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_triv_axioms} *}
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
lemma 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
lemma 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
lemma 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
subsubsection{*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
lemma 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{*Instantiation of the locale @{text M_triv_axioms}*}
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];
ML
{*
val transL = thm "transL";
val nonempty = thm "nonempty";
val upair_ax = thm "upair_ax";
val Union_ax = thm "Union_ax";
val power_ax = thm "power_ax";
val replacement = thm "replacement";
val L_nat = thm "L_nat";
fun kill_flex_triv_prems st = Seq.hd ((REPEAT_FIRST assume_tac) st);
fun trivaxL th =
kill_flex_triv_prems
([transL, nonempty, upair_ax, Union_ax, power_ax, replacement, L_nat]
MRS (inst "M" "L" th));
bind_thm ("ball_abs", trivaxL (thm "M_triv_axioms.ball_abs"));
bind_thm ("rall_abs", trivaxL (thm "M_triv_axioms.rall_abs"));
bind_thm ("bex_abs", trivaxL (thm "M_triv_axioms.bex_abs"));
bind_thm ("rex_abs", trivaxL (thm "M_triv_axioms.rex_abs"));
bind_thm ("ball_iff_equiv", trivaxL (thm "M_triv_axioms.ball_iff_equiv"));
bind_thm ("M_equalityI", trivaxL (thm "M_triv_axioms.M_equalityI"));
bind_thm ("empty_abs", trivaxL (thm "M_triv_axioms.empty_abs"));
bind_thm ("subset_abs", trivaxL (thm "M_triv_axioms.subset_abs"));
bind_thm ("upair_abs", trivaxL (thm "M_triv_axioms.upair_abs"));
bind_thm ("upair_in_M_iff", trivaxL (thm "M_triv_axioms.upair_in_M_iff"));
bind_thm ("singleton_in_M_iff", trivaxL (thm "M_triv_axioms.singleton_in_M_iff"));
bind_thm ("pair_abs", trivaxL (thm "M_triv_axioms.pair_abs"));
bind_thm ("pair_in_M_iff", trivaxL (thm "M_triv_axioms.pair_in_M_iff"));
bind_thm ("pair_components_in_M", trivaxL (thm "M_triv_axioms.pair_components_in_M"));
bind_thm ("cartprod_abs", trivaxL (thm "M_triv_axioms.cartprod_abs"));
bind_thm ("union_abs", trivaxL (thm "M_triv_axioms.union_abs"));
bind_thm ("inter_abs", trivaxL (thm "M_triv_axioms.inter_abs"));
bind_thm ("setdiff_abs", trivaxL (thm "M_triv_axioms.setdiff_abs"));
bind_thm ("Union_abs", trivaxL (thm "M_triv_axioms.Union_abs"));
bind_thm ("Union_closed", trivaxL (thm "M_triv_axioms.Union_closed"));
bind_thm ("Un_closed", trivaxL (thm "M_triv_axioms.Un_closed"));
bind_thm ("cons_closed", trivaxL (thm "M_triv_axioms.cons_closed"));
bind_thm ("successor_abs", trivaxL (thm "M_triv_axioms.successor_abs"));
bind_thm ("succ_in_M_iff", trivaxL (thm "M_triv_axioms.succ_in_M_iff"));
bind_thm ("separation_closed", trivaxL (thm "M_triv_axioms.separation_closed"));
bind_thm ("strong_replacementI", trivaxL (thm "M_triv_axioms.strong_replacementI"));
bind_thm ("strong_replacement_closed", trivaxL (thm "M_triv_axioms.strong_replacement_closed"));
bind_thm ("RepFun_closed", trivaxL (thm "M_triv_axioms.RepFun_closed"));
bind_thm ("lam_closed", trivaxL (thm "M_triv_axioms.lam_closed"));
bind_thm ("image_abs", trivaxL (thm "M_triv_axioms.image_abs"));
bind_thm ("powerset_Pow", trivaxL (thm "M_triv_axioms.powerset_Pow"));
bind_thm ("powerset_imp_subset_Pow", trivaxL (thm "M_triv_axioms.powerset_imp_subset_Pow"));
bind_thm ("nat_into_M", trivaxL (thm "M_triv_axioms.nat_into_M"));
bind_thm ("nat_case_closed", trivaxL (thm "M_triv_axioms.nat_case_closed"));
bind_thm ("Inl_in_M_iff", trivaxL (thm "M_triv_axioms.Inl_in_M_iff"));
bind_thm ("Inr_in_M_iff", trivaxL (thm "M_triv_axioms.Inr_in_M_iff"));
bind_thm ("lt_closed", trivaxL (thm "M_triv_axioms.lt_closed"));
bind_thm ("transitive_set_abs", trivaxL (thm "M_triv_axioms.transitive_set_abs"));
bind_thm ("ordinal_abs", trivaxL (thm "M_triv_axioms.ordinal_abs"));
bind_thm ("limit_ordinal_abs", trivaxL (thm "M_triv_axioms.limit_ordinal_abs"));
bind_thm ("successor_ordinal_abs", trivaxL (thm "M_triv_axioms.successor_ordinal_abs"));
bind_thm ("finite_ordinal_abs", trivaxL (thm "M_triv_axioms.finite_ordinal_abs"));
bind_thm ("omega_abs", trivaxL (thm "M_triv_axioms.omega_abs"));
bind_thm ("number1_abs", trivaxL (thm "M_triv_axioms.number1_abs"));
bind_thm ("number1_abs", trivaxL (thm "M_triv_axioms.number1_abs"));
bind_thm ("number3_abs", trivaxL (thm "M_triv_axioms.number3_abs"));
*}
declare ball_abs [simp]
declare rall_abs [simp]
declare bex_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_replacementI
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 number1_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
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 (rule reflection.Ex_reflection [OF Lset_mono_le Lset_cont Pair_in_Lset],
assumption+)
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 (rule reflection.All_reflection [OF Lset_mono_le Lset_cont Pair_in_Lset],
assumption+)
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
lemmas FOL_reflections =
Triv_reflection Not_reflection And_reflection Or_reflection
Imp_reflection Iff_reflection Ex_reflection All_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
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{*Function Application, Internalized*}
constdefs fun_apply_fm :: "[i,i,i]=>i"
"fun_apply_fm(f,x,y) ==
Forall(Iff(Exists(And(Member(0,succ(succ(f))),
pair_fm(succ(succ(x)), 1, 0))),
Equal(succ(y),0)))"
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 pair_reflection)
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{*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 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 upair_reflection pair_reflection 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 upair_iff_sats pair_iff_sats 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 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)
lemma arity_injection_fm [simp]:
"[| x \<in> nat; y \<in> nat; z \<in> nat |]
==> arity(injection_fm(x,y,z)) = succ(x) \<union> succ(y) \<union> succ(z)"
by (simp add: injection_fm_def succ_Un_distrib [symmetric] Un_ac)
lemma sats_injection_fm [simp]:
"[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> 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 \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
==> injection(**A, x, y, z) <-> sats(A, injection_fm(i,j,k), env)"
by simp
theorem injection_reflection:
"REFLECTS[\<lambda>x. injection(L,f(x),g(x),h(x)),
\<lambda>i x. injection(**Lset(i),f(x),g(x),h(x))]"
apply (simp only: injection_def setclass_simps)
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) &
(\<forall>y[M]. y\<in>B --> (\<exists>x[M]. x\<in>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 \<in> nat; y \<in> nat; z \<in> nat |] ==> surjection_fm(x,y,z) \<in> formula"
by (simp add: surjection_fm_def)
lemma arity_surjection_fm [simp]:
"[| x \<in> nat; y \<in> nat; z \<in> nat |]
==> arity(surjection_fm(x,y,z)) = succ(x) \<union> succ(y) \<union> succ(z)"
by (simp add: surjection_fm_def succ_Un_distrib [symmetric] Un_ac)
lemma sats_surjection_fm [simp]:
"[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> 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 \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
==> surjection(**A, x, y, z) <-> sats(A, surjection_fm(i,j,k), env)"
by simp
theorem surjection_reflection:
"REFLECTS[\<lambda>x. surjection(L,f(x),g(x),h(x)),
\<lambda>i x. surjection(**Lset(i),f(x),g(x),h(x))]"
apply (simp only: surjection_def setclass_simps)
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 \<in> nat; y \<in> nat; z \<in> nat |] ==> bijection_fm(x,y,z) \<in> formula"
by (simp add: bijection_fm_def)
lemma arity_bijection_fm [simp]:
"[| x \<in> nat; y \<in> nat; z \<in> nat |]
==> arity(bijection_fm(x,y,z)) = succ(x) \<union> succ(y) \<union> succ(z)"
by (simp add: bijection_fm_def succ_Un_distrib [symmetric] Un_ac)
lemma sats_bijection_fm [simp]:
"[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> 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 \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
==> bijection(**A, x, y, z) <-> sats(A, bijection_fm(i,j,k), env)"
by simp
theorem bijection_reflection:
"REFLECTS[\<lambda>x. bijection(L,f(x),g(x),h(x)),
\<lambda>i x. bijection(**Lset(i),f(x),g(x),h(x))]"
apply (simp only: bijection_def setclass_simps)
apply (intro And_reflection injection_reflection surjection_reflection)
done
subsubsection{*Restriction of a Relation, Internalized*}
(* "restriction(M,r,A,z) ==
\<forall>x[M]. x \<in> z <-> (x \<in> r & (\<exists>u[M]. u\<in>A & (\<exists>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 \<in> nat; y \<in> nat; z \<in> nat |] ==> restriction_fm(x,y,z) \<in> formula"
by (simp add: restriction_fm_def)
lemma arity_restriction_fm [simp]:
"[| x \<in> nat; y \<in> nat; z \<in> nat |]
==> arity(restriction_fm(x,y,z)) = succ(x) \<union> succ(y) \<union> succ(z)"
by (simp add: restriction_fm_def succ_Un_distrib [symmetric] Un_ac)
lemma sats_restriction_fm [simp]:
"[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> 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 \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
==> restriction(**A, x, y, z) <-> sats(A, restriction_fm(i,j,k), env)"
by simp
theorem restriction_reflection:
"REFLECTS[\<lambda>x. restriction(L,f(x),g(x),h(x)),
\<lambda>i x. restriction(**Lset(i),f(x),g(x),h(x))]"
apply (simp only: restriction_def setclass_simps)
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) &
(\<forall>x[M]. x\<in>A --> (\<forall>y[M]. y\<in>A -->
(\<forall>p[M]. \<forall>fx[M]. \<forall>fy[M]. \<forall>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\<in>r <-> q\<in>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 \<in> nat; r \<in> nat; B \<in> nat; s \<in> nat; f \<in> nat |]
==> order_isomorphism_fm(A,r,B,s,f) \<in> formula"
by (simp add: order_isomorphism_fm_def)
lemma arity_order_isomorphism_fm [simp]:
"[| A \<in> nat; r \<in> nat; B \<in> nat; s \<in> nat; f \<in> nat |]
==> arity(order_isomorphism_fm(A,r,B,s,f)) =
succ(A) \<union> succ(r) \<union> succ(B) \<union> succ(s) \<union> succ(f)"
by (simp add: order_isomorphism_fm_def succ_Un_distrib [symmetric] Un_ac)
lemma sats_order_isomorphism_fm [simp]:
"[| U \<in> nat; r \<in> nat; B \<in> nat; s \<in> nat; f \<in> nat; env \<in> 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 \<in> nat; j \<in> nat; k \<in> nat; j' \<in> nat; k' \<in> nat; env \<in> 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[\<lambda>x. order_isomorphism(L,f(x),g(x),h(x),g'(x),h'(x)),
\<lambda>i x. order_isomorphism(**Lset(i),f(x),g(x),h(x),g'(x),h'(x))]"
apply (simp only: order_isomorphism_def setclass_simps)
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) &
(\<forall>x[M]. x\<in>a --> (\<exists>y[M]. y\<in>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 \<in> nat ==> limit_ordinal_fm(x) \<in> formula"
by (simp add: limit_ordinal_fm_def)
lemma arity_limit_ordinal_fm [simp]:
"x \<in> nat ==> arity(limit_ordinal_fm(x)) = succ(x)"
by (simp add: limit_ordinal_fm_def succ_Un_distrib [symmetric] Un_ac)
lemma sats_limit_ordinal_fm [simp]:
"[| x \<in> nat; env \<in> 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 \<in> nat; env \<in> list(A)|]
==> limit_ordinal(**A, x) <-> sats(A, limit_ordinal_fm(i), env)"
by simp
theorem limit_ordinal_reflection:
"REFLECTS[\<lambda>x. limit_ordinal(L,f(x)),
\<lambda>i x. limit_ordinal(**Lset(i),f(x))]"
apply (simp only: limit_ordinal_def setclass_simps)
apply (intro FOL_reflections ordinal_reflection
empty_reflection successor_reflection)
done
subsubsection{*Omega: The Set of Natural Numbers*}
(* omega(M,a) == limit_ordinal(M,a) & (\<forall>x[M]. x\<in>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 \<in> nat ==> omega_fm(x) \<in> formula"
by (simp add: omega_fm_def)
lemma arity_omega_fm [simp]:
"x \<in> nat ==> arity(omega_fm(x)) = succ(x)"
by (simp add: omega_fm_def succ_Un_distrib [symmetric] Un_ac)
lemma sats_omega_fm [simp]:
"[| x \<in> nat; env \<in> 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 \<in> nat; env \<in> list(A)|]
==> omega(**A, x) <-> sats(A, omega_fm(i), env)"
by simp
theorem omega_reflection:
"REFLECTS[\<lambda>x. omega(L,f(x)),
\<lambda>i x. omega(**Lset(i),f(x))]"
apply (simp only: omega_def setclass_simps)
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
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
ordinal_iff_sats limit_ordinal_iff_sats omega_iff_sats
end