Theory Internalize

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

theory Internalize imports L_axioms Datatype_absolute begin

subsection‹Internalized Forms of Data Structuring Operators›

subsubsection‹The Formula @{term is_Inl}, Internalized›

(*  is_Inl(M,a,z) == ∃zero[M]. empty(M,zero) & pair(M,zero,a,z) *)
definition
  Inl_fm :: "[i,i]=>i" where
    "Inl_fm(a,z) == Exists(And(empty_fm(0), pair_fm(0,succ(a),succ(z))))"

lemma Inl_type [TC]:
     "[| x ∈ nat; z ∈ nat |] ==> Inl_fm(x,z) ∈ formula"
by (simp add: Inl_fm_def)

lemma sats_Inl_fm [simp]:
   "[| x ∈ nat; z ∈ nat; env ∈ list(A)|]
    ==> sats(A, Inl_fm(x,z), env) ⟷ is_Inl(##A, nth(x,env), nth(z,env))"
by (simp add: Inl_fm_def is_Inl_def)

lemma Inl_iff_sats:
      "[| nth(i,env) = x; nth(k,env) = z;
          i ∈ nat; k ∈ nat; env ∈ list(A)|]
       ==> is_Inl(##A, x, z) ⟷ sats(A, Inl_fm(i,k), env)"
by simp

theorem Inl_reflection:
     "REFLECTS[λx. is_Inl(L,f(x),h(x)),
               λi x. is_Inl(##Lset(i),f(x),h(x))]"
apply (simp only: is_Inl_def)
apply (intro FOL_reflections function_reflections)
done


subsubsection‹The Formula @{term is_Inr}, Internalized›

(*  is_Inr(M,a,z) == ∃n1[M]. number1(M,n1) & pair(M,n1,a,z) *)
definition
  Inr_fm :: "[i,i]=>i" where
    "Inr_fm(a,z) == Exists(And(number1_fm(0), pair_fm(0,succ(a),succ(z))))"

lemma Inr_type [TC]:
     "[| x ∈ nat; z ∈ nat |] ==> Inr_fm(x,z) ∈ formula"
by (simp add: Inr_fm_def)

lemma sats_Inr_fm [simp]:
   "[| x ∈ nat; z ∈ nat; env ∈ list(A)|]
    ==> sats(A, Inr_fm(x,z), env) ⟷ is_Inr(##A, nth(x,env), nth(z,env))"
by (simp add: Inr_fm_def is_Inr_def)

lemma Inr_iff_sats:
      "[| nth(i,env) = x; nth(k,env) = z;
          i ∈ nat; k ∈ nat; env ∈ list(A)|]
       ==> is_Inr(##A, x, z) ⟷ sats(A, Inr_fm(i,k), env)"
by simp

theorem Inr_reflection:
     "REFLECTS[λx. is_Inr(L,f(x),h(x)),
               λi x. is_Inr(##Lset(i),f(x),h(x))]"
apply (simp only: is_Inr_def)
apply (intro FOL_reflections function_reflections)
done


subsubsection‹The Formula @{term is_Nil}, Internalized›

(* is_Nil(M,xs) == ∃zero[M]. empty(M,zero) & is_Inl(M,zero,xs) *)

definition
  Nil_fm :: "i=>i" where
    "Nil_fm(x) == Exists(And(empty_fm(0), Inl_fm(0,succ(x))))"

lemma Nil_type [TC]: "x ∈ nat ==> Nil_fm(x) ∈ formula"
by (simp add: Nil_fm_def)

lemma sats_Nil_fm [simp]:
   "[| x ∈ nat; env ∈ list(A)|]
    ==> sats(A, Nil_fm(x), env) ⟷ is_Nil(##A, nth(x,env))"
by (simp add: Nil_fm_def is_Nil_def)

lemma Nil_iff_sats:
      "[| nth(i,env) = x; i ∈ nat; env ∈ list(A)|]
       ==> is_Nil(##A, x) ⟷ sats(A, Nil_fm(i), env)"
by simp

theorem Nil_reflection:
     "REFLECTS[λx. is_Nil(L,f(x)),
               λi x. is_Nil(##Lset(i),f(x))]"
apply (simp only: is_Nil_def)
apply (intro FOL_reflections function_reflections Inl_reflection)
done


subsubsection‹The Formula @{term is_Cons}, Internalized›


(*  "is_Cons(M,a,l,Z) == ∃p[M]. pair(M,a,l,p) & is_Inr(M,p,Z)" *)
definition
  Cons_fm :: "[i,i,i]=>i" where
    "Cons_fm(a,l,Z) ==
       Exists(And(pair_fm(succ(a),succ(l),0), Inr_fm(0,succ(Z))))"

lemma Cons_type [TC]:
     "[| x ∈ nat; y ∈ nat; z ∈ nat |] ==> Cons_fm(x,y,z) ∈ formula"
by (simp add: Cons_fm_def)

lemma sats_Cons_fm [simp]:
   "[| x ∈ nat; y ∈ nat; z ∈ nat; env ∈ list(A)|]
    ==> sats(A, Cons_fm(x,y,z), env) ⟷
       is_Cons(##A, nth(x,env), nth(y,env), nth(z,env))"
by (simp add: Cons_fm_def is_Cons_def)

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

theorem Cons_reflection:
     "REFLECTS[λx. is_Cons(L,f(x),g(x),h(x)),
               λi x. is_Cons(##Lset(i),f(x),g(x),h(x))]"
apply (simp only: is_Cons_def)
apply (intro FOL_reflections pair_reflection Inr_reflection)
done

subsubsection‹The Formula @{term is_quasilist}, Internalized›

(* is_quasilist(M,xs) == is_Nil(M,z) | (∃x[M]. ∃l[M]. is_Cons(M,x,l,z))" *)

definition
  quasilist_fm :: "i=>i" where
    "quasilist_fm(x) ==
       Or(Nil_fm(x), Exists(Exists(Cons_fm(1,0,succ(succ(x))))))"

lemma quasilist_type [TC]: "x ∈ nat ==> quasilist_fm(x) ∈ formula"
by (simp add: quasilist_fm_def)

lemma sats_quasilist_fm [simp]:
   "[| x ∈ nat; env ∈ list(A)|]
    ==> sats(A, quasilist_fm(x), env) ⟷ is_quasilist(##A, nth(x,env))"
by (simp add: quasilist_fm_def is_quasilist_def)

lemma quasilist_iff_sats:
      "[| nth(i,env) = x; i ∈ nat; env ∈ list(A)|]
       ==> is_quasilist(##A, x) ⟷ sats(A, quasilist_fm(i), env)"
by simp

theorem quasilist_reflection:
     "REFLECTS[λx. is_quasilist(L,f(x)),
               λi x. is_quasilist(##Lset(i),f(x))]"
apply (simp only: is_quasilist_def)
apply (intro FOL_reflections Nil_reflection Cons_reflection)
done


subsection‹Absoluteness for the Function @{term nth}›


subsubsection‹The Formula @{term is_hd}, Internalized›

(*   "is_hd(M,xs,H) == 
       (is_Nil(M,xs) ⟶ empty(M,H)) &
       (∀x[M]. ∀l[M]. ~ is_Cons(M,x,l,xs) | H=x) &
       (is_quasilist(M,xs) | empty(M,H))" *)
definition
  hd_fm :: "[i,i]=>i" where
    "hd_fm(xs,H) == 
       And(Implies(Nil_fm(xs), empty_fm(H)),
           And(Forall(Forall(Or(Neg(Cons_fm(1,0,xs#+2)), Equal(H#+2,1)))),
               Or(quasilist_fm(xs), empty_fm(H))))"

lemma hd_type [TC]:
     "[| x ∈ nat; y ∈ nat |] ==> hd_fm(x,y) ∈ formula"
by (simp add: hd_fm_def) 

lemma sats_hd_fm [simp]:
   "[| x ∈ nat; y ∈ nat; env ∈ list(A)|]
    ==> sats(A, hd_fm(x,y), env) ⟷ is_hd(##A, nth(x,env), nth(y,env))"
by (simp add: hd_fm_def is_hd_def)

lemma hd_iff_sats:
      "[| nth(i,env) = x; nth(j,env) = y;
          i ∈ nat; j ∈ nat; env ∈ list(A)|]
       ==> is_hd(##A, x, y) ⟷ sats(A, hd_fm(i,j), env)"
by simp

theorem hd_reflection:
     "REFLECTS[λx. is_hd(L,f(x),g(x)), 
               λi x. is_hd(##Lset(i),f(x),g(x))]"
apply (simp only: is_hd_def)
apply (intro FOL_reflections Nil_reflection Cons_reflection
             quasilist_reflection empty_reflection)  
done


subsubsection‹The Formula @{term is_tl}, Internalized›

(*     "is_tl(M,xs,T) ==
       (is_Nil(M,xs) ⟶ T=xs) &
       (∀x[M]. ∀l[M]. ~ is_Cons(M,x,l,xs) | T=l) &
       (is_quasilist(M,xs) | empty(M,T))" *)
definition
  tl_fm :: "[i,i]=>i" where
    "tl_fm(xs,T) ==
       And(Implies(Nil_fm(xs), Equal(T,xs)),
           And(Forall(Forall(Or(Neg(Cons_fm(1,0,xs#+2)), Equal(T#+2,0)))),
               Or(quasilist_fm(xs), empty_fm(T))))"

lemma tl_type [TC]:
     "[| x ∈ nat; y ∈ nat |] ==> tl_fm(x,y) ∈ formula"
by (simp add: tl_fm_def)

lemma sats_tl_fm [simp]:
   "[| x ∈ nat; y ∈ nat; env ∈ list(A)|]
    ==> sats(A, tl_fm(x,y), env) ⟷ is_tl(##A, nth(x,env), nth(y,env))"
by (simp add: tl_fm_def is_tl_def)

lemma tl_iff_sats:
      "[| nth(i,env) = x; nth(j,env) = y;
          i ∈ nat; j ∈ nat; env ∈ list(A)|]
       ==> is_tl(##A, x, y) ⟷ sats(A, tl_fm(i,j), env)"
by simp

theorem tl_reflection:
     "REFLECTS[λx. is_tl(L,f(x),g(x)),
               λi x. is_tl(##Lset(i),f(x),g(x))]"
apply (simp only: is_tl_def)
apply (intro FOL_reflections Nil_reflection Cons_reflection
             quasilist_reflection empty_reflection)
done


subsubsection‹The Operator @{term is_bool_of_o}›

(*   is_bool_of_o :: "[i=>o, o, i] => o"
   "is_bool_of_o(M,P,z) == (P & number1(M,z)) | (~P & empty(M,z))" *)

text‹The formula @{term p} has no free variables.›
definition
  bool_of_o_fm :: "[i, i]=>i" where
  "bool_of_o_fm(p,z) == 
    Or(And(p,number1_fm(z)),
       And(Neg(p),empty_fm(z)))"

lemma is_bool_of_o_type [TC]:
     "[| p ∈ formula; z ∈ nat |] ==> bool_of_o_fm(p,z) ∈ formula"
by (simp add: bool_of_o_fm_def)

lemma sats_bool_of_o_fm:
  assumes p_iff_sats: "P ⟷ sats(A, p, env)"
  shows 
      "[|z ∈ nat; env ∈ list(A)|]
       ==> sats(A, bool_of_o_fm(p,z), env) ⟷
           is_bool_of_o(##A, P, nth(z,env))"
by (simp add: bool_of_o_fm_def is_bool_of_o_def p_iff_sats [THEN iff_sym])

lemma is_bool_of_o_iff_sats:
  "[| P ⟷ sats(A, p, env); nth(k,env) = z; k ∈ nat; env ∈ list(A)|]
   ==> is_bool_of_o(##A, P, z) ⟷ sats(A, bool_of_o_fm(p,k), env)"
by (simp add: sats_bool_of_o_fm)

theorem bool_of_o_reflection:
     "REFLECTS [P(L), λi. P(##Lset(i))] ==>
      REFLECTS[λx. is_bool_of_o(L, P(L,x), f(x)),  
               λi x. is_bool_of_o(##Lset(i), P(##Lset(i),x), f(x))]"
apply (simp (no_asm) only: is_bool_of_o_def)
apply (intro FOL_reflections function_reflections, assumption+)
done


subsection‹More Internalizations›

subsubsection‹The Operator @{term is_lambda}›

text‹The two arguments of @{term p} are always 1, 0. Remember that
 @{term p} will be enclosed by three quantifiers.›

(* is_lambda :: "[i=>o, i, [i,i]=>o, i] => o"
    "is_lambda(M, A, is_b, z) == 
       ∀p[M]. p ∈ z ⟷
        (∃u[M]. ∃v[M]. u∈A & pair(M,u,v,p) & is_b(u,v))" *)
definition
  lambda_fm :: "[i, i, i]=>i" where
  "lambda_fm(p,A,z) == 
    Forall(Iff(Member(0,succ(z)),
            Exists(Exists(And(Member(1,A#+3),
                           And(pair_fm(1,0,2), p))))))"

text‹We call @{term p} with arguments x, y by equating them with 
  the corresponding quantified variables with de Bruijn indices 1, 0.›

lemma is_lambda_type [TC]:
     "[| p ∈ formula; x ∈ nat; y ∈ nat |] 
      ==> lambda_fm(p,x,y) ∈ formula"
by (simp add: lambda_fm_def) 

lemma sats_lambda_fm:
  assumes is_b_iff_sats: 
      "!!a0 a1 a2. 
        [|a0∈A; a1∈A; a2∈A|] 
        ==> is_b(a1, a0) ⟷ sats(A, p, Cons(a0,Cons(a1,Cons(a2,env))))"
  shows 
      "[|x ∈ nat; y ∈ nat; env ∈ list(A)|]
       ==> sats(A, lambda_fm(p,x,y), env) ⟷ 
           is_lambda(##A, nth(x,env), is_b, nth(y,env))"
by (simp add: lambda_fm_def is_lambda_def is_b_iff_sats [THEN iff_sym]) 

theorem is_lambda_reflection:
  assumes is_b_reflection:
    "!!f g h. REFLECTS[λx. is_b(L, f(x), g(x), h(x)), 
                     λi x. is_b(##Lset(i), f(x), g(x), h(x))]"
  shows "REFLECTS[λx. is_lambda(L, A(x), is_b(L,x), f(x)), 
               λi x. is_lambda(##Lset(i), A(x), is_b(##Lset(i),x), f(x))]"
apply (simp (no_asm_use) only: is_lambda_def)
apply (intro FOL_reflections is_b_reflection pair_reflection)
done

subsubsection‹The Operator @{term is_Member}, Internalized›

(*    "is_Member(M,x,y,Z) ==
        ∃p[M]. ∃u[M]. pair(M,x,y,p) & is_Inl(M,p,u) & is_Inl(M,u,Z)" *)
definition
  Member_fm :: "[i,i,i]=>i" where
    "Member_fm(x,y,Z) ==
       Exists(Exists(And(pair_fm(x#+2,y#+2,1), 
                      And(Inl_fm(1,0), Inl_fm(0,Z#+2)))))"

lemma is_Member_type [TC]:
     "[| x ∈ nat; y ∈ nat; z ∈ nat |] ==> Member_fm(x,y,z) ∈ formula"
by (simp add: Member_fm_def)

lemma sats_Member_fm [simp]:
   "[| x ∈ nat; y ∈ nat; z ∈ nat; env ∈ list(A)|]
    ==> sats(A, Member_fm(x,y,z), env) ⟷
        is_Member(##A, nth(x,env), nth(y,env), nth(z,env))"
by (simp add: Member_fm_def is_Member_def)

lemma Member_iff_sats:
      "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z;
          i ∈ nat; j ∈ nat; k ∈ nat; env ∈ list(A)|]
       ==> is_Member(##A, x, y, z) ⟷ sats(A, Member_fm(i,j,k), env)"
by (simp add: sats_Member_fm)

theorem Member_reflection:
     "REFLECTS[λx. is_Member(L,f(x),g(x),h(x)),
               λi x. is_Member(##Lset(i),f(x),g(x),h(x))]"
apply (simp only: is_Member_def)
apply (intro FOL_reflections pair_reflection Inl_reflection)
done

subsubsection‹The Operator @{term is_Equal}, Internalized›

(*    "is_Equal(M,x,y,Z) ==
        ∃p[M]. ∃u[M]. pair(M,x,y,p) & is_Inr(M,p,u) & is_Inl(M,u,Z)" *)
definition
  Equal_fm :: "[i,i,i]=>i" where
    "Equal_fm(x,y,Z) ==
       Exists(Exists(And(pair_fm(x#+2,y#+2,1), 
                      And(Inr_fm(1,0), Inl_fm(0,Z#+2)))))"

lemma is_Equal_type [TC]:
     "[| x ∈ nat; y ∈ nat; z ∈ nat |] ==> Equal_fm(x,y,z) ∈ formula"
by (simp add: Equal_fm_def)

lemma sats_Equal_fm [simp]:
   "[| x ∈ nat; y ∈ nat; z ∈ nat; env ∈ list(A)|]
    ==> sats(A, Equal_fm(x,y,z), env) ⟷
        is_Equal(##A, nth(x,env), nth(y,env), nth(z,env))"
by (simp add: Equal_fm_def is_Equal_def)

lemma Equal_iff_sats:
      "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z;
          i ∈ nat; j ∈ nat; k ∈ nat; env ∈ list(A)|]
       ==> is_Equal(##A, x, y, z) ⟷ sats(A, Equal_fm(i,j,k), env)"
by (simp add: sats_Equal_fm)

theorem Equal_reflection:
     "REFLECTS[λx. is_Equal(L,f(x),g(x),h(x)),
               λi x. is_Equal(##Lset(i),f(x),g(x),h(x))]"
apply (simp only: is_Equal_def)
apply (intro FOL_reflections pair_reflection Inl_reflection Inr_reflection)
done

subsubsection‹The Operator @{term is_Nand}, Internalized›

(*    "is_Nand(M,x,y,Z) ==
        ∃p[M]. ∃u[M]. pair(M,x,y,p) & is_Inl(M,p,u) & is_Inr(M,u,Z)" *)
definition
  Nand_fm :: "[i,i,i]=>i" where
    "Nand_fm(x,y,Z) ==
       Exists(Exists(And(pair_fm(x#+2,y#+2,1), 
                      And(Inl_fm(1,0), Inr_fm(0,Z#+2)))))"

lemma is_Nand_type [TC]:
     "[| x ∈ nat; y ∈ nat; z ∈ nat |] ==> Nand_fm(x,y,z) ∈ formula"
by (simp add: Nand_fm_def)

lemma sats_Nand_fm [simp]:
   "[| x ∈ nat; y ∈ nat; z ∈ nat; env ∈ list(A)|]
    ==> sats(A, Nand_fm(x,y,z), env) ⟷
        is_Nand(##A, nth(x,env), nth(y,env), nth(z,env))"
by (simp add: Nand_fm_def is_Nand_def)

lemma Nand_iff_sats:
      "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z;
          i ∈ nat; j ∈ nat; k ∈ nat; env ∈ list(A)|]
       ==> is_Nand(##A, x, y, z) ⟷ sats(A, Nand_fm(i,j,k), env)"
by (simp add: sats_Nand_fm)

theorem Nand_reflection:
     "REFLECTS[λx. is_Nand(L,f(x),g(x),h(x)),
               λi x. is_Nand(##Lset(i),f(x),g(x),h(x))]"
apply (simp only: is_Nand_def)
apply (intro FOL_reflections pair_reflection Inl_reflection Inr_reflection)
done

subsubsection‹The Operator @{term is_Forall}, Internalized›

(* "is_Forall(M,p,Z) == ∃u[M]. is_Inr(M,p,u) & is_Inr(M,u,Z)" *)
definition
  Forall_fm :: "[i,i]=>i" where
    "Forall_fm(x,Z) ==
       Exists(And(Inr_fm(succ(x),0), Inr_fm(0,succ(Z))))"

lemma is_Forall_type [TC]:
     "[| x ∈ nat; y ∈ nat |] ==> Forall_fm(x,y) ∈ formula"
by (simp add: Forall_fm_def)

lemma sats_Forall_fm [simp]:
   "[| x ∈ nat; y ∈ nat; env ∈ list(A)|]
    ==> sats(A, Forall_fm(x,y), env) ⟷
        is_Forall(##A, nth(x,env), nth(y,env))"
by (simp add: Forall_fm_def is_Forall_def)

lemma Forall_iff_sats:
      "[| nth(i,env) = x; nth(j,env) = y; 
          i ∈ nat; j ∈ nat; env ∈ list(A)|]
       ==> is_Forall(##A, x, y) ⟷ sats(A, Forall_fm(i,j), env)"
by (simp add: sats_Forall_fm)

theorem Forall_reflection:
     "REFLECTS[λx. is_Forall(L,f(x),g(x)),
               λi x. is_Forall(##Lset(i),f(x),g(x))]"
apply (simp only: is_Forall_def)
apply (intro FOL_reflections pair_reflection Inr_reflection)
done


subsubsection‹The Operator @{term is_and}, Internalized›

(* is_and(M,a,b,z) == (number1(M,a)  & z=b) | 
                       (~number1(M,a) & empty(M,z)) *)
definition
  and_fm :: "[i,i,i]=>i" where
    "and_fm(a,b,z) ==
       Or(And(number1_fm(a), Equal(z,b)),
          And(Neg(number1_fm(a)),empty_fm(z)))"

lemma is_and_type [TC]:
     "[| x ∈ nat; y ∈ nat; z ∈ nat |] ==> and_fm(x,y,z) ∈ formula"
by (simp add: and_fm_def)

lemma sats_and_fm [simp]:
   "[| x ∈ nat; y ∈ nat; z ∈ nat; env ∈ list(A)|]
    ==> sats(A, and_fm(x,y,z), env) ⟷
        is_and(##A, nth(x,env), nth(y,env), nth(z,env))"
by (simp add: and_fm_def is_and_def)

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

theorem is_and_reflection:
     "REFLECTS[λx. is_and(L,f(x),g(x),h(x)),
               λi x. is_and(##Lset(i),f(x),g(x),h(x))]"
apply (simp only: is_and_def)
apply (intro FOL_reflections function_reflections)
done


subsubsection‹The Operator @{term is_or}, Internalized›

(* is_or(M,a,b,z) == (number1(M,a)  & number1(M,z)) | 
                     (~number1(M,a) & z=b) *)

definition
  or_fm :: "[i,i,i]=>i" where
    "or_fm(a,b,z) ==
       Or(And(number1_fm(a), number1_fm(z)),
          And(Neg(number1_fm(a)), Equal(z,b)))"

lemma is_or_type [TC]:
     "[| x ∈ nat; y ∈ nat; z ∈ nat |] ==> or_fm(x,y,z) ∈ formula"
by (simp add: or_fm_def)

lemma sats_or_fm [simp]:
   "[| x ∈ nat; y ∈ nat; z ∈ nat; env ∈ list(A)|]
    ==> sats(A, or_fm(x,y,z), env) ⟷
        is_or(##A, nth(x,env), nth(y,env), nth(z,env))"
by (simp add: or_fm_def is_or_def)

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

theorem is_or_reflection:
     "REFLECTS[λx. is_or(L,f(x),g(x),h(x)),
               λi x. is_or(##Lset(i),f(x),g(x),h(x))]"
apply (simp only: is_or_def)
apply (intro FOL_reflections function_reflections)
done



subsubsection‹The Operator @{term is_not}, Internalized›

(* is_not(M,a,z) == (number1(M,a)  & empty(M,z)) | 
                     (~number1(M,a) & number1(M,z)) *)
definition
  not_fm :: "[i,i]=>i" where
    "not_fm(a,z) ==
       Or(And(number1_fm(a), empty_fm(z)),
          And(Neg(number1_fm(a)), number1_fm(z)))"

lemma is_not_type [TC]:
     "[| x ∈ nat; z ∈ nat |] ==> not_fm(x,z) ∈ formula"
by (simp add: not_fm_def)

lemma sats_is_not_fm [simp]:
   "[| x ∈ nat; z ∈ nat; env ∈ list(A)|]
    ==> sats(A, not_fm(x,z), env) ⟷ is_not(##A, nth(x,env), nth(z,env))"
by (simp add: not_fm_def is_not_def)

lemma is_not_iff_sats:
      "[| nth(i,env) = x; nth(k,env) = z;
          i ∈ nat; k ∈ nat; env ∈ list(A)|]
       ==> is_not(##A, x, z) ⟷ sats(A, not_fm(i,k), env)"
by simp

theorem is_not_reflection:
     "REFLECTS[λx. is_not(L,f(x),g(x)),
               λi x. is_not(##Lset(i),f(x),g(x))]"
apply (simp only: is_not_def)
apply (intro FOL_reflections function_reflections)
done


lemmas extra_reflections = 
    Inl_reflection Inr_reflection Nil_reflection Cons_reflection
    quasilist_reflection hd_reflection tl_reflection bool_of_o_reflection
    is_lambda_reflection Member_reflection Equal_reflection Nand_reflection
    Forall_reflection is_and_reflection is_or_reflection is_not_reflection

subsection‹Well-Founded Recursion!›

subsubsection‹The Operator @{term M_is_recfun}›

text‹Alternative definition, minimizing nesting of quantifiers around MH›
lemma M_is_recfun_iff:
   "M_is_recfun(M,MH,r,a,f) ⟷
    (∀z[M]. z ∈ f ⟷ 
     (∃x[M]. ∃f_r_sx[M]. ∃y[M]. 
             MH(x, f_r_sx, y) & pair(M,x,y,z) &
             (∃xa[M]. ∃sx[M]. ∃r_sx[M]. 
                pair(M,x,a,xa) & upair(M,x,x,sx) &
               pre_image(M,r,sx,r_sx) & restriction(M,f,r_sx,f_r_sx) &
               xa ∈ r)))"
apply (simp add: M_is_recfun_def)
apply (rule rall_cong, blast) 
done


(* M_is_recfun :: "[i=>o, [i,i,i]=>o, i, i, i] => o"
   "M_is_recfun(M,MH,r,a,f) ==
     ∀z[M]. z ∈ f ⟷
               2      1           0
new def     (∃x[M]. ∃f_r_sx[M]. ∃y[M]. 
             MH(x, f_r_sx, y) & pair(M,x,y,z) &
             (∃xa[M]. ∃sx[M]. ∃r_sx[M]. 
                pair(M,x,a,xa) & upair(M,x,x,sx) &
               pre_image(M,r,sx,r_sx) & restriction(M,f,r_sx,f_r_sx) &
               xa ∈ r)"
*)

text‹The three arguments of @{term p} are always 2, 1, 0 and z›
definition
  is_recfun_fm :: "[i, i, i, i]=>i" where
  "is_recfun_fm(p,r,a,f) == 
   Forall(Iff(Member(0,succ(f)),
    Exists(Exists(Exists(
     And(p, 
      And(pair_fm(2,0,3),
       Exists(Exists(Exists(
        And(pair_fm(5,a#+7,2),
         And(upair_fm(5,5,1),
          And(pre_image_fm(r#+7,1,0),
           And(restriction_fm(f#+7,0,4), Member(2,r#+7)))))))))))))))"

lemma is_recfun_type [TC]:
     "[| p ∈ formula; x ∈ nat; y ∈ nat; z ∈ nat |] 
      ==> is_recfun_fm(p,x,y,z) ∈ formula"
by (simp add: is_recfun_fm_def)


lemma sats_is_recfun_fm:
  assumes MH_iff_sats: 
      "!!a0 a1 a2 a3. 
        [|a0∈A; a1∈A; a2∈A; a3∈A|] 
        ==> MH(a2, a1, a0) ⟷ sats(A, p, Cons(a0,Cons(a1,Cons(a2,Cons(a3,env)))))"
  shows 
      "[|x ∈ nat; y ∈ nat; z ∈ nat; env ∈ list(A)|]
       ==> sats(A, is_recfun_fm(p,x,y,z), env) ⟷
           M_is_recfun(##A, MH, nth(x,env), nth(y,env), nth(z,env))"
by (simp add: is_recfun_fm_def M_is_recfun_iff MH_iff_sats [THEN iff_sym])

lemma is_recfun_iff_sats:
  assumes MH_iff_sats: 
      "!!a0 a1 a2 a3. 
        [|a0∈A; a1∈A; a2∈A; a3∈A|] 
        ==> MH(a2, a1, a0) ⟷ sats(A, p, Cons(a0,Cons(a1,Cons(a2,Cons(a3,env)))))"
  shows
  "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z; 
      i ∈ nat; j ∈ nat; k ∈ nat; env ∈ list(A)|]
   ==> M_is_recfun(##A, MH, x, y, z) ⟷ sats(A, is_recfun_fm(p,i,j,k), env)"
by (simp add: sats_is_recfun_fm [OF MH_iff_sats]) 

text‹The additional variable in the premise, namely @{term f'}, is essential.
It lets @{term MH} depend upon @{term x}, which seems often necessary.
The same thing occurs in ‹is_wfrec_reflection›.›
theorem is_recfun_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. M_is_recfun(L, MH(L,x), f(x), g(x), h(x)), 
             λi x. M_is_recfun(##Lset(i), MH(##Lset(i),x), f(x), g(x), h(x))]"
apply (simp (no_asm_use) only: M_is_recfun_def)
apply (intro FOL_reflections function_reflections
             restriction_reflection MH_reflection)
done

subsubsection‹The Operator @{term is_wfrec}›

text‹The three arguments of @{term p} are always 2, 1, 0;
      @{term p} is enclosed by 5 quantifiers.›

(* is_wfrec :: "[i=>o, i, [i,i,i]=>o, i, i] => o"
    "is_wfrec(M,MH,r,a,z) == 
      ∃f[M]. M_is_recfun(M,MH,r,a,f) & MH(a,f,z)" *)
definition
  is_wfrec_fm :: "[i, i, i, i]=>i" where
  "is_wfrec_fm(p,r,a,z) == 
    Exists(And(is_recfun_fm(p, succ(r), succ(a), 0),
           Exists(Exists(Exists(Exists(
             And(Equal(2,a#+5), And(Equal(1,4), And(Equal(0,z#+5), p)))))))))"

text‹We call @{term p} with arguments a, f, z by equating them with 
  the corresponding quantified variables with de Bruijn indices 2, 1, 0.›

text‹There's an additional existential quantifier to ensure that the
      environments in both calls to MH have the same length.›

lemma is_wfrec_type [TC]:
     "[| p ∈ formula; x ∈ nat; y ∈ nat; z ∈ nat |] 
      ==> is_wfrec_fm(p,x,y,z) ∈ formula"
by (simp add: is_wfrec_fm_def) 

lemma sats_is_wfrec_fm:
  assumes MH_iff_sats: 
      "!!a0 a1 a2 a3 a4. 
        [|a0∈A; a1∈A; a2∈A; a3∈A; a4∈A|] 
        ==> MH(a2, a1, a0) ⟷ sats(A, p, Cons(a0,Cons(a1,Cons(a2,Cons(a3,Cons(a4,env))))))"
  shows 
      "[|x ∈ nat; y < length(env); z < length(env); env ∈ list(A)|]
       ==> sats(A, is_wfrec_fm(p,x,y,z), env) ⟷ 
           is_wfrec(##A, MH, nth(x,env), nth(y,env), nth(z,env))"
apply (frule_tac x=z in lt_length_in_nat, assumption)  
apply (frule lt_length_in_nat, assumption)  
apply (simp add: is_wfrec_fm_def sats_is_recfun_fm is_wfrec_def MH_iff_sats [THEN iff_sym], blast) 
done


lemma is_wfrec_iff_sats:
  assumes MH_iff_sats: 
      "!!a0 a1 a2 a3 a4. 
        [|a0∈A; a1∈A; a2∈A; a3∈A; a4∈A|] 
        ==> MH(a2, a1, a0) ⟷ sats(A, p, Cons(a0,Cons(a1,Cons(a2,Cons(a3,Cons(a4,env))))))"
  shows
  "[|nth(i,env) = x; nth(j,env) = y; nth(k,env) = z; 
      i ∈ nat; j < length(env); k < length(env); env ∈ list(A)|]
   ==> is_wfrec(##A, MH, x, y, z) ⟷ sats(A, is_wfrec_fm(p,i,j,k), env)" 
by (simp add: sats_is_wfrec_fm [OF MH_iff_sats])

theorem is_wfrec_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_wfrec(L, MH(L,x), f(x), g(x), h(x)), 
               λi x. is_wfrec(##Lset(i), MH(##Lset(i),x), f(x), g(x), h(x))]"
apply (simp (no_asm_use) only: is_wfrec_def)
apply (intro FOL_reflections MH_reflection is_recfun_reflection)
done


subsection‹For Datatypes›

subsubsection‹Binary Products, Internalized›

definition
  cartprod_fm :: "[i,i,i]=>i" where
(* "cartprod(M,A,B,z) ==
        ∀u[M]. u ∈ z ⟷ (∃x[M]. x∈A & (∃y[M]. y∈B & pair(M,x,y,u)))" *)
    "cartprod_fm(A,B,z) ==
       Forall(Iff(Member(0,succ(z)),
                  Exists(And(Member(0,succ(succ(A))),
                         Exists(And(Member(0,succ(succ(succ(B)))),
                                    pair_fm(1,0,2)))))))"

lemma cartprod_type [TC]:
     "[| x ∈ nat; y ∈ nat; z ∈ nat |] ==> cartprod_fm(x,y,z) ∈ formula"
by (simp add: cartprod_fm_def)

lemma sats_cartprod_fm [simp]:
   "[| x ∈ nat; y ∈ nat; z ∈ nat; env ∈ list(A)|]
    ==> sats(A, cartprod_fm(x,y,z), env) ⟷
        cartprod(##A, nth(x,env), nth(y,env), nth(z,env))"
by (simp add: cartprod_fm_def cartprod_def)

lemma cartprod_iff_sats:
      "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z;
          i ∈ nat; j ∈ nat; k ∈ nat; env ∈ list(A)|]
       ==> cartprod(##A, x, y, z) ⟷ sats(A, cartprod_fm(i,j,k), env)"
by (simp add: sats_cartprod_fm)

theorem cartprod_reflection:
     "REFLECTS[λx. cartprod(L,f(x),g(x),h(x)),
               λi x. cartprod(##Lset(i),f(x),g(x),h(x))]"
apply (simp only: cartprod_def)
apply (intro FOL_reflections pair_reflection)
done


subsubsection‹Binary Sums, Internalized›

(* "is_sum(M,A,B,Z) ==
       ∃A0[M]. ∃n1[M]. ∃s1[M]. ∃B1[M].
         3      2       1        0
       number1(M,n1) & cartprod(M,n1,A,A0) & upair(M,n1,n1,s1) &
       cartprod(M,s1,B,B1) & union(M,A0,B1,Z)"  *)
definition
  sum_fm :: "[i,i,i]=>i" where
    "sum_fm(A,B,Z) ==
       Exists(Exists(Exists(Exists(
        And(number1_fm(2),
            And(cartprod_fm(2,A#+4,3),
                And(upair_fm(2,2,1),
                    And(cartprod_fm(1,B#+4,0), union_fm(3,0,Z#+4)))))))))"

lemma sum_type [TC]:
     "[| x ∈ nat; y ∈ nat; z ∈ nat |] ==> sum_fm(x,y,z) ∈ formula"
by (simp add: sum_fm_def)

lemma sats_sum_fm [simp]:
   "[| x ∈ nat; y ∈ nat; z ∈ nat; env ∈ list(A)|]
    ==> sats(A, sum_fm(x,y,z), env) ⟷
        is_sum(##A, nth(x,env), nth(y,env), nth(z,env))"
by (simp add: sum_fm_def is_sum_def)

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

theorem sum_reflection:
     "REFLECTS[λx. is_sum(L,f(x),g(x),h(x)),
               λi x. is_sum(##Lset(i),f(x),g(x),h(x))]"
apply (simp only: is_sum_def)
apply (intro FOL_reflections function_reflections cartprod_reflection)
done


subsubsection‹The Operator @{term quasinat}›

(* "is_quasinat(M,z) == empty(M,z) | (∃m[M]. successor(M,m,z))" *)
definition
  quasinat_fm :: "i=>i" where
    "quasinat_fm(z) == Or(empty_fm(z), Exists(succ_fm(0,succ(z))))"

lemma quasinat_type [TC]:
     "x ∈ nat ==> quasinat_fm(x) ∈ formula"
by (simp add: quasinat_fm_def)

lemma sats_quasinat_fm [simp]:
   "[| x ∈ nat; env ∈ list(A)|]
    ==> sats(A, quasinat_fm(x), env) ⟷ is_quasinat(##A, nth(x,env))"
by (simp add: quasinat_fm_def is_quasinat_def)

lemma quasinat_iff_sats:
      "[| nth(i,env) = x; nth(j,env) = y;
          i ∈ nat; env ∈ list(A)|]
       ==> is_quasinat(##A, x) ⟷ sats(A, quasinat_fm(i), env)"
by simp

theorem quasinat_reflection:
     "REFLECTS[λx. is_quasinat(L,f(x)),
               λi x. is_quasinat(##Lset(i),f(x))]"
apply (simp only: is_quasinat_def)
apply (intro FOL_reflections function_reflections)
done


subsubsection‹The Operator @{term is_nat_case}›
text‹I could not get it to work with the more natural assumption that 
 @{term is_b} takes two arguments.  Instead it must be a formula where 1 and 0
 stand for @{term m} and @{term b}, respectively.›

(* is_nat_case :: "[i=>o, i, [i,i]=>o, i, i] => o"
    "is_nat_case(M, a, is_b, k, z) ==
       (empty(M,k) ⟶ z=a) &
       (∀m[M]. successor(M,m,k) ⟶ is_b(m,z)) &
       (is_quasinat(M,k) | empty(M,z))" *)
text‹The formula @{term is_b} has free variables 1 and 0.›
definition
  is_nat_case_fm :: "[i, i, i, i]=>i" where
 "is_nat_case_fm(a,is_b,k,z) == 
    And(Implies(empty_fm(k), Equal(z,a)),
        And(Forall(Implies(succ_fm(0,succ(k)), 
                   Forall(Implies(Equal(0,succ(succ(z))), is_b)))),
            Or(quasinat_fm(k), empty_fm(z))))"

lemma is_nat_case_type [TC]:
     "[| is_b ∈ formula;  
         x ∈ nat; y ∈ nat; z ∈ nat |] 
      ==> is_nat_case_fm(x,is_b,y,z) ∈ formula"
by (simp add: is_nat_case_fm_def)

lemma sats_is_nat_case_fm:
  assumes is_b_iff_sats: 
      "!!a. a ∈ A ==> is_b(a,nth(z, env)) ⟷ 
                      sats(A, p, Cons(nth(z,env), Cons(a, env)))"
  shows 
      "[|x ∈ nat; y ∈ nat; z < length(env); env ∈ list(A)|]
       ==> sats(A, is_nat_case_fm(x,p,y,z), env) ⟷
           is_nat_case(##A, nth(x,env), is_b, nth(y,env), nth(z,env))"
apply (frule lt_length_in_nat, assumption)
apply (simp add: is_nat_case_fm_def is_nat_case_def is_b_iff_sats [THEN iff_sym])
done

lemma is_nat_case_iff_sats:
  "[| (!!a. a ∈ A ==> is_b(a,z) ⟷
                      sats(A, p, Cons(z, Cons(a,env))));
      nth(i,env) = x; nth(j,env) = y; nth(k,env) = z; 
      i ∈ nat; j ∈ nat; k < length(env); env ∈ list(A)|]
   ==> is_nat_case(##A, x, is_b, y, z) ⟷ sats(A, is_nat_case_fm(i,p,j,k), env)"
by (simp add: sats_is_nat_case_fm [of A is_b])


text‹The second argument of @{term is_b} gives it direct access to @{term x},
  which is essential for handling free variable references.  Without this
  argument, we cannot prove reflection for @{term iterates_MH}.›
theorem is_nat_case_reflection:
  assumes is_b_reflection:
    "!!h f g. REFLECTS[λx. is_b(L, h(x), f(x), g(x)),
                     λi x. is_b(##Lset(i), h(x), f(x), g(x))]"
  shows "REFLECTS[λx. is_nat_case(L, f(x), is_b(L,x), g(x), h(x)),
               λi x. is_nat_case(##Lset(i), f(x), is_b(##Lset(i), x), g(x), h(x))]"
apply (simp (no_asm_use) only: is_nat_case_def)
apply (intro FOL_reflections function_reflections
             restriction_reflection is_b_reflection quasinat_reflection)
done


subsection‹The Operator @{term iterates_MH}, Needed for Iteration›

(*  iterates_MH :: "[i=>o, [i,i]=>o, i, i, i, i] => o"
   "iterates_MH(M,isF,v,n,g,z) ==
        is_nat_case(M, v, λm u. ∃gm[M]. fun_apply(M,g,m,gm) & isF(gm,u),
                    n, z)" *)
definition
  iterates_MH_fm :: "[i, i, i, i, i]=>i" where
 "iterates_MH_fm(isF,v,n,g,z) == 
    is_nat_case_fm(v, 
      Exists(And(fun_apply_fm(succ(succ(succ(g))),2,0), 
                     Forall(Implies(Equal(0,2), isF)))), 
      n, z)"

lemma iterates_MH_type [TC]:
     "[| p ∈ formula;  
         v ∈ nat; x ∈ nat; y ∈ nat; z ∈ nat |] 
      ==> iterates_MH_fm(p,v,x,y,z) ∈ formula"
by (simp add: iterates_MH_fm_def)

lemma sats_iterates_MH_fm:
  assumes is_F_iff_sats:
      "!!a b c d. [| a ∈ A; b ∈ A; c ∈ A; d ∈ A|]
              ==> is_F(a,b) ⟷
                  sats(A, p, Cons(b, Cons(a, Cons(c, Cons(d,env)))))"
  shows 
      "[|v ∈ nat; x ∈ nat; y ∈ nat; z < length(env); env ∈ list(A)|]
       ==> sats(A, iterates_MH_fm(p,v,x,y,z), env) ⟷
           iterates_MH(##A, is_F, nth(v,env), nth(x,env), nth(y,env), nth(z,env))"
apply (frule lt_length_in_nat, assumption)  
apply (simp add: iterates_MH_fm_def iterates_MH_def sats_is_nat_case_fm 
              is_F_iff_sats [symmetric])
apply (rule is_nat_case_cong) 
apply (simp_all add: setclass_def)
done

lemma iterates_MH_iff_sats:
  assumes is_F_iff_sats:
      "!!a b c d. [| a ∈ A; b ∈ A; c ∈ A; d ∈ A|]
              ==> is_F(a,b) ⟷
                  sats(A, p, Cons(b, Cons(a, Cons(c, Cons(d,env)))))"
  shows 
  "[| nth(i',env) = v; nth(i,env) = x; nth(j,env) = y; nth(k,env) = z; 
      i' ∈ nat; i ∈ nat; j ∈ nat; k < length(env); env ∈ list(A)|]
   ==> iterates_MH(##A, is_F, v, x, y, z) ⟷
       sats(A, iterates_MH_fm(p,i',i,j,k), env)"
by (simp add: sats_iterates_MH_fm [OF is_F_iff_sats]) 

text‹The second argument of @{term p} gives it direct access to @{term x},
  which is essential for handling free variable references.  Without this
  argument, we cannot prove reflection for @{term list_N}.›
theorem iterates_MH_reflection:
  assumes p_reflection:
    "!!f g h. REFLECTS[λx. p(L, h(x), f(x), g(x)),
                     λi x. p(##Lset(i), h(x), f(x), g(x))]"
 shows "REFLECTS[λx. iterates_MH(L, p(L,x), e(x), f(x), g(x), h(x)),
               λi x. iterates_MH(##Lset(i), p(##Lset(i),x), e(x), f(x), g(x), h(x))]"
apply (simp (no_asm_use) only: iterates_MH_def)
apply (intro FOL_reflections function_reflections is_nat_case_reflection
             restriction_reflection p_reflection)
done


subsubsection‹The Operator @{term is_iterates}›

text‹The three arguments of @{term p} are always 2, 1, 0;
      @{term p} is enclosed by 9 (??) quantifiers.›

(*    "is_iterates(M,isF,v,n,Z) == 
      ∃sn[M]. ∃msn[M]. successor(M,n,sn) & membership(M,sn,msn) &
       1       0       is_wfrec(M, iterates_MH(M,isF,v), msn, n, Z)"*)

definition
  is_iterates_fm :: "[i, i, i, i]=>i" where
  "is_iterates_fm(p,v,n,Z) == 
     Exists(Exists(
      And(succ_fm(n#+2,1),
       And(Memrel_fm(1,0),
              is_wfrec_fm(iterates_MH_fm(p, v#+7, 2, 1, 0), 
                          0, n#+2, Z#+2)))))"

text‹We call @{term p} with arguments a, f, z by equating them with 
  the corresponding quantified variables with de Bruijn indices 2, 1, 0.›


lemma is_iterates_type [TC]:
     "[| p ∈ formula; x ∈ nat; y ∈ nat; z ∈ nat |] 
      ==> is_iterates_fm(p,x,y,z) ∈ formula"
by (simp add: is_iterates_fm_def) 

lemma sats_is_iterates_fm:
  assumes is_F_iff_sats:
      "!!a b c d e f g h i j k. 
              [| a ∈ A; b ∈ A; c ∈ A; d ∈ A; e ∈ A; f ∈ A; 
                 g ∈ A; h ∈ A; i ∈ A; j ∈ A; k ∈ A|]
              ==> is_F(a,b) ⟷
                  sats(A, p, Cons(b, Cons(a, Cons(c, Cons(d, Cons(e, Cons(f, 
                      Cons(g, Cons(h, Cons(i, Cons(j, Cons(k, env))))))))))))"
  shows 
      "[|x ∈ nat; y < length(env); z < length(env); env ∈ list(A)|]
       ==> sats(A, is_iterates_fm(p,x,y,z), env) ⟷
           is_iterates(##A, is_F, nth(x,env), nth(y,env), nth(z,env))"
apply (frule_tac x=z in lt_length_in_nat, assumption)  
apply (frule lt_length_in_nat, assumption)  
apply (simp add: is_iterates_fm_def is_iterates_def sats_is_nat_case_fm 
              is_F_iff_sats [symmetric] sats_is_wfrec_fm sats_iterates_MH_fm)
done


lemma is_iterates_iff_sats:
  assumes is_F_iff_sats:
      "!!a b c d e f g h i j k. 
              [| a ∈ A; b ∈ A; c ∈ A; d ∈ A; e ∈ A; f ∈ A; 
                 g ∈ A; h ∈ A; i ∈ A; j ∈ A; k ∈ A|]
              ==> is_F(a,b) ⟷
                  sats(A, p, Cons(b, Cons(a, Cons(c, Cons(d, Cons(e, Cons(f, 
                      Cons(g, Cons(h, Cons(i, Cons(j, Cons(k, env))))))))))))"
  shows 
  "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z; 
      i ∈ nat; j < length(env); k < length(env); env ∈ list(A)|]
   ==> is_iterates(##A, is_F, x, y, z) ⟷
       sats(A, is_iterates_fm(p,i,j,k), env)"
by (simp add: sats_is_iterates_fm [OF is_F_iff_sats]) 

text‹The second argument of @{term p} gives it direct access to @{term x},
  which is essential for handling free variable references.  Without this
  argument, we cannot prove reflection for @{term list_N}.›
theorem is_iterates_reflection:
  assumes p_reflection:
    "!!f g h. REFLECTS[λx. p(L, h(x), f(x), g(x)),
                     λi x. p(##Lset(i), h(x), f(x), g(x))]"
 shows "REFLECTS[λx. is_iterates(L, p(L,x), f(x), g(x), h(x)),
               λi x. is_iterates(##Lset(i), p(##Lset(i),x), f(x), g(x), h(x))]"
apply (simp (no_asm_use) only: is_iterates_def)
apply (intro FOL_reflections function_reflections p_reflection
             is_wfrec_reflection iterates_MH_reflection)
done


subsubsection‹The Formula @{term is_eclose_n}, Internalized›

(* is_eclose_n(M,A,n,Z) == is_iterates(M, big_union(M), A, n, Z) *)

definition
  eclose_n_fm :: "[i,i,i]=>i" where
  "eclose_n_fm(A,n,Z) == is_iterates_fm(big_union_fm(1,0), A, n, Z)"

lemma eclose_n_fm_type [TC]:
 "[| x ∈ nat; y ∈ nat; z ∈ nat |] ==> eclose_n_fm(x,y,z) ∈ formula"
by (simp add: eclose_n_fm_def)

lemma sats_eclose_n_fm [simp]:
   "[| x ∈ nat; y < length(env); z < length(env); env ∈ list(A)|]
    ==> sats(A, eclose_n_fm(x,y,z), env) ⟷
        is_eclose_n(##A, nth(x,env), nth(y,env), nth(z,env))"
apply (frule_tac x=z in lt_length_in_nat, assumption)  
apply (frule_tac x=y in lt_length_in_nat, assumption)  
apply (simp add: eclose_n_fm_def is_eclose_n_def 
                 sats_is_iterates_fm) 
done

lemma eclose_n_iff_sats:
      "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z;
          i ∈ nat; j < length(env); k < length(env); env ∈ list(A)|]
       ==> is_eclose_n(##A, x, y, z) ⟷ sats(A, eclose_n_fm(i,j,k), env)"
by (simp add: sats_eclose_n_fm)

theorem eclose_n_reflection:
     "REFLECTS[λx. is_eclose_n(L, f(x), g(x), h(x)),  
               λi x. is_eclose_n(##Lset(i), f(x), g(x), h(x))]"
apply (simp only: is_eclose_n_def)
apply (intro FOL_reflections function_reflections is_iterates_reflection) 
done


subsubsection‹Membership in @{term "eclose(A)"}›

(* mem_eclose(M,A,l) == 
      ∃n[M]. ∃eclosen[M]. 
       finite_ordinal(M,n) & is_eclose_n(M,A,n,eclosen) & l ∈ eclosen *)
definition
  mem_eclose_fm :: "[i,i]=>i" where
    "mem_eclose_fm(x,y) ==
       Exists(Exists(
         And(finite_ordinal_fm(1),
           And(eclose_n_fm(x#+2,1,0), Member(y#+2,0)))))"

lemma mem_eclose_type [TC]:
     "[| x ∈ nat; y ∈ nat |] ==> mem_eclose_fm(x,y) ∈ formula"
by (simp add: mem_eclose_fm_def)

lemma sats_mem_eclose_fm [simp]:
   "[| x ∈ nat; y ∈ nat; env ∈ list(A)|]
    ==> sats(A, mem_eclose_fm(x,y), env) ⟷ mem_eclose(##A, nth(x,env), nth(y,env))"
by (simp add: mem_eclose_fm_def mem_eclose_def)

lemma mem_eclose_iff_sats:
      "[| nth(i,env) = x; nth(j,env) = y;
          i ∈ nat; j ∈ nat; env ∈ list(A)|]
       ==> mem_eclose(##A, x, y) ⟷ sats(A, mem_eclose_fm(i,j), env)"
by simp

theorem mem_eclose_reflection:
     "REFLECTS[λx. mem_eclose(L,f(x),g(x)),
               λi x. mem_eclose(##Lset(i),f(x),g(x))]"
apply (simp only: mem_eclose_def)
apply (intro FOL_reflections finite_ordinal_reflection eclose_n_reflection)
done


subsubsection‹The Predicate ``Is @{term "eclose(A)"}''›

(* is_eclose(M,A,Z) == ∀l[M]. l ∈ Z ⟷ mem_eclose(M,A,l) *)
definition
  is_eclose_fm :: "[i,i]=>i" where
    "is_eclose_fm(A,Z) ==
       Forall(Iff(Member(0,succ(Z)), mem_eclose_fm(succ(A),0)))"

lemma is_eclose_type [TC]:
     "[| x ∈ nat; y ∈ nat |] ==> is_eclose_fm(x,y) ∈ formula"
by (simp add: is_eclose_fm_def)

lemma sats_is_eclose_fm [simp]:
   "[| x ∈ nat; y ∈ nat; env ∈ list(A)|]
    ==> sats(A, is_eclose_fm(x,y), env) ⟷ is_eclose(##A, nth(x,env), nth(y,env))"
by (simp add: is_eclose_fm_def is_eclose_def)

lemma is_eclose_iff_sats:
      "[| nth(i,env) = x; nth(j,env) = y;
          i ∈ nat; j ∈ nat; env ∈ list(A)|]
       ==> is_eclose(##A, x, y) ⟷ sats(A, is_eclose_fm(i,j), env)"
by simp

theorem is_eclose_reflection:
     "REFLECTS[λx. is_eclose(L,f(x),g(x)),
               λi x. is_eclose(##Lset(i),f(x),g(x))]"
apply (simp only: is_eclose_def)
apply (intro FOL_reflections mem_eclose_reflection)
done


subsubsection‹The List Functor, Internalized›

definition
  list_functor_fm :: "[i,i,i]=>i" where
(* "is_list_functor(M,A,X,Z) ==
        ∃n1[M]. ∃AX[M].
         number1(M,n1) & cartprod(M,A,X,AX) & is_sum(M,n1,AX,Z)" *)
    "list_functor_fm(A,X,Z) ==
       Exists(Exists(
        And(number1_fm(1),
            And(cartprod_fm(A#+2,X#+2,0), sum_fm(1,0,Z#+2)))))"

lemma list_functor_type [TC]:
     "[| x ∈ nat; y ∈ nat; z ∈ nat |] ==> list_functor_fm(x,y,z) ∈ formula"
by (simp add: list_functor_fm_def)

lemma sats_list_functor_fm [simp]:
   "[| x ∈ nat; y ∈ nat; z ∈ nat; env ∈ list(A)|]
    ==> sats(A, list_functor_fm(x,y,z), env) ⟷
        is_list_functor(##A, nth(x,env), nth(y,env), nth(z,env))"
by (simp add: list_functor_fm_def is_list_functor_def)

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

theorem list_functor_reflection:
     "REFLECTS[λx. is_list_functor(L,f(x),g(x),h(x)),
               λi x. is_list_functor(##Lset(i),f(x),g(x),h(x))]"
apply (simp only: is_list_functor_def)
apply (intro FOL_reflections number1_reflection
             cartprod_reflection sum_reflection)
done


subsubsection‹The Formula @{term is_list_N}, Internalized›

(* "is_list_N(M,A,n,Z) == 
      ∃zero[M]. empty(M,zero) & 
                is_iterates(M, is_list_functor(M,A), zero, n, Z)" *)

definition
  list_N_fm :: "[i,i,i]=>i" where
  "list_N_fm(A,n,Z) == 
     Exists(
       And(empty_fm(0),
           is_iterates_fm(list_functor_fm(A#+9#+3,1,0), 0, n#+1, Z#+1)))"

lemma list_N_fm_type [TC]:
 "[| x ∈ nat; y ∈ nat; z ∈ nat |] ==> list_N_fm(x,y,z) ∈ formula"
by (simp add: list_N_fm_def)

lemma sats_list_N_fm [simp]:
   "[| x ∈ nat; y < length(env); z < length(env); env ∈ list(A)|]
    ==> sats(A, list_N_fm(x,y,z), env) ⟷
        is_list_N(##A, nth(x,env), nth(y,env), nth(z,env))"
apply (frule_tac x=z in lt_length_in_nat, assumption)  
apply (frule_tac x=y in lt_length_in_nat, assumption)  
apply (simp add: list_N_fm_def is_list_N_def sats_is_iterates_fm) 
done

lemma list_N_iff_sats:
      "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z;
          i ∈ nat; j < length(env); k < length(env); env ∈ list(A)|]
       ==> is_list_N(##A, x, y, z) ⟷ sats(A, list_N_fm(i,j,k), env)"
by (simp add: sats_list_N_fm)

theorem list_N_reflection:
     "REFLECTS[λx. is_list_N(L, f(x), g(x), h(x)),  
               λi x. is_list_N(##Lset(i), f(x), g(x), h(x))]"
apply (simp only: is_list_N_def)
apply (intro FOL_reflections function_reflections 
             is_iterates_reflection list_functor_reflection) 
done



subsubsection‹The Predicate ``Is A List''›

(* mem_list(M,A,l) == 
      ∃n[M]. ∃listn[M]. 
       finite_ordinal(M,n) & is_list_N(M,A,n,listn) & l ∈ listn *)
definition
  mem_list_fm :: "[i,i]=>i" where
    "mem_list_fm(x,y) ==
       Exists(Exists(
         And(finite_ordinal_fm(1),
           And(list_N_fm(x#+2,1,0), Member(y#+2,0)))))"

lemma mem_list_type [TC]:
     "[| x ∈ nat; y ∈ nat |] ==> mem_list_fm(x,y) ∈ formula"
by (simp add: mem_list_fm_def)

lemma sats_mem_list_fm [simp]:
   "[| x ∈ nat; y ∈ nat; env ∈ list(A)|]
    ==> sats(A, mem_list_fm(x,y), env) ⟷ mem_list(##A, nth(x,env), nth(y,env))"
by (simp add: mem_list_fm_def mem_list_def)

lemma mem_list_iff_sats:
      "[| nth(i,env) = x; nth(j,env) = y;
          i ∈ nat; j ∈ nat; env ∈ list(A)|]
       ==> mem_list(##A, x, y) ⟷ sats(A, mem_list_fm(i,j), env)"
by simp

theorem mem_list_reflection:
     "REFLECTS[λx. mem_list(L,f(x),g(x)),
               λi x. mem_list(##Lset(i),f(x),g(x))]"
apply (simp only: mem_list_def)
apply (intro FOL_reflections finite_ordinal_reflection list_N_reflection)
done


subsubsection‹The Predicate ``Is @{term "list(A)"}''›

(* is_list(M,A,Z) == ∀l[M]. l ∈ Z ⟷ mem_list(M,A,l) *)
definition
  is_list_fm :: "[i,i]=>i" where
    "is_list_fm(A,Z) ==
       Forall(Iff(Member(0,succ(Z)), mem_list_fm(succ(A),0)))"

lemma is_list_type [TC]:
     "[| x ∈ nat; y ∈ nat |] ==> is_list_fm(x,y) ∈ formula"
by (simp add: is_list_fm_def)

lemma sats_is_list_fm [simp]:
   "[| x ∈ nat; y ∈ nat; env ∈ list(A)|]
    ==> sats(A, is_list_fm(x,y), env) ⟷ is_list(##A, nth(x,env), nth(y,env))"
by (simp add: is_list_fm_def is_list_def)

lemma is_list_iff_sats:
      "[| nth(i,env) = x; nth(j,env) = y;
          i ∈ nat; j ∈ nat; env ∈ list(A)|]
       ==> is_list(##A, x, y) ⟷ sats(A, is_list_fm(i,j), env)"
by simp

theorem is_list_reflection:
     "REFLECTS[λx. is_list(L,f(x),g(x)),
               λi x. is_list(##Lset(i),f(x),g(x))]"
apply (simp only: is_list_def)
apply (intro FOL_reflections mem_list_reflection)
done


subsubsection‹The Formula Functor, Internalized›

definition formula_functor_fm :: "[i,i]=>i" where
(*     "is_formula_functor(M,X,Z) ==
        ∃nat'[M]. ∃natnat[M]. ∃natnatsum[M]. ∃XX[M]. ∃X3[M].
           4           3               2       1       0
          omega(M,nat') & cartprod(M,nat',nat',natnat) &
          is_sum(M,natnat,natnat,natnatsum) &
          cartprod(M,X,X,XX) & is_sum(M,XX,X,X3) &
          is_sum(M,natnatsum,X3,Z)" *)
    "formula_functor_fm(X,Z) ==
       Exists(Exists(Exists(Exists(Exists(
        And(omega_fm(4),
         And(cartprod_fm(4,4,3),
          And(sum_fm(3,3,2),
           And(cartprod_fm(X#+5,X#+5,1),
            And(sum_fm(1,X#+5,0), sum_fm(2,0,Z#+5)))))))))))"

lemma formula_functor_type [TC]:
     "[| x ∈ nat; y ∈ nat |] ==> formula_functor_fm(x,y) ∈ formula"
by (simp add: formula_functor_fm_def)

lemma sats_formula_functor_fm [simp]:
   "[| x ∈ nat; y ∈ nat; env ∈ list(A)|]
    ==> sats(A, formula_functor_fm(x,y), env) ⟷
        is_formula_functor(##A, nth(x,env), nth(y,env))"
by (simp add: formula_functor_fm_def is_formula_functor_def)

lemma formula_functor_iff_sats:
  "[| nth(i,env) = x; nth(j,env) = y;
      i ∈ nat; j ∈ nat; env ∈ list(A)|]
   ==> is_formula_functor(##A, x, y) ⟷ sats(A, formula_functor_fm(i,j), env)"
by simp

theorem formula_functor_reflection:
     "REFLECTS[λx. is_formula_functor(L,f(x),g(x)),
               λi x. is_formula_functor(##Lset(i),f(x),g(x))]"
apply (simp only: is_formula_functor_def)
apply (intro FOL_reflections omega_reflection
             cartprod_reflection sum_reflection)
done


subsubsection‹The Formula @{term is_formula_N}, Internalized›

(*  "is_formula_N(M,n,Z) == 
      ∃zero[M]. empty(M,zero) & 
                is_iterates(M, is_formula_functor(M), zero, n, Z)" *) 
definition
  formula_N_fm :: "[i,i]=>i" where
  "formula_N_fm(n,Z) == 
     Exists(
       And(empty_fm(0),
           is_iterates_fm(formula_functor_fm(1,0), 0, n#+1, Z#+1)))"

lemma formula_N_fm_type [TC]:
 "[| x ∈ nat; y ∈ nat |] ==> formula_N_fm(x,y) ∈ formula"
by (simp add: formula_N_fm_def)

lemma sats_formula_N_fm [simp]:
   "[| x < length(env); y < length(env); env ∈ list(A)|]
    ==> sats(A, formula_N_fm(x,y), env) ⟷
        is_formula_N(##A, nth(x,env), nth(y,env))"
apply (frule_tac x=y in lt_length_in_nat, assumption)  
apply (frule lt_length_in_nat, assumption)  
apply (simp add: formula_N_fm_def is_formula_N_def sats_is_iterates_fm) 
done

lemma formula_N_iff_sats:
      "[| nth(i,env) = x; nth(j,env) = y; 
          i < length(env); j < length(env); env ∈ list(A)|]
       ==> is_formula_N(##A, x, y) ⟷ sats(A, formula_N_fm(i,j), env)"
by (simp add: sats_formula_N_fm)

theorem formula_N_reflection:
     "REFLECTS[λx. is_formula_N(L, f(x), g(x)),  
               λi x. is_formula_N(##Lset(i), f(x), g(x))]"
apply (simp only: is_formula_N_def)
apply (intro FOL_reflections function_reflections 
             is_iterates_reflection formula_functor_reflection) 
done



subsubsection‹The Predicate ``Is A Formula''›

(*  mem_formula(M,p) == 
      ∃n[M]. ∃formn[M]. 
       finite_ordinal(M,n) & is_formula_N(M,n,formn) & p ∈ formn *)
definition
  mem_formula_fm :: "i=>i" where
    "mem_formula_fm(x) ==
       Exists(Exists(
         And(finite_ordinal_fm(1),
           And(formula_N_fm(1,0), Member(x#+2,0)))))"

lemma mem_formula_type [TC]:
     "x ∈ nat ==> mem_formula_fm(x) ∈ formula"
by (simp add: mem_formula_fm_def)

lemma sats_mem_formula_fm [simp]:
   "[| x ∈ nat; env ∈ list(A)|]
    ==> sats(A, mem_formula_fm(x), env) ⟷ mem_formula(##A, nth(x,env))"
by (simp add: mem_formula_fm_def mem_formula_def)

lemma mem_formula_iff_sats:
      "[| nth(i,env) = x; i ∈ nat; env ∈ list(A)|]
       ==> mem_formula(##A, x) ⟷ sats(A, mem_formula_fm(i), env)"
by simp

theorem mem_formula_reflection:
     "REFLECTS[λx. mem_formula(L,f(x)),
               λi x. mem_formula(##Lset(i),f(x))]"
apply (simp only: mem_formula_def)
apply (intro FOL_reflections finite_ordinal_reflection formula_N_reflection)
done



subsubsection‹The Predicate ``Is @{term "formula"}''›

(* is_formula(M,Z) == ∀p[M]. p ∈ Z ⟷ mem_formula(M,p) *)
definition
  is_formula_fm :: "i=>i" where
    "is_formula_fm(Z) == Forall(Iff(Member(0,succ(Z)), mem_formula_fm(0)))"

lemma is_formula_type [TC]:
     "x ∈ nat ==> is_formula_fm(x) ∈ formula"
by (simp add: is_formula_fm_def)

lemma sats_is_formula_fm [simp]:
   "[| x ∈ nat; env ∈ list(A)|]
    ==> sats(A, is_formula_fm(x), env) ⟷ is_formula(##A, nth(x,env))"
by (simp add: is_formula_fm_def is_formula_def)

lemma is_formula_iff_sats:
      "[| nth(i,env) = x; i ∈ nat; env ∈ list(A)|]
       ==> is_formula(##A, x) ⟷ sats(A, is_formula_fm(i), env)"
by simp

theorem is_formula_reflection:
     "REFLECTS[λx. is_formula(L,f(x)),
               λi x. is_formula(##Lset(i),f(x))]"
apply (simp only: is_formula_def)
apply (intro FOL_reflections mem_formula_reflection)
done


subsubsection‹The Operator @{term is_transrec}›

text‹The three arguments of @{term p} are always 2, 1, 0.  It is buried
   within eight quantifiers!
   We call @{term p} with arguments a, f, z by equating them with 
  the corresponding quantified variables with de Bruijn indices 2, 1, 0.›

(* is_transrec :: "[i=>o, [i,i,i]=>o, i, i] => o"
   "is_transrec(M,MH,a,z) == 
      ∃sa[M]. ∃esa[M]. ∃mesa[M]. 
       2       1         0
       upair(M,a,a,sa) & is_eclose(M,sa,esa) & membership(M,esa,mesa) &
       is_wfrec(M,MH,mesa,a,z)" *)
definition
  is_transrec_fm :: "[i, i, i]=>i" where
 "is_transrec_fm(p,a,z) == 
    Exists(Exists(Exists(
      And(upair_fm(a#+3,a#+3,2),
       And(is_eclose_fm(2,1),
        And(Memrel_fm(1,0), is_wfrec_fm(p,0,a#+3,z#+3)))))))"


lemma is_transrec_type [TC]:
     "[| p ∈ formula; x ∈ nat; z ∈ nat |] 
      ==> is_transrec_fm(p,x,z) ∈ formula"
by (simp add: is_transrec_fm_def) 

lemma sats_is_transrec_fm:
  assumes MH_iff_sats: 
      "!!a0 a1 a2 a3 a4 a5 a6 a7. 
        [|a0∈A; a1∈A; a2∈A; a3∈A; a4∈A; a5∈A; a6∈A; a7∈A|] 
        ==> MH(a2, a1, a0) ⟷ 
            sats(A, p, Cons(a0,Cons(a1,Cons(a2,Cons(a3,
                          Cons(a4,Cons(a5,Cons(a6,Cons(a7,env)))))))))"
  shows 
      "[|x < length(env); z < length(env); env ∈ list(A)|]
       ==> sats(A, is_transrec_fm(p,x,z), env) ⟷ 
           is_transrec(##A, MH, nth(x,env), nth(z,env))"
apply (frule_tac x=z in lt_length_in_nat, assumption)  
apply (frule_tac x=x in lt_length_in_nat, assumption)  
apply (simp add: is_transrec_fm_def sats_is_wfrec_fm is_transrec_def MH_iff_sats [THEN iff_sym]) 
done


lemma is_transrec_iff_sats:
  assumes MH_iff_sats: 
      "!!a0 a1 a2 a3 a4 a5 a6 a7. 
        [|a0∈A; a1∈A; a2∈A; a3∈A; a4∈A; a5∈A; a6∈A; a7∈A|] 
        ==> MH(a2, a1, a0) ⟷ 
            sats(A, p, Cons(a0,Cons(a1,Cons(a2,Cons(a3,
                          Cons(a4,Cons(a5,Cons(a6,Cons(a7,env)))))))))"
  shows
  "[|nth(i,env) = x; nth(k,env) = z; 
      i < length(env); k < length(env); env ∈ list(A)|]
   ==> is_transrec(##A, MH, x, z) ⟷ sats(A, is_transrec_fm(p,i,k), env)" 
by (simp add: sats_is_transrec_fm [OF MH_iff_sats])

theorem is_transrec_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_transrec(L, MH(L,x), f(x), h(x)), 
               λi x. is_transrec(##Lset(i), MH(##Lset(i),x), f(x), h(x))]"
apply (simp (no_asm_use) only: is_transrec_def)
apply (intro FOL_reflections function_reflections MH_reflection 
             is_wfrec_reflection is_eclose_reflection)
done

end