# 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)"

lemma (in M_basic) transitive_rel_abs [simp]:
"M(A) ⟹ transitive_rel(M,A,r) ⟷ trans[A](r)"

lemma (in M_basic) linear_rel_abs [simp]:
"M(A) ⟹ linear_rel(M,A,r) ⟷ linear(A,r)"

lemma (in M_basic) wellordered_is_trans_on:
"⟦wellordered(M,A,r); M(A)⟧ ⟹ trans[A](r)"

lemma (in M_basic) wellordered_is_linear:
"⟦wellordered(M,A,r); M(A)⟧ ⟹ linear(A,r)"

lemma (in M_basic) wellordered_is_wellfounded_on:
"⟦wellordered(M,A,r); M(A)⟧ ⟹ wellfounded_on(M,A,r)"

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)"

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)"

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 (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 (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)"

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)"

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 (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)"
unfolding 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"