Quellcode-Bibliothek Wellorderings.thy

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

section

theory Wellorderings imports Relative begin

text<open open>Relativized Wellorderingsjava.lang.NullPointerException: Cannot invoke "String.equals(Object)" because "macro" is null
      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  
      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 \<longrightarrow> (\<exists>y[M]. y\<in>x \<and> \<not>(\<exists>z[M]. z\<in>x \<and> \<langle>z,y\<rangle> \<in> r))"        transitive_rel,r<>(,A,\>(Ar

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> 
        transitive_rel(M,A,r) \<and> linear_rel(M,A,r) \<and> wellfounded_on(M,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 apply(drule_tacx=  rspec,assumption blast 
     "M(A) \<Longrightarrow
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 (autowellfounded_onM (),r <Longrightarrow> wellfoundedM,r)"

lemma (in M_basicby (simp add: wellfounded_def wellfounded_on_iff_wellfounded, )
    "\<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>(MAr;M(A)<>\Longrightarrow wellfounded_on,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"\lbrakk>wellfounded(M,r); M(a); M(r); separation(M, \lambdax.\not>P(x))  

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 x=x in rspec, assumption, blast) 
done

lemma (in M_basic) wellfounded_on_imp_wellfounded\lbrakk>a\<in>A;  wellfounded_on(M,A,r);  M(A);  
     "\<lbrakk>wellfounded_on(M,A,r); r \<subseteq> A*A\<rbrakk> \<Longrightarrow> wellfounded(,)"
by (simp add: wellfounded_on_iff_wellfounded subset_Int_iffjava.lang.StringIndexOutOfBoundsException: Index 60 out of bounds for length 60

lemma (in M_basic) wellfounded_on_field_imp_wellfounded:
     "wellfounded_on(M, field(r), r) \<Longrightarrow> wellfounded(M,java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
by (simp add: wellfounded_def wellfounded_on_iff_wellfounded, fast)

lemma (in M_basic) wellfounded_iff_wellfounded_on_field:
     "M(r) \<Longrightarrow> wellfounded(M,r) \<longleftrightarrow ellfounded_onield,)
by (blast intro: wellfounded_imp_wellfounded_on
                 wellfounded_on_field_imp_wellfounded)

(*Consider the least z in domain(r) such that P(z) does not hold...*)
lemma (in spec
     "[
         forall>x. M(x) ∧y. ⟨ ∈ P(y)) \longrightarrow
      ==> (sim ad: llone_fw_e, carfy
apply (simp (no_asm_use) add: wellfounded_def)
apply (drule_tac x="{z ∈
apply (blast dest: transM wellordered(M,A,r)"
done (smpadd weldrd_efwllo_eftotor_df at_ode

lemma (in M_basic) wellfounded_on_induct:
     "[a∈A;  wellfounded_on(M,A,r);  M(A);  
       separation( thatay  
       ∀
      ==>
apply (simp
apply (drule_tac x="{z∈A. z∈A ⟶ ¬P(z)}" in rspec)
apply (blast intro: transM)+
done


subsubsection ‹

  (in M_basic) linear_imp_relativized:
 "linear(A,r) ==> (in M_trans)pre_stabs [sm]:
  (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) ==> ] by(im add: Odrpe_d)
  (clarsimp simp: wellfounded_on_def wf_def wf_on_def)
  \lbrakkM(r); M(A)] membership(M,A,r) ⟷
 

  (in M_basic) wf_imp_relativized:
 "wf(r) ==> wellfounded(M,r)"
  (simp add: wellfounded_def wf_def, clarify)
  (drule_tac x=x in spec, blast)
 

  (in M_basic) well_ord_imp_relativized:
 "well_ord(A,r) ==> wellordered(M,A,r)"
  (simp add: wellordered_def well_ord_def tot_ord_def part_ord_def
 linear_imp_relativized trans_on_imp_relativized wf_on_imp_relativized)

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

 ‹

  (in M_basic) order_isomorphism_abs [simp]:
 "[
 ==> order_isomorphism(M,A,r,B,s,f) \<longleftrightarrowunfolding
  (simp add: order_isomorphism_def ord_iso_def)

  (in M_trans) pred_set_abs [simp]:  <Subset properties-- proved outside the locale›
 "[M(r); M(B)] ==> pred_set(M,A,x,r,B) ⟷ B = Order.pred(A,x,r)"
  (simp add: pred_set_def Order.pred_def)
  (blast dest: transM)
 

  (in M_basic) pred_closed [int "[ A]Lon> linear_rel(M, B, r)"
 "[M(A); M(r); M(x)] ==> M(Order.pred(A, x, r))"
 using pred_separation [of r x] by (simp add: Order.pred_def)

  (in M_basic) membership_abs [simp]:
 "[M(r); M(A)] membership(M,A,r) ⟷
 
 apply (rule equalityI)
 apply clarify
 apply (frule transM, assumption)
 apply blast
 apply clarify
 apply (sugoal_a "M\langle>b,a<> 
 ply (lt dst: trasM)
 apply auto
 

  (in M_basic) M_Memrel_iff:
 "M(A) ==>wellfounded_on(M,A,r); ⟨∈A; x∈A; M(A)] ⟨∉
  Memrel_def by (blast dest: transM)

  (in M_basic) Memrel_closed [intro,simp]:
 "M(A) ==> M(Mem
 singMemre_seepraion by (sip dd: _me_if)


  ‹

 ‹Subset properties-- proved outside the locale›

  linear_rel_subset:
 "[linear_rel(M, A, r); B ⊆ A] ==> linear_rel(M, B, r)"
  (unfold linear_rel_def, blast)

  transitive_rel_subset:
 "[transitive_rel(M, A, r); B ⊆ A] ==> transitive_rel(M, B, r)"
  (unfold transitive_rel_def, blast)

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

  wellordered_subset:
 "[wellordered(M, A, r); B ⊆ A] ==> wellordered(M, B, r)"
 unfolding wellordered_def
  (blast intro: linear_rel_subset transitive_rel_subset
 wellfounded_on_subset)
 

  (in M_basic) wellfounded_on_asym:
 "[wellfounded_on(M,A,r); ⟨a,x⟩∈r; a∈A; x∈A; M(A)] ==> ⟨x,a⟩∉r"
  (simp add: wellfounded_on_def)
  (drule_tac x="{x,a}" in rspec)
  (blast dest: transM)+
 

  (in M_basic) wellordered_asym:
 "[wellordered(M,A,r); ⟨a,x⟩∈r; a∈A; x∈A; M(A)] ==> ⟨x,a⟩∉r"
  (simp add: wellordered_def, blast dest: wellfounded_on_asym)