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 \<^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=>o,i,i]=>o" where
    "irreflexive(M,A,r) == ∀x[M]. x∈A ⟶ <x,x> ∉ r"
  
definition
  transitive_rel :: "[i=>o,i,i]=>o" where
    "transitive_rel(M,A,r) == 
        ∀x[M]. x∈A ⟶ (∀y[M]. y∈A ⟶ (∀z[M]. z∈A ⟶ 
                          <x,y>∈r ⟶ <y,z>∈r ⟶ <x,z>∈r))"

definition
  linear_rel :: "[i=>o,i,i]=>o" where
    "linear_rel(M,A,r) == 
        ∀x[M]. x∈A ⟶ (∀y[M]. y∈A ⟶ <x,y>∈r | x=y | <y,x>∈r)"

definition
  wellfounded :: "[i=>o,i]=>o" where
    ― ‹EVERY non-empty set has an ‹r›-minimal element›
    "wellfounded(M,r) == 
        ∀x[M]. x≠0 ⟶ (∃y[M]. y∈x & ~(∃z[M]. z∈x & <z,y> ∈ r))"
definition
  wellfounded_on :: "[i=>o,i,i]=>o" where
    ― ‹every non-empty SUBSET OF ‹A› has an ‹r›-minimal element›
    "wellfounded_on(M,A,r) == 
        ∀x[M]. x≠0 ⟶ x⊆A ⟶ (∃y[M]. y∈x & ~(∃z[M]. z∈x & <z,y> ∈ r))"

definition
  wellordered :: "[i=>o,i,i]=>o" where
    ― ‹linear and wellfounded on ‹A››
    "wellordered(M,A,r) == 
        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) 
done

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
                 wellfounded_on_field_imp_wellfounded)

(*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) |]
      ==> P(a)"
apply (simp (no_asm_use) add: wellfounded_def)
apply (drule_tac x="{z ∈ domain(r). ~P(z)}" in rspec)
apply (blast dest: transM)+
done

lemma (in M_basic) wellfounded_on_induct: 
     "[| a∈A;  wellfounded_on(M,A,r);  M(A);  
       separation(M, λx. x∈A ⟶ ~P(x));  
       ∀x∈A. M(x) & (∀y∈A. <y,x> ∈ r ⟶ P(y)) ⟶ P(x) |]
      ==> P(a)"
apply (simp (no_asm_use) add: wellfounded_on_def)
apply (drule_tac x="{z∈A. z∈A ⟶ ~P(z)}" in rspec)
apply (blast intro: transM)+
done


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) 
done

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) 
done

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) 
done

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 
done

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)"
apply (unfold wellordered_def)
apply (blast intro: linear_rel_subset transitive_rel_subset 
                    wellfounded_on_subset)
done

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

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

end