
   202
    ultimately show ?case by(auto 4 3 dest: leq_antisym)
507b39df1a57 add formalisation of Bourbaki-Witt fixpoint theorem
Andreas Lochbihler
parents:
diff changeset
   203
  qed
507b39df1a57 add formalisation of Bourbaki-Witt fixpoint theorem
Andreas Lochbihler
parents:
diff changeset
   204
qed
507b39df1a57 add formalisation of Bourbaki-Witt fixpoint theorem
Andreas Lochbihler
parents:
diff changeset
   205
507b39df1a57 add formalisation of Bourbaki-Witt fixpoint theorem
Andreas Lochbihler
parents:
diff changeset
   206
lemma chain_iterates_above: 
507b39df1a57 add formalisation of Bourbaki-Witt fixpoint theorem
Andreas Lochbihler
parents:
diff changeset
   207
  assumes a: "a \<in> Field leq"
507b39df1a57 add formalisation of Bourbaki-Witt fixpoint theorem
Andreas Lochbihler
parents:
diff changeset
   208
  shows "iterates_above a \<in> Chains leq" (is "?C \<in> _")
507b39df1a57 add formalisation of Bourbaki-Witt fixpoint theorem
Andreas Lochbihler
parents:
diff changeset
   209
proof (rule ChainsI)
507b39df1a57 add formalisation of Bourbaki-Witt fixpoint theorem
Andreas Lochbihler
parents:
diff changeset
   210
  fix x y
507b39df1a57 add formalisation of Bourbaki-Witt fixpoint theorem
Andreas Lochbihler
parents:
diff changeset
   211
  assume "x \<in> ?C" "y \<in> ?C"
507b39df1a57 add formalisation of Bourbaki-Witt fixpoint theorem
Andreas Lochbihler
parents:
diff changeset
   212
  hence "x \<in> iterates_above y \<or> y \<in> iterates_above x" using a by(rule iterates_above_triangle)
507b39df1a57 add formalisation of Bourbaki-Witt fixpoint theorem
Andreas Lochbihler
parents:
diff changeset
   213
  moreover from \<open>x \<in> ?C\<close> a have "x \<in> Field leq" by(rule iterates_above_Field)
507b39df1a57 add formalisation of Bourbaki-Witt fixpoint theorem
Andreas Lochbihler
parents:
diff changeset
   214
  moreover from \<open>y \<in> ?C\<close> a have "y \<in> Field leq" by(rule iterates_above_Field)
507b39df1a57 add formalisation of Bourbaki-Witt fixpoint theorem
Andreas Lochbihler
parents:
diff changeset
   215
  ultimately show "(x, y) \<in> leq \<or> (y, x) \<in> leq" by(auto dest: iterates_above_ge)
507b39df1a57 add formalisation of Bourbaki-Witt fixpoint theorem
Andreas Lochbihler
parents:
diff changeset
   216
qed
507b39df1a57 add formalisation of Bourbaki-Witt fixpoint theorem
Andreas Lochbihler
parents:
diff changeset
   217
507b39df1a57 add formalisation of Bourbaki-Witt fixpoint theorem
Andreas Lochbihler
parents:
diff changeset
   218
lemma fixp_iterates_above: "a \<in> Field leq \<Longrightarrow> fixp_above a \<in> iterates_above a"
507b39df1a57 add formalisation of Bourbaki-Witt fixpoint theorem
Andreas Lochbihler
parents:
diff changeset
   219
unfolding fixp_above_def by(rule iterates_above.Sup)(blast intro: chain_iterates_above)+
507b39df1a57 add formalisation of Bourbaki-Witt fixpoint theorem
Andreas Lochbihler
parents:
diff changeset
   220
507b39df1a57 add formalisation of Bourbaki-Witt fixpoint theorem
Andreas Lochbihler
parents:
diff changeset
   221
lemma
507b39df1a57 add formalisation of Bourbaki-Witt fixpoint theorem
Andreas Lochbihler
parents:
diff changeset
   222
  assumes b: "b \<in> iterates_above a"
507b39df1a57 add formalisation of Bourbaki-Witt fixpoint theorem
Andreas Lochbihler
parents:
diff changeset
   223
  and fb: "f b = b"
507b39df1a57 add formalisation of Bourbaki-Witt fixpoint theorem
Andreas Lochbihler
parents:
diff changeset
   224
  and x: "x \<in> iterates_above a"
507b39df1a57 add formalisation of Bourbaki-Witt fixpoint theorem
Andreas Lochbihler
parents:
diff changeset
   225
  and a: "a \<in> Field leq"
507b39df1a57 add formalisation of Bourbaki-Witt fixpoint theorem
Andreas Lochbihler
parents:
diff changeset
   226
  shows "b \<in> iterates_above x"
507b39df1a57 add formalisation of Bourbaki-Witt fixpoint theorem
Andreas Lochbihler
parents:
diff changeset
   227
using x
507b39df1a57 add formalisation of Bourbaki-Witt fixpoint theorem
Andreas Lochbihler
parents:
diff changeset
   228
proof(induction)
507b39df1a57 add formalisation of Bourbaki-Witt fixpoint theorem
Andreas Lochbihler
parents:
diff changeset
   229
  case base show ?case using b by simp
507b39df1a57 add formalisation of Bourbaki-Witt fixpoint theorem
Andreas Lochbihler
parents:
diff changeset
   230
next
507b39df1a57 add formalisation of Bourbaki-Witt fixpoint theorem
Andreas Lochbihler
parents:
diff changeset
   231
  case (step x)
507b39df1a57 add formalisation of Bourbaki-Witt fixpoint theorem
Andreas Lochbihler
parents:
diff changeset
   232
  from step.hyps a have "x \<in> Field leq" by(rule iterates_above_Field)
507b39df1a57 add formalisation of Bourbaki-Witt fixpoint theorem
Andreas Lochbihler
parents:
diff changeset
   233
  from iterates_above_successor[OF step.IH this] fb
507b39df1a57 add formalisation of Bourbaki-Witt fixpoint theorem
Andreas Lochbihler
parents:
diff changeset
   234
  show ?case by(auto)
507b39df1a57 add formalisation of Bourbaki-Witt fixpoint theorem
Andreas Lochbihler
parents:
diff changeset
   235
next
507b39df1a57 add formalisation of Bourbaki-Witt fixpoint theorem
Andreas Lochbihler
parents:
diff changeset
   236
  case (Sup M)
507b39df1a57 add formalisation of Bourbaki-Witt fixpoint theorem
Andreas Lochbihler
parents:
diff changeset
   237
  oops
507b39df1a57 add formalisation of Bourbaki-Witt fixpoint theorem
Andreas Lochbihler
parents:
diff changeset
   238
507b39df1a57 add formalisation of Bourbaki-Witt fixpoint theorem
Andreas Lochbihler
parents:
diff changeset
   239
lemma fixp_above_Field: "a \<in> Field leq \<Longrightarrow> fixp_above a \<in> Field leq"
507b39df1a57 add formalisation of Bourbaki-Witt fixpoint theorem
Andreas Lochbihler
parents:
diff changeset
   240
using fixp_iterates_above by(rule iterates_above_Field)
507b39df1a57 add formalisation of Bourbaki-Witt fixpoint theorem
Andreas Lochbihler
parents:
diff changeset
   241
507b39df1a57 add formalisation of Bourbaki-Witt fixpoint theorem
Andreas Lochbihler
parents:
diff changeset
   242
lemma fixp_above_unfold:
507b39df1a57 add formalisation of Bourbaki-Witt fixpoint theorem
Andreas Lochbihler
parents:
diff changeset
   243
  assumes a: "a \<in> Field leq"
507b39df1a57 add formalisation of Bourbaki-Witt fixpoint theorem
Andreas Lochbihler
parents:
diff changeset
   244
  shows "fixp_above a = f (fixp_above a)" (is "?a = f ?a")
507b39df1a57 add formalisation of Bourbaki-Witt fixpoint theorem
Andreas Lochbihler
parents:
diff changeset
   245
proof(rule leq_antisym)
507b39df1a57 add formalisation of Bourbaki-Witt fixpoint theorem
Andreas Lochbihler
parents:
diff changeset
   246
  show "(?a, f ?a) \<in> leq" using fixp_above_Field[OF a] by(rule increasing)
507b39df1a57 add formalisation of Bourbaki-Witt fixpoint theorem
Andreas Lochbihler
parents:
diff changeset
   247
  
507b39df1a57 add formalisation of Bourbaki-Witt fixpoint theorem
Andreas Lochbihler
parents:
diff changeset
   248
  have "f ?a \<in> iterates_above a" using fixp_iterates_above[OF a] by(rule iterates_above.step)
507b39df1a57 add formalisation of Bourbaki-Witt fixpoint theorem
Andreas Lochbihler
parents:
diff changeset
   249
  with chain_iterates_above[OF a] show "(f ?a, ?a) \<in> leq" unfolding fixp_above_def by(rule lub_upper)
507b39df1a57 add formalisation of Bourbaki-Witt fixpoint theorem
Andreas Lochbihler
parents:
diff changeset
   250
qed
507b39df1a57 add formalisation of Bourbaki-Witt fixpoint theorem
Andreas Lochbihler
parents:
diff changeset
   251
507b39df1a57 add formalisation of Bourbaki-Witt fixpoint theorem
Andreas Lochbihler
parents:
diff changeset
   252
end
507b39df1a57 add formalisation of Bourbaki-Witt fixpoint theorem
Andreas Lochbihler
parents:
diff changeset
   253
507b39df1a57 add formalisation of Bourbaki-Witt fixpoint theorem
Andreas Lochbihler
parents:
diff changeset
   254
end
507b39df1a57 add formalisation of Bourbaki-Witt fixpoint theorem
Andreas Lochbihler
parents:
diff changeset
   255
507b39df1a57 add formalisation of Bourbaki-Witt fixpoint theorem
Andreas Lochbihler
parents:
diff changeset
   256
end