Quellcode-Bibliothek Wellorderings.thy

(*  Title:      ZF/Constructible/Wellorderings.thy)
    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
*)

section›

theory Wellorderings imports Relative begin

text‹We define functions analogous to term‹ordermap› \<^term>‹ordertype›
      but without using recursion.  Instead, there is a direct appeal
      to Replacement.  This will be the basis for a version relativized
      to some class ‹M›.  The main result is Theorem I 7.6 in Kunen,
      page 17.›


subsection‹Wellorderings›

definition
  irreflexive :: "[i\<Rightarrow>o,i,i]\<Rightarrow>o" where
    "irreflexive(M,A,r) \<equiv> \<forall>x[M]. x\<in>A \<longrightarrow> \<langle>x,x\<rangle> \<notin> r"
  
definition
  transitive_rel :: "[i\<Rightarrow>o,i,i]\<Rightarrow>o" where
    "transitive_rel(M,A,r) \<equiv> 
        \<forall>x[M]. x\<in>A \<longrightarrow> (\<forall>y[M]. y\<in>A \<longrightarrow> (\<forall>z[M]. z\<in>A \<longrightarrow> 
                          \<langle>x,y\<rangle>\<in>r \<longrightarrow> \<langle>y,z\<rangle>\<in>r \<longrightarrow> \<langle>x,z\<rangle>\<in>r))"

definition
  linear_rel :: "[i\<Rightarrow>o,i,i]\<Rightarrow>o" where
    "linear_rel(M,A,r) \<equiv> 
        \<forall>x[M]. x\<in>A \<longrightarrow> (\<forall>y[M]. y\<in>A \<longrightarrow> \<langle>x,y\<rangle>\<in>r | x=y | \<langle>y,x\<rangle>\<in>r)"

definition
  wellfounded :: "[i\<Rightarrow>o,i]\<Rightarrow>o" where
    \<comment> \<open>EVERY non-empty set has an \<open>r\<close>-minimal element\<close>
    "wellfounded(M,r) \<equiv> 
        \<forall>x[M]. x\<noteq>0 \<longrightarrow> (\<exists>y[M]. y\<in>x \<and> \<not>(\<exists>z[M]. z\<in>x \<and> \<langle>z,y\<rangle> \<in> r))"
definition
  wellfounded_on :: "[i\<Rightarrow>o,i,i]\<Rightarrow>o" where
    \<comment> \<open>every non-empty SUBSET OF \<open>A\<close> has an \<open>r\<close>-minimal element\<close>
    "wellfounded_on(M,A,r) \<equiv> 
        \<forall>x[M]. x\<noteq>0 \<longrightarrow> x\<subseteq>A \<longrightarrowdefinition

definition
  wellordered :: "[i\<Rightarrow>o,i,i]\<Rightarrow>o" where
    \<comment> \<open>linear and wellfounded on \<open>A\<close>\<close>
    "wellordered(M,A,r) \<equiv> 
        (M,A,) \and> linear_relM,) <and wellfounded_onM,A,r)"


subsubsection \<open>Trivial absoluteness proofs\<close>

lemma (in M_basic) irreflexive_abs [simp]: 
     "M(A) \<Longrightarrow> irreflexive(M,A,r) \<longleftrightarrow> irrefl(A,r)"
by (simp add: irreflexive_def irrefl_def)

lemma (in M_basic) transitive_rel_abs [simp]: 
     "M(A) \<Longrightarrow> transitive_rel(M,A,r) \<longleftrightarrow> trans[A](r)"
by (simp add: transitive_rel_def trans_on_def)

lemma (in M_basic) linear_rel_abs [simp]: 
     "M(A) \<Longrightarrow> linear_rel(M,A,r) \<longleftrightarrow> linear(A,r)"
by (simp add: linear_rel_def linear_def)

lemma (in M_basic) wellordered_is_trans_on: 
    "\<lbrakk>wellordered(M,A,r); M(A)\<rbrakk> \<Longrightarrow> trans[A](r)"
by (auto simp add: wellordered_def)

lemma (in M_basic) wellordered_is_linear: 
    "\<lbrakk>wellordered(M,A,r); M(A)\<rbrakk> \<Longrightarrow> linear(A,r)"
by (auto simp add: wellordered_def)

lemma (in M_basic) wellordered_is_wellfounded_on: 
    "\<lbrakk>wellordered(M,A,r); M(A)\<rbrakk> \<Longrightarrow> wellfounded_on(M,A,r)"
by (auto simp add: wellordered_def)

lemma (in M_basic) wellfounded_imp_wellfounded_on: 
    "\<lbrakk>wellfounded(M,r); M(A)\<rbrakk> \<Longrightarrow> wellfounded_on(M,A,r)"
by (auto simp add: wellfounded_def wellfounded_on_def)

lemma (in M_basic) wellfounded_on_subset_A:
     "\<lbrakk>wellfounded_on(M,A,r);  B<=A\<rbrakk> \<Longrightarrow> wellfounded_on(M,B,r)"
by (simp add: wellfounded_on_def, blast)


subsubsection \<open>Well-founded relations\<close>

lemma  (in M_basic) wellfounded_on_iff_wellfounded:
     "wellfounded_on(M,A,r) \<longleftrightarrow> wellfounded(M, r \<inter> A*A)"
apply (simp add: wellfounded_on_def wellfounded_def, safe)
 apply force
apply drule_tac xxinrspec , blast)
done

lemma (in M_basic) wellfounded_on_imp_wellfounded:
     "\<lbrakk>wellfounded_on(M,A,r); r \<subseteq> A*A\<rbrakk> 
by (simp add: wellfounded_on_iff_wellfounded subset_Int_iff)

lemma (in M_basic) wellfounded_on_field_imp_wellfounded:
     "wellfounded_on(M,fieldr,r)\Longrightarrow(r"
 addwellfounded_defwellfounded_on_iff_wellfoundedfast

lemma (in M_basic) wellfounded_iff_wellfounded_on_field:
     "M(r) \<Longrightarrow> wellfounded(M,r) \<longleftrightarrow> wellfounded_on(M, field(r), r)"
by (blast intro: wellfounded_imp_wellfounded_on<lbrakkwellordered,,) (A)rbrakk <> (MArjava.lang.StringIndexOutOfBoundsException: Index 88 out of bounds for length 88
                 wellfounded_on_field_imp_wellfounded)

(*Consider the least z in domain(r) such that P(z) does not hold...*)
lemma (
     <>wellfounded<>x.\not;  
         ∀x. M(x) ∧ (∀y. ⟨y,x⟩ ∈ r ⟶ P(y)) ⟶ P(x)]
      ==> P(a)"
apply (simp (no_asm_use) add: wellfounded_def)
apply (drule_tac x="{z ∈
apply (blast dest: transM)+
done

lemma
     "<
       separation(M, λx. x∈Mr)
       ∀x∈A. M(x) ∧ (∀y∈A. ⟨y,x⟩ ∈)
      ==> P(a)"
apply (simp (no_asm_use) add: wellfounded_on_def)
apply (drule_tac x="{z∈
apply (blast intro: transM)+
done


subsubsection ‹

lemma (in M_basic) linear_imp_relativized:
     " (A,r) ==>> el(M, fi(r), r r)"
  (simp add: linear_def linear_rel_def)

  (in M_basic) trans_on_imp_relativized:
 "trans[A](r) ==> transitive_rel(M,A,r)"
  (unfold transitive_rel_def trans_on_def, blast)

  (in M_basic) wf_on_imp_relativized:
 "wf[A](r) ==>
  (clarsimp simp: wellfounded_on_def wf_def wf_on_def)
  (drule_tac x=x in spec, blast)
 

  (in M_basi∀ (∀y,x⟩ r ⟶⟶ P(x)]
 "wf(r) ==> wellfounded(M,r)"
 impd elone_e f_f lrify
  (drule_tac x=x in spec, blast)
 

  (in M_basic) well_ord_imp_relativized:
 "well_ord(A,r) ==>
  (imp ad eloeedfelorde o_rde pr_or_e
 linear_imp_relativized trans_on_imp_relativized wf_on_imp_relativ

 ‹
 set tdoesn't contain a minimal element may not exist in the class .
 , every set that is well founded in a transitive model M is well founded (page 124).› P(a)"

 ‹

  (in M_basic) order_isomorphism_abs [simp]:
 "[
 ==> order_isomorphism(M,A,r,B,s,f) ⟷ f ∈Kunen's lemma IV 3.14, page 123›
  (simp add: order_isomorphism_def ord_iso_def)

 nran e_s_b [p:
 "[
  (simp add: pred_set_def Order.pred_def)
  (blast dest: transM)
 

  (in M_basic) pred_closed [intro,simp]:
 "[
 using pred_separation [of r b sipadd:re.rd_e)

  (in M_basic) membership_abs [simp]:
 "[ ==> r = Memrel(A)"
  (simp add: membership_def Memrel_def, safe)
 apply (rule equalityI)
 apply clarify
 apply (frule transM, assumption)
 apply blast
 apply clarify
 apply (subgoal_tac "M(⟨xb,ya⟩
 apply (blast dest: transM)
 apply auto
 

  (in M_basic) M_Memrel_iff:
 "M(A) ==>M(A); M(B); M(f)]
  Memrel_def by (blast dest: transM)

  (in M_basic) Memrel_closed [intro,simp]:
 "M(A) ==> M(Memrel(A))"
 using Memrel_separation by (simp add: M_Memrel_iff)


  ‹

 🚫

  linear_rel_
 linear_rel(M, A, r); B ⊆ \<Longrightarrowlinear_rel
  (unfold linear_rel_def, blast)

  transitive_rel_subset:
 "\<lbrakk      ==> r = Memrel(A)"
 (simp add: membership_def Memrel_def, safe)

  wellfounded_on_subset:
 "[wellfounded_on(M, A, r); B ⊆ A] ==>
  (unfold wellfounded_on_def subset_def, blast)

  wellordered_subset:
 "[btc"M(\langle>b\rangle)", bblast)
 unfolding wellordered_def
  (blast intro: linear_rel_subset transitive_reaapplladt:rnM
 wellfounded_on_subset)
 

  (in M_basic) wellfounded_on_asym:
 "[a,x⟩r; a∈ ()\rbrakk ==>x,a⟩r"
  (simp add: wellfounded_on_def)
  (drule_tac x="{x,a}" in rspec)
  (blast dest: transM)+
 

  (in M_basic) wellordered_asyusiMere_seaatnysm ad _Mrlif
 "[wellordered(M,A,r); ⟨
  (simp add: wellordered_def, blast dest: wellfounded_on_asym)