src/ZF/Constructible/L_axioms.thy
author paulson
Tue Jul 16 16:29:36 2002 +0200 (2002-07-16)
changeset 13363 c26eeb000470
parent 13352 3cd767f8d78b
child 13385 31df66ca0780
permissions -rw-r--r--
instantiation of locales M_trancl and M_wfrank;
proofs of list_replacement{1,2}
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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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ML
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{*
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val transL = thm "transL";
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val nonempty = thm "nonempty";
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val upair_ax = thm "upair_ax";
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val Union_ax = thm "Union_ax";
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val power_ax = thm "power_ax";
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val replacement = thm "replacement";
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val L_nat = thm "L_nat";
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fun kill_flex_triv_prems st = Seq.hd ((REPEAT_FIRST assume_tac) st);
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fun triv_axioms_L th =
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    kill_flex_triv_prems 
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       ([transL, nonempty, upair_ax, Union_ax, power_ax, replacement, L_nat] 
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        MRS (inst "M" "L" th));
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bind_thm ("ball_abs", triv_axioms_L (thm "M_triv_axioms.ball_abs"));
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bind_thm ("rall_abs", triv_axioms_L (thm "M_triv_axioms.rall_abs"));
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bind_thm ("bex_abs", triv_axioms_L (thm "M_triv_axioms.bex_abs"));
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bind_thm ("rex_abs", triv_axioms_L (thm "M_triv_axioms.rex_abs"));
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bind_thm ("ball_iff_equiv", triv_axioms_L (thm "M_triv_axioms.ball_iff_equiv"));
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bind_thm ("M_equalityI", triv_axioms_L (thm "M_triv_axioms.M_equalityI"));
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bind_thm ("empty_abs", triv_axioms_L (thm "M_triv_axioms.empty_abs"));
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bind_thm ("subset_abs", triv_axioms_L (thm "M_triv_axioms.subset_abs"));
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bind_thm ("upair_abs", triv_axioms_L (thm "M_triv_axioms.upair_abs"));
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bind_thm ("upair_in_M_iff", triv_axioms_L (thm "M_triv_axioms.upair_in_M_iff"));
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bind_thm ("singleton_in_M_iff", triv_axioms_L (thm "M_triv_axioms.singleton_in_M_iff"));
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bind_thm ("pair_abs", triv_axioms_L (thm "M_triv_axioms.pair_abs"));
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bind_thm ("pair_in_M_iff", triv_axioms_L (thm "M_triv_axioms.pair_in_M_iff"));
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bind_thm ("pair_components_in_M", triv_axioms_L (thm "M_triv_axioms.pair_components_in_M"));
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bind_thm ("cartprod_abs", triv_axioms_L (thm "M_triv_axioms.cartprod_abs"));
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bind_thm ("union_abs", triv_axioms_L (thm "M_triv_axioms.union_abs"));
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bind_thm ("inter_abs", triv_axioms_L (thm "M_triv_axioms.inter_abs"));
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bind_thm ("setdiff_abs", triv_axioms_L (thm "M_triv_axioms.setdiff_abs"));
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bind_thm ("Union_abs", triv_axioms_L (thm "M_triv_axioms.Union_abs"));
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bind_thm ("Union_closed", triv_axioms_L (thm "M_triv_axioms.Union_closed"));
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bind_thm ("Un_closed", triv_axioms_L (thm "M_triv_axioms.Un_closed"));
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bind_thm ("cons_closed", triv_axioms_L (thm "M_triv_axioms.cons_closed"));
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bind_thm ("successor_abs", triv_axioms_L (thm "M_triv_axioms.successor_abs"));
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bind_thm ("succ_in_M_iff", triv_axioms_L (thm "M_triv_axioms.succ_in_M_iff"));
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bind_thm ("separation_closed", triv_axioms_L (thm "M_triv_axioms.separation_closed"));
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bind_thm ("strong_replacementI", triv_axioms_L (thm "M_triv_axioms.strong_replacementI"));
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bind_thm ("strong_replacement_closed", triv_axioms_L (thm "M_triv_axioms.strong_replacement_closed"));
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bind_thm ("RepFun_closed", triv_axioms_L (thm "M_triv_axioms.RepFun_closed"));
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bind_thm ("lam_closed", triv_axioms_L (thm "M_triv_axioms.lam_closed"));
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bind_thm ("image_abs", triv_axioms_L (thm "M_triv_axioms.image_abs"));
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bind_thm ("powerset_Pow", triv_axioms_L (thm "M_triv_axioms.powerset_Pow"));
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bind_thm ("powerset_imp_subset_Pow", triv_axioms_L (thm "M_triv_axioms.powerset_imp_subset_Pow"));
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bind_thm ("nat_into_M", triv_axioms_L (thm "M_triv_axioms.nat_into_M"));
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bind_thm ("nat_case_closed", triv_axioms_L (thm "M_triv_axioms.nat_case_closed"));
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bind_thm ("Inl_in_M_iff", triv_axioms_L (thm "M_triv_axioms.Inl_in_M_iff"));
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bind_thm ("Inr_in_M_iff", triv_axioms_L (thm "M_triv_axioms.Inr_in_M_iff"));
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bind_thm ("lt_closed", triv_axioms_L (thm "M_triv_axioms.lt_closed"));
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bind_thm ("transitive_set_abs", triv_axioms_L (thm "M_triv_axioms.transitive_set_abs"));
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bind_thm ("ordinal_abs", triv_axioms_L (thm "M_triv_axioms.ordinal_abs"));
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bind_thm ("limit_ordinal_abs", triv_axioms_L (thm "M_triv_axioms.limit_ordinal_abs"));
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bind_thm ("successor_ordinal_abs", triv_axioms_L (thm "M_triv_axioms.successor_ordinal_abs"));
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bind_thm ("finite_ordinal_abs", triv_axioms_L (thm "M_triv_axioms.finite_ordinal_abs"));
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bind_thm ("omega_abs", triv_axioms_L (thm "M_triv_axioms.omega_abs"));
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bind_thm ("number1_abs", triv_axioms_L (thm "M_triv_axioms.number1_abs"));
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bind_thm ("number1_abs", triv_axioms_L (thm "M_triv_axioms.number1_abs"));
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bind_thm ("number3_abs", triv_axioms_L (thm "M_triv_axioms.number3_abs"));
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*}
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declare ball_abs [simp] 
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declare rall_abs [simp] 
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declare bex_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 number1_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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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 (rule reflection.Ex_reflection [OF Lset_mono_le Lset_cont Pair_in_Lset],
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       assumption+)
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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)]" 
paulson@13291
   274
apply (unfold L_Reflects_def L_ClEx_def L_FF_def L_F0_def L_def) 
paulson@13314
   275
apply (elim meta_exE) 
paulson@13314
   276
apply (rule meta_exI)
paulson@13291
   277
apply (rule reflection.All_reflection [OF Lset_mono_le Lset_cont Pair_in_Lset],
paulson@13291
   278
       assumption+)
paulson@13291
   279
done
paulson@13291
   280
paulson@13314
   281
theorem Rex_reflection:
paulson@13314
   282
     "REFLECTS[ \<lambda>x. P(fst(x),snd(x)), \<lambda>a x. Q(a,fst(x),snd(x))]
paulson@13314
   283
      ==> REFLECTS[\<lambda>x. \<exists>z[L]. P(x,z), \<lambda>a x. \<exists>z\<in>Lset(a). Q(a,x,z)]"
paulson@13314
   284
apply (unfold rex_def) 
paulson@13314
   285
apply (intro And_reflection Ex_reflection, assumption)
paulson@13314
   286
done
paulson@13291
   287
paulson@13314
   288
theorem Rall_reflection:
paulson@13314
   289
     "REFLECTS[\<lambda>x. P(fst(x),snd(x)), \<lambda>a x. Q(a,fst(x),snd(x))]
paulson@13314
   290
      ==> REFLECTS[\<lambda>x. \<forall>z[L]. P(x,z), \<lambda>a x. \<forall>z\<in>Lset(a). Q(a,x,z)]" 
paulson@13314
   291
apply (unfold rall_def) 
paulson@13314
   292
apply (intro Imp_reflection All_reflection, assumption)
paulson@13314
   293
done
paulson@13314
   294
paulson@13323
   295
lemmas FOL_reflections = 
paulson@13314
   296
        Triv_reflection Not_reflection And_reflection Or_reflection
paulson@13314
   297
        Imp_reflection Iff_reflection Ex_reflection All_reflection
paulson@13314
   298
        Rex_reflection Rall_reflection
paulson@13291
   299
paulson@13291
   300
lemma ReflectsD:
paulson@13314
   301
     "[|REFLECTS[P,Q]; Ord(i)|] 
paulson@13291
   302
      ==> \<exists>j. i<j & (\<forall>x \<in> Lset(j). P(x) <-> Q(j,x))"
paulson@13314
   303
apply (unfold L_Reflects_def Closed_Unbounded_def) 
paulson@13314
   304
apply (elim meta_exE, clarify) 
paulson@13291
   305
apply (blast dest!: UnboundedD) 
paulson@13291
   306
done
paulson@13291
   307
paulson@13291
   308
lemma ReflectsE:
paulson@13314
   309
     "[| REFLECTS[P,Q]; Ord(i);
paulson@13291
   310
         !!j. [|i<j;  \<forall>x \<in> Lset(j). P(x) <-> Q(j,x)|] ==> R |]
paulson@13291
   311
      ==> R"
paulson@13316
   312
apply (drule ReflectsD, assumption, blast) 
paulson@13314
   313
done
paulson@13291
   314
paulson@13291
   315
lemma Collect_mem_eq: "{x\<in>A. x\<in>B} = A \<inter> B";
paulson@13291
   316
by blast
paulson@13291
   317
paulson@13291
   318
paulson@13339
   319
subsection{*Internalized Formulas for some Set-Theoretic Concepts*}
paulson@13298
   320
paulson@13306
   321
lemmas setclass_simps = rall_setclass_is_ball rex_setclass_is_bex
paulson@13306
   322
paulson@13306
   323
subsubsection{*Some numbers to help write de Bruijn indices*}
paulson@13306
   324
paulson@13306
   325
syntax
paulson@13306
   326
    "3" :: i   ("3")
paulson@13306
   327
    "4" :: i   ("4")
paulson@13306
   328
    "5" :: i   ("5")
paulson@13306
   329
    "6" :: i   ("6")
paulson@13306
   330
    "7" :: i   ("7")
paulson@13306
   331
    "8" :: i   ("8")
paulson@13306
   332
    "9" :: i   ("9")
paulson@13306
   333
paulson@13306
   334
translations
paulson@13306
   335
   "3"  == "succ(2)"
paulson@13306
   336
   "4"  == "succ(3)"
paulson@13306
   337
   "5"  == "succ(4)"
paulson@13306
   338
   "6"  == "succ(5)"
paulson@13306
   339
   "7"  == "succ(6)"
paulson@13306
   340
   "8"  == "succ(7)"
paulson@13306
   341
   "9"  == "succ(8)"
paulson@13306
   342
paulson@13323
   343
paulson@13339
   344
subsubsection{*The Empty Set, Internalized*}
paulson@13323
   345
paulson@13323
   346
constdefs empty_fm :: "i=>i"
paulson@13323
   347
    "empty_fm(x) == Forall(Neg(Member(0,succ(x))))"
paulson@13323
   348
paulson@13323
   349
lemma empty_type [TC]:
paulson@13323
   350
     "x \<in> nat ==> empty_fm(x) \<in> formula"
paulson@13323
   351
by (simp add: empty_fm_def) 
paulson@13323
   352
paulson@13323
   353
lemma arity_empty_fm [simp]:
paulson@13323
   354
     "x \<in> nat ==> arity(empty_fm(x)) = succ(x)"
paulson@13323
   355
by (simp add: empty_fm_def succ_Un_distrib [symmetric] Un_ac) 
paulson@13323
   356
paulson@13323
   357
lemma sats_empty_fm [simp]:
paulson@13323
   358
   "[| x \<in> nat; env \<in> list(A)|]
paulson@13323
   359
    ==> sats(A, empty_fm(x), env) <-> empty(**A, nth(x,env))"
paulson@13323
   360
by (simp add: empty_fm_def empty_def)
paulson@13323
   361
paulson@13323
   362
lemma empty_iff_sats:
paulson@13323
   363
      "[| nth(i,env) = x; nth(j,env) = y; 
paulson@13323
   364
          i \<in> nat; env \<in> list(A)|]
paulson@13323
   365
       ==> empty(**A, x) <-> sats(A, empty_fm(i), env)"
paulson@13323
   366
by simp
paulson@13323
   367
paulson@13323
   368
theorem empty_reflection:
paulson@13323
   369
     "REFLECTS[\<lambda>x. empty(L,f(x)), 
paulson@13323
   370
               \<lambda>i x. empty(**Lset(i),f(x))]"
paulson@13323
   371
apply (simp only: empty_def setclass_simps)
paulson@13323
   372
apply (intro FOL_reflections)  
paulson@13323
   373
done
paulson@13323
   374
paulson@13323
   375
paulson@13339
   376
subsubsection{*Unordered Pairs, Internalized*}
paulson@13298
   377
paulson@13298
   378
constdefs upair_fm :: "[i,i,i]=>i"
paulson@13298
   379
    "upair_fm(x,y,z) == 
paulson@13298
   380
       And(Member(x,z), 
paulson@13298
   381
           And(Member(y,z),
paulson@13298
   382
               Forall(Implies(Member(0,succ(z)), 
paulson@13298
   383
                              Or(Equal(0,succ(x)), Equal(0,succ(y)))))))"
paulson@13298
   384
paulson@13298
   385
lemma upair_type [TC]:
paulson@13298
   386
     "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> upair_fm(x,y,z) \<in> formula"
paulson@13298
   387
by (simp add: upair_fm_def) 
paulson@13298
   388
paulson@13298
   389
lemma arity_upair_fm [simp]:
paulson@13298
   390
     "[| x \<in> nat; y \<in> nat; z \<in> nat |] 
paulson@13298
   391
      ==> arity(upair_fm(x,y,z)) = succ(x) \<union> succ(y) \<union> succ(z)"
paulson@13298
   392
by (simp add: upair_fm_def succ_Un_distrib [symmetric] Un_ac) 
paulson@13298
   393
paulson@13298
   394
lemma sats_upair_fm [simp]:
paulson@13298
   395
   "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
paulson@13298
   396
    ==> sats(A, upair_fm(x,y,z), env) <-> 
paulson@13298
   397
            upair(**A, nth(x,env), nth(y,env), nth(z,env))"
paulson@13298
   398
by (simp add: upair_fm_def upair_def)
paulson@13298
   399
paulson@13298
   400
lemma upair_iff_sats:
paulson@13298
   401
      "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z; 
paulson@13298
   402
          i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
paulson@13298
   403
       ==> upair(**A, x, y, z) <-> sats(A, upair_fm(i,j,k), env)"
paulson@13298
   404
by (simp add: sats_upair_fm)
paulson@13298
   405
paulson@13298
   406
text{*Useful? At least it refers to "real" unordered pairs*}
paulson@13298
   407
lemma sats_upair_fm2 [simp]:
paulson@13298
   408
   "[| x \<in> nat; y \<in> nat; z < length(env); env \<in> list(A); Transset(A)|]
paulson@13298
   409
    ==> sats(A, upair_fm(x,y,z), env) <-> 
paulson@13298
   410
        nth(z,env) = {nth(x,env), nth(y,env)}"
paulson@13298
   411
apply (frule lt_length_in_nat, assumption)  
paulson@13298
   412
apply (simp add: upair_fm_def Transset_def, auto) 
paulson@13298
   413
apply (blast intro: nth_type) 
paulson@13298
   414
done
paulson@13298
   415
paulson@13314
   416
theorem upair_reflection:
paulson@13314
   417
     "REFLECTS[\<lambda>x. upair(L,f(x),g(x),h(x)), 
paulson@13314
   418
               \<lambda>i x. upair(**Lset(i),f(x),g(x),h(x))]" 
paulson@13314
   419
apply (simp add: upair_def)
paulson@13323
   420
apply (intro FOL_reflections)  
paulson@13314
   421
done
paulson@13306
   422
paulson@13339
   423
subsubsection{*Ordered pairs, Internalized*}
paulson@13298
   424
paulson@13298
   425
constdefs pair_fm :: "[i,i,i]=>i"
paulson@13298
   426
    "pair_fm(x,y,z) == 
paulson@13298
   427
       Exists(And(upair_fm(succ(x),succ(x),0),
paulson@13298
   428
              Exists(And(upair_fm(succ(succ(x)),succ(succ(y)),0),
paulson@13298
   429
                         upair_fm(1,0,succ(succ(z)))))))"
paulson@13298
   430
paulson@13298
   431
lemma pair_type [TC]:
paulson@13298
   432
     "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> pair_fm(x,y,z) \<in> formula"
paulson@13298
   433
by (simp add: pair_fm_def) 
paulson@13298
   434
paulson@13298
   435
lemma arity_pair_fm [simp]:
paulson@13298
   436
     "[| x \<in> nat; y \<in> nat; z \<in> nat |] 
paulson@13298
   437
      ==> arity(pair_fm(x,y,z)) = succ(x) \<union> succ(y) \<union> succ(z)"
paulson@13298
   438
by (simp add: pair_fm_def succ_Un_distrib [symmetric] Un_ac) 
paulson@13298
   439
paulson@13298
   440
lemma sats_pair_fm [simp]:
paulson@13298
   441
   "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
paulson@13298
   442
    ==> sats(A, pair_fm(x,y,z), env) <-> 
paulson@13298
   443
        pair(**A, nth(x,env), nth(y,env), nth(z,env))"
paulson@13298
   444
by (simp add: pair_fm_def pair_def)
paulson@13298
   445
paulson@13298
   446
lemma pair_iff_sats:
paulson@13298
   447
      "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z; 
paulson@13298
   448
          i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
paulson@13298
   449
       ==> pair(**A, x, y, z) <-> sats(A, pair_fm(i,j,k), env)"
paulson@13298
   450
by (simp add: sats_pair_fm)
paulson@13298
   451
paulson@13314
   452
theorem pair_reflection:
paulson@13314
   453
     "REFLECTS[\<lambda>x. pair(L,f(x),g(x),h(x)), 
paulson@13314
   454
               \<lambda>i x. pair(**Lset(i),f(x),g(x),h(x))]"
paulson@13314
   455
apply (simp only: pair_def setclass_simps)
paulson@13323
   456
apply (intro FOL_reflections upair_reflection)  
paulson@13314
   457
done
paulson@13306
   458
paulson@13306
   459
paulson@13339
   460
subsubsection{*Binary Unions, Internalized*}
paulson@13298
   461
paulson@13306
   462
constdefs union_fm :: "[i,i,i]=>i"
paulson@13306
   463
    "union_fm(x,y,z) == 
paulson@13306
   464
       Forall(Iff(Member(0,succ(z)),
paulson@13306
   465
                  Or(Member(0,succ(x)),Member(0,succ(y)))))"
paulson@13306
   466
paulson@13306
   467
lemma union_type [TC]:
paulson@13306
   468
     "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> union_fm(x,y,z) \<in> formula"
paulson@13306
   469
by (simp add: union_fm_def) 
paulson@13306
   470
paulson@13306
   471
lemma arity_union_fm [simp]:
paulson@13306
   472
     "[| x \<in> nat; y \<in> nat; z \<in> nat |] 
paulson@13306
   473
      ==> arity(union_fm(x,y,z)) = succ(x) \<union> succ(y) \<union> succ(z)"
paulson@13306
   474
by (simp add: union_fm_def succ_Un_distrib [symmetric] Un_ac) 
paulson@13298
   475
paulson@13306
   476
lemma sats_union_fm [simp]:
paulson@13306
   477
   "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
paulson@13306
   478
    ==> sats(A, union_fm(x,y,z), env) <-> 
paulson@13306
   479
        union(**A, nth(x,env), nth(y,env), nth(z,env))"
paulson@13306
   480
by (simp add: union_fm_def union_def)
paulson@13306
   481
paulson@13306
   482
lemma union_iff_sats:
paulson@13306
   483
      "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z; 
paulson@13306
   484
          i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
paulson@13306
   485
       ==> union(**A, x, y, z) <-> sats(A, union_fm(i,j,k), env)"
paulson@13306
   486
by (simp add: sats_union_fm)
paulson@13298
   487
paulson@13314
   488
theorem union_reflection:
paulson@13314
   489
     "REFLECTS[\<lambda>x. union(L,f(x),g(x),h(x)), 
paulson@13314
   490
               \<lambda>i x. union(**Lset(i),f(x),g(x),h(x))]"
paulson@13314
   491
apply (simp only: union_def setclass_simps)
paulson@13323
   492
apply (intro FOL_reflections)  
paulson@13314
   493
done
paulson@13306
   494
paulson@13298
   495
paulson@13339
   496
subsubsection{*Set ``Cons,'' Internalized*}
paulson@13306
   497
paulson@13306
   498
constdefs cons_fm :: "[i,i,i]=>i"
paulson@13306
   499
    "cons_fm(x,y,z) == 
paulson@13306
   500
       Exists(And(upair_fm(succ(x),succ(x),0),
paulson@13306
   501
                  union_fm(0,succ(y),succ(z))))"
paulson@13298
   502
paulson@13298
   503
paulson@13306
   504
lemma cons_type [TC]:
paulson@13306
   505
     "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> cons_fm(x,y,z) \<in> formula"
paulson@13306
   506
by (simp add: cons_fm_def) 
paulson@13306
   507
paulson@13306
   508
lemma arity_cons_fm [simp]:
paulson@13306
   509
     "[| x \<in> nat; y \<in> nat; z \<in> nat |] 
paulson@13306
   510
      ==> arity(cons_fm(x,y,z)) = succ(x) \<union> succ(y) \<union> succ(z)"
paulson@13306
   511
by (simp add: cons_fm_def succ_Un_distrib [symmetric] Un_ac) 
paulson@13306
   512
paulson@13306
   513
lemma sats_cons_fm [simp]:
paulson@13306
   514
   "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
paulson@13306
   515
    ==> sats(A, cons_fm(x,y,z), env) <-> 
paulson@13306
   516
        is_cons(**A, nth(x,env), nth(y,env), nth(z,env))"
paulson@13306
   517
by (simp add: cons_fm_def is_cons_def)
paulson@13306
   518
paulson@13306
   519
lemma cons_iff_sats:
paulson@13306
   520
      "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z; 
paulson@13306
   521
          i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
paulson@13306
   522
       ==> is_cons(**A, x, y, z) <-> sats(A, cons_fm(i,j,k), env)"
paulson@13306
   523
by simp
paulson@13306
   524
paulson@13314
   525
theorem cons_reflection:
paulson@13314
   526
     "REFLECTS[\<lambda>x. is_cons(L,f(x),g(x),h(x)), 
paulson@13314
   527
               \<lambda>i x. is_cons(**Lset(i),f(x),g(x),h(x))]"
paulson@13314
   528
apply (simp only: is_cons_def setclass_simps)
paulson@13323
   529
apply (intro FOL_reflections upair_reflection union_reflection)  
paulson@13323
   530
done
paulson@13323
   531
paulson@13323
   532
paulson@13339
   533
subsubsection{*Successor Function, Internalized*}
paulson@13323
   534
paulson@13323
   535
constdefs succ_fm :: "[i,i]=>i"
paulson@13323
   536
    "succ_fm(x,y) == cons_fm(x,x,y)"
paulson@13323
   537
paulson@13323
   538
lemma succ_type [TC]:
paulson@13323
   539
     "[| x \<in> nat; y \<in> nat |] ==> succ_fm(x,y) \<in> formula"
paulson@13323
   540
by (simp add: succ_fm_def) 
paulson@13323
   541
paulson@13323
   542
lemma arity_succ_fm [simp]:
paulson@13323
   543
     "[| x \<in> nat; y \<in> nat |] 
paulson@13323
   544
      ==> arity(succ_fm(x,y)) = succ(x) \<union> succ(y)"
paulson@13323
   545
by (simp add: succ_fm_def)
paulson@13323
   546
paulson@13323
   547
lemma sats_succ_fm [simp]:
paulson@13323
   548
   "[| x \<in> nat; y \<in> nat; env \<in> list(A)|]
paulson@13323
   549
    ==> sats(A, succ_fm(x,y), env) <-> 
paulson@13323
   550
        successor(**A, nth(x,env), nth(y,env))"
paulson@13323
   551
by (simp add: succ_fm_def successor_def)
paulson@13323
   552
paulson@13323
   553
lemma successor_iff_sats:
paulson@13323
   554
      "[| nth(i,env) = x; nth(j,env) = y; 
paulson@13323
   555
          i \<in> nat; j \<in> nat; env \<in> list(A)|]
paulson@13323
   556
       ==> successor(**A, x, y) <-> sats(A, succ_fm(i,j), env)"
paulson@13323
   557
by simp
paulson@13323
   558
paulson@13323
   559
theorem successor_reflection:
paulson@13323
   560
     "REFLECTS[\<lambda>x. successor(L,f(x),g(x)), 
paulson@13323
   561
               \<lambda>i x. successor(**Lset(i),f(x),g(x))]"
paulson@13323
   562
apply (simp only: successor_def setclass_simps)
paulson@13323
   563
apply (intro cons_reflection)  
paulson@13314
   564
done
paulson@13298
   565
paulson@13298
   566
paulson@13363
   567
subsubsection{*The Number 1, Internalized*}
paulson@13363
   568
paulson@13363
   569
(* "number1(M,a) == (\<exists>x[M]. empty(M,x) & successor(M,x,a))" *)
paulson@13363
   570
constdefs number1_fm :: "i=>i"
paulson@13363
   571
    "number1_fm(a) == Exists(And(empty_fm(0), succ_fm(0,succ(a))))"
paulson@13363
   572
paulson@13363
   573
lemma number1_type [TC]:
paulson@13363
   574
     "x \<in> nat ==> number1_fm(x) \<in> formula"
paulson@13363
   575
by (simp add: number1_fm_def) 
paulson@13363
   576
paulson@13363
   577
lemma arity_number1_fm [simp]:
paulson@13363
   578
     "x \<in> nat ==> arity(number1_fm(x)) = succ(x)"
paulson@13363
   579
by (simp add: number1_fm_def succ_Un_distrib [symmetric] Un_ac) 
paulson@13363
   580
paulson@13363
   581
lemma sats_number1_fm [simp]:
paulson@13363
   582
   "[| x \<in> nat; env \<in> list(A)|]
paulson@13363
   583
    ==> sats(A, number1_fm(x), env) <-> number1(**A, nth(x,env))"
paulson@13363
   584
by (simp add: number1_fm_def number1_def)
paulson@13363
   585
paulson@13363
   586
lemma number1_iff_sats:
paulson@13363
   587
      "[| nth(i,env) = x; nth(j,env) = y; 
paulson@13363
   588
          i \<in> nat; env \<in> list(A)|]
paulson@13363
   589
       ==> number1(**A, x) <-> sats(A, number1_fm(i), env)"
paulson@13363
   590
by simp
paulson@13363
   591
paulson@13363
   592
theorem number1_reflection:
paulson@13363
   593
     "REFLECTS[\<lambda>x. number1(L,f(x)), 
paulson@13363
   594
               \<lambda>i x. number1(**Lset(i),f(x))]"
paulson@13363
   595
apply (simp only: number1_def setclass_simps)
paulson@13363
   596
apply (intro FOL_reflections empty_reflection successor_reflection)
paulson@13363
   597
done
paulson@13363
   598
paulson@13363
   599
paulson@13352
   600
subsubsection{*Big Union, Internalized*}
paulson@13306
   601
paulson@13352
   602
(*  "big_union(M,A,z) == \<forall>x[M]. x \<in> z <-> (\<exists>y[M]. y\<in>A & x \<in> y)" *)
paulson@13352
   603
constdefs big_union_fm :: "[i,i]=>i"
paulson@13352
   604
    "big_union_fm(A,z) == 
paulson@13352
   605
       Forall(Iff(Member(0,succ(z)),
paulson@13352
   606
                  Exists(And(Member(0,succ(succ(A))), Member(1,0)))))"
paulson@13298
   607
paulson@13352
   608
lemma big_union_type [TC]:
paulson@13352
   609
     "[| x \<in> nat; y \<in> nat |] ==> big_union_fm(x,y) \<in> formula"
paulson@13352
   610
by (simp add: big_union_fm_def) 
paulson@13306
   611
paulson@13352
   612
lemma arity_big_union_fm [simp]:
paulson@13352
   613
     "[| x \<in> nat; y \<in> nat |] 
paulson@13352
   614
      ==> arity(big_union_fm(x,y)) = succ(x) \<union> succ(y)"
paulson@13352
   615
by (simp add: big_union_fm_def succ_Un_distrib [symmetric] Un_ac)
paulson@13298
   616
paulson@13352
   617
lemma sats_big_union_fm [simp]:
paulson@13352
   618
   "[| x \<in> nat; y \<in> nat; env \<in> list(A)|]
paulson@13352
   619
    ==> sats(A, big_union_fm(x,y), env) <-> 
paulson@13352
   620
        big_union(**A, nth(x,env), nth(y,env))"
paulson@13352
   621
by (simp add: big_union_fm_def big_union_def)
paulson@13306
   622
paulson@13352
   623
lemma big_union_iff_sats:
paulson@13352
   624
      "[| nth(i,env) = x; nth(j,env) = y; 
paulson@13352
   625
          i \<in> nat; j \<in> nat; env \<in> list(A)|]
paulson@13352
   626
       ==> big_union(**A, x, y) <-> sats(A, big_union_fm(i,j), env)"
paulson@13306
   627
by simp
paulson@13306
   628
paulson@13352
   629
theorem big_union_reflection:
paulson@13352
   630
     "REFLECTS[\<lambda>x. big_union(L,f(x),g(x)), 
paulson@13352
   631
               \<lambda>i x. big_union(**Lset(i),f(x),g(x))]"
paulson@13352
   632
apply (simp only: big_union_def setclass_simps)
paulson@13352
   633
apply (intro FOL_reflections)  
paulson@13314
   634
done
paulson@13298
   635
paulson@13298
   636
paulson@13306
   637
subsubsection{*Variants of Satisfaction Definitions for Ordinals, etc.*}
paulson@13306
   638
paulson@13306
   639
text{*Differs from the one in Formula by using "ordinal" rather than "Ord"*}
paulson@13306
   640
paulson@13306
   641
paulson@13306
   642
lemma sats_subset_fm':
paulson@13306
   643
   "[|x \<in> nat; y \<in> nat; env \<in> list(A)|]
paulson@13306
   644
    ==> sats(A, subset_fm(x,y), env) <-> subset(**A, nth(x,env), nth(y,env))" 
paulson@13323
   645
by (simp add: subset_fm_def Relative.subset_def) 
paulson@13298
   646
paulson@13314
   647
theorem subset_reflection:
paulson@13314
   648
     "REFLECTS[\<lambda>x. subset(L,f(x),g(x)), 
paulson@13314
   649
               \<lambda>i x. subset(**Lset(i),f(x),g(x))]" 
paulson@13323
   650
apply (simp only: Relative.subset_def setclass_simps)
paulson@13323
   651
apply (intro FOL_reflections)  
paulson@13314
   652
done
paulson@13306
   653
paulson@13306
   654
lemma sats_transset_fm':
paulson@13306
   655
   "[|x \<in> nat; env \<in> list(A)|]
paulson@13306
   656
    ==> sats(A, transset_fm(x), env) <-> transitive_set(**A, nth(x,env))"
paulson@13306
   657
by (simp add: sats_subset_fm' transset_fm_def transitive_set_def) 
paulson@13298
   658
paulson@13314
   659
theorem transitive_set_reflection:
paulson@13314
   660
     "REFLECTS[\<lambda>x. transitive_set(L,f(x)),
paulson@13314
   661
               \<lambda>i x. transitive_set(**Lset(i),f(x))]"
paulson@13314
   662
apply (simp only: transitive_set_def setclass_simps)
paulson@13323
   663
apply (intro FOL_reflections subset_reflection)  
paulson@13314
   664
done
paulson@13306
   665
paulson@13306
   666
lemma sats_ordinal_fm':
paulson@13306
   667
   "[|x \<in> nat; env \<in> list(A)|]
paulson@13306
   668
    ==> sats(A, ordinal_fm(x), env) <-> ordinal(**A,nth(x,env))"
paulson@13306
   669
by (simp add: sats_transset_fm' ordinal_fm_def ordinal_def)
paulson@13306
   670
paulson@13306
   671
lemma ordinal_iff_sats:
paulson@13306
   672
      "[| nth(i,env) = x;  i \<in> nat; env \<in> list(A)|]
paulson@13306
   673
       ==> ordinal(**A, x) <-> sats(A, ordinal_fm(i), env)"
paulson@13306
   674
by (simp add: sats_ordinal_fm')
paulson@13306
   675
paulson@13314
   676
theorem ordinal_reflection:
paulson@13314
   677
     "REFLECTS[\<lambda>x. ordinal(L,f(x)), \<lambda>i x. ordinal(**Lset(i),f(x))]"
paulson@13314
   678
apply (simp only: ordinal_def setclass_simps)
paulson@13323
   679
apply (intro FOL_reflections transitive_set_reflection)  
paulson@13314
   680
done
paulson@13298
   681
paulson@13298
   682
paulson@13339
   683
subsubsection{*Membership Relation, Internalized*}
paulson@13298
   684
paulson@13306
   685
constdefs Memrel_fm :: "[i,i]=>i"
paulson@13306
   686
    "Memrel_fm(A,r) == 
paulson@13306
   687
       Forall(Iff(Member(0,succ(r)),
paulson@13306
   688
                  Exists(And(Member(0,succ(succ(A))),
paulson@13306
   689
                             Exists(And(Member(0,succ(succ(succ(A)))),
paulson@13306
   690
                                        And(Member(1,0),
paulson@13306
   691
                                            pair_fm(1,0,2))))))))"
paulson@13306
   692
paulson@13306
   693
lemma Memrel_type [TC]:
paulson@13306
   694
     "[| x \<in> nat; y \<in> nat |] ==> Memrel_fm(x,y) \<in> formula"
paulson@13306
   695
by (simp add: Memrel_fm_def) 
paulson@13298
   696
paulson@13306
   697
lemma arity_Memrel_fm [simp]:
paulson@13306
   698
     "[| x \<in> nat; y \<in> nat |] 
paulson@13306
   699
      ==> arity(Memrel_fm(x,y)) = succ(x) \<union> succ(y)"
paulson@13306
   700
by (simp add: Memrel_fm_def succ_Un_distrib [symmetric] Un_ac) 
paulson@13306
   701
paulson@13306
   702
lemma sats_Memrel_fm [simp]:
paulson@13306
   703
   "[| x \<in> nat; y \<in> nat; env \<in> list(A)|]
paulson@13306
   704
    ==> sats(A, Memrel_fm(x,y), env) <-> 
paulson@13306
   705
        membership(**A, nth(x,env), nth(y,env))"
paulson@13306
   706
by (simp add: Memrel_fm_def membership_def)
paulson@13298
   707
paulson@13306
   708
lemma Memrel_iff_sats:
paulson@13306
   709
      "[| nth(i,env) = x; nth(j,env) = y; 
paulson@13306
   710
          i \<in> nat; j \<in> nat; env \<in> list(A)|]
paulson@13306
   711
       ==> membership(**A, x, y) <-> sats(A, Memrel_fm(i,j), env)"
paulson@13306
   712
by simp
paulson@13304
   713
paulson@13314
   714
theorem membership_reflection:
paulson@13314
   715
     "REFLECTS[\<lambda>x. membership(L,f(x),g(x)), 
paulson@13314
   716
               \<lambda>i x. membership(**Lset(i),f(x),g(x))]"
paulson@13314
   717
apply (simp only: membership_def setclass_simps)
paulson@13323
   718
apply (intro FOL_reflections pair_reflection)  
paulson@13314
   719
done
paulson@13304
   720
paulson@13339
   721
subsubsection{*Predecessor Set, Internalized*}
paulson@13304
   722
paulson@13306
   723
constdefs pred_set_fm :: "[i,i,i,i]=>i"
paulson@13306
   724
    "pred_set_fm(A,x,r,B) == 
paulson@13306
   725
       Forall(Iff(Member(0,succ(B)),
paulson@13306
   726
                  Exists(And(Member(0,succ(succ(r))),
paulson@13306
   727
                             And(Member(1,succ(succ(A))),
paulson@13306
   728
                                 pair_fm(1,succ(succ(x)),0))))))"
paulson@13306
   729
paulson@13306
   730
paulson@13306
   731
lemma pred_set_type [TC]:
paulson@13306
   732
     "[| A \<in> nat; x \<in> nat; r \<in> nat; B \<in> nat |] 
paulson@13306
   733
      ==> pred_set_fm(A,x,r,B) \<in> formula"
paulson@13306
   734
by (simp add: pred_set_fm_def) 
paulson@13304
   735
paulson@13306
   736
lemma arity_pred_set_fm [simp]:
paulson@13306
   737
   "[| A \<in> nat; x \<in> nat; r \<in> nat; B \<in> nat |] 
paulson@13306
   738
    ==> arity(pred_set_fm(A,x,r,B)) = succ(A) \<union> succ(x) \<union> succ(r) \<union> succ(B)"
paulson@13306
   739
by (simp add: pred_set_fm_def succ_Un_distrib [symmetric] Un_ac) 
paulson@13306
   740
paulson@13306
   741
lemma sats_pred_set_fm [simp]:
paulson@13306
   742
   "[| U \<in> nat; x \<in> nat; r \<in> nat; B \<in> nat; env \<in> list(A)|]
paulson@13306
   743
    ==> sats(A, pred_set_fm(U,x,r,B), env) <-> 
paulson@13306
   744
        pred_set(**A, nth(U,env), nth(x,env), nth(r,env), nth(B,env))"
paulson@13306
   745
by (simp add: pred_set_fm_def pred_set_def)
paulson@13306
   746
paulson@13306
   747
lemma pred_set_iff_sats:
paulson@13306
   748
      "[| nth(i,env) = U; nth(j,env) = x; nth(k,env) = r; nth(l,env) = B; 
paulson@13306
   749
          i \<in> nat; j \<in> nat; k \<in> nat; l \<in> nat; env \<in> list(A)|]
paulson@13306
   750
       ==> pred_set(**A,U,x,r,B) <-> sats(A, pred_set_fm(i,j,k,l), env)"
paulson@13306
   751
by (simp add: sats_pred_set_fm)
paulson@13306
   752
paulson@13314
   753
theorem pred_set_reflection:
paulson@13314
   754
     "REFLECTS[\<lambda>x. pred_set(L,f(x),g(x),h(x),b(x)), 
paulson@13314
   755
               \<lambda>i x. pred_set(**Lset(i),f(x),g(x),h(x),b(x))]" 
paulson@13314
   756
apply (simp only: pred_set_def setclass_simps)
paulson@13323
   757
apply (intro FOL_reflections pair_reflection)  
paulson@13314
   758
done
paulson@13304
   759
paulson@13304
   760
paulson@13298
   761
paulson@13339
   762
subsubsection{*Domain of a Relation, Internalized*}
paulson@13306
   763
paulson@13306
   764
(* "is_domain(M,r,z) == 
paulson@13306
   765
	\<forall>x[M]. (x \<in> z <-> (\<exists>w[M]. w\<in>r & (\<exists>y[M]. pair(M,x,y,w))))" *)
paulson@13306
   766
constdefs domain_fm :: "[i,i]=>i"
paulson@13306
   767
    "domain_fm(r,z) == 
paulson@13306
   768
       Forall(Iff(Member(0,succ(z)),
paulson@13306
   769
                  Exists(And(Member(0,succ(succ(r))),
paulson@13306
   770
                             Exists(pair_fm(2,0,1))))))"
paulson@13306
   771
paulson@13306
   772
lemma domain_type [TC]:
paulson@13306
   773
     "[| x \<in> nat; y \<in> nat |] ==> domain_fm(x,y) \<in> formula"
paulson@13306
   774
by (simp add: domain_fm_def) 
paulson@13306
   775
paulson@13306
   776
lemma arity_domain_fm [simp]:
paulson@13306
   777
     "[| x \<in> nat; y \<in> nat |] 
paulson@13306
   778
      ==> arity(domain_fm(x,y)) = succ(x) \<union> succ(y)"
paulson@13306
   779
by (simp add: domain_fm_def succ_Un_distrib [symmetric] Un_ac) 
paulson@13306
   780
paulson@13306
   781
lemma sats_domain_fm [simp]:
paulson@13306
   782
   "[| x \<in> nat; y \<in> nat; env \<in> list(A)|]
paulson@13306
   783
    ==> sats(A, domain_fm(x,y), env) <-> 
paulson@13306
   784
        is_domain(**A, nth(x,env), nth(y,env))"
paulson@13306
   785
by (simp add: domain_fm_def is_domain_def)
paulson@13306
   786
paulson@13306
   787
lemma domain_iff_sats:
paulson@13306
   788
      "[| nth(i,env) = x; nth(j,env) = y; 
paulson@13306
   789
          i \<in> nat; j \<in> nat; env \<in> list(A)|]
paulson@13306
   790
       ==> is_domain(**A, x, y) <-> sats(A, domain_fm(i,j), env)"
paulson@13306
   791
by simp
paulson@13306
   792
paulson@13314
   793
theorem domain_reflection:
paulson@13314
   794
     "REFLECTS[\<lambda>x. is_domain(L,f(x),g(x)), 
paulson@13314
   795
               \<lambda>i x. is_domain(**Lset(i),f(x),g(x))]"
paulson@13314
   796
apply (simp only: is_domain_def setclass_simps)
paulson@13323
   797
apply (intro FOL_reflections pair_reflection)  
paulson@13314
   798
done
paulson@13306
   799
paulson@13306
   800
paulson@13339
   801
subsubsection{*Range of a Relation, Internalized*}
paulson@13306
   802
paulson@13306
   803
(* "is_range(M,r,z) == 
paulson@13306
   804
	\<forall>y[M]. (y \<in> z <-> (\<exists>w[M]. w\<in>r & (\<exists>x[M]. pair(M,x,y,w))))" *)
paulson@13306
   805
constdefs range_fm :: "[i,i]=>i"
paulson@13306
   806
    "range_fm(r,z) == 
paulson@13306
   807
       Forall(Iff(Member(0,succ(z)),
paulson@13306
   808
                  Exists(And(Member(0,succ(succ(r))),
paulson@13306
   809
                             Exists(pair_fm(0,2,1))))))"
paulson@13306
   810
paulson@13306
   811
lemma range_type [TC]:
paulson@13306
   812
     "[| x \<in> nat; y \<in> nat |] ==> range_fm(x,y) \<in> formula"
paulson@13306
   813
by (simp add: range_fm_def) 
paulson@13306
   814
paulson@13306
   815
lemma arity_range_fm [simp]:
paulson@13306
   816
     "[| x \<in> nat; y \<in> nat |] 
paulson@13306
   817
      ==> arity(range_fm(x,y)) = succ(x) \<union> succ(y)"
paulson@13306
   818
by (simp add: range_fm_def succ_Un_distrib [symmetric] Un_ac) 
paulson@13306
   819
paulson@13306
   820
lemma sats_range_fm [simp]:
paulson@13306
   821
   "[| x \<in> nat; y \<in> nat; env \<in> list(A)|]
paulson@13306
   822
    ==> sats(A, range_fm(x,y), env) <-> 
paulson@13306
   823
        is_range(**A, nth(x,env), nth(y,env))"
paulson@13306
   824
by (simp add: range_fm_def is_range_def)
paulson@13306
   825
paulson@13306
   826
lemma range_iff_sats:
paulson@13306
   827
      "[| nth(i,env) = x; nth(j,env) = y; 
paulson@13306
   828
          i \<in> nat; j \<in> nat; env \<in> list(A)|]
paulson@13306
   829
       ==> is_range(**A, x, y) <-> sats(A, range_fm(i,j), env)"
paulson@13306
   830
by simp
paulson@13306
   831
paulson@13314
   832
theorem range_reflection:
paulson@13314
   833
     "REFLECTS[\<lambda>x. is_range(L,f(x),g(x)), 
paulson@13314
   834
               \<lambda>i x. is_range(**Lset(i),f(x),g(x))]"
paulson@13314
   835
apply (simp only: is_range_def setclass_simps)
paulson@13323
   836
apply (intro FOL_reflections pair_reflection)  
paulson@13314
   837
done
paulson@13306
   838
paulson@13306
   839
 
paulson@13339
   840
subsubsection{*Field of a Relation, Internalized*}
paulson@13323
   841
paulson@13323
   842
(* "is_field(M,r,z) == 
paulson@13323
   843
	\<exists>dr[M]. is_domain(M,r,dr) & 
paulson@13323
   844
            (\<exists>rr[M]. is_range(M,r,rr) & union(M,dr,rr,z))" *)
paulson@13323
   845
constdefs field_fm :: "[i,i]=>i"
paulson@13323
   846
    "field_fm(r,z) == 
paulson@13323
   847
       Exists(And(domain_fm(succ(r),0), 
paulson@13323
   848
              Exists(And(range_fm(succ(succ(r)),0), 
paulson@13323
   849
                         union_fm(1,0,succ(succ(z)))))))"
paulson@13323
   850
paulson@13323
   851
lemma field_type [TC]:
paulson@13323
   852
     "[| x \<in> nat; y \<in> nat |] ==> field_fm(x,y) \<in> formula"
paulson@13323
   853
by (simp add: field_fm_def) 
paulson@13323
   854
paulson@13323
   855
lemma arity_field_fm [simp]:
paulson@13323
   856
     "[| x \<in> nat; y \<in> nat |] 
paulson@13323
   857
      ==> arity(field_fm(x,y)) = succ(x) \<union> succ(y)"
paulson@13323
   858
by (simp add: field_fm_def succ_Un_distrib [symmetric] Un_ac) 
paulson@13323
   859
paulson@13323
   860
lemma sats_field_fm [simp]:
paulson@13323
   861
   "[| x \<in> nat; y \<in> nat; env \<in> list(A)|]
paulson@13323
   862
    ==> sats(A, field_fm(x,y), env) <-> 
paulson@13323
   863
        is_field(**A, nth(x,env), nth(y,env))"
paulson@13323
   864
by (simp add: field_fm_def is_field_def)
paulson@13323
   865
paulson@13323
   866
lemma field_iff_sats:
paulson@13323
   867
      "[| nth(i,env) = x; nth(j,env) = y; 
paulson@13323
   868
          i \<in> nat; j \<in> nat; env \<in> list(A)|]
paulson@13323
   869
       ==> is_field(**A, x, y) <-> sats(A, field_fm(i,j), env)"
paulson@13323
   870
by simp
paulson@13323
   871
paulson@13323
   872
theorem field_reflection:
paulson@13323
   873
     "REFLECTS[\<lambda>x. is_field(L,f(x),g(x)), 
paulson@13323
   874
               \<lambda>i x. is_field(**Lset(i),f(x),g(x))]"
paulson@13323
   875
apply (simp only: is_field_def setclass_simps)
paulson@13323
   876
apply (intro FOL_reflections domain_reflection range_reflection
paulson@13323
   877
             union_reflection)
paulson@13323
   878
done
paulson@13323
   879
paulson@13323
   880
paulson@13339
   881
subsubsection{*Image under a Relation, Internalized*}
paulson@13306
   882
paulson@13306
   883
(* "image(M,r,A,z) == 
paulson@13306
   884
        \<forall>y[M]. (y \<in> z <-> (\<exists>w[M]. w\<in>r & (\<exists>x[M]. x\<in>A & pair(M,x,y,w))))" *)
paulson@13306
   885
constdefs image_fm :: "[i,i,i]=>i"
paulson@13306
   886
    "image_fm(r,A,z) == 
paulson@13306
   887
       Forall(Iff(Member(0,succ(z)),
paulson@13306
   888
                  Exists(And(Member(0,succ(succ(r))),
paulson@13306
   889
                             Exists(And(Member(0,succ(succ(succ(A)))),
paulson@13306
   890
	 			        pair_fm(0,2,1)))))))"
paulson@13306
   891
paulson@13306
   892
lemma image_type [TC]:
paulson@13306
   893
     "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> image_fm(x,y,z) \<in> formula"
paulson@13306
   894
by (simp add: image_fm_def) 
paulson@13306
   895
paulson@13306
   896
lemma arity_image_fm [simp]:
paulson@13306
   897
     "[| x \<in> nat; y \<in> nat; z \<in> nat |] 
paulson@13306
   898
      ==> arity(image_fm(x,y,z)) = succ(x) \<union> succ(y) \<union> succ(z)"
paulson@13306
   899
by (simp add: image_fm_def succ_Un_distrib [symmetric] Un_ac) 
paulson@13306
   900
paulson@13306
   901
lemma sats_image_fm [simp]:
paulson@13306
   902
   "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
paulson@13306
   903
    ==> sats(A, image_fm(x,y,z), env) <-> 
paulson@13306
   904
        image(**A, nth(x,env), nth(y,env), nth(z,env))"
paulson@13323
   905
by (simp add: image_fm_def Relative.image_def)
paulson@13306
   906
paulson@13306
   907
lemma image_iff_sats:
paulson@13306
   908
      "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z; 
paulson@13306
   909
          i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
paulson@13306
   910
       ==> image(**A, x, y, z) <-> sats(A, image_fm(i,j,k), env)"
paulson@13306
   911
by (simp add: sats_image_fm)
paulson@13306
   912
paulson@13314
   913
theorem image_reflection:
paulson@13314
   914
     "REFLECTS[\<lambda>x. image(L,f(x),g(x),h(x)), 
paulson@13314
   915
               \<lambda>i x. image(**Lset(i),f(x),g(x),h(x))]"
paulson@13323
   916
apply (simp only: Relative.image_def setclass_simps)
paulson@13323
   917
apply (intro FOL_reflections pair_reflection)  
paulson@13314
   918
done
paulson@13306
   919
paulson@13306
   920
paulson@13348
   921
subsubsection{*Pre-Image under a Relation, Internalized*}
paulson@13348
   922
paulson@13348
   923
(* "pre_image(M,r,A,z) == 
paulson@13348
   924
	\<forall>x[M]. x \<in> z <-> (\<exists>w[M]. w\<in>r & (\<exists>y[M]. y\<in>A & pair(M,x,y,w)))" *)
paulson@13348
   925
constdefs pre_image_fm :: "[i,i,i]=>i"
paulson@13348
   926
    "pre_image_fm(r,A,z) == 
paulson@13348
   927
       Forall(Iff(Member(0,succ(z)),
paulson@13348
   928
                  Exists(And(Member(0,succ(succ(r))),
paulson@13348
   929
                             Exists(And(Member(0,succ(succ(succ(A)))),
paulson@13348
   930
	 			        pair_fm(2,0,1)))))))"
paulson@13348
   931
paulson@13348
   932
lemma pre_image_type [TC]:
paulson@13348
   933
     "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> pre_image_fm(x,y,z) \<in> formula"
paulson@13348
   934
by (simp add: pre_image_fm_def) 
paulson@13348
   935
paulson@13348
   936
lemma arity_pre_image_fm [simp]:
paulson@13348
   937
     "[| x \<in> nat; y \<in> nat; z \<in> nat |] 
paulson@13348
   938
      ==> arity(pre_image_fm(x,y,z)) = succ(x) \<union> succ(y) \<union> succ(z)"
paulson@13348
   939
by (simp add: pre_image_fm_def succ_Un_distrib [symmetric] Un_ac) 
paulson@13348
   940
paulson@13348
   941
lemma sats_pre_image_fm [simp]:
paulson@13348
   942
   "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
paulson@13348
   943
    ==> sats(A, pre_image_fm(x,y,z), env) <-> 
paulson@13348
   944
        pre_image(**A, nth(x,env), nth(y,env), nth(z,env))"
paulson@13348
   945
by (simp add: pre_image_fm_def Relative.pre_image_def)
paulson@13348
   946
paulson@13348
   947
lemma pre_image_iff_sats:
paulson@13348
   948
      "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z; 
paulson@13348
   949
          i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
paulson@13348
   950
       ==> pre_image(**A, x, y, z) <-> sats(A, pre_image_fm(i,j,k), env)"
paulson@13348
   951
by (simp add: sats_pre_image_fm)
paulson@13348
   952
paulson@13348
   953
theorem pre_image_reflection:
paulson@13348
   954
     "REFLECTS[\<lambda>x. pre_image(L,f(x),g(x),h(x)), 
paulson@13348
   955
               \<lambda>i x. pre_image(**Lset(i),f(x),g(x),h(x))]"
paulson@13348
   956
apply (simp only: Relative.pre_image_def setclass_simps)
paulson@13348
   957
apply (intro FOL_reflections pair_reflection)  
paulson@13348
   958
done
paulson@13348
   959
paulson@13348
   960
paulson@13352
   961
subsubsection{*Function Application, Internalized*}
paulson@13352
   962
paulson@13352
   963
(* "fun_apply(M,f,x,y) == 
paulson@13352
   964
        (\<exists>xs[M]. \<exists>fxs[M]. 
paulson@13352
   965
         upair(M,x,x,xs) & image(M,f,xs,fxs) & big_union(M,fxs,y))" *)
paulson@13352
   966
constdefs fun_apply_fm :: "[i,i,i]=>i"
paulson@13352
   967
    "fun_apply_fm(f,x,y) == 
paulson@13352
   968
       Exists(Exists(And(upair_fm(succ(succ(x)), succ(succ(x)), 1),
paulson@13352
   969
                         And(image_fm(succ(succ(f)), 1, 0), 
paulson@13352
   970
                             big_union_fm(0,succ(succ(y)))))))"
paulson@13352
   971
paulson@13352
   972
lemma fun_apply_type [TC]:
paulson@13352
   973
     "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> fun_apply_fm(x,y,z) \<in> formula"
paulson@13352
   974
by (simp add: fun_apply_fm_def) 
paulson@13352
   975
paulson@13352
   976
lemma arity_fun_apply_fm [simp]:
paulson@13352
   977
     "[| x \<in> nat; y \<in> nat; z \<in> nat |] 
paulson@13352
   978
      ==> arity(fun_apply_fm(x,y,z)) = succ(x) \<union> succ(y) \<union> succ(z)"
paulson@13352
   979
by (simp add: fun_apply_fm_def succ_Un_distrib [symmetric] Un_ac) 
paulson@13352
   980
paulson@13352
   981
lemma sats_fun_apply_fm [simp]:
paulson@13352
   982
   "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
paulson@13352
   983
    ==> sats(A, fun_apply_fm(x,y,z), env) <-> 
paulson@13352
   984
        fun_apply(**A, nth(x,env), nth(y,env), nth(z,env))"
paulson@13352
   985
by (simp add: fun_apply_fm_def fun_apply_def)
paulson@13352
   986
paulson@13352
   987
lemma fun_apply_iff_sats:
paulson@13352
   988
      "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z; 
paulson@13352
   989
          i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
paulson@13352
   990
       ==> fun_apply(**A, x, y, z) <-> sats(A, fun_apply_fm(i,j,k), env)"
paulson@13352
   991
by simp
paulson@13352
   992
paulson@13352
   993
theorem fun_apply_reflection:
paulson@13352
   994
     "REFLECTS[\<lambda>x. fun_apply(L,f(x),g(x),h(x)), 
paulson@13352
   995
               \<lambda>i x. fun_apply(**Lset(i),f(x),g(x),h(x))]" 
paulson@13352
   996
apply (simp only: fun_apply_def setclass_simps)
paulson@13352
   997
apply (intro FOL_reflections upair_reflection image_reflection
paulson@13352
   998
             big_union_reflection)  
paulson@13352
   999
done
paulson@13352
  1000
paulson@13352
  1001
paulson@13339
  1002
subsubsection{*The Concept of Relation, Internalized*}
paulson@13306
  1003
paulson@13306
  1004
(* "is_relation(M,r) == 
paulson@13306
  1005
        (\<forall>z[M]. z\<in>r --> (\<exists>x[M]. \<exists>y[M]. pair(M,x,y,z)))" *)
paulson@13306
  1006
constdefs relation_fm :: "i=>i"
paulson@13306
  1007
    "relation_fm(r) == 
paulson@13306
  1008
       Forall(Implies(Member(0,succ(r)), Exists(Exists(pair_fm(1,0,2)))))"
paulson@13306
  1009
paulson@13306
  1010
lemma relation_type [TC]:
paulson@13306
  1011
     "[| x \<in> nat |] ==> relation_fm(x) \<in> formula"
paulson@13306
  1012
by (simp add: relation_fm_def) 
paulson@13306
  1013
paulson@13306
  1014
lemma arity_relation_fm [simp]:
paulson@13306
  1015
     "x \<in> nat ==> arity(relation_fm(x)) = succ(x)"
paulson@13306
  1016
by (simp add: relation_fm_def succ_Un_distrib [symmetric] Un_ac) 
paulson@13306
  1017
paulson@13306
  1018
lemma sats_relation_fm [simp]:
paulson@13306
  1019
   "[| x \<in> nat; env \<in> list(A)|]
paulson@13306
  1020
    ==> sats(A, relation_fm(x), env) <-> is_relation(**A, nth(x,env))"
paulson@13306
  1021
by (simp add: relation_fm_def is_relation_def)
paulson@13306
  1022
paulson@13306
  1023
lemma relation_iff_sats:
paulson@13306
  1024
      "[| nth(i,env) = x; nth(j,env) = y; 
paulson@13306
  1025
          i \<in> nat; env \<in> list(A)|]
paulson@13306
  1026
       ==> is_relation(**A, x) <-> sats(A, relation_fm(i), env)"
paulson@13306
  1027
by simp
paulson@13306
  1028
paulson@13314
  1029
theorem is_relation_reflection:
paulson@13314
  1030
     "REFLECTS[\<lambda>x. is_relation(L,f(x)), 
paulson@13314
  1031
               \<lambda>i x. is_relation(**Lset(i),f(x))]"
paulson@13314
  1032
apply (simp only: is_relation_def setclass_simps)
paulson@13323
  1033
apply (intro FOL_reflections pair_reflection)  
paulson@13314
  1034
done
paulson@13306
  1035
paulson@13306
  1036
paulson@13339
  1037
subsubsection{*The Concept of Function, Internalized*}
paulson@13306
  1038
paulson@13306
  1039
(* "is_function(M,r) == 
paulson@13306
  1040
	\<forall>x[M]. \<forall>y[M]. \<forall>y'[M]. \<forall>p[M]. \<forall>p'[M]. 
paulson@13306
  1041
           pair(M,x,y,p) --> pair(M,x,y',p') --> p\<in>r --> p'\<in>r --> y=y'" *)
paulson@13306
  1042
constdefs function_fm :: "i=>i"
paulson@13306
  1043
    "function_fm(r) == 
paulson@13306
  1044
       Forall(Forall(Forall(Forall(Forall(
paulson@13306
  1045
         Implies(pair_fm(4,3,1),
paulson@13306
  1046
                 Implies(pair_fm(4,2,0),
paulson@13306
  1047
                         Implies(Member(1,r#+5),
paulson@13306
  1048
                                 Implies(Member(0,r#+5), Equal(3,2))))))))))"
paulson@13306
  1049
paulson@13306
  1050
lemma function_type [TC]:
paulson@13306
  1051
     "[| x \<in> nat |] ==> function_fm(x) \<in> formula"
paulson@13306
  1052
by (simp add: function_fm_def) 
paulson@13306
  1053
paulson@13306
  1054
lemma arity_function_fm [simp]:
paulson@13306
  1055
     "x \<in> nat ==> arity(function_fm(x)) = succ(x)"
paulson@13306
  1056
by (simp add: function_fm_def succ_Un_distrib [symmetric] Un_ac) 
paulson@13306
  1057
paulson@13306
  1058
lemma sats_function_fm [simp]:
paulson@13306
  1059
   "[| x \<in> nat; env \<in> list(A)|]
paulson@13306
  1060
    ==> sats(A, function_fm(x), env) <-> is_function(**A, nth(x,env))"
paulson@13306
  1061
by (simp add: function_fm_def is_function_def)
paulson@13306
  1062
paulson@13306
  1063
lemma function_iff_sats:
paulson@13306
  1064
      "[| nth(i,env) = x; nth(j,env) = y; 
paulson@13306
  1065
          i \<in> nat; env \<in> list(A)|]
paulson@13306
  1066
       ==> is_function(**A, x) <-> sats(A, function_fm(i), env)"
paulson@13306
  1067
by simp
paulson@13306
  1068
paulson@13314
  1069
theorem is_function_reflection:
paulson@13314
  1070
     "REFLECTS[\<lambda>x. is_function(L,f(x)), 
paulson@13314
  1071
               \<lambda>i x. is_function(**Lset(i),f(x))]"
paulson@13314
  1072
apply (simp only: is_function_def setclass_simps)
paulson@13323
  1073
apply (intro FOL_reflections pair_reflection)  
paulson@13314
  1074
done
paulson@13298
  1075
paulson@13298
  1076
paulson@13339
  1077
subsubsection{*Typed Functions, Internalized*}
paulson@13309
  1078
paulson@13309
  1079
(* "typed_function(M,A,B,r) == 
paulson@13309
  1080
        is_function(M,r) & is_relation(M,r) & is_domain(M,r,A) &
paulson@13309
  1081
        (\<forall>u[M]. u\<in>r --> (\<forall>x[M]. \<forall>y[M]. pair(M,x,y,u) --> y\<in>B))" *)
paulson@13309
  1082
paulson@13309
  1083
constdefs typed_function_fm :: "[i,i,i]=>i"
paulson@13309
  1084
    "typed_function_fm(A,B,r) == 
paulson@13309
  1085
       And(function_fm(r),
paulson@13309
  1086
         And(relation_fm(r),
paulson@13309
  1087
           And(domain_fm(r,A),
paulson@13309
  1088
             Forall(Implies(Member(0,succ(r)),
paulson@13309
  1089
                  Forall(Forall(Implies(pair_fm(1,0,2),Member(0,B#+3)))))))))"
paulson@13309
  1090
paulson@13309
  1091
lemma typed_function_type [TC]:
paulson@13309
  1092
     "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> typed_function_fm(x,y,z) \<in> formula"
paulson@13309
  1093
by (simp add: typed_function_fm_def) 
paulson@13309
  1094
paulson@13309
  1095
lemma arity_typed_function_fm [simp]:
paulson@13309
  1096
     "[| x \<in> nat; y \<in> nat; z \<in> nat |] 
paulson@13309
  1097
      ==> arity(typed_function_fm(x,y,z)) = succ(x) \<union> succ(y) \<union> succ(z)"
paulson@13309
  1098
by (simp add: typed_function_fm_def succ_Un_distrib [symmetric] Un_ac) 
paulson@13309
  1099
paulson@13309
  1100
lemma sats_typed_function_fm [simp]:
paulson@13309
  1101
   "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
paulson@13309
  1102
    ==> sats(A, typed_function_fm(x,y,z), env) <-> 
paulson@13309
  1103
        typed_function(**A, nth(x,env), nth(y,env), nth(z,env))"
paulson@13309
  1104
by (simp add: typed_function_fm_def typed_function_def)
paulson@13309
  1105
paulson@13309
  1106
lemma typed_function_iff_sats:
paulson@13309
  1107
  "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z; 
paulson@13309
  1108
      i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
paulson@13309
  1109
   ==> typed_function(**A, x, y, z) <-> sats(A, typed_function_fm(i,j,k), env)"
paulson@13309
  1110
by simp
paulson@13309
  1111
paulson@13323
  1112
lemmas function_reflections = 
paulson@13363
  1113
        empty_reflection number1_reflection
paulson@13363
  1114
	upair_reflection pair_reflection union_reflection
paulson@13352
  1115
	big_union_reflection cons_reflection successor_reflection 
paulson@13323
  1116
        fun_apply_reflection subset_reflection
paulson@13323
  1117
	transitive_set_reflection membership_reflection
paulson@13323
  1118
	pred_set_reflection domain_reflection range_reflection field_reflection
paulson@13348
  1119
        image_reflection pre_image_reflection
paulson@13314
  1120
	is_relation_reflection is_function_reflection
paulson@13309
  1121
paulson@13323
  1122
lemmas function_iff_sats = 
paulson@13363
  1123
        empty_iff_sats number1_iff_sats 
paulson@13363
  1124
	upair_iff_sats pair_iff_sats union_iff_sats
paulson@13323
  1125
	cons_iff_sats successor_iff_sats
paulson@13323
  1126
        fun_apply_iff_sats  Memrel_iff_sats
paulson@13323
  1127
	pred_set_iff_sats domain_iff_sats range_iff_sats field_iff_sats
paulson@13348
  1128
        image_iff_sats pre_image_iff_sats 
paulson@13323
  1129
	relation_iff_sats function_iff_sats
paulson@13323
  1130
paulson@13309
  1131
paulson@13314
  1132
theorem typed_function_reflection:
paulson@13314
  1133
     "REFLECTS[\<lambda>x. typed_function(L,f(x),g(x),h(x)), 
paulson@13314
  1134
               \<lambda>i x. typed_function(**Lset(i),f(x),g(x),h(x))]"
paulson@13314
  1135
apply (simp only: typed_function_def setclass_simps)
paulson@13323
  1136
apply (intro FOL_reflections function_reflections)  
paulson@13323
  1137
done
paulson@13323
  1138
paulson@13323
  1139
paulson@13339
  1140
subsubsection{*Composition of Relations, Internalized*}
paulson@13323
  1141
paulson@13323
  1142
(* "composition(M,r,s,t) == 
paulson@13323
  1143
        \<forall>p[M]. p \<in> t <-> 
paulson@13323
  1144
               (\<exists>x[M]. \<exists>y[M]. \<exists>z[M]. \<exists>xy[M]. \<exists>yz[M]. 
paulson@13323
  1145
                pair(M,x,z,p) & pair(M,x,y,xy) & pair(M,y,z,yz) & 
paulson@13323
  1146
                xy \<in> s & yz \<in> r)" *)
paulson@13323
  1147
constdefs composition_fm :: "[i,i,i]=>i"
paulson@13323
  1148
  "composition_fm(r,s,t) == 
paulson@13323
  1149
     Forall(Iff(Member(0,succ(t)),
paulson@13323
  1150
             Exists(Exists(Exists(Exists(Exists( 
paulson@13323
  1151
              And(pair_fm(4,2,5),
paulson@13323
  1152
               And(pair_fm(4,3,1),
paulson@13323
  1153
                And(pair_fm(3,2,0),
paulson@13323
  1154
                 And(Member(1,s#+6), Member(0,r#+6))))))))))))"
paulson@13323
  1155
paulson@13323
  1156
lemma composition_type [TC]:
paulson@13323
  1157
     "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> composition_fm(x,y,z) \<in> formula"
paulson@13323
  1158
by (simp add: composition_fm_def) 
paulson@13323
  1159
paulson@13323
  1160
lemma arity_composition_fm [simp]:
paulson@13323
  1161
     "[| x \<in> nat; y \<in> nat; z \<in> nat |] 
paulson@13323
  1162
      ==> arity(composition_fm(x,y,z)) = succ(x) \<union> succ(y) \<union> succ(z)"
paulson@13323
  1163
by (simp add: composition_fm_def succ_Un_distrib [symmetric] Un_ac) 
paulson@13323
  1164
paulson@13323
  1165
lemma sats_composition_fm [simp]:
paulson@13323
  1166
   "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
paulson@13323
  1167
    ==> sats(A, composition_fm(x,y,z), env) <-> 
paulson@13323
  1168
        composition(**A, nth(x,env), nth(y,env), nth(z,env))"
paulson@13323
  1169
by (simp add: composition_fm_def composition_def)
paulson@13323
  1170
paulson@13323
  1171
lemma composition_iff_sats:
paulson@13323
  1172
      "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z; 
paulson@13323
  1173
          i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
paulson@13323
  1174
       ==> composition(**A, x, y, z) <-> sats(A, composition_fm(i,j,k), env)"
paulson@13323
  1175
by simp
paulson@13323
  1176
paulson@13323
  1177
theorem composition_reflection:
paulson@13323
  1178
     "REFLECTS[\<lambda>x. composition(L,f(x),g(x),h(x)), 
paulson@13323
  1179
               \<lambda>i x. composition(**Lset(i),f(x),g(x),h(x))]"
paulson@13323
  1180
apply (simp only: composition_def setclass_simps)
paulson@13323
  1181
apply (intro FOL_reflections pair_reflection)  
paulson@13314
  1182
done
paulson@13314
  1183
paulson@13309
  1184
paulson@13339
  1185
subsubsection{*Injections, Internalized*}
paulson@13309
  1186
paulson@13309
  1187
(* "injection(M,A,B,f) == 
paulson@13309
  1188
	typed_function(M,A,B,f) &
paulson@13309
  1189
        (\<forall>x[M]. \<forall>x'[M]. \<forall>y[M]. \<forall>p[M]. \<forall>p'[M]. 
paulson@13309
  1190
          pair(M,x,y,p) --> pair(M,x',y,p') --> p\<in>f --> p'\<in>f --> x=x')" *)
paulson@13309
  1191
constdefs injection_fm :: "[i,i,i]=>i"
paulson@13309
  1192
 "injection_fm(A,B,f) == 
paulson@13309
  1193
    And(typed_function_fm(A,B,f),
paulson@13309
  1194
       Forall(Forall(Forall(Forall(Forall(
paulson@13309
  1195
         Implies(pair_fm(4,2,1),
paulson@13309
  1196
                 Implies(pair_fm(3,2,0),
paulson@13309
  1197
                         Implies(Member(1,f#+5),
paulson@13309
  1198
                                 Implies(Member(0,f#+5), Equal(4,3)))))))))))"
paulson@13309
  1199
paulson@13309
  1200
paulson@13309
  1201
lemma injection_type [TC]:
paulson@13309
  1202
     "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> injection_fm(x,y,z) \<in> formula"
paulson@13309
  1203
by (simp add: injection_fm_def) 
paulson@13309
  1204
paulson@13309
  1205
lemma arity_injection_fm [simp]:
paulson@13309
  1206
     "[| x \<in> nat; y \<in> nat; z \<in> nat |] 
paulson@13309
  1207
      ==> arity(injection_fm(x,y,z)) = succ(x) \<union> succ(y) \<union> succ(z)"
paulson@13309
  1208
by (simp add: injection_fm_def succ_Un_distrib [symmetric] Un_ac) 
paulson@13309
  1209
paulson@13309
  1210
lemma sats_injection_fm [simp]:
paulson@13309
  1211
   "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
paulson@13309
  1212
    ==> sats(A, injection_fm(x,y,z), env) <-> 
paulson@13309
  1213
        injection(**A, nth(x,env), nth(y,env), nth(z,env))"
paulson@13309
  1214
by (simp add: injection_fm_def injection_def)
paulson@13309
  1215
paulson@13309
  1216
lemma injection_iff_sats:
paulson@13309
  1217
  "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z; 
paulson@13309
  1218
      i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
paulson@13309
  1219
   ==> injection(**A, x, y, z) <-> sats(A, injection_fm(i,j,k), env)"
paulson@13309
  1220
by simp
paulson@13309
  1221
paulson@13314
  1222
theorem injection_reflection:
paulson@13314
  1223
     "REFLECTS[\<lambda>x. injection(L,f(x),g(x),h(x)), 
paulson@13314
  1224
               \<lambda>i x. injection(**Lset(i),f(x),g(x),h(x))]"
paulson@13314
  1225
apply (simp only: injection_def setclass_simps)
paulson@13323
  1226
apply (intro FOL_reflections function_reflections typed_function_reflection)  
paulson@13314
  1227
done
paulson@13309
  1228
paulson@13309
  1229
paulson@13339
  1230
subsubsection{*Surjections, Internalized*}
paulson@13309
  1231
paulson@13309
  1232
(*  surjection :: "[i=>o,i,i,i] => o"
paulson@13309
  1233
    "surjection(M,A,B,f) == 
paulson@13309
  1234
        typed_function(M,A,B,f) &
paulson@13309
  1235
        (\<forall>y[M]. y\<in>B --> (\<exists>x[M]. x\<in>A & fun_apply(M,f,x,y)))" *)
paulson@13309
  1236
constdefs surjection_fm :: "[i,i,i]=>i"
paulson@13309
  1237
 "surjection_fm(A,B,f) == 
paulson@13309
  1238
    And(typed_function_fm(A,B,f),
paulson@13309
  1239
       Forall(Implies(Member(0,succ(B)),
paulson@13309
  1240
                      Exists(And(Member(0,succ(succ(A))),
paulson@13309
  1241
                                 fun_apply_fm(succ(succ(f)),0,1))))))"
paulson@13309
  1242
paulson@13309
  1243
lemma surjection_type [TC]:
paulson@13309
  1244
     "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> surjection_fm(x,y,z) \<in> formula"
paulson@13309
  1245
by (simp add: surjection_fm_def) 
paulson@13309
  1246
paulson@13309
  1247
lemma arity_surjection_fm [simp]:
paulson@13309
  1248
     "[| x \<in> nat; y \<in> nat; z \<in> nat |] 
paulson@13309
  1249
      ==> arity(surjection_fm(x,y,z)) = succ(x) \<union> succ(y) \<union> succ(z)"
paulson@13309
  1250
by (simp add: surjection_fm_def succ_Un_distrib [symmetric] Un_ac) 
paulson@13309
  1251
paulson@13309
  1252
lemma sats_surjection_fm [simp]:
paulson@13309
  1253
   "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
paulson@13309
  1254
    ==> sats(A, surjection_fm(x,y,z), env) <-> 
paulson@13309
  1255
        surjection(**A, nth(x,env), nth(y,env), nth(z,env))"
paulson@13309
  1256
by (simp add: surjection_fm_def surjection_def)
paulson@13309
  1257
paulson@13309
  1258
lemma surjection_iff_sats:
paulson@13309
  1259
  "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z; 
paulson@13309
  1260
      i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
paulson@13309
  1261
   ==> surjection(**A, x, y, z) <-> sats(A, surjection_fm(i,j,k), env)"
paulson@13309
  1262
by simp
paulson@13309
  1263
paulson@13314
  1264
theorem surjection_reflection:
paulson@13314
  1265
     "REFLECTS[\<lambda>x. surjection(L,f(x),g(x),h(x)), 
paulson@13314
  1266
               \<lambda>i x. surjection(**Lset(i),f(x),g(x),h(x))]"
paulson@13314
  1267
apply (simp only: surjection_def setclass_simps)
paulson@13323
  1268
apply (intro FOL_reflections function_reflections typed_function_reflection)  
paulson@13314
  1269
done
paulson@13309
  1270
paulson@13309
  1271
paulson@13309
  1272
paulson@13339
  1273
subsubsection{*Bijections, Internalized*}
paulson@13309
  1274
paulson@13309
  1275
(*   bijection :: "[i=>o,i,i,i] => o"
paulson@13309
  1276
    "bijection(M,A,B,f) == injection(M,A,B,f) & surjection(M,A,B,f)" *)
paulson@13309
  1277
constdefs bijection_fm :: "[i,i,i]=>i"
paulson@13309
  1278
 "bijection_fm(A,B,f) == And(injection_fm(A,B,f), surjection_fm(A,B,f))"
paulson@13309
  1279
paulson@13309
  1280
lemma bijection_type [TC]:
paulson@13309
  1281
     "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> bijection_fm(x,y,z) \<in> formula"
paulson@13309
  1282
by (simp add: bijection_fm_def) 
paulson@13309
  1283
paulson@13309
  1284
lemma arity_bijection_fm [simp]:
paulson@13309
  1285
     "[| x \<in> nat; y \<in> nat; z \<in> nat |] 
paulson@13309
  1286
      ==> arity(bijection_fm(x,y,z)) = succ(x) \<union> succ(y) \<union> succ(z)"
paulson@13309
  1287
by (simp add: bijection_fm_def succ_Un_distrib [symmetric] Un_ac) 
paulson@13309
  1288
paulson@13309
  1289
lemma sats_bijection_fm [simp]:
paulson@13309
  1290
   "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
paulson@13309
  1291
    ==> sats(A, bijection_fm(x,y,z), env) <-> 
paulson@13309
  1292
        bijection(**A, nth(x,env), nth(y,env), nth(z,env))"
paulson@13309
  1293
by (simp add: bijection_fm_def bijection_def)
paulson@13309
  1294
paulson@13309
  1295
lemma bijection_iff_sats:
paulson@13309
  1296
  "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z; 
paulson@13309
  1297
      i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
paulson@13309
  1298
   ==> bijection(**A, x, y, z) <-> sats(A, bijection_fm(i,j,k), env)"
paulson@13309
  1299
by simp
paulson@13309
  1300
paulson@13314
  1301
theorem bijection_reflection:
paulson@13314
  1302
     "REFLECTS[\<lambda>x. bijection(L,f(x),g(x),h(x)), 
paulson@13314
  1303
               \<lambda>i x. bijection(**Lset(i),f(x),g(x),h(x))]"
paulson@13314
  1304
apply (simp only: bijection_def setclass_simps)
paulson@13314
  1305
apply (intro And_reflection injection_reflection surjection_reflection)  
paulson@13314
  1306
done
paulson@13309
  1307
paulson@13309
  1308
paulson@13348
  1309
subsubsection{*Restriction of a Relation, Internalized*}
paulson@13348
  1310
paulson@13348
  1311
paulson@13348
  1312
(* "restriction(M,r,A,z) == 
paulson@13348
  1313
	\<forall>x[M]. x \<in> z <-> (x \<in> r & (\<exists>u[M]. u\<in>A & (\<exists>v[M]. pair(M,u,v,x))))" *)
paulson@13348
  1314
constdefs restriction_fm :: "[i,i,i]=>i"
paulson@13348
  1315
    "restriction_fm(r,A,z) == 
paulson@13348
  1316
       Forall(Iff(Member(0,succ(z)),
paulson@13348
  1317
                  And(Member(0,succ(r)),
paulson@13348
  1318
                      Exists(And(Member(0,succ(succ(A))),
paulson@13348
  1319
                                 Exists(pair_fm(1,0,2)))))))"
paulson@13348
  1320
paulson@13348
  1321
lemma restriction_type [TC]:
paulson@13348
  1322
     "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> restriction_fm(x,y,z) \<in> formula"
paulson@13348
  1323
by (simp add: restriction_fm_def) 
paulson@13348
  1324
paulson@13348
  1325
lemma arity_restriction_fm [simp]:
paulson@13348
  1326
     "[| x \<in> nat; y \<in> nat; z \<in> nat |] 
paulson@13348
  1327
      ==> arity(restriction_fm(x,y,z)) = succ(x) \<union> succ(y) \<union> succ(z)"
paulson@13348
  1328
by (simp add: restriction_fm_def succ_Un_distrib [symmetric] Un_ac) 
paulson@13348
  1329
paulson@13348
  1330
lemma sats_restriction_fm [simp]:
paulson@13348
  1331
   "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
paulson@13348
  1332
    ==> sats(A, restriction_fm(x,y,z), env) <-> 
paulson@13348
  1333
        restriction(**A, nth(x,env), nth(y,env), nth(z,env))"
paulson@13348
  1334
by (simp add: restriction_fm_def restriction_def)
paulson@13348
  1335
paulson@13348
  1336
lemma restriction_iff_sats:
paulson@13348
  1337
      "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z; 
paulson@13348
  1338
          i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
paulson@13348
  1339
       ==> restriction(**A, x, y, z) <-> sats(A, restriction_fm(i,j,k), env)"
paulson@13348
  1340
by simp
paulson@13348
  1341
paulson@13348
  1342
theorem restriction_reflection:
paulson@13348
  1343
     "REFLECTS[\<lambda>x. restriction(L,f(x),g(x),h(x)), 
paulson@13348
  1344
               \<lambda>i x. restriction(**Lset(i),f(x),g(x),h(x))]"
paulson@13348
  1345
apply (simp only: restriction_def setclass_simps)
paulson@13348
  1346
apply (intro FOL_reflections pair_reflection)  
paulson@13348
  1347
done
paulson@13348
  1348
paulson@13339
  1349
subsubsection{*Order-Isomorphisms, Internalized*}
paulson@13309
  1350
paulson@13309
  1351
(*  order_isomorphism :: "[i=>o,i,i,i,i,i] => o"
paulson@13309
  1352
   "order_isomorphism(M,A,r,B,s,f) == 
paulson@13309
  1353
        bijection(M,A,B,f) & 
paulson@13309
  1354
        (\<forall>x[M]. x\<in>A --> (\<forall>y[M]. y\<in>A -->
paulson@13309
  1355
          (\<forall>p[M]. \<forall>fx[M]. \<forall>fy[M]. \<forall>q[M].
paulson@13309
  1356
            pair(M,x,y,p) --> fun_apply(M,f,x,fx) --> fun_apply(M,f,y,fy) --> 
paulson@13309
  1357
            pair(M,fx,fy,q) --> (p\<in>r <-> q\<in>s))))"
paulson@13309
  1358
  *)
paulson@13309
  1359
paulson@13309
  1360
constdefs order_isomorphism_fm :: "[i,i,i,i,i]=>i"
paulson@13309
  1361
 "order_isomorphism_fm(A,r,B,s,f) == 
paulson@13309
  1362
   And(bijection_fm(A,B,f), 
paulson@13309
  1363
     Forall(Implies(Member(0,succ(A)),
paulson@13309
  1364
       Forall(Implies(Member(0,succ(succ(A))),
paulson@13309
  1365
         Forall(Forall(Forall(Forall(
paulson@13309
  1366
           Implies(pair_fm(5,4,3),
paulson@13309
  1367
             Implies(fun_apply_fm(f#+6,5,2),
paulson@13309
  1368
               Implies(fun_apply_fm(f#+6,4,1),
paulson@13309
  1369
                 Implies(pair_fm(2,1,0), 
paulson@13309
  1370
                   Iff(Member(3,r#+6), Member(0,s#+6)))))))))))))))"
paulson@13309
  1371
paulson@13309
  1372
lemma order_isomorphism_type [TC]:
paulson@13309
  1373
     "[| A \<in> nat; r \<in> nat; B \<in> nat; s \<in> nat; f \<in> nat |]  
paulson@13309
  1374
      ==> order_isomorphism_fm(A,r,B,s,f) \<in> formula"
paulson@13309
  1375
by (simp add: order_isomorphism_fm_def) 
paulson@13309
  1376
paulson@13309
  1377
lemma arity_order_isomorphism_fm [simp]:
paulson@13309
  1378
     "[| A \<in> nat; r \<in> nat; B \<in> nat; s \<in> nat; f \<in> nat |] 
paulson@13309
  1379
      ==> arity(order_isomorphism_fm(A,r,B,s,f)) = 
paulson@13309
  1380
          succ(A) \<union> succ(r) \<union> succ(B) \<union> succ(s) \<union> succ(f)" 
paulson@13309
  1381
by (simp add: order_isomorphism_fm_def succ_Un_distrib [symmetric] Un_ac) 
paulson@13309
  1382
paulson@13309
  1383
lemma sats_order_isomorphism_fm [simp]:
paulson@13309
  1384
   "[| U \<in> nat; r \<in> nat; B \<in> nat; s \<in> nat; f \<in> nat; env \<in> list(A)|]
paulson@13309
  1385
    ==> sats(A, order_isomorphism_fm(U,r,B,s,f), env) <-> 
paulson@13309
  1386
        order_isomorphism(**A, nth(U,env), nth(r,env), nth(B,env), 
paulson@13309
  1387
                               nth(s,env), nth(f,env))"
paulson@13309
  1388
by (simp add: order_isomorphism_fm_def order_isomorphism_def)
paulson@13309
  1389
paulson@13309
  1390
lemma order_isomorphism_iff_sats:
paulson@13309
  1391
  "[| nth(i,env) = U; nth(j,env) = r; nth(k,env) = B; nth(j',env) = s; 
paulson@13309
  1392
      nth(k',env) = f; 
paulson@13309
  1393
      i \<in> nat; j \<in> nat; k \<in> nat; j' \<in> nat; k' \<in> nat; env \<in> list(A)|]
paulson@13309
  1394
   ==> order_isomorphism(**A,U,r,B,s,f) <-> 
paulson@13309
  1395
       sats(A, order_isomorphism_fm(i,j,k,j',k'), env)" 
paulson@13309
  1396
by simp
paulson@13309
  1397
paulson@13314
  1398
theorem order_isomorphism_reflection:
paulson@13314
  1399
     "REFLECTS[\<lambda>x. order_isomorphism(L,f(x),g(x),h(x),g'(x),h'(x)), 
paulson@13314
  1400
               \<lambda>i x. order_isomorphism(**Lset(i),f(x),g(x),h(x),g'(x),h'(x))]"
paulson@13314
  1401
apply (simp only: order_isomorphism_def setclass_simps)
paulson@13323
  1402
apply (intro FOL_reflections function_reflections bijection_reflection)  
paulson@13323
  1403
done
paulson@13323
  1404
paulson@13339
  1405
subsubsection{*Limit Ordinals, Internalized*}
paulson@13323
  1406
paulson@13323
  1407
text{*A limit ordinal is a non-empty, successor-closed ordinal*}
paulson@13323
  1408
paulson@13323
  1409
(* "limit_ordinal(M,a) == 
paulson@13323
  1410
	ordinal(M,a) & ~ empty(M,a) & 
paulson@13323
  1411
        (\<forall>x[M]. x\<in>a --> (\<exists>y[M]. y\<in>a & successor(M,x,y)))" *)
paulson@13323
  1412
paulson@13323
  1413
constdefs limit_ordinal_fm :: "i=>i"
paulson@13323
  1414
    "limit_ordinal_fm(x) == 
paulson@13323
  1415
        And(ordinal_fm(x),
paulson@13323
  1416
            And(Neg(empty_fm(x)),
paulson@13323
  1417
	        Forall(Implies(Member(0,succ(x)),
paulson@13323
  1418
                               Exists(And(Member(0,succ(succ(x))),
paulson@13323
  1419
                                          succ_fm(1,0)))))))"
paulson@13323
  1420
paulson@13323
  1421
lemma limit_ordinal_type [TC]:
paulson@13323
  1422
     "x \<in> nat ==> limit_ordinal_fm(x) \<in> formula"
paulson@13323
  1423
by (simp add: limit_ordinal_fm_def) 
paulson@13323
  1424
paulson@13323
  1425
lemma arity_limit_ordinal_fm [simp]:
paulson@13323
  1426
     "x \<in> nat ==> arity(limit_ordinal_fm(x)) = succ(x)"
paulson@13323
  1427
by (simp add: limit_ordinal_fm_def succ_Un_distrib [symmetric] Un_ac) 
paulson@13323
  1428
paulson@13323
  1429
lemma sats_limit_ordinal_fm [simp]:
paulson@13323
  1430
   "[| x \<in> nat; env \<in> list(A)|]
paulson@13323
  1431
    ==> sats(A, limit_ordinal_fm(x), env) <-> limit_ordinal(**A, nth(x,env))"
paulson@13323
  1432
by (simp add: limit_ordinal_fm_def limit_ordinal_def sats_ordinal_fm')
paulson@13323
  1433
paulson@13323
  1434
lemma limit_ordinal_iff_sats:
paulson@13323
  1435
      "[| nth(i,env) = x; nth(j,env) = y; 
paulson@13323
  1436
          i \<in> nat; env \<in> list(A)|]
paulson@13323
  1437
       ==> limit_ordinal(**A, x) <-> sats(A, limit_ordinal_fm(i), env)"
paulson@13323
  1438
by simp
paulson@13323
  1439
paulson@13323
  1440
theorem limit_ordinal_reflection:
paulson@13323
  1441
     "REFLECTS[\<lambda>x. limit_ordinal(L,f(x)), 
paulson@13323
  1442
               \<lambda>i x. limit_ordinal(**Lset(i),f(x))]"
paulson@13323
  1443
apply (simp only: limit_ordinal_def setclass_simps)
paulson@13323
  1444
apply (intro FOL_reflections ordinal_reflection 
paulson@13323
  1445
             empty_reflection successor_reflection)  
paulson@13314
  1446
done
paulson@13309
  1447
paulson@13323
  1448
subsubsection{*Omega: The Set of Natural Numbers*}
paulson@13323
  1449
paulson@13323
  1450
(* omega(M,a) == limit_ordinal(M,a) & (\<forall>x[M]. x\<in>a --> ~ limit_ordinal(M,x)) *)
paulson@13323
  1451
constdefs omega_fm :: "i=>i"
paulson@13323
  1452
    "omega_fm(x) == 
paulson@13323
  1453
       And(limit_ordinal_fm(x),
paulson@13323
  1454
           Forall(Implies(Member(0,succ(x)),
paulson@13323
  1455
                          Neg(limit_ordinal_fm(0)))))"
paulson@13323
  1456
paulson@13323
  1457
lemma omega_type [TC]:
paulson@13323
  1458
     "x \<in> nat ==> omega_fm(x) \<in> formula"
paulson@13323
  1459
by (simp add: omega_fm_def) 
paulson@13323
  1460
paulson@13323
  1461
lemma arity_omega_fm [simp]:
paulson@13323
  1462
     "x \<in> nat ==> arity(omega_fm(x)) = succ(x)"
paulson@13323
  1463
by (simp add: omega_fm_def succ_Un_distrib [symmetric] Un_ac) 
paulson@13323
  1464
paulson@13323
  1465
lemma sats_omega_fm [simp]:
paulson@13323
  1466
   "[| x \<in> nat; env \<in> list(A)|]
paulson@13323
  1467
    ==> sats(A, omega_fm(x), env) <-> omega(**A, nth(x,env))"
paulson@13323
  1468
by (simp add: omega_fm_def omega_def)
paulson@13316
  1469
paulson@13323
  1470
lemma omega_iff_sats:
paulson@13323
  1471
      "[| nth(i,env) = x; nth(j,env) = y; 
paulson@13323
  1472
          i \<in> nat; env \<in> list(A)|]
paulson@13323
  1473
       ==> omega(**A, x) <-> sats(A, omega_fm(i), env)"
paulson@13323
  1474
by simp
paulson@13323
  1475
paulson@13323
  1476
theorem omega_reflection:
paulson@13323
  1477
     "REFLECTS[\<lambda>x. omega(L,f(x)), 
paulson@13323
  1478
               \<lambda>i x. omega(**Lset(i),f(x))]"
paulson@13323
  1479
apply (simp only: omega_def setclass_simps)
paulson@13323
  1480
apply (intro FOL_reflections limit_ordinal_reflection)  
paulson@13323
  1481
done
paulson@13323
  1482
paulson@13323
  1483
paulson@13323
  1484
lemmas fun_plus_reflections =
paulson@13323
  1485
        typed_function_reflection composition_reflection
paulson@13323
  1486
        injection_reflection surjection_reflection
paulson@13348
  1487
        bijection_reflection restriction_reflection
paulson@13348
  1488
        order_isomorphism_reflection
paulson@13323
  1489
        ordinal_reflection limit_ordinal_reflection omega_reflection
paulson@13323
  1490
paulson@13323
  1491
lemmas fun_plus_iff_sats = 
paulson@13323
  1492
	typed_function_iff_sats composition_iff_sats
paulson@13348
  1493
        injection_iff_sats surjection_iff_sats 
paulson@13348
  1494
        bijection_iff_sats restriction_iff_sats 
paulson@13316
  1495
        order_isomorphism_iff_sats
paulson@13323
  1496
        ordinal_iff_sats limit_ordinal_iff_sats omega_iff_sats
paulson@13316
  1497
paulson@13223
  1498
end