src/ZF/Constructible/L_axioms.thy
author haftmann
Fri, 17 Jun 2005 16:12:49 +0200
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(*  Title:      ZF/Constructible/L_axioms.thy
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    ID:         $Id$
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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 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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constdefs
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  L_F0 :: "[i=>o,i] => i"
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    "L_F0(P,y) == \<mu> b. (\<exists>z. L(z) \<and> P(<y,z>)) --> (\<exists>z\<in>Lset(b). P(<y,z>))"
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  L_FF :: "[i=>o,i] => i"
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    "L_FF(P)   == \<lambda>a. \<Union>y\<in>Lset(a). L_F0(P,y)"
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  L_ClEx :: "[i=>o,i] => o"
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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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constdefs
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  L_Reflects :: "[i=>o,[i,i]=>o] => prop"      ("(3REFLECTS/ [_,/ _])")
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    "REFLECTS[P,Q] == (??Cl. Closed_Unbounded(Cl) &
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                           (\<forall>a. Cl(a) --> (\<forall>x \<in> Lset(a). P(x) <-> 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) --> P'(x), \<lambda>a x. Q(a,x) --> 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) <-> P'(x), \<lambda>a x. Q(a,x) <-> 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) --> 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) <-> 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) <-> 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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syntax
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    "3" :: i   ("3")
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    "4" :: i   ("4")
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    "5" :: i   ("5")
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    "6" :: i   ("6")
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    "7" :: i   ("7")
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    "8" :: i   ("8")
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    "9" :: i   ("9")
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translations
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   "3"  == "succ(2)"
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   "4"  == "succ(3)"
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   "5"  == "succ(4)"
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   "6"  == "succ(5)"
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   "7"  == "succ(6)"
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   "8"  == "succ(7)"
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   "9"  == "succ(8)"
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subsubsection{*The Empty Set, Internalized*}
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constdefs empty_fm :: "i=>i"
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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) <-> 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) <-> 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) <-> 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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subsubsection{*Unordered Pairs, Internalized*}
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constdefs upair_fm :: "[i,i,i]=>i"
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    "upair_fm(x,y,z) ==
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       And(Member(x,z),
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           And(Member(y,z),
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               Forall(Implies(Member(0,succ(z)),
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                              Or(Equal(0,succ(x)), Equal(0,succ(y)))))))"
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   334
lemma upair_type [TC]:
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   335
     "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> upair_fm(x,y,z) \<in> formula"
13429
wenzelm
parents: 13428
diff changeset
   336
by (simp add: upair_fm_def)
13298
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   337
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   338
lemma sats_upair_fm [simp]:
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   339
   "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
13429
wenzelm
parents: 13428
diff changeset
   340
    ==> sats(A, upair_fm(x,y,z), env) <->
13807
a28a8fbc76d4 changed ** to ## to avoid conflict with new comment syntax
paulson
parents: 13655
diff changeset
   341
            upair(##A, nth(x,env), nth(y,env), nth(z,env))"
13298
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   342
by (simp add: upair_fm_def upair_def)
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   343
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   344
lemma upair_iff_sats:
13429
wenzelm
parents: 13428
diff changeset
   345
      "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z;
13298
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   346
          i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
13807
a28a8fbc76d4 changed ** to ## to avoid conflict with new comment syntax
paulson
parents: 13655
diff changeset
   347
       ==> upair(##A, x, y, z) <-> sats(A, upair_fm(i,j,k), env)"
13298
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   348
by (simp add: sats_upair_fm)
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   349
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   350
text{*Useful? At least it refers to "real" unordered pairs*}
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   351
lemma sats_upair_fm2 [simp]:
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   352
   "[| x \<in> nat; y \<in> nat; z < length(env); env \<in> list(A); Transset(A)|]
13429
wenzelm
parents: 13428
diff changeset
   353
    ==> sats(A, upair_fm(x,y,z), env) <->
13298
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   354
        nth(z,env) = {nth(x,env), nth(y,env)}"
13429
wenzelm
parents: 13428
diff changeset
   355
apply (frule lt_length_in_nat, assumption)
wenzelm
parents: 13428
diff changeset
   356
apply (simp add: upair_fm_def Transset_def, auto)
wenzelm
parents: 13428
diff changeset
   357
apply (blast intro: nth_type)
13298
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   358
done
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   359
13314
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   360
theorem upair_reflection:
13429
wenzelm
parents: 13428
diff changeset
   361
     "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
   362
               \<lambda>i x. upair(##Lset(i),f(x),g(x),h(x))]"
13314
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   363
apply (simp add: upair_def)
13429
wenzelm
parents: 13428
diff changeset
   364
apply (intro FOL_reflections)
13314
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   365
done
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   366
13339
0f89104dd377 Fixed quantified variable name preservation for ball and bex (bounded quants)
paulson
parents: 13323
diff changeset
   367
subsubsection{*Ordered pairs, Internalized*}
13298
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   368
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   369
constdefs pair_fm :: "[i,i,i]=>i"
13429
wenzelm
parents: 13428
diff changeset
   370
    "pair_fm(x,y,z) ==
13298
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   371
       Exists(And(upair_fm(succ(x),succ(x),0),
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   372
              Exists(And(upair_fm(succ(succ(x)),succ(succ(y)),0),
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   373
                         upair_fm(1,0,succ(succ(z)))))))"
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   374
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   375
lemma pair_type [TC]:
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   376
     "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> pair_fm(x,y,z) \<in> formula"
13429
wenzelm
parents: 13428
diff changeset
   377
by (simp add: pair_fm_def)
13298
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   378
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   379
lemma sats_pair_fm [simp]:
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   380
   "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
13429
wenzelm
parents: 13428
diff changeset
   381
    ==> sats(A, pair_fm(x,y,z), env) <->
13807
a28a8fbc76d4 changed ** to ## to avoid conflict with new comment syntax
paulson
parents: 13655
diff changeset
   382
        pair(##A, nth(x,env), nth(y,env), nth(z,env))"
13298
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   383
by (simp add: pair_fm_def pair_def)
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   384
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   385
lemma pair_iff_sats:
13429
wenzelm
parents: 13428
diff changeset
   386
      "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z;
13298
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   387
          i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
13807
a28a8fbc76d4 changed ** to ## to avoid conflict with new comment syntax
paulson
parents: 13655
diff changeset
   388
       ==> pair(##A, x, y, z) <-> sats(A, pair_fm(i,j,k), env)"
13298
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   389
by (simp add: sats_pair_fm)
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   390
13314
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   391
theorem pair_reflection:
13429
wenzelm
parents: 13428
diff changeset
   392
     "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
   393
               \<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
   394
apply (simp only: pair_def)
13429
wenzelm
parents: 13428
diff changeset
   395
apply (intro FOL_reflections upair_reflection)
13314
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   396
done
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   397
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   398
13339
0f89104dd377 Fixed quantified variable name preservation for ball and bex (bounded quants)
paulson
parents: 13323
diff changeset
   399
subsubsection{*Binary Unions, Internalized*}
13298
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   400
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   401
constdefs union_fm :: "[i,i,i]=>i"
13429
wenzelm
parents: 13428
diff changeset
   402
    "union_fm(x,y,z) ==
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   403
       Forall(Iff(Member(0,succ(z)),
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   404
                  Or(Member(0,succ(x)),Member(0,succ(y)))))"
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
lemma union_type [TC]:
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   407
     "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> union_fm(x,y,z) \<in> formula"
13429
wenzelm
parents: 13428
diff changeset
   408
by (simp add: union_fm_def)
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   409
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   410
lemma sats_union_fm [simp]:
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   411
   "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
13429
wenzelm
parents: 13428
diff changeset
   412
    ==> sats(A, union_fm(x,y,z), env) <->
13807
a28a8fbc76d4 changed ** to ## to avoid conflict with new comment syntax
paulson
parents: 13655
diff changeset
   413
        union(##A, nth(x,env), nth(y,env), nth(z,env))"
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   414
by (simp add: union_fm_def union_def)
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   415
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   416
lemma union_iff_sats:
13429
wenzelm
parents: 13428
diff changeset
   417
      "[| 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
   418
          i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
13807
a28a8fbc76d4 changed ** to ## to avoid conflict with new comment syntax
paulson
parents: 13655
diff changeset
   419
       ==> union(##A, x, y, z) <-> sats(A, union_fm(i,j,k), env)"
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   420
by (simp add: sats_union_fm)
13298
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   421
13314
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   422
theorem union_reflection:
13429
wenzelm
parents: 13428
diff changeset
   423
     "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
   424
               \<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
   425
apply (simp only: union_def)
13429
wenzelm
parents: 13428
diff changeset
   426
apply (intro FOL_reflections)
13314
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   427
done
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   428
13298
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   429
13339
0f89104dd377 Fixed quantified variable name preservation for ball and bex (bounded quants)
paulson
parents: 13323
diff changeset
   430
subsubsection{*Set ``Cons,'' Internalized*}
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   431
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   432
constdefs cons_fm :: "[i,i,i]=>i"
13429
wenzelm
parents: 13428
diff changeset
   433
    "cons_fm(x,y,z) ==
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   434
       Exists(And(upair_fm(succ(x),succ(x),0),
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   435
                  union_fm(0,succ(y),succ(z))))"
13298
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   436
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   437
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   438
lemma cons_type [TC]:
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   439
     "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> cons_fm(x,y,z) \<in> formula"
13429
wenzelm
parents: 13428
diff changeset
   440
by (simp add: cons_fm_def)
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   441
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   442
lemma sats_cons_fm [simp]:
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   443
   "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
13429
wenzelm
parents: 13428
diff changeset
   444
    ==> sats(A, cons_fm(x,y,z), env) <->
13807
a28a8fbc76d4 changed ** to ## to avoid conflict with new comment syntax
paulson
parents: 13655
diff changeset
   445
        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
   446
by (simp add: cons_fm_def is_cons_def)
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   447
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   448
lemma cons_iff_sats:
13429
wenzelm
parents: 13428
diff changeset
   449
      "[| 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
   450
          i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
13807
a28a8fbc76d4 changed ** to ## to avoid conflict with new comment syntax
paulson
parents: 13655
diff changeset
   451
       ==> is_cons(##A, x, y, z) <-> sats(A, cons_fm(i,j,k), env)"
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   452
by simp
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   453
13314
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   454
theorem cons_reflection:
13429
wenzelm
parents: 13428
diff changeset
   455
     "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
   456
               \<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
   457
apply (simp only: is_cons_def)
13429
wenzelm
parents: 13428
diff changeset
   458
apply (intro FOL_reflections upair_reflection union_reflection)
13323
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   459
done
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   460
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   461
13339
0f89104dd377 Fixed quantified variable name preservation for ball and bex (bounded quants)
paulson
parents: 13323
diff changeset
   462
subsubsection{*Successor Function, Internalized*}
13323
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   463
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   464
constdefs succ_fm :: "[i,i]=>i"
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   465
    "succ_fm(x,y) == cons_fm(x,x,y)"
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   466
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   467
lemma succ_type [TC]:
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   468
     "[| x \<in> nat; y \<in> nat |] ==> succ_fm(x,y) \<in> formula"
13429
wenzelm
parents: 13428
diff changeset
   469
by (simp add: succ_fm_def)
13323
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
lemma sats_succ_fm [simp]:
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   472
   "[| x \<in> nat; y \<in> nat; env \<in> list(A)|]
13429
wenzelm
parents: 13428
diff changeset
   473
    ==> sats(A, succ_fm(x,y), env) <->
13807
a28a8fbc76d4 changed ** to ## to avoid conflict with new comment syntax
paulson
parents: 13655
diff changeset
   474
        successor(##A, nth(x,env), nth(y,env))"
13323
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   475
by (simp add: succ_fm_def successor_def)
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   476
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   477
lemma successor_iff_sats:
13429
wenzelm
parents: 13428
diff changeset
   478
      "[| nth(i,env) = x; nth(j,env) = y;
13323
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   479
          i \<in> nat; j \<in> nat; env \<in> list(A)|]
13807
a28a8fbc76d4 changed ** to ## to avoid conflict with new comment syntax
paulson
parents: 13655
diff changeset
   480
       ==> successor(##A, x, y) <-> sats(A, succ_fm(i,j), env)"
13323
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   481
by simp
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   482
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   483
theorem successor_reflection:
13429
wenzelm
parents: 13428
diff changeset
   484
     "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
   485
               \<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
   486
apply (simp only: successor_def)
13429
wenzelm
parents: 13428
diff changeset
   487
apply (intro cons_reflection)
13314
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   488
done
13298
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   489
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   490
13363
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   491
subsubsection{*The Number 1, Internalized*}
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   492
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   493
(* "number1(M,a) == (\<exists>x[M]. empty(M,x) & successor(M,x,a))" *)
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   494
constdefs number1_fm :: "i=>i"
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   495
    "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
   496
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   497
lemma number1_type [TC]:
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   498
     "x \<in> nat ==> number1_fm(x) \<in> formula"
13429
wenzelm
parents: 13428
diff changeset
   499
by (simp add: number1_fm_def)
13363
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   500
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   501
lemma sats_number1_fm [simp]:
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   502
   "[| x \<in> nat; env \<in> list(A)|]
13807
a28a8fbc76d4 changed ** to ## to avoid conflict with new comment syntax
paulson
parents: 13655
diff changeset
   503
    ==> sats(A, number1_fm(x), env) <-> number1(##A, nth(x,env))"
13363
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   504
by (simp add: number1_fm_def number1_def)
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   505
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   506
lemma number1_iff_sats:
13429
wenzelm
parents: 13428
diff changeset
   507
      "[| nth(i,env) = x; nth(j,env) = y;
13363
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   508
          i \<in> nat; env \<in> list(A)|]
13807
a28a8fbc76d4 changed ** to ## to avoid conflict with new comment syntax
paulson
parents: 13655
diff changeset
   509
       ==> number1(##A, x) <-> sats(A, number1_fm(i), env)"
13363
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   510
by simp
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   511
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   512
theorem number1_reflection:
13429
wenzelm
parents: 13428
diff changeset
   513
     "REFLECTS[\<lambda>x. number1(L,f(x)),
13807
a28a8fbc76d4 changed ** to ## to avoid conflict with new comment syntax
paulson
parents: 13655
diff changeset
   514
               \<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
   515
apply (simp only: number1_def)
13363
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   516
apply (intro FOL_reflections empty_reflection successor_reflection)
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   517
done
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   518
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   519
13352
3cd767f8d78b new definitions of fun_apply and M_is_recfun
paulson
parents: 13348
diff changeset
   520
subsubsection{*Big Union, Internalized*}
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   521
13352
3cd767f8d78b new definitions of fun_apply and M_is_recfun
paulson
parents: 13348
diff changeset
   522
(*  "big_union(M,A,z) == \<forall>x[M]. x \<in> z <-> (\<exists>y[M]. y\<in>A & x \<in> y)" *)
3cd767f8d78b new definitions of fun_apply and M_is_recfun
paulson
parents: 13348
diff changeset
   523
constdefs big_union_fm :: "[i,i]=>i"
13429
wenzelm
parents: 13428
diff changeset
   524
    "big_union_fm(A,z) ==
13352
3cd767f8d78b new definitions of fun_apply and M_is_recfun
paulson
parents: 13348
diff changeset
   525
       Forall(Iff(Member(0,succ(z)),
3cd767f8d78b new definitions of fun_apply and M_is_recfun
paulson
parents: 13348
diff changeset
   526
                  Exists(And(Member(0,succ(succ(A))), Member(1,0)))))"
13298
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   527
13352
3cd767f8d78b new definitions of fun_apply and M_is_recfun
paulson
parents: 13348
diff changeset
   528
lemma big_union_type [TC]:
3cd767f8d78b new definitions of fun_apply and M_is_recfun
paulson
parents: 13348
diff changeset
   529
     "[| x \<in> nat; y \<in> nat |] ==> big_union_fm(x,y) \<in> formula"
13429
wenzelm
parents: 13428
diff changeset
   530
by (simp add: big_union_fm_def)
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   531
13352
3cd767f8d78b new definitions of fun_apply and M_is_recfun
paulson
parents: 13348
diff changeset
   532
lemma sats_big_union_fm [simp]:
3cd767f8d78b new definitions of fun_apply and M_is_recfun
paulson
parents: 13348
diff changeset
   533
   "[| x \<in> nat; y \<in> nat; env \<in> list(A)|]
13429
wenzelm
parents: 13428
diff changeset
   534
    ==> sats(A, big_union_fm(x,y), env) <->
13807
a28a8fbc76d4 changed ** to ## to avoid conflict with new comment syntax
paulson
parents: 13655
diff changeset
   535
        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
   536
by (simp add: big_union_fm_def big_union_def)
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   537
13352
3cd767f8d78b new definitions of fun_apply and M_is_recfun
paulson
parents: 13348
diff changeset
   538
lemma big_union_iff_sats:
13429
wenzelm
parents: 13428
diff changeset
   539
      "[| nth(i,env) = x; nth(j,env) = y;
13352
3cd767f8d78b new definitions of fun_apply and M_is_recfun
paulson
parents: 13348
diff changeset
   540
          i \<in> nat; j \<in> nat; env \<in> list(A)|]
13807
a28a8fbc76d4 changed ** to ## to avoid conflict with new comment syntax
paulson
parents: 13655
diff changeset
   541
       ==> big_union(##A, x, y) <-> sats(A, big_union_fm(i,j), env)"
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   542
by simp
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   543
13352
3cd767f8d78b new definitions of fun_apply and M_is_recfun
paulson
parents: 13348
diff changeset
   544
theorem big_union_reflection:
13429
wenzelm
parents: 13428
diff changeset
   545
     "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
   546
               \<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
   547
apply (simp only: big_union_def)
13429
wenzelm
parents: 13428
diff changeset
   548
apply (intro FOL_reflections)
13314
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   549
done
13298
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   550
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   551
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   552
subsubsection{*Variants of Satisfaction Definitions for Ordinals, etc.*}
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   553
13651
ac80e101306a Cosmetic changes suggested by writing the paper. Deleted some
paulson
parents: 13634
diff changeset
   554
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
   555
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
   556
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
   557
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
   558
@{text M_trivial}, we no longer require the earlier versions.*}
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   559
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   560
lemma sats_subset_fm':
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   561
   "[|x \<in> nat; y \<in> nat; env \<in> list(A)|]
13807
a28a8fbc76d4 changed ** to ## to avoid conflict with new comment syntax
paulson
parents: 13655
diff changeset
   562
    ==> sats(A, subset_fm(x,y), env) <-> subset(##A, nth(x,env), nth(y,env))"
13429
wenzelm
parents: 13428
diff changeset
   563
by (simp add: subset_fm_def Relative.subset_def)
13298
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   564
13314
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   565
theorem subset_reflection:
13429
wenzelm
parents: 13428
diff changeset
   566
     "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
   567
               \<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
   568
apply (simp only: Relative.subset_def)
13429
wenzelm
parents: 13428
diff changeset
   569
apply (intro FOL_reflections)
13314
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   570
done
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   571
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   572
lemma sats_transset_fm':
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   573
   "[|x \<in> nat; env \<in> list(A)|]
13807
a28a8fbc76d4 changed ** to ## to avoid conflict with new comment syntax
paulson
parents: 13655
diff changeset
   574
    ==> sats(A, transset_fm(x), env) <-> transitive_set(##A, nth(x,env))"
13429
wenzelm
parents: 13428
diff changeset
   575
by (simp add: sats_subset_fm' transset_fm_def transitive_set_def)
13298
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   576
13314
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   577
theorem transitive_set_reflection:
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   578
     "REFLECTS[\<lambda>x. transitive_set(L,f(x)),
13807
a28a8fbc76d4 changed ** to ## to avoid conflict with new comment syntax
paulson
parents: 13655
diff changeset
   579
               \<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
   580
apply (simp only: transitive_set_def)
13429
wenzelm
parents: 13428
diff changeset
   581
apply (intro FOL_reflections subset_reflection)
13314
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   582
done
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   583
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   584
lemma sats_ordinal_fm':
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   585
   "[|x \<in> nat; env \<in> list(A)|]
13807
a28a8fbc76d4 changed ** to ## to avoid conflict with new comment syntax
paulson
parents: 13655
diff changeset
   586
    ==> sats(A, ordinal_fm(x), env) <-> ordinal(##A,nth(x,env))"
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   587
by (simp add: sats_transset_fm' ordinal_fm_def ordinal_def)
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   588
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   589
lemma ordinal_iff_sats:
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   590
      "[| nth(i,env) = x;  i \<in> nat; env \<in> list(A)|]
13807
a28a8fbc76d4 changed ** to ## to avoid conflict with new comment syntax
paulson
parents: 13655
diff changeset
   591
       ==> ordinal(##A, x) <-> sats(A, ordinal_fm(i), env)"
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   592
by (simp add: sats_ordinal_fm')
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   593
13314
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   594
theorem ordinal_reflection:
13807
a28a8fbc76d4 changed ** to ## to avoid conflict with new comment syntax
paulson
parents: 13655
diff changeset
   595
     "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
   596
apply (simp only: ordinal_def)
13429
wenzelm
parents: 13428
diff changeset
   597
apply (intro FOL_reflections transitive_set_reflection)
13314
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   598
done
13298
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   599
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   600
13339
0f89104dd377 Fixed quantified variable name preservation for ball and bex (bounded quants)
paulson
parents: 13323
diff changeset
   601
subsubsection{*Membership Relation, Internalized*}
13298
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   602
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   603
constdefs Memrel_fm :: "[i,i]=>i"
13429
wenzelm
parents: 13428
diff changeset
   604
    "Memrel_fm(A,r) ==
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   605
       Forall(Iff(Member(0,succ(r)),
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   606
                  Exists(And(Member(0,succ(succ(A))),
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   607
                             Exists(And(Member(0,succ(succ(succ(A)))),
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   608
                                        And(Member(1,0),
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   609
                                            pair_fm(1,0,2))))))))"
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   610
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   611
lemma Memrel_type [TC]:
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   612
     "[| x \<in> nat; y \<in> nat |] ==> Memrel_fm(x,y) \<in> formula"
13429
wenzelm
parents: 13428
diff changeset
   613
by (simp add: Memrel_fm_def)
13298
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   614
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   615
lemma sats_Memrel_fm [simp]:
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   616
   "[| x \<in> nat; y \<in> nat; env \<in> list(A)|]
13429
wenzelm
parents: 13428
diff changeset
   617
    ==> sats(A, Memrel_fm(x,y), env) <->
13807
a28a8fbc76d4 changed ** to ## to avoid conflict with new comment syntax
paulson
parents: 13655
diff changeset
   618
        membership(##A, nth(x,env), nth(y,env))"
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   619
by (simp add: Memrel_fm_def membership_def)
13298
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   620
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   621
lemma Memrel_iff_sats:
13429
wenzelm
parents: 13428
diff changeset
   622
      "[| nth(i,env) = x; nth(j,env) = y;
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   623
          i \<in> nat; j \<in> nat; env \<in> list(A)|]
13807
a28a8fbc76d4 changed ** to ## to avoid conflict with new comment syntax
paulson
parents: 13655
diff changeset
   624
       ==> membership(##A, x, y) <-> sats(A, Memrel_fm(i,j), env)"
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   625
by simp
13304
43ef6c6dd906 more separation instances
paulson
parents: 13299
diff changeset
   626
13314
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   627
theorem membership_reflection:
13429
wenzelm
parents: 13428
diff changeset
   628
     "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
   629
               \<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
   630
apply (simp only: membership_def)
13429
wenzelm
parents: 13428
diff changeset
   631
apply (intro FOL_reflections pair_reflection)
13314
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   632
done
13304
43ef6c6dd906 more separation instances
paulson
parents: 13299
diff changeset
   633
13339
0f89104dd377 Fixed quantified variable name preservation for ball and bex (bounded quants)
paulson
parents: 13323
diff changeset
   634
subsubsection{*Predecessor Set, Internalized*}
13304
43ef6c6dd906 more separation instances
paulson
parents: 13299
diff changeset
   635
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   636
constdefs pred_set_fm :: "[i,i,i,i]=>i"
13429
wenzelm
parents: 13428
diff changeset
   637
    "pred_set_fm(A,x,r,B) ==
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   638
       Forall(Iff(Member(0,succ(B)),
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   639
                  Exists(And(Member(0,succ(succ(r))),
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   640
                             And(Member(1,succ(succ(A))),
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   641
                                 pair_fm(1,succ(succ(x)),0))))))"
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   642
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   643
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   644
lemma pred_set_type [TC]:
13429
wenzelm
parents: 13428
diff changeset
   645
     "[| 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
   646
      ==> pred_set_fm(A,x,r,B) \<in> formula"
13429
wenzelm
parents: 13428
diff changeset
   647
by (simp add: pred_set_fm_def)
13304
43ef6c6dd906 more separation instances
paulson
parents: 13299
diff changeset
   648
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   649
lemma sats_pred_set_fm [simp]:
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   650
   "[| U \<in> nat; x \<in> nat; r \<in> nat; B \<in> nat; env \<in> list(A)|]
13429
wenzelm
parents: 13428
diff changeset
   651
    ==> sats(A, pred_set_fm(U,x,r,B), env) <->
13807
a28a8fbc76d4 changed ** to ## to avoid conflict with new comment syntax
paulson
parents: 13655
diff changeset
   652
        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
   653
by (simp add: pred_set_fm_def pred_set_def)
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   654
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   655
lemma pred_set_iff_sats:
13429
wenzelm
parents: 13428
diff changeset
   656
      "[| 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
   657
          i \<in> nat; j \<in> nat; k \<in> nat; l \<in> nat; env \<in> list(A)|]
13807
a28a8fbc76d4 changed ** to ## to avoid conflict with new comment syntax
paulson
parents: 13655
diff changeset
   658
       ==> pred_set(##A,U,x,r,B) <-> sats(A, pred_set_fm(i,j,k,l), env)"
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   659
by (simp add: sats_pred_set_fm)
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   660
13314
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   661
theorem pred_set_reflection:
13429
wenzelm
parents: 13428
diff changeset
   662
     "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
   663
               \<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
   664
apply (simp only: pred_set_def)
13429
wenzelm
parents: 13428
diff changeset
   665
apply (intro FOL_reflections pair_reflection)
13314
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   666
done
13304
43ef6c6dd906 more separation instances
paulson
parents: 13299
diff changeset
   667
43ef6c6dd906 more separation instances
paulson
parents: 13299
diff changeset
   668
13298
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   669
13339
0f89104dd377 Fixed quantified variable name preservation for ball and bex (bounded quants)
paulson
parents: 13323
diff changeset
   670
subsubsection{*Domain of a Relation, Internalized*}
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   671
13429
wenzelm
parents: 13428
diff changeset
   672
(* "is_domain(M,r,z) ==
wenzelm
parents: 13428
diff changeset
   673
        \<forall>x[M]. (x \<in> z <-> (\<exists>w[M]. w\<in>r & (\<exists>y[M]. pair(M,x,y,w))))" *)
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   674
constdefs domain_fm :: "[i,i]=>i"
13429
wenzelm
parents: 13428
diff changeset
   675
    "domain_fm(r,z) ==
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   676
       Forall(Iff(Member(0,succ(z)),
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   677
                  Exists(And(Member(0,succ(succ(r))),
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   678
                             Exists(pair_fm(2,0,1))))))"
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   679
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   680
lemma domain_type [TC]:
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   681
     "[| x \<in> nat; y \<in> nat |] ==> domain_fm(x,y) \<in> formula"
13429
wenzelm
parents: 13428
diff changeset
   682
by (simp add: domain_fm_def)
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   683
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   684
lemma sats_domain_fm [simp]:
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   685
   "[| x \<in> nat; y \<in> nat; env \<in> list(A)|]
13429
wenzelm
parents: 13428
diff changeset
   686
    ==> sats(A, domain_fm(x,y), env) <->
13807
a28a8fbc76d4 changed ** to ## to avoid conflict with new comment syntax
paulson
parents: 13655
diff changeset
   687
        is_domain(##A, nth(x,env), nth(y,env))"
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   688
by (simp add: domain_fm_def is_domain_def)
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   689
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   690
lemma domain_iff_sats:
13429
wenzelm
parents: 13428
diff changeset
   691
      "[| nth(i,env) = x; nth(j,env) = y;
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   692
          i \<in> nat; j \<in> nat; env \<in> list(A)|]
13807
a28a8fbc76d4 changed ** to ## to avoid conflict with new comment syntax
paulson
parents: 13655
diff changeset
   693
       ==> is_domain(##A, x, y) <-> sats(A, domain_fm(i,j), env)"
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   694
by simp
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   695
13314
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   696
theorem domain_reflection:
13429
wenzelm
parents: 13428
diff changeset
   697
     "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
   698
               \<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
   699
apply (simp only: is_domain_def)
13429
wenzelm
parents: 13428
diff changeset
   700
apply (intro FOL_reflections pair_reflection)
13314
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   701
done
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   702
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   703
13339
0f89104dd377 Fixed quantified variable name preservation for ball and bex (bounded quants)
paulson
parents: 13323
diff changeset
   704
subsubsection{*Range of a Relation, Internalized*}
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   705
13429
wenzelm
parents: 13428
diff changeset
   706
(* "is_range(M,r,z) ==
wenzelm
parents: 13428
diff changeset
   707
        \<forall>y[M]. (y \<in> z <-> (\<exists>w[M]. w\<in>r & (\<exists>x[M]. pair(M,x,y,w))))" *)
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   708
constdefs range_fm :: "[i,i]=>i"
13429
wenzelm
parents: 13428
diff changeset
   709
    "range_fm(r,z) ==
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   710
       Forall(Iff(Member(0,succ(z)),
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   711
                  Exists(And(Member(0,succ(succ(r))),
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   712
                             Exists(pair_fm(0,2,1))))))"
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   713
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   714
lemma range_type [TC]:
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   715
     "[| x \<in> nat; y \<in> nat |] ==> range_fm(x,y) \<in> formula"
13429
wenzelm
parents: 13428
diff changeset
   716
by (simp add: range_fm_def)
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   717
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   718
lemma sats_range_fm [simp]:
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   719
   "[| x \<in> nat; y \<in> nat; env \<in> list(A)|]
13429
wenzelm
parents: 13428
diff changeset
   720
    ==> sats(A, range_fm(x,y), env) <->
13807
a28a8fbc76d4 changed ** to ## to avoid conflict with new comment syntax
paulson
parents: 13655
diff changeset
   721
        is_range(##A, nth(x,env), nth(y,env))"
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   722
by (simp add: range_fm_def is_range_def)
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   723
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   724
lemma range_iff_sats:
13429
wenzelm
parents: 13428
diff changeset
   725
      "[| nth(i,env) = x; nth(j,env) = y;
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   726
          i \<in> nat; j \<in> nat; env \<in> list(A)|]
13807
a28a8fbc76d4 changed ** to ## to avoid conflict with new comment syntax
paulson
parents: 13655
diff changeset
   727
       ==> is_range(##A, x, y) <-> sats(A, range_fm(i,j), env)"
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   728
by simp
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   729
13314
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   730
theorem range_reflection:
13429
wenzelm
parents: 13428
diff changeset
   731
     "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
   732
               \<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
   733
apply (simp only: is_range_def)
13429
wenzelm
parents: 13428
diff changeset
   734
apply (intro FOL_reflections pair_reflection)
13314
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   735
done
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   736
13429
wenzelm
parents: 13428
diff changeset
   737
13339
0f89104dd377 Fixed quantified variable name preservation for ball and bex (bounded quants)
paulson
parents: 13323
diff changeset
   738
subsubsection{*Field of a Relation, Internalized*}
13323
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   739
13429
wenzelm
parents: 13428
diff changeset
   740
(* "is_field(M,r,z) ==
wenzelm
parents: 13428
diff changeset
   741
        \<exists>dr[M]. is_domain(M,r,dr) &
13323
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   742
            (\<exists>rr[M]. is_range(M,r,rr) & union(M,dr,rr,z))" *)
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   743
constdefs field_fm :: "[i,i]=>i"
13429
wenzelm
parents: 13428
diff changeset
   744
    "field_fm(r,z) ==
wenzelm
parents: 13428
diff changeset
   745
       Exists(And(domain_fm(succ(r),0),
wenzelm
parents: 13428
diff changeset
   746
              Exists(And(range_fm(succ(succ(r)),0),
13323
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   747
                         union_fm(1,0,succ(succ(z)))))))"
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   748
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   749
lemma field_type [TC]:
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   750
     "[| x \<in> nat; y \<in> nat |] ==> field_fm(x,y) \<in> formula"
13429
wenzelm
parents: 13428
diff changeset
   751
by (simp add: field_fm_def)
13323
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   752
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   753
lemma sats_field_fm [simp]:
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   754
   "[| x \<in> nat; y \<in> nat; env \<in> list(A)|]
13429
wenzelm
parents: 13428
diff changeset
   755
    ==> sats(A, field_fm(x,y), env) <->
13807
a28a8fbc76d4 changed ** to ## to avoid conflict with new comment syntax
paulson
parents: 13655
diff changeset
   756
        is_field(##A, nth(x,env), nth(y,env))"
13323
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   757
by (simp add: field_fm_def is_field_def)
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   758
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   759
lemma field_iff_sats:
13429
wenzelm
parents: 13428
diff changeset
   760
      "[| nth(i,env) = x; nth(j,env) = y;
13323
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   761
          i \<in> nat; j \<in> nat; env \<in> list(A)|]
13807
a28a8fbc76d4 changed ** to ## to avoid conflict with new comment syntax
paulson
parents: 13655
diff changeset
   762
       ==> is_field(##A, x, y) <-> sats(A, field_fm(i,j), env)"
13323
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   763
by simp
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   764
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   765
theorem field_reflection:
13429
wenzelm
parents: 13428
diff changeset
   766
     "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
   767
               \<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
   768
apply (simp only: is_field_def)
13323
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   769
apply (intro FOL_reflections domain_reflection range_reflection
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   770
             union_reflection)
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   771
done
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   772
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   773
13339
0f89104dd377 Fixed quantified variable name preservation for ball and bex (bounded quants)
paulson
parents: 13323
diff changeset
   774
subsubsection{*Image under a Relation, Internalized*}
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   775
13429
wenzelm
parents: 13428
diff changeset
   776
(* "image(M,r,A,z) ==
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   777
        \<forall>y[M]. (y \<in> z <-> (\<exists>w[M]. w\<in>r & (\<exists>x[M]. x\<in>A & pair(M,x,y,w))))" *)
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   778
constdefs image_fm :: "[i,i,i]=>i"
13429
wenzelm
parents: 13428
diff changeset
   779
    "image_fm(r,A,z) ==
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   780
       Forall(Iff(Member(0,succ(z)),
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   781
                  Exists(And(Member(0,succ(succ(r))),
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   782
                             Exists(And(Member(0,succ(succ(succ(A)))),
13429
wenzelm
parents: 13428
diff changeset
   783
                                        pair_fm(0,2,1)))))))"
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   784
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   785
lemma image_type [TC]:
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   786
     "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> image_fm(x,y,z) \<in> formula"
13429
wenzelm
parents: 13428
diff changeset
   787
by (simp add: image_fm_def)
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   788
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   789
lemma sats_image_fm [simp]:
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   790
   "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
13429
wenzelm
parents: 13428
diff changeset
   791
    ==> sats(A, image_fm(x,y,z), env) <->
13807
a28a8fbc76d4 changed ** to ## to avoid conflict with new comment syntax
paulson
parents: 13655
diff changeset
   792
        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
   793
by (simp add: image_fm_def Relative.image_def)
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   794
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   795
lemma image_iff_sats:
13429
wenzelm
parents: 13428
diff changeset
   796
      "[| 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
   797
          i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
13807
a28a8fbc76d4 changed ** to ## to avoid conflict with new comment syntax
paulson
parents: 13655
diff changeset
   798
       ==> image(##A, x, y, z) <-> sats(A, image_fm(i,j,k), env)"
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   799
by (simp add: sats_image_fm)
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   800
13314
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   801
theorem image_reflection:
13429
wenzelm
parents: 13428
diff changeset
   802
     "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
   803
               \<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
   804
apply (simp only: Relative.image_def)
13429
wenzelm
parents: 13428
diff changeset
   805
apply (intro FOL_reflections pair_reflection)
13314
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   806
done
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
13348
374d05460db4 Separation/Replacement up to M_wfrank!
paulson
parents: 13339
diff changeset
   809
subsubsection{*Pre-Image under a Relation, Internalized*}
374d05460db4 Separation/Replacement up to M_wfrank!
paulson
parents: 13339
diff changeset
   810
13429
wenzelm
parents: 13428
diff changeset
   811
(* "pre_image(M,r,A,z) ==
wenzelm
parents: 13428
diff changeset
   812
        \<forall>x[M]. x \<in> z <-> (\<exists>w[M]. w\<in>r & (\<exists>y[M]. y\<in>A & pair(M,x,y,w)))" *)
13348
374d05460db4 Separation/Replacement up to M_wfrank!
paulson
parents: 13339
diff changeset
   813
constdefs pre_image_fm :: "[i,i,i]=>i"
13429
wenzelm
parents: 13428
diff changeset
   814
    "pre_image_fm(r,A,z) ==
13348
374d05460db4 Separation/Replacement up to M_wfrank!
paulson
parents: 13339
diff changeset
   815
       Forall(Iff(Member(0,succ(z)),
374d05460db4 Separation/Replacement up to M_wfrank!
paulson
parents: 13339
diff changeset
   816
                  Exists(And(Member(0,succ(succ(r))),
374d05460db4 Separation/Replacement up to M_wfrank!
paulson
parents: 13339
diff changeset
   817
                             Exists(And(Member(0,succ(succ(succ(A)))),
13429
wenzelm
parents: 13428
diff changeset
   818
                                        pair_fm(2,0,1)))))))"
13348
374d05460db4 Separation/Replacement up to M_wfrank!
paulson
parents: 13339
diff changeset
   819
374d05460db4 Separation/Replacement up to M_wfrank!
paulson
parents: 13339
diff changeset
   820
lemma pre_image_type [TC]:
374d05460db4 Separation/Replacement up to M_wfrank!
paulson
parents: 13339
diff changeset
   821
     "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> pre_image_fm(x,y,z) \<in> formula"
13429
wenzelm
parents: 13428
diff changeset
   822
by (simp add: pre_image_fm_def)
13348
374d05460db4 Separation/Replacement up to M_wfrank!
paulson
parents: 13339
diff changeset
   823
374d05460db4 Separation/Replacement up to M_wfrank!
paulson
parents: 13339
diff changeset
   824
lemma sats_pre_image_fm [simp]:
374d05460db4 Separation/Replacement up to M_wfrank!
paulson
parents: 13339
diff changeset
   825
   "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
13429
wenzelm
parents: 13428
diff changeset
   826
    ==> sats(A, pre_image_fm(x,y,z), env) <->
13807
a28a8fbc76d4 changed ** to ## to avoid conflict with new comment syntax
paulson
parents: 13655
diff changeset
   827
        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
   828
by (simp add: pre_image_fm_def Relative.pre_image_def)
374d05460db4 Separation/Replacement up to M_wfrank!
paulson
parents: 13339
diff changeset
   829
374d05460db4 Separation/Replacement up to M_wfrank!
paulson
parents: 13339
diff changeset
   830
lemma pre_image_iff_sats:
13429
wenzelm
parents: 13428
diff changeset
   831
      "[| 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
   832
          i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
13807
a28a8fbc76d4 changed ** to ## to avoid conflict with new comment syntax
paulson
parents: 13655
diff changeset
   833
       ==> pre_image(##A, x, y, z) <-> sats(A, pre_image_fm(i,j,k), env)"
13348
374d05460db4 Separation/Replacement up to M_wfrank!
paulson
parents: 13339
diff changeset
   834
by (simp add: sats_pre_image_fm)
374d05460db4 Separation/Replacement up to M_wfrank!
paulson
parents: 13339
diff changeset
   835
374d05460db4 Separation/Replacement up to M_wfrank!
paulson
parents: 13339
diff changeset
   836
theorem pre_image_reflection:
13429
wenzelm
parents: 13428
diff changeset
   837
     "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
   838
               \<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
   839
apply (simp only: Relative.pre_image_def)
13429
wenzelm
parents: 13428
diff changeset
   840
apply (intro FOL_reflections pair_reflection)
13348
374d05460db4 Separation/Replacement up to M_wfrank!
paulson
parents: 13339
diff changeset
   841
done
374d05460db4 Separation/Replacement up to M_wfrank!
paulson
parents: 13339
diff changeset
   842
374d05460db4 Separation/Replacement up to M_wfrank!
paulson
parents: 13339
diff changeset
   843
13352
3cd767f8d78b new definitions of fun_apply and M_is_recfun
paulson
parents: 13348
diff changeset
   844
subsubsection{*Function Application, Internalized*}
3cd767f8d78b new definitions of fun_apply and M_is_recfun
paulson
parents: 13348
diff changeset
   845
13429
wenzelm
parents: 13428
diff changeset
   846
(* "fun_apply(M,f,x,y) ==
wenzelm
parents: 13428
diff changeset
   847
        (\<exists>xs[M]. \<exists>fxs[M].
13352
3cd767f8d78b new definitions of fun_apply and M_is_recfun
paulson
parents: 13348
diff changeset
   848
         upair(M,x,x,xs) & image(M,f,xs,fxs) & big_union(M,fxs,y))" *)
3cd767f8d78b new definitions of fun_apply and M_is_recfun
paulson
parents: 13348
diff changeset
   849
constdefs fun_apply_fm :: "[i,i,i]=>i"
13429
wenzelm
parents: 13428
diff changeset
   850
    "fun_apply_fm(f,x,y) ==
13352
3cd767f8d78b new definitions of fun_apply and M_is_recfun
paulson
parents: 13348
diff changeset
   851
       Exists(Exists(And(upair_fm(succ(succ(x)), succ(succ(x)), 1),
13429
wenzelm
parents: 13428
diff changeset
   852
                         And(image_fm(succ(succ(f)), 1, 0),
13352
3cd767f8d78b new definitions of fun_apply and M_is_recfun
paulson
parents: 13348
diff changeset
   853
                             big_union_fm(0,succ(succ(y)))))))"
3cd767f8d78b new definitions of fun_apply and M_is_recfun
paulson
parents: 13348
diff changeset
   854
3cd767f8d78b new definitions of fun_apply and M_is_recfun
paulson
parents: 13348
diff changeset
   855
lemma fun_apply_type [TC]:
3cd767f8d78b new definitions of fun_apply and M_is_recfun
paulson
parents: 13348
diff changeset
   856
     "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> fun_apply_fm(x,y,z) \<in> formula"
13429
wenzelm
parents: 13428
diff changeset
   857
by (simp add: fun_apply_fm_def)
13352
3cd767f8d78b new definitions of fun_apply and M_is_recfun
paulson
parents: 13348
diff changeset
   858
3cd767f8d78b new definitions of fun_apply and M_is_recfun
paulson
parents: 13348
diff changeset
   859
lemma sats_fun_apply_fm [simp]:
3cd767f8d78b new definitions of fun_apply and M_is_recfun
paulson
parents: 13348
diff changeset
   860
   "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
13429
wenzelm
parents: 13428
diff changeset
   861
    ==> sats(A, fun_apply_fm(x,y,z), env) <->
13807
a28a8fbc76d4 changed ** to ## to avoid conflict with new comment syntax
paulson
parents: 13655
diff changeset
   862
        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
   863
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
   864
3cd767f8d78b new definitions of fun_apply and M_is_recfun
paulson
parents: 13348
diff changeset
   865
lemma fun_apply_iff_sats:
13429
wenzelm
parents: 13428
diff changeset
   866
      "[| 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
   867
          i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
13807
a28a8fbc76d4 changed ** to ## to avoid conflict with new comment syntax
paulson
parents: 13655
diff changeset
   868
       ==> fun_apply(##A, x, y, z) <-> 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
   869
by simp
3cd767f8d78b new definitions of fun_apply and M_is_recfun
paulson
parents: 13348
diff changeset
   870
3cd767f8d78b new definitions of fun_apply and M_is_recfun
paulson
parents: 13348
diff changeset
   871
theorem fun_apply_reflection:
13429
wenzelm
parents: 13428
diff changeset
   872
     "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
   873
               \<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
   874
apply (simp only: fun_apply_def)
13352
3cd767f8d78b new definitions of fun_apply and M_is_recfun
paulson
parents: 13348
diff changeset
   875
apply (intro FOL_reflections upair_reflection image_reflection
13429
wenzelm
parents: 13428
diff changeset
   876
             big_union_reflection)
13352
3cd767f8d78b new definitions of fun_apply and M_is_recfun
paulson
parents: 13348
diff changeset
   877
done
3cd767f8d78b new definitions of fun_apply and M_is_recfun
paulson
parents: 13348
diff changeset
   878
3cd767f8d78b new definitions of fun_apply and M_is_recfun
paulson
parents: 13348
diff changeset
   879
13339
0f89104dd377 Fixed quantified variable name preservation for ball and bex (bounded quants)
paulson
parents: 13323
diff changeset
   880
subsubsection{*The Concept of Relation, Internalized*}
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   881
13429
wenzelm
parents: 13428
diff changeset
   882
(* "is_relation(M,r) ==
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   883
        (\<forall>z[M]. z\<in>r --> (\<exists>x[M]. \<exists>y[M]. pair(M,x,y,z)))" *)
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   884
constdefs relation_fm :: "i=>i"
13429
wenzelm
parents: 13428
diff changeset
   885
    "relation_fm(r) ==
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   886
       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
   887
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   888
lemma relation_type [TC]:
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   889
     "[| x \<in> nat |] ==> relation_fm(x) \<in> formula"
13429
wenzelm
parents: 13428
diff changeset
   890
by (simp add: relation_fm_def)
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   891
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   892
lemma sats_relation_fm [simp]:
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   893
   "[| x \<in> nat; env \<in> list(A)|]
13807
a28a8fbc76d4 changed ** to ## to avoid conflict with new comment syntax
paulson
parents: 13655
diff changeset
   894
    ==> sats(A, relation_fm(x), env) <-> is_relation(##A, nth(x,env))"
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   895
by (simp add: relation_fm_def is_relation_def)
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   896
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   897
lemma relation_iff_sats:
13429
wenzelm
parents: 13428
diff changeset
   898
      "[| nth(i,env) = x; nth(j,env) = y;
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   899
          i \<in> nat; env \<in> list(A)|]
13807
a28a8fbc76d4 changed ** to ## to avoid conflict with new comment syntax
paulson
parents: 13655
diff changeset
   900
       ==> is_relation(##A, x) <-> sats(A, relation_fm(i), env)"
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   901
by simp
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   902
13314
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   903
theorem is_relation_reflection:
13429
wenzelm
parents: 13428
diff changeset
   904
     "REFLECTS[\<lambda>x. is_relation(L,f(x)),
13807
a28a8fbc76d4 changed ** to ## to avoid conflict with new comment syntax
paulson
parents: 13655
diff changeset
   905
               \<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
   906
apply (simp only: is_relation_def)
13429
wenzelm
parents: 13428
diff changeset
   907
apply (intro FOL_reflections pair_reflection)
13314
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   908
done
13306
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
13339
0f89104dd377 Fixed quantified variable name preservation for ball and bex (bounded quants)
paulson
parents: 13323
diff changeset
   911
subsubsection{*The Concept of Function, Internalized*}
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   912
13429
wenzelm
parents: 13428
diff changeset
   913
(* "is_function(M,r) ==
wenzelm
parents: 13428
diff changeset
   914
        \<forall>x[M]. \<forall>y[M]. \<forall>y'[M]. \<forall>p[M]. \<forall>p'[M].
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   915
           pair(M,x,y,p) --> pair(M,x,y',p') --> p\<in>r --> p'\<in>r --> y=y'" *)
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   916
constdefs function_fm :: "i=>i"
13429
wenzelm
parents: 13428
diff changeset
   917
    "function_fm(r) ==
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   918
       Forall(Forall(Forall(Forall(Forall(
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   919
         Implies(pair_fm(4,3,1),
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   920
                 Implies(pair_fm(4,2,0),
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   921
                         Implies(Member(1,r#+5),
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   922
                                 Implies(Member(0,r#+5), Equal(3,2))))))))))"
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   923
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   924
lemma function_type [TC]:
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   925
     "[| x \<in> nat |] ==> function_fm(x) \<in> formula"
13429
wenzelm
parents: 13428
diff changeset
   926
by (simp add: function_fm_def)
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   927
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   928
lemma sats_function_fm [simp]:
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   929
   "[| x \<in> nat; env \<in> list(A)|]
13807
a28a8fbc76d4 changed ** to ## to avoid conflict with new comment syntax
paulson
parents: 13655
diff changeset
   930
    ==> sats(A, function_fm(x), env) <-> is_function(##A, nth(x,env))"
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   931
by (simp add: function_fm_def is_function_def)
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   932
13505
52a16cb7fefb Relativized right up to L satisfies V=L!
paulson
parents: 13496
diff changeset
   933
lemma is_function_iff_sats:
13429
wenzelm
parents: 13428
diff changeset
   934
      "[| nth(i,env) = x; nth(j,env) = y;
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   935
          i \<in> nat; env \<in> list(A)|]
13807
a28a8fbc76d4 changed ** to ## to avoid conflict with new comment syntax
paulson
parents: 13655
diff changeset
   936
       ==> is_function(##A, x) <-> sats(A, function_fm(i), env)"
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   937
by simp
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   938
13314
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   939
theorem is_function_reflection:
13429
wenzelm
parents: 13428
diff changeset
   940
     "REFLECTS[\<lambda>x. is_function(L,f(x)),
13807
a28a8fbc76d4 changed ** to ## to avoid conflict with new comment syntax
paulson
parents: 13655
diff changeset
   941
               \<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
   942
apply (simp only: is_function_def)
13429
wenzelm
parents: 13428
diff changeset
   943
apply (intro FOL_reflections pair_reflection)
13314
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   944
done
13298
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   945
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   946
13339
0f89104dd377 Fixed quantified variable name preservation for ball and bex (bounded quants)
paulson
parents: 13323
diff changeset
   947
subsubsection{*Typed Functions, Internalized*}
13309
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
   948
13429
wenzelm
parents: 13428
diff changeset
   949
(* "typed_function(M,A,B,r) ==
13309
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
   950
        is_function(M,r) & is_relation(M,r) & is_domain(M,r,A) &
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
   951
        (\<forall>u[M]. u\<in>r --> (\<forall>x[M]. \<forall>y[M]. pair(M,x,y,u) --> y\<in>B))" *)
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
   952
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
   953
constdefs typed_function_fm :: "[i,i,i]=>i"
13429
wenzelm
parents: 13428
diff changeset
   954
    "typed_function_fm(A,B,r) ==
13309
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
   955
       And(function_fm(r),
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
   956
         And(relation_fm(r),
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
   957
           And(domain_fm(r,A),
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
   958
             Forall(Implies(Member(0,succ(r)),
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
   959
                  Forall(Forall(Implies(pair_fm(1,0,2),Member(0,B#+3)))))))))"
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
   960
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
   961
lemma typed_function_type [TC]:
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
   962
     "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> typed_function_fm(x,y,z) \<in> formula"
13429
wenzelm
parents: 13428
diff changeset
   963
by (simp add: typed_function_fm_def)
13309
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
   964
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
   965
lemma sats_typed_function_fm [simp]:
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
   966
   "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
13429
wenzelm
parents: 13428
diff changeset
   967
    ==> sats(A, typed_function_fm(x,y,z), env) <->
13807
a28a8fbc76d4 changed ** to ## to avoid conflict with new comment syntax
paulson
parents: 13655
diff changeset
   968
        typed_function(##A, nth(x,env), nth(y,env), nth(z,env))"
13309
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
   969
by (simp add: typed_function_fm_def typed_function_def)
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
   970
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
   971
lemma typed_function_iff_sats:
13429
wenzelm
parents: 13428
diff changeset
   972
  "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z;
13309
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
   973
      i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
13807
a28a8fbc76d4 changed ** to ## to avoid conflict with new comment syntax
paulson
parents: 13655
diff changeset
   974
   ==> typed_function(##A, x, y, z) <-> sats(A, typed_function_fm(i,j,k), env)"
13309
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
   975
by simp
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
   976
13429
wenzelm
parents: 13428
diff changeset
   977
lemmas function_reflections =
13363
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   978
        empty_reflection number1_reflection
13429
wenzelm
parents: 13428
diff changeset
   979
        upair_reflection pair_reflection union_reflection
wenzelm
parents: 13428
diff changeset
   980
        big_union_reflection cons_reflection successor_reflection
13323
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   981
        fun_apply_reflection subset_reflection
13429
wenzelm
parents: 13428
diff changeset
   982
        transitive_set_reflection membership_reflection
wenzelm
parents: 13428
diff changeset
   983
        pred_set_reflection domain_reflection range_reflection field_reflection
13348
374d05460db4 Separation/Replacement up to M_wfrank!
paulson
parents: 13339
diff changeset
   984
        image_reflection pre_image_reflection
13429
wenzelm
parents: 13428
diff changeset
   985
        is_relation_reflection is_function_reflection
13309
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
   986
13429
wenzelm
parents: 13428
diff changeset
   987
lemmas function_iff_sats =
wenzelm
parents: 13428
diff changeset
   988
        empty_iff_sats number1_iff_sats
wenzelm
parents: 13428
diff changeset
   989
        upair_iff_sats pair_iff_sats union_iff_sats
13505
52a16cb7fefb Relativized right up to L satisfies V=L!
paulson
parents: 13496
diff changeset
   990
        big_union_iff_sats cons_iff_sats successor_iff_sats
13323
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   991
        fun_apply_iff_sats  Memrel_iff_sats
13429
wenzelm
parents: 13428
diff changeset
   992
        pred_set_iff_sats domain_iff_sats range_iff_sats field_iff_sats
wenzelm
parents: 13428
diff changeset
   993
        image_iff_sats pre_image_iff_sats
13505
52a16cb7fefb Relativized right up to L satisfies V=L!
paulson
parents: 13496
diff changeset
   994
        relation_iff_sats is_function_iff_sats
13323
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   995
13309
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
   996
13314
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   997
theorem typed_function_reflection:
13429
wenzelm
parents: 13428
diff changeset
   998
     "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
   999
               \<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
  1000
apply (simp only: typed_function_def)
13429
wenzelm
parents: 13428
diff changeset
  1001
apply (intro FOL_reflections function_reflections)
13323
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
  1002
done
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
  1003
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
  1004
13339
0f89104dd377 Fixed quantified variable name preservation for ball and bex (bounded quants)
paulson
parents: 13323
diff changeset
  1005
subsubsection{*Composition of Relations, Internalized*}
13323
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
  1006
13429
wenzelm
parents: 13428
diff changeset
  1007
(* "composition(M,r,s,t) ==
wenzelm
parents: 13428
diff changeset
  1008
        \<forall>p[M]. p \<in> t <->
wenzelm
parents: 13428
diff changeset
  1009
               (\<exists>x[M]. \<exists>y[M]. \<exists>z[M]. \<exists>xy[M]. \<exists>yz[M].
wenzelm
parents: 13428
diff changeset
  1010
                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
  1011
                xy \<in> s & yz \<in> r)" *)
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
  1012
constdefs composition_fm :: "[i,i,i]=>i"
13429
wenzelm
parents: 13428
diff changeset
  1013
  "composition_fm(r,s,t) ==
13323
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
  1014
     Forall(Iff(Member(0,succ(t)),
13429
wenzelm
parents: 13428
diff changeset
  1015
             Exists(Exists(Exists(Exists(Exists(
13323
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
  1016
              And(pair_fm(4,2,5),
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
  1017
               And(pair_fm(4,3,1),
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
  1018
                And(pair_fm(3,2,0),
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
  1019
                 And(Member(1,s#+6), Member(0,r#+6))))))))))))"
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
  1020
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
  1021
lemma composition_type [TC]:
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
  1022
     "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> composition_fm(x,y,z) \<in> formula"
13429
wenzelm
parents: 13428
diff changeset
  1023
by (simp add: composition_fm_def)
13323
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
  1024
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
  1025
lemma sats_composition_fm [simp]:
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
  1026
   "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
13429
wenzelm
parents: 13428
diff changeset
  1027
    ==> sats(A, composition_fm(x,y,z), env) <->
13807
a28a8fbc76d4 changed ** to ## to avoid conflict with new comment syntax
paulson
parents: 13655
diff changeset
  1028
        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
  1029
by (simp add: composition_fm_def composition_def)
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
  1030
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
  1031
lemma composition_iff_sats:
13429
wenzelm
parents: 13428
diff changeset
  1032
      "[| 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
  1033
          i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
13807
a28a8fbc76d4 changed ** to ## to avoid conflict with new comment syntax
paulson
parents: 13655
diff changeset
  1034
       ==> composition(##A, x, y, z) <-> sats(A, composition_fm(i,j,k), env)"
13323
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
  1035
by simp
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
  1036
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
  1037
theorem composition_reflection:
13429
wenzelm
parents: 13428
diff changeset
  1038
     "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
  1039
               \<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
  1040
apply (simp only: composition_def)
13429
wenzelm
parents: 13428
diff changeset
  1041
apply (intro FOL_reflections pair_reflection)
13314
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
  1042
done
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
  1043
13309
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1044
13339
0f89104dd377 Fixed quantified variable name preservation for ball and bex (bounded quants)
paulson
parents: 13323
diff changeset
  1045
subsubsection{*Injections, Internalized*}
13309
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1046
13429
wenzelm
parents: 13428
diff changeset
  1047
(* "injection(M,A,B,f) ==
wenzelm
parents: 13428
diff changeset
  1048
        typed_function(M,A,B,f) &
wenzelm
parents: 13428
diff changeset
  1049
        (\<forall>x[M]. \<forall>x'[M]. \<forall>y[M]. \<forall>p[M]. \<forall>p'[M].
13309
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1050
          pair(M,x,y,p) --> pair(M,x',y,p') --> p\<in>f --> p'\<in>f --> x=x')" *)
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1051
constdefs injection_fm :: "[i,i,i]=>i"
13429
wenzelm
parents: 13428
diff changeset
  1052
 "injection_fm(A,B,f) ==
13309
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1053
    And(typed_function_fm(A,B,f),
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1054
       Forall(Forall(Forall(Forall(Forall(
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1055
         Implies(pair_fm(4,2,1),
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1056
                 Implies(pair_fm(3,2,0),
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1057
                         Implies(Member(1,f#+5),
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1058
                                 Implies(Member(0,f#+5), Equal(4,3)))))))))))"
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1059
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1060
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1061
lemma injection_type [TC]:
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1062
     "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> injection_fm(x,y,z) \<in> formula"
13429
wenzelm
parents: 13428
diff changeset
  1063
by (simp add: injection_fm_def)
13309
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1064
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1065
lemma sats_injection_fm [simp]:
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1066
   "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
13429
wenzelm
parents: 13428
diff changeset
  1067
    ==> sats(A, injection_fm(x,y,z), env) <->
13807
a28a8fbc76d4 changed ** to ## to avoid conflict with new comment syntax
paulson
parents: 13655
diff changeset
  1068
        injection(##A, nth(x,env), nth(y,env), nth(z,env))"
13309
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1069
by (simp add: injection_fm_def injection_def)
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1070
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1071
lemma injection_iff_sats:
13429
wenzelm
parents: 13428
diff changeset
  1072
  "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z;
13309
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1073
      i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
13807
a28a8fbc76d4 changed ** to ## to avoid conflict with new comment syntax
paulson
parents: 13655
diff changeset
  1074
   ==> injection(##A, x, y, z) <-> sats(A, injection_fm(i,j,k), env)"
13309
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1075
by simp
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1076
13314
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
  1077
theorem injection_reflection:
13429
wenzelm
parents: 13428
diff changeset
  1078
     "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
  1079
               \<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
  1080
apply (simp only: injection_def)
13429
wenzelm
parents: 13428
diff changeset
  1081
apply (intro FOL_reflections function_reflections typed_function_reflection)
13314
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
  1082
done
13309
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1083
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1084
13339
0f89104dd377 Fixed quantified variable name preservation for ball and bex (bounded quants)
paulson
parents: 13323
diff changeset
  1085
subsubsection{*Surjections, Internalized*}
13309
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1086
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1087
(*  surjection :: "[i=>o,i,i,i] => o"
13429
wenzelm
parents: 13428
diff changeset
  1088
    "surjection(M,A,B,f) ==
13309
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1089
        typed_function(M,A,B,f) &
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1090
        (\<forall>y[M]. y\<in>B --> (\<exists>x[M]. x\<in>A & fun_apply(M,f,x,y)))" *)
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1091
constdefs surjection_fm :: "[i,i,i]=>i"
13429
wenzelm
parents: 13428
diff changeset
  1092
 "surjection_fm(A,B,f) ==
13309
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1093
    And(typed_function_fm(A,B,f),
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1094
       Forall(Implies(Member(0,succ(B)),
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1095
                      Exists(And(Member(0,succ(succ(A))),
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1096
                                 fun_apply_fm(succ(succ(f)),0,1))))))"
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1097
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1098
lemma surjection_type [TC]:
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1099
     "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> surjection_fm(x,y,z) \<in> formula"
13429
wenzelm
parents: 13428
diff changeset
  1100
by (simp add: surjection_fm_def)
13309
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1101
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1102
lemma sats_surjection_fm [simp]:
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1103
   "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
13429
wenzelm
parents: 13428
diff changeset
  1104
    ==> sats(A, surjection_fm(x,y,z), env) <->
13807
a28a8fbc76d4 changed ** to ## to avoid conflict with new comment syntax
paulson
parents: 13655
diff changeset
  1105
        surjection(##A, nth(x,env), nth(y,env), nth(z,env))"
13309
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1106
by (simp add: surjection_fm_def surjection_def)
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1107
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1108
lemma surjection_iff_sats:
13429
wenzelm
parents: 13428
diff changeset
  1109
  "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z;
13309
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1110
      i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
13807
a28a8fbc76d4 changed ** to ## to avoid conflict with new comment syntax
paulson
parents: 13655
diff changeset
  1111
   ==> surjection(##A, x, y, z) <-> sats(A, surjection_fm(i,j,k), env)"
13309
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1112
by simp
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1113
13314
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
  1114
theorem surjection_reflection:
13429
wenzelm
parents: 13428
diff changeset
  1115
     "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
  1116
               \<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
  1117
apply (simp only: surjection_def)
13429
wenzelm
parents: 13428
diff changeset
  1118
apply (intro FOL_reflections function_reflections typed_function_reflection)
13314
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
  1119
done
13309
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1120
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1121
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1122
13339
0f89104dd377 Fixed quantified variable name preservation for ball and bex (bounded quants)
paulson
parents: 13323
diff changeset
  1123
subsubsection{*Bijections, Internalized*}
13309
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1124
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1125
(*   bijection :: "[i=>o,i,i,i] => o"
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1126
    "bijection(M,A,B,f) == injection(M,A,B,f) & surjection(M,A,B,f)" *)
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1127
constdefs bijection_fm :: "[i,i,i]=>i"
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1128
 "bijection_fm(A,B,f) == And(injection_fm(A,B,f), surjection_fm(A,B,f))"
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1129
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1130
lemma bijection_type [TC]:
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1131
     "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> bijection_fm(x,y,z) \<in> formula"
13429
wenzelm
parents: 13428
diff changeset
  1132
by (simp add: bijection_fm_def)
13309
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1133
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1134
lemma sats_bijection_fm [simp]:
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1135
   "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
13429
wenzelm
parents: 13428
diff changeset
  1136
    ==> sats(A, bijection_fm(x,y,z), env) <->
13807
a28a8fbc76d4 changed ** to ## to avoid conflict with new comment syntax
paulson
parents: 13655
diff changeset
  1137
        bijection(##A, nth(x,env), nth(y,env), nth(z,env))"
13309
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1138
by (simp add: bijection_fm_def bijection_def)
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1139
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1140
lemma bijection_iff_sats:
13429
wenzelm
parents: 13428
diff changeset
  1141
  "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z;
13309
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1142
      i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
13807
a28a8fbc76d4 changed ** to ## to avoid conflict with new comment syntax
paulson
parents: 13655
diff changeset
  1143
   ==> bijection(##A, x, y, z) <-> sats(A, bijection_fm(i,j,k), env)"
13309
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1144
by simp
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1145
13314
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
  1146
theorem bijection_reflection:
13429
wenzelm
parents: 13428
diff changeset
  1147
     "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
  1148
               \<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
  1149
apply (simp only: bijection_def)
13429
wenzelm
parents: 13428
diff changeset
  1150
apply (intro And_reflection injection_reflection surjection_reflection)
13314
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
  1151
done
13309
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1152
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1153
13348
374d05460db4 Separation/Replacement up to M_wfrank!
paulson
parents: 13339
diff changeset
  1154
subsubsection{*Restriction of a Relation, Internalized*}
374d05460db4 Separation/Replacement up to M_wfrank!
paulson
parents: 13339
diff changeset
  1155
374d05460db4 Separation/Replacement up to M_wfrank!
paulson
parents: 13339
diff changeset
  1156
13429
wenzelm
parents: 13428
diff changeset
  1157
(* "restriction(M,r,A,z) ==
wenzelm
parents: 13428
diff changeset
  1158
        \<forall>x[M]. x \<in> z <-> (x \<in> r & (\<exists>u[M]. u\<in>A & (\<exists>v[M]. pair(M,u,v,x))))" *)
13348
374d05460db4 Separation/Replacement up to M_wfrank!
paulson
parents: 13339
diff changeset
  1159
constdefs restriction_fm :: "[i,i,i]=>i"
13429
wenzelm
parents: 13428
diff changeset
  1160
    "restriction_fm(r,A,z) ==
13348
374d05460db4 Separation/Replacement up to M_wfrank!
paulson
parents: 13339
diff changeset
  1161
       Forall(Iff(Member(0,succ(z)),
374d05460db4 Separation/Replacement up to M_wfrank!
paulson
parents: 13339
diff changeset
  1162
                  And(Member(0,succ(r)),
374d05460db4 Separation/Replacement up to M_wfrank!
paulson
parents: 13339
diff changeset
  1163
                      Exists(And(Member(0,succ(succ(A))),
374d05460db4 Separation/Replacement up to M_wfrank!
paulson
parents: 13339
diff changeset
  1164
                                 Exists(pair_fm(1,0,2)))))))"
374d05460db4 Separation/Replacement up to M_wfrank!
paulson
parents: 13339
diff changeset
  1165
374d05460db4 Separation/Replacement up to M_wfrank!
paulson
parents: 13339
diff changeset
  1166
lemma restriction_type [TC]:
374d05460db4 Separation/Replacement up to M_wfrank!
paulson
parents: 13339
diff changeset
  1167
     "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> restriction_fm(x,y,z) \<in> formula"
13429
wenzelm
parents: 13428
diff changeset
  1168
by (simp add: restriction_fm_def)
13348
374d05460db4 Separation/Replacement up to M_wfrank!
paulson
parents: 13339
diff changeset
  1169
374d05460db4 Separation/Replacement up to M_wfrank!
paulson
parents: 13339
diff changeset
  1170
lemma sats_restriction_fm [simp]:
374d05460db4 Separation/Replacement up to M_wfrank!
paulson
parents: 13339
diff changeset
  1171
   "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
13429
wenzelm
parents: 13428
diff changeset
  1172
    ==> sats(A, restriction_fm(x,y,z), env) <->
13807
a28a8fbc76d4 changed ** to ## to avoid conflict with new comment syntax
paulson
parents: 13655
diff changeset
  1173
        restriction(##A, nth(x,env), nth(y,env), nth(z,env))"
13348
374d05460db4 Separation/Replacement up to M_wfrank!
paulson
parents: 13339
diff changeset
  1174
by (simp add: restriction_fm_def restriction_def)
374d05460db4 Separation/Replacement up to M_wfrank!
paulson
parents: 13339
diff changeset
  1175
374d05460db4 Separation/Replacement up to M_wfrank!
paulson
parents: 13339
diff changeset
  1176
lemma restriction_iff_sats:
13429
wenzelm
parents: 13428
diff changeset
  1177
      "[| 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
  1178
          i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
13807
a28a8fbc76d4 changed ** to ## to avoid conflict with new comment syntax
paulson
parents: 13655
diff changeset
  1179
       ==> restriction(##A, x, y, z) <-> sats(A, restriction_fm(i,j,k), env)"
13348
374d05460db4 Separation/Replacement up to M_wfrank!
paulson
parents: 13339
diff changeset
  1180
by simp
374d05460db4 Separation/Replacement up to M_wfrank!
paulson
parents: 13339
diff changeset
  1181
374d05460db4 Separation/Replacement up to M_wfrank!
paulson
parents: 13339
diff changeset
  1182
theorem restriction_reflection:
13429
wenzelm
parents: 13428
diff changeset
  1183
     "REFLECTS[\<lambda>x. restriction(L,f(x),g(x),h(x)),
13807
a28a8fbc76d4 changed ** to ## to avoid conflict with new comment syntax
paulson
parents: 13655
diff changeset
  1184
               \<lambda>i x. restriction(##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
  1185
apply (simp only: restriction_def)
13429
wenzelm
parents: 13428
diff changeset
  1186
apply (intro FOL_reflections pair_reflection)
13348
374d05460db4 Separation/Replacement up to M_wfrank!
paulson
parents: 13339
diff changeset
  1187
done
374d05460db4 Separation/Replacement up to M_wfrank!
paulson
parents: 13339
diff changeset
  1188
13339
0f89104dd377 Fixed quantified variable name preservation for ball and bex (bounded quants)
paulson
parents: 13323
diff changeset
  1189
subsubsection{*Order-Isomorphisms, Internalized*}
13309
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1190
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1191
(*  order_isomorphism :: "[i=>o,i,i,i,i,i] => o"
13429
wenzelm
parents: 13428
diff changeset
  1192
   "order_isomorphism(M,A,r,B,s,f) ==
wenzelm
parents: 13428
diff changeset
  1193
        bijection(M,A,B,f) &
13309
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1194
        (\<forall>x[M]. x\<in>A --> (\<forall>y[M]. y\<in>A -->
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1195
          (\<forall>p[M]. \<forall>fx[M]. \<forall>fy[M]. \<forall>q[M].
13429
wenzelm
parents: 13428
diff changeset
  1196
            pair(M,x,y,p) --> fun_apply(M,f,x,fx) --> fun_apply(M,f,y,fy) -->
13309
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1197
            pair(M,fx,fy,q) --> (p\<in>r <-> q\<in>s))))"
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1198
  *)
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1199
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1200
constdefs order_isomorphism_fm :: "[i,i,i,i,i]=>i"
13429
wenzelm
parents: 13428
diff changeset
  1201
 "order_isomorphism_fm(A,r,B,s,f) ==
wenzelm
parents: 13428
diff changeset
  1202
   And(bijection_fm(A,B,f),
13309
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1203
     Forall(Implies(Member(0,succ(A)),
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1204
       Forall(Implies(Member(0,succ(succ(A))),
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1205
         Forall(Forall(Forall(Forall(
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1206
           Implies(pair_fm(5,4,3),
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1207
             Implies(fun_apply_fm(f#+6,5,2),
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1208
               Implies(fun_apply_fm(f#+6,4,1),
13429
wenzelm
parents: 13428
diff changeset
  1209
                 Implies(pair_fm(2,1,0),
13309
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1210
                   Iff(Member(3,r#+6), Member(0,s#+6)))))))))))))))"
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1211
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1212
lemma order_isomorphism_type [TC]:
13429
wenzelm
parents: 13428
diff changeset
  1213
     "[| A \<in> nat; r \<in> nat; B \<in> nat; s \<in> nat; f \<in> nat |]
13309
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1214
      ==> order_isomorphism_fm(A,r,B,s,f) \<in> formula"
13429
wenzelm
parents: 13428
diff changeset
  1215
by (simp add: order_isomorphism_fm_def)
13309
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1216
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1217
lemma sats_order_isomorphism_fm [simp]:
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1218
   "[| U \<in> nat; r \<in> nat; B \<in> nat; s \<in> nat; f \<in> nat; env \<in> list(A)|]
13429
wenzelm
parents: 13428
diff changeset
  1219
    ==> sats(A, order_isomorphism_fm(U,r,B,s,f), env) <->
13807