(* Title: ZF/Constructible/DPow_absolute.thy
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
*)
section \<open>Absoluteness for the Definable Powerset Function\<close>
theory DPow_absolute imports Satisfies_absolute begin
subsection\<open>Preliminary Internalizations\<close>
subsubsection\<open>The Operator \<^term>\<open>is_formula_rec\<close>\<close>
text\<open>The three arguments of \<^term>\<open>p\<close> are always 2, 1, 0. It is buried
within 11 quantifiers\<And>\<close>
(* is_formula_rec :: "[i\<Rightarrow>o, [i,i,i]\<Rightarrow>o, i, i] \<Rightarrow> o"
"is_formula_rec(M,MH,p,z) \<equiv>
\<exists>dp[M]. \<exists>i[M]. \<exists>f[M]. finite_ordinal(M,dp) \<and> is_depth(M,p,dp) \<and>
2 1 0
successor(M,dp,i) \<and> fun_apply(M,f,p,z) \<and> is_transrec(M,MH,i,f)"
*)
definition
formula_rec_fm :: "[i, i, i]\<Rightarrow>i" where
"formula_rec_fm(mh,p,z) \<equiv>
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]:
"\<lbrakk>p \<in> formula; x \<in> nat; z \<in> nat\<rbrakk>
\<Longrightarrow> formula_rec_fm(p,x,z) \<in> formula"
by (simp add: formula_rec_fm_def)
lemma sats_formula_rec_fm:
assumes MH_iff_sats:
"\<And>a0 a1 a2 a3 a4 a5 a6 a7 a8 a9 a10.
\<lbrakk>a0\<in>A; a1\<in>A; a2\<in>A; a3\<in>A; a4\<in>A; a5\<in>A; a6\<in>A; a7\<in>A; a8\<in>A; a9\<in>A; a10\<in>A\<rbrakk>
\<Longrightarrow> MH(a2, a1, a0) \<longleftrightarrow>
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
"\<lbrakk>x \<in> nat; z \<in> nat; env \<in> list(A)\<rbrakk>
\<Longrightarrow> sats(A, formula_rec_fm(p,x,z), env) \<longleftrightarrow>
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:
"\<And>a0 a1 a2 a3 a4 a5 a6 a7 a8 a9 a10.
\<lbrakk>a0\<in>A; a1\<in>A; a2\<in>A; a3\<in>A; a4\<in>A; a5\<in>A; a6\<in>A; a7\<in>A; a8\<in>A; a9\<in>A; a10\<in>A\<rbrakk>
\<Longrightarrow> MH(a2, a1, a0) \<longleftrightarrow>
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
"\<lbrakk>nth(i,env) = x; nth(k,env) = z;
i \<in> nat; k \<in> nat; env \<in> list(A)\<rbrakk>
\<Longrightarrow> is_formula_rec(##A, MH, x, z) \<longleftrightarrow> 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:
"\<And>f' f g h. REFLECTS[\<lambda>x. MH(L, f'(x), f(x), g(x), h(x)),
\<lambda>i x. MH(##Lset(i), f'(x), f(x), g(x), h(x))]"
shows "REFLECTS[\<lambda>x. is_formula_rec(L, MH(L,x), f(x), h(x)),
\<lambda>i 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\<open>The Operator \<^term>\<open>is_satisfies\<close>\<close>
(* is_satisfies(M,A,p,z) \<equiv> is_formula_rec (M, satisfies_MH(M,A), p, z) *)
definition
satisfies_fm :: "[i,i,i]\<Rightarrow>i" where
"satisfies_fm(x) \<equiv> formula_rec_fm (satisfies_MH_fm(x#+5#+6, 2, 1, 0))"
lemma is_satisfies_type [TC]:
"\<lbrakk>x \<in> nat; y \<in> nat; z \<in> nat\<rbrakk> \<Longrightarrow> satisfies_fm(x,y,z) \<in> formula"
by (simp add: satisfies_fm_def)
lemma sats_satisfies_fm [simp]:
"\<lbrakk>x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)\<rbrakk>
\<Longrightarrow> sats(A, satisfies_fm(x,y,z), env) \<longleftrightarrow>
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:
"\<lbrakk>nth(i,env) = x; nth(j,env) = y; nth(k,env) = z;
i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)\<rbrakk>
\<Longrightarrow> is_satisfies(##A, x, y, z) \<longleftrightarrow> sats(A, satisfies_fm(i,j,k), env)"
by (simp)
theorem satisfies_reflection:
"REFLECTS[\<lambda>x. is_satisfies(L,f(x),g(x),h(x)),
\<lambda>i 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 \<open>Relativization of the Operator \<^term>\<open>DPow'\<close>\<close>
lemma DPow'_eq:
"DPow'(A) = {z . ep \<in> list(A) * formula,
\<exists>env \<in> list(A). \<exists>p \<in> formula.
ep = \<langle>env,p\<rangle> \<and> z = {x\<in>A. sats(A, p, Cons(x,env))}}"
by (simp add: DPow'_def, blast)
text\<open>Relativize the use of \<^term>\<open>sats\<close> within \<^term>\<open>DPow'\<close>
(the comprehension).\<close>
definition
is_DPow_sats :: "[i\<Rightarrow>o,i,i,i,i] \<Rightarrow> o" where
"is_DPow_sats(M,A,env,p,x) \<equiv>
\<forall>n1[M]. \<forall>e[M]. \<forall>sp[M].
is_satisfies(M,A,p,sp) \<longrightarrow> is_Cons(M,x,env,e) \<longrightarrow>
fun_apply(M, sp, e, n1) \<longrightarrow> number1(M, n1)"
lemma (in M_satisfies) DPow_sats_abs:
"\<lbrakk>M(A); env \<in> list(A); p \<in> formula; M(x)\<rbrakk>
\<Longrightarrow> is_DPow_sats(M,A,env,p,x) \<longleftrightarrow> 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:
"\<lbrakk>M(A); env \<in> list(A); p \<in> formula\<rbrakk>
\<Longrightarrow> Collect(A, is_DPow_sats(M,A,env,p)) =
{x \<in> A. sats(A, p, Cons(x,env))}"
by (simp add: DPow_sats_abs transM [of _ A])
subsubsection\<open>The Operator \<^term>\<open>is_DPow_sats\<close>, Internalized\<close>
(* is_DPow_sats(M,A,env,p,x) \<equiv>
\<forall>n1[M]. \<forall>e[M]. \<forall>sp[M].
is_satisfies(M,A,p,sp) \<longrightarrow> is_Cons(M,x,env,e) \<longrightarrow>
fun_apply(M, sp, e, n1) \<longrightarrow> number1(M, n1) *)
definition
DPow_sats_fm :: "[i,i,i,i]\<Rightarrow>i" where
"DPow_sats_fm(A,env,p,x) \<equiv>
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]:
"\<lbrakk>A \<in> nat; x \<in> nat; y \<in> nat; z \<in> nat\<rbrakk>
\<Longrightarrow> DPow_sats_fm(A,x,y,z) \<in> formula"
by (simp add: DPow_sats_fm_def)
lemma sats_DPow_sats_fm [simp]:
"\<lbrakk>u \<in> nat; x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)\<rbrakk>
\<Longrightarrow> sats(A, DPow_sats_fm(u,x,y,z), env) \<longleftrightarrow>
is_DPow_sats(##A, nth(u,env), nth(x,env), nth(y,env), nth(z,env))"
by (simp add: DPow_sats_fm_def is_DPow_sats_def)
lemma DPow_sats_iff_sats:
"\<lbrakk>nth(u,env) = nu; nth(x,env) = nx; nth(y,env) = ny; nth(z,env) = nz;
u \<in> nat; x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)\<rbrakk>
\<Longrightarrow> is_DPow_sats(##A,nu,nx,ny,nz) \<longleftrightarrow>
sats(A, DPow_sats_fm(u,x,y,z), env)"
by simp
theorem DPow_sats_reflection:
"REFLECTS[\<lambda>x. is_DPow_sats(L,f(x),g(x),h(x),g'(x)),
\<lambda>i x. is_DPow_sats(##Lset(i),f(x),g(x),h(x),g'(x))]"
unfolding is_DPow_sats_def
apply (intro FOL_reflections function_reflections extra_reflections
satisfies_reflection)
done
subsection\<open>A Locale for Relativizing the Operator \<^term>\<open>DPow'\<close>\<close>
locale M_DPow = M_satisfies +
assumes sep:
"\<lbrakk>M(A); env \<in> list(A); p \<in> formula\<rbrakk>
\<Longrightarrow> separation(M, \<lambda>x. is_DPow_sats(M,A,env,p,x))"
and rep:
"M(A)
\<Longrightarrow> strong_replacement (M,
\<lambda>ep z. \<exists>env[M]. \<exists>p[M]. mem_formula(M,p) \<and> mem_list(M,A,env) \<and>
pair(M,env,p,ep) \<and>
is_Collect(M, A, \<lambda>x. is_DPow_sats(M,A,env,p,x), z))"
lemma (in M_DPow) sep':
"\<lbrakk>M(A); env \<in> list(A); p \<in> formula\<rbrakk>
\<Longrightarrow> separation(M, \<lambda>x. 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)
\<Longrightarrow> strong_replacement (M,
\<lambda>ep z. \<exists>env\<in>list(A). \<exists>p\<in>formula.
ep = \<langle>env,p\<rangle> \<and> z = {x \<in> 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, \<lambda>xy z. \<exists>x\<in>B. \<exists>y\<in>C. xy = \<langle>x,y\<rangle> \<and> z = f(x,y))"
by (simp add: univalent_def, blast)
lemma (in M_DPow) DPow'_closed: "M(A) \<Longrightarrow> M(DPow'(A))"
apply (simp add: DPow'_eq)
apply (fast intro: rep' sep' univalent_pair_eq)
done
text\<open>Relativization of the Operator \<^term>\<open>DPow'\<close>\<close>
definition
is_DPow' :: "[i\<Rightarrow>o,i,i] \<Rightarrow> o" where
"is_DPow'(M,A,Z) \<equiv>
\<forall>X[M]. X \<in> Z \<longleftrightarrow>
subset(M,X,A) \<and>
(\<exists>env[M]. \<exists>p[M]. mem_formula(M,p) \<and> mem_list(M,A,env) \<and>
is_Collect(M, A, is_DPow_sats(M,A,env,p), X))"
lemma (in M_DPow) DPow'_abs:
"\<lbrakk>M(A); M(Z)\<rbrakk> \<Longrightarrow> is_DPow'(M,A,Z) \<longleftrightarrow> 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\<open>Instantiating the Locale \<open>M_DPow\<close>\<close>
subsubsection\<open>The Instance of Separation\<close>
lemma DPow_separation:
"\<lbrakk>L(A); env \<in> list(A); p \<in> formula\<rbrakk>
\<Longrightarrow> separation(L, \<lambda>x. 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\<open>The Instance of Replacement\<close>
lemma DPow_replacement_Reflects:
"REFLECTS [\<lambda>x. \<exists>u[L]. u \<in> B \<and>
(\<exists>env[L]. \<exists>p[L].
mem_formula(L,p) \<and> mem_list(L,A,env) \<and> pair(L,env,p,u) \<and>
is_Collect (L, A, is_DPow_sats(L,A,env,p), x)),
\<lambda>i x. \<exists>u \<in> Lset(i). u \<in> B \<and>
(\<exists>env \<in> Lset(i). \<exists>p \<in> Lset(i).
mem_formula(##Lset(i),p) \<and> mem_list(##Lset(i),A,env) \<and>
pair(##Lset(i),env,p,u) \<and>
is_Collect (##Lset(i), A, is_DPow_sats(##Lset(i),A,env,p), x))]"
unfolding is_Collect_def
apply (intro FOL_reflections function_reflections mem_formula_reflection
mem_list_reflection DPow_sats_reflection)
done
lemma DPow_replacement:
"L(A)
\<Longrightarrow> strong_replacement (L,
\<lambda>ep z. \<exists>env[L]. \<exists>p[L]. mem_formula(L,p) \<and> mem_list(L,A,env) \<and>
pair(L,env,p,ep) \<and>
is_Collect(L, A, \<lambda>x. 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)
unfolding 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\<open>Actually Instantiating the Locale\<close>
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\<open>The Operator \<^term>\<open>is_Collect\<close>\<close>
text\<open>The formula \<^term>\<open>is_P\<close> has one free variable, 0, and it is
enclosed within a single quantifier.\<close>
(* is_Collect :: "[i\<Rightarrow>o,i,i\<Rightarrow>o,i] \<Rightarrow> o"
"is_Collect(M,A,P,z) \<equiv> \<forall>x[M]. x \<in> z \<longleftrightarrow> x \<in> A \<and> P(x)" *)
definition
Collect_fm :: "[i, i, i]\<Rightarrow>i" where
"Collect_fm(A,is_P,z) \<equiv>
Forall(Iff(Member(0,succ(z)),
And(Member(0,succ(A)), is_P)))"
lemma is_Collect_type [TC]:
"\<lbrakk>is_P \<in> formula; x \<in> nat; y \<in> nat\<rbrakk>
\<Longrightarrow> Collect_fm(x,is_P,y) \<in> formula"
by (simp add: Collect_fm_def)
lemma sats_Collect_fm:
assumes is_P_iff_sats:
"\<And>a. a \<in> A \<Longrightarrow> is_P(a) \<longleftrightarrow> sats(A, p, Cons(a, env))"
shows
"\<lbrakk>x \<in> nat; y \<in> nat; env \<in> list(A)\<rbrakk>
\<Longrightarrow> sats(A, Collect_fm(x,p,y), env) \<longleftrightarrow>
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:
"\<And>a. a \<in> A \<Longrightarrow> is_P(a) \<longleftrightarrow> sats(A, p, Cons(a, env))"
shows
"\<lbrakk>nth(i,env) = x; nth(j,env) = y;
i \<in> nat; j \<in> nat; env \<in> list(A)\<rbrakk>
\<Longrightarrow> is_Collect(##A, x, is_P, y) \<longleftrightarrow> sats(A, Collect_fm(i,p,j), env)"
by (simp add: sats_Collect_fm [OF is_P_iff_sats])
text\<open>The second argument of \<^term>\<open>is_P\<close> gives it direct access to \<^term>\<open>x\<close>,
which is essential for handling free variable references.\<close>
theorem Collect_reflection:
assumes is_P_reflection:
"\<And>h f g. REFLECTS[\<lambda>x. is_P(L, f(x), g(x)),
\<lambda>i x. is_P(##Lset(i), f(x), g(x))]"
shows "REFLECTS[\<lambda>x. is_Collect(L, f(x), is_P(L,x), g(x)),
\<lambda>i 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\<open>The Operator \<^term>\<open>is_Replace\<close>\<close>
text\<open>BEWARE! The formula \<^term>\<open>is_P\<close> has free variables 0, 1
and not the usual 1, 0! It is enclosed within two quantifiers.\<close>
(* is_Replace :: "[i\<Rightarrow>o,i,[i,i]\<Rightarrow>o,i] \<Rightarrow> o"
"is_Replace(M,A,P,z) \<equiv> \<forall>u[M]. u \<in> z \<longleftrightarrow> (\<exists>x[M]. x\<in>A \<and> P(x,u))" *)
definition
Replace_fm :: "[i, i, i]\<Rightarrow>i" where
"Replace_fm(A,is_P,z) \<equiv>
Forall(Iff(Member(0,succ(z)),
Exists(And(Member(0,A#+2), is_P))))"
lemma is_Replace_type [TC]:
"\<lbrakk>is_P \<in> formula; x \<in> nat; y \<in> nat\<rbrakk>
\<Longrightarrow> Replace_fm(x,is_P,y) \<in> formula"
by (simp add: Replace_fm_def)
lemma sats_Replace_fm:
assumes is_P_iff_sats:
"\<And>a b. \<lbrakk>a \<in> A; b \<in> A\<rbrakk>
\<Longrightarrow> is_P(a,b) \<longleftrightarrow> sats(A, p, Cons(a,Cons(b,env)))"
shows
"\<lbrakk>x \<in> nat; y \<in> nat; env \<in> list(A)\<rbrakk>
\<Longrightarrow> sats(A, Replace_fm(x,p,y), env) \<longleftrightarrow>
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:
"\<And>a b. \<lbrakk>a \<in> A; b \<in> A\<rbrakk>
\<Longrightarrow> is_P(a,b) \<longleftrightarrow> sats(A, p, Cons(a,Cons(b,env)))"
shows
"\<lbrakk>nth(i,env) = x; nth(j,env) = y;
i \<in> nat; j \<in> nat; env \<in> list(A)\<rbrakk>
\<Longrightarrow> is_Replace(##A, x, is_P, y) \<longleftrightarrow> sats(A, Replace_fm(i,p,j), env)"
by (simp add: sats_Replace_fm [OF is_P_iff_sats])
text\<open>The second argument of \<^term>\<open>is_P\<close> gives it direct access to \<^term>\<open>x\<close>,
which is essential for handling free variable references.\<close>
theorem Replace_reflection:
assumes is_P_reflection:
"\<And>h f g. REFLECTS[\<lambda>x. is_P(L, f(x), g(x), h(x)),
\<lambda>i x. is_P(##Lset(i), f(x), g(x), h(x))]"
shows "REFLECTS[\<lambda>x. is_Replace(L, f(x), is_P(L,x), g(x)),
\<lambda>i 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\<open>The Operator \<^term>\<open>is_DPow'\<close>, Internalized\<close>
(* "is_DPow'(M,A,Z) \<equiv>
\<forall>X[M]. X \<in> Z \<longleftrightarrow>
subset(M,X,A) \<and>
(\<exists>env[M]. \<exists>p[M]. mem_formula(M,p) \<and> mem_list(M,A,env) \<and>
is_Collect(M, A, is_DPow_sats(M,A,env,p), X))" *)
definition
DPow'_fm :: "[i,i]\<Rightarrow>i" where
"DPow'_fm(A,Z) \<equiv>
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]:
"\<lbrakk>x \<in> nat; y \<in> nat\<rbrakk> \<Longrightarrow> DPow'_fm(x,y) \<in> formula"
by (simp add: DPow'_fm_def)
lemma sats_DPow'_fm [simp]:
"\<lbrakk>x \<in> nat; y \<in> nat; env \<in> list(A)\<rbrakk>
\<Longrightarrow> sats(A, DPow'_fm(x,y), env) \<longleftrightarrow>
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:
"\<lbrakk>nth(i,env) = x; nth(j,env) = y;
i \<in> nat; j \<in> nat; env \<in> list(A)\<rbrakk>
\<Longrightarrow> is_DPow'(##A, x, y) \<longleftrightarrow> sats(A, DPow'_fm(i,j), env)"
by (simp)
theorem DPow'_reflection:
"REFLECTS[\<lambda>x. is_DPow'(L,f(x),g(x)),
\<lambda>i 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\<open>A Locale for Relativizing the Operator \<^term>\<open>Lset\<close>\<close>
definition
transrec_body :: "[i\<Rightarrow>o,i,i,i,i] \<Rightarrow> o" where
"transrec_body(M,g,x) \<equiv>
\<lambda>y z. \<exists>gy[M]. y \<in> x \<and> fun_apply(M,g,y,gy) \<and> is_DPow'(M,gy,z)"
lemma (in M_DPow) transrec_body_abs:
"\<lbrakk>M(x); M(g); M(z)\<rbrakk>
\<Longrightarrow> transrec_body(M,g,x,y,z) \<longleftrightarrow> y \<in> x \<and> z = DPow'(g`y)"
by (simp add: transrec_body_def DPow'_abs transM [of _ x])
locale M_Lset = M_DPow +
assumes strong_rep:
"\<lbrakk>M(x); M(g)\<rbrakk> \<Longrightarrow> strong_replacement(M, \<lambda>y z. transrec_body(M,g,x,y,z))"
and transrec_rep:
"M(i) \<Longrightarrow> transrec_replacement(M, \<lambda>x f u.
\<exists>r[M]. is_Replace(M, x, transrec_body(M,f,x), r) \<and>
big_union(M, r, u), i)"
lemma (in M_Lset) strong_rep':
"\<lbrakk>M(x); M(g)\<rbrakk>
\<Longrightarrow> strong_replacement(M, \<lambda>y z. y \<in> x \<and> z = DPow'(g`y))"
by (insert strong_rep [of x g], simp add: transrec_body_abs)
lemma (in M_Lset) DPow_apply_closed:
"\<lbrakk>M(f); M(x); y\<in>x\<rbrakk> \<Longrightarrow> M(DPow'(f`y))"
by (blast intro: DPow'_closed dest: transM)
lemma (in M_Lset) RepFun_DPow_apply_closed:
"\<lbrakk>M(f); M(x)\<rbrakk> \<Longrightarrow> M({DPow'(f`y). y\<in>x})"
by (blast intro: DPow_apply_closed RepFun_closed2 strong_rep')
lemma (in M_Lset) RepFun_DPow_abs:
"\<lbrakk>M(x); M(f); M(r)\<rbrakk>
\<Longrightarrow> is_Replace(M, x, \<lambda>y z. transrec_body(M,f,x,y,z), r) \<longleftrightarrow>
r = {DPow'(f`y). y\<in>x}"
apply (simp add: transrec_body_abs RepFun_def)
apply (rule iff_trans)
apply (rule Replace_abs)
apply (simp_all add: DPow_apply_closed strong_rep')
done
lemma (in M_Lset) transrec_rep':
"M(i) \<Longrightarrow> transrec_replacement(M, \<lambda>x f u. u = (\<Union>y\<in>x. DPow'(f ` y)), i)"
apply (insert transrec_rep [of i])
apply (simp add: RepFun_DPow_apply_closed RepFun_DPow_abs
transrec_replacement_def)
done
text\<open>Relativization of the Operator \<^term>\<open>Lset\<close>\<close>
definition
is_Lset :: "[i\<Rightarrow>o, i, i] \<Rightarrow> o" where
\<comment> \<open>We can use the term language below because \<^term>\<open>is_Lset\<close> will
not have to be internalized: it isn't used in any instance of
separation.\<close>
"is_Lset(M,a,z) \<equiv> is_transrec(M, \<lambda>x f u. u = (\<Union>y\<in>x. DPow'(f`y)), a, z)"
lemma (in M_Lset) Lset_abs:
"\<lbrakk>Ord(i); M(i); M(z)\<rbrakk>
\<Longrightarrow> is_Lset(M,i,z) \<longleftrightarrow> z = Lset(i)"
apply (simp add: is_Lset_def Lset_eq_transrec_DPow')
apply (rule transrec_abs)
apply (simp_all add: transrec_rep' relation2_def RepFun_DPow_apply_closed)
done
lemma (in M_Lset) Lset_closed:
"\<lbrakk>Ord(i); M(i)\<rbrakk> \<Longrightarrow> M(Lset(i))"
apply (simp add: Lset_eq_transrec_DPow')
apply (rule transrec_closed [OF transrec_rep'])
apply (simp_all add: relation2_def RepFun_DPow_apply_closed)
done
subsection\<open>Instantiating the Locale \<open>M_Lset\<close>\<close>
subsubsection\<open>The First Instance of Replacement\<close>
lemma strong_rep_Reflects:
"REFLECTS [\<lambda>u. \<exists>v[L]. v \<in> B \<and> (\<exists>gy[L].
v \<in> x \<and> fun_apply(L,g,v,gy) \<and> is_DPow'(L,gy,u)),
\<lambda>i u. \<exists>v \<in> Lset(i). v \<in> B \<and> (\<exists>gy \<in> Lset(i).
v \<in> x \<and> fun_apply(##Lset(i),g,v,gy) \<and> is_DPow'(##Lset(i),gy,u))]"
by (intro FOL_reflections function_reflections DPow'_reflection)
lemma strong_rep:
"\<lbrakk>L(x); L(g)\<rbrakk> \<Longrightarrow> strong_replacement(L, \<lambda>y z. transrec_body(L,g,x,y,z))"
unfolding 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\<open>The Second Instance of Replacement\<close>
lemma transrec_rep_Reflects:
"REFLECTS [\<lambda>x. \<exists>v[L]. v \<in> B \<and>
(\<exists>y[L]. pair(L,v,y,x) \<and>
is_wfrec (L, \<lambda>x f u. \<exists>r[L].
is_Replace (L, x, \<lambda>y z.
\<exists>gy[L]. y \<in> x \<and> fun_apply(L,f,y,gy) \<and>
is_DPow'(L,gy,z), r) \<and> big_union(L,r,u), mr, v, y)),
\<lambda>i x. \<exists>v \<in> Lset(i). v \<in> B \<and>
(\<exists>y \<in> Lset(i). pair(##Lset(i),v,y,x) \<and>
is_wfrec (##Lset(i), \<lambda>x f u. \<exists>r \<in> Lset(i).
is_Replace (##Lset(i), x, \<lambda>y z.
\<exists>gy \<in> Lset(i). y \<in> x \<and> fun_apply(##Lset(i),f,y,gy) \<and>
is_DPow'(##Lset(i),gy,z), r) \<and>
big_union(##Lset(i),r,u), mr, v, y))]"
apply (simp only: rex_setclass_is_bex [symmetric])
\<comment> \<open>Convert \<open>\<exists>y\<in>Lset(i)\<close> to \<open>\<exists>y[##Lset(i)]\<close> within the body
of the \<^term>\<open>is_wfrec\<close> application.\<close>
apply (intro FOL_reflections function_reflections
is_wfrec_reflection Replace_reflection DPow'_reflection)
done
lemma transrec_rep:
"\<lbrakk>L(j)\<rbrakk>
\<Longrightarrow> transrec_replacement(L, \<lambda>x f u.
\<exists>r[L]. is_Replace(L, x, transrec_body(L,f,x), r) \<and>
big_union(L, r, u), j)"
apply (rule L.transrec_replacementI, assumption)
unfolding 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\<open>Actually Instantiating \<open>M_Lset\<close>\<close>
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\<open>Finally: the point of the whole theory!\<close>
lemmas Lset_closed = M_Lset.Lset_closed [OF M_Lset_L]
and Lset_abs = M_Lset.Lset_abs [OF M_Lset_L]
subsection\<open>The Notion of Constructible Set\<close>
definition
constructible :: "[i\<Rightarrow>o,i] \<Rightarrow> o" where
"constructible(M,x) \<equiv>
\<exists>i[M]. \<exists>Li[M]. ordinal(M,i) \<and> is_Lset(M,i,Li) \<and> x \<in> Li"
theorem V_equals_L_in_L:
"L(x) \<longleftrightarrow> constructible(L,x)"
apply (simp add: constructible_def Lset_abs Lset_closed)
apply (simp add: L_def)
apply (blast intro: Ord_in_L)
done
end