# Theory DPow_absolute

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

section ‹Absoluteness for the Definable Powerset Function›

theory DPow_absolute imports Satisfies_absolute begin

subsection‹Preliminary Internalizations›

subsubsection‹The Operator \<^term>‹is_formula_rec››

text‹The three arguments of \<^term>‹p› are always 2, 1, 0.  It is buried
within 11 quantifiers!!›

(* is_formula_rec :: "[i=>o, [i,i,i]=>o, i, i] => o"
"is_formula_rec(M,MH,p,z)  ==
∃dp[M]. ∃i[M]. ∃f[M]. finite_ordinal(M,dp) & is_depth(M,p,dp) &
2       1      0
successor(M,dp,i) & fun_apply(M,f,p,z) & is_transrec(M,MH,i,f)"
*)

definition
formula_rec_fm :: "[i, i, i]=>i" where
"formula_rec_fm(mh,p,z) ==
Exists(Exists(Exists(
And(finite_ordinal_fm(2),
And(depth_fm(p#+3,2),
And(succ_fm(2,1),
And(fun_apply_fm(0,p#+3,z#+3), is_transrec_fm(mh,1,0))))))))"

lemma is_formula_rec_type [TC]:
"[| p ∈ formula; x ∈ nat; z ∈ nat |]
==> formula_rec_fm(p,x,z) ∈ formula"

lemma sats_formula_rec_fm:
assumes MH_iff_sats:
"!!a0 a1 a2 a3 a4 a5 a6 a7 a8 a9 a10.
[|a0∈A; a1∈A; a2∈A; a3∈A; a4∈A; a5∈A; a6∈A; a7∈A; a8∈A; a9∈A; a10∈A|]
==> MH(a2, a1, a0) ⟷
sats(A, p, Cons(a0,Cons(a1,Cons(a2,Cons(a3,
Cons(a4,Cons(a5,Cons(a6,Cons(a7,
Cons(a8,Cons(a9,Cons(a10,env))))))))))))"
shows
"[|x ∈ nat; z ∈ nat; env ∈ list(A)|]
==> sats(A, formula_rec_fm(p,x,z), env) ⟷
is_formula_rec(##A, MH, nth(x,env), nth(z,env))"
by (simp add: formula_rec_fm_def sats_is_transrec_fm is_formula_rec_def
MH_iff_sats [THEN iff_sym])

lemma formula_rec_iff_sats:
assumes MH_iff_sats:
"!!a0 a1 a2 a3 a4 a5 a6 a7 a8 a9 a10.
[|a0∈A; a1∈A; a2∈A; a3∈A; a4∈A; a5∈A; a6∈A; a7∈A; a8∈A; a9∈A; a10∈A|]
==> MH(a2, a1, a0) ⟷
sats(A, p, Cons(a0,Cons(a1,Cons(a2,Cons(a3,
Cons(a4,Cons(a5,Cons(a6,Cons(a7,
Cons(a8,Cons(a9,Cons(a10,env))))))))))))"
shows
"[|nth(i,env) = x; nth(k,env) = z;
i ∈ nat; k ∈ nat; env ∈ list(A)|]
==> is_formula_rec(##A, MH, x, z) ⟷ sats(A, formula_rec_fm(p,i,k), env)"
by (simp add: sats_formula_rec_fm [OF MH_iff_sats])

theorem formula_rec_reflection:
assumes MH_reflection:
"!!f' f g h. REFLECTS[λx. MH(L, f'(x), f(x), g(x), h(x)),
λi x. MH(##Lset(i), f'(x), f(x), g(x), h(x))]"
shows "REFLECTS[λx. is_formula_rec(L, MH(L,x), f(x), h(x)),
λi x. is_formula_rec(##Lset(i), MH(##Lset(i),x), f(x), h(x))]"
apply (simp (no_asm_use) only: is_formula_rec_def)
apply (intro FOL_reflections function_reflections fun_plus_reflections
depth_reflection is_transrec_reflection MH_reflection)
done

subsubsection‹The Operator \<^term>‹is_satisfies››

(* is_satisfies(M,A,p,z) == is_formula_rec (M, satisfies_MH(M,A), p, z) *)
definition
satisfies_fm :: "[i,i,i]=>i" where
"satisfies_fm(x) == formula_rec_fm (satisfies_MH_fm(x#+5#+6, 2, 1, 0))"

lemma is_satisfies_type [TC]:
"[| x ∈ nat; y ∈ nat; z ∈ nat |] ==> satisfies_fm(x,y,z) ∈ formula"

lemma sats_satisfies_fm [simp]:
"[| x ∈ nat; y ∈ nat; z ∈ nat; env ∈ list(A)|]
==> sats(A, satisfies_fm(x,y,z), env) ⟷
is_satisfies(##A, nth(x,env), nth(y,env), nth(z,env))"
by (simp add: satisfies_fm_def is_satisfies_def sats_formula_rec_fm)

lemma satisfies_iff_sats:
"[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z;
i ∈ nat; j ∈ nat; k ∈ nat; env ∈ list(A)|]
==> is_satisfies(##A, x, y, z) ⟷ sats(A, satisfies_fm(i,j,k), env)"
by (simp)

theorem satisfies_reflection:
"REFLECTS[λx. is_satisfies(L,f(x),g(x),h(x)),
λi x. is_satisfies(##Lset(i),f(x),g(x),h(x))]"
apply (simp only: is_satisfies_def)
apply (intro formula_rec_reflection satisfies_MH_reflection)
done

subsection ‹Relativization of the Operator \<^term>‹DPow'››

lemma DPow'_eq:
"DPow'(A) = {z . ep ∈ list(A) * formula,
∃env ∈ list(A). ∃p ∈ formula.
ep = <env,p> & z = {x∈A. sats(A, p, Cons(x,env))}}"

text‹Relativize the use of \<^term>‹sats› within \<^term>‹DPow'›
(the comprehension).›
definition
is_DPow_sats :: "[i=>o,i,i,i,i] => o" where
"is_DPow_sats(M,A,env,p,x) ==
∀n1[M]. ∀e[M]. ∀sp[M].
is_satisfies(M,A,p,sp) ⟶ is_Cons(M,x,env,e) ⟶
fun_apply(M, sp, e, n1) ⟶ number1(M, n1)"

lemma (in M_satisfies) DPow_sats_abs:
"[| M(A); env ∈ list(A); p ∈ formula; M(x) |]
==> is_DPow_sats(M,A,env,p,x) ⟷ sats(A, p, Cons(x,env))"
apply (subgoal_tac "M(env)")
apply (simp add: is_DPow_sats_def satisfies_closed satisfies_abs)
apply (blast dest: transM)
done

lemma (in M_satisfies) Collect_DPow_sats_abs:
"[| M(A); env ∈ list(A); p ∈ formula |]
==> Collect(A, is_DPow_sats(M,A,env,p)) =
{x ∈ A. sats(A, p, Cons(x,env))}"
by (simp add: DPow_sats_abs transM [of _ A])

subsubsection‹The Operator \<^term>‹is_DPow_sats›, Internalized›

(* is_DPow_sats(M,A,env,p,x) ==
∀n1[M]. ∀e[M]. ∀sp[M].
is_satisfies(M,A,p,sp) ⟶ is_Cons(M,x,env,e) ⟶
fun_apply(M, sp, e, n1) ⟶ number1(M, n1) *)

definition
DPow_sats_fm :: "[i,i,i,i]=>i" where
"DPow_sats_fm(A,env,p,x) ==
Forall(Forall(Forall(
Implies(satisfies_fm(A#+3,p#+3,0),
Implies(Cons_fm(x#+3,env#+3,1),
Implies(fun_apply_fm(0,1,2), number1_fm(2)))))))"

lemma is_DPow_sats_type [TC]:
"[| A ∈ nat; x ∈ nat; y ∈ nat; z ∈ nat |]
==> DPow_sats_fm(A,x,y,z) ∈ formula"

lemma sats_DPow_sats_fm [simp]:
"[| u ∈ nat; x ∈ nat; y ∈ nat; z ∈ nat; env ∈ list(A)|]
==> sats(A, DPow_sats_fm(u,x,y,z), env) ⟷
is_DPow_sats(##A, nth(u,env), nth(x,env), nth(y,env), nth(z,env))"

lemma DPow_sats_iff_sats:
"[| nth(u,env) = nu; nth(x,env) = nx; nth(y,env) = ny; nth(z,env) = nz;
u ∈ nat; x ∈ nat; y ∈ nat; z ∈ nat; env ∈ list(A)|]
==> is_DPow_sats(##A,nu,nx,ny,nz) ⟷
sats(A, DPow_sats_fm(u,x,y,z), env)"
by simp

theorem DPow_sats_reflection:
"REFLECTS[λx. is_DPow_sats(L,f(x),g(x),h(x),g'(x)),
λi x. is_DPow_sats(##Lset(i),f(x),g(x),h(x),g'(x))]"
apply (unfold is_DPow_sats_def)
apply (intro FOL_reflections function_reflections extra_reflections
satisfies_reflection)
done

subsection‹A Locale for Relativizing the Operator \<^term>‹DPow'››

locale M_DPow = M_satisfies +
assumes sep:
"[| M(A); env ∈ list(A); p ∈ formula |]
==> separation(M, λx. is_DPow_sats(M,A,env,p,x))"
and rep:
"M(A)
==> strong_replacement (M,
λep z. ∃env[M]. ∃p[M]. mem_formula(M,p) & mem_list(M,A,env) &
pair(M,env,p,ep) &
is_Collect(M, A, λx. is_DPow_sats(M,A,env,p,x), z))"

lemma (in M_DPow) sep':
"[| M(A); env ∈ list(A); p ∈ formula |]
==> separation(M, λx. sats(A, p, Cons(x,env)))"
by (insert sep [of A env p], simp add: DPow_sats_abs)

lemma (in M_DPow) rep':
"M(A)
==> strong_replacement (M,
λep z. ∃env∈list(A). ∃p∈formula.
ep = <env,p> & z = {x ∈ A . sats(A, p, Cons(x, env))})"
by (insert rep [of A], simp add: Collect_DPow_sats_abs)

lemma univalent_pair_eq:
"univalent (M, A, λxy z. ∃x∈B. ∃y∈C. xy = ⟨x,y⟩ ∧ z = f(x,y))"

lemma (in M_DPow) DPow'_closed: "M(A) ==> M(DPow'(A))"
apply (fast intro: rep' sep' univalent_pair_eq)
done

text‹Relativization of the Operator \<^term>‹DPow'››
definition
is_DPow' :: "[i=>o,i,i] => o" where
"is_DPow'(M,A,Z) ==
∀X[M]. X ∈ Z ⟷
subset(M,X,A) &
(∃env[M]. ∃p[M]. mem_formula(M,p) & mem_list(M,A,env) &
is_Collect(M, A, is_DPow_sats(M,A,env,p), X))"

lemma (in M_DPow) DPow'_abs:
"[|M(A); M(Z)|] ==> is_DPow'(M,A,Z) ⟷ Z = DPow'(A)"
apply (rule iffI)
prefer 2 apply (simp add: is_DPow'_def DPow'_def Collect_DPow_sats_abs)
apply (rule M_equalityI)
apply (simp add: is_DPow'_def DPow'_def Collect_DPow_sats_abs, assumption)
apply (erule DPow'_closed)
done

subsection‹Instantiating the Locale ‹M_DPow››

subsubsection‹The Instance of Separation›

lemma DPow_separation:
"[| L(A); env ∈ list(A); p ∈ formula |]
==> separation(L, λx. is_DPow_sats(L,A,env,p,x))"
apply (rule gen_separation_multi [OF DPow_sats_reflection, of "{A,env,p}"],
auto intro: transL)
apply (rule_tac env="[A,env,p]" in DPow_LsetI)
apply (rule DPow_sats_iff_sats sep_rules | simp)+
done

subsubsection‹The Instance of Replacement›

lemma DPow_replacement_Reflects:
"REFLECTS [λx. ∃u[L]. u ∈ B &
(∃env[L]. ∃p[L].
mem_formula(L,p) & mem_list(L,A,env) & pair(L,env,p,u) &
is_Collect (L, A, is_DPow_sats(L,A,env,p), x)),
λi x. ∃u ∈ Lset(i). u ∈ B &
(∃env ∈ Lset(i). ∃p ∈ Lset(i).
mem_formula(##Lset(i),p) & mem_list(##Lset(i),A,env) &
pair(##Lset(i),env,p,u) &
is_Collect (##Lset(i), A, is_DPow_sats(##Lset(i),A,env,p), x))]"
apply (unfold is_Collect_def)
apply (intro FOL_reflections function_reflections mem_formula_reflection
mem_list_reflection DPow_sats_reflection)
done

lemma DPow_replacement:
"L(A)
==> strong_replacement (L,
λep z. ∃env[L]. ∃p[L]. mem_formula(L,p) & mem_list(L,A,env) &
pair(L,env,p,ep) &
is_Collect(L, A, λx. is_DPow_sats(L,A,env,p,x), z))"
apply (rule strong_replacementI)
apply (rule_tac u="{A,B}"
in gen_separation_multi [OF DPow_replacement_Reflects],
auto)
apply (unfold is_Collect_def)
apply (rule_tac env="[A,B]" in DPow_LsetI)
apply (rule sep_rules mem_formula_iff_sats mem_list_iff_sats
DPow_sats_iff_sats | simp)+
done

subsubsection‹Actually Instantiating the Locale›

lemma M_DPow_axioms_L: "M_DPow_axioms(L)"
apply (rule M_DPow_axioms.intro)
apply (assumption | rule DPow_separation DPow_replacement)+
done

theorem M_DPow_L: "M_DPow(L)"
apply (rule M_DPow.intro)
apply (rule M_satisfies_L)
apply (rule M_DPow_axioms_L)
done

lemmas DPow'_closed [intro, simp] = M_DPow.DPow'_closed [OF M_DPow_L]
and DPow'_abs [intro, simp] = M_DPow.DPow'_abs [OF M_DPow_L]

subsubsection‹The Operator \<^term>‹is_Collect››

text‹The formula \<^term>‹is_P› has one free variable, 0, and it is
enclosed within a single quantifier.›

(* is_Collect :: "[i=>o,i,i=>o,i] => o"
"is_Collect(M,A,P,z) == ∀x[M]. x ∈ z ⟷ x ∈ A & P(x)" *)

definition
Collect_fm :: "[i, i, i]=>i" where
"Collect_fm(A,is_P,z) ==
Forall(Iff(Member(0,succ(z)),
And(Member(0,succ(A)), is_P)))"

lemma is_Collect_type [TC]:
"[| is_P ∈ formula; x ∈ nat; y ∈ nat |]
==> Collect_fm(x,is_P,y) ∈ formula"

lemma sats_Collect_fm:
assumes is_P_iff_sats:
"!!a. a ∈ A ==> is_P(a) ⟷ sats(A, p, Cons(a, env))"
shows
"[|x ∈ nat; y ∈ nat; env ∈ list(A)|]
==> sats(A, Collect_fm(x,p,y), env) ⟷
is_Collect(##A, nth(x,env), is_P, nth(y,env))"
by (simp add: Collect_fm_def is_Collect_def is_P_iff_sats [THEN iff_sym])

lemma Collect_iff_sats:
assumes is_P_iff_sats:
"!!a. a ∈ A ==> is_P(a) ⟷ sats(A, p, Cons(a, env))"
shows
"[| nth(i,env) = x; nth(j,env) = y;
i ∈ nat; j ∈ nat; env ∈ list(A)|]
==> is_Collect(##A, x, is_P, y) ⟷ sats(A, Collect_fm(i,p,j), env)"
by (simp add: sats_Collect_fm [OF is_P_iff_sats])

which is essential for handling free variable references.›
theorem Collect_reflection:
assumes is_P_reflection:
"!!h f g. REFLECTS[λx. is_P(L, f(x), g(x)),
λi x. is_P(##Lset(i), f(x), g(x))]"
shows "REFLECTS[λx. is_Collect(L, f(x), is_P(L,x), g(x)),
λi x. is_Collect(##Lset(i), f(x), is_P(##Lset(i), x), g(x))]"
apply (simp (no_asm_use) only: is_Collect_def)
apply (intro FOL_reflections is_P_reflection)
done

subsubsection‹The Operator \<^term>‹is_Replace››

text‹BEWARE!  The formula \<^term>‹is_P› has free variables 0, 1
and not the usual 1, 0!  It is enclosed within two quantifiers.›

(*  is_Replace :: "[i=>o,i,[i,i]=>o,i] => o"
"is_Replace(M,A,P,z) == ∀u[M]. u ∈ z ⟷ (∃x[M]. x∈A & P(x,u))" *)

definition
Replace_fm :: "[i, i, i]=>i" where
"Replace_fm(A,is_P,z) ==
Forall(Iff(Member(0,succ(z)),
Exists(And(Member(0,A#+2), is_P))))"

lemma is_Replace_type [TC]:
"[| is_P ∈ formula; x ∈ nat; y ∈ nat |]
==> Replace_fm(x,is_P,y) ∈ formula"

lemma sats_Replace_fm:
assumes is_P_iff_sats:
"!!a b. [|a ∈ A; b ∈ A|]
==> is_P(a,b) ⟷ sats(A, p, Cons(a,Cons(b,env)))"
shows
"[|x ∈ nat; y ∈ nat; env ∈ list(A)|]
==> sats(A, Replace_fm(x,p,y), env) ⟷
is_Replace(##A, nth(x,env), is_P, nth(y,env))"
by (simp add: Replace_fm_def is_Replace_def is_P_iff_sats [THEN iff_sym])

lemma Replace_iff_sats:
assumes is_P_iff_sats:
"!!a b. [|a ∈ A; b ∈ A|]
==> is_P(a,b) ⟷ sats(A, p, Cons(a,Cons(b,env)))"
shows
"[| nth(i,env) = x; nth(j,env) = y;
i ∈ nat; j ∈ nat; env ∈ list(A)|]
==> is_Replace(##A, x, is_P, y) ⟷ sats(A, Replace_fm(i,p,j), env)"
by (simp add: sats_Replace_fm [OF is_P_iff_sats])

which is essential for handling free variable references.›
theorem Replace_reflection:
assumes is_P_reflection:
"!!h f g. REFLECTS[λx. is_P(L, f(x), g(x), h(x)),
λi x. is_P(##Lset(i), f(x), g(x), h(x))]"
shows "REFLECTS[λx. is_Replace(L, f(x), is_P(L,x), g(x)),
λi x. is_Replace(##Lset(i), f(x), is_P(##Lset(i), x), g(x))]"
apply (simp (no_asm_use) only: is_Replace_def)
apply (intro FOL_reflections is_P_reflection)
done

subsubsection‹The Operator \<^term>‹is_DPow'›, Internalized›

(*  "is_DPow'(M,A,Z) ==
∀X[M]. X ∈ Z ⟷
subset(M,X,A) &
(∃env[M]. ∃p[M]. mem_formula(M,p) & mem_list(M,A,env) &
is_Collect(M, A, is_DPow_sats(M,A,env,p), X))" *)

definition
DPow'_fm :: "[i,i]=>i" where
"DPow'_fm(A,Z) ==
Forall(
Iff(Member(0,succ(Z)),
And(subset_fm(0,succ(A)),
Exists(Exists(
And(mem_formula_fm(0),
And(mem_list_fm(A#+3,1),
Collect_fm(A#+3,
DPow_sats_fm(A#+4, 2, 1, 0), 2))))))))"

lemma is_DPow'_type [TC]:
"[| x ∈ nat; y ∈ nat |] ==> DPow'_fm(x,y) ∈ formula"

lemma sats_DPow'_fm [simp]:
"[| x ∈ nat; y ∈ nat; env ∈ list(A)|]
==> sats(A, DPow'_fm(x,y), env) ⟷
is_DPow'(##A, nth(x,env), nth(y,env))"
by (simp add: DPow'_fm_def is_DPow'_def sats_subset_fm' sats_Collect_fm)

lemma DPow'_iff_sats:
"[| nth(i,env) = x; nth(j,env) = y;
i ∈ nat; j ∈ nat; env ∈ list(A)|]
==> is_DPow'(##A, x, y) ⟷ sats(A, DPow'_fm(i,j), env)"
by (simp)

theorem DPow'_reflection:
"REFLECTS[λx. is_DPow'(L,f(x),g(x)),
λi x. is_DPow'(##Lset(i),f(x),g(x))]"
apply (simp only: is_DPow'_def)
apply (intro FOL_reflections function_reflections mem_formula_reflection
mem_list_reflection Collect_reflection DPow_sats_reflection)
done

subsection‹A Locale for Relativizing the Operator \<^term>‹Lset››

definition
transrec_body :: "[i=>o,i,i,i,i] => o" where
"transrec_body(M,g,x) ==
λy z. ∃gy[M]. y ∈ x & fun_apply(M,g,y,gy) & is_DPow'(M,gy,z)"

lemma (in M_DPow) transrec_body_abs:
"[|M(x); M(g); M(z)|]
==> transrec_body(M,g,x,y,z) ⟷ y ∈ x & z = DPow'(g`y)"
by (simp add: transrec_body_def DPow'_abs transM [of _ x])

locale M_Lset = M_DPow +
assumes strong_rep:
"[|M(x); M(g)|] ==> strong_replacement(M, λy z. transrec_body(M,g,x,y,z))"
and transrec_rep:
"M(i) ==> transrec_replacement(M, λx f u.
∃r[M]. is_Replace(M, x, transrec_body(M,f,x), r) &
big_union(M, r, u), i)"

lemma (in M_Lset) strong_rep':
"[|M(x); M(g)|]
==> strong_replacement(M, λy z. y ∈ x & z = DPow'(g`y))"
by (insert strong_rep [of x g], simp add: transrec_body_abs)

lemma (in M_Lset) DPow_apply_closed:
"[|M(f); M(x); y∈x|] ==> M(DPow'(f`y))"
by (blast intro: DPow'_closed dest: transM)

lemma (in M_Lset) RepFun_DPow_apply_closed:
"[|M(f); M(x)|] ==> M({DPow'(f`y). y∈x})"
by (blast intro: DPow_apply_closed RepFun_closed2 strong_rep')

lemma (in M_Lset) RepFun_DPow_abs:
"[|M(x); M(f); M(r) |]
==> is_Replace(M, x, λy z. transrec_body(M,f,x,y,z), r) ⟷
r =  {DPow'(f`y). y∈x}"
apply (rule iff_trans)
apply (rule Replace_abs)
done

lemma (in M_Lset) transrec_rep':
"M(i) ==> transrec_replacement(M, λx f u. u = (⋃y∈x. DPow'(f ` y)), i)"
apply (insert transrec_rep [of i])
transrec_replacement_def)
done

text‹Relativization of the Operator \<^term>‹Lset››

definition
is_Lset :: "[i=>o, i, i] => o" where
― ‹We can use the term language below because \<^term>‹is_Lset› will
not have to be internalized: it isn't used in any instance of
separation.›
"is_Lset(M,a,z) == is_transrec(M, %x f u. u = (⋃y∈x. DPow'(f`y)), a, z)"

lemma (in M_Lset) Lset_abs:
"[|Ord(i);  M(i);  M(z)|]
==> is_Lset(M,i,z) ⟷ z = Lset(i)"
apply (rule transrec_abs)
apply (simp_all add: transrec_rep' relation2_def RepFun_DPow_apply_closed)
done

lemma (in M_Lset) Lset_closed:
"[|Ord(i);  M(i)|] ==> M(Lset(i))"
apply (rule transrec_closed [OF transrec_rep'])
done

subsection‹Instantiating the Locale ‹M_Lset››

subsubsection‹The First Instance of Replacement›

lemma strong_rep_Reflects:
"REFLECTS [λu. ∃v[L]. v ∈ B & (∃gy[L].
v ∈ x & fun_apply(L,g,v,gy) & is_DPow'(L,gy,u)),
λi u. ∃v ∈ Lset(i). v ∈ B & (∃gy ∈ Lset(i).
v ∈ x & fun_apply(##Lset(i),g,v,gy) & is_DPow'(##Lset(i),gy,u))]"
by (intro FOL_reflections function_reflections DPow'_reflection)

lemma strong_rep:
"[|L(x); L(g)|] ==> strong_replacement(L, λy z. transrec_body(L,g,x,y,z))"
apply (unfold transrec_body_def)
apply (rule strong_replacementI)
apply (rule_tac u="{x,g,B}"
in gen_separation_multi [OF strong_rep_Reflects], auto)
apply (rule_tac env="[x,g,B]" in DPow_LsetI)
apply (rule sep_rules DPow'_iff_sats | simp)+
done

subsubsection‹The Second Instance of Replacement›

lemma transrec_rep_Reflects:
"REFLECTS [λx. ∃v[L]. v ∈ B &
(∃y[L]. pair(L,v,y,x) &
is_wfrec (L, λx f u. ∃r[L].
is_Replace (L, x, λy z.
∃gy[L]. y ∈ x & fun_apply(L,f,y,gy) &
is_DPow'(L,gy,z), r) & big_union(L,r,u), mr, v, y)),
λi x. ∃v ∈ Lset(i). v ∈ B &
(∃y ∈ Lset(i). pair(##Lset(i),v,y,x) &
is_wfrec (##Lset(i), λx f u. ∃r ∈ Lset(i).
is_Replace (##Lset(i), x, λy z.
∃gy ∈ Lset(i). y ∈ x & fun_apply(##Lset(i),f,y,gy) &
is_DPow'(##Lset(i),gy,z), r) &
big_union(##Lset(i),r,u), mr, v, y))]"
apply (simp only: rex_setclass_is_bex [symmetric])
― ‹Convert ‹∃y∈Lset(i)› to ‹∃y[##Lset(i)]› within the body
of the \<^term>‹is_wfrec› application.›
apply (intro FOL_reflections function_reflections
is_wfrec_reflection Replace_reflection DPow'_reflection)
done

lemma transrec_rep:
"[|L(j)|]
==> transrec_replacement(L, λx f u.
∃r[L]. is_Replace(L, x, transrec_body(L,f,x), r) &
big_union(L, r, u), j)"
apply (rule L.transrec_replacementI, assumption)
apply (unfold transrec_body_def)
apply (rule strong_replacementI)
apply (rule_tac u="{j,B,Memrel(eclose({j}))}"
in gen_separation_multi [OF transrec_rep_Reflects], auto)
apply (rule_tac env="[j,B,Memrel(eclose({j}))]" in DPow_LsetI)
apply (rule sep_rules is_wfrec_iff_sats Replace_iff_sats DPow'_iff_sats |
simp)+
done

subsubsection‹Actually Instantiating ‹M_Lset››

lemma M_Lset_axioms_L: "M_Lset_axioms(L)"
apply (rule M_Lset_axioms.intro)
apply (assumption | rule strong_rep transrec_rep)+
done

theorem M_Lset_L: "M_Lset(L)"
apply (rule M_Lset.intro)
apply (rule M_DPow_L)
apply (rule M_Lset_axioms_L)
done

text‹Finally: the point of the whole theory!›
lemmas Lset_closed = M_Lset.Lset_closed [OF M_Lset_L]
and Lset_abs = M_Lset.Lset_abs [OF M_Lset_L]

subsection‹The Notion of Constructible Set›

definition
constructible :: "[i=>o,i] => o" where
"constructible(M,x) ==
∃i[M]. ∃Li[M]. ordinal(M,i) & is_Lset(M,i,Li) & x ∈ Li"

theorem V_equals_L_in_L:
"L(x) ⟷ constructible(L,x)"
apply (simp add: constructible_def Lset_abs Lset_closed)