src/ZF/Constructible/L_axioms.thy

Various simplifications of the Constructible theories

(*  Title:      ZF/Constructible/L_axioms.thy
    ID:         $Id$
    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
    Copyright   2002  University of Cambridge
*)

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

theory L_axioms = Formula + Relative + Reflection + MetaExists:

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

lemma transL: "[| y\<in>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 \<in> 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 \<in> Lset(i); univalent(L,X,Q); Ord(i)|]
      ==> \<exists>j. Ord(j) & Replace(X, %x y. Q(x,y) & L(y)) \<subseteq> Lset(j)"
apply (rule_tac x="\<Union>y \<in> 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)|]
      ==> \<exists>Y. L(Y) & Replace(X, %x y. Q(x,y) & L(y)) \<subseteq> 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 Pair_in_Lset = Formula.Pair_in_LLimit

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

lemmas rall_abs = M_trivial.rall_abs [OF M_trivial_L]
  and rex_abs = M_trivial.rex_abs [OF M_trivial_L]
  and ball_iff_equiv = M_trivial.ball_iff_equiv [OF M_trivial_L]
  and M_equalityI = M_trivial.M_equalityI [OF M_trivial_L]
  and empty_abs = M_trivial.empty_abs [OF M_trivial_L]
  and subset_abs = M_trivial.subset_abs [OF M_trivial_L]
  and upair_abs = M_trivial.upair_abs [OF M_trivial_L]
  and upair_in_M_iff = M_trivial.upair_in_M_iff [OF M_trivial_L]
  and singleton_in_M_iff = M_trivial.singleton_in_M_iff [OF M_trivial_L]
  and pair_abs = M_trivial.pair_abs [OF M_trivial_L]
  and pair_in_M_iff = M_trivial.pair_in_M_iff [OF M_trivial_L]
  and pair_components_in_M = M_trivial.pair_components_in_M [OF M_trivial_L]
  and cartprod_abs = M_trivial.cartprod_abs [OF M_trivial_L]
  and union_abs = M_trivial.union_abs [OF M_trivial_L]
  and inter_abs = M_trivial.inter_abs [OF M_trivial_L]
  and setdiff_abs = M_trivial.setdiff_abs [OF M_trivial_L]
  and Union_abs = M_trivial.Union_abs [OF M_trivial_L]
  and Union_closed = M_trivial.Union_closed [OF M_trivial_L]
  and Un_closed = M_trivial.Un_closed [OF M_trivial_L]
  and cons_closed = M_trivial.cons_closed [OF M_trivial_L]
  and successor_abs = M_trivial.successor_abs [OF M_trivial_L]
  and succ_in_M_iff = M_trivial.succ_in_M_iff [OF M_trivial_L]
  and separation_closed = M_trivial.separation_closed [OF M_trivial_L]
  and strong_replacementI = 
      M_trivial.strong_replacementI [OF M_trivial_L, rule_format]
  and strong_replacement_closed = M_trivial.strong_replacement_closed [OF M_trivial_L]
  and RepFun_closed = M_trivial.RepFun_closed [OF M_trivial_L]
  and lam_closed = M_trivial.lam_closed [OF M_trivial_L]
  and image_abs = M_trivial.image_abs [OF M_trivial_L]
  and powerset_Pow = M_trivial.powerset_Pow [OF M_trivial_L]
  and powerset_imp_subset_Pow = M_trivial.powerset_imp_subset_Pow [OF M_trivial_L]
  and nat_into_M = M_trivial.nat_into_M [OF M_trivial_L]
  and nat_case_closed = M_trivial.nat_case_closed [OF M_trivial_L]
  and Inl_in_M_iff = M_trivial.Inl_in_M_iff [OF M_trivial_L]
  and Inr_in_M_iff = M_trivial.Inr_in_M_iff [OF M_trivial_L]
  and lt_closed = M_trivial.lt_closed [OF M_trivial_L]
  and transitive_set_abs = M_trivial.transitive_set_abs [OF M_trivial_L]
  and ordinal_abs = M_trivial.ordinal_abs [OF M_trivial_L]
  and limit_ordinal_abs = M_trivial.limit_ordinal_abs [OF M_trivial_L]
  and successor_ordinal_abs = M_trivial.successor_ordinal_abs [OF M_trivial_L]
  and finite_ordinal_abs = M_trivial.finite_ordinal_abs [OF M_trivial_L]
  and omega_abs = M_trivial.omega_abs [OF M_trivial_L]
  and number1_abs = M_trivial.number1_abs [OF M_trivial_L]
  and number2_abs = M_trivial.number2_abs [OF M_trivial_L]
  and number3_abs = M_trivial.number3_abs [OF M_trivial_L]

declare rall_abs [simp]
declare rex_abs [simp]
declare empty_abs [simp]
declare subset_abs [simp]
declare upair_abs [simp]
declare upair_in_M_iff [iff]
declare singleton_in_M_iff [iff]
declare pair_abs [simp]
declare pair_in_M_iff [iff]
declare cartprod_abs [simp]
declare union_abs [simp]
declare inter_abs [simp]
declare setdiff_abs [simp]
declare Union_abs [simp]
declare Union_closed [intro, simp]
declare Un_closed [intro, simp]
declare cons_closed [intro, simp]
declare successor_abs [simp]
declare succ_in_M_iff [iff]
declare separation_closed [intro, simp]
declare strong_replacement_closed [intro, simp]
declare RepFun_closed [intro, simp]
declare lam_closed [intro, simp]
declare image_abs [simp]
declare nat_into_M [intro]
declare Inl_in_M_iff [iff]
declare Inr_in_M_iff [iff]
declare transitive_set_abs [simp]
declare ordinal_abs [simp]
declare limit_ordinal_abs [simp]
declare successor_ordinal_abs [simp]
declare finite_ordinal_abs [simp]
declare omega_abs [simp]
declare number1_abs [simp]
declare number2_abs [simp]
declare number3_abs [simp]


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

text{*instances of locale constants*}
constdefs
  L_F0 :: "[i=>o,i] => i"
    "L_F0(P,y) == \<mu>b. (\<exists>z. L(z) \<and> P(<y,z>)) --> (\<exists>z\<in>Lset(b). P(<y,z>))"

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

  L_ClEx :: "[i=>o,i] => o"
    "L_ClEx(P) == \<lambda>a. Limit(a) \<and> 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) &
                           (\<forall>a. Cl(a) --> (\<forall>x \<in> Lset(a). P(x) <-> Q(a,x))))"


theorem Triv_reflection:
     "REFLECTS[P, \<lambda>a 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[\<lambda>x. ~P(x), \<lambda>a 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[\<lambda>x. P(x) \<and> P'(x), \<lambda>a x. Q(a,x) \<and> Q'(a,x)]"
apply (unfold L_Reflects_def)
apply (elim meta_exE)
apply (rule_tac x="\<lambda>a. Cl(a) \<and> Cla(a)" in meta_exI)
apply (simp add: Closed_Unbounded_Int, blast)
done

theorem Or_reflection:
     "[| REFLECTS[P,Q]; REFLECTS[P',Q'] |]
      ==> REFLECTS[\<lambda>x. P(x) \<or> P'(x), \<lambda>a x. Q(a,x) \<or> Q'(a,x)]"
apply (unfold L_Reflects_def)
apply (elim meta_exE)
apply (rule_tac x="\<lambda>a. Cl(a) \<and> Cla(a)" in meta_exI)
apply (simp add: Closed_Unbounded_Int, blast)
done

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

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


lemma reflection_Lset: "reflection(Lset)"
apply (blast intro: reflection.intro Lset_mono_le Lset_cont Pair_in_Lset) +
done

theorem Ex_reflection:
     "REFLECTS[\<lambda>x. P(fst(x),snd(x)), \<lambda>a x. Q(a,fst(x),snd(x))]
      ==> REFLECTS[\<lambda>x. \<exists>z. L(z) \<and> P(x,z), \<lambda>a x. \<exists>z\<in>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[\<lambda>x. P(fst(x),snd(x)), \<lambda>a x. Q(a,fst(x),snd(x))]
      ==> REFLECTS[\<lambda>x. \<forall>z. L(z) --> P(x,z), \<lambda>a x. \<forall>z\<in>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[ \<lambda>x. P(fst(x),snd(x)), \<lambda>a x. Q(a,fst(x),snd(x))]
      ==> REFLECTS[\<lambda>x. \<exists>z[L]. P(x,z), \<lambda>a x. \<exists>z\<in>Lset(a). Q(a,x,z)]"
apply (unfold rex_def)
apply (intro And_reflection Ex_reflection, assumption)
done

theorem Rall_reflection:
     "REFLECTS[\<lambda>x. P(fst(x),snd(x)), \<lambda>a x. Q(a,fst(x),snd(x))]
      ==> REFLECTS[\<lambda>x. \<forall>z[L]. P(x,z), \<lambda>a x. \<forall>z\<in>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[ \<lambda>x. P(fst(x),snd(x)), \<lambda>a x. Q(a,fst(x),snd(x))]
      ==> REFLECTS[\<lambda>x. \<exists>z[L]. P(x,z), \<lambda>a x. \<exists>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[\<lambda>x. P(fst(x),snd(x)), \<lambda>a x. Q(a,fst(x),snd(x))]
      ==> REFLECTS[\<lambda>x. \<forall>z[L]. P(x,z), \<lambda>a x. \<forall>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)|]
      ==> \<exists>j. i<j & (\<forall>x \<in> 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;  \<forall>x \<in> Lset(j). P(x) <-> Q(j,x)|] ==> R |]
      ==> R"
apply (drule ReflectsD, assumption, blast)
done

lemma Collect_mem_eq: "{x\<in>A. x\<in>B} = A \<inter> B"
by blast


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

lemmas setclass_simps = rall_setclass_is_ball rex_setclass_is_bex

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 \<in> nat ==> empty_fm(x) \<in> formula"
by (simp add: empty_fm_def)

lemma arity_empty_fm [simp]:
     "x \<in> nat ==> arity(empty_fm(x)) = succ(x)"
by (simp add: empty_fm_def succ_Un_distrib [symmetric] Un_ac)

lemma sats_empty_fm [simp]:
   "[| x \<in> nat; env \<in> 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 \<in> nat; env \<in> list(A)|]
       ==> empty(**A, x) <-> sats(A, empty_fm(i), env)"
by simp

theorem empty_reflection:
     "REFLECTS[\<lambda>x. empty(L,f(x)),
               \<lambda>i x. empty(**Lset(i),f(x))]"
apply (simp only: empty_def setclass_simps)
apply (intro FOL_reflections)
done

text{*Not used.  But maybe useful?*}
lemma Transset_sats_empty_fm_eq_0:
   "[| n \<in> nat; env \<in> 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 \<in> nat; y \<in> nat; z \<in> nat |] ==> upair_fm(x,y,z) \<in> formula"
by (simp add: upair_fm_def)

lemma arity_upair_fm [simp]:
     "[| x \<in> nat; y \<in> nat; z \<in> nat |]
      ==> arity(upair_fm(x,y,z)) = succ(x) \<union> succ(y) \<union> succ(z)"
by (simp add: upair_fm_def succ_Un_distrib [symmetric] Un_ac)

lemma sats_upair_fm [simp]:
   "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> 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 \<in> nat; j \<in> nat; k \<in> nat; env \<in> 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 \<in> nat; y \<in> nat; z < length(env); env \<in> 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[\<lambda>x. upair(L,f(x),g(x),h(x)),
               \<lambda>i 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 \<in> nat; y \<in> nat; z \<in> nat |] ==> pair_fm(x,y,z) \<in> formula"
by (simp add: pair_fm_def)

lemma arity_pair_fm [simp]:
     "[| x \<in> nat; y \<in> nat; z \<in> nat |]
      ==> arity(pair_fm(x,y,z)) = succ(x) \<union> succ(y) \<union> succ(z)"
by (simp add: pair_fm_def succ_Un_distrib [symmetric] Un_ac)

lemma sats_pair_fm [simp]:
   "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> 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 \<in> nat; j \<in> nat; k \<in> nat; env \<in> 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[\<lambda>x. pair(L,f(x),g(x),h(x)),
               \<lambda>i x. pair(**Lset(i),f(x),g(x),h(x))]"
apply (simp only: pair_def setclass_simps)
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 \<in> nat; y \<in> nat; z \<in> nat |] ==> union_fm(x,y,z) \<in> formula"
by (simp add: union_fm_def)

lemma arity_union_fm [simp]:
     "[| x \<in> nat; y \<in> nat; z \<in> nat |]
      ==> arity(union_fm(x,y,z)) = succ(x) \<union> succ(y) \<union> succ(z)"
by (simp add: union_fm_def succ_Un_distrib [symmetric] Un_ac)

lemma sats_union_fm [simp]:
   "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> 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 \<in> nat; j \<in> nat; k \<in> nat; env \<in> 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[\<lambda>x. union(L,f(x),g(x),h(x)),
               \<lambda>i x. union(**Lset(i),f(x),g(x),h(x))]"
apply (simp only: union_def setclass_simps)
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 \<in> nat; y \<in> nat; z \<in> nat |] ==> cons_fm(x,y,z) \<in> formula"
by (simp add: cons_fm_def)

lemma arity_cons_fm [simp]:
     "[| x \<in> nat; y \<in> nat; z \<in> nat |]
      ==> arity(cons_fm(x,y,z)) = succ(x) \<union> succ(y) \<union> succ(z)"
by (simp add: cons_fm_def succ_Un_distrib [symmetric] Un_ac)

lemma sats_cons_fm [simp]:
   "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> 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 \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
       ==> is_cons(**A, x, y, z) <-> sats(A, cons_fm(i,j,k), env)"
by simp

theorem cons_reflection:
     "REFLECTS[\<lambda>x. is_cons(L,f(x),g(x),h(x)),
               \<lambda>i x. is_cons(**Lset(i),f(x),g(x),h(x))]"
apply (simp only: is_cons_def setclass_simps)
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 \<in> nat; y \<in> nat |] ==> succ_fm(x,y) \<in> formula"
by (simp add: succ_fm_def)

lemma arity_succ_fm [simp]:
     "[| x \<in> nat; y \<in> nat |]
      ==> arity(succ_fm(x,y)) = succ(x) \<union> succ(y)"
by (simp add: succ_fm_def)

lemma sats_succ_fm [simp]:
   "[| x \<in> nat; y \<in> nat; env \<in> 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 \<in> nat; j \<in> nat; env \<in> list(A)|]
       ==> successor(**A, x, y) <-> sats(A, succ_fm(i,j), env)"
by simp

theorem successor_reflection:
     "REFLECTS[\<lambda>x. successor(L,f(x),g(x)),
               \<lambda>i x. successor(**Lset(i),f(x),g(x))]"
apply (simp only: successor_def setclass_simps)
apply (intro cons_reflection)
done


subsubsection{*The Number 1, Internalized*}

(* "number1(M,a) == (\<exists>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 \<in> nat ==> number1_fm(x) \<in> formula"
by (simp add: number1_fm_def)

lemma arity_number1_fm [simp]:
     "x \<in> nat ==> arity(number1_fm(x)) = succ(x)"
by (simp add: number1_fm_def succ_Un_distrib [symmetric] Un_ac)

lemma sats_number1_fm [simp]:
   "[| x \<in> nat; env \<in> 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 \<in> nat; env \<in> list(A)|]
       ==> number1(**A, x) <-> sats(A, number1_fm(i), env)"
by simp

theorem number1_reflection:
     "REFLECTS[\<lambda>x. number1(L,f(x)),
               \<lambda>i x. number1(**Lset(i),f(x))]"
apply (simp only: number1_def setclass_simps)
apply (intro FOL_reflections empty_reflection successor_reflection)
done


subsubsection{*Big Union, Internalized*}

(*  "big_union(M,A,z) == \<forall>x[M]. x \<in> z <-> (\<exists>y[M]. y\<in>A & x \<in> 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 \<in> nat; y \<in> nat |] ==> big_union_fm(x,y) \<in> formula"
by (simp add: big_union_fm_def)

lemma arity_big_union_fm [simp]:
     "[| x \<in> nat; y \<in> nat |]
      ==> arity(big_union_fm(x,y)) = succ(x) \<union> succ(y)"
by (simp add: big_union_fm_def succ_Un_distrib [symmetric] Un_ac)

lemma sats_big_union_fm [simp]:
   "[| x \<in> nat; y \<in> nat; env \<in> 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 \<in> nat; j \<in> nat; env \<in> list(A)|]
       ==> big_union(**A, x, y) <-> sats(A, big_union_fm(i,j), env)"
by simp

theorem big_union_reflection:
     "REFLECTS[\<lambda>x. big_union(L,f(x),g(x)),
               \<lambda>i x. big_union(**Lset(i),f(x),g(x))]"
apply (simp only: big_union_def setclass_simps)
apply (intro FOL_reflections)
done


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

text{*Differs from the one in Formula by using "ordinal" rather than "Ord"*}


lemma sats_subset_fm':
   "[|x \<in> nat; y \<in> nat; env \<in> 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[\<lambda>x. subset(L,f(x),g(x)),
               \<lambda>i x. subset(**Lset(i),f(x),g(x))]"
apply (simp only: Relative.subset_def setclass_simps)
apply (intro FOL_reflections)
done

lemma sats_transset_fm':
   "[|x \<in> nat; env \<in> 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[\<lambda>x. transitive_set(L,f(x)),
               \<lambda>i x. transitive_set(**Lset(i),f(x))]"
apply (simp only: transitive_set_def setclass_simps)
apply (intro FOL_reflections subset_reflection)
done

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

theorem ordinal_reflection:
     "REFLECTS[\<lambda>x. ordinal(L,f(x)), \<lambda>i x. ordinal(**Lset(i),f(x))]"
apply (simp only: ordinal_def setclass_simps)
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 \<in> nat; y \<in> nat |] ==> Memrel_fm(x,y) \<in> formula"
by (simp add: Memrel_fm_def)

lemma arity_Memrel_fm [simp]:
     "[| x \<in> nat; y \<in> nat |]
      ==> arity(Memrel_fm(x,y)) = succ(x) \<union> succ(y)"
by (simp add: Memrel_fm_def succ_Un_distrib [symmetric] Un_ac)

lemma sats_Memrel_fm [simp]:
   "[| x \<in> nat; y \<in> nat; env \<in> 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 \<in> nat; j \<in> nat; env \<in> list(A)|]
       ==> membership(**A, x, y) <-> sats(A, Memrel_fm(i,j), env)"
by simp

theorem membership_reflection:
     "REFLECTS[\<lambda>x. membership(L,f(x),g(x)),
               \<lambda>i x. membership(**Lset(i),f(x),g(x))]"
apply (simp only: membership_def setclass_simps)
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 \<in> nat; x \<in> nat; r \<in> nat; B \<in> nat |]
      ==> pred_set_fm(A,x,r,B) \<in> formula"
by (simp add: pred_set_fm_def)

lemma arity_pred_set_fm [simp]:
   "[| A \<in> nat; x \<in> nat; r \<in> nat; B \<in> nat |]
    ==> arity(pred_set_fm(A,x,r,B)) = succ(A) \<union> succ(x) \<union> succ(r) \<union> succ(B)"
by (simp add: pred_set_fm_def succ_Un_distrib [symmetric] Un_ac)

lemma sats_pred_set_fm [simp]:
   "[| U \<in> nat; x \<in> nat; r \<in> nat; B \<in> nat; env \<in> 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 \<in> nat; j \<in> nat; k \<in> nat; l \<in> nat; env \<in> 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[\<lambda>x. pred_set(L,f(x),g(x),h(x),b(x)),
               \<lambda>i x. pred_set(**Lset(i),f(x),g(x),h(x),b(x))]"
apply (simp only: pred_set_def setclass_simps)
apply (intro FOL_reflections pair_reflection)
done



subsubsection{*Domain of a Relation, Internalized*}

(* "is_domain(M,r,z) ==
        \<forall>x[M]. (x \<in> z <-> (\<exists>w[M]. w\<in>r & (\<exists>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 \<in> nat; y \<in> nat |] ==> domain_fm(x,y) \<in> formula"
by (simp add: domain_fm_def)

lemma arity_domain_fm [simp]:
     "[| x \<in> nat; y \<in> nat |]
      ==> arity(domain_fm(x,y)) = succ(x) \<union> succ(y)"
by (simp add: domain_fm_def succ_Un_distrib [symmetric] Un_ac)

lemma sats_domain_fm [simp]:
   "[| x \<in> nat; y \<in> nat; env \<in> 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 \<in> nat; j \<in> nat; env \<in> list(A)|]
       ==> is_domain(**A, x, y) <-> sats(A, domain_fm(i,j), env)"
by simp

theorem domain_reflection:
     "REFLECTS[\<lambda>x. is_domain(L,f(x),g(x)),
               \<lambda>i x. is_domain(**Lset(i),f(x),g(x))]"
apply (simp only: is_domain_def setclass_simps)
apply (intro FOL_reflections pair_reflection)
done


subsubsection{*Range of a Relation, Internalized*}

(* "is_range(M,r,z) ==
        \<forall>y[M]. (y \<in> z <-> (\<exists>w[M]. w\<in>r & (\<exists>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 \<in> nat; y \<in> nat |] ==> range_fm(x,y) \<in> formula"
by (simp add: range_fm_def)

lemma arity_range_fm [simp]:
     "[| x \<in> nat; y \<in> nat |]
      ==> arity(range_fm(x,y)) = succ(x) \<union> succ(y)"
by (simp add: range_fm_def succ_Un_distrib [symmetric] Un_ac)

lemma sats_range_fm [simp]:
   "[| x \<in> nat; y \<in> nat; env \<in> 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 \<in> nat; j \<in> nat; env \<in> list(A)|]
       ==> is_range(**A, x, y) <-> sats(A, range_fm(i,j), env)"
by simp

theorem range_reflection:
     "REFLECTS[\<lambda>x. is_range(L,f(x),g(x)),
               \<lambda>i x. is_range(**Lset(i),f(x),g(x))]"
apply (simp only: is_range_def setclass_simps)
apply (intro FOL_reflections pair_reflection)
done


subsubsection{*Field of a Relation, Internalized*}

(* "is_field(M,r,z) ==
        \<exists>dr[M]. is_domain(M,r,dr) &
            (\<exists>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 \<in> nat; y \<in> nat |] ==> field_fm(x,y) \<in> formula"
by (simp add: field_fm_def)

lemma arity_field_fm [simp]:
     "[| x \<in> nat; y \<in> nat |]
      ==> arity(field_fm(x,y)) = succ(x) \<union> succ(y)"
by (simp add: field_fm_def succ_Un_distrib [symmetric] Un_ac)

lemma sats_field_fm [simp]:
   "[| x \<in> nat; y \<in> nat; env \<in> 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 \<in> nat; j \<in> nat; env \<in> list(A)|]
       ==> is_field(**A, x, y) <-> sats(A, field_fm(i,j), env)"
by simp

theorem field_reflection:
     "REFLECTS[\<lambda>x. is_field(L,f(x),g(x)),
               \<lambda>i x. is_field(**Lset(i),f(x),g(x))]"
apply (simp only: is_field_def setclass_simps)
apply (intro FOL_reflections domain_reflection range_reflection
             union_reflection)
done


subsubsection{*Image under a Relation, Internalized*}

(* "image(M,r,A,z) ==
        \<forall>y[M]. (y \<in> z <-> (\<exists>w[M]. w\<in>r & (\<exists>x[M]. x\<in>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 \<in> nat; y \<in> nat; z \<in> nat |] ==> image_fm(x,y,z) \<in> formula"
by (simp add: image_fm_def)

lemma arity_image_fm [simp]:
     "[| x \<in> nat; y \<in> nat; z \<in> nat |]
      ==> arity(image_fm(x,y,z)) = succ(x) \<union> succ(y) \<union> succ(z)"
by (simp add: image_fm_def succ_Un_distrib [symmetric] Un_ac)

lemma sats_image_fm [simp]:
   "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> 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 \<in> nat; j \<in> nat; k \<in> nat; env \<in> 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[\<lambda>x. image(L,f(x),g(x),h(x)),
               \<lambda>i x. image(**Lset(i),f(x),g(x),h(x))]"
apply (simp only: Relative.image_def setclass_simps)
apply (intro FOL_reflections pair_reflection)
done


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

(* "pre_image(M,r,A,z) ==
        \<forall>x[M]. x \<in> z <-> (\<exists>w[M]. w\<in>r & (\<exists>y[M]. y\<in>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 \<in> nat; y \<in> nat; z \<in> nat |] ==> pre_image_fm(x,y,z) \<in> formula"
by (simp add: pre_image_fm_def)

lemma arity_pre_image_fm [simp]:
     "[| x \<in> nat; y \<in> nat; z \<in> nat |]
      ==> arity(pre_image_fm(x,y,z)) = succ(x) \<union> succ(y) \<union> succ(z)"
by (simp add: pre_image_fm_def succ_Un_distrib [symmetric] Un_ac)

lemma sats_pre_image_fm [simp]:
   "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> 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 \<in> nat; j \<in> nat; k \<in> nat; env \<in> 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[\<lambda>x. pre_image(L,f(x),g(x),h(x)),
               \<lambda>i x. pre_image(**Lset(i),f(x),g(x),h(x))]"
apply (simp only: Relative.pre_image_def setclass_simps)
apply (intro FOL_reflections pair_reflection)
done


subsubsection{*Function Application, Internalized*}

(* "fun_apply(M,f,x,y) ==
        (\<exists>xs[M]. \<exists>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 \<in> nat; y \<in> nat; z \<in> nat |] ==> fun_apply_fm(x,y,z) \<in> formula"
by (simp add: fun_apply_fm_def)

lemma arity_fun_apply_fm [simp]:
     "[| x \<in> nat; y \<in> nat; z \<in> nat |]
      ==> arity(fun_apply_fm(x,y,z)) = succ(x) \<union> succ(y) \<union> succ(z)"
by (simp add: fun_apply_fm_def succ_Un_distrib [symmetric] Un_ac)

lemma sats_fun_apply_fm [simp]:
   "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> 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 \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
       ==> fun_apply(**A, x, y, z) <-> sats(A, fun_apply_fm(i,j,k), env)"
by simp

theorem fun_apply_reflection:
     "REFLECTS[\<lambda>x. fun_apply(L,f(x),g(x),h(x)),
               \<lambda>i x. fun_apply(**Lset(i),f(x),g(x),h(x))]"
apply (simp only: fun_apply_def setclass_simps)
apply (intro FOL_reflections upair_reflection image_reflection
             big_union_reflection)
done


subsubsection{*The Concept of Relation, Internalized*}

(* "is_relation(M,r) ==
        (\<forall>z[M]. z\<in>r --> (\<exists>x[M]. \<exists>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 \<in> nat |] ==> relation_fm(x) \<in> formula"
by (simp add: relation_fm_def)

lemma arity_relation_fm [simp]:
     "x \<in> nat ==> arity(relation_fm(x)) = succ(x)"
by (simp add: relation_fm_def succ_Un_distrib [symmetric] Un_ac)

lemma sats_relation_fm [simp]:
   "[| x \<in> nat; env \<in> 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 \<in> nat; env \<in> list(A)|]
       ==> is_relation(**A, x) <-> sats(A, relation_fm(i), env)"
by simp

theorem is_relation_reflection:
     "REFLECTS[\<lambda>x. is_relation(L,f(x)),
               \<lambda>i x. is_relation(**Lset(i),f(x))]"
apply (simp only: is_relation_def setclass_simps)
apply (intro FOL_reflections pair_reflection)
done


subsubsection{*The Concept of Function, Internalized*}

(* "is_function(M,r) ==
        \<forall>x[M]. \<forall>y[M]. \<forall>y'[M]. \<forall>p[M]. \<forall>p'[M].
           pair(M,x,y,p) --> pair(M,x,y',p') --> p\<in>r --> p'\<in>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 \<in> nat |] ==> function_fm(x) \<in> formula"
by (simp add: function_fm_def)

lemma arity_function_fm [simp]:
     "x \<in> nat ==> arity(function_fm(x)) = succ(x)"
by (simp add: function_fm_def succ_Un_distrib [symmetric] Un_ac)

lemma sats_function_fm [simp]:
   "[| x \<in> nat; env \<in> 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 \<in> nat; env \<in> list(A)|]
       ==> is_function(**A, x) <-> sats(A, function_fm(i), env)"
by simp

theorem is_function_reflection:
     "REFLECTS[\<lambda>x. is_function(L,f(x)),
               \<lambda>i x. is_function(**Lset(i),f(x))]"
apply (simp only: is_function_def setclass_simps)
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) &
        (\<forall>u[M]. u\<in>r --> (\<forall>x[M]. \<forall>y[M]. pair(M,x,y,u) --> y\<in>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 \<in> nat; y \<in> nat; z \<in> nat |] ==> typed_function_fm(x,y,z) \<in> formula"
by (simp add: typed_function_fm_def)

lemma arity_typed_function_fm [simp]:
     "[| x \<in> nat; y \<in> nat; z \<in> nat |]
      ==> arity(typed_function_fm(x,y,z)) = succ(x) \<union> succ(y) \<union> succ(z)"
by (simp add: typed_function_fm_def succ_Un_distrib [symmetric] Un_ac)

lemma sats_typed_function_fm [simp]:
   "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> 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 \<in> nat; j \<in> nat; k \<in> nat; env \<in> 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[\<lambda>x. typed_function(L,f(x),g(x),h(x)),
               \<lambda>i x. typed_function(**Lset(i),f(x),g(x),h(x))]"
apply (simp only: typed_function_def setclass_simps)
apply (intro FOL_reflections function_reflections)
done


subsubsection{*Composition of Relations, Internalized*}

(* "composition(M,r,s,t) ==
        \<forall>p[M]. p \<in> t <->
               (\<exists>x[M]. \<exists>y[M]. \<exists>z[M]. \<exists>xy[M]. \<exists>yz[M].
                pair(M,x,z,p) & pair(M,x,y,xy) & pair(M,y,z,yz) &
                xy \<in> s & yz \<in> 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 \<in> nat; y \<in> nat; z \<in> nat |] ==> composition_fm(x,y,z) \<in> formula"
by (simp add: composition_fm_def)

lemma arity_composition_fm [simp]:
     "[| x \<in> nat; y \<in> nat; z \<in> nat |]
      ==> arity(composition_fm(x,y,z)) = succ(x) \<union> succ(y) \<union> succ(z)"
by (simp add: composition_fm_def succ_Un_distrib [symmetric] Un_ac)

lemma sats_composition_fm [simp]:
   "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> 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 \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
       ==> composition(**A, x, y, z) <-> sats(A, composition_fm(i,j,k), env)"
by simp

theorem composition_reflection:
     "REFLECTS[\<lambda>x. composition(L,f(x),g(x),h(x)),
               \<lambda>i x. composition(**Lset(i),f(x),g(x),h(x))]"
apply (simp only: composition_def setclass_simps)
apply (intro FOL_reflections pair_reflection)
done


subsubsection{*Injections, Internalized*}

(* "injection(M,A,B,f) ==
        typed_function(M,A,B,f) &
        (\<forall>x[M]. \<forall>x'[M]. \<forall>y[M]. \<forall>p[M]. \<forall>p'[M].
          pair(M,x,y,p) --> pair(M,x',y,p') --> p\<in>f --> p'\<in>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 \<in> nat; y \<in> nat; z \<in> nat |] ==> injection_fm(x,y,z) \<in> formula"
by (simp add: injection_fm_def)