src/ZF/Constructible/L_axioms.thy
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Fri, 16 Aug 2002 16:41:48 +0200
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Relativized right up to L satisfies V=L!
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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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    Copyright   2002  University of Cambridge
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*)
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header {* The ZF Axioms (Except Separation) in L *}
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theory L_axioms = Formula + Relative + Reflection + MetaExists:
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text {* The class L satisfies the premises of locale @{text M_triv_axioms} *}
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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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lemma 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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lemma 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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lemma 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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subsubsection{*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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lemma 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_triv_axioms}*}
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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 Pair_in_Lset = Formula.Pair_in_LLimit
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lemmas L_nat = Ord_in_L [OF Ord_nat]
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theorem M_triv_axioms_L: "PROP M_triv_axioms(L)"
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  apply (rule M_triv_axioms.intro)
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        apply (erule (1) transL)
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       apply (rule nonempty)
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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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lemmas rall_abs = M_triv_axioms.rall_abs [OF M_triv_axioms_L]
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  and rex_abs = M_triv_axioms.rex_abs [OF M_triv_axioms_L]
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  and ball_iff_equiv = M_triv_axioms.ball_iff_equiv [OF M_triv_axioms_L]
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  and M_equalityI = M_triv_axioms.M_equalityI [OF M_triv_axioms_L]
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  and empty_abs = M_triv_axioms.empty_abs [OF M_triv_axioms_L]
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  and subset_abs = M_triv_axioms.subset_abs [OF M_triv_axioms_L]
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  and upair_abs = M_triv_axioms.upair_abs [OF M_triv_axioms_L]
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  and upair_in_M_iff = M_triv_axioms.upair_in_M_iff [OF M_triv_axioms_L]
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  and singleton_in_M_iff = M_triv_axioms.singleton_in_M_iff [OF M_triv_axioms_L]
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  and pair_abs = M_triv_axioms.pair_abs [OF M_triv_axioms_L]
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  and pair_in_M_iff = M_triv_axioms.pair_in_M_iff [OF M_triv_axioms_L]
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  and pair_components_in_M = M_triv_axioms.pair_components_in_M [OF M_triv_axioms_L]
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  and cartprod_abs = M_triv_axioms.cartprod_abs [OF M_triv_axioms_L]
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  and union_abs = M_triv_axioms.union_abs [OF M_triv_axioms_L]
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  and inter_abs = M_triv_axioms.inter_abs [OF M_triv_axioms_L]
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  and setdiff_abs = M_triv_axioms.setdiff_abs [OF M_triv_axioms_L]
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  and Union_abs = M_triv_axioms.Union_abs [OF M_triv_axioms_L]
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  and Union_closed = M_triv_axioms.Union_closed [OF M_triv_axioms_L]
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  and Un_closed = M_triv_axioms.Un_closed [OF M_triv_axioms_L]
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  and cons_closed = M_triv_axioms.cons_closed [OF M_triv_axioms_L]
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  and successor_abs = M_triv_axioms.successor_abs [OF M_triv_axioms_L]
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  and succ_in_M_iff = M_triv_axioms.succ_in_M_iff [OF M_triv_axioms_L]
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  and separation_closed = M_triv_axioms.separation_closed [OF M_triv_axioms_L]
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  and strong_replacementI = M_triv_axioms.strong_replacementI [OF M_triv_axioms_L]
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  and strong_replacement_closed = M_triv_axioms.strong_replacement_closed [OF M_triv_axioms_L]
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  and RepFun_closed = M_triv_axioms.RepFun_closed [OF M_triv_axioms_L]
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  and lam_closed = M_triv_axioms.lam_closed [OF M_triv_axioms_L]
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  and image_abs = M_triv_axioms.image_abs [OF M_triv_axioms_L]
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  and powerset_Pow = M_triv_axioms.powerset_Pow [OF M_triv_axioms_L]
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  and powerset_imp_subset_Pow = M_triv_axioms.powerset_imp_subset_Pow [OF M_triv_axioms_L]
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  and nat_into_M = M_triv_axioms.nat_into_M [OF M_triv_axioms_L]
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  and nat_case_closed = M_triv_axioms.nat_case_closed [OF M_triv_axioms_L]
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  and Inl_in_M_iff = M_triv_axioms.Inl_in_M_iff [OF M_triv_axioms_L]
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  and Inr_in_M_iff = M_triv_axioms.Inr_in_M_iff [OF M_triv_axioms_L]
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  and lt_closed = M_triv_axioms.lt_closed [OF M_triv_axioms_L]
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  and transitive_set_abs = M_triv_axioms.transitive_set_abs [OF M_triv_axioms_L]
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  and ordinal_abs = M_triv_axioms.ordinal_abs [OF M_triv_axioms_L]
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  and limit_ordinal_abs = M_triv_axioms.limit_ordinal_abs [OF M_triv_axioms_L]
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  and successor_ordinal_abs = M_triv_axioms.successor_ordinal_abs [OF M_triv_axioms_L]
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  and finite_ordinal_abs = M_triv_axioms.finite_ordinal_abs [OF M_triv_axioms_L]
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  and omega_abs = M_triv_axioms.omega_abs [OF M_triv_axioms_L]
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  and number1_abs = M_triv_axioms.number1_abs [OF M_triv_axioms_L]
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  and number2_abs = M_triv_axioms.number2_abs [OF M_triv_axioms_L]
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  and number3_abs = M_triv_axioms.number3_abs [OF M_triv_axioms_L]
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declare rall_abs [simp]
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declare rex_abs [simp]
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declare empty_abs [simp]
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declare subset_abs [simp]
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declare upair_abs [simp]
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declare upair_in_M_iff [iff]
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declare singleton_in_M_iff [iff]
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declare pair_abs [simp]
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declare pair_in_M_iff [iff]
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declare cartprod_abs [simp]
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declare union_abs [simp]
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declare inter_abs [simp]
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declare setdiff_abs [simp]
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declare Union_abs [simp]
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declare Union_closed [intro, simp]
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declare Un_closed [intro, simp]
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declare cons_closed [intro, simp]
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declare successor_abs [simp]
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declare succ_in_M_iff [iff]
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declare separation_closed [intro, simp]
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declare strong_replacementI
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declare strong_replacement_closed [intro, simp]
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declare RepFun_closed [intro, simp]
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declare lam_closed [intro, simp]
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declare image_abs [simp]
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declare nat_into_M [intro]
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declare Inl_in_M_iff [iff]
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declare Inr_in_M_iff [iff]
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declare transitive_set_abs [simp]
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declare ordinal_abs [simp]
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declare limit_ordinal_abs [simp]
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declare successor_ordinal_abs [simp]
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declare finite_ordinal_abs [simp]
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declare omega_abs [simp]
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declare number1_abs [simp]
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declare number2_abs [simp]
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declare number3_abs [simp]
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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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apply (blast intro: reflection.intro Lset_mono_le Lset_cont Pair_in_Lset) +
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done
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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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apply (drule ReflectsD, assumption, blast)
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done
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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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0f89104dd377 Fixed quantified variable name preservation for ball and bex (bounded quants)
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subsection{*Internalized Formulas for some Set-Theoretic Concepts*}
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lemmas setclass_simps = rall_setclass_is_ball rex_setclass_is_bex
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subsubsection{*Some numbers to help write de Bruijn indices*}
6eebcddee32b more internalized formulas and separation proofs
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parents: 13304
diff changeset
   338
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   339
syntax
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   340
    "3" :: i   ("3")
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   341
    "4" :: i   ("4")
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   342
    "5" :: i   ("5")
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   343
    "6" :: i   ("6")
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   344
    "7" :: i   ("7")
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   345
    "8" :: i   ("8")
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   346
    "9" :: i   ("9")
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   347
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   348
translations
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   349
   "3"  == "succ(2)"
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   350
   "4"  == "succ(3)"
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   351
   "5"  == "succ(4)"
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   352
   "6"  == "succ(5)"
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   353
   "7"  == "succ(6)"
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   354
   "8"  == "succ(7)"
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   355
   "9"  == "succ(8)"
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   356
13323
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   357
13339
0f89104dd377 Fixed quantified variable name preservation for ball and bex (bounded quants)
paulson
parents: 13323
diff changeset
   358
subsubsection{*The Empty Set, Internalized*}
13323
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   359
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   360
constdefs empty_fm :: "i=>i"
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   361
    "empty_fm(x) == Forall(Neg(Member(0,succ(x))))"
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   362
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   363
lemma empty_type [TC]:
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   364
     "x \<in> nat ==> empty_fm(x) \<in> formula"
13429
wenzelm
parents: 13428
diff changeset
   365
by (simp add: empty_fm_def)
13323
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   366
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   367
lemma arity_empty_fm [simp]:
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   368
     "x \<in> nat ==> arity(empty_fm(x)) = succ(x)"
13429
wenzelm
parents: 13428
diff changeset
   369
by (simp add: empty_fm_def succ_Un_distrib [symmetric] Un_ac)
13323
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   370
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   371
lemma sats_empty_fm [simp]:
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   372
   "[| x \<in> nat; env \<in> list(A)|]
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   373
    ==> sats(A, empty_fm(x), env) <-> empty(**A, nth(x,env))"
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   374
by (simp add: empty_fm_def empty_def)
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   375
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   376
lemma empty_iff_sats:
13429
wenzelm
parents: 13428
diff changeset
   377
      "[| nth(i,env) = x; nth(j,env) = y;
13323
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   378
          i \<in> nat; env \<in> list(A)|]
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   379
       ==> empty(**A, x) <-> sats(A, empty_fm(i), env)"
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   380
by simp
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   381
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   382
theorem empty_reflection:
13429
wenzelm
parents: 13428
diff changeset
   383
     "REFLECTS[\<lambda>x. empty(L,f(x)),
13323
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   384
               \<lambda>i x. empty(**Lset(i),f(x))]"
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   385
apply (simp only: empty_def setclass_simps)
13429
wenzelm
parents: 13428
diff changeset
   386
apply (intro FOL_reflections)
13323
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   387
done
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   388
13385
31df66ca0780 Expressing Lset and L without using length and arity; simplifies Separation
paulson
parents: 13363
diff changeset
   389
text{*Not used.  But maybe useful?*}
31df66ca0780 Expressing Lset and L without using length and arity; simplifies Separation
paulson
parents: 13363
diff changeset
   390
lemma Transset_sats_empty_fm_eq_0:
31df66ca0780 Expressing Lset and L without using length and arity; simplifies Separation
paulson
parents: 13363
diff changeset
   391
   "[| n \<in> nat; env \<in> list(A); Transset(A)|]
31df66ca0780 Expressing Lset and L without using length and arity; simplifies Separation
paulson
parents: 13363
diff changeset
   392
    ==> sats(A, empty_fm(n), env) <-> nth(n,env) = 0"
31df66ca0780 Expressing Lset and L without using length and arity; simplifies Separation
paulson
parents: 13363
diff changeset
   393
apply (simp add: empty_fm_def empty_def Transset_def, auto)
13429
wenzelm
parents: 13428
diff changeset
   394
apply (case_tac "n < length(env)")
wenzelm
parents: 13428
diff changeset
   395
apply (frule nth_type, assumption+, blast)
wenzelm
parents: 13428
diff changeset
   396
apply (simp_all add: not_lt_iff_le nth_eq_0)
13385
31df66ca0780 Expressing Lset and L without using length and arity; simplifies Separation
paulson
parents: 13363
diff changeset
   397
done
31df66ca0780 Expressing Lset and L without using length and arity; simplifies Separation
paulson
parents: 13363
diff changeset
   398
13323
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   399
13339
0f89104dd377 Fixed quantified variable name preservation for ball and bex (bounded quants)
paulson
parents: 13323
diff changeset
   400
subsubsection{*Unordered Pairs, Internalized*}
13298
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   401
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   402
constdefs upair_fm :: "[i,i,i]=>i"
13429
wenzelm
parents: 13428
diff changeset
   403
    "upair_fm(x,y,z) ==
wenzelm
parents: 13428
diff changeset
   404
       And(Member(x,z),
13298
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   405
           And(Member(y,z),
13429
wenzelm
parents: 13428
diff changeset
   406
               Forall(Implies(Member(0,succ(z)),
13298
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   407
                              Or(Equal(0,succ(x)), Equal(0,succ(y)))))))"
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   408
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   409
lemma upair_type [TC]:
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   410
     "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> upair_fm(x,y,z) \<in> formula"
13429
wenzelm
parents: 13428
diff changeset
   411
by (simp add: upair_fm_def)
13298
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   412
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   413
lemma arity_upair_fm [simp]:
13429
wenzelm
parents: 13428
diff changeset
   414
     "[| x \<in> nat; y \<in> nat; z \<in> nat |]
13298
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   415
      ==> arity(upair_fm(x,y,z)) = succ(x) \<union> succ(y) \<union> succ(z)"
13429
wenzelm
parents: 13428
diff changeset
   416
by (simp add: upair_fm_def succ_Un_distrib [symmetric] Un_ac)
13298
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   417
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   418
lemma sats_upair_fm [simp]:
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   419
   "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
13429
wenzelm
parents: 13428
diff changeset
   420
    ==> sats(A, upair_fm(x,y,z), env) <->
13298
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   421
            upair(**A, nth(x,env), nth(y,env), nth(z,env))"
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   422
by (simp add: upair_fm_def upair_def)
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   423
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   424
lemma upair_iff_sats:
13429
wenzelm
parents: 13428
diff changeset
   425
      "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z;
13298
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   426
          i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   427
       ==> upair(**A, x, y, z) <-> sats(A, upair_fm(i,j,k), env)"
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   428
by (simp add: sats_upair_fm)
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   429
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   430
text{*Useful? At least it refers to "real" unordered pairs*}
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   431
lemma sats_upair_fm2 [simp]:
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   432
   "[| x \<in> nat; y \<in> nat; z < length(env); env \<in> list(A); Transset(A)|]
13429
wenzelm
parents: 13428
diff changeset
   433
    ==> sats(A, upair_fm(x,y,z), env) <->
13298
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   434
        nth(z,env) = {nth(x,env), nth(y,env)}"
13429
wenzelm
parents: 13428
diff changeset
   435
apply (frule lt_length_in_nat, assumption)
wenzelm
parents: 13428
diff changeset
   436
apply (simp add: upair_fm_def Transset_def, auto)
wenzelm
parents: 13428
diff changeset
   437
apply (blast intro: nth_type)
13298
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   438
done
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   439
13314
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   440
theorem upair_reflection:
13429
wenzelm
parents: 13428
diff changeset
   441
     "REFLECTS[\<lambda>x. upair(L,f(x),g(x),h(x)),
wenzelm
parents: 13428
diff changeset
   442
               \<lambda>i x. upair(**Lset(i),f(x),g(x),h(x))]"
13314
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   443
apply (simp add: upair_def)
13429
wenzelm
parents: 13428
diff changeset
   444
apply (intro FOL_reflections)
13314
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   445
done
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   446
13339
0f89104dd377 Fixed quantified variable name preservation for ball and bex (bounded quants)
paulson
parents: 13323
diff changeset
   447
subsubsection{*Ordered pairs, Internalized*}
13298
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   448
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   449
constdefs pair_fm :: "[i,i,i]=>i"
13429
wenzelm
parents: 13428
diff changeset
   450
    "pair_fm(x,y,z) ==
13298
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   451
       Exists(And(upair_fm(succ(x),succ(x),0),
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   452
              Exists(And(upair_fm(succ(succ(x)),succ(succ(y)),0),
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   453
                         upair_fm(1,0,succ(succ(z)))))))"
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   454
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   455
lemma pair_type [TC]:
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   456
     "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> pair_fm(x,y,z) \<in> formula"
13429
wenzelm
parents: 13428
diff changeset
   457
by (simp add: pair_fm_def)
13298
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   458
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   459
lemma arity_pair_fm [simp]:
13429
wenzelm
parents: 13428
diff changeset
   460
     "[| x \<in> nat; y \<in> nat; z \<in> nat |]
13298
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   461
      ==> arity(pair_fm(x,y,z)) = succ(x) \<union> succ(y) \<union> succ(z)"
13429
wenzelm
parents: 13428
diff changeset
   462
by (simp add: pair_fm_def succ_Un_distrib [symmetric] Un_ac)
13298
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   463
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   464
lemma sats_pair_fm [simp]:
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   465
   "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
13429
wenzelm
parents: 13428
diff changeset
   466
    ==> sats(A, pair_fm(x,y,z), env) <->
13298
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   467
        pair(**A, nth(x,env), nth(y,env), nth(z,env))"
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   468
by (simp add: pair_fm_def pair_def)
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   469
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   470
lemma pair_iff_sats:
13429
wenzelm
parents: 13428
diff changeset
   471
      "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z;
13298
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   472
          i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   473
       ==> pair(**A, x, y, z) <-> sats(A, pair_fm(i,j,k), env)"
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   474
by (simp add: sats_pair_fm)
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   475
13314
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   476
theorem pair_reflection:
13429
wenzelm
parents: 13428
diff changeset
   477
     "REFLECTS[\<lambda>x. pair(L,f(x),g(x),h(x)),
13314
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   478
               \<lambda>i x. pair(**Lset(i),f(x),g(x),h(x))]"
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   479
apply (simp only: pair_def setclass_simps)
13429
wenzelm
parents: 13428
diff changeset
   480
apply (intro FOL_reflections upair_reflection)
13314
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   481
done
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   482
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   483
13339
0f89104dd377 Fixed quantified variable name preservation for ball and bex (bounded quants)
paulson
parents: 13323
diff changeset
   484
subsubsection{*Binary Unions, Internalized*}
13298
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   485
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   486
constdefs union_fm :: "[i,i,i]=>i"
13429
wenzelm
parents: 13428
diff changeset
   487
    "union_fm(x,y,z) ==
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   488
       Forall(Iff(Member(0,succ(z)),
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   489
                  Or(Member(0,succ(x)),Member(0,succ(y)))))"
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   490
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   491
lemma union_type [TC]:
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   492
     "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> union_fm(x,y,z) \<in> formula"
13429
wenzelm
parents: 13428
diff changeset
   493
by (simp add: union_fm_def)
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   494
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   495
lemma arity_union_fm [simp]:
13429
wenzelm
parents: 13428
diff changeset
   496
     "[| x \<in> nat; y \<in> nat; z \<in> nat |]
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   497
      ==> arity(union_fm(x,y,z)) = succ(x) \<union> succ(y) \<union> succ(z)"
13429
wenzelm
parents: 13428
diff changeset
   498
by (simp add: union_fm_def succ_Un_distrib [symmetric] Un_ac)
13298
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   499
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   500
lemma sats_union_fm [simp]:
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   501
   "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
13429
wenzelm
parents: 13428
diff changeset
   502
    ==> sats(A, union_fm(x,y,z), env) <->
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   503
        union(**A, nth(x,env), nth(y,env), nth(z,env))"
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   504
by (simp add: union_fm_def union_def)
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   505
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   506
lemma union_iff_sats:
13429
wenzelm
parents: 13428
diff changeset
   507
      "[| 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
   508
          i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   509
       ==> union(**A, x, y, z) <-> sats(A, union_fm(i,j,k), env)"
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   510
by (simp add: sats_union_fm)
13298
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   511
13314
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   512
theorem union_reflection:
13429
wenzelm
parents: 13428
diff changeset
   513
     "REFLECTS[\<lambda>x. union(L,f(x),g(x),h(x)),
13314
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   514
               \<lambda>i x. union(**Lset(i),f(x),g(x),h(x))]"
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   515
apply (simp only: union_def setclass_simps)
13429
wenzelm
parents: 13428
diff changeset
   516
apply (intro FOL_reflections)
13314
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   517
done
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   518
13298
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   519
13339
0f89104dd377 Fixed quantified variable name preservation for ball and bex (bounded quants)
paulson
parents: 13323
diff changeset
   520
subsubsection{*Set ``Cons,'' Internalized*}
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   521
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   522
constdefs cons_fm :: "[i,i,i]=>i"
13429
wenzelm
parents: 13428
diff changeset
   523
    "cons_fm(x,y,z) ==
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   524
       Exists(And(upair_fm(succ(x),succ(x),0),
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   525
                  union_fm(0,succ(y),succ(z))))"
13298
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   526
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   527
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   528
lemma cons_type [TC]:
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   529
     "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> cons_fm(x,y,z) \<in> formula"
13429
wenzelm
parents: 13428
diff changeset
   530
by (simp add: cons_fm_def)
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   531
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   532
lemma arity_cons_fm [simp]:
13429
wenzelm
parents: 13428
diff changeset
   533
     "[| x \<in> nat; y \<in> nat; z \<in> nat |]
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   534
      ==> arity(cons_fm(x,y,z)) = succ(x) \<union> succ(y) \<union> succ(z)"
13429
wenzelm
parents: 13428
diff changeset
   535
by (simp add: cons_fm_def succ_Un_distrib [symmetric] Un_ac)
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   536
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   537
lemma sats_cons_fm [simp]:
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   538
   "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
13429
wenzelm
parents: 13428
diff changeset
   539
    ==> sats(A, cons_fm(x,y,z), env) <->
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   540
        is_cons(**A, nth(x,env), nth(y,env), nth(z,env))"
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   541
by (simp add: cons_fm_def is_cons_def)
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   542
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   543
lemma cons_iff_sats:
13429
wenzelm
parents: 13428
diff changeset
   544
      "[| 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
   545
          i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   546
       ==> is_cons(**A, x, y, z) <-> sats(A, cons_fm(i,j,k), env)"
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   547
by simp
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   548
13314
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   549
theorem cons_reflection:
13429
wenzelm
parents: 13428
diff changeset
   550
     "REFLECTS[\<lambda>x. is_cons(L,f(x),g(x),h(x)),
13314
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   551
               \<lambda>i x. is_cons(**Lset(i),f(x),g(x),h(x))]"
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   552
apply (simp only: is_cons_def setclass_simps)
13429
wenzelm
parents: 13428
diff changeset
   553
apply (intro FOL_reflections upair_reflection union_reflection)
13323
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   554
done
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   555
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   556
13339
0f89104dd377 Fixed quantified variable name preservation for ball and bex (bounded quants)
paulson
parents: 13323
diff changeset
   557
subsubsection{*Successor Function, Internalized*}
13323
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   558
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   559
constdefs succ_fm :: "[i,i]=>i"
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   560
    "succ_fm(x,y) == cons_fm(x,x,y)"
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   561
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   562
lemma succ_type [TC]:
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   563
     "[| x \<in> nat; y \<in> nat |] ==> succ_fm(x,y) \<in> formula"
13429
wenzelm
parents: 13428
diff changeset
   564
by (simp add: succ_fm_def)
13323
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   565
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   566
lemma arity_succ_fm [simp]:
13429
wenzelm
parents: 13428
diff changeset
   567
     "[| x \<in> nat; y \<in> nat |]
13323
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   568
      ==> arity(succ_fm(x,y)) = succ(x) \<union> succ(y)"
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   569
by (simp add: succ_fm_def)
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   570
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   571
lemma sats_succ_fm [simp]:
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   572
   "[| x \<in> nat; y \<in> nat; env \<in> list(A)|]
13429
wenzelm
parents: 13428
diff changeset
   573
    ==> sats(A, succ_fm(x,y), env) <->
13323
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   574
        successor(**A, nth(x,env), nth(y,env))"
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   575
by (simp add: succ_fm_def successor_def)
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   576
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   577
lemma successor_iff_sats:
13429
wenzelm
parents: 13428
diff changeset
   578
      "[| nth(i,env) = x; nth(j,env) = y;
13323
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   579
          i \<in> nat; j \<in> nat; env \<in> list(A)|]
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   580
       ==> successor(**A, x, y) <-> sats(A, succ_fm(i,j), env)"
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   581
by simp
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   582
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   583
theorem successor_reflection:
13429
wenzelm
parents: 13428
diff changeset
   584
     "REFLECTS[\<lambda>x. successor(L,f(x),g(x)),
13323
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   585
               \<lambda>i x. successor(**Lset(i),f(x),g(x))]"
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   586
apply (simp only: successor_def setclass_simps)
13429
wenzelm
parents: 13428
diff changeset
   587
apply (intro cons_reflection)
13314
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   588
done
13298
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   589
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   590
13363
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   591
subsubsection{*The Number 1, Internalized*}
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   592
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   593
(* "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
   594
constdefs number1_fm :: "i=>i"
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   595
    "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
   596
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   597
lemma number1_type [TC]:
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   598
     "x \<in> nat ==> number1_fm(x) \<in> formula"
13429
wenzelm
parents: 13428
diff changeset
   599
by (simp add: number1_fm_def)
13363
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   600
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   601
lemma arity_number1_fm [simp]:
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   602
     "x \<in> nat ==> arity(number1_fm(x)) = succ(x)"
13429
wenzelm
parents: 13428
diff changeset
   603
by (simp add: number1_fm_def succ_Un_distrib [symmetric] Un_ac)
13363
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   604
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   605
lemma sats_number1_fm [simp]:
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   606
   "[| x \<in> nat; env \<in> list(A)|]
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   607
    ==> sats(A, number1_fm(x), env) <-> number1(**A, nth(x,env))"
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   608
by (simp add: number1_fm_def number1_def)
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   609
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   610
lemma number1_iff_sats:
13429
wenzelm
parents: 13428
diff changeset
   611
      "[| nth(i,env) = x; nth(j,env) = y;
13363
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   612
          i \<in> nat; env \<in> list(A)|]
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   613
       ==> number1(**A, x) <-> sats(A, number1_fm(i), env)"
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   614
by simp
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   615
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   616
theorem number1_reflection:
13429
wenzelm
parents: 13428
diff changeset
   617
     "REFLECTS[\<lambda>x. number1(L,f(x)),
13363
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   618
               \<lambda>i x. number1(**Lset(i),f(x))]"
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   619
apply (simp only: number1_def setclass_simps)
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   620
apply (intro FOL_reflections empty_reflection successor_reflection)
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   621
done
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   622
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   623
13352
3cd767f8d78b new definitions of fun_apply and M_is_recfun
paulson
parents: 13348
diff changeset
   624
subsubsection{*Big Union, Internalized*}
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   625
13352
3cd767f8d78b new definitions of fun_apply and M_is_recfun
paulson
parents: 13348
diff changeset
   626
(*  "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
   627
constdefs big_union_fm :: "[i,i]=>i"
13429
wenzelm
parents: 13428
diff changeset
   628
    "big_union_fm(A,z) ==
13352
3cd767f8d78b new definitions of fun_apply and M_is_recfun
paulson
parents: 13348
diff changeset
   629
       Forall(Iff(Member(0,succ(z)),
3cd767f8d78b new definitions of fun_apply and M_is_recfun
paulson
parents: 13348
diff changeset
   630
                  Exists(And(Member(0,succ(succ(A))), Member(1,0)))))"
13298
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   631
13352
3cd767f8d78b new definitions of fun_apply and M_is_recfun
paulson
parents: 13348
diff changeset
   632
lemma big_union_type [TC]:
3cd767f8d78b new definitions of fun_apply and M_is_recfun
paulson
parents: 13348
diff changeset
   633
     "[| x \<in> nat; y \<in> nat |] ==> big_union_fm(x,y) \<in> formula"
13429
wenzelm
parents: 13428
diff changeset
   634
by (simp add: big_union_fm_def)
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   635
13352
3cd767f8d78b new definitions of fun_apply and M_is_recfun
paulson
parents: 13348
diff changeset
   636
lemma arity_big_union_fm [simp]:
13429
wenzelm
parents: 13428
diff changeset
   637
     "[| x \<in> nat; y \<in> nat |]
13352
3cd767f8d78b new definitions of fun_apply and M_is_recfun
paulson
parents: 13348
diff changeset
   638
      ==> arity(big_union_fm(x,y)) = succ(x) \<union> succ(y)"
3cd767f8d78b new definitions of fun_apply and M_is_recfun
paulson
parents: 13348
diff changeset
   639
by (simp add: big_union_fm_def succ_Un_distrib [symmetric] Un_ac)
13298
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   640
13352
3cd767f8d78b new definitions of fun_apply and M_is_recfun
paulson
parents: 13348
diff changeset
   641
lemma sats_big_union_fm [simp]:
3cd767f8d78b new definitions of fun_apply and M_is_recfun
paulson
parents: 13348
diff changeset
   642
   "[| x \<in> nat; y \<in> nat; env \<in> list(A)|]
13429
wenzelm
parents: 13428
diff changeset
   643
    ==> sats(A, big_union_fm(x,y), env) <->
13352
3cd767f8d78b new definitions of fun_apply and M_is_recfun
paulson
parents: 13348
diff changeset
   644
        big_union(**A, nth(x,env), nth(y,env))"
3cd767f8d78b new definitions of fun_apply and M_is_recfun
paulson
parents: 13348
diff changeset
   645
by (simp add: big_union_fm_def big_union_def)
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   646
13352
3cd767f8d78b new definitions of fun_apply and M_is_recfun
paulson
parents: 13348
diff changeset
   647
lemma big_union_iff_sats:
13429
wenzelm
parents: 13428
diff changeset
   648
      "[| nth(i,env) = x; nth(j,env) = y;
13352
3cd767f8d78b new definitions of fun_apply and M_is_recfun
paulson
parents: 13348
diff changeset
   649
          i \<in> nat; j \<in> nat; env \<in> list(A)|]
3cd767f8d78b new definitions of fun_apply and M_is_recfun
paulson
parents: 13348
diff changeset
   650
       ==> 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
   651
by simp
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   652
13352
3cd767f8d78b new definitions of fun_apply and M_is_recfun
paulson
parents: 13348
diff changeset
   653
theorem big_union_reflection:
13429
wenzelm
parents: 13428
diff changeset
   654
     "REFLECTS[\<lambda>x. big_union(L,f(x),g(x)),
13352
3cd767f8d78b new definitions of fun_apply and M_is_recfun
paulson
parents: 13348
diff changeset
   655
               \<lambda>i x. big_union(**Lset(i),f(x),g(x))]"
3cd767f8d78b new definitions of fun_apply and M_is_recfun
paulson
parents: 13348
diff changeset
   656
apply (simp only: big_union_def setclass_simps)
13429
wenzelm
parents: 13428
diff changeset
   657
apply (intro FOL_reflections)
13314
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   658
done
13298
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   659
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   660
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   661
subsubsection{*Variants of Satisfaction Definitions for Ordinals, etc.*}
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   662
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   663
text{*Differs from the one in Formula by using "ordinal" rather than "Ord"*}
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   664
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   665
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   666
lemma sats_subset_fm':
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   667
   "[|x \<in> nat; y \<in> nat; env \<in> list(A)|]
13429
wenzelm
parents: 13428
diff changeset
   668
    ==> sats(A, subset_fm(x,y), env) <-> subset(**A, nth(x,env), nth(y,env))"
wenzelm
parents: 13428
diff changeset
   669
by (simp add: subset_fm_def Relative.subset_def)
13298
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   670
13314
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   671
theorem subset_reflection:
13429
wenzelm
parents: 13428
diff changeset
   672
     "REFLECTS[\<lambda>x. subset(L,f(x),g(x)),
wenzelm
parents: 13428
diff changeset
   673
               \<lambda>i x. subset(**Lset(i),f(x),g(x))]"
13323
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   674
apply (simp only: Relative.subset_def setclass_simps)
13429
wenzelm
parents: 13428
diff changeset
   675
apply (intro FOL_reflections)
13314
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   676
done
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   677
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   678
lemma sats_transset_fm':
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   679
   "[|x \<in> nat; env \<in> list(A)|]
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   680
    ==> sats(A, transset_fm(x), env) <-> transitive_set(**A, nth(x,env))"
13429
wenzelm
parents: 13428
diff changeset
   681
by (simp add: sats_subset_fm' transset_fm_def transitive_set_def)
13298
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   682
13314
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   683
theorem transitive_set_reflection:
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   684
     "REFLECTS[\<lambda>x. transitive_set(L,f(x)),
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   685
               \<lambda>i x. transitive_set(**Lset(i),f(x))]"
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   686
apply (simp only: transitive_set_def setclass_simps)
13429
wenzelm
parents: 13428
diff changeset
   687
apply (intro FOL_reflections subset_reflection)
13314
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   688
done
13306
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 sats_ordinal_fm':
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   691
   "[|x \<in> nat; env \<in> list(A)|]
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   692
    ==> sats(A, ordinal_fm(x), env) <-> ordinal(**A,nth(x,env))"
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   693
by (simp add: sats_transset_fm' ordinal_fm_def ordinal_def)
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   694
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   695
lemma ordinal_iff_sats:
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   696
      "[| nth(i,env) = x;  i \<in> nat; env \<in> list(A)|]
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   697
       ==> ordinal(**A, x) <-> sats(A, ordinal_fm(i), env)"
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   698
by (simp add: sats_ordinal_fm')
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   699
13314
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   700
theorem ordinal_reflection:
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   701
     "REFLECTS[\<lambda>x. ordinal(L,f(x)), \<lambda>i x. ordinal(**Lset(i),f(x))]"
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   702
apply (simp only: ordinal_def setclass_simps)
13429
wenzelm
parents: 13428
diff changeset
   703
apply (intro FOL_reflections transitive_set_reflection)
13314
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   704
done
13298
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   705
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   706
13339
0f89104dd377 Fixed quantified variable name preservation for ball and bex (bounded quants)
paulson
parents: 13323
diff changeset
   707
subsubsection{*Membership Relation, Internalized*}
13298
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   708
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   709
constdefs Memrel_fm :: "[i,i]=>i"
13429
wenzelm
parents: 13428
diff changeset
   710
    "Memrel_fm(A,r) ==
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   711
       Forall(Iff(Member(0,succ(r)),
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   712
                  Exists(And(Member(0,succ(succ(A))),
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   713
                             Exists(And(Member(0,succ(succ(succ(A)))),
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   714
                                        And(Member(1,0),
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   715
                                            pair_fm(1,0,2))))))))"
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   716
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   717
lemma Memrel_type [TC]:
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   718
     "[| x \<in> nat; y \<in> nat |] ==> Memrel_fm(x,y) \<in> formula"
13429
wenzelm
parents: 13428
diff changeset
   719
by (simp add: Memrel_fm_def)
13298
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   720
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   721
lemma arity_Memrel_fm [simp]:
13429
wenzelm
parents: 13428
diff changeset
   722
     "[| x \<in> nat; y \<in> nat |]
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   723
      ==> arity(Memrel_fm(x,y)) = succ(x) \<union> succ(y)"
13429
wenzelm
parents: 13428
diff changeset
   724
by (simp add: Memrel_fm_def succ_Un_distrib [symmetric] Un_ac)
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   725
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   726
lemma sats_Memrel_fm [simp]:
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   727
   "[| x \<in> nat; y \<in> nat; env \<in> list(A)|]
13429
wenzelm
parents: 13428
diff changeset
   728
    ==> sats(A, Memrel_fm(x,y), env) <->
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   729
        membership(**A, nth(x,env), nth(y,env))"
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   730
by (simp add: Memrel_fm_def membership_def)
13298
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   731
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   732
lemma Memrel_iff_sats:
13429
wenzelm
parents: 13428
diff changeset
   733
      "[| nth(i,env) = x; nth(j,env) = y;
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   734
          i \<in> nat; j \<in> nat; env \<in> list(A)|]
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   735
       ==> membership(**A, x, y) <-> sats(A, Memrel_fm(i,j), env)"
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   736
by simp
13304
43ef6c6dd906 more separation instances
paulson
parents: 13299
diff changeset
   737
13314
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   738
theorem membership_reflection:
13429
wenzelm
parents: 13428
diff changeset
   739
     "REFLECTS[\<lambda>x. membership(L,f(x),g(x)),
13314
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   740
               \<lambda>i x. membership(**Lset(i),f(x),g(x))]"
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   741
apply (simp only: membership_def setclass_simps)
13429
wenzelm
parents: 13428
diff changeset
   742
apply (intro FOL_reflections pair_reflection)
13314
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   743
done
13304
43ef6c6dd906 more separation instances
paulson
parents: 13299
diff changeset
   744
13339
0f89104dd377 Fixed quantified variable name preservation for ball and bex (bounded quants)
paulson
parents: 13323
diff changeset
   745
subsubsection{*Predecessor Set, Internalized*}
13304
43ef6c6dd906 more separation instances
paulson
parents: 13299
diff changeset
   746
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   747
constdefs pred_set_fm :: "[i,i,i,i]=>i"
13429
wenzelm
parents: 13428
diff changeset
   748
    "pred_set_fm(A,x,r,B) ==
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   749
       Forall(Iff(Member(0,succ(B)),
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   750
                  Exists(And(Member(0,succ(succ(r))),
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   751
                             And(Member(1,succ(succ(A))),
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   752
                                 pair_fm(1,succ(succ(x)),0))))))"
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   753
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   754
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   755
lemma pred_set_type [TC]:
13429
wenzelm
parents: 13428
diff changeset
   756
     "[| 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
   757
      ==> pred_set_fm(A,x,r,B) \<in> formula"
13429
wenzelm
parents: 13428
diff changeset
   758
by (simp add: pred_set_fm_def)
13304
43ef6c6dd906 more separation instances
paulson
parents: 13299
diff changeset
   759
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   760
lemma arity_pred_set_fm [simp]:
13429
wenzelm
parents: 13428
diff changeset
   761
   "[| 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
   762
    ==> arity(pred_set_fm(A,x,r,B)) = succ(A) \<union> succ(x) \<union> succ(r) \<union> succ(B)"
13429
wenzelm
parents: 13428
diff changeset
   763
by (simp add: pred_set_fm_def succ_Un_distrib [symmetric] Un_ac)
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   764
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   765
lemma sats_pred_set_fm [simp]:
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   766
   "[| U \<in> nat; x \<in> nat; r \<in> nat; B \<in> nat; env \<in> list(A)|]
13429
wenzelm
parents: 13428
diff changeset
   767
    ==> sats(A, pred_set_fm(U,x,r,B), env) <->
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   768
        pred_set(**A, nth(U,env), nth(x,env), nth(r,env), nth(B,env))"
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   769
by (simp add: pred_set_fm_def pred_set_def)
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   770
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   771
lemma pred_set_iff_sats:
13429
wenzelm
parents: 13428
diff changeset
   772
      "[| 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
   773
          i \<in> nat; j \<in> nat; k \<in> nat; l \<in> nat; env \<in> list(A)|]
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   774
       ==> pred_set(**A,U,x,r,B) <-> sats(A, pred_set_fm(i,j,k,l), env)"
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   775
by (simp add: sats_pred_set_fm)
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   776
13314
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   777
theorem pred_set_reflection:
13429
wenzelm
parents: 13428
diff changeset
   778
     "REFLECTS[\<lambda>x. pred_set(L,f(x),g(x),h(x),b(x)),
wenzelm
parents: 13428
diff changeset
   779
               \<lambda>i x. pred_set(**Lset(i),f(x),g(x),h(x),b(x))]"
13314
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   780
apply (simp only: pred_set_def setclass_simps)
13429
wenzelm
parents: 13428
diff changeset
   781
apply (intro FOL_reflections pair_reflection)
13314
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   782
done
13304
43ef6c6dd906 more separation instances
paulson
parents: 13299
diff changeset
   783
43ef6c6dd906 more separation instances
paulson
parents: 13299
diff changeset
   784
13298
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   785
13339
0f89104dd377 Fixed quantified variable name preservation for ball and bex (bounded quants)
paulson
parents: 13323
diff changeset
   786
subsubsection{*Domain of a Relation, Internalized*}
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   787
13429
wenzelm
parents: 13428
diff changeset
   788
(* "is_domain(M,r,z) ==
wenzelm
parents: 13428
diff changeset
   789
        \<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
   790
constdefs domain_fm :: "[i,i]=>i"
13429
wenzelm
parents: 13428
diff changeset
   791
    "domain_fm(r,z) ==
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   792
       Forall(Iff(Member(0,succ(z)),
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   793
                  Exists(And(Member(0,succ(succ(r))),
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   794
                             Exists(pair_fm(2,0,1))))))"
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   795
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   796
lemma domain_type [TC]:
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   797
     "[| x \<in> nat; y \<in> nat |] ==> domain_fm(x,y) \<in> formula"
13429
wenzelm
parents: 13428
diff changeset
   798
by (simp add: domain_fm_def)
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   799
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   800
lemma arity_domain_fm [simp]:
13429
wenzelm
parents: 13428
diff changeset
   801
     "[| x \<in> nat; y \<in> nat |]
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   802
      ==> arity(domain_fm(x,y)) = succ(x) \<union> succ(y)"
13429
wenzelm
parents: 13428
diff changeset
   803
by (simp add: domain_fm_def succ_Un_distrib [symmetric] Un_ac)
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   804
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   805
lemma sats_domain_fm [simp]:
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   806
   "[| x \<in> nat; y \<in> nat; env \<in> list(A)|]
13429
wenzelm
parents: 13428
diff changeset
   807
    ==> sats(A, domain_fm(x,y), env) <->
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   808
        is_domain(**A, nth(x,env), nth(y,env))"
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   809
by (simp add: domain_fm_def is_domain_def)
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   810
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   811
lemma domain_iff_sats:
13429
wenzelm
parents: 13428
diff changeset
   812
      "[| nth(i,env) = x; nth(j,env) = y;
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   813
          i \<in> nat; j \<in> nat; env \<in> list(A)|]
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   814
       ==> is_domain(**A, x, y) <-> sats(A, domain_fm(i,j), env)"
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   815
by simp
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   816
13314
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   817
theorem domain_reflection:
13429
wenzelm
parents: 13428
diff changeset
   818
     "REFLECTS[\<lambda>x. is_domain(L,f(x),g(x)),
13314
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   819
               \<lambda>i x. is_domain(**Lset(i),f(x),g(x))]"
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   820
apply (simp only: is_domain_def setclass_simps)
13429
wenzelm
parents: 13428
diff changeset
   821
apply (intro FOL_reflections pair_reflection)
13314
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   822
done
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   823
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   824
13339
0f89104dd377 Fixed quantified variable name preservation for ball and bex (bounded quants)
paulson
parents: 13323
diff changeset
   825
subsubsection{*Range of a Relation, Internalized*}
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   826
13429
wenzelm
parents: 13428
diff changeset
   827
(* "is_range(M,r,z) ==
wenzelm
parents: 13428
diff changeset
   828
        \<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
   829
constdefs range_fm :: "[i,i]=>i"
13429
wenzelm
parents: 13428
diff changeset
   830
    "range_fm(r,z) ==
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   831
       Forall(Iff(Member(0,succ(z)),
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   832
                  Exists(And(Member(0,succ(succ(r))),
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   833
                             Exists(pair_fm(0,2,1))))))"
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   834
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   835
lemma range_type [TC]:
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   836
     "[| x \<in> nat; y \<in> nat |] ==> range_fm(x,y) \<in> formula"
13429
wenzelm
parents: 13428
diff changeset
   837
by (simp add: range_fm_def)
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   838
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   839
lemma arity_range_fm [simp]:
13429
wenzelm
parents: 13428
diff changeset
   840
     "[| x \<in> nat; y \<in> nat |]
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   841
      ==> arity(range_fm(x,y)) = succ(x) \<union> succ(y)"
13429
wenzelm
parents: 13428
diff changeset
   842
by (simp add: range_fm_def succ_Un_distrib [symmetric] Un_ac)
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   843
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   844
lemma sats_range_fm [simp]:
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   845
   "[| x \<in> nat; y \<in> nat; env \<in> list(A)|]
13429
wenzelm
parents: 13428
diff changeset
   846
    ==> sats(A, range_fm(x,y), env) <->
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   847
        is_range(**A, nth(x,env), nth(y,env))"
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   848
by (simp add: range_fm_def is_range_def)
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   849
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   850
lemma range_iff_sats:
13429
wenzelm
parents: 13428
diff changeset
   851
      "[| nth(i,env) = x; nth(j,env) = y;
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   852
          i \<in> nat; j \<in> nat; env \<in> list(A)|]
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   853
       ==> is_range(**A, x, y) <-> sats(A, range_fm(i,j), env)"
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   854
by simp
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   855
13314
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   856
theorem range_reflection:
13429
wenzelm
parents: 13428
diff changeset
   857
     "REFLECTS[\<lambda>x. is_range(L,f(x),g(x)),
13314
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   858
               \<lambda>i x. is_range(**Lset(i),f(x),g(x))]"
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   859
apply (simp only: is_range_def setclass_simps)
13429
wenzelm
parents: 13428
diff changeset
   860
apply (intro FOL_reflections pair_reflection)
13314
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   861
done
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   862
13429
wenzelm
parents: 13428
diff changeset
   863
13339
0f89104dd377 Fixed quantified variable name preservation for ball and bex (bounded quants)
paulson
parents: 13323
diff changeset
   864
subsubsection{*Field of a Relation, Internalized*}
13323
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   865
13429
wenzelm
parents: 13428
diff changeset
   866
(* "is_field(M,r,z) ==
wenzelm
parents: 13428
diff changeset
   867
        \<exists>dr[M]. is_domain(M,r,dr) &
13323
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   868
            (\<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
   869
constdefs field_fm :: "[i,i]=>i"
13429
wenzelm
parents: 13428
diff changeset
   870
    "field_fm(r,z) ==
wenzelm
parents: 13428
diff changeset
   871
       Exists(And(domain_fm(succ(r),0),
wenzelm
parents: 13428
diff changeset
   872
              Exists(And(range_fm(succ(succ(r)),0),
13323
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   873
                         union_fm(1,0,succ(succ(z)))))))"
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   874
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   875
lemma field_type [TC]:
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   876
     "[| x \<in> nat; y \<in> nat |] ==> field_fm(x,y) \<in> formula"
13429
wenzelm
parents: 13428
diff changeset
   877
by (simp add: field_fm_def)
13323
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   878
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   879
lemma arity_field_fm [simp]:
13429
wenzelm
parents: 13428
diff changeset
   880
     "[| x \<in> nat; y \<in> nat |]
13323
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   881
      ==> arity(field_fm(x,y)) = succ(x) \<union> succ(y)"
13429
wenzelm
parents: 13428
diff changeset
   882
by (simp add: field_fm_def succ_Un_distrib [symmetric] Un_ac)
13323
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   883
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   884
lemma sats_field_fm [simp]:
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   885
   "[| x \<in> nat; y \<in> nat; env \<in> list(A)|]
13429
wenzelm
parents: 13428
diff changeset
   886
    ==> sats(A, field_fm(x,y), env) <->
13323
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   887
        is_field(**A, nth(x,env), nth(y,env))"
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   888
by (simp add: field_fm_def is_field_def)
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   889
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   890
lemma field_iff_sats:
13429
wenzelm
parents: 13428
diff changeset
   891
      "[| nth(i,env) = x; nth(j,env) = y;
13323
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   892
          i \<in> nat; j \<in> nat; env \<in> list(A)|]
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   893
       ==> is_field(**A, x, y) <-> sats(A, field_fm(i,j), env)"
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   894
by simp
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   895
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   896
theorem field_reflection:
13429
wenzelm
parents: 13428
diff changeset
   897
     "REFLECTS[\<lambda>x. is_field(L,f(x),g(x)),
13323
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   898
               \<lambda>i x. is_field(**Lset(i),f(x),g(x))]"
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   899
apply (simp only: is_field_def setclass_simps)
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   900
apply (intro FOL_reflections domain_reflection range_reflection
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   901
             union_reflection)
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   902
done
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   903
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   904
13339
0f89104dd377 Fixed quantified variable name preservation for ball and bex (bounded quants)
paulson
parents: 13323
diff changeset
   905
subsubsection{*Image under a Relation, Internalized*}
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   906
13429
wenzelm
parents: 13428
diff changeset
   907
(* "image(M,r,A,z) ==
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   908
        \<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
   909
constdefs image_fm :: "[i,i,i]=>i"
13429
wenzelm
parents: 13428
diff changeset
   910
    "image_fm(r,A,z) ==
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   911
       Forall(Iff(Member(0,succ(z)),
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   912
                  Exists(And(Member(0,succ(succ(r))),
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   913
                             Exists(And(Member(0,succ(succ(succ(A)))),
13429
wenzelm
parents: 13428
diff changeset
   914
                                        pair_fm(0,2,1)))))))"
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   915
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   916
lemma image_type [TC]:
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   917
     "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> image_fm(x,y,z) \<in> formula"
13429
wenzelm
parents: 13428
diff changeset
   918
by (simp add: image_fm_def)
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   919
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   920
lemma arity_image_fm [simp]:
13429
wenzelm
parents: 13428
diff changeset
   921
     "[| x \<in> nat; y \<in> nat; z \<in> nat |]
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   922
      ==> arity(image_fm(x,y,z)) = succ(x) \<union> succ(y) \<union> succ(z)"
13429
wenzelm
parents: 13428
diff changeset
   923
by (simp add: image_fm_def succ_Un_distrib [symmetric] Un_ac)
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   924
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   925
lemma sats_image_fm [simp]:
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   926
   "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
13429
wenzelm
parents: 13428
diff changeset
   927
    ==> sats(A, image_fm(x,y,z), env) <->
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   928
        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
   929
by (simp add: image_fm_def Relative.image_def)
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   930
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   931
lemma image_iff_sats:
13429
wenzelm
parents: 13428
diff changeset
   932
      "[| 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
   933
          i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   934
       ==> image(**A, x, y, z) <-> sats(A, image_fm(i,j,k), env)"
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   935
by (simp add: sats_image_fm)
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   936
13314
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   937
theorem image_reflection:
13429
wenzelm
parents: 13428
diff changeset
   938
     "REFLECTS[\<lambda>x. image(L,f(x),g(x),h(x)),
13314
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   939
               \<lambda>i x. image(**Lset(i),f(x),g(x),h(x))]"
13323
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   940
apply (simp only: Relative.image_def setclass_simps)
13429
wenzelm
parents: 13428
diff changeset
   941
apply (intro FOL_reflections pair_reflection)
13314
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   942
done
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   943
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   944
13348
374d05460db4 Separation/Replacement up to M_wfrank!
paulson
parents: 13339
diff changeset
   945
subsubsection{*Pre-Image under a Relation, Internalized*}
374d05460db4 Separation/Replacement up to M_wfrank!
paulson
parents: 13339
diff changeset
   946
13429
wenzelm
parents: 13428
diff changeset
   947
(* "pre_image(M,r,A,z) ==
wenzelm
parents: 13428
diff changeset
   948
        \<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
   949
constdefs pre_image_fm :: "[i,i,i]=>i"
13429
wenzelm
parents: 13428
diff changeset
   950
    "pre_image_fm(r,A,z) ==
13348
374d05460db4 Separation/Replacement up to M_wfrank!
paulson
parents: 13339
diff changeset
   951
       Forall(Iff(Member(0,succ(z)),
374d05460db4 Separation/Replacement up to M_wfrank!
paulson
parents: 13339
diff changeset
   952
                  Exists(And(Member(0,succ(succ(r))),
374d05460db4 Separation/Replacement up to M_wfrank!
paulson
parents: 13339
diff changeset
   953
                             Exists(And(Member(0,succ(succ(succ(A)))),
13429
wenzelm
parents: 13428
diff changeset
   954
                                        pair_fm(2,0,1)))))))"
13348
374d05460db4 Separation/Replacement up to M_wfrank!
paulson
parents: 13339
diff changeset
   955
374d05460db4 Separation/Replacement up to M_wfrank!
paulson
parents: 13339
diff changeset
   956
lemma pre_image_type [TC]:
374d05460db4 Separation/Replacement up to M_wfrank!
paulson
parents: 13339
diff changeset
   957
     "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> pre_image_fm(x,y,z) \<in> formula"
13429
wenzelm
parents: 13428
diff changeset
   958
by (simp add: pre_image_fm_def)
13348
374d05460db4 Separation/Replacement up to M_wfrank!
paulson
parents: 13339
diff changeset
   959
374d05460db4 Separation/Replacement up to M_wfrank!
paulson
parents: 13339
diff changeset
   960
lemma arity_pre_image_fm [simp]:
13429
wenzelm
parents: 13428
diff changeset
   961
     "[| x \<in> nat; y \<in> nat; z \<in> nat |]
13348
374d05460db4 Separation/Replacement up to M_wfrank!
paulson
parents: 13339
diff changeset
   962
      ==> arity(pre_image_fm(x,y,z)) = succ(x) \<union> succ(y) \<union> succ(z)"
13429
wenzelm
parents: 13428
diff changeset
   963
by (simp add: pre_image_fm_def succ_Un_distrib [symmetric] Un_ac)
13348
374d05460db4 Separation/Replacement up to M_wfrank!
paulson
parents: 13339
diff changeset
   964
374d05460db4 Separation/Replacement up to M_wfrank!
paulson
parents: 13339
diff changeset
   965
lemma sats_pre_image_fm [simp]:
374d05460db4 Separation/Replacement up to M_wfrank!
paulson
parents: 13339
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, pre_image_fm(x,y,z), env) <->
13348
374d05460db4 Separation/Replacement up to M_wfrank!
paulson
parents: 13339
diff changeset
   968
        pre_image(**A, nth(x,env), nth(y,env), nth(z,env))"
374d05460db4 Separation/Replacement up to M_wfrank!
paulson
parents: 13339
diff changeset
   969
by (simp add: pre_image_fm_def Relative.pre_image_def)
374d05460db4 Separation/Replacement up to M_wfrank!
paulson
parents: 13339
diff changeset
   970
374d05460db4 Separation/Replacement up to M_wfrank!
paulson
parents: 13339
diff changeset
   971
lemma pre_image_iff_sats:
13429
wenzelm
parents: 13428
diff changeset
   972
      "[| 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
   973
          i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
374d05460db4 Separation/Replacement up to M_wfrank!
paulson
parents: 13339
diff changeset
   974
       ==> pre_image(**A, x, y, z) <-> sats(A, pre_image_fm(i,j,k), env)"
374d05460db4 Separation/Replacement up to M_wfrank!
paulson
parents: 13339
diff changeset
   975
by (simp add: sats_pre_image_fm)
374d05460db4 Separation/Replacement up to M_wfrank!
paulson
parents: 13339
diff changeset
   976
374d05460db4 Separation/Replacement up to M_wfrank!
paulson
parents: 13339
diff changeset
   977
theorem pre_image_reflection:
13429
wenzelm
parents: 13428
diff changeset
   978
     "REFLECTS[\<lambda>x. pre_image(L,f(x),g(x),h(x)),
13348
374d05460db4 Separation/Replacement up to M_wfrank!
paulson
parents: 13339
diff changeset
   979
               \<lambda>i x. pre_image(**Lset(i),f(x),g(x),h(x))]"
374d05460db4 Separation/Replacement up to M_wfrank!
paulson
parents: 13339
diff changeset
   980
apply (simp only: Relative.pre_image_def setclass_simps)
13429
wenzelm
parents: 13428
diff changeset
   981
apply (intro FOL_reflections pair_reflection)
13348
374d05460db4 Separation/Replacement up to M_wfrank!
paulson
parents: 13339
diff changeset
   982
done
374d05460db4 Separation/Replacement up to M_wfrank!
paulson
parents: 13339
diff changeset
   983
374d05460db4 Separation/Replacement up to M_wfrank!
paulson
parents: 13339
diff changeset
   984
13352
3cd767f8d78b new definitions of fun_apply and M_is_recfun
paulson
parents: 13348
diff changeset
   985
subsubsection{*Function Application, Internalized*}
3cd767f8d78b new definitions of fun_apply and M_is_recfun
paulson
parents: 13348
diff changeset
   986
13429
wenzelm
parents: 13428
diff changeset
   987
(* "fun_apply(M,f,x,y) ==
wenzelm
parents: 13428
diff changeset
   988
        (\<exists>xs[M]. \<exists>fxs[M].
13352
3cd767f8d78b new definitions of fun_apply and M_is_recfun
paulson
parents: 13348
diff changeset
   989
         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
   990
constdefs fun_apply_fm :: "[i,i,i]=>i"
13429
wenzelm
parents: 13428
diff changeset
   991
    "fun_apply_fm(f,x,y) ==
13352
3cd767f8d78b new definitions of fun_apply and M_is_recfun
paulson
parents: 13348
diff changeset
   992
       Exists(Exists(And(upair_fm(succ(succ(x)), succ(succ(x)), 1),
13429
wenzelm
parents: 13428
diff changeset
   993
                         And(image_fm(succ(succ(f)), 1, 0),
13352
3cd767f8d78b new definitions of fun_apply and M_is_recfun
paulson
parents: 13348
diff changeset
   994
                             big_union_fm(0,succ(succ(y)))))))"
3cd767f8d78b new definitions of fun_apply and M_is_recfun
paulson
parents: 13348
diff changeset
   995
3cd767f8d78b new definitions of fun_apply and M_is_recfun
paulson
parents: 13348
diff changeset
   996
lemma fun_apply_type [TC]:
3cd767f8d78b new definitions of fun_apply and M_is_recfun
paulson
parents: 13348
diff changeset
   997
     "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> fun_apply_fm(x,y,z) \<in> formula"
13429
wenzelm
parents: 13428
diff changeset
   998
by (simp add: fun_apply_fm_def)
13352
3cd767f8d78b new definitions of fun_apply and M_is_recfun
paulson
parents: 13348
diff changeset
   999
3cd767f8d78b new definitions of fun_apply and M_is_recfun
paulson
parents: 13348
diff changeset
  1000
lemma arity_fun_apply_fm [simp]:
13429
wenzelm
parents: 13428
diff changeset
  1001
     "[| x \<in> nat; y \<in> nat; z \<in> nat |]
13352
3cd767f8d78b new definitions of fun_apply and M_is_recfun
paulson
parents: 13348
diff changeset
  1002
      ==> arity(fun_apply_fm(x,y,z)) = succ(x) \<union> succ(y) \<union> succ(z)"
13429
wenzelm
parents: 13428
diff changeset
  1003
by (simp add: fun_apply_fm_def succ_Un_distrib [symmetric] Un_ac)
13352
3cd767f8d78b new definitions of fun_apply and M_is_recfun
paulson
parents: 13348
diff changeset
  1004
3cd767f8d78b new definitions of fun_apply and M_is_recfun
paulson
parents: 13348
diff changeset
  1005
lemma sats_fun_apply_fm [simp]:
3cd767f8d78b new definitions of fun_apply and M_is_recfun
paulson
parents: 13348
diff changeset
  1006
   "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
13429
wenzelm
parents: 13428
diff changeset
  1007
    ==> sats(A, fun_apply_fm(x,y,z), env) <->
13352
3cd767f8d78b new definitions of fun_apply and M_is_recfun
paulson
parents: 13348
diff changeset
  1008
        fun_apply(**A, nth(x,env), nth(y,env), nth(z,env))"
3cd767f8d78b new definitions of fun_apply and M_is_recfun
paulson
parents: 13348
diff changeset
  1009
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
  1010
3cd767f8d78b new definitions of fun_apply and M_is_recfun
paulson
parents: 13348
diff changeset
  1011
lemma fun_apply_iff_sats:
13429
wenzelm
parents: 13428
diff changeset
  1012
      "[| 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
  1013
          i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
3cd767f8d78b new definitions of fun_apply and M_is_recfun
paulson
parents: 13348
diff changeset
  1014
       ==> fun_apply(**A, x, y, z) <-> sats(A, fun_apply_fm(i,j,k), env)"
3cd767f8d78b new definitions of fun_apply and M_is_recfun
paulson
parents: 13348
diff changeset
  1015
by simp
3cd767f8d78b new definitions of fun_apply and M_is_recfun
paulson
parents: 13348
diff changeset
  1016
3cd767f8d78b new definitions of fun_apply and M_is_recfun
paulson
parents: 13348
diff changeset
  1017
theorem fun_apply_reflection:
13429
wenzelm
parents: 13428
diff changeset
  1018
     "REFLECTS[\<lambda>x. fun_apply(L,f(x),g(x),h(x)),
wenzelm
parents: 13428
diff changeset
  1019
               \<lambda>i x. fun_apply(**Lset(i),f(x),g(x),h(x))]"
13352
3cd767f8d78b new definitions of fun_apply and M_is_recfun
paulson
parents: 13348
diff changeset
  1020
apply (simp only: fun_apply_def setclass_simps)
3cd767f8d78b new definitions of fun_apply and M_is_recfun
paulson
parents: 13348
diff changeset
  1021
apply (intro FOL_reflections upair_reflection image_reflection
13429
wenzelm
parents: 13428
diff changeset
  1022
             big_union_reflection)
13352
3cd767f8d78b new definitions of fun_apply and M_is_recfun
paulson
parents: 13348
diff changeset
  1023
done
3cd767f8d78b new definitions of fun_apply and M_is_recfun
paulson
parents: 13348
diff changeset
  1024
3cd767f8d78b new definitions of fun_apply and M_is_recfun
paulson
parents: 13348
diff changeset
  1025
13339
0f89104dd377 Fixed quantified variable name preservation for ball and bex (bounded quants)
paulson
parents: 13323
diff changeset
  1026
subsubsection{*The Concept of Relation, Internalized*}
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
  1027
13429
wenzelm
parents: 13428
diff changeset
  1028
(* "is_relation(M,r) ==
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
  1029
        (\<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
  1030
constdefs relation_fm :: "i=>i"
13429
wenzelm
parents: 13428
diff changeset
  1031
    "relation_fm(r) ==
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
  1032
       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
  1033
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
  1034
lemma relation_type [TC]:
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
  1035
     "[| x \<in> nat |] ==> relation_fm(x) \<in> formula"
13429
wenzelm
parents: 13428
diff changeset
  1036
by (simp add: relation_fm_def)
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
  1037
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
  1038
lemma arity_relation_fm [simp]:
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
  1039
     "x \<in> nat ==> arity(relation_fm(x)) = succ(x)"
13429
wenzelm
parents: 13428
diff changeset
  1040
by (simp add: relation_fm_def succ_Un_distrib [symmetric] Un_ac)
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
  1041
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
  1042
lemma sats_relation_fm [simp]:
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
  1043
   "[| x \<in> nat; env \<in> list(A)|]
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
  1044
    ==> sats(A, relation_fm(x), env) <-> is_relation(**A, nth(x,env))"
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
  1045
by (simp add: relation_fm_def is_relation_def)
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
  1046
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
  1047
lemma relation_iff_sats:
13429
wenzelm
parents: 13428
diff changeset
  1048
      "[| nth(i,env) = x; nth(j,env) = y;
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
  1049
          i \<in> nat; env \<in> list(A)|]
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
  1050
       ==> is_relation(**A, x) <-> sats(A, relation_fm(i), env)"
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
  1051
by simp
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
  1052
13314
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
  1053
theorem is_relation_reflection:
13429
wenzelm
parents: 13428
diff changeset
  1054
     "REFLECTS[\<lambda>x. is_relation(L,f(x)),
13314
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
  1055
               \<lambda>i x. is_relation(**Lset(i),f(x))]"
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
  1056
apply (simp only: is_relation_def setclass_simps)
13429
wenzelm
parents: 13428
diff changeset
  1057
apply (intro FOL_reflections pair_reflection)
13314
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
  1058
done
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
  1059
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
  1060
13339
0f89104dd377 Fixed quantified variable name preservation for ball and bex (bounded quants)
paulson
parents: 13323
diff changeset
  1061
subsubsection{*The Concept of Function, Internalized*}
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
  1062
13429
wenzelm
parents: 13428
diff changeset
  1063
(* "is_function(M,r) ==
wenzelm
parents: 13428
diff changeset
  1064
        \<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
  1065
           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
  1066
constdefs function_fm :: "i=>i"
13429
wenzelm
parents: 13428
diff changeset
  1067
    "function_fm(r) ==
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
  1068
       Forall(Forall(Forall(Forall(Forall(
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
  1069
         Implies(pair_fm(4,3,1),
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
  1070
                 Implies(pair_fm(4,2,0),
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
  1071
                         Implies(Member(1,r#+5),
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
  1072
                                 Implies(Member(0,r#+5), Equal(3,2))))))))))"
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
  1073
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
  1074
lemma function_type [TC]:
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
  1075
     "[| x \<in> nat |] ==> function_fm(x) \<in> formula"
13429
wenzelm
parents: 13428
diff changeset
  1076
by (simp add: function_fm_def)
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
  1077
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
  1078
lemma arity_function_fm [simp]:
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
  1079
     "x \<in> nat ==> arity(function_fm(x)) = succ(x)"
13429
wenzelm
parents: 13428
diff changeset
  1080
by (simp add: function_fm_def succ_Un_distrib [symmetric] Un_ac)
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
  1081
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
  1082
lemma sats_function_fm [simp]:
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
  1083
   "[| x \<in> nat; env \<in> list(A)|]
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
  1084
    ==> sats(A, function_fm(x), env) <-> is_function(**A, nth(x,env))"
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
  1085
by (simp add: function_fm_def is_function_def)
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
  1086
13505
52a16cb7fefb Relativized right up to L satisfies V=L!
paulson
parents: 13496
diff changeset
  1087
lemma is_function_iff_sats:
13429
wenzelm
parents: 13428
diff changeset
  1088
      "[| nth(i,env) = x; nth(j,env) = y;
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
  1089
          i \<in> nat; env \<in> list(A)|]
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
  1090
       ==> is_function(**A, x) <-> sats(A, function_fm(i), env)"
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
  1091
by simp
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
  1092
13314
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
  1093
theorem is_function_reflection:
13429
wenzelm
parents: 13428
diff changeset
  1094
     "REFLECTS[\<lambda>x. is_function(L,f(x)),
13314
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
  1095
               \<lambda>i x. is_function(**Lset(i),f(x))]"
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
  1096
apply (simp only: is_function_def setclass_simps)
13429
wenzelm
parents: 13428
diff changeset
  1097
apply (intro FOL_reflections pair_reflection)
13314
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
  1098
done
13298
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
  1099
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
  1100
13339
0f89104dd377 Fixed quantified variable name preservation for ball and bex (bounded quants)
paulson
parents: 13323
diff changeset
  1101
subsubsection{*Typed Functions, Internalized*}
13309
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1102
13429
wenzelm
parents: 13428
diff changeset
  1103
(* "typed_function(M,A,B,r) ==
13309
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1104
        is_function(M,r) & is_relation(M,r) & is_domain(M,r,A) &
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1105
        (\<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
  1106
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1107
constdefs typed_function_fm :: "[i,i,i]=>i"
13429
wenzelm
parents: 13428
diff changeset
  1108
    "typed_function_fm(A,B,r) ==
13309
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1109
       And(function_fm(r),
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1110
         And(relation_fm(r),
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1111
           And(domain_fm(r,A),
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1112
             Forall(Implies(Member(0,succ(r)),
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1113
                  Forall(Forall(Implies(pair_fm(1,0,2),Member(0,B#+3)))))))))"
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1114
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1115
lemma typed_function_type [TC]:
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1116
     "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> typed_function_fm(x,y,z) \<in> formula"
13429
wenzelm
parents: 13428
diff changeset
  1117
by (simp add: typed_function_fm_def)
13309
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1118
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1119
lemma arity_typed_function_fm [simp]:
13429
wenzelm
parents: 13428
diff changeset
  1120
     "[| x \<in> nat; y \<in> nat; z \<in> nat |]
13309
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1121
      ==> arity(typed_function_fm(x,y,z)) = succ(x) \<union> succ(y) \<union> succ(z)"
13429
wenzelm
parents: 13428
diff changeset
  1122
by (simp add: typed_function_fm_def succ_Un_distrib [symmetric] Un_ac)
13309
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1123
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1124
lemma sats_typed_function_fm [simp]:
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1125
   "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
13429
wenzelm
parents: 13428
diff changeset
  1126
    ==> sats(A, typed_function_fm(x,y,z), env) <->
13309
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1127
        typed_function(**A, nth(x,env), nth(y,env), nth(z,env))"
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1128
by (simp add: typed_function_fm_def typed_function_def)
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 typed_function_iff_sats:
13429
wenzelm
parents: 13428
diff changeset
  1131
  "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z;
13309
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1132
      i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1133
   ==> typed_function(**A, x, y, z) <-> sats(A, typed_function_fm(i,j,k), env)"
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1134
by simp
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1135
13429
wenzelm
parents: 13428
diff changeset
  1136
lemmas function_reflections =
13363
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
  1137
        empty_reflection number1_reflection
13429
wenzelm
parents: 13428
diff changeset
  1138
        upair_reflection pair_reflection union_reflection
wenzelm
parents: 13428
diff changeset
  1139
        big_union_reflection cons_reflection successor_reflection
13323
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
  1140
        fun_apply_reflection subset_reflection
13429
wenzelm
parents: 13428
diff changeset
  1141
        transitive_set_reflection membership_reflection
wenzelm
parents: 13428
diff changeset
  1142
        pred_set_reflection domain_reflection range_reflection field_reflection
13348
374d05460db4 Separation/Replacement up to M_wfrank!
paulson
parents: 13339
diff changeset
  1143
        image_reflection pre_image_reflection
13429
wenzelm
parents: 13428
diff changeset
  1144
        is_relation_reflection is_function_reflection
13309
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1145
13429
wenzelm
parents: 13428
diff changeset
  1146
lemmas function_iff_sats =
wenzelm
parents: 13428
diff changeset
  1147
        empty_iff_sats number1_iff_sats
wenzelm
parents: 13428
diff changeset
  1148
        upair_iff_sats pair_iff_sats union_iff_sats
13505
52a16cb7fefb Relativized right up to L satisfies V=L!
paulson
parents: 13496
diff changeset
  1149
        big_union_iff_sats cons_iff_sats successor_iff_sats
13323
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
  1150
        fun_apply_iff_sats  Memrel_iff_sats
13429
wenzelm
parents: 13428
diff changeset
  1151
        pred_set_iff_sats domain_iff_sats range_iff_sats field_iff_sats
wenzelm
parents: 13428
diff changeset
  1152
        image_iff_sats pre_image_iff_sats
13505
52a16cb7fefb Relativized right up to L satisfies V=L!
paulson
parents: 13496
diff changeset
  1153
        relation_iff_sats is_function_iff_sats
13323
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
  1154
13309
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1155
13314
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
  1156
theorem typed_function_reflection:
13429
wenzelm
parents: 13428
diff changeset
  1157
     "REFLECTS[\<lambda>x. typed_function(L,f(x),g(x),h(x)),
13314
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
  1158
               \<lambda>i x. typed_function(**Lset(i),f(x),g(x),h(x))]"
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
  1159
apply (simp only: typed_function_def setclass_simps)
13429
wenzelm
parents: 13428
diff changeset
  1160
apply (intro FOL_reflections function_reflections)
13323
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
  1161
done
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
  1162
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
  1163
13339
0f89104dd377 Fixed quantified variable name preservation for ball and bex (bounded quants)
paulson
parents: 13323
diff changeset
  1164
subsubsection{*Composition of Relations, Internalized*}
13323
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
  1165
13429
wenzelm
parents: 13428
diff changeset
  1166
(* "composition(M,r,s,t) ==
wenzelm
parents: 13428
diff changeset
  1167
        \<forall>p[M]. p \<in> t <->
wenzelm
parents: 13428
diff changeset
  1168
               (\<exists>x[M]. \<exists>y[M]. \<exists>z[M]. \<exists>xy[M]. \<exists>yz[M].
wenzelm
parents: 13428
diff changeset
  1169
                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
  1170
                xy \<in> s & yz \<in> r)" *)
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
  1171
constdefs composition_fm :: "[i,i,i]=>i"
13429
wenzelm
parents: 13428
diff changeset
  1172
  "composition_fm(r,s,t) ==
13323
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
  1173
     Forall(Iff(Member(0,succ(t)),
13429
wenzelm
parents: 13428
diff changeset
  1174
             Exists(Exists(Exists(Exists(Exists(
13323
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
  1175
              And(pair_fm(4,2,5),
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
  1176
               And(pair_fm(4,3,1),
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
  1177
                And(pair_fm(3,2,0),
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
  1178
                 And(Member(1,s#+6), Member(0,r#+6))))))))))))"
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
  1179
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
  1180
lemma composition_type [TC]:
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
  1181
     "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> composition_fm(x,y,z) \<in> formula"
13429
wenzelm
parents: 13428
diff changeset
  1182
by (simp add: composition_fm_def)
13323
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
  1183
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
  1184
lemma arity_composition_fm [simp]:
13429
wenzelm
parents: 13428
diff changeset
  1185
     "[| x \<in> nat; y \<in> nat; z \<in> nat |]
13323
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
  1186
      ==> arity(composition_fm(x,y,z)) = succ(x) \<union> succ(y) \<union> succ(z)"
13429
wenzelm
parents: 13428
diff changeset
  1187
by (simp add: composition_fm_def succ_Un_distrib [symmetric] Un_ac)
13323
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
  1188
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
  1189
lemma sats_composition_fm [simp]:
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
  1190
   "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
13429
wenzelm
parents: 13428
diff changeset
  1191
    ==> sats(A, composition_fm(x,y,z), env) <->
13323
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
  1192
        composition(**A, nth(x,env), nth(y,env), nth(z,env))"
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
  1193
by (simp add: composition_fm_def composition_def)
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
  1194
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
  1195
lemma composition_iff_sats:
13429
wenzelm
parents: 13428
diff changeset
  1196
      "[| 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
  1197
          i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
  1198
       ==> composition(**A, x, y, z) <-> sats(A, composition_fm(i,j,k), env)"
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
  1199
by simp
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
  1200
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
  1201
theorem composition_reflection:
13429
wenzelm
parents: 13428
diff changeset
  1202
     "REFLECTS[\<lambda>x. composition(L,f(x),g(x),h(x)),
13323
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
  1203
               \<lambda>i x. composition(**Lset(i),f(x),g(x),h(x))]"
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
  1204
apply (simp only: composition_def setclass_simps)
13429
wenzelm
parents: 13428
diff changeset
  1205
apply (intro FOL_reflections pair_reflection)
13314
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
  1206
done
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
  1207
13309
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1208
13339
0f89104dd377 Fixed quantified variable name preservation for ball and bex (bounded quants)
paulson
parents: 13323
diff changeset
  1209
subsubsection{*Injections, Internalized*}
13309
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1210
13429
wenzelm
parents: 13428
diff changeset
  1211
(* "injection(M,A,B,f) ==
wenzelm
parents: 13428
diff changeset
  1212
        typed_function(M,A,B,f) &
wenzelm
parents: 13428
diff changeset
  1213
        (\<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
  1214
          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
  1215
constdefs injection_fm :: "[i,i,i]=>i"
13429
wenzelm
parents: 13428
diff changeset
  1216
 "injection_fm(A,B,f) ==
13309
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1217
    And(typed_function_fm(A,B,f),
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1218
       Forall(Forall(Forall(Forall(Forall(
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1219
         Implies(pair_fm(4,2,1),
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1220
                 Implies(pair_fm(3,2,0),
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1221
                         Implies(Member(1,f#+5),
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1222
                                 Implies(Member(0,f#+5), Equal(4,3)))))))))))"
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1223
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1224
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1225
lemma injection_type [TC]:
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1226
     "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> injection_fm(x,y,z) \<in> formula"
13429
wenzelm
parents: 13428
diff changeset
  1227
by (simp add: injection_fm_def)
13309
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1228
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1229
lemma arity_injection_fm [simp]:
13429
wenzelm
parents: 13428
diff changeset
  1230
     "[| x \<in> nat; y \<in> nat; z \<in> nat |]
13309
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1231
      ==> arity(injection_fm(x,y,z)) = succ(x) \<union> succ(y) \<union> succ(z)"
13429
wenzelm
parents: 13428
diff changeset
  1232
by (simp add: injection_fm_def succ_Un_distrib [symmetric] Un_ac)
13309
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1233
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1234
lemma sats_injection_fm [simp]:
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1235
   "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
13429
wenzelm
parents: 13428
diff changeset
  1236
    ==> sats(A, injection_fm(x,y,z), env) <->
13309
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1237
        injection(**A, nth(x,env), nth(y,env), nth(z,env))&qu