# Theory Rec_Separation

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

section ‹Separation for Facts About Recursion›

theory Rec_Separation imports Separation Internalize begin

text‹This theory proves all instances needed for locales ‹M_trancl› and ‹M_datatypes››

lemma eq_succ_imp_lt: "[|i = succ(j); Ord(i)|] ==> j<i"
by simp

subsection‹The Locale ‹M_trancl››

subsubsection‹Separation for Reflexive/Transitive Closure›

text‹First, The Defining Formula›

(* "rtran_closure_mem(M,A,r,p) ==
∃nnat[M]. ∃n[M]. ∃n'[M].
omega(M,nnat) & n∈nnat & successor(M,n,n') &
(∃f[M]. typed_function(M,n',A,f) &
(∃x[M]. ∃y[M]. ∃zero[M]. pair(M,x,y,p) & empty(M,zero) &
fun_apply(M,f,zero,x) & fun_apply(M,f,n,y)) &
(∀j[M]. j∈n ⟶
(∃fj[M]. ∃sj[M]. ∃fsj[M]. ∃ffp[M].
fun_apply(M,f,j,fj) & successor(M,j,sj) &
fun_apply(M,f,sj,fsj) & pair(M,fj,fsj,ffp) & ffp ∈ 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) ⟷
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 ∈ nat; j ∈ nat; k ∈ nat; env ∈ 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[λx. rtran_closure_mem(L,f(x),g(x),h(x)),
λi 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‹Separation for @{term "rtrancl(r)"}.›
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‹Reflexive/Transitive Closure, Internalized›

(*  "rtran_closure(M,r,s) ==
∀A[M]. is_field(M,r,A) ⟶
(∀p[M]. p ∈ s ⟷ 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) ⟷
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 ∈ nat; j ∈ nat; env ∈ list(A)|]
==> rtran_closure(##A, x, y) ⟷ sats(A, rtran_closure_fm(i,j), env)"
by simp

theorem rtran_closure_reflection:
"REFLECTS[λx. rtran_closure(L,f(x),g(x)),
λi 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‹Transitive Closure of a Relation, Internalized›

(*  "tran_closure(M,r,t) ==
∃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) ⟷
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 ∈ nat; j ∈ nat; env ∈ list(A)|]
==> tran_closure(##A, x, y) ⟷ sats(A, tran_closure_fm(i,j), env)"
by simp

theorem tran_closure_reflection:
"REFLECTS[λx. tran_closure(L,f(x),g(x)),
λi 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‹Separation for the Proof of ‹wellfounded_on_trancl››

lemma wellfounded_trancl_reflects:
"REFLECTS[λx. ∃w[L]. ∃wx[L]. ∃rp[L].
w ∈ Z & pair(L,w,x,wx) & tran_closure(L,r,rp) & wx ∈ rp,
λi x. ∃w ∈ Lset(i). ∃wx ∈ Lset(i). ∃rp ∈ Lset(i).
w ∈ Z & pair(##Lset(i),w,x,wx) & tran_closure(##Lset(i),r,rp) &
wx ∈ rp]"
by (intro FOL_reflections function_reflections fun_plus_reflections
tran_closure_reflection)

lemma wellfounded_trancl_separation:
"[| L(r); L(Z) |] ==>
separation (L, λx.
∃w[L]. ∃wx[L]. ∃rp[L].
w ∈ Z & pair(L,w,x,wx) & tran_closure(L,r,rp) & wx ∈ 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‹Instantiating the locale ‹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_basic_L M_trancl_axioms_L])

interpretation L?: M_trancl L by (rule M_trancl_L)

subsection‹@{term L} is Closed Under the Operator @{term list}›

subsubsection‹Instances of Replacement for Lists›

lemma list_replacement1_Reflects:
"REFLECTS
[λx. ∃u[L]. u ∈ B ∧ (∃y[L]. pair(L,u,y,x) ∧
is_wfrec(L, iterates_MH(L, is_list_functor(L,A), 0), memsn, u, y)),
λi x. ∃u ∈ Lset(i). u ∈ B ∧ (∃y ∈ Lset(i). pair(##Lset(i), u, y, x) ∧
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 simp add: nonempty)
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
[λx. ∃u[L]. u ∈ B & u ∈ nat &
is_iterates(L, is_list_functor(L, A), 0, u, x),
λi x. ∃u ∈ Lset(i). u ∈ B & u ∈ 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,
λn y. n∈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 simp add: L_nat nonempty)
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‹@{term L} is Closed Under the Operator @{term formula}›

subsubsection‹Instances of Replacement for Formulas›

(*FIXME: could prove a lemma iterates_replacementI to eliminate the
need to expand iterates_replacement and wfrec_replacement*)
lemma formula_replacement1_Reflects:
"REFLECTS
[λx. ∃u[L]. u ∈ B & (∃y[L]. pair(L,u,y,x) &
is_wfrec(L, iterates_MH(L, is_formula_functor(L), 0), memsn, u, y)),
λi x. ∃u ∈ Lset(i). u ∈ B & (∃y ∈ Lset(i). pair(##Lset(i), u, y, x) &
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 simp add: nonempty)
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
[λx. ∃u[L]. u ∈ B & u ∈ nat &
is_iterates(L, is_formula_functor(L), 0, u, x),
λi x. ∃u ∈ Lset(i). u ∈ B & u ∈ 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,
λn y. n∈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 simp add: nonempty L_nat)
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‹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) ==
∃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) ⟷
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 ∈ nat; k ∈ nat; env ∈ 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[λx. is_nth(L, f(x), g(x), h(x)),
λi 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‹An Instance of Replacement for @{term nth}›

(*FIXME: could prove a lemma iterates_replacementI to eliminate the
need to expand iterates_replacement and wfrec_replacement*)
lemma nth_replacement_Reflects:
"REFLECTS
[λx. ∃u[L]. u ∈ B & (∃y[L]. pair(L,u,y,x) &
is_wfrec(L, iterates_MH(L, is_tl(L), z), memsn, u, y)),
λi x. ∃u ∈ Lset(i). u ∈ B & (∃y ∈ Lset(i). pair(##Lset(i), u, y, x) &
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‹Instantiating the locale ‹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_L)
apply (rule M_datatypes_axioms_L)
done

interpretation L?: M_datatypes L by (rule M_datatypes_L)

subsection‹@{term L} is Closed Under the Operator @{term eclose}›

subsubsection‹Instances of Replacement for @{term eclose}›

lemma eclose_replacement1_Reflects:
"REFLECTS
[λx. ∃u[L]. u ∈ B & (∃y[L]. pair(L,u,y,x) &
is_wfrec(L, iterates_MH(L, big_union(L), A), memsn, u, y)),
λi x. ∃u ∈ Lset(i). u ∈ B & (∃y ∈ Lset(i). pair(##Lset(i), u, y, x) &
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
[λx. ∃u[L]. u ∈ B & u ∈ nat &
is_iterates(L, big_union(L), A, u, x),
λi x. ∃u ∈ Lset(i). u ∈ B & u ∈ 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,
λn y. n∈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 simp add: L_nat)
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‹Instantiating the locale ‹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_L)
apply (rule M_eclose_axioms_L)
done

interpretation L?: M_eclose L by (rule M_eclose_L)

end
```