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(* Title: ZF/Constructible/Rec_Separation.thy
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
*)
section \<open>Separation for Facts About Recursion\<close>
theory Rec_Separation imports Separation Internalize begin
text\<open>This theory proves all instances needed for locales \<open>M_trancl\<close> and \<open>M_datatypes\<close>\<close>
lemma eq_succ_imp_lt: "[|i = succ(j); Ord(i)|] ==> j
by simp
subsection\<open>The Locale \<open>M_trancl\<close>\<close>
subsubsection\<open>Separation for Reflexive/Transitive Closure\<close>
text\<open>First, The Defining Formula\<close>
(* "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 \<longrightarrow>
(\<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)))"*)
definition
rtran_closure_mem_fm :: "[i,i,i]=>i" where
"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 \ nat; y \ nat; z \ nat |] ==> rtran_closure_mem_fm(x,y,z) \ formula"
by (simp add: rtran_closure_mem_fm_def)
lemma sats_rtran_closure_mem_fm [simp]:
"[| x \ nat; y \ nat; z \ nat; env \ list(A)|]
==> sats(A, rtran_closure_mem_fm(x,y,z), env) \<longleftrightarrow>
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) \<longleftrightarrow> sats(A, rtran_closure_mem_fm(i,j,k), env)"
by (simp)
lemma rtran_closure_mem_reflection:
"REFLECTS[\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)
apply (intro FOL_reflections function_reflections fun_plus_reflections)
done
text\<open>Separation for \<^term>\<open>rtrancl(r)\<close>.\<close>
lemma rtrancl_separation:
"[| L(r); L(A) |] ==> separation (L, rtran_closure_mem(L,A,r))"
apply (rule gen_separation_multi [OF rtran_closure_mem_reflection, of "{r,A}"],
auto)
apply (rule_tac env="[r,A]" in DPow_LsetI)
apply (rule rtran_closure_mem_iff_sats sep_rules | simp)+
done
subsubsection\<open>Reflexive/Transitive Closure, Internalized\<close>
(* "rtran_closure(M,r,s) ==
\<forall>A[M]. is_field(M,r,A) \<longrightarrow>
(\<forall>p[M]. p \<in> s \<longleftrightarrow> rtran_closure_mem(M,A,r,p))" *)
definition
rtran_closure_fm :: "[i,i]=>i" where
"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 \ nat; y \ nat |] ==> rtran_closure_fm(x,y) \ formula"
by (simp add: rtran_closure_fm_def)
lemma sats_rtran_closure_fm [simp]:
"[| x \ nat; y \ nat; env \ list(A)|]
==> sats(A, rtran_closure_fm(x,y), env) \<longleftrightarrow>
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) \<longleftrightarrow> sats(A, rtran_closure_fm(i,j), env)"
by simp
theorem rtran_closure_reflection:
"REFLECTS[\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)
apply (intro FOL_reflections function_reflections rtran_closure_mem_reflection)
done
subsubsection\<open>Transitive Closure of a Relation, Internalized\<close>
(* "tran_closure(M,r,t) ==
\<exists>s[M]. rtran_closure(M,r,s) & composition(M,r,s,t)" *)
definition
tran_closure_fm :: "[i,i]=>i" where
"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 \ nat; y \ nat |] ==> tran_closure_fm(x,y) \ formula"
by (simp add: tran_closure_fm_def)
lemma sats_tran_closure_fm [simp]:
"[| x \ nat; y \ nat; env \ list(A)|]
==> sats(A, tran_closure_fm(x,y), env) \<longleftrightarrow>
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) \<longleftrightarrow> sats(A, tran_closure_fm(i,j), env)"
by simp
theorem tran_closure_reflection:
"REFLECTS[\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)
apply (intro FOL_reflections function_reflections
rtran_closure_reflection composition_reflection)
done
subsubsection\<open>Separation for the Proof of \<open>wellfounded_on_trancl\<close>\<close>
lemma wellfounded_trancl_reflects:
"REFLECTS[\x. \w[L]. \wx[L]. \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_multi [OF wellfounded_trancl_reflects, of "{r,Z}"],
auto)
apply (rule_tac env="[r,Z]" in DPow_LsetI)
apply (rule sep_rules tran_closure_iff_sats | simp)+
done
subsubsection\<open>Instantiating the locale \<open>M_trancl\<close>\<close>
lemma M_trancl_axioms_L: "M_trancl_axioms(L)"
apply (rule M_trancl_axioms.intro)
apply (assumption | rule rtrancl_separation wellfounded_trancl_separation L_nat)+
done
theorem M_trancl_L: "M_trancl(L)"
by (rule M_trancl.intro [OF M_basic_L M_trancl_axioms_L])
interpretation L: M_trancl L by (rule M_trancl_L)
subsection\<open>\<^term>\<open>L\<close> is Closed Under the Operator \<^term>\<open>list\<close>\<close>
subsubsection\<open>Instances of Replacement for Lists\<close>
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 (rule_tac u="{B,A,n,0,Memrel(succ(n))}"
in gen_separation_multi [OF list_replacement1_Reflects],
auto)
apply (rule_tac env="[B,A,n,0,Memrel(succ(n))]" in DPow_LsetI)
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 & u \<in> nat &
is_iterates(L, is_list_functor(L, A), 0, u, x),
\<lambda>i x. \<exists>u \<in> Lset(i). u \<in> B & u \<in> nat &
is_iterates(##Lset(i), is_list_functor(##Lset(i), A), 0, u, x)]"
by (intro FOL_reflections
is_iterates_reflection list_functor_reflection)
lemma list_replacement2:
"L(A) ==> strong_replacement(L,
\<lambda>n y. n\<in>nat & is_iterates(L, is_list_functor(L,A), 0, n, y))"
apply (rule strong_replacementI)
apply (rule_tac u="{A,B,0,nat}"
in gen_separation_multi [OF list_replacement2_Reflects],
auto)
apply (rule_tac env="[A,B,0,nat]" in DPow_LsetI)
apply (rule sep_rules list_functor_iff_sats is_iterates_iff_sats | simp)+
done
subsection\<open>\<^term>\<open>L\<close> is Closed Under the Operator \<^term>\<open>formula\<close>\<close>
subsubsection\<open>Instances of Replacement for Formulas\<close>
(*FIXME: could prove a lemma iterates_replacementI to eliminate the
need to expand iterates_replacement and wfrec_replacement*)
lemma formula_replacement1_Reflects:
"REFLECTS
[\<lambda>x. \<exists>u[L]. u \<in> B & (\<exists>y[L]. pair(L,u,y,x) &
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 & (\<exists>y \<in> Lset(i). pair(##Lset(i), u, y, x) &</span>
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 (rule_tac u="{B,n,0,Memrel(succ(n))}"
in gen_separation_multi [OF formula_replacement1_Reflects],
auto)
apply (rule_tac env="[n,B,0,Memrel(succ(n))]" in DPow_LsetI)
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 & u \<in> nat &
is_iterates(L, is_formula_functor(L), 0, u, x),
\<lambda>i x. \<exists>u \<in> Lset(i). u \<in> B & u \<in> nat &
is_iterates(##Lset(i), is_formula_functor(##Lset(i)), 0, u, x)]"
by (intro FOL_reflections
is_iterates_reflection formula_functor_reflection)
lemma formula_replacement2:
"strong_replacement(L,
\<lambda>n y. n\<in>nat & is_iterates(L, is_formula_functor(L), 0, n, y))"
apply (rule strong_replacementI)
apply (rule_tac u="{B,0,nat}"
in gen_separation_multi [OF formula_replacement2_Reflects],
auto)
apply (rule_tac env="[B,0,nat]" in DPow_LsetI)
apply (rule sep_rules formula_functor_iff_sats is_iterates_iff_sats | simp)+
done
text\<open>NB The proofs for type \<^term>\<open>formula\<close> are virtually identical to those
for \<^term>\<open>list(A)\<close>. It was a cut-and-paste job!\<close>
subsubsection\<open>The Formula \<^term>\<open>is_nth\<close>, Internalized\<close>
(* "is_nth(M,n,l,Z) ==
\<exists>X[M]. is_iterates(M, is_tl(M), l, n, X) & is_hd(M,X,Z)" *)
definition
nth_fm :: "[i,i,i]=>i" where
"nth_fm(n,l,Z) ==
Exists(And(is_iterates_fm(tl_fm(1,0), succ(l), succ(n), 0),
hd_fm(0,succ(Z))))"
lemma nth_fm_type [TC]:
"[| x \ nat; y \ nat; z \ nat |] ==> nth_fm(x,y,z) \ formula"
by (simp add: nth_fm_def)
lemma sats_nth_fm [simp]:
"[| x < length(env); y \ nat; z \ nat; env \ list(A)|]
==> sats(A, nth_fm(x,y,z), env) \<longleftrightarrow>
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_iterates_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) \<longleftrightarrow> sats(A, nth_fm(i,j,k), env)"
by (simp)
theorem nth_reflection:
"REFLECTS[\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)
apply (intro FOL_reflections is_iterates_reflection
hd_reflection tl_reflection)
done
subsubsection\<open>An Instance of Replacement for \<^term>\<open>nth\<close>\<close>
(*FIXME: could prove a lemma iterates_replacementI to eliminate the
need to expand iterates_replacement and wfrec_replacement*)
lemma nth_replacement_Reflects:
"REFLECTS
[\<lambda>x. \<exists>u[L]. u \<in> B & (\<exists>y[L]. pair(L,u,y,x) &
is_wfrec(L, iterates_MH(L, is_tl(L), z), memsn, u, y)),
\<lambda>i x. \<exists>u \<in> Lset(i). u \<in> B & (\<exists>y \<in> Lset(i). pair(##Lset(i), u, y, x) &</span>
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 tl_reflection)
lemma nth_replacement:
"L(w) ==> iterates_replacement(L, is_tl(L), w)"
apply (unfold iterates_replacement_def wfrec_replacement_def, clarify)
apply (rule strong_replacementI)
apply (rule_tac u="{B,w,Memrel(succ(n))}"
in gen_separation_multi [OF nth_replacement_Reflects],
auto)
apply (rule_tac env="[B,w,Memrel(succ(n))]" in DPow_LsetI)
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\<open>Instantiating the locale \<open>M_datatypes\<close>\<close>
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: "M_datatypes(L)"
apply (rule M_datatypes.intro)
apply (rule M_trancl_L)
apply (rule M_datatypes_axioms_L)
done
interpretation L: M_datatypes L by (rule M_datatypes_L)
subsection\<open>\<^term>\<open>L\<close> is Closed Under the Operator \<^term>\<open>eclose\<close>\<close>
subsubsection\<open>Instances of Replacement for \<^term>\<open>eclose\<close>\<close>
lemma eclose_replacement1_Reflects:
"REFLECTS
[\<lambda>x. \<exists>u[L]. u \<in> B & (\<exists>y[L]. pair(L,u,y,x) &
is_wfrec(L, iterates_MH(L, big_union(L), A), memsn, u, y)),
\<lambda>i x. \<exists>u \<in> Lset(i). u \<in> B & (\<exists>y \<in> Lset(i). pair(##Lset(i), u, y, x) &</span>
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 (rule_tac u="{B,A,n,Memrel(succ(n))}"
in gen_separation_multi [OF eclose_replacement1_Reflects], auto)
apply (rule_tac env="[B,A,n,Memrel(succ(n))]" in DPow_LsetI)
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 & u \<in> nat &
is_iterates(L, big_union(L), A, u, x),
\<lambda>i x. \<exists>u \<in> Lset(i). u \<in> B & u \<in> nat &
is_iterates(##Lset(i), big_union(##Lset(i)), A, u, x)]"
by (intro FOL_reflections function_reflections is_iterates_reflection)
lemma eclose_replacement2:
"L(A) ==> strong_replacement(L,
\<lambda>n y. n\<in>nat & is_iterates(L, big_union(L), A, n, y))"
apply (rule strong_replacementI)
apply (rule_tac u="{A,B,nat}"
in gen_separation_multi [OF eclose_replacement2_Reflects],
auto)
apply (rule_tac env="[A,B,nat]" in DPow_LsetI)
apply (rule sep_rules is_iterates_iff_sats big_union_iff_sats | simp)+
done
subsubsection\<open>Instantiating the locale \<open>M_eclose\<close>\<close>
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: "M_eclose(L)"
apply (rule M_eclose.intro)
apply (rule M_datatypes_L)
apply (rule M_eclose_axioms_L)
done
interpretation L: M_eclose L by (rule M_eclose_L)
end
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