Theory Wellorderings

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

section ‹Relativized Wellorderings›

theory Wellorderings imports Relative begin

text‹We define functions analogous to termordermap termordertype 
      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.›


  irreflexive :: "[io,i,i]o" where
    "irreflexive(M,A,r)  x[M]. xA  x,x  r"
  transitive_rel :: "[io,i,i]o" where
        x[M]. xA  (y[M]. yA  (z[M]. zA  
                          x,yr  y,zr  x,zr))"

  linear_rel :: "[io,i,i]o" where
        x[M]. xA  (y[M]. yA  x,yr | x=y | y,xr)"

  wellfounded :: "[io,i]o" where
    ― ‹EVERY non-empty set has an r›-minimal element›
        x[M]. x0  (y[M]. yx  ¬(z[M]. zx  z,y  r))"
  wellfounded_on :: "[io,i,i]o" where
    ― ‹every non-empty SUBSET OF A› has an r›-minimal element›
        x[M]. x0  xA  (y[M]. yx  ¬(z[M]. zx  z,y  r))"

  wellordered :: "[io,i,i]o" where
    ― ‹linear and wellfounded on A›
        transitive_rel(M,A,r)  linear_rel(M,A,r)  wellfounded_on(M,A,r)"

subsubsection ‹Trivial absoluteness proofs›

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

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

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

lemma (in M_basic) wellordered_is_trans_on: 
    "wellordered(M,A,r); M(A)  trans[A](r)"
by (auto simp add: wellordered_def)

lemma (in M_basic) wellordered_is_linear: 
    "wellordered(M,A,r); M(A)  linear(A,r)"
by (auto simp add: wellordered_def)

lemma (in M_basic) wellordered_is_wellfounded_on: 
    "wellordered(M,A,r); M(A)  wellfounded_on(M,A,r)"
by (auto simp add: wellordered_def)

lemma (in M_basic) wellfounded_imp_wellfounded_on: 
    "wellfounded(M,r); M(A)  wellfounded_on(M,A,r)"
by (auto simp add: wellfounded_def wellfounded_on_def)

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

subsubsection ‹Well-founded relations›

lemma  (in M_basic) wellfounded_on_iff_wellfounded:
     "wellfounded_on(M,A,r)  wellfounded(M, r  A*A)"
apply (simp add: wellfounded_on_def wellfounded_def, safe)
 apply force
apply (drule_tac x=x in rspec, assumption, blast) 

lemma (in M_basic) wellfounded_on_imp_wellfounded:
     "wellfounded_on(M,A,r); r  A*A  wellfounded(M,r)"
by (simp add: wellfounded_on_iff_wellfounded subset_Int_iff)

lemma (in M_basic) wellfounded_on_field_imp_wellfounded:
     "wellfounded_on(M, field(r), r)  wellfounded(M,r)"
by (simp add: wellfounded_def wellfounded_on_iff_wellfounded, fast)

lemma (in M_basic) wellfounded_iff_wellfounded_on_field:
     "M(r)  wellfounded(M,r)  wellfounded_on(M, field(r), r)"
by (blast intro: wellfounded_imp_wellfounded_on

(*Consider the least z in domain(r) such that P(z) does not hold...*)
lemma (in M_basic) wellfounded_induct: 
     "wellfounded(M,r); M(a); M(r); separation(M, λx. ¬P(x));  
         x. M(x)  (y. y,x  r  P(y))  P(x)
apply (simp (no_asm_use) add: wellfounded_def)
apply (drule_tac x="{z  domain(r). ¬P(z)}" in rspec)
apply (blast dest: transM)+

lemma (in M_basic) wellfounded_on_induct: 
     "aA;  wellfounded_on(M,A,r);  M(A);  
       separation(M, λx. xA  ¬P(x));  
       xA. M(x)  (yA. y,x  r  P(y))  P(x)
apply (simp (no_asm_use) add: wellfounded_on_def)
apply (drule_tac x="{zA. zA  ¬P(z)}" in rspec)
apply (blast intro: transM)+

subsubsection ‹Kunen's lemma IV 3.14, page 123›

lemma (in M_basic) linear_imp_relativized: 
     "linear(A,r)  linear_rel(M,A,r)" 
by (simp add: linear_def linear_rel_def) 

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

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

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

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

text‹The property being well founded (and hence of being well ordered) is not absolute: 
the set that doesn't contain a minimal element may not exist in the class M. 
However, every set that is well founded in a transitive model M is well founded (page 124).›

subsection‹Relativized versions of order-isomorphisms and order types›

lemma (in M_basic) order_isomorphism_abs [simp]: 
     "M(A); M(B); M(f) 
       order_isomorphism(M,A,r,B,s,f)  f  ord_iso(A,r,B,s)"
by (simp add: order_isomorphism_def ord_iso_def)

lemma (in M_trans) pred_set_abs [simp]: 
     "M(r); M(B)  pred_set(M,A,x,r,B)  B = Order.pred(A,x,r)"
apply (simp add: pred_set_def Order.pred_def)
apply (blast dest: transM) 

lemma (in M_basic) pred_closed [intro,simp]: 
  "M(A); M(r); M(x)  M(Order.pred(A, x, r))"
  using pred_separation [of r x] by (simp add: Order.pred_def) 

lemma (in M_basic) membership_abs [simp]: 
     "M(r); M(A)  membership(M,A,r)  r = Memrel(A)"
apply (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)", blast) 
  apply (blast dest: transM) 
 apply auto 

lemma (in M_basic) M_Memrel_iff:
     "M(A)  Memrel(A) = {z  A*A. x[M]. y[M]. z = x,y  x  y}"
unfolding Memrel_def by (blast dest: transM)

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

subsection ‹Main results of Kunen, Chapter 1 section 6›

text‹Subset properties-- proved outside the locale›

lemma linear_rel_subset: 
    "linear_rel(M, A, r); B  A  linear_rel(M, B, r)"
by (unfold linear_rel_def, blast)

lemma transitive_rel_subset: 
    "transitive_rel(M, A, r); B  A  transitive_rel(M, B, r)"
by (unfold transitive_rel_def, blast)

lemma wellfounded_on_subset: 
    "wellfounded_on(M, A, r); B  A  wellfounded_on(M, B, r)"
by (unfold wellfounded_on_def subset_def, blast)

lemma wellordered_subset: 
    "wellordered(M, A, r); B  A  wellordered(M, B, r)"
  unfolding wellordered_def
apply (blast intro: linear_rel_subset transitive_rel_subset 

lemma (in M_basic) wellfounded_on_asym:
     "wellfounded_on(M,A,r);  a,xr;  aA; xA;  M(A)  x,ar"
apply (simp add: wellfounded_on_def) 
apply (drule_tac x="{x,a}" in rspec) 
apply (blast dest: transM)+

lemma (in M_basic) wellordered_asym:
     "wellordered(M,A,r);  a,xr;  aA; xA;  M(A)  x,ar"
by (simp add: wellordered_def, blast dest: wellfounded_on_asym)