src/ZF/Constructible/L_axioms.thy
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(*  Title:      ZF/Constructible/L_axioms.thy
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    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
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*)
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header {* The ZF Axioms (Except Separation) in L *}
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theory L_axioms imports Formula Relative Reflection MetaExists begin
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text {* The class L satisfies the premises of locale @{text M_trivial} *}
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lemma transL: "[| y\<in>x; L(x) |] ==> L(y)"
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apply (insert Transset_Lset)
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apply (simp add: Transset_def L_def, blast)
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done
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lemma nonempty: "L(0)"
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apply (simp add: L_def)
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apply (blast intro: zero_in_Lset)
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done
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theorem upair_ax: "upair_ax(L)"
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apply (simp add: upair_ax_def upair_def, clarify)
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apply (rule_tac x="{x,y}" in rexI)
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apply (simp_all add: doubleton_in_L)
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done
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theorem Union_ax: "Union_ax(L)"
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apply (simp add: Union_ax_def big_union_def, clarify)
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apply (rule_tac x="\<Union>(x)" in rexI)
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apply (simp_all add: Union_in_L, auto)
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apply (blast intro: transL)
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done
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theorem power_ax: "power_ax(L)"
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apply (simp add: power_ax_def powerset_def Relative.subset_def, clarify)
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apply (rule_tac x="{y \<in> Pow(x). L(y)}" in rexI)
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apply (simp_all add: LPow_in_L, auto)
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apply (blast intro: transL)
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done
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text{*We don't actually need @{term L} to satisfy the foundation axiom.*}
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theorem foundation_ax: "foundation_ax(L)"
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apply (simp add: foundation_ax_def)
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apply (rule rallI) 
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apply (cut_tac A=x in foundation)
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apply (blast intro: transL)
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done
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subsection{*For L to satisfy Replacement *}
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(*Can't move these to Formula unless the definition of univalent is moved
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there too!*)
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lemma LReplace_in_Lset:
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     "[|X \<in> Lset(i); univalent(L,X,Q); Ord(i)|]
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      ==> \<exists>j. Ord(j) & Replace(X, %x y. Q(x,y) & L(y)) \<subseteq> Lset(j)"
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apply (rule_tac x="\<Union>y \<in> Replace(X, %x y. Q(x,y) & L(y)). succ(lrank(y))"
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       in exI)
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apply simp
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apply clarify
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apply (rule_tac a=x in UN_I)
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 apply (simp_all add: Replace_iff univalent_def)
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apply (blast dest: transL L_I)
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done
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lemma LReplace_in_L:
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     "[|L(X); univalent(L,X,Q)|]
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      ==> \<exists>Y. L(Y) & Replace(X, %x y. Q(x,y) & L(y)) \<subseteq> Y"
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apply (drule L_D, clarify)
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apply (drule LReplace_in_Lset, assumption+)
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apply (blast intro: L_I Lset_in_Lset_succ)
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done
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theorem replacement: "replacement(L,P)"
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apply (simp add: replacement_def, clarify)
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apply (frule LReplace_in_L, assumption+, clarify)
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apply (rule_tac x=Y in rexI)
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apply (simp_all add: Replace_iff univalent_def, blast)
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done
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subsection{*Instantiating the locale @{text M_trivial}*}
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text{*No instances of Separation yet.*}
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lemma Lset_mono_le: "mono_le_subset(Lset)"
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by (simp add: mono_le_subset_def le_imp_subset Lset_mono)
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lemma Lset_cont: "cont_Ord(Lset)"
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by (simp add: cont_Ord_def Limit_Lset_eq OUnion_def Limit_is_Ord)
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lemmas L_nat = Ord_in_L [OF Ord_nat]
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theorem M_trivial_L: "PROP M_trivial(L)"
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  apply (rule M_trivial.intro)
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       apply (erule (1) transL)
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      apply (rule upair_ax)
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     apply (rule Union_ax)
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    apply (rule power_ax)
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   apply (rule replacement)
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  apply (rule L_nat)
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  done
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interpretation L?: M_trivial L by (rule M_trivial_L)
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(* Replaces the following declarations...
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lemmas rall_abs = M_trivial.rall_abs [OF M_trivial_L]
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  and rex_abs = M_trivial.rex_abs [OF M_trivial_L]
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...
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declare rall_abs [simp]
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declare rex_abs [simp]
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...and dozens of similar ones.
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*)
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subsection{*Instantiation of the locale @{text reflection}*}
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text{*instances of locale constants*}
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definition
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  L_F0 :: "[i=>o,i] => i" where
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    "L_F0(P,y) == \<mu> b. (\<exists>z. L(z) \<and> P(<y,z>)) \<longrightarrow> (\<exists>z\<in>Lset(b). P(<y,z>))"
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definition
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  L_FF :: "[i=>o,i] => i" where
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    "L_FF(P)   == \<lambda>a. \<Union>y\<in>Lset(a). L_F0(P,y)"
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definition
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  L_ClEx :: "[i=>o,i] => o" where
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    "L_ClEx(P) == \<lambda>a. Limit(a) \<and> normalize(L_FF(P),a) = a"
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text{*We must use the meta-existential quantifier; otherwise the reflection
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      terms become enormous!*}
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definition
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  L_Reflects :: "[i=>o,[i,i]=>o] => prop"  ("(3REFLECTS/ [_,/ _])") where
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    "REFLECTS[P,Q] == (??Cl. Closed_Unbounded(Cl) &
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                           (\<forall>a. Cl(a) \<longrightarrow> (\<forall>x \<in> Lset(a). P(x) \<longleftrightarrow> Q(a,x))))"
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theorem Triv_reflection:
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     "REFLECTS[P, \<lambda>a x. P(x)]"
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apply (simp add: L_Reflects_def)
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apply (rule meta_exI)
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apply (rule Closed_Unbounded_Ord)
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done
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theorem Not_reflection:
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     "REFLECTS[P,Q] ==> REFLECTS[\<lambda>x. ~P(x), \<lambda>a x. ~Q(a,x)]"
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apply (unfold L_Reflects_def)
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apply (erule meta_exE)
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apply (rule_tac x=Cl in meta_exI, simp)
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done
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theorem And_reflection:
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     "[| REFLECTS[P,Q]; REFLECTS[P',Q'] |]
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      ==> REFLECTS[\<lambda>x. P(x) \<and> P'(x), \<lambda>a x. Q(a,x) \<and> Q'(a,x)]"
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apply (unfold L_Reflects_def)
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apply (elim meta_exE)
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apply (rule_tac x="\<lambda>a. Cl(a) \<and> Cla(a)" in meta_exI)
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apply (simp add: Closed_Unbounded_Int, blast)
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done
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theorem Or_reflection:
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     "[| REFLECTS[P,Q]; REFLECTS[P',Q'] |]
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      ==> REFLECTS[\<lambda>x. P(x) \<or> P'(x), \<lambda>a x. Q(a,x) \<or> Q'(a,x)]"
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apply (unfold L_Reflects_def)
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apply (elim meta_exE)
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apply (rule_tac x="\<lambda>a. Cl(a) \<and> Cla(a)" in meta_exI)
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apply (simp add: Closed_Unbounded_Int, blast)
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done
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theorem Imp_reflection:
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     "[| REFLECTS[P,Q]; REFLECTS[P',Q'] |]
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      ==> REFLECTS[\<lambda>x. P(x) \<longrightarrow> P'(x), \<lambda>a x. Q(a,x) \<longrightarrow> Q'(a,x)]"
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apply (unfold L_Reflects_def)
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apply (elim meta_exE)
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apply (rule_tac x="\<lambda>a. Cl(a) \<and> Cla(a)" in meta_exI)
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apply (simp add: Closed_Unbounded_Int, blast)
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done
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theorem Iff_reflection:
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     "[| REFLECTS[P,Q]; REFLECTS[P',Q'] |]
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      ==> REFLECTS[\<lambda>x. P(x) \<longleftrightarrow> P'(x), \<lambda>a x. Q(a,x) \<longleftrightarrow> Q'(a,x)]"
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apply (unfold L_Reflects_def)
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apply (elim meta_exE)
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apply (rule_tac x="\<lambda>a. Cl(a) \<and> Cla(a)" in meta_exI)
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apply (simp add: Closed_Unbounded_Int, blast)
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done
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lemma reflection_Lset: "reflection(Lset)"
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by (blast intro: reflection.intro Lset_mono_le Lset_cont 
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                 Formula.Pair_in_LLimit)+
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theorem Ex_reflection:
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     "REFLECTS[\<lambda>x. P(fst(x),snd(x)), \<lambda>a x. Q(a,fst(x),snd(x))]
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      ==> REFLECTS[\<lambda>x. \<exists>z. L(z) \<and> P(x,z), \<lambda>a x. \<exists>z\<in>Lset(a). Q(a,x,z)]"
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apply (unfold L_Reflects_def L_ClEx_def L_FF_def L_F0_def L_def)
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apply (elim meta_exE)
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apply (rule meta_exI)
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apply (erule reflection.Ex_reflection [OF reflection_Lset])
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done
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theorem All_reflection:
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     "REFLECTS[\<lambda>x. P(fst(x),snd(x)), \<lambda>a x. Q(a,fst(x),snd(x))]
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      ==> REFLECTS[\<lambda>x. \<forall>z. L(z) \<longrightarrow> P(x,z), \<lambda>a x. \<forall>z\<in>Lset(a). Q(a,x,z)]"
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apply (unfold L_Reflects_def L_ClEx_def L_FF_def L_F0_def L_def)
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apply (elim meta_exE)
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apply (rule meta_exI)
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apply (erule reflection.All_reflection [OF reflection_Lset])
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done
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theorem Rex_reflection:
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     "REFLECTS[ \<lambda>x. P(fst(x),snd(x)), \<lambda>a x. Q(a,fst(x),snd(x))]
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      ==> REFLECTS[\<lambda>x. \<exists>z[L]. P(x,z), \<lambda>a x. \<exists>z\<in>Lset(a). Q(a,x,z)]"
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apply (unfold rex_def)
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apply (intro And_reflection Ex_reflection, assumption)
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done
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theorem Rall_reflection:
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     "REFLECTS[\<lambda>x. P(fst(x),snd(x)), \<lambda>a x. Q(a,fst(x),snd(x))]
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      ==> REFLECTS[\<lambda>x. \<forall>z[L]. P(x,z), \<lambda>a x. \<forall>z\<in>Lset(a). Q(a,x,z)]"
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apply (unfold rall_def)
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apply (intro Imp_reflection All_reflection, assumption)
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done
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text{*This version handles an alternative form of the bounded quantifier
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      in the second argument of @{text REFLECTS}.*}
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theorem Rex_reflection':
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     "REFLECTS[\<lambda>x. P(fst(x),snd(x)), \<lambda>a x. Q(a,fst(x),snd(x))]
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      ==> REFLECTS[\<lambda>x. \<exists>z[L]. P(x,z), \<lambda>a x. \<exists>z[##Lset(a)]. Q(a,x,z)]"
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apply (unfold setclass_def rex_def)
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apply (erule Rex_reflection [unfolded rex_def Bex_def]) 
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done
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text{*As above.*}
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theorem Rall_reflection':
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     "REFLECTS[\<lambda>x. P(fst(x),snd(x)), \<lambda>a x. Q(a,fst(x),snd(x))]
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      ==> REFLECTS[\<lambda>x. \<forall>z[L]. P(x,z), \<lambda>a x. \<forall>z[##Lset(a)]. Q(a,x,z)]"
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apply (unfold setclass_def rall_def)
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apply (erule Rall_reflection [unfolded rall_def Ball_def]) 
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done
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lemmas FOL_reflections =
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        Triv_reflection Not_reflection And_reflection Or_reflection
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        Imp_reflection Iff_reflection Ex_reflection All_reflection
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        Rex_reflection Rall_reflection Rex_reflection' Rall_reflection'
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lemma ReflectsD:
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     "[|REFLECTS[P,Q]; Ord(i)|]
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      ==> \<exists>j. i<j & (\<forall>x \<in> Lset(j). P(x) \<longleftrightarrow> Q(j,x))"
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apply (unfold L_Reflects_def Closed_Unbounded_def)
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apply (elim meta_exE, clarify)
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apply (blast dest!: UnboundedD)
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done
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lemma ReflectsE:
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     "[| REFLECTS[P,Q]; Ord(i);
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         !!j. [|i<j;  \<forall>x \<in> Lset(j). P(x) \<longleftrightarrow> Q(j,x)|] ==> R |]
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      ==> R"
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by (drule ReflectsD, assumption, blast)
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lemma Collect_mem_eq: "{x\<in>A. x\<in>B} = A \<inter> B"
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by blast
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subsection{*Internalized Formulas for some Set-Theoretic Concepts*}
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subsubsection{*Some numbers to help write de Bruijn indices*}
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abbreviation
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  digit3 :: i   ("3") where "3 == succ(2)"
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abbreviation
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  digit4 :: i   ("4") where "4 == succ(3)"
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abbreviation
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  digit5 :: i   ("5") where "5 == succ(4)"
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abbreviation
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  digit6 :: i   ("6") where "6 == succ(5)"
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abbreviation
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  digit7 :: i   ("7") where "7 == succ(6)"
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abbreviation
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  digit8 :: i   ("8") where "8 == succ(7)"
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abbreviation
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  digit9 :: i   ("9") where "9 == succ(8)"
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subsubsection{*The Empty Set, Internalized*}
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definition
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  empty_fm :: "i=>i" where
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    "empty_fm(x) == Forall(Neg(Member(0,succ(x))))"
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lemma empty_type [TC]:
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     "x \<in> nat ==> empty_fm(x) \<in> formula"
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by (simp add: empty_fm_def)
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lemma sats_empty_fm [simp]:
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   "[| x \<in> nat; env \<in> list(A)|]
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    ==> sats(A, empty_fm(x), env) \<longleftrightarrow> empty(##A, nth(x,env))"
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by (simp add: empty_fm_def empty_def)
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lemma empty_iff_sats:
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      "[| nth(i,env) = x; nth(j,env) = y;
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          i \<in> nat; env \<in> list(A)|]
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       ==> empty(##A, x) \<longleftrightarrow> sats(A, empty_fm(i), env)"
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by simp
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theorem empty_reflection:
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     "REFLECTS[\<lambda>x. empty(L,f(x)),
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               \<lambda>i x. empty(##Lset(i),f(x))]"
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apply (simp only: empty_def)
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apply (intro FOL_reflections)
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done
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text{*Not used.  But maybe useful?*}
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lemma Transset_sats_empty_fm_eq_0:
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   "[| n \<in> nat; env \<in> list(A); Transset(A)|]
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    ==> sats(A, empty_fm(n), env) \<longleftrightarrow> nth(n,env) = 0"
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apply (simp add: empty_fm_def empty_def Transset_def, auto)
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apply (case_tac "n < length(env)")
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apply (frule nth_type, assumption+, blast)
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apply (simp_all add: not_lt_iff_le nth_eq_0)
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done
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0f89104dd377 Fixed quantified variable name preservation for ball and bex (bounded quants)
paulson
parents: 13323
diff changeset
   331
subsubsection{*Unordered Pairs, Internalized*}
13298
b4f370679c65 Constructible: some separation axioms
paulson
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   332
21404
eb85850d3eb7 more robust syntax for definition/abbreviation/notation;
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   333
definition
eb85850d3eb7 more robust syntax for definition/abbreviation/notation;
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parents: 21233
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   334
  upair_fm :: "[i,i,i]=>i" where
13429
wenzelm
parents: 13428
diff changeset
   335
    "upair_fm(x,y,z) ==
wenzelm
parents: 13428
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   336
       And(Member(x,z),
13298
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
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   337
           And(Member(y,z),
13429
wenzelm
parents: 13428
diff changeset
   338
               Forall(Implies(Member(0,succ(z)),
13298
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   339
                              Or(Equal(0,succ(x)), Equal(0,succ(y)))))))"
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   340
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   341
lemma upair_type [TC]:
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   342
     "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> upair_fm(x,y,z) \<in> formula"
13429
wenzelm
parents: 13428
diff changeset
   343
by (simp add: upair_fm_def)
13298
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   344
b4f370679c65 Constructible: some separation axioms
paulson
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diff changeset
   345
lemma sats_upair_fm [simp]:
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   346
   "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
46823
57bf0cecb366 More mathematical symbols for ZF examples
paulson
parents: 32960
diff changeset
   347
    ==> sats(A, upair_fm(x,y,z), env) \<longleftrightarrow>
13807
a28a8fbc76d4 changed ** to ## to avoid conflict with new comment syntax
paulson
parents: 13655
diff changeset
   348
            upair(##A, nth(x,env), nth(y,env), nth(z,env))"
13298
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   349
by (simp add: upair_fm_def upair_def)
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   350
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   351
lemma upair_iff_sats:
13429
wenzelm
parents: 13428
diff changeset
   352
      "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z;
13298
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   353
          i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
46823
57bf0cecb366 More mathematical symbols for ZF examples
paulson
parents: 32960
diff changeset
   354
       ==> upair(##A, x, y, z) \<longleftrightarrow> sats(A, upair_fm(i,j,k), env)"
13298
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   355
by (simp add: sats_upair_fm)
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   356
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   357
text{*Useful? At least it refers to "real" unordered pairs*}
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   358
lemma sats_upair_fm2 [simp]:
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   359
   "[| x \<in> nat; y \<in> nat; z < length(env); env \<in> list(A); Transset(A)|]
46823
57bf0cecb366 More mathematical symbols for ZF examples
paulson
parents: 32960
diff changeset
   360
    ==> sats(A, upair_fm(x,y,z), env) \<longleftrightarrow>
13298
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   361
        nth(z,env) = {nth(x,env), nth(y,env)}"
13429
wenzelm
parents: 13428
diff changeset
   362
apply (frule lt_length_in_nat, assumption)
wenzelm
parents: 13428
diff changeset
   363
apply (simp add: upair_fm_def Transset_def, auto)
wenzelm
parents: 13428
diff changeset
   364
apply (blast intro: nth_type)
13298
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   365
done
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   366
13314
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   367
theorem upair_reflection:
13429
wenzelm
parents: 13428
diff changeset
   368
     "REFLECTS[\<lambda>x. upair(L,f(x),g(x),h(x)),
13807
a28a8fbc76d4 changed ** to ## to avoid conflict with new comment syntax
paulson
parents: 13655
diff changeset
   369
               \<lambda>i x. upair(##Lset(i),f(x),g(x),h(x))]"
13314
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   370
apply (simp add: upair_def)
13429
wenzelm
parents: 13428
diff changeset
   371
apply (intro FOL_reflections)
13314
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   372
done
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   373
13339
0f89104dd377 Fixed quantified variable name preservation for ball and bex (bounded quants)
paulson
parents: 13323
diff changeset
   374
subsubsection{*Ordered pairs, Internalized*}
13298
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   375
21404
eb85850d3eb7 more robust syntax for definition/abbreviation/notation;
wenzelm
parents: 21233
diff changeset
   376
definition
eb85850d3eb7 more robust syntax for definition/abbreviation/notation;
wenzelm
parents: 21233
diff changeset
   377
  pair_fm :: "[i,i,i]=>i" where
13429
wenzelm
parents: 13428
diff changeset
   378
    "pair_fm(x,y,z) ==
13298
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   379
       Exists(And(upair_fm(succ(x),succ(x),0),
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   380
              Exists(And(upair_fm(succ(succ(x)),succ(succ(y)),0),
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   381
                         upair_fm(1,0,succ(succ(z)))))))"
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   382
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   383
lemma pair_type [TC]:
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   384
     "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> pair_fm(x,y,z) \<in> formula"
13429
wenzelm
parents: 13428
diff changeset
   385
by (simp add: pair_fm_def)
13298
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   386
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   387
lemma sats_pair_fm [simp]:
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   388
   "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
46823
57bf0cecb366 More mathematical symbols for ZF examples
paulson
parents: 32960
diff changeset
   389
    ==> sats(A, pair_fm(x,y,z), env) \<longleftrightarrow>
13807
a28a8fbc76d4 changed ** to ## to avoid conflict with new comment syntax
paulson
parents: 13655
diff changeset
   390
        pair(##A, nth(x,env), nth(y,env), nth(z,env))"
13298
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   391
by (simp add: pair_fm_def pair_def)
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   392
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   393
lemma pair_iff_sats:
13429
wenzelm
parents: 13428
diff changeset
   394
      "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z;
13298
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   395
          i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
46823
57bf0cecb366 More mathematical symbols for ZF examples
paulson
parents: 32960
diff changeset
   396
       ==> pair(##A, x, y, z) \<longleftrightarrow> sats(A, pair_fm(i,j,k), env)"
13298
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   397
by (simp add: sats_pair_fm)
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   398
13314
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   399
theorem pair_reflection:
13429
wenzelm
parents: 13428
diff changeset
   400
     "REFLECTS[\<lambda>x. pair(L,f(x),g(x),h(x)),
13807
a28a8fbc76d4 changed ** to ## to avoid conflict with new comment syntax
paulson
parents: 13655
diff changeset
   401
               \<lambda>i x. pair(##Lset(i),f(x),g(x),h(x))]"
13655
95b95cdb4704 Tidying up. New primitives is_iterates and is_iterates_fm.
paulson
parents: 13651
diff changeset
   402
apply (simp only: pair_def)
13429
wenzelm
parents: 13428
diff changeset
   403
apply (intro FOL_reflections upair_reflection)
13314
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   404
done
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   405
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   406
13339
0f89104dd377 Fixed quantified variable name preservation for ball and bex (bounded quants)
paulson
parents: 13323
diff changeset
   407
subsubsection{*Binary Unions, Internalized*}
13298
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   408
21404
eb85850d3eb7 more robust syntax for definition/abbreviation/notation;
wenzelm
parents: 21233
diff changeset
   409
definition
eb85850d3eb7 more robust syntax for definition/abbreviation/notation;
wenzelm
parents: 21233
diff changeset
   410
  union_fm :: "[i,i,i]=>i" where
13429
wenzelm
parents: 13428
diff changeset
   411
    "union_fm(x,y,z) ==
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   412
       Forall(Iff(Member(0,succ(z)),
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   413
                  Or(Member(0,succ(x)),Member(0,succ(y)))))"
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   414
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   415
lemma union_type [TC]:
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   416
     "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> union_fm(x,y,z) \<in> formula"
13429
wenzelm
parents: 13428
diff changeset
   417
by (simp add: union_fm_def)
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   418
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   419
lemma sats_union_fm [simp]:
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   420
   "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
46823
57bf0cecb366 More mathematical symbols for ZF examples
paulson
parents: 32960
diff changeset
   421
    ==> sats(A, union_fm(x,y,z), env) \<longleftrightarrow>
13807
a28a8fbc76d4 changed ** to ## to avoid conflict with new comment syntax
paulson
parents: 13655
diff changeset
   422
        union(##A, nth(x,env), nth(y,env), nth(z,env))"
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   423
by (simp add: union_fm_def union_def)
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   424
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   425
lemma union_iff_sats:
13429
wenzelm
parents: 13428
diff changeset
   426
      "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z;
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   427
          i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
46823
57bf0cecb366 More mathematical symbols for ZF examples
paulson
parents: 32960
diff changeset
   428
       ==> union(##A, x, y, z) \<longleftrightarrow> sats(A, union_fm(i,j,k), env)"
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   429
by (simp add: sats_union_fm)
13298
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   430
13314
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   431
theorem union_reflection:
13429
wenzelm
parents: 13428
diff changeset
   432
     "REFLECTS[\<lambda>x. union(L,f(x),g(x),h(x)),
13807
a28a8fbc76d4 changed ** to ## to avoid conflict with new comment syntax
paulson
parents: 13655
diff changeset
   433
               \<lambda>i x. union(##Lset(i),f(x),g(x),h(x))]"
13655
95b95cdb4704 Tidying up. New primitives is_iterates and is_iterates_fm.
paulson
parents: 13651
diff changeset
   434
apply (simp only: union_def)
13429
wenzelm
parents: 13428
diff changeset
   435
apply (intro FOL_reflections)
13314
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   436
done
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   437
13298
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   438
13339
0f89104dd377 Fixed quantified variable name preservation for ball and bex (bounded quants)
paulson
parents: 13323
diff changeset
   439
subsubsection{*Set ``Cons,'' Internalized*}
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   440
21404
eb85850d3eb7 more robust syntax for definition/abbreviation/notation;
wenzelm
parents: 21233
diff changeset
   441
definition
eb85850d3eb7 more robust syntax for definition/abbreviation/notation;
wenzelm
parents: 21233
diff changeset
   442
  cons_fm :: "[i,i,i]=>i" where
13429
wenzelm
parents: 13428
diff changeset
   443
    "cons_fm(x,y,z) ==
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   444
       Exists(And(upair_fm(succ(x),succ(x),0),
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   445
                  union_fm(0,succ(y),succ(z))))"
13298
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   446
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   447
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   448
lemma cons_type [TC]:
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   449
     "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> cons_fm(x,y,z) \<in> formula"
13429
wenzelm
parents: 13428
diff changeset
   450
by (simp add: cons_fm_def)
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   451
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   452
lemma sats_cons_fm [simp]:
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   453
   "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
46823
57bf0cecb366 More mathematical symbols for ZF examples
paulson
parents: 32960
diff changeset
   454
    ==> sats(A, cons_fm(x,y,z), env) \<longleftrightarrow>
13807
a28a8fbc76d4 changed ** to ## to avoid conflict with new comment syntax
paulson
parents: 13655
diff changeset
   455
        is_cons(##A, nth(x,env), nth(y,env), nth(z,env))"
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   456
by (simp add: cons_fm_def is_cons_def)
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   457
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   458
lemma cons_iff_sats:
13429
wenzelm
parents: 13428
diff changeset
   459
      "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z;
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   460
          i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
46823
57bf0cecb366 More mathematical symbols for ZF examples
paulson
parents: 32960
diff changeset
   461
       ==> is_cons(##A, x, y, z) \<longleftrightarrow> sats(A, cons_fm(i,j,k), env)"
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   462
by simp
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   463
13314
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   464
theorem cons_reflection:
13429
wenzelm
parents: 13428
diff changeset
   465
     "REFLECTS[\<lambda>x. is_cons(L,f(x),g(x),h(x)),
13807
a28a8fbc76d4 changed ** to ## to avoid conflict with new comment syntax
paulson
parents: 13655
diff changeset
   466
               \<lambda>i x. is_cons(##Lset(i),f(x),g(x),h(x))]"
13655
95b95cdb4704 Tidying up. New primitives is_iterates and is_iterates_fm.
paulson
parents: 13651
diff changeset
   467
apply (simp only: is_cons_def)
13429
wenzelm
parents: 13428
diff changeset
   468
apply (intro FOL_reflections upair_reflection union_reflection)
13323
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   469
done
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   470
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   471
13339
0f89104dd377 Fixed quantified variable name preservation for ball and bex (bounded quants)
paulson
parents: 13323
diff changeset
   472
subsubsection{*Successor Function, Internalized*}
13323
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   473
21404
eb85850d3eb7 more robust syntax for definition/abbreviation/notation;
wenzelm
parents: 21233
diff changeset
   474
definition
eb85850d3eb7 more robust syntax for definition/abbreviation/notation;
wenzelm
parents: 21233
diff changeset
   475
  succ_fm :: "[i,i]=>i" where
13323
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   476
    "succ_fm(x,y) == cons_fm(x,x,y)"
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   477
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   478
lemma succ_type [TC]:
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   479
     "[| x \<in> nat; y \<in> nat |] ==> succ_fm(x,y) \<in> formula"
13429
wenzelm
parents: 13428
diff changeset
   480
by (simp add: succ_fm_def)
13323
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   481
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   482
lemma sats_succ_fm [simp]:
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   483
   "[| x \<in> nat; y \<in> nat; env \<in> list(A)|]
46823
57bf0cecb366 More mathematical symbols for ZF examples
paulson
parents: 32960
diff changeset
   484
    ==> sats(A, succ_fm(x,y), env) \<longleftrightarrow>
13807
a28a8fbc76d4 changed ** to ## to avoid conflict with new comment syntax
paulson
parents: 13655
diff changeset
   485
        successor(##A, nth(x,env), nth(y,env))"
13323
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   486
by (simp add: succ_fm_def successor_def)
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   487
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   488
lemma successor_iff_sats:
13429
wenzelm
parents: 13428
diff changeset
   489
      "[| nth(i,env) = x; nth(j,env) = y;
13323
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   490
          i \<in> nat; j \<in> nat; env \<in> list(A)|]
46823
57bf0cecb366 More mathematical symbols for ZF examples
paulson
parents: 32960
diff changeset
   491
       ==> successor(##A, x, y) \<longleftrightarrow> sats(A, succ_fm(i,j), env)"
13323
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   492
by simp
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   493
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   494
theorem successor_reflection:
13429
wenzelm
parents: 13428
diff changeset
   495
     "REFLECTS[\<lambda>x. successor(L,f(x),g(x)),
13807
a28a8fbc76d4 changed ** to ## to avoid conflict with new comment syntax
paulson
parents: 13655
diff changeset
   496
               \<lambda>i x. successor(##Lset(i),f(x),g(x))]"
13655
95b95cdb4704 Tidying up. New primitives is_iterates and is_iterates_fm.
paulson
parents: 13651
diff changeset
   497
apply (simp only: successor_def)
13429
wenzelm
parents: 13428
diff changeset
   498
apply (intro cons_reflection)
13314
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   499
done
13298
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   500
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   501
13363
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   502
subsubsection{*The Number 1, Internalized*}
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   503
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   504
(* "number1(M,a) == (\<exists>x[M]. empty(M,x) & successor(M,x,a))" *)
21404
eb85850d3eb7 more robust syntax for definition/abbreviation/notation;
wenzelm
parents: 21233
diff changeset
   505
definition
eb85850d3eb7 more robust syntax for definition/abbreviation/notation;
wenzelm
parents: 21233
diff changeset
   506
  number1_fm :: "i=>i" where
13363
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   507
    "number1_fm(a) == Exists(And(empty_fm(0), succ_fm(0,succ(a))))"
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   508
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   509
lemma number1_type [TC]:
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   510
     "x \<in> nat ==> number1_fm(x) \<in> formula"
13429
wenzelm
parents: 13428
diff changeset
   511
by (simp add: number1_fm_def)
13363
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   512
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   513
lemma sats_number1_fm [simp]:
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   514
   "[| x \<in> nat; env \<in> list(A)|]
46823
57bf0cecb366 More mathematical symbols for ZF examples
paulson
parents: 32960
diff changeset
   515
    ==> sats(A, number1_fm(x), env) \<longleftrightarrow> number1(##A, nth(x,env))"
13363
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   516
by (simp add: number1_fm_def number1_def)
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   517
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   518
lemma number1_iff_sats:
13429
wenzelm
parents: 13428
diff changeset
   519
      "[| nth(i,env) = x; nth(j,env) = y;
13363
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   520
          i \<in> nat; env \<in> list(A)|]
46823
57bf0cecb366 More mathematical symbols for ZF examples
paulson
parents: 32960
diff changeset
   521
       ==> number1(##A, x) \<longleftrightarrow> sats(A, number1_fm(i), env)"
13363
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   522
by simp
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   523
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   524
theorem number1_reflection:
13429
wenzelm
parents: 13428
diff changeset
   525
     "REFLECTS[\<lambda>x. number1(L,f(x)),
13807
a28a8fbc76d4 changed ** to ## to avoid conflict with new comment syntax
paulson
parents: 13655
diff changeset
   526
               \<lambda>i x. number1(##Lset(i),f(x))]"
13655
95b95cdb4704 Tidying up. New primitives is_iterates and is_iterates_fm.
paulson
parents: 13651
diff changeset
   527
apply (simp only: number1_def)
13363
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   528
apply (intro FOL_reflections empty_reflection successor_reflection)
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   529
done
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   530
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   531
13352
3cd767f8d78b new definitions of fun_apply and M_is_recfun
paulson
parents: 13348
diff changeset
   532
subsubsection{*Big Union, Internalized*}
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   533
46823
57bf0cecb366 More mathematical symbols for ZF examples
paulson
parents: 32960
diff changeset
   534
(*  "big_union(M,A,z) == \<forall>x[M]. x \<in> z \<longleftrightarrow> (\<exists>y[M]. y\<in>A & x \<in> y)" *)
21404
eb85850d3eb7 more robust syntax for definition/abbreviation/notation;
wenzelm
parents: 21233
diff changeset
   535
definition
eb85850d3eb7 more robust syntax for definition/abbreviation/notation;
wenzelm
parents: 21233
diff changeset
   536
  big_union_fm :: "[i,i]=>i" where
13429
wenzelm
parents: 13428
diff changeset
   537
    "big_union_fm(A,z) ==
13352
3cd767f8d78b new definitions of fun_apply and M_is_recfun
paulson
parents: 13348
diff changeset
   538
       Forall(Iff(Member(0,succ(z)),
3cd767f8d78b new definitions of fun_apply and M_is_recfun
paulson
parents: 13348
diff changeset
   539
                  Exists(And(Member(0,succ(succ(A))), Member(1,0)))))"
13298
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   540
13352
3cd767f8d78b new definitions of fun_apply and M_is_recfun
paulson
parents: 13348
diff changeset
   541
lemma big_union_type [TC]:
3cd767f8d78b new definitions of fun_apply and M_is_recfun
paulson
parents: 13348
diff changeset
   542
     "[| x \<in> nat; y \<in> nat |] ==> big_union_fm(x,y) \<in> formula"
13429
wenzelm
parents: 13428
diff changeset
   543
by (simp add: big_union_fm_def)
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   544
13352
3cd767f8d78b new definitions of fun_apply and M_is_recfun
paulson
parents: 13348
diff changeset
   545
lemma sats_big_union_fm [simp]:
3cd767f8d78b new definitions of fun_apply and M_is_recfun
paulson
parents: 13348
diff changeset
   546
   "[| x \<in> nat; y \<in> nat; env \<in> list(A)|]
46823
57bf0cecb366 More mathematical symbols for ZF examples
paulson
parents: 32960
diff changeset
   547
    ==> sats(A, big_union_fm(x,y), env) \<longleftrightarrow>
13807
a28a8fbc76d4 changed ** to ## to avoid conflict with new comment syntax
paulson
parents: 13655
diff changeset
   548
        big_union(##A, nth(x,env), nth(y,env))"
13352
3cd767f8d78b new definitions of fun_apply and M_is_recfun
paulson
parents: 13348
diff changeset
   549
by (simp add: big_union_fm_def big_union_def)
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   550
13352
3cd767f8d78b new definitions of fun_apply and M_is_recfun
paulson
parents: 13348
diff changeset
   551
lemma big_union_iff_sats:
13429
wenzelm
parents: 13428
diff changeset
   552
      "[| nth(i,env) = x; nth(j,env) = y;
13352
3cd767f8d78b new definitions of fun_apply and M_is_recfun
paulson
parents: 13348
diff changeset
   553
          i \<in> nat; j \<in> nat; env \<in> list(A)|]
46823
57bf0cecb366 More mathematical symbols for ZF examples
paulson
parents: 32960
diff changeset
   554
       ==> big_union(##A, x, y) \<longleftrightarrow> sats(A, big_union_fm(i,j), env)"
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   555
by simp
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   556
13352
3cd767f8d78b new definitions of fun_apply and M_is_recfun
paulson
parents: 13348
diff changeset
   557
theorem big_union_reflection:
13429
wenzelm
parents: 13428
diff changeset
   558
     "REFLECTS[\<lambda>x. big_union(L,f(x),g(x)),
13807
a28a8fbc76d4 changed ** to ## to avoid conflict with new comment syntax
paulson
parents: 13655
diff changeset
   559
               \<lambda>i x. big_union(##Lset(i),f(x),g(x))]"
13655
95b95cdb4704 Tidying up. New primitives is_iterates and is_iterates_fm.
paulson
parents: 13651
diff changeset
   560
apply (simp only: big_union_def)
13429
wenzelm
parents: 13428
diff changeset
   561
apply (intro FOL_reflections)
13314
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   562
done
13298
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   563
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   564
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   565
subsubsection{*Variants of Satisfaction Definitions for Ordinals, etc.*}
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   566
13651
ac80e101306a Cosmetic changes suggested by writing the paper. Deleted some
paulson
parents: 13634
diff changeset
   567
text{*The @{text sats} theorems below are standard versions of the ones proved
ac80e101306a Cosmetic changes suggested by writing the paper. Deleted some
paulson
parents: 13634
diff changeset
   568
in theory @{text Formula}.  They relate elements of type @{term formula} to
ac80e101306a Cosmetic changes suggested by writing the paper. Deleted some
paulson
parents: 13634
diff changeset
   569
relativized concepts such as @{term subset} or @{term ordinal} rather than to
ac80e101306a Cosmetic changes suggested by writing the paper. Deleted some
paulson
parents: 13634
diff changeset
   570
real concepts such as @{term Ord}.  Now that we have instantiated the locale
ac80e101306a Cosmetic changes suggested by writing the paper. Deleted some
paulson
parents: 13634
diff changeset
   571
@{text M_trivial}, we no longer require the earlier versions.*}
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   572
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   573
lemma sats_subset_fm':
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   574
   "[|x \<in> nat; y \<in> nat; env \<in> list(A)|]
46823
57bf0cecb366 More mathematical symbols for ZF examples
paulson
parents: 32960
diff changeset
   575
    ==> sats(A, subset_fm(x,y), env) \<longleftrightarrow> subset(##A, nth(x,env), nth(y,env))"
13429
wenzelm
parents: 13428
diff changeset
   576
by (simp add: subset_fm_def Relative.subset_def)
13298
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   577
13314
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   578
theorem subset_reflection:
13429
wenzelm
parents: 13428
diff changeset
   579
     "REFLECTS[\<lambda>x. subset(L,f(x),g(x)),
13807
a28a8fbc76d4 changed ** to ## to avoid conflict with new comment syntax
paulson
parents: 13655
diff changeset
   580
               \<lambda>i x. subset(##Lset(i),f(x),g(x))]"
13655
95b95cdb4704 Tidying up. New primitives is_iterates and is_iterates_fm.
paulson
parents: 13651
diff changeset
   581
apply (simp only: Relative.subset_def)
13429
wenzelm
parents: 13428
diff changeset
   582
apply (intro FOL_reflections)
13314
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   583
done
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   584
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   585
lemma sats_transset_fm':
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   586
   "[|x \<in> nat; env \<in> list(A)|]
46823
57bf0cecb366 More mathematical symbols for ZF examples
paulson
parents: 32960
diff changeset
   587
    ==> sats(A, transset_fm(x), env) \<longleftrightarrow> transitive_set(##A, nth(x,env))"
13429
wenzelm
parents: 13428
diff changeset
   588
by (simp add: sats_subset_fm' transset_fm_def transitive_set_def)
13298
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   589
13314
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   590
theorem transitive_set_reflection:
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   591
     "REFLECTS[\<lambda>x. transitive_set(L,f(x)),
13807
a28a8fbc76d4 changed ** to ## to avoid conflict with new comment syntax
paulson
parents: 13655
diff changeset
   592
               \<lambda>i x. transitive_set(##Lset(i),f(x))]"
13655
95b95cdb4704 Tidying up. New primitives is_iterates and is_iterates_fm.
paulson
parents: 13651
diff changeset
   593
apply (simp only: transitive_set_def)
13429
wenzelm
parents: 13428
diff changeset
   594
apply (intro FOL_reflections subset_reflection)
13314
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   595
done
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   596
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   597
lemma sats_ordinal_fm':
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   598
   "[|x \<in> nat; env \<in> list(A)|]
46823
57bf0cecb366 More mathematical symbols for ZF examples
paulson
parents: 32960
diff changeset
   599
    ==> sats(A, ordinal_fm(x), env) \<longleftrightarrow> ordinal(##A,nth(x,env))"
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   600
by (simp add: sats_transset_fm' ordinal_fm_def ordinal_def)
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   601
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   602
lemma ordinal_iff_sats:
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   603
      "[| nth(i,env) = x;  i \<in> nat; env \<in> list(A)|]
46823
57bf0cecb366 More mathematical symbols for ZF examples
paulson
parents: 32960
diff changeset
   604
       ==> ordinal(##A, x) \<longleftrightarrow> sats(A, ordinal_fm(i), env)"
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   605
by (simp add: sats_ordinal_fm')
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   606
13314
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   607
theorem ordinal_reflection:
13807
a28a8fbc76d4 changed ** to ## to avoid conflict with new comment syntax
paulson
parents: 13655
diff changeset
   608
     "REFLECTS[\<lambda>x. ordinal(L,f(x)), \<lambda>i x. ordinal(##Lset(i),f(x))]"
13655
95b95cdb4704 Tidying up. New primitives is_iterates and is_iterates_fm.
paulson
parents: 13651
diff changeset
   609
apply (simp only: ordinal_def)
13429
wenzelm
parents: 13428
diff changeset
   610
apply (intro FOL_reflections transitive_set_reflection)
13314
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   611
done
13298
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   612
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   613
13339
0f89104dd377 Fixed quantified variable name preservation for ball and bex (bounded quants)
paulson
parents: 13323
diff changeset
   614
subsubsection{*Membership Relation, Internalized*}
13298
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   615
21404
eb85850d3eb7 more robust syntax for definition/abbreviation/notation;
wenzelm
parents: 21233
diff changeset
   616
definition
eb85850d3eb7 more robust syntax for definition/abbreviation/notation;
wenzelm
parents: 21233
diff changeset
   617
  Memrel_fm :: "[i,i]=>i" where
13429
wenzelm
parents: 13428
diff changeset
   618
    "Memrel_fm(A,r) ==
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   619
       Forall(Iff(Member(0,succ(r)),
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   620
                  Exists(And(Member(0,succ(succ(A))),
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   621
                             Exists(And(Member(0,succ(succ(succ(A)))),
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   622
                                        And(Member(1,0),
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   623
                                            pair_fm(1,0,2))))))))"
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   624
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   625
lemma Memrel_type [TC]:
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   626
     "[| x \<in> nat; y \<in> nat |] ==> Memrel_fm(x,y) \<in> formula"
13429
wenzelm
parents: 13428
diff changeset
   627
by (simp add: Memrel_fm_def)
13298
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   628
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   629
lemma sats_Memrel_fm [simp]:
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   630
   "[| x \<in> nat; y \<in> nat; env \<in> list(A)|]
46823
57bf0cecb366 More mathematical symbols for ZF examples
paulson
parents: 32960
diff changeset
   631
    ==> sats(A, Memrel_fm(x,y), env) \<longleftrightarrow>
13807
a28a8fbc76d4 changed ** to ## to avoid conflict with new comment syntax
paulson
parents: 13655
diff changeset
   632
        membership(##A, nth(x,env), nth(y,env))"
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   633
by (simp add: Memrel_fm_def membership_def)
13298
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   634
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   635
lemma Memrel_iff_sats:
13429
wenzelm
parents: 13428
diff changeset
   636
      "[| nth(i,env) = x; nth(j,env) = y;
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   637
          i \<in> nat; j \<in> nat; env \<in> list(A)|]
46823
57bf0cecb366 More mathematical symbols for ZF examples
paulson
parents: 32960
diff changeset
   638
       ==> membership(##A, x, y) \<longleftrightarrow> sats(A, Memrel_fm(i,j), env)"
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   639
by simp
13304
43ef6c6dd906 more separation instances
paulson
parents: 13299
diff changeset
   640
13314
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   641
theorem membership_reflection:
13429
wenzelm
parents: 13428
diff changeset
   642
     "REFLECTS[\<lambda>x. membership(L,f(x),g(x)),
13807
a28a8fbc76d4 changed ** to ## to avoid conflict with new comment syntax
paulson
parents: 13655
diff changeset
   643
               \<lambda>i x. membership(##Lset(i),f(x),g(x))]"
13655
95b95cdb4704 Tidying up. New primitives is_iterates and is_iterates_fm.
paulson
parents: 13651
diff changeset
   644
apply (simp only: membership_def)
13429
wenzelm
parents: 13428
diff changeset
   645
apply (intro FOL_reflections pair_reflection)
13314
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   646
done
13304
43ef6c6dd906 more separation instances
paulson
parents: 13299
diff changeset
   647
13339
0f89104dd377 Fixed quantified variable name preservation for ball and bex (bounded quants)
paulson
parents: 13323
diff changeset
   648
subsubsection{*Predecessor Set, Internalized*}
13304
43ef6c6dd906 more separation instances
paulson
parents: 13299
diff changeset
   649
21404
eb85850d3eb7 more robust syntax for definition/abbreviation/notation;
wenzelm
parents: 21233
diff changeset
   650
definition
eb85850d3eb7 more robust syntax for definition/abbreviation/notation;
wenzelm
parents: 21233
diff changeset
   651
  pred_set_fm :: "[i,i,i,i]=>i" where
13429
wenzelm
parents: 13428
diff changeset
   652
    "pred_set_fm(A,x,r,B) ==
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   653
       Forall(Iff(Member(0,succ(B)),
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   654
                  Exists(And(Member(0,succ(succ(r))),
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   655
                             And(Member(1,succ(succ(A))),
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   656
                                 pair_fm(1,succ(succ(x)),0))))))"
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   657
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   658
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   659
lemma pred_set_type [TC]:
13429
wenzelm
parents: 13428
diff changeset
   660
     "[| A \<in> nat; x \<in> nat; r \<in> nat; B \<in> nat |]
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   661
      ==> pred_set_fm(A,x,r,B) \<in> formula"
13429
wenzelm
parents: 13428
diff changeset
   662
by (simp add: pred_set_fm_def)
13304
43ef6c6dd906 more separation instances
paulson
parents: 13299
diff changeset
   663
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   664
lemma sats_pred_set_fm [simp]:
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   665
   "[| U \<in> nat; x \<in> nat; r \<in> nat; B \<in> nat; env \<in> list(A)|]
46823
57bf0cecb366 More mathematical symbols for ZF examples
paulson
parents: 32960
diff changeset
   666
    ==> sats(A, pred_set_fm(U,x,r,B), env) \<longleftrightarrow>
13807
a28a8fbc76d4 changed ** to ## to avoid conflict with new comment syntax
paulson
parents: 13655
diff changeset
   667
        pred_set(##A, nth(U,env), nth(x,env), nth(r,env), nth(B,env))"
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   668
by (simp add: pred_set_fm_def pred_set_def)
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   669
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   670
lemma pred_set_iff_sats:
13429
wenzelm
parents: 13428
diff changeset
   671
      "[| nth(i,env) = U; nth(j,env) = x; nth(k,env) = r; nth(l,env) = B;
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   672
          i \<in> nat; j \<in> nat; k \<in> nat; l \<in> nat; env \<in> list(A)|]
46823
57bf0cecb366 More mathematical symbols for ZF examples
paulson
parents: 32960
diff changeset
   673
       ==> pred_set(##A,U,x,r,B) \<longleftrightarrow> sats(A, pred_set_fm(i,j,k,l), env)"
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   674
by (simp add: sats_pred_set_fm)
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   675
13314
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   676
theorem pred_set_reflection:
13429
wenzelm
parents: 13428
diff changeset
   677
     "REFLECTS[\<lambda>x. pred_set(L,f(x),g(x),h(x),b(x)),
13807
a28a8fbc76d4 changed ** to ## to avoid conflict with new comment syntax
paulson
parents: 13655
diff changeset
   678
               \<lambda>i x. pred_set(##Lset(i),f(x),g(x),h(x),b(x))]"
13655
95b95cdb4704 Tidying up. New primitives is_iterates and is_iterates_fm.
paulson
parents: 13651
diff changeset
   679
apply (simp only: pred_set_def)
13429
wenzelm
parents: 13428
diff changeset
   680
apply (intro FOL_reflections pair_reflection)
13314
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   681
done
13304
43ef6c6dd906 more separation instances
paulson
parents: 13299
diff changeset
   682
43ef6c6dd906 more separation instances
paulson
parents: 13299
diff changeset
   683
13298
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   684
13339
0f89104dd377 Fixed quantified variable name preservation for ball and bex (bounded quants)
paulson
parents: 13323
diff changeset
   685
subsubsection{*Domain of a Relation, Internalized*}
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   686
13429
wenzelm
parents: 13428
diff changeset
   687
(* "is_domain(M,r,z) ==
46823
57bf0cecb366 More mathematical symbols for ZF examples
paulson
parents: 32960
diff changeset
   688
        \<forall>x[M]. (x \<in> z \<longleftrightarrow> (\<exists>w[M]. w\<in>r & (\<exists>y[M]. pair(M,x,y,w))))" *)
21404
eb85850d3eb7 more robust syntax for definition/abbreviation/notation;
wenzelm
parents: 21233
diff changeset
   689
definition
eb85850d3eb7 more robust syntax for definition/abbreviation/notation;
wenzelm
parents: 21233
diff changeset
   690
  domain_fm :: "[i,i]=>i" where
13429
wenzelm
parents: 13428
diff changeset
   691
    "domain_fm(r,z) ==
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   692
       Forall(Iff(Member(0,succ(z)),
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   693
                  Exists(And(Member(0,succ(succ(r))),
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   694
                             Exists(pair_fm(2,0,1))))))"
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   695
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   696
lemma domain_type [TC]:
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   697
     "[| x \<in> nat; y \<in> nat |] ==> domain_fm(x,y) \<in> formula"
13429
wenzelm
parents: 13428
diff changeset
   698
by (simp add: domain_fm_def)
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   699
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   700
lemma sats_domain_fm [simp]:
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   701
   "[| x \<in> nat; y \<in> nat; env \<in> list(A)|]
46823
57bf0cecb366 More mathematical symbols for ZF examples
paulson
parents: 32960
diff changeset
   702
    ==> sats(A, domain_fm(x,y), env) \<longleftrightarrow>
13807
a28a8fbc76d4 changed ** to ## to avoid conflict with new comment syntax
paulson
parents: 13655
diff changeset
   703
        is_domain(##A, nth(x,env), nth(y,env))"
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   704
by (simp add: domain_fm_def is_domain_def)
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   705
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   706
lemma domain_iff_sats:
13429
wenzelm
parents: 13428
diff changeset
   707
      "[| nth(i,env) = x; nth(j,env) = y;
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   708
          i \<in> nat; j \<in> nat; env \<in> list(A)|]
46823
57bf0cecb366 More mathematical symbols for ZF examples
paulson
parents: 32960
diff changeset
   709
       ==> is_domain(##A, x, y) \<longleftrightarrow> sats(A, domain_fm(i,j), env)"
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   710
by simp
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   711
13314
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   712
theorem domain_reflection:
13429
wenzelm
parents: 13428
diff changeset
   713
     "REFLECTS[\<lambda>x. is_domain(L,f(x),g(x)),
13807
a28a8fbc76d4 changed ** to ## to avoid conflict with new comment syntax
paulson
parents: 13655
diff changeset
   714
               \<lambda>i x. is_domain(##Lset(i),f(x),g(x))]"
13655
95b95cdb4704 Tidying up. New primitives is_iterates and is_iterates_fm.
paulson
parents: 13651
diff changeset
   715
apply (simp only: is_domain_def)
13429
wenzelm
parents: 13428
diff changeset
   716
apply (intro FOL_reflections pair_reflection)
13314
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   717
done
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   718
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   719
13339
0f89104dd377 Fixed quantified variable name preservation for ball and bex (bounded quants)
paulson
parents: 13323
diff changeset
   720
subsubsection{*Range of a Relation, Internalized*}
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   721
13429
wenzelm
parents: 13428
diff changeset
   722
(* "is_range(M,r,z) ==
46823
57bf0cecb366 More mathematical symbols for ZF examples
paulson
parents: 32960
diff changeset
   723
        \<forall>y[M]. (y \<in> z \<longleftrightarrow> (\<exists>w[M]. w\<in>r & (\<exists>x[M]. pair(M,x,y,w))))" *)
21404
eb85850d3eb7 more robust syntax for definition/abbreviation/notation;
wenzelm
parents: 21233
diff changeset
   724
definition
eb85850d3eb7 more robust syntax for definition/abbreviation/notation;
wenzelm
parents: 21233
diff changeset
   725
  range_fm :: "[i,i]=>i" where
13429
wenzelm
parents: 13428
diff changeset
   726
    "range_fm(r,z) ==
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   727
       Forall(Iff(Member(0,succ(z)),
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   728
                  Exists(And(Member(0,succ(succ(r))),
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   729
                             Exists(pair_fm(0,2,1))))))"
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   730
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   731
lemma range_type [TC]:
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   732
     "[| x \<in> nat; y \<in> nat |] ==> range_fm(x,y) \<in> formula"
13429
wenzelm
parents: 13428
diff changeset
   733
by (simp add: range_fm_def)
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   734
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   735
lemma sats_range_fm [simp]:
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   736
   "[| x \<in> nat; y \<in> nat; env \<in> list(A)|]
46823
57bf0cecb366 More mathematical symbols for ZF examples
paulson
parents: 32960
diff changeset
   737
    ==> sats(A, range_fm(x,y), env) \<longleftrightarrow>
13807
a28a8fbc76d4 changed ** to ## to avoid conflict with new comment syntax
paulson
parents: 13655
diff changeset
   738
        is_range(##A, nth(x,env), nth(y,env))"
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   739
by (simp add: range_fm_def is_range_def)
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   740
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   741
lemma range_iff_sats:
13429
wenzelm
parents: 13428
diff changeset
   742
      "[| nth(i,env) = x; nth(j,env) = y;
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   743
          i \<in> nat; j \<in> nat; env \<in> list(A)|]
46823
57bf0cecb366 More mathematical symbols for ZF examples
paulson
parents: 32960
diff changeset
   744
       ==> is_range(##A, x, y) \<longleftrightarrow> sats(A, range_fm(i,j), env)"
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   745
by simp
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   746
13314
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   747
theorem range_reflection:
13429
wenzelm
parents: 13428
diff changeset
   748
     "REFLECTS[\<lambda>x. is_range(L,f(x),g(x)),
13807
a28a8fbc76d4 changed ** to ## to avoid conflict with new comment syntax
paulson
parents: 13655
diff changeset
   749
               \<lambda>i x. is_range(##Lset(i),f(x),g(x))]"
13655
95b95cdb4704 Tidying up. New primitives is_iterates and is_iterates_fm.
paulson
parents: 13651
diff changeset
   750
apply (simp only: is_range_def)
13429
wenzelm
parents: 13428
diff changeset
   751
apply (intro FOL_reflections pair_reflection)
13314
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   752
done
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   753
13429
wenzelm
parents: 13428
diff changeset
   754
13339
0f89104dd377 Fixed quantified variable name preservation for ball and bex (bounded quants)
paulson
parents: 13323
diff changeset
   755
subsubsection{*Field of a Relation, Internalized*}
13323
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   756
13429
wenzelm
parents: 13428
diff changeset
   757
(* "is_field(M,r,z) ==
wenzelm
parents: 13428
diff changeset
   758
        \<exists>dr[M]. is_domain(M,r,dr) &
13323
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   759
            (\<exists>rr[M]. is_range(M,r,rr) & union(M,dr,rr,z))" *)
21404
eb85850d3eb7 more robust syntax for definition/abbreviation/notation;
wenzelm
parents: 21233
diff changeset
   760
definition
eb85850d3eb7 more robust syntax for definition/abbreviation/notation;
wenzelm
parents: 21233
diff changeset
   761
  field_fm :: "[i,i]=>i" where
13429
wenzelm
parents: 13428
diff changeset
   762
    "field_fm(r,z) ==
wenzelm
parents: 13428
diff changeset
   763
       Exists(And(domain_fm(succ(r),0),
wenzelm
parents: 13428
diff changeset
   764
              Exists(And(range_fm(succ(succ(r)),0),
13323
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   765
                         union_fm(1,0,succ(succ(z)))))))"
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   766
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   767
lemma field_type [TC]:
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   768
     "[| x \<in> nat; y \<in> nat |] ==> field_fm(x,y) \<in> formula"
13429
wenzelm
parents: 13428
diff changeset
   769
by (simp add: field_fm_def)
13323
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   770
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   771
lemma sats_field_fm [simp]:
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   772
   "[| x \<in> nat; y \<in> nat; env \<in> list(A)|]
46823
57bf0cecb366 More mathematical symbols for ZF examples
paulson
parents: 32960
diff changeset
   773
    ==> sats(A, field_fm(x,y), env) \<longleftrightarrow>
13807
a28a8fbc76d4 changed ** to ## to avoid conflict with new comment syntax
paulson
parents: 13655
diff changeset
   774
        is_field(##A, nth(x,env), nth(y,env))"
13323
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   775
by (simp add: field_fm_def is_field_def)
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   776
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   777
lemma field_iff_sats:
13429
wenzelm
parents: 13428
diff changeset
   778
      "[| nth(i,env) = x; nth(j,env) = y;
13323
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   779
          i \<in> nat; j \<in> nat; env \<in> list(A)|]
46823
57bf0cecb366 More mathematical symbols for ZF examples
paulson
parents: 32960
diff changeset
   780
       ==> is_field(##A, x, y) \<longleftrightarrow> sats(A, field_fm(i,j), env)"
13323
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   781
by simp
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   782
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   783
theorem field_reflection:
13429
wenzelm
parents: 13428
diff changeset
   784
     "REFLECTS[\<lambda>x. is_field(L,f(x),g(x)),
13807
a28a8fbc76d4 changed ** to ## to avoid conflict with new comment syntax
paulson
parents: 13655
diff changeset
   785
               \<lambda>i x. is_field(##Lset(i),f(x),g(x))]"
13655
95b95cdb4704 Tidying up. New primitives is_iterates and is_iterates_fm.
paulson
parents: 13651
diff changeset
   786
apply (simp only: is_field_def)
13323
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   787
apply (intro FOL_reflections domain_reflection range_reflection
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   788
             union_reflection)
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   789
done
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   790
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   791
13339
0f89104dd377 Fixed quantified variable name preservation for ball and bex (bounded quants)
paulson
parents: 13323
diff changeset
   792
subsubsection{*Image under a Relation, Internalized*}
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   793
13429
wenzelm
parents: 13428
diff changeset
   794
(* "image(M,r,A,z) ==
46823
57bf0cecb366 More mathematical symbols for ZF examples
paulson
parents: 32960
diff changeset
   795
        \<forall>y[M]. (y \<in> z \<longleftrightarrow> (\<exists>w[M]. w\<in>r & (\<exists>x[M]. x\<in>A & pair(M,x,y,w))))" *)
21404
eb85850d3eb7 more robust syntax for definition/abbreviation/notation;
wenzelm
parents: 21233
diff changeset
   796
definition
eb85850d3eb7 more robust syntax for definition/abbreviation/notation;
wenzelm
parents: 21233
diff changeset
   797
  image_fm :: "[i,i,i]=>i" where
13429
wenzelm
parents: 13428
diff changeset
   798
    "image_fm(r,A,z) ==
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   799
       Forall(Iff(Member(0,succ(z)),
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   800
                  Exists(And(Member(0,succ(succ(r))),
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   801
                             Exists(And(Member(0,succ(succ(succ(A)))),
13429
wenzelm
parents: 13428
diff changeset
   802
                                        pair_fm(0,2,1)))))))"
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   803
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   804
lemma image_type [TC]:
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   805
     "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> image_fm(x,y,z) \<in> formula"
13429
wenzelm
parents: 13428
diff changeset
   806
by (simp add: image_fm_def)
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   807
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   808
lemma sats_image_fm [simp]:
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   809
   "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
46823
57bf0cecb366 More mathematical symbols for ZF examples
paulson
parents: 32960
diff changeset
   810
    ==> sats(A, image_fm(x,y,z), env) \<longleftrightarrow>
13807
a28a8fbc76d4 changed ** to ## to avoid conflict with new comment syntax
paulson
parents: 13655
diff changeset
   811
        image(##A, nth(x,env), nth(y,env), nth(z,env))"
13323
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   812
by (simp add: image_fm_def Relative.image_def)
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   813
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   814
lemma image_iff_sats:
13429
wenzelm
parents: 13428
diff changeset
   815
      "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z;
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   816
          i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
46823
57bf0cecb366 More mathematical symbols for ZF examples
paulson
parents: 32960
diff changeset
   817
       ==> image(##A, x, y, z) \<longleftrightarrow> sats(A, image_fm(i,j,k), env)"
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   818
by (simp add: sats_image_fm)
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   819
13314
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   820
theorem image_reflection:
13429
wenzelm
parents: 13428
diff changeset
   821
     "REFLECTS[\<lambda>x. image(L,f(x),g(x),h(x)),
13807
a28a8fbc76d4 changed ** to ## to avoid conflict with new comment syntax
paulson
parents: 13655
diff changeset
   822
               \<lambda>i x. image(##Lset(i),f(x),g(x),h(x))]"
13655
95b95cdb4704 Tidying up. New primitives is_iterates and is_iterates_fm.
paulson
parents: 13651
diff changeset
   823
apply (simp only: Relative.image_def)
13429
wenzelm
parents: 13428
diff changeset
   824
apply (intro FOL_reflections pair_reflection)
13314
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   825
done
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   826
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   827
13348
374d05460db4 Separation/Replacement up to M_wfrank!
paulson
parents: 13339
diff changeset
   828
subsubsection{*Pre-Image under a Relation, Internalized*}
374d05460db4 Separation/Replacement up to M_wfrank!
paulson
parents: 13339
diff changeset
   829
13429
wenzelm
parents: 13428
diff changeset
   830
(* "pre_image(M,r,A,z) ==
46823
57bf0cecb366 More mathematical symbols for ZF examples
paulson
parents: 32960
diff changeset
   831
        \<forall>x[M]. x \<in> z \<longleftrightarrow> (\<exists>w[M]. w\<in>r & (\<exists>y[M]. y\<in>A & pair(M,x,y,w)))" *)
21404
eb85850d3eb7 more robust syntax for definition/abbreviation/notation;
wenzelm
parents: 21233
diff changeset
   832
definition
eb85850d3eb7 more robust syntax for definition/abbreviation/notation;
wenzelm
parents: 21233
diff changeset
   833
  pre_image_fm :: "[i,i,i]=>i" where
13429
wenzelm
parents: 13428
diff changeset
   834
    "pre_image_fm(r,A,z) ==
13348
374d05460db4 Separation/Replacement up to M_wfrank!
paulson
parents: 13339
diff changeset
   835
       Forall(Iff(Member(0,succ(z)),
374d05460db4 Separation/Replacement up to M_wfrank!
paulson
parents: 13339
diff changeset
   836
                  Exists(And(Member(0,succ(succ(r))),
374d05460db4 Separation/Replacement up to M_wfrank!
paulson
parents: 13339
diff changeset
   837
                             Exists(And(Member(0,succ(succ(succ(A)))),
13429
wenzelm
parents: 13428
diff changeset
   838
                                        pair_fm(2,0,1)))))))"
13348
374d05460db4 Separation/Replacement up to M_wfrank!
paulson
parents: 13339
diff changeset
   839
374d05460db4 Separation/Replacement up to M_wfrank!
paulson
parents: 13339
diff changeset
   840
lemma pre_image_type [TC]:
374d05460db4 Separation/Replacement up to M_wfrank!
paulson
parents: 13339
diff changeset
   841
     "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> pre_image_fm(x,y,z) \<in> formula"
13429
wenzelm
parents: 13428
diff changeset
   842
by (simp add: pre_image_fm_def)
13348
374d05460db4 Separation/Replacement up to M_wfrank!
paulson
parents: 13339
diff changeset
   843
374d05460db4 Separation/Replacement up to M_wfrank!
paulson
parents: 13339
diff changeset
   844
lemma sats_pre_image_fm [simp]:
374d05460db4 Separation/Replacement up to M_wfrank!
paulson
parents: 13339
diff changeset
   845
   "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
46823
57bf0cecb366 More mathematical symbols for ZF examples
paulson
parents: 32960
diff changeset
   846
    ==> sats(A, pre_image_fm(x,y,z), env) \<longleftrightarrow>
13807
a28a8fbc76d4 changed ** to ## to avoid conflict with new comment syntax
paulson
parents: 13655
diff changeset
   847
        pre_image(##A, nth(x,env), nth(y,env), nth(z,env))"
13348
374d05460db4 Separation/Replacement up to M_wfrank!
paulson
parents: 13339
diff changeset
   848
by (simp add: pre_image_fm_def Relative.pre_image_def)
374d05460db4 Separation/Replacement up to M_wfrank!
paulson
parents: 13339
diff changeset
   849
374d05460db4 Separation/Replacement up to M_wfrank!
paulson
parents: 13339
diff changeset
   850
lemma pre_image_iff_sats:
13429
wenzelm
parents: 13428
diff changeset
   851
      "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z;
13348
374d05460db4 Separation/Replacement up to M_wfrank!
paulson
parents: 13339
diff changeset
   852
          i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
46823
57bf0cecb366 More mathematical symbols for ZF examples
paulson
parents: 32960
diff changeset
   853
       ==> pre_image(##A, x, y, z) \<longleftrightarrow> sats(A, pre_image_fm(i,j,k), env)"
13348
374d05460db4 Separation/Replacement up to M_wfrank!
paulson
parents: 13339
diff changeset
   854
by (simp add: sats_pre_image_fm)
374d05460db4 Separation/Replacement up to M_wfrank!
paulson
parents: 13339
diff changeset
   855
374d05460db4 Separation/Replacement up to M_wfrank!
paulson
parents: 13339
diff changeset
   856
theorem pre_image_reflection:
13429
wenzelm
parents: 13428
diff changeset
   857
     "REFLECTS[\<lambda>x. pre_image(L,f(x),g(x),h(x)),
13807
a28a8fbc76d4 changed ** to ## to avoid conflict with new comment syntax
paulson
parents: 13655
diff changeset
   858
               \<lambda>i x. pre_image(##Lset(i),f(x),g(x),h(x))]"
13655
95b95cdb4704 Tidying up. New primitives is_iterates and is_iterates_fm.
paulson
parents: 13651
diff changeset
   859
apply (simp only: Relative.pre_image_def)
13429
wenzelm
parents: 13428
diff changeset
   860
apply (intro FOL_reflections pair_reflection)
13348
374d05460db4 Separation/Replacement up to M_wfrank!
paulson
parents: 13339
diff changeset
   861
done
374d05460db4 Separation/Replacement up to M_wfrank!
paulson
parents: 13339
diff changeset
   862
374d05460db4 Separation/Replacement up to M_wfrank!
paulson
parents: 13339
diff changeset
   863
13352
3cd767f8d78b new definitions of fun_apply and M_is_recfun
paulson
parents: 13348
diff changeset
   864
subsubsection{*Function Application, Internalized*}
3cd767f8d78b new definitions of fun_apply and M_is_recfun
paulson
parents: 13348
diff changeset
   865
13429
wenzelm
parents: 13428
diff changeset
   866
(* "fun_apply(M,f,x,y) ==
wenzelm
parents: 13428
diff changeset
   867
        (\<exists>xs[M]. \<exists>fxs[M].
13352
3cd767f8d78b new definitions of fun_apply and M_is_recfun
paulson
parents: 13348
diff changeset
   868
         upair(M,x,x,xs) & image(M,f,xs,fxs) & big_union(M,fxs,y))" *)
21404
eb85850d3eb7 more robust syntax for definition/abbreviation/notation;
wenzelm
parents: 21233
diff changeset
   869
definition
eb85850d3eb7 more robust syntax for definition/abbreviation/notation;
wenzelm
parents: 21233
diff changeset
   870
  fun_apply_fm :: "[i,i,i]=>i" where
13429
wenzelm
parents: 13428
diff changeset
   871
    "fun_apply_fm(f,x,y) ==
13352
3cd767f8d78b new definitions of fun_apply and M_is_recfun
paulson
parents: 13348
diff changeset
   872
       Exists(Exists(And(upair_fm(succ(succ(x)), succ(succ(x)), 1),
13429
wenzelm
parents: 13428
diff changeset
   873
                         And(image_fm(succ(succ(f)), 1, 0),
13352
3cd767f8d78b new definitions of fun_apply and M_is_recfun
paulson
parents: 13348
diff changeset
   874
                             big_union_fm(0,succ(succ(y)))))))"
3cd767f8d78b new definitions of fun_apply and M_is_recfun
paulson
parents: 13348
diff changeset
   875
3cd767f8d78b new definitions of fun_apply and M_is_recfun
paulson
parents: 13348
diff changeset
   876
lemma fun_apply_type [TC]:
3cd767f8d78b new definitions of fun_apply and M_is_recfun
paulson
parents: 13348
diff changeset
   877
     "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> fun_apply_fm(x,y,z) \<in> formula"
13429
wenzelm
parents: 13428
diff changeset
   878
by (simp add: fun_apply_fm_def)
13352
3cd767f8d78b new definitions of fun_apply and M_is_recfun
paulson
parents: 13348
diff changeset
   879
3cd767f8d78b new definitions of fun_apply and M_is_recfun
paulson
parents: 13348
diff changeset
   880
lemma sats_fun_apply_fm [simp]:
3cd767f8d78b new definitions of fun_apply and M_is_recfun
paulson
parents: 13348
diff changeset
   881
   "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
46823
57bf0cecb366 More mathematical symbols for ZF examples
paulson
parents: 32960
diff changeset
   882
    ==> sats(A, fun_apply_fm(x,y,z), env) \<longleftrightarrow>
13807
a28a8fbc76d4 changed ** to ## to avoid conflict with new comment syntax
paulson
parents: 13655
diff changeset
   883
        fun_apply(##A, nth(x,env), nth(y,env), nth(z,env))"
13352
3cd767f8d78b new definitions of fun_apply and M_is_recfun
paulson
parents: 13348
diff changeset
   884
by (simp add: fun_apply_fm_def fun_apply_def)
3cd767f8d78b new definitions of fun_apply and M_is_recfun
paulson
parents: 13348
diff changeset
   885
3cd767f8d78b new definitions of fun_apply and M_is_recfun
paulson
parents: 13348
diff changeset
   886
lemma fun_apply_iff_sats:
13429
wenzelm
parents: 13428
diff changeset
   887
      "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z;
13352
3cd767f8d78b new definitions of fun_apply and M_is_recfun
paulson
parents: 13348
diff changeset
   888
          i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
46823
57bf0cecb366 More mathematical symbols for ZF examples
paulson
parents: 32960
diff changeset
   889
       ==> fun_apply(##A, x, y, z) \<longleftrightarrow> sats(A, fun_apply_fm(i,j,k), env)"
13352
3cd767f8d78b new definitions of fun_apply and M_is_recfun
paulson
parents: 13348
diff changeset
   890
by simp
3cd767f8d78b new definitions of fun_apply and M_is_recfun
paulson
parents: 13348
diff changeset
   891
3cd767f8d78b new definitions of fun_apply and M_is_recfun
paulson
parents: 13348
diff changeset
   892
theorem fun_apply_reflection:
13429
wenzelm
parents: 13428
diff changeset
   893
     "REFLECTS[\<lambda>x. fun_apply(L,f(x),g(x),h(x)),
13807
a28a8fbc76d4 changed ** to ## to avoid conflict with new comment syntax
paulson
parents: 13655
diff changeset
   894
               \<lambda>i x. fun_apply(##Lset(i),f(x),g(x),h(x))]"
13655
95b95cdb4704 Tidying up. New primitives is_iterates and is_iterates_fm.
paulson
parents: 13651
diff changeset
   895
apply (simp only: fun_apply_def)
13352
3cd767f8d78b new definitions of fun_apply and M_is_recfun
paulson
parents: 13348
diff changeset
   896
apply (intro FOL_reflections upair_reflection image_reflection
13429
wenzelm
parents: 13428
diff changeset
   897
             big_union_reflection)
13352
3cd767f8d78b new definitions of fun_apply and M_is_recfun
paulson
parents: 13348
diff changeset
   898
done
3cd767f8d78b new definitions of fun_apply and M_is_recfun
paulson
parents: 13348
diff changeset
   899
3cd767f8d78b new definitions of fun_apply and M_is_recfun
paulson
parents: 13348
diff changeset
   900
13339
0f89104dd377 Fixed quantified variable name preservation for ball and bex (bounded quants)
paulson
parents: 13323
diff changeset
   901
subsubsection{*The Concept of Relation, Internalized*}
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   902
13429
wenzelm
parents: 13428
diff changeset
   903
(* "is_relation(M,r) ==
46823
57bf0cecb366 More mathematical symbols for ZF examples
paulson
parents: 32960
diff changeset
   904
        (\<forall>z[M]. z\<in>r \<longrightarrow> (\<exists>x[M]. \<exists>y[M]. pair(M,x,y,z)))" *)
21404
eb85850d3eb7 more robust syntax for definition/abbreviation/notation;
wenzelm
parents: 21233
diff changeset
   905
definition
eb85850d3eb7 more robust syntax for definition/abbreviation/notation;
wenzelm
parents: 21233
diff changeset
   906
  relation_fm :: "i=>i" where
13429
wenzelm
parents: 13428
diff changeset
   907
    "relation_fm(r) ==
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   908
       Forall(Implies(Member(0,succ(r)), Exists(Exists(pair_fm(1,0,2)))))"
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   909
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   910
lemma relation_type [TC]:
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   911
     "[| x \<in> nat |] ==> relation_fm(x) \<in> formula"
13429
wenzelm
parents: 13428
diff changeset
   912
by (simp add: relation_fm_def)
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   913
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   914
lemma sats_relation_fm [simp]:
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   915
   "[| x \<in> nat; env \<in> list(A)|]
46823
57bf0cecb366 More mathematical symbols for ZF examples
paulson
parents: 32960
diff changeset
   916
    ==> sats(A, relation_fm(x), env) \<longleftrightarrow> is_relation(##A, nth(x,env))"
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   917
by (simp add: relation_fm_def is_relation_def)
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   918
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   919
lemma relation_iff_sats:
13429
wenzelm
parents: 13428
diff changeset
   920
      "[| nth(i,env) = x; nth(j,env) = y;
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   921
          i \<in> nat; env \<in> list(A)|]
46823
57bf0cecb366 More mathematical symbols for ZF examples
paulson
parents: 32960
diff changeset
   922
       ==> is_relation(##A, x) \<longleftrightarrow> sats(A, relation_fm(i), env)"
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   923
by simp
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   924
13314
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   925
theorem is_relation_reflection:
13429
wenzelm
parents: 13428
diff changeset
   926
     "REFLECTS[\<lambda>x. is_relation(L,f(x)),
13807
a28a8fbc76d4 changed ** to ## to avoid conflict with new comment syntax
paulson
parents: 13655
diff changeset
   927
               \<lambda>i x. is_relation(##Lset(i),f(x))]"
13655
95b95cdb4704 Tidying up. New primitives is_iterates and is_iterates_fm.
paulson
parents: 13651
diff changeset
   928
apply (simp only: is_relation_def)
13429
wenzelm
parents: 13428
diff changeset
   929
apply (intro FOL_reflections pair_reflection)
13314
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   930
done
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   931
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   932
13339
0f89104dd377 Fixed quantified variable name preservation for ball and bex (bounded quants)
paulson
parents: 13323
diff changeset
   933
subsubsection{*The Concept of Function, Internalized*}
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   934
13429
wenzelm
parents: 13428
diff changeset
   935
(* "is_function(M,r) ==
wenzelm
parents: 13428
diff changeset
   936
        \<forall>x[M]. \<forall>y[M]. \<forall>y'[M]. \<forall>p[M]. \<forall>p'[M].
46823
57bf0cecb366 More mathematical symbols for ZF examples
paulson
parents: 32960
diff changeset
   937
           pair(M,x,y,p) \<longrightarrow> pair(M,x,y',p') \<longrightarrow> p\<in>r \<longrightarrow> p'\<in>r \<longrightarrow> y=y'" *)
21404
eb85850d3eb7 more robust syntax for definition/abbreviation/notation;
wenzelm
parents: 21233
diff changeset
   938
definition
eb85850d3eb7 more robust syntax for definition/abbreviation/notation;
wenzelm
parents: 21233
diff changeset
   939
  function_fm :: "i=>i" where
13429
wenzelm
parents: 13428
diff changeset
   940
    "function_fm(r) ==
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   941
       Forall(Forall(Forall(Forall(Forall(
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   942
         Implies(pair_fm(4,3,1),
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   943
                 Implies(pair_fm(4,2,0),
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   944
                         Implies(Member(1,r#+5),
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   945
                                 Implies(Member(0,r#+5), Equal(3,2))))))))))"
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   946
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   947
lemma function_type [TC]:
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   948
     "[| x \<in> nat |] ==> function_fm(x) \<in> formula"
13429
wenzelm
parents: 13428
diff changeset
   949
by (simp add: function_fm_def)
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   950
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   951
lemma sats_function_fm [simp]:
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   952
   "[| x \<in> nat; env \<in> list(A)|]
46823
57bf0cecb366 More mathematical symbols for ZF examples
paulson
parents: 32960
diff changeset
   953
    ==> sats(A, function_fm(x), env) \<longleftrightarrow> is_function(##A, nth(x,env))"
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   954
by (simp add: function_fm_def is_function_def)
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   955
13505
52a16cb7fefb Relativized right up to L satisfies V=L!
paulson
parents: 13496
diff changeset
   956
lemma is_function_iff_sats:
13429
wenzelm
parents: 13428
diff changeset
   957
      "[| nth(i,env) = x; nth(j,env) = y;
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   958
          i \<in> nat; env \<in> list(A)|]
46823
57bf0cecb366 More mathematical symbols for ZF examples
paulson
parents: 32960
diff changeset
   959
       ==> is_function(##A, x) \<longleftrightarrow> sats(A, function_fm(i), env)"
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   960
by simp
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   961
13314
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   962
theorem is_function_reflection:
13429
wenzelm
parents: 13428
diff changeset
   963
     "REFLECTS[\<lambda>x. is_function(L,f(x)),
13807
a28a8fbc76d4 changed ** to ## to avoid conflict with new comment syntax
paulson
parents: 13655
diff changeset
   964
               \<lambda>i x. is_function(##Lset(i),f(x))]"
13655
95b95cdb4704 Tidying up. New primitives is_iterates and is_iterates_fm.
paulson
parents: 13651
diff changeset
   965
apply (simp only: is_function_def)
13429
wenzelm
parents: 13428
diff changeset
   966
apply (intro FOL_reflections pair_reflection)
13314
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   967
done
13298
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   968
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   969
13339
0f89104dd377 Fixed quantified variable name preservation for ball and bex (bounded quants)
paulson
parents: 13323
diff changeset
   970
subsubsection{*Typed Functions, Internalized*}
13309
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
   971
13429
wenzelm
parents: 13428
diff changeset
   972
(* "typed_function(M,A,B,r) ==
13309
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
   973
        is_function(M,r) & is_relation(M,r) & is_domain(M,r,A) &
46823
57bf0cecb366 More mathematical symbols for ZF examples
paulson
parents: 32960
diff changeset
   974
        (\<forall>u[M]. u\<in>r \<longrightarrow> (\<forall>x[M]. \<forall>y[M]. pair(M,x,y,u) \<longrightarrow> y\<in>B))" *)
13309
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
   975
21404
eb85850d3eb7 more robust syntax for definition/abbreviation/notation;
wenzelm
parents: 21233
diff changeset
   976
definition
eb85850d3eb7 more robust syntax for definition/abbreviation/notation;
wenzelm
parents: 21233
diff changeset
   977
  typed_function_fm :: "[i,i,i]=>i" where
13429
wenzelm
parents: 13428
diff changeset
   978
    "typed_function_fm(A,B,r) ==
13309
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
   979
       And(function_fm(r),
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
   980
         And(relation_fm(r),
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
   981
           And(domain_fm(r,A),
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
   982
             Forall(Implies(Member(0,succ(r)),
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
   983
                  Forall(Forall(Implies(pair_fm(1,0,2),Member(0,B#+3)))))))))"
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
   984
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
   985
lemma typed_function_type [TC]:
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
   986
     "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> typed_function_fm(x,y,z) \<in> formula"
13429
wenzelm
parents: 13428
diff changeset
   987
by (simp add: typed_function_fm_def)
13309
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
   988
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
   989
lemma sats_typed_function_fm [simp]:
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
   990
   "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
46823
57bf0cecb366 More mathematical symbols for ZF examples
paulson
parents: 32960
diff changeset
   991
    ==> sats(A, typed_function_fm(x,y,z), env) \<longleftrightarrow>
13807
a28a8fbc76d4 changed ** to ## to avoid conflict with new comment syntax
paulson
parents: 13655
diff changeset
   992
        typed_function(##A, nth(x,env), nth(y,env), nth(z,env))"
13309
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
   993
by (simp add: typed_function_fm_def typed_function_def)
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
   994
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
   995
lemma typed_function_iff_sats:
13429
wenzelm
parents: 13428
diff changeset
   996
  "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z;
13309
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
   997
      i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
46823
57bf0cecb366 More mathematical symbols for ZF examples
paulson
parents: 32960
diff changeset
   998
   ==> typed_function(##A, x, y, z) \<longleftrightarrow> sats(A, typed_function_fm(i,j,k), env)"
13309
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
   999
by simp
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1000
13429
wenzelm
parents: 13428
diff changeset
  1001
lemmas function_reflections =
13363
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
  1002
        empty_reflection number1_reflection
13429
wenzelm
parents: 13428
diff changeset
  1003
        upair_reflection pair_reflection union_reflection
wenzelm
parents: 13428
diff changeset
  1004
        big_union_reflection cons_reflection successor_reflection
13323
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
  1005
        fun_apply_reflection subset_reflection
13429
wenzelm
parents: 13428
diff changeset
  1006
        transitive_set_reflection membership_reflection
wenzelm
parents: 13428
diff changeset
  1007
        pred_set_reflection domain_reflection range_reflection field_reflection
13348
374d05460db4 Separation/Replacement up to M_wfrank!
paulson
parents: 13339
diff changeset
  1008
        image_reflection pre_image_reflection
13429
wenzelm
parents: 13428
diff changeset
  1009
        is_relation_reflection is_function_reflection
13309
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1010
13429
wenzelm
parents: 13428
diff changeset
  1011
lemmas function_iff_sats =
wenzelm
parents: 13428
diff changeset
  1012
        empty_iff_sats number1_iff_sats
wenzelm
parents: 13428
diff changeset
  1013
        upair_iff_sats pair_iff_sats union_iff_sats
13505
52a16cb7fefb Relativized right up to L satisfies V=L!
paulson
parents: 13496
diff changeset
  1014
        big_union_iff_sats cons_iff_sats successor_iff_sats
13323
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
  1015
        fun_apply_iff_sats  Memrel_iff_sats
13429
wenzelm
parents: 13428
diff changeset
  1016
        pred_set_iff_sats domain_iff_sats range_iff_sats field_iff_sats
wenzelm
parents: 13428
diff changeset
  1017
        image_iff_sats pre_image_iff_sats
13505
52a16cb7fefb Relativized right up to L satisfies V=L!
paulson
parents: 13496
diff changeset
  1018
        relation_iff_sats is_function_iff_sats
13323
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
  1019
13309
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1020
13314
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
  1021
theorem typed_function_reflection:
13429
wenzelm
parents: 13428
diff changeset
  1022
     "REFLECTS[\<lambda>x. typed_function(L,f(x),g(x),h(x)),
13807
a28a8fbc76d4 changed ** to ## to avoid conflict with new comment syntax
paulson
parents: 13655
diff changeset
  1023
               \<lambda>i x. typed_function(##Lset(i),f(x),g(x),h(x))]"
13655
95b95cdb4704 Tidying up. New primitives is_iterates and is_iterates_fm.
paulson
parents: 13651
diff changeset
  1024
apply (simp only: typed_function_def)
13429
wenzelm
parents: 13428
diff changeset
  1025
apply (intro FOL_reflections function_reflections)
13323
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
  1026
done
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
  1027
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
  1028
13339
0f89104dd377 Fixed quantified variable name preservation for ball and bex (bounded quants)
paulson
parents: 13323
diff changeset
  1029
subsubsection{*Composition of Relations, Internalized*}
13323
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
  1030
13429
wenzelm
parents: 13428
diff changeset
  1031
(* "composition(M,r,s,t) ==
46823
57bf0cecb366 More mathematical symbols for ZF examples
paulson
parents: 32960
diff changeset
  1032
        \<forall>p[M]. p \<in> t \<longleftrightarrow>
13429
wenzelm
parents: 13428
diff changeset
  1033
               (\<exists>x[M]. \<exists>y[M]. \<exists>z[M]. \<exists>xy[M]. \<exists>yz[M].
wenzelm
parents: 13428
diff changeset
  1034
                pair(M,x,z,p) & pair(M,x,y,xy) & pair(M,y,z,yz) &
13323
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
  1035
                xy \<in> s & yz \<in> r)" *)
21404
eb85850d3eb7 more robust syntax for definition/abbreviation/notation;
wenzelm
parents: 21233
diff changeset
  1036
definition
eb85850d3eb7 more robust syntax for definition/abbreviation/notation;
wenzelm
parents: 21233
diff changeset
  1037
  composition_fm :: "[i,i,i]=>i" where
13429
wenzelm
parents: 13428
diff changeset
  1038
  "composition_fm(r,s,t) ==
13323
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
  1039
     Forall(Iff(Member(0,succ(t)),
13429
wenzelm
parents: 13428
diff changeset
  1040
             Exists(Exists(Exists(Exists(Exists(
13323
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
  1041
              And(pair_fm(4,2,5),
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
  1042
               And(pair_fm(4,3,1),
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
  1043
                And(pair_fm(3,2,0),
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
  1044
                 And(Member(1,s#+6), Member(0,r#+6))))))))))))"
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
  1045
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
  1046
lemma composition_type [TC]:
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
  1047
     "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> composition_fm(x,y,z) \<in> formula"
13429
wenzelm
parents: 13428
diff changeset
  1048
by (simp add: composition_fm_def)
13323
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
  1049
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
  1050
lemma sats_composition_fm [simp]:
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
  1051
   "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
46823
57bf0cecb366 More mathematical symbols for ZF examples
paulson
parents: 32960
diff changeset
  1052
    ==> sats(A, composition_fm(x,y,z), env) \<longleftrightarrow>
13807
a28a8fbc76d4 changed ** to ## to avoid conflict with new comment syntax
paulson
parents: 13655
diff changeset
  1053
        composition(##A, nth(x,env), nth(y,env), nth(z,env))"
13323
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
  1054
by (simp add: composition_fm_def composition_def)
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
  1055
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
  1056
lemma composition_iff_sats:
13429
wenzelm
parents: 13428
diff changeset
  1057
      "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z;
13323
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
  1058
          i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
46823
57bf0cecb366 More mathematical symbols for ZF examples
paulson
parents: 32960
diff changeset
  1059
       ==> composition(##A, x, y, z) \<longleftrightarrow> sats(A, composition_fm(i,j,k), env)"
13323
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
  1060
by simp
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
  1061
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
  1062
theorem composition_reflection:
13429
wenzelm
parents: 13428
diff changeset
  1063
     "REFLECTS[\<lambda>x. composition(L,f(x),g(x),h(x)),
13807
a28a8fbc76d4 changed ** to ## to avoid conflict with new comment syntax
paulson
parents: 13655
diff changeset
  1064
               \<lambda>i x. composition(##Lset(i),f(x),g(x),h(x))]"
13655
95b95cdb4704 Tidying up. New primitives is_iterates and is_iterates_fm.
paulson
parents: 13651
diff changeset
  1065
apply (simp only: composition_def)
13429
wenzelm
parents: 13428
diff changeset
  1066
apply (intro FOL_reflections pair_reflection)
13314
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
  1067
done
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
  1068
13309
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1069
13339
0f89104dd377 Fixed quantified variable name preservation for ball and bex (bounded quants)
paulson
parents: 13323
diff changeset
  1070
subsubsection{*Injections, Internalized*}
13309
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1071
13429
wenzelm
parents: 13428
diff changeset
  1072
(* "injection(M,A,B,f) ==
wenzelm
parents: 13428
diff changeset
  1073
        typed_function(M,A,B,f) &
wenzelm
parents: 13428
diff changeset
  1074
        (\<forall>x[M]. \<forall>x'[M]. \<forall>y[M]. \<forall>p[M]. \<forall>p'[M].
46823
57bf0cecb366 More mathematical symbols for ZF examples
paulson
parents: 32960
diff changeset
  1075
          pair(M,x,y,p) \<longrightarrow> pair(M,x',y,p') \<longrightarrow> p\<in>f \<longrightarrow> p'\<in>f \<longrightarrow> x=x')" *)
21404
eb85850d3eb7 more robust syntax for definition/abbreviation/notation;
wenzelm
parents: 21233
diff changeset
  1076
definition
eb85850d3eb7 more robust syntax for definition/abbreviation/notation;
wenzelm
parents: 21233
diff changeset
  1077
  injection_fm :: "[i,i,i]=>i" where
eb85850d3eb7 more robust syntax for definition/abbreviation/notation;
wenzelm
parents: 21233
diff changeset
  1078
  "injection_fm(A,B,f) ==
13309
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1079
    And(typed_function_fm(A,B,f),
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1080
       Forall(Forall(Forall(Forall(Forall(
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1081
         Implies(pair_fm(4,2,1),
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1082
                 Implies(pair_fm(3,2,0),
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1083
                         Implies(Member(1,f#+5),
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1084
                                 Implies(Member(0,f#+5), Equal(4,3)))))))))))"
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1085
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1086
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1087
lemma injection_type [TC]:
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1088
     "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> injection_fm(x,y,z) \<in> formula"
13429
wenzelm
parents: 13428
diff changeset
  1089
by (simp add: injection_fm_def)
13309
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1090
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1091
lemma sats_injection_fm [simp]:
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1092
   "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
46823
57bf0cecb366 More mathematical symbols for ZF examples
paulson
parents: 32960
diff changeset
  1093
    ==> sats(A, injection_fm(x,y,z), env) \<longleftrightarrow>
13807
a28a8fbc76d4 changed ** to ## to avoid conflict with new comment syntax
paulson
parents: 13655
diff changeset
  1094
        injection(##A, nth(x,env), nth(y,env), nth(z,env))"
13309
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1095
by (simp add: injection_fm_def injection_def)
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1096
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1097
lemma injection_iff_sats:
13429
wenzelm
parents: 13428
diff changeset
  1098
  "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z;
13309
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1099
      i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
46823
57bf0cecb366 More mathematical symbols for ZF examples
paulson
parents: 32960
diff changeset
  1100
   ==> injection(##A, x, y, z) \<longleftrightarrow> sats(A, injection_fm(i,j,k), env)"
13309
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1101
by simp
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1102
13314
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
  1103
theorem injection_reflection:
13429
wenzelm
parents: 13428
diff changeset
  1104
     "REFLECTS[\<lambda>x. injection(L,f(x),g(x),h(x)),
13807
a28a8fbc76d4 changed ** to ## to avoid conflict with new comment syntax
paulson
parents: 13655
diff changeset
  1105
               \<lambda>i x. injection(##Lset(i),f(x),g(x),h(x))]"
13655
95b95cdb4704 Tidying up. New primitives is_iterates and is_iterates_fm.
paulson
parents: 13651
diff changeset
  1106
apply (simp only: injection_def)
13429
wenzelm
parents: 13428
diff changeset
  1107
apply (intro FOL_reflections function_reflections typed_function_reflection)
13314
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
  1108
done
13309
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1109
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1110
13339
0f89104dd377 Fixed quantified variable name preservation for ball and bex (bounded quants)
paulson
parents: 13323
diff changeset
  1111
subsubsection{*Surjections, Internalized*}
13309
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1112
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1113
(*  surjection :: "[i=>o,i,i,i] => o"
13429
wenzelm
parents: 13428
diff changeset
  1114
    "surjection(M,A,B,f) ==
13309
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1115
        typed_function(M,A,B,f) &
46823
57bf0cecb366 More mathematical symbols for ZF examples
paulson
parents: 32960
diff changeset
  1116
        (\<forall>y[M]. y\<in>B \<longrightarrow> (\<exists>x[M]. x\<in>A & fun_apply(M,f,x,y)))" *)
21404
eb85850d3eb7 more robust syntax for definition/abbreviation/notation;
wenzelm
parents: 21233
diff changeset
  1117
definition
eb85850d3eb7 more robust syntax for definition/abbreviation/notation;
wenzelm
parents: 21233
diff changeset
  1118
  surjection_fm :: "[i,i,i]=>i" where
eb85850d3eb7 more robust syntax for definition/abbreviation/notation;
wenzelm
parents: 21233
diff changeset
  1119
  "surjection_fm(A,B,f) ==
13309
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1120
    And(typed_function_fm(A,B,f),
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1121
       Forall(Implies(Member(0,succ(B)),
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1122
                      Exists(And(Member(0,succ(succ(A))),
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1123
                                 fun_apply_fm(succ(succ(f)),0,1))))))"
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1124
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1125
lemma surjection_type [TC]:
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1126
     "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> surjection_fm(x,y,z) \<in> formula"
13429
wenzelm
parents: 13428
diff changeset
  1127
by (simp add: surjection_fm_def)
13309
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1128
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1129
lemma sats_surjection_fm [simp]:
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1130
   "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
46823
57bf0cecb366 More mathematical symbols for ZF examples
paulson
parents: 32960
diff changeset
  1131
    ==> sats(A, surjection_fm(x,y,z), env) \<longleftrightarrow>
13807
a28a8fbc76d4 changed ** to ## to avoid conflict with new comment syntax
paulson
parents: 13655
diff changeset
  1132
        surjection(##A, nth(x,env), nth(y,env), nth(z,env))"
13309
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1133
by (simp add: surjection_fm_def surjection_def)
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1134
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1135
lemma surjection_iff_sats:
13429
wenzelm
parents: 13428
diff changeset
  1136
  "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z;
13309
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1137
      i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
46823
57bf0cecb366 More mathematical symbols for ZF examples
paulson
parents: 32960
diff changeset
  1138
   ==> surjection(##A, x, y, z) \<longleftrightarrow> sats(A, surjection_fm(i,j,k), env)"
13309
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1139
by simp
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1140
13314
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
  1141
theorem surjection_reflection:
13429
wenzelm
parents: 13428
diff changeset
  1142
     "REFLECTS[\<lambda>x. surjection(L,f(x),g(x),h(x)),
13807
a28a8fbc76d4 changed ** to ## to avoid conflict with new comment syntax
paulson
parents: 13655
diff changeset
  1143
               \<lambda>i x. surjection(##Lset(i),f(x),g(x),h(x))]"
13655
95b95cdb4704 Tidying up. New primitives is_iterates and is_iterates_fm.
paulson
parents: 13651
diff changeset
  1144
apply (simp only: surjection_def)
13429
wenzelm
parents: 13428
diff changeset
  1145
apply (intro FOL_reflections function_reflections typed_function_reflection)
13314
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
  1146
done
13309
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1147
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1148
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1149
13339
0f89104dd377 Fixed quantified variable name preservation for ball and bex (bounded quants)
paulson
parents: 13323
diff changeset
  1150
subsubsection{*Bijections, Internalized*}
13309
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1151
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1152
(*   bijection :: "[i=>o,i,i,i] => o"
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1153
    "bijection(M,A,B,f) == injection(M,A,B,f) & surjection(M,A,B,f)" *)
21404
eb85850d3eb7 more robust syntax for definition/abbreviation/notation;
wenzelm
parents: 21233
diff changeset
  1154
definition
eb85850d3eb7 more robust syntax for definition/abbreviation/notation;
wenzelm
parents: 21233
diff changeset
  1155
  bijection_fm :: "[i,i,i]=>i" where
eb85850d3eb7 more robust syntax for definition/abbreviation/notation;
wenzelm
parents: 21233
diff changeset
  1156
  "bijection_fm(A,B,f) == And(injection_fm(A,B,f), surjection_fm(A,B,f))"
13309
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1157
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1158
lemma bijection_type [TC]:
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1159
     "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> bijection_fm(x,y,z) \<in> formula"
13429
wenzelm
parents: 13428
diff changeset
  1160
by (simp add: bijection_fm_def)
13309
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1161
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1162
lemma sats_bijection_fm [simp]:
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1163
   "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
46823
57bf0cecb366 More mathematical symbols for ZF examples
paulson
parents: 32960
diff changeset
  1164
    ==> sats(A, bijection_fm(x,y,z), env) \<longleftrightarrow>
13807
a28a8fbc76d4 changed ** to ## to avoid conflict with new comment syntax
paulson
parents: 13655
diff changeset
  1165
        bijection(##A, nth(x,env), nth(y,env), nth(z,env))"
13309
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1166
by (simp add: bijection_fm_def bijection_def)
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1167
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1168
lemma bijection_iff_sats:
13429
wenzelm
parents: 13428
diff changeset
  1169
  "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z;
13309
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1170
      i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
46823
57bf0cecb366 More mathematical symbols for ZF examples
paulson
parents: 32960
diff changeset
  1171
   ==> bijection(##A, x, y, z) \<longleftrightarrow> sats(A, bijection_fm(i,j,k), env)"
13309
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1172
by simp
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1173
13314
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
  1174
theorem bijection_reflection:
13429
wenzelm
parents: 13428
diff changeset
  1175
     "REFLECTS[\<lambda>x. bijection(L,f(x),g(x),h(x)),
13807
a28a8fbc76d4 changed ** to ## to avoid conflict with new comment syntax
paulson
parents: 13655
diff changeset
  1176
               \<lambda>i x. bijection(##Lset(i),f(x),g(x),h(x))]"
13655
95b95cdb4704 Tidying up. New primitives is_iterates and is_iterates_fm.
paulson
parents: 13651
diff changeset
  1177
apply (simp only: bijection_def)
13429
wenzelm
parents: 13428
diff changeset
  1178
apply (intro And_reflection injection_reflection surjection_reflection)
13314
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
  1179
done
13309
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1180
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1181
13348
374d05460db4 Separation/Replacement up to M_wfrank!
paulson
parents: 13339
diff changeset
  1182
subsubsection{*Restriction of a Relation, Internalized*}
374d05460db4 Separation/Replacement up to M_wfrank!
paulson
parents: 13339
diff changeset
  1183
374d05460db4 Separation/Replacement up to M_wfrank!
paulson
parents: 13339
diff changeset
  1184
13429
wenzelm
parents: 13428
diff changeset
  1185
(* "restriction(M,r,A,z) ==
46823
57bf0cecb366 More mathematical symbols for ZF examples
paulson
parents: 32960
diff changeset
  1186
        \<forall>x[M]. x \<in> z \<longleftrightarrow> (x \<in> r & (\<exists>u[M]. u\<in>A & (\<exists>v[M]. pair(M,u,v,x))))" *)
21404
eb85850d3eb7 more robust syntax for definition/abbreviation/notation;
wenzelm
parents: 21233
diff changeset
  1187
definition
eb85850d3eb7 more robust syntax for definition/abbreviation/notation;
wenzelm
parents: 21233
diff changeset
  1188
  restriction_fm :: "[i,i,i]=>i" where
13429
wenzelm
parents: 13428
diff changeset
  1189
    "restriction_fm(r,A,z) ==
13348
374d05460db4 Separation/Replacement up to M_wfrank!
paulson
parents: 13339
diff changeset
  1190
       Forall(Iff(Member(0,succ(z)),
374d05460db4 Separation/Replacement up to M_wfrank!
paulson
parents: 13339
diff changeset
  1191
                  And(Member(0,succ(r)),
374d05460db4 Separation/Replacement up to M_wfrank!
paulson
parents: 13339
diff changeset
  1192
                      Exists(And(Member(0,succ(succ(A))),
374d05460db4 Separation/Replacement up to M_wfrank!
paulson
parents: 13339
diff changeset
  1193
                                 Exists(pair_fm(1,0,2)))))))"
374d05460db4 Separation/Replacement up to M_wfrank!
paulson
parents: 13339
diff changeset
  1194
374d05460db4 Separation/Replacement up to M_wfrank!
paulson
parents: 13339
diff changeset
  1195
lemma restriction_type [TC]:
374d05460db4 Separation/Replacement up to M_wfrank!
paulson
parents: 13339
diff changeset
  1196
     "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> restriction_fm(x,y,z) \<in> formula"
13429
wenzelm
parents: 13428
diff changeset
  1197
by (simp add: restriction_fm_def)
13348
374d05460db4 Separation/Replacement up to M_wfrank!
paulson
parents: 13339
diff changeset
  1198
374d05460db4 Separation/Replacement up to M_wfrank!
paulson
parents: 13339
diff changeset
  1199
lemma sats_restriction_fm [simp]:
374d05460db4 Separation/Replacement up to M_wfrank!
paulson
parents: 13339
diff changeset
  1200
   "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
46823
57bf0cecb366 More mathematical symbols for ZF examples
paulson
parents: 32960
diff changeset
  1201
    ==> sats(A, restriction_fm(x,y,z), env) \<longleftrightarrow>
13807
a28a8fbc76d4 changed ** to ## to avoid conflict with new comment syntax
paulson
parents: 13655
diff changeset
  1202
        restriction(##A, nth(x,env), nth(y,env), nth(z,env))"
13348
374d05460db4 Separation/Replacement up to M_wfrank!
paulson
parents: 13339
diff changeset
  1203
by (simp add: restriction_fm_def restriction_def)
374d05460db4 Separation/Replacement up to M_wfrank!
paulson
parents: 13339
diff changeset
  1204
374d05460db4 Separation/Replacement up to M_wfrank!
paulson
parents: 13339
diff changeset
  1205
lemma restriction_iff_sats:
13429
wenzelm
parents: 13428
diff changeset
  1206
      "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z;
13348
374d05460db4 Separation/Replacement up to M_wfrank!
paulson
parents: 13339
diff changeset
  1207