(* Title: ZF/Constructible/Rec_Separation.thy
ID: $Id$
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
*)
header {*Separation for Facts About Recursion*}
theory Rec_Separation = Separation + Internalize:
text{*This theory proves all instances needed for locales @{text
"M_trancl"} and @{text "M_datatypes"}*}
lemma eq_succ_imp_lt: "[|i = succ(j); Ord(i)|] ==> j<i"
by simp
subsection{*The Locale @{text "M_trancl"}*}
subsubsection{*Separation for Reflexive/Transitive Closure*}
text{*First, The Defining Formula*}
(* "rtran_closure_mem(M,A,r,p) ==
\<exists>nnat[M]. \<exists>n[M]. \<exists>n'[M].
omega(M,nnat) & n\<in>nnat & successor(M,n,n') &
(\<exists>f[M]. typed_function(M,n',A,f) &
(\<exists>x[M]. \<exists>y[M]. \<exists>zero[M]. pair(M,x,y,p) & empty(M,zero) &
fun_apply(M,f,zero,x) & fun_apply(M,f,n,y)) &
(\<forall>j[M]. j\<in>n -->
(\<exists>fj[M]. \<exists>sj[M]. \<exists>fsj[M]. \<exists>ffp[M].
fun_apply(M,f,j,fj) & successor(M,j,sj) &
fun_apply(M,f,sj,fsj) & pair(M,fj,fsj,ffp) & ffp \<in> r)))"*)
constdefs rtran_closure_mem_fm :: "[i,i,i]=>i"
"rtran_closure_mem_fm(A,r,p) ==
Exists(Exists(Exists(
And(omega_fm(2),
And(Member(1,2),
And(succ_fm(1,0),
Exists(And(typed_function_fm(1, A#+4, 0),
And(Exists(Exists(Exists(
And(pair_fm(2,1,p#+7),
And(empty_fm(0),
And(fun_apply_fm(3,0,2), fun_apply_fm(3,5,1))))))),
Forall(Implies(Member(0,3),
Exists(Exists(Exists(Exists(
And(fun_apply_fm(5,4,3),
And(succ_fm(4,2),
And(fun_apply_fm(5,2,1),
And(pair_fm(3,1,0), Member(0,r#+9))))))))))))))))))))"
lemma rtran_closure_mem_type [TC]:
"[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> rtran_closure_mem_fm(x,y,z) \<in> formula"
by (simp add: rtran_closure_mem_fm_def)
lemma arity_rtran_closure_mem_fm [simp]:
"[| x \<in> nat; y \<in> nat; z \<in> nat |]
==> arity(rtran_closure_mem_fm(x,y,z)) = succ(x) \<union> succ(y) \<union> succ(z)"
by (simp add: rtran_closure_mem_fm_def succ_Un_distrib [symmetric] Un_ac)
lemma sats_rtran_closure_mem_fm [simp]:
"[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
==> sats(A, rtran_closure_mem_fm(x,y,z), env) <->
rtran_closure_mem(**A, nth(x,env), nth(y,env), nth(z,env))"
by (simp add: rtran_closure_mem_fm_def rtran_closure_mem_def)
lemma rtran_closure_mem_iff_sats:
"[| 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)|]
==> rtran_closure_mem(**A, x, y, z) <-> sats(A, rtran_closure_mem_fm(i,j,k), env)"
by (simp add: sats_rtran_closure_mem_fm)
lemma rtran_closure_mem_reflection:
"REFLECTS[\<lambda>x. rtran_closure_mem(L,f(x),g(x),h(x)),
\<lambda>i x. rtran_closure_mem(**Lset(i),f(x),g(x),h(x))]"
apply (simp only: rtran_closure_mem_def setclass_simps)
apply (intro FOL_reflections function_reflections fun_plus_reflections)
done
text{*Separation for @{term "rtrancl(r)"}.*}
lemma rtrancl_separation:
"[| L(r); L(A) |] ==> separation (L, rtran_closure_mem(L,A,r))"
apply (rule gen_separation [OF rtran_closure_mem_reflection, of "{r,A}"], simp)
apply (drule mem_Lset_imp_subset_Lset, clarsimp)
apply (rule DPow_LsetI)
apply (rule_tac env = "[x,r,A]" in rtran_closure_mem_iff_sats)
apply (rule sep_rules | simp)+
done
subsubsection{*Reflexive/Transitive Closure, Internalized*}
(* "rtran_closure(M,r,s) ==
\<forall>A[M]. is_field(M,r,A) -->
(\<forall>p[M]. p \<in> s <-> rtran_closure_mem(M,A,r,p))" *)
constdefs rtran_closure_fm :: "[i,i]=>i"
"rtran_closure_fm(r,s) ==
Forall(Implies(field_fm(succ(r),0),
Forall(Iff(Member(0,succ(succ(s))),
rtran_closure_mem_fm(1,succ(succ(r)),0)))))"
lemma rtran_closure_type [TC]:
"[| x \<in> nat; y \<in> nat |] ==> rtran_closure_fm(x,y) \<in> formula"
by (simp add: rtran_closure_fm_def)
lemma arity_rtran_closure_fm [simp]:
"[| x \<in> nat; y \<in> nat |]
==> arity(rtran_closure_fm(x,y)) = succ(x) \<union> succ(y)"
by (simp add: rtran_closure_fm_def succ_Un_distrib [symmetric] Un_ac)
lemma sats_rtran_closure_fm [simp]:
"[| x \<in> nat; y \<in> nat; env \<in> list(A)|]
==> sats(A, rtran_closure_fm(x,y), env) <->
rtran_closure(**A, nth(x,env), nth(y,env))"
by (simp add: rtran_closure_fm_def rtran_closure_def)
lemma rtran_closure_iff_sats:
"[| nth(i,env) = x; nth(j,env) = y;
i \<in> nat; j \<in> nat; env \<in> list(A)|]
==> rtran_closure(**A, x, y) <-> sats(A, rtran_closure_fm(i,j), env)"
by simp
theorem rtran_closure_reflection:
"REFLECTS[\<lambda>x. rtran_closure(L,f(x),g(x)),
\<lambda>i x. rtran_closure(**Lset(i),f(x),g(x))]"
apply (simp only: rtran_closure_def setclass_simps)
apply (intro FOL_reflections function_reflections rtran_closure_mem_reflection)
done
subsubsection{*Transitive Closure of a Relation, Internalized*}
(* "tran_closure(M,r,t) ==
\<exists>s[M]. rtran_closure(M,r,s) & composition(M,r,s,t)" *)
constdefs tran_closure_fm :: "[i,i]=>i"
"tran_closure_fm(r,s) ==
Exists(And(rtran_closure_fm(succ(r),0), composition_fm(succ(r),0,succ(s))))"
lemma tran_closure_type [TC]:
"[| x \<in> nat; y \<in> nat |] ==> tran_closure_fm(x,y) \<in> formula"
by (simp add: tran_closure_fm_def)
lemma arity_tran_closure_fm [simp]:
"[| x \<in> nat; y \<in> nat |]
==> arity(tran_closure_fm(x,y)) = succ(x) \<union> succ(y)"
by (simp add: tran_closure_fm_def succ_Un_distrib [symmetric] Un_ac)
lemma sats_tran_closure_fm [simp]:
"[| x \<in> nat; y \<in> nat; env \<in> list(A)|]
==> sats(A, tran_closure_fm(x,y), env) <->
tran_closure(**A, nth(x,env), nth(y,env))"
by (simp add: tran_closure_fm_def tran_closure_def)
lemma tran_closure_iff_sats:
"[| nth(i,env) = x; nth(j,env) = y;
i \<in> nat; j \<in> nat; env \<in> list(A)|]
==> tran_closure(**A, x, y) <-> sats(A, tran_closure_fm(i,j), env)"
by simp
theorem tran_closure_reflection:
"REFLECTS[\<lambda>x. tran_closure(L,f(x),g(x)),
\<lambda>i x. tran_closure(**Lset(i),f(x),g(x))]"
apply (simp only: tran_closure_def setclass_simps)
apply (intro FOL_reflections function_reflections
rtran_closure_reflection composition_reflection)
done
subsubsection{*Separation for the Proof of @{text "wellfounded_on_trancl"}*}
lemma wellfounded_trancl_reflects:
"REFLECTS[\<lambda>x. \<exists>w[L]. \<exists>wx[L]. \<exists>rp[L].
w \<in> Z & pair(L,w,x,wx) & tran_closure(L,r,rp) & wx \<in> rp,
\<lambda>i x. \<exists>w \<in> Lset(i). \<exists>wx \<in> Lset(i). \<exists>rp \<in> Lset(i).
w \<in> Z & pair(**Lset(i),w,x,wx) & tran_closure(**Lset(i),r,rp) &
wx \<in> rp]"
by (intro FOL_reflections function_reflections fun_plus_reflections
tran_closure_reflection)
lemma wellfounded_trancl_separation:
"[| L(r); L(Z) |] ==>
separation (L, \<lambda>x.
\<exists>w[L]. \<exists>wx[L]. \<exists>rp[L].
w \<in> Z & pair(L,w,x,wx) & tran_closure(L,r,rp) & wx \<in> rp)"
apply (rule gen_separation [OF wellfounded_trancl_reflects, of "{r,Z}"], simp)
apply (drule mem_Lset_imp_subset_Lset, clarsimp)
apply (rule DPow_LsetI)
apply (rule bex_iff_sats conj_iff_sats)+
apply (rule_tac env = "[w,x,r,Z]" in mem_iff_sats)
apply (rule sep_rules tran_closure_iff_sats | simp)+
done
subsubsection{*Instantiating the locale @{text M_trancl}*}
lemma M_trancl_axioms_L: "M_trancl_axioms(L)"
apply (rule M_trancl_axioms.intro)
apply (assumption | rule rtrancl_separation wellfounded_trancl_separation)+
done
theorem M_trancl_L: "PROP M_trancl(L)"
by (rule M_trancl.intro
[OF M_trivial_L M_basic_axioms_L M_trancl_axioms_L])
lemmas iterates_abs = M_trancl.iterates_abs [OF M_trancl_L]
and rtran_closure_rtrancl = M_trancl.rtran_closure_rtrancl [OF M_trancl_L]
and rtrancl_closed = M_trancl.rtrancl_closed [OF M_trancl_L]
and rtrancl_abs = M_trancl.rtrancl_abs [OF M_trancl_L]
and trancl_closed = M_trancl.trancl_closed [OF M_trancl_L]
and trancl_abs = M_trancl.trancl_abs [OF M_trancl_L]
and wellfounded_on_trancl = M_trancl.wellfounded_on_trancl [OF M_trancl_L]
and wellfounded_trancl = M_trancl.wellfounded_trancl [OF M_trancl_L]
and wfrec_relativize = M_trancl.wfrec_relativize [OF M_trancl_L]
and trans_wfrec_relativize = M_trancl.trans_wfrec_relativize [OF M_trancl_L]
and trans_wfrec_abs = M_trancl.trans_wfrec_abs [OF M_trancl_L]
and trans_eq_pair_wfrec_iff = M_trancl.trans_eq_pair_wfrec_iff [OF M_trancl_L]
and eq_pair_wfrec_iff = M_trancl.eq_pair_wfrec_iff [OF M_trancl_L]
declare rtrancl_closed [intro,simp]
declare rtrancl_abs [simp]
declare trancl_closed [intro,simp]
declare trancl_abs [simp]
subsection{*@{term L} is Closed Under the Operator @{term list}*}
subsubsection{*Instances of Replacement for Lists*}
lemma list_replacement1_Reflects:
"REFLECTS
[\<lambda>x. \<exists>u[L]. u \<in> B \<and> (\<exists>y[L]. pair(L,u,y,x) \<and>
is_wfrec(L, iterates_MH(L, is_list_functor(L,A), 0), memsn, u, y)),
\<lambda>i x. \<exists>u \<in> Lset(i). u \<in> B \<and> (\<exists>y \<in> Lset(i). pair(**Lset(i), u, y, x) \<and>
is_wfrec(**Lset(i),
iterates_MH(**Lset(i),
is_list_functor(**Lset(i), A), 0), memsn, u, y))]"
by (intro FOL_reflections function_reflections is_wfrec_reflection
iterates_MH_reflection list_functor_reflection)
lemma list_replacement1:
"L(A) ==> iterates_replacement(L, is_list_functor(L,A), 0)"
apply (unfold iterates_replacement_def wfrec_replacement_def, clarify)
apply (rule strong_replacementI)
apply (rename_tac B)
apply (rule_tac u="{B,A,n,0,Memrel(succ(n))}"
in gen_separation [OF list_replacement1_Reflects],
simp add: nonempty)
apply (drule mem_Lset_imp_subset_Lset, clarsimp)
apply (rule DPow_LsetI)
apply (rule bex_iff_sats conj_iff_sats)+
apply (rule_tac env = "[u,x,A,n,B,0,Memrel(succ(n))]" in mem_iff_sats)
apply (rule sep_rules is_nat_case_iff_sats list_functor_iff_sats
is_wfrec_iff_sats iterates_MH_iff_sats quasinat_iff_sats | simp)+
done
lemma list_replacement2_Reflects:
"REFLECTS
[\<lambda>x. \<exists>u[L]. u \<in> B \<and> u \<in> nat \<and>
(\<exists>sn[L]. \<exists>msn[L]. successor(L, u, sn) \<and> membership(L, sn, msn) \<and>
is_wfrec (L, iterates_MH (L, is_list_functor(L, A), 0),
msn, u, x)),
\<lambda>i x. \<exists>u \<in> Lset(i). u \<in> B \<and> u \<in> nat \<and>
(\<exists>sn \<in> Lset(i). \<exists>msn \<in> Lset(i).
successor(**Lset(i), u, sn) \<and> membership(**Lset(i), sn, msn) \<and>
is_wfrec (**Lset(i),
iterates_MH (**Lset(i), is_list_functor(**Lset(i), A), 0),
msn, u, x))]"
by (intro FOL_reflections function_reflections is_wfrec_reflection
iterates_MH_reflection list_functor_reflection)
lemma list_replacement2:
"L(A) ==> strong_replacement(L,
\<lambda>n y. n\<in>nat &
(\<exists>sn[L]. \<exists>msn[L]. successor(L,n,sn) & membership(L,sn,msn) &
is_wfrec(L, iterates_MH(L,is_list_functor(L,A), 0),
msn, n, y)))"
apply (rule strong_replacementI)
apply (rename_tac B)
apply (rule_tac u="{A,B,0,nat}"
in gen_separation [OF list_replacement2_Reflects],
simp add: L_nat nonempty)
apply (drule mem_Lset_imp_subset_Lset, clarsimp)
apply (rule DPow_LsetI)
apply (rule bex_iff_sats conj_iff_sats)+
apply (rule_tac env = "[u,x,A,B,0,nat]" in mem_iff_sats)
apply (rule sep_rules is_nat_case_iff_sats list_functor_iff_sats
is_wfrec_iff_sats iterates_MH_iff_sats quasinat_iff_sats | simp)+
done
subsection{*@{term L} is Closed Under the Operator @{term formula}*}
subsubsection{*Instances of Replacement for Formulas*}
lemma formula_replacement1_Reflects:
"REFLECTS
[\<lambda>x. \<exists>u[L]. u \<in> B \<and> (\<exists>y[L]. pair(L,u,y,x) \<and>
is_wfrec(L, iterates_MH(L, is_formula_functor(L), 0), memsn, u, y)),
\<lambda>i x. \<exists>u \<in> Lset(i). u \<in> B \<and> (\<exists>y \<in> Lset(i). pair(**Lset(i), u, y, x) \<and>
is_wfrec(**Lset(i),
iterates_MH(**Lset(i),
is_formula_functor(**Lset(i)), 0), memsn, u, y))]"
by (intro FOL_reflections function_reflections is_wfrec_reflection
iterates_MH_reflection formula_functor_reflection)
lemma formula_replacement1:
"iterates_replacement(L, is_formula_functor(L), 0)"
apply (unfold iterates_replacement_def wfrec_replacement_def, clarify)
apply (rule strong_replacementI)
apply (rename_tac B)
apply (rule_tac u="{B,n,0,Memrel(succ(n))}"
in gen_separation [OF formula_replacement1_Reflects],
simp add: nonempty)
apply (drule mem_Lset_imp_subset_Lset, clarsimp)
apply (rule DPow_LsetI)
apply (rule bex_iff_sats conj_iff_sats)+
apply (rule_tac env = "[u,x,n,B,0,Memrel(succ(n))]" in mem_iff_sats)
apply (rule sep_rules is_nat_case_iff_sats formula_functor_iff_sats
is_wfrec_iff_sats iterates_MH_iff_sats quasinat_iff_sats | simp)+
done
lemma formula_replacement2_Reflects:
"REFLECTS
[\<lambda>x. \<exists>u[L]. u \<in> B \<and> u \<in> nat \<and>
(\<exists>sn[L]. \<exists>msn[L]. successor(L, u, sn) \<and> membership(L, sn, msn) \<and>
is_wfrec (L, iterates_MH (L, is_formula_functor(L), 0),
msn, u, x)),
\<lambda>i x. \<exists>u \<in> Lset(i). u \<in> B \<and> u \<in> nat \<and>
(\<exists>sn \<in> Lset(i). \<exists>msn \<in> Lset(i).
successor(**Lset(i), u, sn) \<and> membership(**Lset(i), sn, msn) \<and>
is_wfrec (**Lset(i),
iterates_MH (**Lset(i), is_formula_functor(**Lset(i)), 0),
msn, u, x))]"
by (intro FOL_reflections function_reflections is_wfrec_reflection
iterates_MH_reflection formula_functor_reflection)
lemma formula_replacement2:
"strong_replacement(L,
\<lambda>n y. n\<in>nat &
(\<exists>sn[L]. \<exists>msn[L]. successor(L,n,sn) & membership(L,sn,msn) &
is_wfrec(L, iterates_MH(L,is_formula_functor(L), 0),
msn, n, y)))"
apply (rule strong_replacementI)
apply (rename_tac B)
apply (rule_tac u="{B,0,nat}"
in gen_separation [OF formula_replacement2_Reflects],
simp add: nonempty L_nat)
apply (drule mem_Lset_imp_subset_Lset, clarsimp)
apply (rule DPow_LsetI)
apply (rule bex_iff_sats conj_iff_sats)+
apply (rule_tac env = "[u,x,B,0,nat]" in mem_iff_sats)
apply (rule sep_rules is_nat_case_iff_sats formula_functor_iff_sats
is_wfrec_iff_sats iterates_MH_iff_sats quasinat_iff_sats | simp)+
done
text{*NB The proofs for type @{term formula} are virtually identical to those
for @{term "list(A)"}. It was a cut-and-paste job! *}
subsubsection{*The Formula @{term is_nth}, Internalized*}
(* "is_nth(M,n,l,Z) ==
\<exists>X[M]. \<exists>sn[M]. \<exists>msn[M].
2 1 0
successor(M,n,sn) & membership(M,sn,msn) &
is_wfrec(M, iterates_MH(M, is_tl(M), l), msn, n, X) &
is_hd(M,X,Z)" *)
constdefs nth_fm :: "[i,i,i]=>i"
"nth_fm(n,l,Z) ==
Exists(Exists(Exists(
And(succ_fm(n#+3,1),
And(Memrel_fm(1,0),
And(is_wfrec_fm(iterates_MH_fm(tl_fm(1,0),l#+8,2,1,0), 0, n#+3, 2), hd_fm(2,Z#+3)))))))"
lemma nth_fm_type [TC]:
"[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> nth_fm(x,y,z) \<in> formula"
by (simp add: nth_fm_def)
lemma sats_nth_fm [simp]:
"[| x < length(env); y \<in> nat; z \<in> nat; env \<in> list(A)|]
==> sats(A, nth_fm(x,y,z), env) <->
is_nth(**A, nth(x,env), nth(y,env), nth(z,env))"
apply (frule lt_length_in_nat, assumption)
apply (simp add: nth_fm_def is_nth_def sats_is_wfrec_fm sats_iterates_MH_fm)
done
lemma nth_iff_sats:
"[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z;
i < length(env); j \<in> nat; k \<in> nat; env \<in> list(A)|]
==> is_nth(**A, x, y, z) <-> sats(A, nth_fm(i,j,k), env)"
by (simp add: sats_nth_fm)
theorem nth_reflection:
"REFLECTS[\<lambda>x. is_nth(L, f(x), g(x), h(x)),
\<lambda>i x. is_nth(**Lset(i), f(x), g(x), h(x))]"
apply (simp only: is_nth_def setclass_simps)
apply (intro FOL_reflections function_reflections is_wfrec_reflection
iterates_MH_reflection hd_reflection tl_reflection)
done
subsubsection{*An Instance of Replacement for @{term nth}*}
lemma nth_replacement_Reflects:
"REFLECTS
[\<lambda>x. \<exists>u[L]. u \<in> B \<and> (\<exists>y[L]. pair(L,u,y,x) \<and>
is_wfrec(L, iterates_MH(L, is_tl(L), z), memsn, u, y)),
\<lambda>i x. \<exists>u \<in> Lset(i). u \<in> B \<and> (\<exists>y \<in> Lset(i). pair(**Lset(i), u, y, x) \<and>
is_wfrec(**Lset(i),
iterates_MH(**Lset(i),
is_tl(**Lset(i)), z), memsn, u, y))]"
by (intro FOL_reflections function_reflections is_wfrec_reflection
iterates_MH_reflection list_functor_reflection tl_reflection)
lemma nth_replacement:
"L(w) ==> iterates_replacement(L, %l t. is_tl(L,l,t), w)"
apply (unfold iterates_replacement_def wfrec_replacement_def, clarify)
apply (rule strong_replacementI)
apply (rule_tac u="{A,n,w,Memrel(succ(n))}"
in gen_separation [OF nth_replacement_Reflects],
simp add: nonempty)
apply (drule mem_Lset_imp_subset_Lset, clarsimp)
apply (rule DPow_LsetI)
apply (rule bex_iff_sats conj_iff_sats)+
apply (rule_tac env = "[u,x,A,w,Memrel(succ(n))]" in mem_iff_sats)
apply (rule sep_rules is_nat_case_iff_sats tl_iff_sats
is_wfrec_iff_sats iterates_MH_iff_sats quasinat_iff_sats | simp)+
done
subsubsection{*Instantiating the locale @{text M_datatypes}*}
lemma M_datatypes_axioms_L: "M_datatypes_axioms(L)"
apply (rule M_datatypes_axioms.intro)
apply (assumption | rule
list_replacement1 list_replacement2
formula_replacement1 formula_replacement2
nth_replacement)+
done
theorem M_datatypes_L: "PROP M_datatypes(L)"
apply (rule M_datatypes.intro)
apply (rule M_trancl.axioms [OF M_trancl_L])+
apply (rule M_datatypes_axioms_L)
done
lemmas list_closed = M_datatypes.list_closed [OF M_datatypes_L]
and formula_closed = M_datatypes.formula_closed [OF M_datatypes_L]
and list_abs = M_datatypes.list_abs [OF M_datatypes_L]
and formula_abs = M_datatypes.formula_abs [OF M_datatypes_L]
and nth_abs = M_datatypes.nth_abs [OF M_datatypes_L]
declare list_closed [intro,simp]
declare formula_closed [intro,simp]
declare list_abs [simp]
declare formula_abs [simp]
declare nth_abs [simp]
subsection{*@{term L} is Closed Under the Operator @{term eclose}*}
subsubsection{*Instances of Replacement for @{term eclose}*}
lemma eclose_replacement1_Reflects:
"REFLECTS
[\<lambda>x. \<exists>u[L]. u \<in> B \<and> (\<exists>y[L]. pair(L,u,y,x) \<and>
is_wfrec(L, iterates_MH(L, big_union(L), A), memsn, u, y)),
\<lambda>i x. \<exists>u \<in> Lset(i). u \<in> B \<and> (\<exists>y \<in> Lset(i). pair(**Lset(i), u, y, x) \<and>
is_wfrec(**Lset(i),
iterates_MH(**Lset(i), big_union(**Lset(i)), A),
memsn, u, y))]"
by (intro FOL_reflections function_reflections is_wfrec_reflection
iterates_MH_reflection)
lemma eclose_replacement1:
"L(A) ==> iterates_replacement(L, big_union(L), A)"
apply (unfold iterates_replacement_def wfrec_replacement_def, clarify)
apply (rule strong_replacementI)
apply (rename_tac B)
apply (rule_tac u="{B,A,n,Memrel(succ(n))}"
in gen_separation [OF eclose_replacement1_Reflects], simp)
apply (drule mem_Lset_imp_subset_Lset, clarsimp)
apply (rule DPow_LsetI)
apply (rule bex_iff_sats conj_iff_sats)+
apply (rule_tac env = "[u,x,A,n,B,Memrel(succ(n))]" in mem_iff_sats)
apply (rule sep_rules iterates_MH_iff_sats is_nat_case_iff_sats
is_wfrec_iff_sats big_union_iff_sats quasinat_iff_sats | simp)+
done
lemma eclose_replacement2_Reflects:
"REFLECTS
[\<lambda>x. \<exists>u[L]. u \<in> B \<and> u \<in> nat \<and>
(\<exists>sn[L]. \<exists>msn[L]. successor(L, u, sn) \<and> membership(L, sn, msn) \<and>
is_wfrec (L, iterates_MH (L, big_union(L), A),
msn, u, x)),
\<lambda>i x. \<exists>u \<in> Lset(i). u \<in> B \<and> u \<in> nat \<and>
(\<exists>sn \<in> Lset(i). \<exists>msn \<in> Lset(i).
successor(**Lset(i), u, sn) \<and> membership(**Lset(i), sn, msn) \<and>
is_wfrec (**Lset(i),
iterates_MH (**Lset(i), big_union(**Lset(i)), A),
msn, u, x))]"
by (intro FOL_reflections function_reflections is_wfrec_reflection
iterates_MH_reflection)
lemma eclose_replacement2:
"L(A) ==> strong_replacement(L,
\<lambda>n y. n\<in>nat &
(\<exists>sn[L]. \<exists>msn[L]. successor(L,n,sn) & membership(L,sn,msn) &
is_wfrec(L, iterates_MH(L,big_union(L), A),
msn, n, y)))"
apply (rule strong_replacementI)
apply (rename_tac B)
apply (rule_tac u="{A,B,nat}"
in gen_separation [OF eclose_replacement2_Reflects], simp add: L_nat)
apply (drule mem_Lset_imp_subset_Lset, clarsimp)
apply (rule DPow_LsetI)
apply (rule bex_iff_sats conj_iff_sats)+
apply (rule_tac env = "[u,x,A,B,nat]" in mem_iff_sats)
apply (rule sep_rules is_nat_case_iff_sats iterates_MH_iff_sats
is_wfrec_iff_sats big_union_iff_sats quasinat_iff_sats | simp)+
done
subsubsection{*Instantiating the locale @{text M_eclose}*}
lemma M_eclose_axioms_L: "M_eclose_axioms(L)"
apply (rule M_eclose_axioms.intro)
apply (assumption | rule eclose_replacement1 eclose_replacement2)+
done
theorem M_eclose_L: "PROP M_eclose(L)"
apply (rule M_eclose.intro)
apply (rule M_datatypes.axioms [OF M_datatypes_L])+
apply (rule M_eclose_axioms_L)
done
lemmas eclose_closed [intro, simp] = M_eclose.eclose_closed [OF M_eclose_L]
and eclose_abs [intro, simp] = M_eclose.eclose_abs [OF M_eclose_L]
and transrec_replacementI = M_eclose.transrec_replacementI [OF M_eclose_L]
end