Theory Wellorderings

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


header {*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 @{text 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 @{text 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 @{text A} has an @{text 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 @{text 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 (simp add: wellfounded_on_def wf_def wf_on_def, clarify)
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: apply_closed order_isomorphism_def ord_iso_def)

lemma (in M_basic) 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))"
apply (simp add: Order.pred_def)
apply (insert pred_separation [of r x], simp)
done

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}"

apply (simp add: Memrel_def)
apply (blast dest: transM)
done

lemma (in M_basic) Memrel_closed [intro,simp]:
"M(A) ==> M(Memrel(A))"
apply (simp add: M_Memrel_iff)
apply (insert Memrel_separation, simp)
done


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