(* 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 🍋‹ordermap
› 🍋‹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, \x. \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)"
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"
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
