Theory L_axioms (Isabelle2005: October 2005)

# Theory L_axioms

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theory L_axioms
imports Formula Relative Reflection MetaExists
begin

    (*  Title:      ZF/Constructible/L_axioms.thy
        ID:         $Id: L_axioms.thy,v 1.38 2005/06/17 14:15:10 haftmann Exp $
        Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
    *)

    header {* The ZF Axioms (Except Separation) in L *}

    theory L_axioms imports Formula Relative Reflection MetaExists begin

    text {* The class L satisfies the premises of locale @{text M_trivial} *}

    lemma transL: "[| y∈x; L(x) |] ==> L(y)"
    apply (insert Transset_Lset)
    apply (simp add: Transset_def L_def, blast)
    done

    lemma nonempty: "L(0)"
    apply (simp add: L_def)
    apply (blast intro: zero_in_Lset)
    done

    theorem upair_ax: "upair_ax(L)"
    apply (simp add: upair_ax_def upair_def, clarify)
    apply (rule_tac x="{x,y}" in rexI)
    apply (simp_all add: doubleton_in_L)
    done

    theorem Union_ax: "Union_ax(L)"
    apply (simp add: Union_ax_def big_union_def, clarify)
    apply (rule_tac x="Union(x)" in rexI)
    apply (simp_all add: Union_in_L, auto)
    apply (blast intro: transL)
    done

    theorem power_ax: "power_ax(L)"
    apply (simp add: power_ax_def powerset_def Relative.subset_def, clarify)
    apply (rule_tac x="{y ∈ Pow(x). L(y)}" in rexI)
    apply (simp_all add: LPow_in_L, auto)
    apply (blast intro: transL)
    done

    text{*We don't actually need @{term L} to satisfy the foundation axiom.*}
    theorem foundation_ax: "foundation_ax(L)"
    apply (simp add: foundation_ax_def)
    apply (rule rallI)
    apply (cut_tac A=x in foundation)
    apply (blast intro: transL)
    done

    subsection{*For L to satisfy Replacement *}

    (*Can't move these to Formula unless the definition of univalent is moved
    there too!*)

    lemma LReplace_in_Lset:
         "[|X ∈ Lset(i); univalent(L,X,Q); Ord(i)|]
          ==> ∃j. Ord(j) & Replace(X, %x y. Q(x,y) & L(y)) ⊆ Lset(j)"
    apply (rule_tac x="\<Union>y ∈ Replace(X, %x y. Q(x,y) & L(y)). succ(lrank(y))"
           in exI)
    apply simp
    apply clarify
    apply (rule_tac a=x in UN_I)
     apply (simp_all add: Replace_iff univalent_def)
    apply (blast dest: transL L_I)
    done

    lemma LReplace_in_L:
         "[|L(X); univalent(L,X,Q)|]
          ==> ∃Y. L(Y) & Replace(X, %x y. Q(x,y) & L(y)) ⊆ Y"
    apply (drule L_D, clarify)
    apply (drule LReplace_in_Lset, assumption+)
    apply (blast intro: L_I Lset_in_Lset_succ)
    done

    theorem replacement: "replacement(L,P)"
    apply (simp add: replacement_def, clarify)
    apply (frule LReplace_in_L, assumption+, clarify)
    apply (rule_tac x=Y in rexI)
    apply (simp_all add: Replace_iff univalent_def, blast)
    done

    subsection{*Instantiating the locale @{text M_trivial}*}
    text{*No instances of Separation yet.*}

    lemma Lset_mono_le: "mono_le_subset(Lset)"
    by (simp add: mono_le_subset_def le_imp_subset Lset_mono)

    lemma Lset_cont: "cont_Ord(Lset)"
    by (simp add: cont_Ord_def Limit_Lset_eq OUnion_def Limit_is_Ord)

    lemmas L_nat = Ord_in_L [OF Ord_nat]

    theorem M_trivial_L: "PROP M_trivial(L)"
      apply (rule M_trivial.intro)
           apply (erule (1) transL)
          apply (rule upair_ax)
         apply (rule Union_ax)
        apply (rule power_ax)
       apply (rule replacement)
      apply (rule L_nat)
      done

    interpretation M_trivial ["L"] by (rule M_trivial_L)

    (* Replaces the following declarations...
    lemmas rall_abs = M_trivial.rall_abs [OF M_trivial_L]
      and rex_abs = M_trivial.rex_abs [OF M_trivial_L]
    ...
    declare rall_abs [simp]
    declare rex_abs [simp]
    ...and dozens of similar ones.
    *)

    subsection{*Instantiation of the locale @{text reflection}*}

    text{*instances of locale constants*}
    constdefs
      L_F0 :: "[i=>o,i] => i"
        "L_F0(P,y) == μ b. (∃z. L(z) ∧ P(<y,z>)) --> (∃z∈Lset(b). P(<y,z>))"

      L_FF :: "[i=>o,i] => i"
        "L_FF(P)   == λa. \<Union>y∈Lset(a). L_F0(P,y)"

      L_ClEx :: "[i=>o,i] => o"
        "L_ClEx(P) == λa. Limit(a) ∧ normalize(L_FF(P),a) = a"

    text{*We must use the meta-existential quantifier; otherwise the reflection
          terms become enormous!*}
    constdefs
      L_Reflects :: "[i=>o,[i,i]=>o] => prop"      ("(3REFLECTS/ [_,/ _])")
        "REFLECTS[P,Q] == (??Cl. Closed_Unbounded(Cl) &
                               (∀a. Cl(a) --> (∀x ∈ Lset(a). P(x) <-> Q(a,x))))"

    theorem Triv_reflection:
         "REFLECTS[P, λa x. P(x)]"
    apply (simp add: L_Reflects_def)
    apply (rule meta_exI)
    apply (rule Closed_Unbounded_Ord)
    done

    theorem Not_reflection:
         "REFLECTS[P,Q] ==> REFLECTS[λx. ~P(x), λa x. ~Q(a,x)]"
    apply (unfold L_Reflects_def)
    apply (erule meta_exE)
    apply (rule_tac x=Cl in meta_exI, simp)
    done

    theorem And_reflection:
         "[| REFLECTS[P,Q]; REFLECTS[P',Q'] |]
          ==> REFLECTS[λx. P(x) ∧ P'(x), λa x. Q(a,x) ∧ Q'(a,x)]"
    apply (unfold L_Reflects_def)
    apply (elim meta_exE)
    apply (rule_tac x="λa. Cl(a) ∧ Cla(a)" in meta_exI)
    apply (simp add: Closed_Unbounded_Int, blast)
    done

    theorem Or_reflection:
         "[| REFLECTS[P,Q]; REFLECTS[P',Q'] |]
          ==> REFLECTS[λx. P(x) ∨ P'(x), λa x. Q(a,x) ∨ Q'(a,x)]"
    apply (unfold L_Reflects_def)
    apply (elim meta_exE)
    apply (rule_tac x="λa. Cl(a) ∧ Cla(a)" in meta_exI)
    apply (simp add: Closed_Unbounded_Int, blast)
    done

    theorem Imp_reflection:
         "[| REFLECTS[P,Q]; REFLECTS[P',Q'] |]
          ==> REFLECTS[λx. P(x) --> P'(x), λa x. Q(a,x) --> Q'(a,x)]"
    apply (unfold L_Reflects_def)
    apply (elim meta_exE)
    apply (rule_tac x="λa. Cl(a) ∧ Cla(a)" in meta_exI)
    apply (simp add: Closed_Unbounded_Int, blast)
    done

    theorem Iff_reflection:
         "[| REFLECTS[P,Q]; REFLECTS[P',Q'] |]
          ==> REFLECTS[λx. P(x) <-> P'(x), λa x. Q(a,x) <-> Q'(a,x)]"
    apply (unfold L_Reflects_def)
    apply (elim meta_exE)
    apply (rule_tac x="λa. Cl(a) ∧ Cla(a)" in meta_exI)
    apply (simp add: Closed_Unbounded_Int, blast)
    done

    lemma reflection_Lset: "reflection(Lset)"
    by (blast intro: reflection.intro Lset_mono_le Lset_cont
                     Formula.Pair_in_LLimit)+

    theorem Ex_reflection:
         "REFLECTS[λx. P(fst(x),snd(x)), λa x. Q(a,fst(x),snd(x))]
          ==> REFLECTS[λx. ∃z. L(z) ∧ P(x,z), λa x. ∃z∈Lset(a). Q(a,x,z)]"
    apply (unfold L_Reflects_def L_ClEx_def L_FF_def L_F0_def L_def)
    apply (elim meta_exE)
    apply (rule meta_exI)
    apply (erule reflection.Ex_reflection [OF reflection_Lset])
    done

    theorem All_reflection:
         "REFLECTS[λx. P(fst(x),snd(x)), λa x. Q(a,fst(x),snd(x))]
          ==> REFLECTS[λx. ∀z. L(z) --> P(x,z), λa x. ∀z∈Lset(a). Q(a,x,z)]"
    apply (unfold L_Reflects_def L_ClEx_def L_FF_def L_F0_def L_def)
    apply (elim meta_exE)
    apply (rule meta_exI)
    apply (erule reflection.All_reflection [OF reflection_Lset])
    done

    theorem Rex_reflection:
         "REFLECTS[ λx. P(fst(x),snd(x)), λa x. Q(a,fst(x),snd(x))]
          ==> REFLECTS[λx. ∃z[L]. P(x,z), λa x. ∃z∈Lset(a). Q(a,x,z)]"
    apply (unfold rex_def)
    apply (intro And_reflection Ex_reflection, assumption)
    done

    theorem Rall_reflection:
         "REFLECTS[λx. P(fst(x),snd(x)), λa x. Q(a,fst(x),snd(x))]
          ==> REFLECTS[λx. ∀z[L]. P(x,z), λa x. ∀z∈Lset(a). Q(a,x,z)]"
    apply (unfold rall_def)
    apply (intro Imp_reflection All_reflection, assumption)
    done

    text{*This version handles an alternative form of the bounded quantifier
          in the second argument of @{text REFLECTS}.*}
    theorem Rex_reflection':
         "REFLECTS[λx. P(fst(x),snd(x)), λa x. Q(a,fst(x),snd(x))]
          ==> REFLECTS[λx. ∃z[L]. P(x,z), λa x. ∃z[##Lset(a)]. Q(a,x,z)]"
    apply (unfold setclass_def rex_def)
    apply (erule Rex_reflection [unfolded rex_def Bex_def])
    done

    text{*As above.*}
    theorem Rall_reflection':
         "REFLECTS[λx. P(fst(x),snd(x)), λa x. Q(a,fst(x),snd(x))]
          ==> REFLECTS[λx. ∀z[L]. P(x,z), λa x. ∀z[##Lset(a)]. Q(a,x,z)]"
    apply (unfold setclass_def rall_def)
    apply (erule Rall_reflection [unfolded rall_def Ball_def])
    done

    lemmas FOL_reflections =
            Triv_reflection Not_reflection And_reflection Or_reflection
            Imp_reflection Iff_reflection Ex_reflection All_reflection
            Rex_reflection Rall_reflection Rex_reflection' Rall_reflection'

    lemma ReflectsD:
         "[|REFLECTS[P,Q]; Ord(i)|]
          ==> ∃j. i<j & (∀x ∈ Lset(j). P(x) <-> Q(j,x))"
    apply (unfold L_Reflects_def Closed_Unbounded_def)
    apply (elim meta_exE, clarify)
    apply (blast dest!: UnboundedD)
    done

    lemma ReflectsE:
         "[| REFLECTS[P,Q]; Ord(i);
             !!j. [|i<j;  ∀x ∈ Lset(j). P(x) <-> Q(j,x)|] ==> R |]
          ==> R"
    by (drule ReflectsD, assumption, blast)

    lemma Collect_mem_eq: "{x∈A. x∈B} = A ∩ B"
    by blast

    subsection{*Internalized Formulas for some Set-Theoretic Concepts*}

    subsubsection{*Some numbers to help write de Bruijn indices*}

    syntax
        "3" :: i   ("3")
        "4" :: i   ("4")
        "5" :: i   ("5")
        "6" :: i   ("6")
        "7" :: i   ("7")
        "8" :: i   ("8")
        "9" :: i   ("9")

    translations
       "3"  == "succ(2)"
       "4"  == "succ(3)"
       "5"  == "succ(4)"
       "6"  == "succ(5)"
       "7"  == "succ(6)"
       "8"  == "succ(7)"
       "9"  == "succ(8)"

    subsubsection{*The Empty Set, Internalized*}

    constdefs empty_fm :: "i=>i"
        "empty_fm(x) == Forall(Neg(Member(0,succ(x))))"

    lemma empty_type [TC]:
         "x ∈ nat ==> empty_fm(x) ∈ formula"
    by (simp add: empty_fm_def)

    lemma sats_empty_fm [simp]:
       "[| x ∈ nat; env ∈ list(A)|]
        ==> sats(A, empty_fm(x), env) <-> empty(##A, nth(x,env))"
    by (simp add: empty_fm_def empty_def)

    lemma empty_iff_sats:
          "[| nth(i,env) = x; nth(j,env) = y;
              i ∈ nat; env ∈ list(A)|]
           ==> empty(##A, x) <-> sats(A, empty_fm(i), env)"
    by simp

    theorem empty_reflection:
         "REFLECTS[λx. empty(L,f(x)),
                   λi x. empty(##Lset(i),f(x))]"
    apply (simp only: empty_def)
    apply (intro FOL_reflections)
    done

    text{*Not used.  But maybe useful?*}
    lemma Transset_sats_empty_fm_eq_0:
       "[| n ∈ nat; env ∈ list(A); Transset(A)|]
        ==> sats(A, empty_fm(n), env) <-> nth(n,env) = 0"
    apply (simp add: empty_fm_def empty_def Transset_def, auto)
    apply (case_tac "n < length(env)")
    apply (frule nth_type, assumption+, blast)
    apply (simp_all add: not_lt_iff_le nth_eq_0)
    done

    subsubsection{*Unordered Pairs, Internalized*}

    constdefs upair_fm :: "[i,i,i]=>i"
        "upair_fm(x,y,z) ==
           And(Member(x,z),
               And(Member(y,z),
                   Forall(Implies(Member(0,succ(z)),
                                  Or(Equal(0,succ(x)), Equal(0,succ(y)))))))"

    lemma upair_type [TC]:
         "[| x ∈ nat; y ∈ nat; z ∈ nat |] ==> upair_fm(x,y,z) ∈ formula"
    by (simp add: upair_fm_def)

    lemma sats_upair_fm [simp]:
       "[| x ∈ nat; y ∈ nat; z ∈ nat; env ∈ list(A)|]
        ==> sats(A, upair_fm(x,y,z), env) <->
                upair(##A, nth(x,env), nth(y,env), nth(z,env))"
    by (simp add: upair_fm_def upair_def)

    lemma upair_iff_sats:
          "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z;
              i ∈ nat; j ∈ nat; k ∈ nat; env ∈ list(A)|]
           ==> upair(##A, x, y, z) <-> sats(A, upair_fm(i,j,k), env)"
    by (simp add: sats_upair_fm)

    text{*Useful? At least it refers to "real" unordered pairs*}
    lemma sats_upair_fm2 [simp]:
       "[| x ∈ nat; y ∈ nat; z < length(env); env ∈ list(A); Transset(A)|]
        ==> sats(A, upair_fm(x,y,z), env) <->
            nth(z,env) = {nth(x,env), nth(y,env)}"
    apply (frule lt_length_in_nat, assumption)
    apply (simp add: upair_fm_def Transset_def, auto)
    apply (blast intro: nth_type)
    done

    theorem upair_reflection:
         "REFLECTS[λx. upair(L,f(x),g(x),h(x)),
                   λi x. upair(##Lset(i),f(x),g(x),h(x))]"
    apply (simp add: upair_def)
    apply (intro FOL_reflections)
    done

    subsubsection{*Ordered pairs, Internalized*}

    constdefs pair_fm :: "[i,i,i]=>i"
        "pair_fm(x,y,z) ==
           Exists(And(upair_fm(succ(x),succ(x),0),
                  Exists(And(upair_fm(succ(succ(x)),succ(succ(y)),0),
                             upair_fm(1,0,succ(succ(z)))))))"

    lemma pair_type [TC]:
         "[| x ∈ nat; y ∈ nat; z ∈ nat |] ==> pair_fm(x,y,z) ∈ formula"
    by (simp add: pair_fm_def)

    lemma sats_pair_fm [simp]:
       "[| x ∈ nat; y ∈ nat; z ∈ nat; env ∈ list(A)|]
        ==> sats(A, pair_fm(x,y,z), env) <->
            pair(##A, nth(x,env), nth(y,env), nth(z,env))"
    by (simp add: pair_fm_def pair_def)

    lemma pair_iff_sats:
          "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z;
              i ∈ nat; j ∈ nat; k ∈ nat; env ∈ list(A)|]
           ==> pair(##A, x, y, z) <-> sats(A, pair_fm(i,j,k), env)"
    by (simp add: sats_pair_fm)

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

    subsubsection{*Binary Unions, Internalized*}

    constdefs union_fm :: "[i,i,i]=>i"
        "union_fm(x,y,z) ==
           Forall(Iff(Member(0,succ(z)),
                      Or(Member(0,succ(x)),Member(0,succ(y)))))"

    lemma union_type [TC]:
         "[| x ∈ nat; y ∈ nat; z ∈ nat |] ==> union_fm(x,y,z) ∈ formula"
    by (simp add: union_fm_def)

    lemma sats_union_fm [simp]:
       "[| x ∈ nat; y ∈ nat; z ∈ nat; env ∈ list(A)|]
        ==> sats(A, union_fm(x,y,z), env) <->
            union(##A, nth(x,env), nth(y,env), nth(z,env))"
    by (simp add: union_fm_def union_def)

    lemma union_iff_sats:
          "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z;
              i ∈ nat; j ∈ nat; k ∈ nat; env ∈ list(A)|]
           ==> union(##A, x, y, z) <-> sats(A, union_fm(i,j,k), env)"
    by (simp add: sats_union_fm)

    theorem union_reflection:
         "REFLECTS[λx. union(L,f(x),g(x),h(x)),
                   λi x. union(##Lset(i),f(x),g(x),h(x))]"
    apply (simp only: union_def)
    apply (intro FOL_reflections)
    done

    subsubsection{*Set ``Cons,'' Internalized*}

    constdefs cons_fm :: "[i,i,i]=>i"
        "cons_fm(x,y,z) ==
           Exists(And(upair_fm(succ(x),succ(x),0),
                      union_fm(0,succ(y),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 upair_reflection union_reflection)
    done

    subsubsection{*Successor Function, Internalized*}

    constdefs succ_fm :: "[i,i]=>i"
        "succ_fm(x,y) == cons_fm(x,x,y)"

    lemma succ_type [TC]:
         "[| x ∈ nat; y ∈ nat |] ==> succ_fm(x,y) ∈ formula"
    by (simp add: succ_fm_def)

    lemma sats_succ_fm [simp]:
       "[| x ∈ nat; y ∈ nat; env ∈ list(A)|]
        ==> sats(A, succ_fm(x,y), env) <->
            successor(##A, nth(x,env), nth(y,env))"
    by (simp add: succ_fm_def successor_def)

    lemma successor_iff_sats:
          "[| nth(i,env) = x; nth(j,env) = y;
              i ∈ nat; j ∈ nat; env ∈ list(A)|]
           ==> successor(##A, x, y) <-> sats(A, succ_fm(i,j), env)"
    by simp

    theorem successor_reflection:
         "REFLECTS[λx. successor(L,f(x),g(x)),
                   λi x. successor(##Lset(i),f(x),g(x))]"
    apply (simp only: successor_def)
    apply (intro cons_reflection)
    done

    subsubsection{*The Number 1, Internalized*}

    (* "number1(M,a) == (∃x[M]. empty(M,x) & successor(M,x,a))" *)
    constdefs number1_fm :: "i=>i"
        "number1_fm(a) == Exists(And(empty_fm(0), succ_fm(0,succ(a))))"

    lemma number1_type [TC]:
         "x ∈ nat ==> number1_fm(x) ∈ formula"
    by (simp add: number1_fm_def)

    lemma sats_number1_fm [simp]:
       "[| x ∈ nat; env ∈ list(A)|]
        ==> sats(A, number1_fm(x), env) <-> number1(##A, nth(x,env))"
    by (simp add: number1_fm_def number1_def)

    lemma number1_iff_sats:
          "[| nth(i,env) = x; nth(j,env) = y;
              i ∈ nat; env ∈ list(A)|]
           ==> number1(##A, x) <-> sats(A, number1_fm(i), env)"
    by simp

    theorem number1_reflection:
         "REFLECTS[λx. number1(L,f(x)),
                   λi x. number1(##Lset(i),f(x))]"
    apply (simp only: number1_def)
    apply (intro FOL_reflections empty_reflection successor_reflection)
    done

    subsubsection{*Big Union, Internalized*}

    (*  "big_union(M,A,z) == ∀x[M]. x ∈ z <-> (∃y[M]. y∈A & x ∈ y)" *)
    constdefs big_union_fm :: "[i,i]=>i"
        "big_union_fm(A,z) ==
           Forall(Iff(Member(0,succ(z)),
                      Exists(And(Member(0,succ(succ(A))), Member(1,0)))))"

    lemma big_union_type [TC]:
         "[| x ∈ nat; y ∈ nat |] ==> big_union_fm(x,y) ∈ formula"
    by (simp add: big_union_fm_def)

    lemma sats_big_union_fm [simp]:
       "[| x ∈ nat; y ∈ nat; env ∈ list(A)|]
        ==> sats(A, big_union_fm(x,y), env) <->
            big_union(##A, nth(x,env), nth(y,env))"
    by (simp add: big_union_fm_def big_union_def)

    lemma big_union_iff_sats:
          "[| nth(i,env) = x; nth(j,env) = y;
              i ∈ nat; j ∈ nat; env ∈ list(A)|]
           ==> big_union(##A, x, y) <-> sats(A, big_union_fm(i,j), env)"
    by simp

    theorem big_union_reflection:
         "REFLECTS[λx. big_union(L,f(x),g(x)),
                   λi x. big_union(##Lset(i),f(x),g(x))]"
    apply (simp only: big_union_def)
    apply (intro FOL_reflections)
    done

    subsubsection{*Variants of Satisfaction Definitions for Ordinals, etc.*}

    text{*The @{text sats} theorems below are standard versions of the ones proved
    in theory @{text Formula}.  They relate elements of type @{term formula} to
    relativized concepts such as @{term subset} or @{term ordinal} rather than to
    real concepts such as @{term Ord}.  Now that we have instantiated the locale
    @{text M_trivial}, we no longer require the earlier versions.*}

    lemma sats_subset_fm':
       "[|x ∈ nat; y ∈ nat; env ∈ list(A)|]
        ==> sats(A, subset_fm(x,y), env) <-> subset(##A, nth(x,env), nth(y,env))"
    by (simp add: subset_fm_def Relative.subset_def)

    theorem subset_reflection:
         "REFLECTS[λx. subset(L,f(x),g(x)),
                   λi x. subset(##Lset(i),f(x),g(x))]"
    apply (simp only: Relative.subset_def)
    apply (intro FOL_reflections)
    done

    lemma sats_transset_fm':
       "[|x ∈ nat; env ∈ list(A)|]
        ==> sats(A, transset_fm(x), env) <-> transitive_set(##A, nth(x,env))"
    by (simp add: sats_subset_fm' transset_fm_def transitive_set_def)

    theorem transitive_set_reflection:
         "REFLECTS[λx. transitive_set(L,f(x)),
                   λi x. transitive_set(##Lset(i),f(x))]"
    apply (simp only: transitive_set_def)
    apply (intro FOL_reflections subset_reflection)
    done

    lemma sats_ordinal_fm':
       "[|x ∈ nat; env ∈ list(A)|]
        ==> sats(A, ordinal_fm(x), env) <-> ordinal(##A,nth(x,env))"
    by (simp add: sats_transset_fm' ordinal_fm_def ordinal_def)

    lemma ordinal_iff_sats:
          "[| nth(i,env) = x;  i ∈ nat; env ∈ list(A)|]
           ==> ordinal(##A, x) <-> sats(A, ordinal_fm(i), env)"
    by (simp add: sats_ordinal_fm')

    theorem ordinal_reflection:
         "REFLECTS[λx. ordinal(L,f(x)), λi x. ordinal(##Lset(i),f(x))]"
    apply (simp only: ordinal_def)
    apply (intro FOL_reflections transitive_set_reflection)
    done

    subsubsection{*Membership Relation, Internalized*}

    constdefs Memrel_fm :: "[i,i]=>i"
        "Memrel_fm(A,r) ==
           Forall(Iff(Member(0,succ(r)),
                      Exists(And(Member(0,succ(succ(A))),
                                 Exists(And(Member(0,succ(succ(succ(A)))),
                                            And(Member(1,0),
                                                pair_fm(1,0,2))))))))"

    lemma Memrel_type [TC]:
         "[| x ∈ nat; y ∈ nat |] ==> Memrel_fm(x,y) ∈ formula"
    by (simp add: Memrel_fm_def)

    lemma sats_Memrel_fm [simp]:
       "[| x ∈ nat; y ∈ nat; env ∈ list(A)|]
        ==> sats(A, Memrel_fm(x,y), env) <->
            membership(##A, nth(x,env), nth(y,env))"
    by (simp add: Memrel_fm_def membership_def)

    lemma Memrel_iff_sats:
          "[| nth(i,env) = x; nth(j,env) = y;
              i ∈ nat; j ∈ nat; env ∈ list(A)|]
           ==> membership(##A, x, y) <-> sats(A, Memrel_fm(i,j), env)"
    by simp

    theorem membership_reflection:
         "REFLECTS[λx. membership(L,f(x),g(x)),
                   λi x. membership(##Lset(i),f(x),g(x))]"
    apply (simp only: membership_def)
    apply (intro FOL_reflections pair_reflection)
    done

    subsubsection{*Predecessor Set, Internalized*}

    constdefs pred_set_fm :: "[i,i,i,i]=>i"
        "pred_set_fm(A,x,r,B) ==
           Forall(Iff(Member(0,succ(B)),
                      Exists(And(Member(0,succ(succ(r))),
                                 And(Member(1,succ(succ(A))),
                                     pair_fm(1,succ(succ(x)),0))))))"

    lemma pred_set_type [TC]:
         "[| A ∈ nat; x ∈ nat; r ∈ nat; B ∈ nat |]
          ==> pred_set_fm(A,x,r,B) ∈ formula"
    by (simp add: pred_set_fm_def)

    lemma sats_pred_set_fm [simp]:
       "[| U ∈ nat; x ∈ nat; r ∈ nat; B ∈ nat; env ∈ list(A)|]
        ==> sats(A, pred_set_fm(U,x,r,B), env) <->
            pred_set(##A, nth(U,env), nth(x,env), nth(r,env), nth(B,env))"
    by (simp add: pred_set_fm_def pred_set_def)

    lemma pred_set_iff_sats:
          "[| nth(i,env) = U; nth(j,env) = x; nth(k,env) = r; nth(l,env) = B;
              i ∈ nat; j ∈ nat; k ∈ nat; l ∈ nat; env ∈ list(A)|]
           ==> pred_set(##A,U,x,r,B) <-> sats(A, pred_set_fm(i,j,k,l), env)"
    by (simp add: sats_pred_set_fm)

    theorem pred_set_reflection:
         "REFLECTS[λx. pred_set(L,f(x),g(x),h(x),b(x)),
                   λi x. pred_set(##Lset(i),f(x),g(x),h(x),b(x))]"
    apply (simp only: pred_set_def)
    apply (intro FOL_reflections pair_reflection)
    done

    subsubsection{*Domain of a Relation, Internalized*}

    (* "is_domain(M,r,z) ==
            ∀x[M]. (x ∈ z <-> (∃w[M]. w∈r & (∃y[M]. pair(M,x,y,w))))" *)
    constdefs domain_fm :: "[i,i]=>i"
        "domain_fm(r,z) ==
           Forall(Iff(Member(0,succ(z)),
                      Exists(And(Member(0,succ(succ(r))),
                                 Exists(pair_fm(2,0,1))))))"

    lemma domain_type [TC]:
         "[| x ∈ nat; y ∈ nat |] ==> domain_fm(x,y) ∈ formula"
    by (simp add: domain_fm_def)

    lemma sats_domain_fm [simp]:
       "[| x ∈ nat; y ∈ nat; env ∈ list(A)|]
        ==> sats(A, domain_fm(x,y), env) <->
            is_domain(##A, nth(x,env), nth(y,env))"
    by (simp add: domain_fm_def is_domain_def)

    lemma domain_iff_sats:
          "[| nth(i,env) = x; nth(j,env) = y;
              i ∈ nat; j ∈ nat; env ∈ list(A)|]
           ==> is_domain(##A, x, y) <-> sats(A, domain_fm(i,j), env)"
    by simp

    theorem domain_reflection:
         "REFLECTS[λx. is_domain(L,f(x),g(x)),
                   λi x. is_domain(##Lset(i),f(x),g(x))]"
    apply (simp only: is_domain_def)
    apply (intro FOL_reflections pair_reflection)
    done

    subsubsection{*Range of a Relation, Internalized*}

    (* "is_range(M,r,z) ==
            ∀y[M]. (y ∈ z <-> (∃w[M]. w∈r & (∃x[M]. pair(M,x,y,w))))" *)
    constdefs range_fm :: "[i,i]=>i"
        "range_fm(r,z) ==
           Forall(Iff(Member(0,succ(z)),
                      Exists(And(Member(0,succ(succ(r))),
                                 Exists(pair_fm(0,2,1))))))"

    lemma range_type [TC]:
         "[| x ∈ nat; y ∈ nat |] ==> range_fm(x,y) ∈ formula"
    by (simp add: range_fm_def)

    lemma sats_range_fm [simp]:
       "[| x ∈ nat; y ∈ nat; env ∈ list(A)|]
        ==> sats(A, range_fm(x,y), env) <->
            is_range(##A, nth(x,env), nth(y,env))"
    by (simp add: range_fm_def is_range_def)

    lemma range_iff_sats:
          "[| nth(i,env) = x; nth(j,env) = y;
              i ∈ nat; j ∈ nat; env ∈ list(A)|]
           ==> is_range(##A, x, y) <-> sats(A, range_fm(i,j), env)"
    by simp

    theorem range_reflection:
         "REFLECTS[λx. is_range(L,f(x),g(x)),
                   λi x. is_range(##Lset(i),f(x),g(x))]"
    apply (simp only: is_range_def)
    apply (intro FOL_reflections pair_reflection)
    done

    subsubsection{*Field of a Relation, Internalized*}

    (* "is_field(M,r,z) ==
            ∃dr[M]. is_domain(M,r,dr) &
                (∃rr[M]. is_range(M,r,rr) & union(M,dr,rr,z))" *)
    constdefs field_fm :: "[i,i]=>i"
        "field_fm(r,z) ==
           Exists(And(domain_fm(succ(r),0),
                  Exists(And(range_fm(succ(succ(r)),0),
                             union_fm(1,0,succ(succ(z)))))))"

    lemma field_type [TC]:
         "[| x ∈ nat; y ∈ nat |] ==> field_fm(x,y) ∈ formula"
    by (simp add: field_fm_def)

    lemma sats_field_fm [simp]:
       "[| x ∈ nat; y ∈ nat; env ∈ list(A)|]
        ==> sats(A, field_fm(x,y), env) <->
            is_field(##A, nth(x,env), nth(y,env))"
    by (simp add: field_fm_def is_field_def)

    lemma field_iff_sats:
          "[| nth(i,env) = x; nth(j,env) = y;
              i ∈ nat; j ∈ nat; env ∈ list(A)|]
           ==> is_field(##A, x, y) <-> sats(A, field_fm(i,j), env)"
    by simp

    theorem field_reflection:
         "REFLECTS[λx. is_field(L,f(x),g(x)),
                   λi x. is_field(##Lset(i),f(x),g(x))]"
    apply (simp only: is_field_def)
    apply (intro FOL_reflections domain_reflection range_reflection
                 union_reflection)
    done

    subsubsection{*Image under a Relation, Internalized*}

    (* "image(M,r,A,z) ==
            ∀y[M]. (y ∈ z <-> (∃w[M]. w∈r & (∃x[M]. x∈A & pair(M,x,y,w))))" *)
    constdefs image_fm :: "[i,i,i]=>i"
        "image_fm(r,A,z) ==
           Forall(Iff(Member(0,succ(z)),
                      Exists(And(Member(0,succ(succ(r))),
                                 Exists(And(Member(0,succ(succ(succ(A)))),
                                            pair_fm(0,2,1)))))))"

    lemma image_type [TC]:
         "[| x ∈ nat; y ∈ nat; z ∈ nat |] ==> image_fm(x,y,z) ∈ formula"
    by (simp add: image_fm_def)

    lemma sats_image_fm [simp]:
       "[| x ∈ nat; y ∈ nat; z ∈ nat; env ∈ list(A)|]
        ==> sats(A, image_fm(x,y,z), env) <->
            image(##A, nth(x,env), nth(y,env), nth(z,env))"
    by (simp add: image_fm_def Relative.image_def)

    lemma image_iff_sats:
          "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z;
              i ∈ nat; j ∈ nat; k ∈ nat; env ∈ list(A)|]
           ==> image(##A, x, y, z) <-> sats(A, image_fm(i,j,k), env)"
    by (simp add: sats_image_fm)

    theorem image_reflection:
         "REFLECTS[λx. image(L,f(x),g(x),h(x)),
                   λi x. image(##Lset(i),f(x),g(x),h(x))]"
    apply (simp only: Relative.image_def)
    apply (intro FOL_reflections pair_reflection)
    done

    subsubsection{*Pre-Image under a Relation, Internalized*}

    (* "pre_image(M,r,A,z) ==
            ∀x[M]. x ∈ z <-> (∃w[M]. w∈r & (∃y[M]. y∈A & pair(M,x,y,w)))" *)
    constdefs pre_image_fm :: "[i,i,i]=>i"
        "pre_image_fm(r,A,z) ==
           Forall(Iff(Member(0,succ(z)),
                      Exists(And(Member(0,succ(succ(r))),
                                 Exists(And(Member(0,succ(succ(succ(A)))),
                                            pair_fm(2,0,1)))))))"

    lemma pre_image_type [TC]:
         "[| x ∈ nat; y ∈ nat; z ∈ nat |] ==> pre_image_fm(x,y,z) ∈ formula"
    by (simp add: pre_image_fm_def)

    lemma sats_pre_image_fm [simp]:
       "[| x ∈ nat; y ∈ nat; z ∈ nat; env ∈ list(A)|]
        ==> sats(A, pre_image_fm(x,y,z), env) <->
            pre_image(##A, nth(x,env), nth(y,env), nth(z,env))"
    by (simp add: pre_image_fm_def Relative.pre_image_def)

    lemma pre_image_iff_sats:
          "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z;
              i ∈ nat; j ∈ nat; k ∈ nat; env ∈ list(A)|]
           ==> pre_image(##A, x, y, z) <-> sats(A, pre_image_fm(i,j,k), env)"
    by (simp add: sats_pre_image_fm)

    theorem pre_image_reflection:
         "REFLECTS[λx. pre_image(L,f(x),g(x),h(x)),
                   λi x. pre_image(##Lset(i),f(x),g(x),h(x))]"
    apply (simp only: Relative.pre_image_def)
    apply (intro FOL_reflections pair_reflection)
    done

    subsubsection{*Function Application, Internalized*}

    (* "fun_apply(M,f,x,y) ==
            (∃xs[M]. ∃fxs[M].
             upair(M,x,x,xs) & image(M,f,xs,fxs) & big_union(M,fxs,y))" *)
    constdefs fun_apply_fm :: "[i,i,i]=>i"
        "fun_apply_fm(f,x,y) ==
           Exists(Exists(And(upair_fm(succ(succ(x)), succ(succ(x)), 1),
                             And(image_fm(succ(succ(f)), 1, 0),
                                 big_union_fm(0,succ(succ(y)))))))"

    lemma fun_apply_type [TC]:
         "[| x ∈ nat; y ∈ nat; z ∈ nat |] ==> fun_apply_fm(x,y,z) ∈ formula"
    by (simp add: fun_apply_fm_def)

    lemma sats_fun_apply_fm [simp]:
       "[| x ∈ nat; y ∈ nat; z ∈ nat; env ∈ list(A)|]
        ==> sats(A, fun_apply_fm(x,y,z), env) <->
            fun_apply(##A, nth(x,env), nth(y,env), nth(z,env))"
    by (simp add: fun_apply_fm_def fun_apply_def)

    lemma fun_apply_iff_sats:
          "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z;
              i ∈ nat; j ∈ nat; k ∈ nat; env ∈ list(A)|]
           ==> fun_apply(##A, x, y, z) <-> sats(A, fun_apply_fm(i,j,k), env)"
    by simp

    theorem fun_apply_reflection:
         "REFLECTS[λx. fun_apply(L,f(x),g(x),h(x)),
                   λi x. fun_apply(##Lset(i),f(x),g(x),h(x))]"
    apply (simp only: fun_apply_def)
    apply (intro FOL_reflections upair_reflection image_reflection
                 big_union_reflection)
    done

    subsubsection{*The Concept of Relation, Internalized*}

    (* "is_relation(M,r) ==
            (∀z[M]. z∈r --> (∃x[M]. ∃y[M]. pair(M,x,y,z)))" *)
    constdefs relation_fm :: "i=>i"
        "relation_fm(r) ==
           Forall(Implies(Member(0,succ(r)), Exists(Exists(pair_fm(1,0,2)))))"

    lemma relation_type [TC]:
         "[| x ∈ nat |] ==> relation_fm(x) ∈ formula"
    by (simp add: relation_fm_def)

    lemma sats_relation_fm [simp]:
       "[| x ∈ nat; env ∈ list(A)|]
        ==> sats(A, relation_fm(x), env) <-> is_relation(##A, nth(x,env))"
    by (simp add: relation_fm_def is_relation_def)

    lemma relation_iff_sats:
          "[| nth(i,env) = x; nth(j,env) = y;
              i ∈ nat; env ∈ list(A)|]
           ==> is_relation(##A, x) <-> sats(A, relation_fm(i), env)"
    by simp

    theorem is_relation_reflection:
         "REFLECTS[λx. is_relation(L,f(x)),
                   λi x. is_relation(##Lset(i),f(x))]"
    apply (simp only: is_relation_def)
    apply (intro FOL_reflections pair_reflection)
    done

    subsubsection{*The Concept of Function, Internalized*}

    (* "is_function(M,r) ==
            ∀x[M]. ∀y[M]. ∀y'[M]. ∀p[M]. ∀p'[M].
               pair(M,x,y,p) --> pair(M,x,y',p') --> p∈r --> p'∈r --> y=y'" *)
    constdefs function_fm :: "i=>i"
        "function_fm(r) ==
           Forall(Forall(Forall(Forall(Forall(
             Implies(pair_fm(4,3,1),
                     Implies(pair_fm(4,2,0),
                             Implies(Member(1,r#+5),
                                     Implies(Member(0,r#+5), Equal(3,2))))))))))"

    lemma function_type [TC]:
         "[| x ∈ nat |] ==> function_fm(x) ∈ formula"
    by (simp add: function_fm_def)

    lemma sats_function_fm [simp]:
       "[| x ∈ nat; env ∈ list(A)|]
        ==> sats(A, function_fm(x), env) <-> is_function(##A, nth(x,env))"
    by (simp add: function_fm_def is_function_def)

    lemma is_function_iff_sats:
          "[| nth(i,env) = x; nth(j,env) = y;
              i ∈ nat; env ∈ list(A)|]
           ==> is_function(##A, x) <-> sats(A, function_fm(i), env)"
    by simp

    theorem is_function_reflection:
         "REFLECTS[λx. is_function(L,f(x)),
                   λi x. is_function(##Lset(i),f(x))]"
    apply (simp only: is_function_def)
    apply (intro FOL_reflections pair_reflection)
    done

    subsubsection{*Typed Functions, Internalized*}

    (* "typed_function(M,A,B,r) ==
            is_function(M,r) & is_relation(M,r) & is_domain(M,r,A) &
            (∀u[M]. u∈r --> (∀x[M]. ∀y[M]. pair(M,x,y,u) --> y∈B))" *)

    constdefs typed_function_fm :: "[i,i,i]=>i"
        "typed_function_fm(A,B,r) ==
           And(function_fm(r),
             And(relation_fm(r),
               And(domain_fm(r,A),
                 Forall(Implies(Member(0,succ(r)),
                      Forall(Forall(Implies(pair_fm(1,0,2),Member(0,B#+3)))))))))"

    lemma typed_function_type [TC]:
         "[| x ∈ nat; y ∈ nat; z ∈ nat |] ==> typed_function_fm(x,y,z) ∈ formula"
    by (simp add: typed_function_fm_def)

    lemma sats_typed_function_fm [simp]:
       "[| x ∈ nat; y ∈ nat; z ∈ nat; env ∈ list(A)|]
        ==> sats(A, typed_function_fm(x,y,z), env) <->
            typed_function(##A, nth(x,env), nth(y,env), nth(z,env))"
    by (simp add: typed_function_fm_def typed_function_def)

    lemma typed_function_iff_sats:
      "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z;
          i ∈ nat; j ∈ nat; k ∈ nat; env ∈ list(A)|]
       ==> typed_function(##A, x, y, z) <-> sats(A, typed_function_fm(i,j,k), env)"
    by simp

    lemmas function_reflections =
            empty_reflection number1_reflection
            upair_reflection pair_reflection union_reflection
            big_union_reflection cons_reflection successor_reflection
            fun_apply_reflection subset_reflection
            transitive_set_reflection membership_reflection
            pred_set_reflection domain_reflection range_reflection field_reflection
            image_reflection pre_image_reflection
            is_relation_reflection is_function_reflection

    lemmas function_iff_sats =
            empty_iff_sats number1_iff_sats
            upair_iff_sats pair_iff_sats union_iff_sats
            big_union_iff_sats cons_iff_sats successor_iff_sats
            fun_apply_iff_sats  Memrel_iff_sats
            pred_set_iff_sats domain_iff_sats range_iff_sats field_iff_sats
            image_iff_sats pre_image_iff_sats
            relation_iff_sats is_function_iff_sats

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

    subsubsection{*Composition of Relations, Internalized*}

    (* "composition(M,r,s,t) ==
            ∀p[M]. p ∈ t <->
                   (∃x[M]. ∃y[M]. ∃z[M]. ∃xy[M]. ∃yz[M].
                    pair(M,x,z,p) & pair(M,x,y,xy) & pair(M,y,z,yz) &
                    xy ∈ s & yz ∈ r)" *)
    constdefs composition_fm :: "[i,i,i]=>i"
      "composition_fm(r,s,t) ==
         Forall(Iff(Member(0,succ(t)),
                 Exists(Exists(Exists(Exists(Exists(
                  And(pair_fm(4,2,5),
                   And(pair_fm(4,3,1),
                    And(pair_fm(3,2,0),
                     And(Member(1,s#+6), Member(0,r#+6))))))))))))"

    lemma composition_type [TC]:
         "[| x ∈ nat; y ∈ nat; z ∈ nat |] ==> composition_fm(x,y,z) ∈ formula"
    by (simp add: composition_fm_def)

    lemma sats_composition_fm [simp]:
       "[| x ∈ nat; y ∈ nat; z ∈ nat; env ∈ list(A)|]
        ==> sats(A, composition_fm(x,y,z), env) <->
            composition(##A, nth(x,env), nth(y,env), nth(z,env))"
    by (simp add: composition_fm_def composition_def)

    lemma composition_iff_sats:
          "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z;
              i ∈ nat; j ∈ nat; k ∈ nat; env ∈ list(A)|]
           ==> composition(##A, x, y, z) <-> sats(A, composition_fm(i,j,k), env)"
    by simp

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

    subsubsection{*Injections, Internalized*}

    (* "injection(M,A,B,f) ==
            typed_function(M,A,B,f) &
            (∀x[M]. ∀x'[M]. ∀y[M]. ∀p[M]. ∀p'[M].
              pair(M,x,y,p) --> pair(M,x',y,p') --> p∈f --> p'∈f --> x=x')" *)
    constdefs injection_fm :: "[i,i,i]=>i"
     "injection_fm(A,B,f) ==
        And(typed_function_fm(A,B,f),
           Forall(Forall(Forall(Forall(Forall(
             Implies(pair_fm(4,2,1),
                     Implies(pair_fm(3,2,0),
                             Implies(Member(1,f#+5),
                                     Implies(Member(0,f#+5), Equal(4,3)))))))))))"

    lemma injection_type [TC]:
         "[| x ∈ nat; y ∈ nat; z ∈ nat |] ==> injection_fm(x,y,z) ∈ formula"
    by (simp add: injection_fm_def)

    lemma sats_injection_fm [simp]:
       "[| x ∈ nat; y ∈ nat; z ∈ nat; env ∈ list(A)|]
        ==> sats(A, injection_fm(x,y,z), env) <->
            injection(##A, nth(x,env), nth(y,env), nth(z,env))"
    by (simp add: injection_fm_def injection_def)

    lemma injection_iff_sats:
      "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z;
          i ∈ nat; j ∈ nat; k ∈ nat; env ∈ list(A)|]
       ==> injection(##A, x, y, z) <-> sats(A, injection_fm(i,j,k), env)"
    by simp

    theorem injection_reflection:
         "REFLECTS[λx. injection(L,f(x),g(x),h(x)),
                   λi x. injection(##Lset(i),f(x),g(x),h(x))]"
    apply (simp only: injection_def)
    apply (intro FOL_reflections function_reflections typed_function_reflection)
    done

    subsubsection{*Surjections, Internalized*}

    (*  surjection :: "[i=>o,i,i,i] => o"
        "surjection(M,A,B,f) ==
            typed_function(M,A,B,f) &
            (∀y[M]. y∈B --> (∃x[M]. x∈A & fun_apply(M,f,x,y)))" *)
    constdefs surjection_fm :: "[i,i,i]=>i"
     "surjection_fm(A,B,f) ==
        And(typed_function_fm(A,B,f),
           Forall(Implies(Member(0,succ(B)),
                          Exists(And(Member(0,succ(succ(A))),
                                     fun_apply_fm(succ(succ(f)),0,1))))))"

    lemma surjection_type [TC]:
         "[| x ∈ nat; y ∈ nat; z ∈ nat |] ==> surjection_fm(x,y,z) ∈ formula"
    by (simp add: surjection_fm_def)

    lemma sats_surjection_fm [simp]:
       "[| x ∈ nat; y ∈ nat; z ∈ nat; env ∈ list(A)|]
        ==> sats(A, surjection_fm(x,y,z), env) <->
            surjection(##A, nth(x,env), nth(y,env), nth(z,env))"
    by (simp add: surjection_fm_def surjection_def)

    lemma surjection_iff_sats:
      "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z;
          i ∈ nat; j ∈ nat; k ∈ nat; env ∈ list(A)|]
       ==> surjection(##A, x, y, z) <-> sats(A, surjection_fm(i,j,k), env)"
    by simp

    theorem surjection_reflection:
         "REFLECTS[λx. surjection(L,f(x),g(x),h(x)),
                   λi x. surjection(##Lset(i),f(x),g(x),h(x))]"
    apply (simp only: surjection_def)
    apply (intro FOL_reflections function_reflections typed_function_reflection)
    done

    subsubsection{*Bijections, Internalized*}

    (*   bijection :: "[i=>o,i,i,i] => o"
        "bijection(M,A,B,f) == injection(M,A,B,f) & surjection(M,A,B,f)" *)
    constdefs bijection_fm :: "[i,i,i]=>i"
     "bijection_fm(A,B,f) == And(injection_fm(A,B,f), surjection_fm(A,B,f))"

    lemma bijection_type [TC]:
         "[| x ∈ nat; y ∈ nat; z ∈ nat |] ==> bijection_fm(x,y,z) ∈ formula"
    by (simp add: bijection_fm_def)

    lemma sats_bijection_fm [simp]:
       "[| x ∈ nat; y ∈ nat; z ∈ nat; env ∈ list(A)|]
        ==> sats(A, bijection_fm(x,y,z), env) <->
            bijection(##A, nth(x,env), nth(y,env), nth(z,env))"
    by (simp add: bijection_fm_def bijection_def)

    lemma bijection_iff_sats:
      "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z;
          i ∈ nat; j ∈ nat; k ∈ nat; env ∈ list(A)|]
       ==> bijection(##A, x, y, z) <-> sats(A, bijection_fm(i,j,k), env)"
    by simp

    theorem bijection_reflection:
         "REFLECTS[λx. bijection(L,f(x),g(x),h(x)),
                   λi x. bijection(##Lset(i),f(x),g(x),h(x))]"
    apply (simp only: bijection_def)
    apply (intro And_reflection injection_reflection surjection_reflection)
    done

    subsubsection{*Restriction of a Relation, Internalized*}

    (* "restriction(M,r,A,z) ==
            ∀x[M]. x ∈ z <-> (x ∈ r & (∃u[M]. u∈A & (∃v[M]. pair(M,u,v,x))))" *)
    constdefs restriction_fm :: "[i,i,i]=>i"
        "restriction_fm(r,A,z) ==
           Forall(Iff(Member(0,succ(z)),
                      And(Member(0,succ(r)),
                          Exists(And(Member(0,succ(succ(A))),
                                     Exists(pair_fm(1,0,2)))))))"

    lemma restriction_type [TC]:
         "[| x ∈ nat; y ∈ nat; z ∈ nat |] ==> restriction_fm(x,y,z) ∈ formula"
    by (simp add: restriction_fm_def)

    lemma sats_restriction_fm [simp]:
       "[| x ∈ nat; y ∈ nat; z ∈ nat; env ∈ list(A)|]
        ==> sats(A, restriction_fm(x,y,z), env) <->
            restriction(##A, nth(x,env), nth(y,env), nth(z,env))"
    by (simp add: restriction_fm_def restriction_def)

    lemma restriction_iff_sats:
          "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z;
              i ∈ nat; j ∈ nat; k ∈ nat; env ∈ list(A)|]
           ==> restriction(##A, x, y, z) <-> sats(A, restriction_fm(i,j,k), env)"
    by simp

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

    subsubsection{*Order-Isomorphisms, Internalized*}

    (*  order_isomorphism :: "[i=>o,i,i,i,i,i] => o"
       "order_isomorphism(M,A,r,B,s,f) ==
            bijection(M,A,B,f) &
            (∀x[M]. x∈A --> (∀y[M]. y∈A -->
              (∀p[M]. ∀fx[M]. ∀fy[M]. ∀q[M].
                pair(M,x,y,p) --> fun_apply(M,f,x,fx) --> fun_apply(M,f,y,fy) -->
                pair(M,fx,fy,q) --> (p∈r <-> q∈s))))"
      *)

    constdefs order_isomorphism_fm :: "[i,i,i,i,i]=>i"
     "order_isomorphism_fm(A,r,B,s,f) ==
       And(bijection_fm(A,B,f),
         Forall(Implies(Member(0,succ(A)),
           Forall(Implies(Member(0,succ(succ(A))),
             Forall(Forall(Forall(Forall(
               Implies(pair_fm(5,4,3),
                 Implies(fun_apply_fm(f#+6,5,2),
                   Implies(fun_apply_fm(f#+6,4,1),
                     Implies(pair_fm(2,1,0),
                       Iff(Member(3,r#+6), Member(0,s#+6)))))))))))))))"

    lemma order_isomorphism_type [TC]:
         "[| A ∈ nat; r ∈ nat; B ∈ nat; s ∈ nat; f ∈ nat |]
          ==> order_isomorphism_fm(A,r,B,s,f) ∈ formula"
    by (simp add: order_isomorphism_fm_def)

    lemma sats_order_isomorphism_fm [simp]:
       "[| U ∈ nat; r ∈ nat; B ∈ nat; s ∈ nat; f ∈ nat; env ∈ list(A)|]
        ==> sats(A, order_isomorphism_fm(U,r,B,s,f), env) <->
            order_isomorphism(##A, nth(U,env), nth(r,env), nth(B,env),
                                   nth(s,env), nth(f,env))"
    by (simp add: order_isomorphism_fm_def order_isomorphism_def)

    lemma order_isomorphism_iff_sats:
      "[| nth(i,env) = U; nth(j,env) = r; nth(k,env) = B; nth(j',env) = s;
          nth(k',env) = f;
          i ∈ nat; j ∈ nat; k ∈ nat; j' ∈ nat; k' ∈ nat; env ∈ list(A)|]
       ==> order_isomorphism(##A,U,r,B,s,f) <->
           sats(A, order_isomorphism_fm(i,j,k,j',k'), env)"
    by simp

    theorem order_isomorphism_reflection:
         "REFLECTS[λx. order_isomorphism(L,f(x),g(x),h(x),g'(x),h'(x)),
                   λi x. order_isomorphism(##Lset(i),f(x),g(x),h(x),g'(x),h'(x))]"
    apply (simp only: order_isomorphism_def)
    apply (intro FOL_reflections function_reflections bijection_reflection)
    done

    subsubsection{*Limit Ordinals, Internalized*}

    text{*A limit ordinal is a non-empty, successor-closed ordinal*}

    (* "limit_ordinal(M,a) ==
            ordinal(M,a) & ~ empty(M,a) &
            (∀x[M]. x∈a --> (∃y[M]. y∈a & successor(M,x,y)))" *)

    constdefs limit_ordinal_fm :: "i=>i"
        "limit_ordinal_fm(x) ==
            And(ordinal_fm(x),
                And(Neg(empty_fm(x)),
                    Forall(Implies(Member(0,succ(x)),
                                   Exists(And(Member(0,succ(succ(x))),
                                              succ_fm(1,0)))))))"

    lemma limit_ordinal_type [TC]:
         "x ∈ nat ==> limit_ordinal_fm(x) ∈ formula"
    by (simp add: limit_ordinal_fm_def)

    lemma sats_limit_ordinal_fm [simp]:
       "[| x ∈ nat; env ∈ list(A)|]
        ==> sats(A, limit_ordinal_fm(x), env) <-> limit_ordinal(##A, nth(x,env))"
    by (simp add: limit_ordinal_fm_def limit_ordinal_def sats_ordinal_fm')

    lemma limit_ordinal_iff_sats:
          "[| nth(i,env) = x; nth(j,env) = y;
              i ∈ nat; env ∈ list(A)|]
           ==> limit_ordinal(##A, x) <-> sats(A, limit_ordinal_fm(i), env)"
    by simp

    theorem limit_ordinal_reflection:
         "REFLECTS[λx. limit_ordinal(L,f(x)),
                   λi x. limit_ordinal(##Lset(i),f(x))]"
    apply (simp only: limit_ordinal_def)
    apply (intro FOL_reflections ordinal_reflection
                 empty_reflection successor_reflection)
    done

    subsubsection{*Finite Ordinals: The Predicate ``Is A Natural Number''*}

    (*     "finite_ordinal(M,a) ==
            ordinal(M,a) & ~ limit_ordinal(M,a) &
            (∀x[M]. x∈a --> ~ limit_ordinal(M,x))" *)
    constdefs finite_ordinal_fm :: "i=>i"
        "finite_ordinal_fm(x) ==
           And(ordinal_fm(x),
              And(Neg(limit_ordinal_fm(x)),
               Forall(Implies(Member(0,succ(x)),
                              Neg(limit_ordinal_fm(0))))))"

    lemma finite_ordinal_type [TC]:
         "x ∈ nat ==> finite_ordinal_fm(x) ∈ formula"
    by (simp add: finite_ordinal_fm_def)

    lemma sats_finite_ordinal_fm [simp]:
       "[| x ∈ nat; env ∈ list(A)|]
        ==> sats(A, finite_ordinal_fm(x), env) <-> finite_ordinal(##A, nth(x,env))"
    by (simp add: finite_ordinal_fm_def sats_ordinal_fm' finite_ordinal_def)

    lemma finite_ordinal_iff_sats:
          "[| nth(i,env) = x; nth(j,env) = y;
              i ∈ nat; env ∈ list(A)|]
           ==> finite_ordinal(##A, x) <-> sats(A, finite_ordinal_fm(i), env)"
    by simp

    theorem finite_ordinal_reflection:
         "REFLECTS[λx. finite_ordinal(L,f(x)),
                   λi x. finite_ordinal(##Lset(i),f(x))]"
    apply (simp only: finite_ordinal_def)
    apply (intro FOL_reflections ordinal_reflection limit_ordinal_reflection)
    done

    subsubsection{*Omega: The Set of Natural Numbers*}

    (* omega(M,a) == limit_ordinal(M,a) & (∀x[M]. x∈a --> ~ limit_ordinal(M,x)) *)
    constdefs omega_fm :: "i=>i"
        "omega_fm(x) ==
           And(limit_ordinal_fm(x),
               Forall(Implies(Member(0,succ(x)),
                              Neg(limit_ordinal_fm(0)))))"

    lemma omega_type [TC]:
         "x ∈ nat ==> omega_fm(x) ∈ formula"
    by (simp add: omega_fm_def)

    lemma sats_omega_fm [simp]:
       "[| x ∈ nat; env ∈ list(A)|]
        ==> sats(A, omega_fm(x), env) <-> omega(##A, nth(x,env))"
    by (simp add: omega_fm_def omega_def)

    lemma omega_iff_sats:
          "[| nth(i,env) = x; nth(j,env) = y;
              i ∈ nat; env ∈ list(A)|]
           ==> omega(##A, x) <-> sats(A, omega_fm(i), env)"
    by simp

    theorem omega_reflection:
         "REFLECTS[λx. omega(L,f(x)),
                   λi x. omega(##Lset(i),f(x))]"
    apply (simp only: omega_def)
    apply (intro FOL_reflections limit_ordinal_reflection)
    done

    lemmas fun_plus_reflections =
            typed_function_reflection composition_reflection
            injection_reflection surjection_reflection
            bijection_reflection restriction_reflection
            order_isomorphism_reflection finite_ordinal_reflection
            ordinal_reflection limit_ordinal_reflection omega_reflection

    lemmas fun_plus_iff_sats =
            typed_function_iff_sats composition_iff_sats
            injection_iff_sats surjection_iff_sats
            bijection_iff_sats restriction_iff_sats
            order_isomorphism_iff_sats finite_ordinal_iff_sats
            ordinal_iff_sats limit_ordinal_iff_sats omega_iff_sats

    end

lemma transL:

      [| y ∈ x; L(x) |] ==> L(y)

lemma nonempty:

      L(0)

theorem upair_ax:

      upair_ax(L)

theorem Union_ax:

      Union_ax(L)

theorem power_ax:

      power_ax(L)

theorem foundation_ax:

      foundation_ax(L)

### For L to satisfy Replacement

lemma LReplace_in_Lset:

      [| X ∈ Lset(i); univalent(L, X, Q); Ord(i) |]
      ==> ∃j. Ord(j) ∧ {y . x ∈ X, Q(x, y) ∧ L(y)} ⊆ Lset(j)

lemma LReplace_in_L:

      [| L(X); univalent(L, X, Q) |] ==> ∃Y. L(Y) ∧ {y . x ∈ X, Q(x, y) ∧ L(y)} ⊆ Y

theorem replacement:

      replacement(L, P)

### Instantiating the locale @{text M_trivial}

lemma Lset_mono_le:

      mono_le_subset(Lset)

lemma Lset_cont:

      cont_Ord(Lset)

lemmas L_nat:

      L(nat)

lemmas L_nat:

      L(nat)

theorem M_trivial_L:

      PROP M_trivial(L)

### Instantiation of the locale @{text reflection}

theorem Triv_reflection:

      REFLECTS [P, %a x. P(x)]

theorem Not_reflection:

      REFLECTS [P, Q] ==> REFLECTS [%x. ¬ P(x), %a x. ¬ Q(a, x)]

theorem And_reflection:

      [| REFLECTS [P, Q]; REFLECTS [P', Q'] |]
      ==> REFLECTS [%x. P(x) ∧ P'(x), %a x. Q(a, x) ∧ Q'(a, x)]

theorem Or_reflection:

      [| REFLECTS [P, Q]; REFLECTS [P', Q'] |]
      ==> REFLECTS [%x. P(x) ∨ P'(x), %a x. Q(a, x) ∨ Q'(a, x)]

theorem Imp_reflection:

      [| REFLECTS [P, Q]; REFLECTS [P', Q'] |]
      ==> REFLECTS [%x. P(x) --> P'(x), %a x. Q(a, x) --> Q'(a, x)]

theorem Iff_reflection:

      [| REFLECTS [P, Q]; REFLECTS [P', Q'] |]
      ==> REFLECTS [%x. P(x) <-> P'(x), %a x. Q(a, x) <-> Q'(a, x)]

lemma reflection_Lset:

      reflection(Lset)

theorem Ex_reflection:

      REFLECTS [%x. P(fst(x), snd(x)), %a x. Q(a, fst(x), snd(x))]
      ==> REFLECTS [%x. ∃z. L(z) ∧ P(x, z), %a x. ∃z∈Lset(a). Q(a, x, z)]

theorem All_reflection:

      REFLECTS [%x. P(fst(x), snd(x)), %a x. Q(a, fst(x), snd(x))]
      ==> REFLECTS [%x. ∀z. L(z) --> P(x, z), %a x. ∀z∈Lset(a). Q(a, x, z)]

theorem Rex_reflection:

      REFLECTS [%x. P(fst(x), snd(x)), %a x. Q(a, fst(x), snd(x))]
      ==> REFLECTS [%x. ∃z[L]. P(x, z), %a x. ∃z∈Lset(a). Q(a, x, z)]

theorem Rall_reflection:

      REFLECTS [%x. P(fst(x), snd(x)), %a x. Q(a, fst(x), snd(x))]
      ==> REFLECTS [%x. ∀z[L]. P(x, z), %a x. ∀z∈Lset(a). Q(a, x, z)]

theorem Rex_reflection':

      REFLECTS [%x. P(fst(x), snd(x)), %a x. Q(a, fst(x), snd(x))]
      ==> REFLECTS [%x. ∃z[L]. P(x, z), %a x. ∃z[##Lset(a)]. Q(a, x, z)]

theorem Rall_reflection':

      REFLECTS [%x. P(fst(x), snd(x)), %a x. Q(a, fst(x), snd(x))]
      ==> REFLECTS [%x. ∀z[L]. P(x, z), %a x. ∀z[##Lset(a)]. Q(a, x, z)]

lemmas FOL_reflections:

      REFLECTS [P, %a x. P(x)]

      REFLECTS [P, Q] ==> REFLECTS [%x. ¬ P(x), %a x. ¬ Q(a, x)]

      [| REFLECTS [P, Q]; REFLECTS [P', Q'] |]
      ==> REFLECTS [%x. P(x) ∧ P'(x), %a x. Q(a, x) ∧ Q'(a, x)]

      [| REFLECTS [P, Q]; REFLECTS [P', Q'] |]
      ==> REFLECTS [%x. P(x) ∨ P'(x), %a x. Q(a, x) ∨ Q'(a, x)]

      [| REFLECTS [P, Q]; REFLECTS [P', Q'] |]
      ==> REFLECTS [%x. P(x) --> P'(x), %a x. Q(a, x) --> Q'(a, x)]

      [| REFLECTS [P, Q]; REFLECTS [P', Q'] |]
      ==> REFLECTS [%x. P(x) <-> P'(x), %a x. Q(a, x) <-> Q'(a, x)]

      REFLECTS [%x. P(fst(x), snd(x)), %a x. Q(a, fst(x), snd(x))]
      ==> REFLECTS [%x. ∃z. L(z) ∧ P(x, z), %a x. ∃z∈Lset(a). Q(a, x, z)]

      REFLECTS [%x. P(fst(x), snd(x)), %a x. Q(a, fst(x), snd(x))]
      ==> REFLECTS [%x. ∀z. L(z) --> P(x, z), %a x. ∀z∈Lset(a). Q(a, x, z)]

      REFLECTS [%x. P(fst(x), snd(x)), %a x. Q(a, fst(x), snd(x))]
      ==> REFLECTS [%x. ∃z[L]. P(x, z), %a x. ∃z∈Lset(a). Q(a, x, z)]

      REFLECTS [%x. P(fst(x), snd(x)), %a x. Q(a, fst(x), snd(x))]
      ==> REFLECTS [%x. ∀z[L]. P(x, z), %a x. ∀z∈Lset(a). Q(a, x, z)]

      REFLECTS [%x. P(fst(x), snd(x)), %a x. Q(a, fst(x), snd(x))]
      ==> REFLECTS [%x. ∃z[L]. P(x, z), %a x. ∃z[##Lset(a)]. Q(a, x, z)]

      REFLECTS [%x. P(fst(x), snd(x)), %a x. Q(a, fst(x), snd(x))]
      ==> REFLECTS [%x. ∀z[L]. P(x, z), %a x. ∀z[##Lset(a)]. Q(a, x, z)]

lemmas FOL_reflections:

      REFLECTS [P, %a x. P(x)]

      REFLECTS [P, Q] ==> REFLECTS [%x. ¬ P(x), %a x. ¬ Q(a, x)]

      [| REFLECTS [P, Q]; REFLECTS [P', Q'] |]
      ==> REFLECTS [%x. P(x) ∧ P'(x), %a x. Q(a, x) ∧ Q'(a, x)]

      [| REFLECTS [P, Q]; REFLECTS [P', Q'] |]
      ==> REFLECTS [%x. P(x) ∨ P'(x), %a x. Q(a, x) ∨ Q'(a, x)]

      [| REFLECTS [P, Q]; REFLECTS [P', Q'] |]
      ==> REFLECTS [%x. P(x) --> P'(x), %a x. Q(a, x) --> Q'(a, x)]

      [| REFLECTS [P, Q]; REFLECTS [P', Q'] |]
      ==> REFLECTS [%x. P(x) <-> P'(x), %a x. Q(a, x) <-> Q'(a, x)]

      REFLECTS [%x. P(fst(x), snd(x)), %a x. Q(a, fst(x), snd(x))]
      ==> REFLECTS [%x. ∃z. L(z) ∧ P(x, z), %a x. ∃z∈Lset(a). Q(a, x, z)]

      REFLECTS [%x. P(fst(x), snd(x)), %a x. Q(a, fst(x), snd(x))]
      ==> REFLECTS [%x. ∀z. L(z) --> P(x, z), %a x. ∀z∈Lset(a). Q(a, x, z)]

      REFLECTS [%x. P(fst(x), snd(x)), %a x. Q(a, fst(x), snd(x))]
      ==> REFLECTS [%x. ∃z[L]. P(x, z), %a x. ∃z∈Lset(a). Q(a, x, z)]

      REFLECTS [%x. P(fst(x), snd(x)), %a x. Q(a, fst(x), snd(x))]
      ==> REFLECTS [%x. ∀z[L]. P(x, z), %a x. ∀z∈Lset(a). Q(a, x, z)]

      REFLECTS [%x. P(fst(x), snd(x)), %a x. Q(a, fst(x), snd(x))]
      ==> REFLECTS [%x. ∃z[L]. P(x, z), %a x. ∃z[##Lset(a)]. Q(a, x, z)]

      REFLECTS [%x. P(fst(x), snd(x)), %a x. Q(a, fst(x), snd(x))]
      ==> REFLECTS [%x. ∀z[L]. P(x, z), %a x. ∀z[##Lset(a)]. Q(a, x, z)]

lemma ReflectsD:

      [| REFLECTS [P, Q]; Ord(i) |] ==> ∃j. i < j ∧ (∀x∈Lset(j). P(x) <-> Q(j, x))

lemma ReflectsE:

      [| REFLECTS [P, Q]; Ord(i);
         !!j. [| i < j; ∀x∈Lset(j). P(x) <-> Q(j, x) |] ==> R |]
      ==> R

lemma Collect_mem_eq:

      {x ∈ A . x ∈ B} = A ∩ B

### Internalized Formulas for some Set-Theoretic Concepts

#### Some numbers to help write de Bruijn indices

#### The Empty Set, Internalized

lemma empty_type:

      x ∈ nat ==> empty_fm(x) ∈ formula

lemma sats_empty_fm:

      [| x ∈ nat; env ∈ list(A) |]
      ==> sats(A, empty_fm(x), env) <-> empty(##A, nth(x, env))

lemma empty_iff_sats:

      [| nth(i, env) = x; nth(j, env) = y; i ∈ nat; env ∈ list(A) |]
      ==> empty(##A, x) <-> sats(A, empty_fm(i), env)

theorem empty_reflection:

      REFLECTS [%x. empty(L, f(x)), %i x. empty(##Lset(i), f(x))]

lemma Transset_sats_empty_fm_eq_0:

      [| n ∈ nat; env ∈ list(A); Transset(A) |]
      ==> sats(A, empty_fm(n), env) <-> nth(n, env) = 0

#### Unordered Pairs, Internalized

lemma upair_type:

      [| x ∈ nat; y ∈ nat; z ∈ nat |] ==> upair_fm(x, y, z) ∈ formula

lemma sats_upair_fm:

      [| x ∈ nat; y ∈ nat; z ∈ nat; env ∈ list(A) |]
      ==> sats(A, upair_fm(x, y, z), env) <->
          upair(##A, nth(x, env), nth(y, env), nth(z, env))

lemma upair_iff_sats:

      [| nth(i, env) = x; nth(j, env) = y; nth(k, env) = z; i ∈ nat; j ∈ nat; k ∈ nat;
         env ∈ list(A) |]
      ==> upair(##A, x, y, z) <-> sats(A, upair_fm(i, j, k), env)

lemma sats_upair_fm2:

      [| x ∈ nat; y ∈ nat; z < length(env); env ∈ list(A); Transset(A) |]
      ==> sats(A, upair_fm(x, y, z), env) <-> nth(z, env) = {nth(x, env), nth(y, env)}

theorem upair_reflection:

      REFLECTS [%x. upair(L, f(x), g(x), h(x)),
         %i x. upair(##Lset(i), f(x), g(x), h(x))]

#### Ordered pairs, Internalized

lemma pair_type:

      [| x ∈ nat; y ∈ nat; z ∈ nat |] ==> pair_fm(x, y, z) ∈ formula

lemma sats_pair_fm:

      [| x ∈ nat; y ∈ nat; z ∈ nat; env ∈ list(A) |]
      ==> sats(A, pair_fm(x, y, z), env) <->
          pair(##A, nth(x, env), nth(y, env), nth(z, env))

lemma pair_iff_sats:

      [| nth(i, env) = x; nth(j, env) = y; nth(k, env) = z; i ∈ nat; j ∈ nat; k ∈ nat;
         env ∈ list(A) |]
      ==> pair(##A, x, y, z) <-> sats(A, pair_fm(i, j, k), env)

theorem pair_reflection:

      REFLECTS [%x. pair(L, f(x), g(x), h(x)),
         %i x. pair(##Lset(i), f(x), g(x), h(x))]

#### Binary Unions, Internalized

lemma union_type:

      [| x ∈ nat; y ∈ nat; z ∈ nat |] ==> union_fm(x, y, z) ∈ formula

lemma sats_union_fm:

      [| x ∈ nat; y ∈ nat; z ∈ nat; env ∈ list(A) |]
      ==> sats(A, union_fm(x, y, z), env) <->
          union(##A, nth(x, env), nth(y, env), nth(z, env))

lemma union_iff_sats:

      [| nth(i, env) = x; nth(j, env) = y; nth(k, env) = z; i ∈ nat; j ∈ nat; k ∈ nat;
         env ∈ list(A) |]
      ==> union(##A, x, y, z) <-> sats(A, union_fm(i, j, k), env)

theorem union_reflection:

      REFLECTS [%x. union(L, f(x), g(x), h(x)),
         %i x. union(##Lset(i), f(x), g(x), h(x))]

#### Set ``Cons,'' Internalized

lemma cons_type:

      [| x ∈ nat; y ∈ nat; z ∈ nat |] ==> cons_fm(x, y, z) ∈ formula

lemma sats_cons_fm:

      [| 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))

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)

theorem cons_reflection:

      REFLECTS [%x. is_cons(L, f(x), g(x), h(x)),
         %i x. is_cons(##Lset(i), f(x), g(x), h(x))]

#### Successor Function, Internalized

lemma succ_type:

      [| x ∈ nat; y ∈ nat |] ==> succ_fm(x, y) ∈ formula

lemma sats_succ_fm:

      [| x ∈ nat; y ∈ nat; env ∈ list(A) |]
      ==> sats(A, succ_fm(x, y), env) <-> successor(##A, nth(x, env), nth(y, env))

lemma successor_iff_sats:

      [| nth(i, env) = x; nth(j, env) = y; i ∈ nat; j ∈ nat; env ∈ list(A) |]
      ==> successor(##A, x, y) <-> sats(A, succ_fm(i, j), env)

theorem successor_reflection:

      REFLECTS [%x. successor(L, f(x), g(x)), %i x. successor(##Lset(i), f(x), g(x))]

#### The Number 1, Internalized

lemma number1_type:

      x ∈ nat ==> number1_fm(x) ∈ formula

lemma sats_number1_fm:

      [| x ∈ nat; env ∈ list(A) |]
      ==> sats(A, number1_fm(x), env) <-> number1(##A, nth(x, env))

lemma number1_iff_sats:

      [| nth(i, env) = x; nth(j, env) = y; i ∈ nat; env ∈ list(A) |]
      ==> number1(##A, x) <-> sats(A, number1_fm(i), env)

theorem number1_reflection:

      REFLECTS [%x. number1(L, f(x)), %i x. number1(##Lset(i), f(x))]

#### Big Union, Internalized

lemma big_union_type:

      [| x ∈ nat; y ∈ nat |] ==> big_union_fm(x, y) ∈ formula

lemma sats_big_union_fm:

      [| x ∈ nat; y ∈ nat; env ∈ list(A) |]
      ==> sats(A, big_union_fm(x, y), env) <->
          big_union(##A, nth(x, env), nth(y, env))

lemma big_union_iff_sats:

      [| nth(i, env) = x; nth(j, env) = y; i ∈ nat; j ∈ nat; env ∈ list(A) |]
      ==> big_union(##A, x, y) <-> sats(A, big_union_fm(i, j), env)

theorem big_union_reflection:

      REFLECTS [%x. big_union(L, f(x), g(x)), %i x. big_union(##Lset(i), f(x), g(x))]

#### Variants of Satisfaction Definitions for Ordinals, etc.

lemma sats_subset_fm':

      [| x ∈ nat; y ∈ nat; env ∈ list(A) |]
      ==> sats(A, subset_fm(x, y), env) <-> subset(##A, nth(x, env), nth(y, env))

theorem subset_reflection:

      REFLECTS [%x. subset(L, f(x), g(x)), %i x. subset(##Lset(i), f(x), g(x))]

lemma sats_transset_fm':

      [| x ∈ nat; env ∈ list(A) |]
      ==> sats(A, transset_fm(x), env) <-> transitive_set(##A, nth(x, env))

theorem transitive_set_reflection:

      REFLECTS [%x. transitive_set(L, f(x)), %i x. transitive_set(##Lset(i), f(x))]

lemma sats_ordinal_fm':

      [| x ∈ nat; env ∈ list(A) |]
      ==> sats(A, ordinal_fm(x), env) <-> ordinal(##A, nth(x, env))

lemma ordinal_iff_sats:

      [| nth(i, env) = x; i ∈ nat; env ∈ list(A) |]
      ==> ordinal(##A, x) <-> sats(A, ordinal_fm(i), env)

theorem ordinal_reflection:

      REFLECTS [%x. ordinal(L, f(x)), %i x. ordinal(##Lset(i), f(x))]

#### Membership Relation, Internalized

lemma Memrel_type:

      [| x ∈ nat; y ∈ nat |] ==> Memrel_fm(x, y) ∈ formula

lemma sats_Memrel_fm:

      [| x ∈ nat; y ∈ nat; env ∈ list(A) |]
      ==> sats(A, Memrel_fm(x, y), env) <-> membership(##A, nth(x, env), nth(y, env))

lemma Memrel_iff_sats:

      [| nth(i, env) = x; nth(j, env) = y; i ∈ nat; j ∈ nat; env ∈ list(A) |]
      ==> membership(##A, x, y) <-> sats(A, Memrel_fm(i, j), env)

theorem membership_reflection:

      REFLECTS [%x. membership(L, f(x), g(x)),
         %i x. membership(##Lset(i), f(x), g(x))]

#### Predecessor Set, Internalized

lemma pred_set_type:

      [| A ∈ nat; x ∈ nat; r ∈ nat; B ∈ nat |] ==> pred_set_fm(A, x, r, B) ∈ formula

lemma sats_pred_set_fm:

      [| U ∈ nat; x ∈ nat; r ∈ nat; B ∈ nat; env ∈ list(A) |]
      ==> sats(A, pred_set_fm(U, x, r, B), env) <->
          pred_set(##A, nth(U, env), nth(x, env), nth(r, env), nth(B, env))

lemma pred_set_iff_sats:

      [| nth(i, env) = U; nth(j, env) = x; nth(k, env) = r; nth(l, env) = B; i ∈ nat;
         j ∈ nat; k ∈ nat; l ∈ nat; env ∈ list(A) |]
      ==> pred_set(##A, U, x, r, B) <-> sats(A, pred_set_fm(i, j, k, l), env)

theorem pred_set_reflection:

      REFLECTS [%x. pred_set(L, f(x), g(x), h(x), b(x)),
         %i x. pred_set(##Lset(i), f(x), g(x), h(x), b(x))]

#### Domain of a Relation, Internalized

lemma domain_type:

      [| x ∈ nat; y ∈ nat |] ==> domain_fm(x, y) ∈ formula

lemma sats_domain_fm:

      [| x ∈ nat; y ∈ nat; env ∈ list(A) |]
      ==> sats(A, domain_fm(x, y), env) <-> is_domain(##A, nth(x, env), nth(y, env))

lemma domain_iff_sats:

      [| nth(i, env) = x; nth(j, env) = y; i ∈ nat; j ∈ nat; env ∈ list(A) |]
      ==> is_domain(##A, x, y) <-> sats(A, domain_fm(i, j), env)

theorem domain_reflection:

      REFLECTS [%x. is_domain(L, f(x), g(x)), %i x. is_domain(##Lset(i), f(x), g(x))]

#### Range of a Relation, Internalized

lemma range_type:

      [| x ∈ nat; y ∈ nat |] ==> range_fm(x, y) ∈ formula

lemma sats_range_fm:

      [| x ∈ nat; y ∈ nat; env ∈ list(A) |]
      ==> sats(A, range_fm(x, y), env) <-> is_range(##A, nth(x, env), nth(y, env))

lemma range_iff_sats:

      [| nth(i, env) = x; nth(j, env) = y; i ∈ nat; j ∈ nat; env ∈ list(A) |]
      ==> is_range(##A, x, y) <-> sats(A, range_fm(i, j), env)

theorem range_reflection:

      REFLECTS [%x. is_range(L, f(x), g(x)), %i x. is_range(##Lset(i), f(x), g(x))]

#### Field of a Relation, Internalized

lemma field_type:

      [| x ∈ nat; y ∈ nat |] ==> field_fm(x, y) ∈ formula

lemma sats_field_fm:

      [| x ∈ nat; y ∈ nat; env ∈ list(A) |]
      ==> sats(A, field_fm(x, y), env) <-> is_field(##A, nth(x, env), nth(y, env))

lemma field_iff_sats:

      [| nth(i, env) = x; nth(j, env) = y; i ∈ nat; j ∈ nat; env ∈ list(A) |]
      ==> is_field(##A, x, y) <-> sats(A, field_fm(i, j), env)

theorem field_reflection:

      REFLECTS [%x. is_field(L, f(x), g(x)), %i x. is_field(##Lset(i), f(x), g(x))]

#### Image under a Relation, Internalized

lemma image_type:

      [| x ∈ nat; y ∈ nat; z ∈ nat |] ==> image_fm(x, y, z) ∈ formula

lemma sats_image_fm:

      [| x ∈ nat; y ∈ nat; z ∈ nat; env ∈ list(A) |]
      ==> sats(A, image_fm(x, y, z), env) <->
          image(##A, nth(x, env), nth(y, env), nth(z, env))

lemma image_iff_sats:

      [| nth(i, env) = x; nth(j, env) = y; nth(k, env) = z; i ∈ nat; j ∈ nat; k ∈ nat;
         env ∈ list(A) |]
      ==> image(##A, x, y, z) <-> sats(A, image_fm(i, j, k), env)

theorem image_reflection:

      REFLECTS [%x. image(L, f(x), g(x), h(x)),
         %i x. image(##Lset(i), f(x), g(x), h(x))]

#### Pre-Image under a Relation, Internalized

lemma pre_image_type:

      [| x ∈ nat; y ∈ nat; z ∈ nat |] ==> pre_image_fm(x, y, z) ∈ formula

lemma sats_pre_image_fm:

      [| x ∈ nat; y ∈ nat; z ∈ nat; env ∈ list(A) |]
      ==> sats(A, pre_image_fm(x, y, z), env) <->
          pre_image(##A, nth(x, env), nth(y, env), nth(z, env))

lemma pre_image_iff_sats:

      [| nth(i, env) = x; nth(j, env) = y; nth(k, env) = z; i ∈ nat; j ∈ nat; k ∈ nat;
         env ∈ list(A) |]
      ==> pre_image(##A, x, y, z) <-> sats(A, pre_image_fm(i, j, k), env)

theorem pre_image_reflection:

      REFLECTS [%x. pre_image(L, f(x), g(x), h(x)),
         %i x. pre_image(##Lset(i), f(x), g(x), h(x))]

#### Function Application, Internalized

lemma fun_apply_type:

      [| x ∈ nat; y ∈ nat; z ∈ nat |] ==> fun_apply_fm(x, y, z) ∈ formula

lemma sats_fun_apply_fm:

      [| x ∈ nat; y ∈ nat; z ∈ nat; env ∈ list(A) |]
      ==> sats(A, fun_apply_fm(x, y, z), env) <->
          fun_apply(##A, nth(x, env), nth(y, env), nth(z, env))

lemma fun_apply_iff_sats:

      [| nth(i, env) = x; nth(j, env) = y; nth(k, env) = z; i ∈ nat; j ∈ nat; k ∈ nat;
         env ∈ list(A) |]
      ==> fun_apply(##A, x, y, z) <-> sats(A, fun_apply_fm(i, j, k), env)

theorem fun_apply_reflection:

      REFLECTS [%x. fun_apply(L, f(x), g(x), h(x)),
         %i x. fun_apply(##Lset(i), f(x), g(x), h(x))]

#### The Concept of Relation, Internalized

lemma relation_type:

      x ∈ nat ==> relation_fm(x) ∈ formula

lemma sats_relation_fm:

      [| x ∈ nat; env ∈ list(A) |]
      ==> sats(A, relation_fm(x), env) <-> is_relation(##A, nth(x, env))

lemma relation_iff_sats:

      [| nth(i, env) = x; nth(j, env) = y; i ∈ nat; env ∈ list(A) |]
      ==> is_relation(##A, x) <-> sats(A, relation_fm(i), env)

theorem is_relation_reflection:

      REFLECTS [%x. is_relation(L, f(x)), %i x. is_relation(##Lset(i), f(x))]

#### The Concept of Function, Internalized

lemma function_type:

      x ∈ nat ==> function_fm(x) ∈ formula

lemma sats_function_fm:

      [| x ∈ nat; env ∈ list(A) |]
      ==> sats(A, function_fm(x), env) <-> is_function(##A, nth(x, env))

lemma is_function_iff_sats:

      [| nth(i, env) = x; nth(j, env) = y; i ∈ nat; env ∈ list(A) |]
      ==> is_function(##A, x) <-> sats(A, function_fm(i), env)

theorem is_function_reflection:

      REFLECTS [%x. is_function(L, f(x)), %i x. is_function(##Lset(i), f(x))]

#### Typed Functions, Internalized

lemma typed_function_type:

      [| x ∈ nat; y ∈ nat; z ∈ nat |] ==> typed_function_fm(x, y, z) ∈ formula

lemma sats_typed_function_fm:

      [| x ∈ nat; y ∈ nat; z ∈ nat; env ∈ list(A) |]
      ==> sats(A, typed_function_fm(x, y, z), env) <->
          typed_function(##A, nth(x, env), nth(y, env), nth(z, env))

lemma typed_function_iff_sats:

      [| nth(i, env) = x; nth(j, env) = y; nth(k, env) = z; i ∈ nat; j ∈ nat; k ∈ nat;
         env ∈ list(A) |]
      ==> typed_function(##A, x, y, z) <-> sats(A, typed_function_fm(i, j, k), env)

lemmas function_reflections:

      REFLECTS [%x. empty(L, f(x)), %i x. empty(##Lset(i), f(x))]

      REFLECTS [%x. number1(L, f(x)), %i x. number1(##Lset(i), f(x))]

      REFLECTS [%x. upair(L, f(x), g(x), h(x)),
         %i x. upair(##Lset(i), f(x), g(x), h(x))]

      REFLECTS [%x. pair(L, f(x), g(x), h(x)),
         %i x. pair(##Lset(i), f(x), g(x), h(x))]

      REFLECTS [%x. union(L, f(x), g(x), h(x)),
         %i x. union(##Lset(i), f(x), g(x), h(x))]

      REFLECTS [%x. big_union(L, f(x), g(x)), %i x. big_union(##Lset(i), f(x), g(x))]

      REFLECTS [%x. is_cons(L, f(x), g(x), h(x)),
         %i x. is_cons(##Lset(i), f(x), g(x), h(x))]

      REFLECTS [%x. successor(L, f(x), g(x)), %i x. successor(##Lset(i), f(x), g(x))]

      REFLECTS [%x. fun_apply(L, f(x), g(x), h(x)),
         %i x. fun_apply(##Lset(i), f(x), g(x), h(x))]

      REFLECTS [%x. subset(L, f(x), g(x)), %i x. subset(##Lset(i), f(x), g(x))]

      REFLECTS [%x. transitive_set(L, f(x)), %i x. transitive_set(##Lset(i), f(x))]

      REFLECTS [%x. membership(L, f(x), g(x)),
         %i x. membership(##Lset(i), f(x), g(x))]

      REFLECTS [%x. pred_set(L, f(x), g(x), h(x), b(x)),
         %i x. pred_set(##Lset(i), f(x), g(x), h(x), b(x))]

      REFLECTS [%x. is_domain(L, f(x), g(x)), %i x. is_domain(##Lset(i), f(x), g(x))]

      REFLECTS [%x. is_range(L, f(x), g(x)), %i x. is_range(##Lset(i), f(x), g(x))]

      REFLECTS [%x. is_field(L, f(x), g(x)), %i x. is_field(##Lset(i), f(x), g(x))]

      REFLECTS [%x. image(L, f(x), g(x), h(x)),
         %i x. image(##Lset(i), f(x), g(x), h(x))]

      REFLECTS [%x. pre_image(L, f(x), g(x), h(x)),
         %i x. pre_image(##Lset(i), f(x), g(x), h(x))]

      REFLECTS [%x. is_relation(L, f(x)), %i x. is_relation(##Lset(i), f(x))]

      REFLECTS [%x. is_function(L, f(x)), %i x. is_function(##Lset(i), f(x))]

lemmas function_reflections:

      REFLECTS [%x. empty(L, f(x)), %i x. empty(##Lset(i), f(x))]

      REFLECTS [%x. number1(L, f(x)), %i x. number1(##Lset(i), f(x))]

      REFLECTS [%x. upair(L, f(x), g(x), h(x)),
         %i x. upair(##Lset(i), f(x), g(x), h(x))]

      REFLECTS [%x. pair(L, f(x), g(x), h(x)),
         %i x. pair(##Lset(i), f(x), g(x), h(x))]

      REFLECTS [%x. union(L, f(x), g(x), h(x)),
         %i x. union(##Lset(i), f(x), g(x), h(x))]

      REFLECTS [%x. big_union(L, f(x), g(x)), %i x. big_union(##Lset(i), f(x), g(x))]

      REFLECTS [%x. is_cons(L, f(x), g(x), h(x)),
         %i x. is_cons(##Lset(i), f(x), g(x), h(x))]

      REFLECTS [%x. successor(L, f(x), g(x)), %i x. successor(##Lset(i), f(x), g(x))]

      REFLECTS [%x. fun_apply(L, f(x), g(x), h(x)),
         %i x. fun_apply(##Lset(i), f(x), g(x), h(x))]

      REFLECTS [%x. subset(L, f(x), g(x)), %i x. subset(##Lset(i), f(x), g(x))]

      REFLECTS [%x. transitive_set(L, f(x)), %i x. transitive_set(##Lset(i), f(x))]

      REFLECTS [%x. membership(L, f(x), g(x)),
         %i x. membership(##Lset(i), f(x), g(x))]

      REFLECTS [%x. pred_set(L, f(x), g(x), h(x), b(x)),
         %i x. pred_set(##Lset(i), f(x), g(x), h(x), b(x))]

      REFLECTS [%x. is_domain(L, f(x), g(x)), %i x. is_domain(##Lset(i), f(x), g(x))]

      REFLECTS [%x. is_range(L, f(x), g(x)), %i x. is_range(##Lset(i), f(x), g(x))]

      REFLECTS [%x. is_field(L, f(x), g(x)), %i x. is_field(##Lset(i), f(x), g(x))]

      REFLECTS [%x. image(L, f(x), g(x), h(x)),
         %i x. image(##Lset(i), f(x), g(x), h(x))]

      REFLECTS [%x. pre_image(L, f(x), g(x), h(x)),
         %i x. pre_image(##Lset(i), f(x), g(x), h(x))]

      REFLECTS [%x. is_relation(L, f(x)), %i x. is_relation(##Lset(i), f(x))]

      REFLECTS [%x. is_function(L, f(x)), %i x. is_function(##Lset(i), f(x))]

lemmas function_iff_sats:

      [| nth(i, env) = x; nth(j, env) = y; i ∈ nat; env ∈ list(A) |]
      ==> empty(##A, x) <-> sats(A, empty_fm(i), env)

      [| nth(i, env) = x; nth(j, env) = y; i ∈ nat; env ∈ list(A) |]
      ==> number1(##A, x) <-> sats(A, number1_fm(i), env)

      [| nth(i, env) = x; nth(j, env) = y; nth(k, env) = z; i ∈ nat; j ∈ nat; k ∈ nat;
         env ∈ list(A) |]
      ==> upair(##A, x, y, z) <-> sats(A, upair_fm(i, j, k), env)

      [| nth(i, env) = x; nth(j, env) = y; nth(k, env) = z; i ∈ nat; j ∈ nat; k ∈ nat;
         env ∈ list(A) |]
      ==> pair(##A, x, y, z) <-> sats(A, pair_fm(i, j, k), env)

      [| nth(i, env) = x; nth(j, env) = y; nth(k, env) = z; i ∈ nat; j ∈ nat; k ∈ nat;
         env ∈ list(A) |]
      ==> union(##A, x, y, z) <-> sats(A, union_fm(i, j, k), env)

      [| nth(i, env) = x; nth(j, env) = y; i ∈ nat; j ∈ nat; env ∈ list(A) |]
      ==> big_union(##A, x, y) <-> sats(A, big_union_fm(i, j), env)

      [| 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)

      [| nth(i, env) = x; nth(j, env) = y; i ∈ nat; j ∈ nat; env ∈ list(A) |]
      ==> successor(##A, x, y) <-> sats(A, succ_fm(i, j), env)

      [| nth(i, env) = x; nth(j, env) = y; nth(k, env) = z; i ∈ nat; j ∈ nat; k ∈ nat;
         env ∈ list(A) |]
      ==> fun_apply(##A, x, y, z) <-> sats(A, fun_apply_fm(i, j, k), env)

      [| nth(i, env) = x; nth(j, env) = y; i ∈ nat; j ∈ nat; env ∈ list(A) |]
      ==> membership(##A, x, y) <-> sats(A, Memrel_fm(i, j), env)

      [| nth(i, env) = U; nth(j, env) = x; nth(k, env) = r; nth(l, env) = B; i ∈ nat;
         j ∈ nat; k ∈ nat; l ∈ nat; env ∈ list(A) |]
      ==> pred_set(##A, U, x, r, B) <-> sats(A, pred_set_fm(i, j, k, l), env)

      [| nth(i, env) = x; nth(j, env) = y; i ∈ nat; j ∈ nat; env ∈ list(A) |]
      ==> is_domain(##A, x, y) <-> sats(A, domain_fm(i, j), env)

      [| nth(i, env) = x; nth(j, env) = y; i ∈ nat; j ∈ nat; env ∈ list(A) |]
      ==> is_range(##A, x, y) <-> sats(A, range_fm(i, j), env)

      [| nth(i, env) = x; nth(j, env) = y; i ∈ nat; j ∈ nat; env ∈ list(A) |]
      ==> is_field(##A, x, y) <-> sats(A, field_fm(i, j), env)

      [| nth(i, env) = x; nth(j, env) = y; nth(k, env) = z; i ∈ nat; j ∈ nat; k ∈ nat;
         env ∈ list(A) |]
      ==> image(##A, x, y, z) <-> sats(A, image_fm(i, j, k), env)

      [| nth(i, env) = x; nth(j, env) = y; nth(k, env) = z; i ∈ nat; j ∈ nat; k ∈ nat;
         env ∈ list(A) |]
      ==> pre_image(##A, x, y, z) <-> sats(A, pre_image_fm(i, j, k), env)

      [| nth(i, env) = x; nth(j, env) = y; i ∈ nat; env ∈ list(A) |]
      ==> is_relation(##A, x) <-> sats(A, relation_fm(i), env)

      [| nth(i, env) = x; nth(j, env) = y; i ∈ nat; env ∈ list(A) |]
      ==> is_function(##A, x) <-> sats(A, function_fm(i), env)

lemmas function_iff_sats:

      [| nth(i, env) = x; nth(j, env) = y; i ∈ nat; env ∈ list(A) |]
      ==> empty(##A, x) <-> sats(A, empty_fm(i), env)

      [| nth(i, env) = x; nth(j, env) = y; i ∈ nat; env ∈ list(A) |]
      ==> number1(##A, x) <-> sats(A, number1_fm(i), env)

      [| nth(i, env) = x; nth(j, env) = y; nth(k, env) = z; i ∈ nat; j ∈ nat; k ∈ nat;
         env ∈ list(A) |]
      ==> upair(##A, x, y, z) <-> sats(A, upair_fm(i, j, k), env)

      [| nth(i, env) = x; nth(j, env) = y; nth(k, env) = z; i ∈ nat; j ∈ nat; k ∈ nat;
         env ∈ list(A) |]
      ==> pair(##A, x, y, z) <-> sats(A, pair_fm(i, j, k), env)

      [| nth(i, env) = x; nth(j, env) = y; nth(k, env) = z; i ∈ nat; j ∈ nat; k ∈ nat;
         env ∈ list(A) |]
      ==> union(##A, x, y, z) <-> sats(A, union_fm(i, j, k), env)

      [| nth(i, env) = x; nth(j, env) = y; i ∈ nat; j ∈ nat; env ∈ list(A) |]
      ==> big_union(##A, x, y) <-> sats(A, big_union_fm(i, j), env)

      [| 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)

      [| nth(i, env) = x; nth(j, env) = y; i ∈ nat; j ∈ nat; env ∈ list(A) |]
      ==> successor(##A, x, y) <-> sats(A, succ_fm(i, j), env)

      [| nth(i, env) = x; nth(j, env) = y; nth(k, env) = z; i ∈ nat; j ∈ nat; k ∈ nat;
         env ∈ list(A) |]
      ==> fun_apply(##A, x, y, z) <-> sats(A, fun_apply_fm(i, j, k), env)

      [| nth(i, env) = x; nth(j, env) = y; i ∈ nat; j ∈ nat; env ∈ list(A) |]
      ==> membership(##A, x, y) <-> sats(A, Memrel_fm(i, j), env)

      [| nth(i, env) = U; nth(j, env) = x; nth(k, env) = r; nth(l, env) = B; i ∈ nat;
         j ∈ nat; k ∈ nat; l ∈ nat; env ∈ list(A) |]
      ==> pred_set(##A, U, x, r, B) <-> sats(A, pred_set_fm(i, j, k, l), env)

      [| nth(i, env) = x; nth(j, env) = y; i ∈ nat; j ∈ nat; env ∈ list(A) |]
      ==> is_domain(##A, x, y) <-> sats(A, domain_fm(i, j), env)

      [| nth(i, env) = x; nth(j, env) = y; i ∈ nat; j ∈ nat; env ∈ list(A) |]
      ==> is_range(##A, x, y) <-> sats(A, range_fm(i, j), env)

      [| nth(i, env) = x; nth(j, env) = y; i ∈ nat; j ∈ nat; env ∈ list(A) |]
      ==> is_field(##A, x, y) <-> sats(A, field_fm(i, j), env)

      [| nth(i, env) = x; nth(j, env) = y; nth(k, env) = z; i ∈ nat; j ∈ nat; k ∈ nat;
         env ∈ list(A) |]
      ==> image(##A, x, y, z) <-> sats(A, image_fm(i, j, k), env)

      [| nth(i, env) = x; nth(j, env) = y; nth(k, env) = z; i ∈ nat; j ∈ nat; k ∈ nat;
         env ∈ list(A) |]
      ==> pre_image(##A, x, y, z) <-> sats(A, pre_image_fm(i, j, k), env)

      [| nth(i, env) = x; nth(j, env) = y; i ∈ nat; env ∈ list(A) |]
      ==> is_relation(##A, x) <-> sats(A, relation_fm(i), env)

      [| nth(i, env) = x; nth(j, env) = y; i ∈ nat; env ∈ list(A) |]
      ==> is_function(##A, x) <-> sats(A, function_fm(i), env)

theorem typed_function_reflection:

      REFLECTS [%x. typed_function(L, f(x), g(x), h(x)),
         %i x. typed_function(##Lset(i), f(x), g(x), h(x))]

#### Composition of Relations, Internalized

lemma composition_type:

      [| x ∈ nat; y ∈ nat; z ∈ nat |] ==> composition_fm(x, y, z) ∈ formula

lemma sats_composition_fm:

      [| x ∈ nat; y ∈ nat; z ∈ nat; env ∈ list(A) |]
      ==> sats(A, composition_fm(x, y, z), env) <->
          composition(##A, nth(x, env), nth(y, env), nth(z, env))

lemma composition_iff_sats:

      [| nth(i, env) = x; nth(j, env) = y; nth(k, env) = z; i ∈ nat; j ∈ nat; k ∈ nat;
         env ∈ list(A) |]
      ==> composition(##A, x, y, z) <-> sats(A, composition_fm(i, j, k), env)

theorem composition_reflection:

      REFLECTS [%x. composition(L, f(x), g(x), h(x)),
         %i x. composition(##Lset(i), f(x), g(x), h(x))]

#### Injections, Internalized

lemma injection_type:

      [| x ∈ nat; y ∈ nat; z ∈ nat |] ==> injection_fm(x, y, z) ∈ formula

lemma sats_injection_fm:

      [| x ∈ nat; y ∈ nat; z ∈ nat; env ∈ list(A) |]
      ==> sats(A, injection_fm(x, y, z), env) <->
          injection(##A, nth(x, env), nth(y, env), nth(z, env))

lemma injection_iff_sats:

      [| nth(i, env) = x; nth(j, env) = y; nth(k, env) = z; i ∈ nat; j ∈ nat; k ∈ nat;
         env ∈ list(A) |]
      ==> injection(##A, x, y, z) <-> sats(A, injection_fm(i, j, k), env)

theorem injection_reflection:

      REFLECTS [%x. injection(L, f(x), g(x), h(x)),
         %i x. injection(##Lset(i), f(x), g(x), h(x))]

#### Surjections, Internalized

lemma surjection_type:

      [| x ∈ nat; y ∈ nat; z ∈ nat |] ==> surjection_fm(x, y, z) ∈ formula

lemma sats_surjection_fm:

      [| x ∈ nat; y ∈ nat; z ∈ nat; env ∈ list(A) |]
      ==> sats(A, surjection_fm(x, y, z), env) <->
          surjection(##A, nth(x, env), nth(y, env), nth(z, env))

lemma surjection_iff_sats:

      [| nth(i, env) = x; nth(j, env) = y; nth(k, env) = z; i ∈ nat; j ∈ nat; k ∈ nat;
         env ∈ list(A) |]
      ==> surjection(##A, x, y, z) <-> sats(A, surjection_fm(i, j, k), env)

theorem surjection_reflection:

      REFLECTS [%x. surjection(L, f(x), g(x), h(x)),
         %i x. surjection(##Lset(i), f(x), g(x), h(x))]

#### Bijections, Internalized

lemma bijection_type:

      [| x ∈ nat; y ∈ nat; z ∈ nat |] ==> bijection_fm(x, y, z) ∈ formula

lemma sats_bijection_fm:

      [| x ∈ nat; y ∈ nat; z ∈ nat; env ∈ list(A) |]
      ==> sats(A, bijection_fm(x, y, z), env) <->
          bijection(##A, nth(x, env), nth(y, env), nth(z, env))

lemma bijection_iff_sats:

      [| nth(i, env) = x; nth(j, env) = y; nth(k, env) = z; i ∈ nat; j ∈ nat; k ∈ nat;
         env ∈ list(A) |]
      ==> bijection(##A, x, y, z) <-> sats(A, bijection_fm(i, j, k), env)

theorem bijection_reflection:

      REFLECTS [%x. bijection(L, f(x), g(x), h(x)),
         %i x. bijection(##Lset(i), f(x), g(x), h(x))]

#### Restriction of a Relation, Internalized

lemma restriction_type:

      [| x ∈ nat; y ∈ nat; z ∈ nat |] ==> restriction_fm(x, y, z) ∈ formula

lemma sats_restriction_fm:

      [| x ∈ nat; y ∈ nat; z ∈ nat; env ∈ list(A) |]
      ==> sats(A, restriction_fm(x, y, z), env) <->
          restriction(##A, nth(x, env), nth(y, env), nth(z, env))

lemma restriction_iff_sats:

      [| nth(i, env) = x; nth(j, env) = y; nth(k, env) = z; i ∈ nat; j ∈ nat; k ∈ nat;
         env ∈ list(A) |]
      ==> restriction(##A, x, y, z) <-> sats(A, restriction_fm(i, j, k), env)

theorem restriction_reflection:

      REFLECTS [%x. restriction(L, f(x), g(x), h(x)),
         %i x. restriction(##Lset(i), f(x), g(x), h(x))]

#### Order-Isomorphisms, Internalized

lemma order_isomorphism_type:

      [| A ∈ nat; r ∈ nat; B ∈ nat; s ∈ nat; f ∈ nat |]
      ==> order_isomorphism_fm(A, r, B, s, f) ∈ formula

lemma sats_order_isomorphism_fm:

      [| U ∈ nat; r ∈ nat; B ∈ nat; s ∈ nat; f ∈ nat; env ∈ list(A) |]
      ==> sats(A, order_isomorphism_fm(U, r, B, s, f), env) <->
          order_isomorphism
           (##A, nth(U, env), nth(r, env), nth(B, env), nth(s, env), nth(f, env))

lemma order_isomorphism_iff_sats:

      [| nth(i, env) = U; nth(j, env) = r; nth(k, env) = B; nth(j', env) = s;
         nth(k', env) = f; i ∈ nat; j ∈ nat; k ∈ nat; j' ∈ nat; k' ∈ nat;
         env ∈ list(A) |]
      ==> order_isomorphism(##A, U, r, B, s, f) <->
          sats(A, order_isomorphism_fm(i, j, k, j', k'), env)

theorem order_isomorphism_reflection:

      REFLECTS [%x. order_isomorphism(L, f(x), g(x), h(x), g'(x), h'(x)),
         %i x. order_isomorphism(##Lset(i), f(x), g(x), h(x), g'(x), h'(x))]

#### Limit Ordinals, Internalized

lemma limit_ordinal_type:

      x ∈ nat ==> limit_ordinal_fm(x) ∈ formula

lemma sats_limit_ordinal_fm:

      [| x ∈ nat; env ∈ list(A) |]
      ==> sats(A, limit_ordinal_fm(x), env) <-> limit_ordinal(##A, nth(x, env))

lemma limit_ordinal_iff_sats:

      [| nth(i, env) = x; nth(j, env) = y; i ∈ nat; env ∈ list(A) |]
      ==> limit_ordinal(##A, x) <-> sats(A, limit_ordinal_fm(i), env)

theorem limit_ordinal_reflection:

      REFLECTS [%x. limit_ordinal(L, f(x)), %i x. limit_ordinal(##Lset(i), f(x))]

#### Finite Ordinals: The Predicate ``Is A Natural Number''

lemma finite_ordinal_type:

      x ∈ nat ==> finite_ordinal_fm(x) ∈ formula

lemma sats_finite_ordinal_fm:

      [| x ∈ nat; env ∈ list(A) |]
      ==> sats(A, finite_ordinal_fm(x), env) <-> finite_ordinal(##A, nth(x, env))

lemma finite_ordinal_iff_sats:

      [| nth(i, env) = x; nth(j, env) = y; i ∈ nat; env ∈ list(A) |]
      ==> finite_ordinal(##A, x) <-> sats(A, finite_ordinal_fm(i), env)

theorem finite_ordinal_reflection:

      REFLECTS [%x. finite_ordinal(L, f(x)), %i x. finite_ordinal(##Lset(i), f(x))]

#### Omega: The Set of Natural Numbers

lemma omega_type:

      x ∈ nat ==> omega_fm(x) ∈ formula

lemma sats_omega_fm:

      [| x ∈ nat; env ∈ list(A) |]
      ==> sats(A, omega_fm(x), env) <-> omega(##A, nth(x, env))

lemma omega_iff_sats:

      [| nth(i, env) = x; nth(j, env) = y; i ∈ nat; env ∈ list(A) |]
      ==> omega(##A, x) <-> sats(A, omega_fm(i), env)

theorem omega_reflection:

      REFLECTS [%x. omega(L, f(x)), %i x. omega(##Lset(i), f(x))]

lemmas fun_plus_reflections:

      REFLECTS [%x. typed_function(L, f(x), g(x), h(x)),
         %i x. typed_function(##Lset(i), f(x), g(x), h(x))]

      REFLECTS [%x. composition(L, f(x), g(x), h(x)),
         %i x. composition(##Lset(i), f(x), g(x), h(x))]

      REFLECTS [%x. injection(L, f(x), g(x), h(x)),
         %i x. injection(##Lset(i), f(x), g(x), h(x))]

      REFLECTS [%x. surjection(L, f(x), g(x), h(x)),
         %i x. surjection(##Lset(i), f(x), g(x), h(x))]

      REFLECTS [%x. bijection(L, f(x), g(x), h(x)),
         %i x. bijection(##Lset(i), f(x), g(x), h(x))]

      REFLECTS [%x. restriction(L, f(x), g(x), h(x)),
         %i x. restriction(##Lset(i), f(x), g(x), h(x))]

      REFLECTS [%x. order_isomorphism(L, f(x), g(x), h(x), g'(x), h'(x)),
         %i x. order_isomorphism(##Lset(i), f(x), g(x), h(x), g'(x), h'(x))]

      REFLECTS [%x. finite_ordinal(L, f(x)), %i x. finite_ordinal(##Lset(i), f(x))]

      REFLECTS [%x. ordinal(L, f(x)), %i x. ordinal(##Lset(i), f(x))]

      REFLECTS [%x. limit_ordinal(L, f(x)), %i x. limit_ordinal(##Lset(i), f(x))]

      REFLECTS [%x. omega(L, f(x)), %i x. omega(##Lset(i), f(x))]

lemmas fun_plus_reflections:

      REFLECTS [%x. typed_function(L, f(x), g(x), h(x)),
         %i x. typed_function(##Lset(i), f(x), g(x), h(x))]

      REFLECTS [%x. composition(L, f(x), g(x), h(x)),
         %i x. composition(##Lset(i), f(x), g(x), h(x))]

      REFLECTS [%x. injection(L, f(x), g(x), h(x)),
         %i x. injection(##Lset(i), f(x), g(x), h(x))]

      REFLECTS [%x. surjection(L, f(x), g(x), h(x)),
         %i x. surjection(##Lset(i), f(x), g(x), h(x))]

      REFLECTS [%x. bijection(L, f(x), g(x), h(x)),
         %i x. bijection(##Lset(i), f(x), g(x), h(x))]

      REFLECTS [%x. restriction(L, f(x), g(x), h(x)),
         %i x. restriction(##Lset(i), f(x), g(x), h(x))]

      REFLECTS [%x. order_isomorphism(L, f(x), g(x), h(x), g'(x), h'(x)),
         %i x. order_isomorphism(##Lset(i), f(x), g(x), h(x), g'(x), h'(x))]

      REFLECTS [%x. finite_ordinal(L, f(x)), %i x. finite_ordinal(##Lset(i), f(x))]

      REFLECTS [%x. ordinal(L, f(x)), %i x. ordinal(##Lset(i), f(x))]

      REFLECTS [%x. limit_ordinal(L, f(x)), %i x. limit_ordinal(##Lset(i), f(x))]

      REFLECTS [%x. omega(L, f(x)), %i x. omega(##Lset(i), f(x))]

lemmas fun_plus_iff_sats:

      [| nth(i, env) = x; nth(j, env) = y; nth(k, env) = z; i ∈ nat; j ∈ nat; k ∈ nat;
         env ∈ list(A) |]
      ==> typed_function(##A, x, y, z) <-> sats(A, typed_function_fm(i, j, k), env)

      [| nth(i, env) = x; nth(j, env) = y; nth(k, env) = z; i ∈ nat; j ∈ nat; k ∈ nat;
         env ∈ list(A) |]
      ==> composition(##A, x, y, z) <-> sats(A, composition_fm(i, j, k), env)

      [| nth(i, env) = x; nth(j, env) = y; nth(k, env) = z; i ∈ nat; j ∈ nat; k ∈ nat;
         env ∈ list(A) |]
      ==> injection(##A, x, y, z) <-> sats(A, injection_fm(i, j, k), env)

      [| nth(i, env) = x; nth(j, env) = y; nth(k, env) = z; i ∈ nat; j ∈ nat; k ∈ nat;
         env ∈ list(A) |]
      ==> surjection(##A, x, y, z) <-> sats(A, surjection_fm(i, j, k), env)

      [| nth(i, env) = x; nth(j, env) = y; nth(k, env) = z; i ∈ nat; j ∈ nat; k ∈ nat;
         env ∈ list(A) |]
      ==> bijection(##A, x, y, z) <-> sats(A, bijection_fm(i, j, k), env)

      [| nth(i, env) = x; nth(j, env) = y; nth(k, env) = z; i ∈ nat; j ∈ nat; k ∈ nat;
         env ∈ list(A) |]
      ==> restriction(##A, x, y, z) <-> sats(A, restriction_fm(i, j, k), env)

      [| nth(i, env) = U; nth(j, env) = r; nth(k, env) = B; nth(j', env) = s;
         nth(k', env) = f; i ∈ nat; j ∈ nat; k ∈ nat; j' ∈ nat; k' ∈ nat;
         env ∈ list(A) |]
      ==> order_isomorphism(##A, U, r, B, s, f) <->
          sats(A, order_isomorphism_fm(i, j, k, j', k'), env)

      [| nth(i, env) = x; nth(j, env) = y; i ∈ nat; env ∈ list(A) |]
      ==> finite_ordinal(##A, x) <-> sats(A, finite_ordinal_fm(i), env)

      [| nth(i, env) = x; i ∈ nat; env ∈ list(A) |]
      ==> ordinal(##A, x) <-> sats(A, ordinal_fm(i), env)

      [| nth(i, env) = x; nth(j, env) = y; i ∈ nat; env ∈ list(A) |]
      ==> limit_ordinal(##A, x) <-> sats(A, limit_ordinal_fm(i), env)

      [| nth(i, env) = x; nth(j, env) = y; i ∈ nat; env ∈ list(A) |]
      ==> omega(##A, x) <-> sats(A, omega_fm(i), env)

lemmas fun_plus_iff_sats:

      [| nth(i, env) = x; nth(j, env) = y; nth(k, env) = z; i ∈ nat; j ∈ nat; k ∈ nat;
         env ∈ list(A) |]
      ==> typed_function(##A, x, y, z) <-> sats(A, typed_function_fm(i, j, k), env)

      [| nth(i, env) = x; nth(j, env) = y; nth(k, env) = z; i ∈ nat; j ∈ nat; k ∈ nat;
         env ∈ list(A) |]
      ==> composition(##A, x, y, z) <-> sats(A, composition_fm(i, j, k), env)

      [| nth(i, env) = x; nth(j, env) = y; nth(k, env) = z; i ∈ nat; j ∈ nat; k ∈ nat;
         env ∈ list(A) |]
      ==> injection(##A, x, y, z) <-> sats(A, injection_fm(i, j, k), env)

      [| nth(i, env) = x; nth(j, env) = y; nth(k, env) = z; i ∈ nat; j ∈ nat; k ∈ nat;
         env ∈ list(A) |]
      ==> surjection(##A, x, y, z) <-> sats(A, surjection_fm(i, j, k), env)

      [| nth(i, env) = x; nth(j, env) = y; nth(k, env) = z; i ∈ nat; j ∈ nat; k ∈ nat;
         env ∈ list(A) |]
      ==> bijection(##A, x, y, z) <-> sats(A, bijection_fm(i, j, k), env)

      [| nth(i, env) = x; nth(j, env) = y; nth(k, env) = z; i ∈ nat; j ∈ nat; k ∈ nat;
         env ∈ list(A) |]
      ==> restriction(##A, x, y, z) <-> sats(A, restriction_fm(i, j, k), env)

      [| nth(i, env) = U; nth(j, env) = r; nth(k, env) = B; nth(j', env) = s;
         nth(k', env) = f; i ∈ nat; j ∈ nat; k ∈ nat; j' ∈ nat; k' ∈ nat;
         env ∈ list(A) |]
      ==> order_isomorphism(##A, U, r, B, s, f) <->
          sats(A, order_isomorphism_fm(i, j, k, j', k'), env)

      [| nth(i, env) = x; nth(j, env) = y; i ∈ nat; env ∈ list(A) |]
      ==> finite_ordinal(##A, x) <-> sats(A, finite_ordinal_fm(i), env)

      [| nth(i, env) = x; i ∈ nat; env ∈ list(A) |]
      ==> ordinal(##A, x) <-> sats(A, ordinal_fm(i), env)

      [| nth(i, env) = x; nth(j, env) = y; i ∈ nat; env ∈ list(A) |]
      ==> limit_ordinal(##A, x) <-> sats(A, limit_ordinal_fm(i), env)

      [| nth(i, env) = x; nth(j, env) = y; i ∈ nat; env ∈ list(A) |]
      ==> omega(##A, x) <-> sats(A, omega_fm(i), env)