Theory AC_in_L

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

section ‹The Axiom of Choice Holds in L!›

theory AC_in_L imports Formula Separation begin

subsection‹Extending a Wellordering over a List -- Lexicographic Power›

text‹This could be moved into a library.›

consts
rlist   :: "[i,i]=>i"

inductive
domains "rlist(A,r)" ⊆ "list(A) * list(A)"
intros
shorterI:
"[| length(l') < length(l); l' ∈ list(A); l ∈ list(A) |]
==> <l', l> ∈ rlist(A,r)"

sameI:
"[| <l',l> ∈ rlist(A,r); a ∈ A |]
==> <Cons(a,l'), Cons(a,l)> ∈ rlist(A,r)"

diffI:
"[| length(l') = length(l); <a',a> ∈ r;
l' ∈ list(A); l ∈ list(A); a' ∈ A; a ∈ A |]
==> <Cons(a',l'), Cons(a,l)> ∈ rlist(A,r)"
type_intros list.intros

subsubsection‹Type checking›

lemmas rlist_type = rlist.dom_subset

lemmas field_rlist = rlist_type [THEN field_rel_subset]

subsubsection‹Linearity›

lemma rlist_Nil_Cons [intro]:
"[|a ∈ A; l ∈ list(A)|] ==> <[], Cons(a,l)> ∈ rlist(A, r)"

lemma linear_rlist:
assumes r: "linear(A,r)" shows "linear(list(A),rlist(A,r))"
proof -
{ fix xs ys
have "xs ∈ list(A) ⟹ ys ∈ list(A) ⟹ ⟨xs,ys⟩ ∈ rlist(A,r) ∨ xs = ys ∨ ⟨ys,xs⟩ ∈ rlist(A, r) "
proof (induct xs arbitrary: ys rule: list.induct)
case Nil
thus ?case by (induct ys rule: list.induct) (auto simp add: shorterI)
next
case (Cons x xs)
{ fix y ys
assume "y ∈ A" and "ys ∈ list(A)"
with Cons
have "⟨Cons(x,xs),Cons(y,ys)⟩ ∈ rlist(A,r) ∨ x=y & xs = ys ∨ ⟨Cons(y,ys), Cons(x,xs)⟩ ∈ rlist(A,r)"
apply (rule_tac i = "length(xs)" and j = "length(ys)" in Ord_linear_lt)
apply (rule linearE [OF r, of x y])
apply (auto simp add: diffI intro: sameI)
done
}
note yConsCase = this
show ?case using ‹ys ∈ list(A)›
by (cases rule: list.cases) (simp_all add: Cons rlist_Nil_Cons yConsCase)
qed
}
thus ?thesis by (simp add: linear_def)
qed

subsubsection‹Well-foundedness›

text‹Nothing preceeds Nil in this ordering.›
inductive_cases rlist_NilE: " <l,[]> ∈ rlist(A,r)"

inductive_cases rlist_ConsE: " <l', Cons(x,l)> ∈ rlist(A,r)"

lemma not_rlist_Nil [simp]: " <l,[]> ∉ rlist(A,r)"
by (blast intro: elim: rlist_NilE)

lemma rlist_imp_length_le: "<l',l> ∈ rlist(A,r) ==> length(l') ≤ length(l)"
apply (erule rlist.induct)
done

lemma wf_on_rlist_n:
"[| n ∈ nat; wf[A](r) |] ==> wf[{l ∈ list(A). length(l) = n}](rlist(A,r))"
apply (induct_tac n)
apply (rule wf_onI2, simp)
apply (rule wf_onI2, clarify)
apply (erule_tac a=y in list.cases, clarify)
apply (simp (no_asm_use))
apply clarify
apply (simp (no_asm_use))
apply (subgoal_tac "∀l2 ∈ list(A). length(l2) = x ⟶ Cons(a,l2) ∈ B", blast)
apply (erule_tac a=a in wf_on_induct, assumption)
apply (rule ballI)
apply (rule impI)
apply (erule_tac a=l2 in wf_on_induct, blast, clarify)
apply (rename_tac a' l2 l')
apply (drule_tac x="Cons(a',l')" in bspec, typecheck)
apply simp
apply (erule mp, clarify)
apply (erule rlist_ConsE, auto)
done

lemma list_eq_UN_length: "list(A) = (⋃n∈nat. {l ∈ list(A). length(l) = n})"
by (blast intro: length_type)

lemma wf_on_rlist: "wf[A](r) ==> wf[list(A)](rlist(A,r))"
apply (subst list_eq_UN_length)
apply (rule wf_on_Union)
apply (rule wf_imp_wf_on [OF wf_Memrel [of nat]])
apply (frule rlist_type [THEN subsetD])
apply (drule rlist_imp_length_le)
apply (erule leE)
done

lemma wf_rlist: "wf(r) ==> wf(rlist(field(r),r))"
apply (rule wf_on_subset_A [OF _ field_rlist])
apply (blast intro: wf_on_rlist)
done

lemma well_ord_rlist:
"well_ord(A,r) ==> well_ord(list(A), rlist(A,r))"
apply (rule well_ordI)
apply (simp add: well_ord_def tot_ord_def linear_rlist)
done

subsection‹An Injection from Formulas into the Natural Numbers›

text‹There is a well-known bijection between @{term "nat*nat"} and @{term
nat} given by the expression f(m,n) = triangle(m+n) + m, where triangle(k)
enumerates the triangular numbers and can be defined by triangle(0)=0,
triangle(succ(k)) = succ(k + triangle(k)).  Some small amount of effort is
needed to show that f is a bijection.  We already know that such a bijection exists by the theorem ‹well_ord_InfCard_square_eq›:
@{thm[display] well_ord_InfCard_square_eq[no_vars]}

However, this result merely states that there is a bijection between the two
sets.  It provides no means of naming a specific bijection.  Therefore, we
conduct the proofs under the assumption that a bijection exists.  The simplest
way to organize this is to use a locale.›

text‹Locale for any arbitrary injection between @{term "nat*nat"}
and @{term nat}›
locale Nat_Times_Nat =
fixes fn
assumes fn_inj: "fn ∈ inj(nat*nat, nat)"

consts   enum :: "[i,i]=>i"
primrec
"enum(f, Member(x,y)) = f ` <0, f ` <x,y>>"
"enum(f, Equal(x,y)) = f ` <1, f ` <x,y>>"
"enum(f, Nand(p,q)) = f ` <2, f ` <enum(f,p), enum(f,q)>>"
"enum(f, Forall(p)) = f ` <succ(2), enum(f,p)>"

lemma (in Nat_Times_Nat) fn_type [TC,simp]:
"[|x ∈ nat; y ∈ nat|] ==> fn`<x,y> ∈ nat"
by (blast intro: inj_is_fun [OF fn_inj] apply_funtype)

lemma (in Nat_Times_Nat) fn_iff:
"[|x ∈ nat; y ∈ nat; u ∈ nat; v ∈ nat|]
==> (fn`<x,y> = fn`<u,v>) ⟷ (x=u & y=v)"
by (blast dest: inj_apply_equality [OF fn_inj])

lemma (in Nat_Times_Nat) enum_type [TC,simp]:
"p ∈ formula ==> enum(fn,p) ∈ nat"
by (induct_tac p, simp_all)

lemma (in Nat_Times_Nat) enum_inject [rule_format]:
"p ∈ formula ==> ∀q∈formula. enum(fn,p) = enum(fn,q) ⟶ p=q"
apply (induct_tac p, simp_all)
apply (rule ballI)
apply (erule formula.cases)
apply (rule ballI)
apply (erule formula.cases)
apply (rule ballI)
apply (erule_tac a=qa in formula.cases)
apply blast
apply (rule ballI)
apply (erule_tac a=q in formula.cases)
done

lemma (in Nat_Times_Nat) inj_formula_nat:
"(λp ∈ formula. enum(fn,p)) ∈ inj(formula, nat)"
apply (blast intro: enum_inject)
done

lemma (in Nat_Times_Nat) well_ord_formula:
"well_ord(formula, measure(formula, enum(fn)))"
apply (rule well_ord_measure, simp)
apply (blast intro: enum_inject)
done

lemmas nat_times_nat_lepoll_nat =
InfCard_nat [THEN InfCard_square_eqpoll, THEN eqpoll_imp_lepoll]

text‹Not needed--but interesting?›
theorem formula_lepoll_nat: "formula ≲ nat"
apply (insert nat_times_nat_lepoll_nat)
apply (unfold lepoll_def)
apply (blast intro: Nat_Times_Nat.inj_formula_nat Nat_Times_Nat.intro)
done

subsection‹Defining the Wellordering on @{term "DPow(A)"}›

text‹The objective is to build a wellordering on @{term "DPow(A)"} from a
given one on @{term A}.  We first introduce wellorderings for environments,
which are lists built over @{term "A"}.  We combine it with the enumeration of
formulas.  The order type of the resulting wellordering gives us a map from
(environment, formula) pairs into the ordinals.  For each member of @{term
"DPow(A)"}, we take the minimum such ordinal.›

definition
env_form_r :: "[i,i,i]=>i" where
―‹wellordering on (environment, formula) pairs›
"env_form_r(f,r,A) ==
rmult(list(A), rlist(A, r),
formula, measure(formula, enum(f)))"

definition
env_form_map :: "[i,i,i,i]=>i" where
―‹map from (environment, formula) pairs to ordinals›
"env_form_map(f,r,A,z)
== ordermap(list(A) * formula, env_form_r(f,r,A)) ` z"

definition
DPow_ord :: "[i,i,i,i,i]=>o" where
―‹predicate that holds if @{term k} is a valid index for @{term X}›
"DPow_ord(f,r,A,X,k) ==
∃env ∈ list(A). ∃p ∈ formula.
arity(p) ≤ succ(length(env)) &
X = {x∈A. sats(A, p, Cons(x,env))} &
env_form_map(f,r,A,<env,p>) = k"

definition
DPow_least :: "[i,i,i,i]=>i" where
―‹function yielding the smallest index for @{term X}›
"DPow_least(f,r,A,X) == μ k. DPow_ord(f,r,A,X,k)"

definition
DPow_r :: "[i,i,i]=>i" where
―‹a wellordering on @{term "DPow(A)"}›
"DPow_r(f,r,A) == measure(DPow(A), DPow_least(f,r,A))"

lemma (in Nat_Times_Nat) well_ord_env_form_r:
"well_ord(A,r)
==> well_ord(list(A) * formula, env_form_r(fn,r,A))"
by (simp add: env_form_r_def well_ord_rmult well_ord_rlist well_ord_formula)

lemma (in Nat_Times_Nat) Ord_env_form_map:
"[|well_ord(A,r); z ∈ list(A) * formula|]
==> Ord(env_form_map(fn,r,A,z))"
by (simp add: env_form_map_def Ord_ordermap well_ord_env_form_r)

lemma DPow_imp_ex_DPow_ord:
"X ∈ DPow(A) ==> ∃k. DPow_ord(fn,r,A,X,k)"
apply (blast dest!: DPowD)
done

lemma (in Nat_Times_Nat) DPow_ord_imp_Ord:
"[|DPow_ord(fn,r,A,X,k); well_ord(A,r)|] ==> Ord(k)"
done

lemma (in Nat_Times_Nat) DPow_imp_DPow_least:
"[|X ∈ DPow(A); well_ord(A,r)|]
==> DPow_ord(fn, r, A, X, DPow_least(fn,r,A,X))"
apply (blast dest: DPow_imp_ex_DPow_ord intro: DPow_ord_imp_Ord LeastI)
done

lemma (in Nat_Times_Nat) env_form_map_inject:
"[|env_form_map(fn,r,A,u) = env_form_map(fn,r,A,v); well_ord(A,r);
u ∈ list(A) * formula;  v ∈ list(A) * formula|]
==> u=v"
apply (rule inj_apply_equality [OF bij_is_inj, OF ordermap_bij,
OF well_ord_env_form_r], assumption+)
done

lemma (in Nat_Times_Nat) DPow_ord_unique:
"[|DPow_ord(fn,r,A,X,k); DPow_ord(fn,r,A,Y,k); well_ord(A,r)|]
==> X=Y"
apply (drule env_form_map_inject, auto)
done

lemma (in Nat_Times_Nat) well_ord_DPow_r:
"well_ord(A,r) ==> well_ord(DPow(A), DPow_r(fn,r,A))"
apply (rule well_ord_measure)
apply (drule DPow_imp_DPow_least, assumption)+
apply simp
apply (blast intro: DPow_ord_unique)
done

lemma (in Nat_Times_Nat) DPow_r_type:
"DPow_r(fn,r,A) ⊆ DPow(A) * DPow(A)"
by (simp add: DPow_r_def measure_def, blast)

subsection‹Limit Construction for Well-Orderings›

text‹Now we work towards the transfinite definition of wellorderings for
@{term "Lset(i)"}.  We assume as an inductive hypothesis that there is a family
of wellorderings for smaller ordinals.›

definition
rlimit :: "[i,i=>i]=>i" where
―‹Expresses the wellordering at limit ordinals.  The conditional
lets us remove the premise @{term "Limit(i)"} from some theorems.›
"rlimit(i,r) ==
if Limit(i) then
{z: Lset(i) * Lset(i).
∃x' x. z = <x',x> &
(lrank(x') < lrank(x) |
(lrank(x') = lrank(x) & <x',x> ∈ r(succ(lrank(x)))))}
else 0"

definition
Lset_new :: "i=>i" where
―‹This constant denotes the set of elements introduced at level
@{term "succ(i)"}›
"Lset_new(i) == {x ∈ Lset(succ(i)). lrank(x) = i}"

lemma Limit_Lset_eq2:
"Limit(i) ==> Lset(i) = (⋃j∈i. Lset_new(j))"
apply (rule equalityI)
apply safe
apply (subgoal_tac "Ord(y)")
prefer 2 apply (blast intro: Ord_in_Ord Limit_is_Ord)
apply (simp_all add: Limit_is_Ord Lset_iff_lrank_lt Lset_new_def
Ord_mem_iff_lt)
apply (blast intro: lt_trans)
apply (rule_tac x = "succ(lrank(x))" in bexI)
apply (blast intro: Limit_has_succ ltD)
done

lemma wf_on_Lset:
"wf[Lset(succ(j))](r(succ(j))) ==> wf[Lset_new(j)](rlimit(i,r))"
apply (erule wf_subset)
done

lemma wf_on_rlimit:
"(∀j<i. wf[Lset(j)](r(j))) ==> wf[Lset(i)](rlimit(i,r))"
apply (case_tac "Limit(i)")
prefer 2
apply (rule wf_on_Union)
apply (rule wf_imp_wf_on [OF wf_Memrel [of i]])
apply (blast intro: wf_on_Lset Limit_has_succ Limit_is_Ord ltI)
apply (force simp add: rlimit_def Limit_is_Ord Lset_iff_lrank_lt Lset_new_def
Ord_mem_iff_lt)
done

lemma linear_rlimit:
"[|Limit(i); ∀j<i. linear(Lset(j), r(j)) |]
==> linear(Lset(i), rlimit(i,r))"
apply (frule Limit_is_Ord)
apply (simp add: linear_def rlimit_def Ball_def lt_Ord Lset_iff_lrank_lt)
apply (rename_tac u v)
apply (rule_tac i="lrank(u)" and j="lrank(v)" in Ord_linear_lt, simp_all)
apply (drule_tac x="succ(lrank(u) ∪ lrank(v))" in ospec)
apply (drule_tac x=u in spec, simp)
apply (drule_tac x=v in spec, simp)
done

lemma well_ord_rlimit:
"[|Limit(i); ∀j<i. well_ord(Lset(j), r(j)) |]
==> well_ord(Lset(i), rlimit(i,r))"
by (blast intro: well_ordI wf_on_rlimit well_ord_is_wf
linear_rlimit well_ord_is_linear)

lemma rlimit_cong:
"(!!j. j<i ==> r'(j) = r(j)) ==> rlimit(i,r) = rlimit(i,r')"
apply (rule refl iff_refl Collect_cong ex_cong conj_cong)+
done

subsection‹Transfinite Definition of the Wellordering on @{term "L"}›

definition
L_r :: "[i, i] => i" where
"L_r(f) == %i.
transrec3(i, 0, λx r. DPow_r(f, r, Lset(x)),
λx r. rlimit(x, λy. r`y))"

subsubsection‹The Corresponding Recursion Equations›
lemma [simp]: "L_r(f,0) = 0"

lemma [simp]: "L_r(f, succ(i)) = DPow_r(f, L_r(f,i), Lset(i))"

text‹The limit case is non-trivial because of the distinction between
object-level and meta-level abstraction.›
lemma [simp]: "Limit(i) ==> L_r(f,i) = rlimit(i, L_r(f))"
by (simp cong: rlimit_cong add: transrec3_Limit L_r_def ltD)

lemma (in Nat_Times_Nat) L_r_type:
"Ord(i) ==> L_r(fn,i) ⊆ Lset(i) * Lset(i)"
apply (induct i rule: trans_induct3)
apply (simp_all add: Lset_succ DPow_r_type well_ord_DPow_r rlimit_def
Transset_subset_DPow [OF Transset_Lset], blast)
done

lemma (in Nat_Times_Nat) well_ord_L_r:
"Ord(i) ==> well_ord(Lset(i), L_r(fn,i))"
apply (induct i rule: trans_induct3)
apply (simp_all add: well_ord0 Lset_succ L_r_type well_ord_DPow_r
well_ord_rlimit ltD)
done

lemma well_ord_L_r:
"Ord(i) ==> ∃r. well_ord(Lset(i), r)"
apply (insert nat_times_nat_lepoll_nat)
apply (unfold lepoll_def)
apply (blast intro: Nat_Times_Nat.well_ord_L_r Nat_Times_Nat.intro)
done

text‹Every constructible set is well-ordered! Therefore the Wellordering Theorem and
the Axiom of Choice hold in @{term L}!!›
theorem L_implies_AC: assumes x: "L(x)" shows "∃r. well_ord(x,r)"
using Transset_Lset x
apply (blast dest!: well_ord_L_r intro: well_ord_subset)
done

interpretation L?: M_basic L by (rule M_basic_L)

theorem "∀x[L]. ∃r. wellordered(L,x,r)"
proof
fix x
assume "L(x)"
then obtain r where "well_ord(x,r)"
by (blast dest: L_implies_AC)
thus "∃r. wellordered(L,x,r)"
by (blast intro: well_ord_imp_relativized)
qed

text‹In order to prove @{term" ∃r[L]. wellordered(L,x,r)"}, it's necessary to know
that @{term r} is actually constructible. It follows from the assumption ``@{term V} equals @{term L''},
but this reasoning doesn't appear to work in Isabelle.›

end
```