src/ZF/Constructible/L_axioms.thy
author paulson
Mon Jul 08 15:56:39 2002 +0200 (2002-07-08)
changeset 13314 84b9de3cbc91
parent 13309 a6adee8ea75a
child 13316 d16629fd0f95
permissions -rw-r--r--
Defining a meta-existential quantifier.
Using it to streamline reflection proofs.
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header {*The Class L Satisfies the ZF Axioms*}
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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_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{*Instantiation of the locale @{text M_triv_axioms}*}
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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 trivaxL 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", trivaxL (thm "M_triv_axioms.ball_abs"));
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bind_thm ("rall_abs", trivaxL (thm "M_triv_axioms.rall_abs"));
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bind_thm ("bex_abs", trivaxL (thm "M_triv_axioms.bex_abs"));
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bind_thm ("rex_abs", trivaxL (thm "M_triv_axioms.rex_abs"));
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bind_thm ("ball_iff_equiv", trivaxL (thm "M_triv_axioms.ball_iff_equiv"));
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bind_thm ("M_equalityI", trivaxL (thm "M_triv_axioms.M_equalityI"));
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bind_thm ("empty_abs", trivaxL (thm "M_triv_axioms.empty_abs"));
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bind_thm ("subset_abs", trivaxL (thm "M_triv_axioms.subset_abs"));
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bind_thm ("upair_abs", trivaxL (thm "M_triv_axioms.upair_abs"));
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bind_thm ("upair_in_M_iff", trivaxL (thm "M_triv_axioms.upair_in_M_iff"));
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bind_thm ("singleton_in_M_iff", trivaxL (thm "M_triv_axioms.singleton_in_M_iff"));
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bind_thm ("pair_abs", trivaxL (thm "M_triv_axioms.pair_abs"));
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bind_thm ("pair_in_M_iff", trivaxL (thm "M_triv_axioms.pair_in_M_iff"));
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bind_thm ("pair_components_in_M", trivaxL (thm "M_triv_axioms.pair_components_in_M"));
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bind_thm ("cartprod_abs", trivaxL (thm "M_triv_axioms.cartprod_abs"));
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bind_thm ("union_abs", trivaxL (thm "M_triv_axioms.union_abs"));
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bind_thm ("inter_abs", trivaxL (thm "M_triv_axioms.inter_abs"));
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bind_thm ("setdiff_abs", trivaxL (thm "M_triv_axioms.setdiff_abs"));
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bind_thm ("Union_abs", trivaxL (thm "M_triv_axioms.Union_abs"));
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bind_thm ("Union_closed", trivaxL (thm "M_triv_axioms.Union_closed"));
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bind_thm ("Un_closed", trivaxL (thm "M_triv_axioms.Un_closed"));
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bind_thm ("cons_closed", trivaxL (thm "M_triv_axioms.cons_closed"));
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bind_thm ("successor_abs", trivaxL (thm "M_triv_axioms.successor_abs"));
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bind_thm ("succ_in_M_iff", trivaxL (thm "M_triv_axioms.succ_in_M_iff"));
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bind_thm ("separation_closed", trivaxL (thm "M_triv_axioms.separation_closed"));
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bind_thm ("strong_replacementI", trivaxL (thm "M_triv_axioms.strong_replacementI"));
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bind_thm ("strong_replacement_closed", trivaxL (thm "M_triv_axioms.strong_replacement_closed"));
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bind_thm ("RepFun_closed", trivaxL (thm "M_triv_axioms.RepFun_closed"));
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bind_thm ("lam_closed", trivaxL (thm "M_triv_axioms.lam_closed"));
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bind_thm ("image_abs", trivaxL (thm "M_triv_axioms.image_abs"));
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bind_thm ("powerset_Pow", trivaxL (thm "M_triv_axioms.powerset_Pow"));
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bind_thm ("powerset_imp_subset_Pow", trivaxL (thm "M_triv_axioms.powerset_imp_subset_Pow"));
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bind_thm ("nat_into_M", trivaxL (thm "M_triv_axioms.nat_into_M"));
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bind_thm ("nat_case_closed", trivaxL (thm "M_triv_axioms.nat_case_closed"));
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bind_thm ("Inl_in_M_iff", trivaxL (thm "M_triv_axioms.Inl_in_M_iff"));
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bind_thm ("Inr_in_M_iff", trivaxL (thm "M_triv_axioms.Inr_in_M_iff"));
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bind_thm ("lt_closed", trivaxL (thm "M_triv_axioms.lt_closed"));
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bind_thm ("transitive_set_abs", trivaxL (thm "M_triv_axioms.transitive_set_abs"));
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bind_thm ("ordinal_abs", trivaxL (thm "M_triv_axioms.ordinal_abs"));
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bind_thm ("limit_ordinal_abs", trivaxL (thm "M_triv_axioms.limit_ordinal_abs"));
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bind_thm ("successor_ordinal_abs", trivaxL (thm "M_triv_axioms.successor_ordinal_abs"));
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bind_thm ("finite_ordinal_abs", trivaxL (thm "M_triv_axioms.finite_ordinal_abs"));
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bind_thm ("omega_abs", trivaxL (thm "M_triv_axioms.omega_abs"));
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bind_thm ("number1_abs", trivaxL (thm "M_triv_axioms.number1_abs"));
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bind_thm ("number1_abs", trivaxL (thm "M_triv_axioms.number1_abs"));
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bind_thm ("number3_abs", trivaxL (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)]" 
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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.All_reflection [OF Lset_mono_le Lset_cont Pair_in_Lset],
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       assumption+)
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done
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   280
paulson@13314
   281
theorem Rex_reflection:
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   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) 
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   285
apply (intro And_reflection Ex_reflection, assumption)
paulson@13314
   286
done
paulson@13291
   287
paulson@13314
   288
theorem Rall_reflection:
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   289
     "REFLECTS[\<lambda>x. P(fst(x),snd(x)), \<lambda>a x. Q(a,fst(x),snd(x))]
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   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@13314
   295
lemmas FOL_reflection = 
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   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@13314
   312
apply (drule ReflectsD, assumption)
paulson@13314
   313
apply blast 
paulson@13314
   314
done
paulson@13291
   315
paulson@13291
   316
lemma Collect_mem_eq: "{x\<in>A. x\<in>B} = A \<inter> B";
paulson@13291
   317
by blast
paulson@13291
   318
paulson@13291
   319
paulson@13298
   320
subsection{*Internalized formulas for some relativized ones*}
paulson@13298
   321
paulson@13306
   322
lemmas setclass_simps = rall_setclass_is_ball rex_setclass_is_bex
paulson@13306
   323
paulson@13306
   324
subsubsection{*Some numbers to help write de Bruijn indices*}
paulson@13306
   325
paulson@13306
   326
syntax
paulson@13306
   327
    "3" :: i   ("3")
paulson@13306
   328
    "4" :: i   ("4")
paulson@13306
   329
    "5" :: i   ("5")
paulson@13306
   330
    "6" :: i   ("6")
paulson@13306
   331
    "7" :: i   ("7")
paulson@13306
   332
    "8" :: i   ("8")
paulson@13306
   333
    "9" :: i   ("9")
paulson@13306
   334
paulson@13306
   335
translations
paulson@13306
   336
   "3"  == "succ(2)"
paulson@13306
   337
   "4"  == "succ(3)"
paulson@13306
   338
   "5"  == "succ(4)"
paulson@13306
   339
   "6"  == "succ(5)"
paulson@13306
   340
   "7"  == "succ(6)"
paulson@13306
   341
   "8"  == "succ(7)"
paulson@13306
   342
   "9"  == "succ(8)"
paulson@13306
   343
paulson@13298
   344
subsubsection{*Unordered pairs*}
paulson@13298
   345
paulson@13298
   346
constdefs upair_fm :: "[i,i,i]=>i"
paulson@13298
   347
    "upair_fm(x,y,z) == 
paulson@13298
   348
       And(Member(x,z), 
paulson@13298
   349
           And(Member(y,z),
paulson@13298
   350
               Forall(Implies(Member(0,succ(z)), 
paulson@13298
   351
                              Or(Equal(0,succ(x)), Equal(0,succ(y)))))))"
paulson@13298
   352
paulson@13298
   353
lemma upair_type [TC]:
paulson@13298
   354
     "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> upair_fm(x,y,z) \<in> formula"
paulson@13298
   355
by (simp add: upair_fm_def) 
paulson@13298
   356
paulson@13298
   357
lemma arity_upair_fm [simp]:
paulson@13298
   358
     "[| x \<in> nat; y \<in> nat; z \<in> nat |] 
paulson@13298
   359
      ==> arity(upair_fm(x,y,z)) = succ(x) \<union> succ(y) \<union> succ(z)"
paulson@13298
   360
by (simp add: upair_fm_def succ_Un_distrib [symmetric] Un_ac) 
paulson@13298
   361
paulson@13298
   362
lemma sats_upair_fm [simp]:
paulson@13298
   363
   "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
paulson@13298
   364
    ==> sats(A, upair_fm(x,y,z), env) <-> 
paulson@13298
   365
            upair(**A, nth(x,env), nth(y,env), nth(z,env))"
paulson@13298
   366
by (simp add: upair_fm_def upair_def)
paulson@13298
   367
paulson@13298
   368
lemma upair_iff_sats:
paulson@13298
   369
      "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z; 
paulson@13298
   370
          i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
paulson@13298
   371
       ==> upair(**A, x, y, z) <-> sats(A, upair_fm(i,j,k), env)"
paulson@13298
   372
by (simp add: sats_upair_fm)
paulson@13298
   373
paulson@13298
   374
text{*Useful? At least it refers to "real" unordered pairs*}
paulson@13298
   375
lemma sats_upair_fm2 [simp]:
paulson@13298
   376
   "[| x \<in> nat; y \<in> nat; z < length(env); env \<in> list(A); Transset(A)|]
paulson@13298
   377
    ==> sats(A, upair_fm(x,y,z), env) <-> 
paulson@13298
   378
        nth(z,env) = {nth(x,env), nth(y,env)}"
paulson@13298
   379
apply (frule lt_length_in_nat, assumption)  
paulson@13298
   380
apply (simp add: upair_fm_def Transset_def, auto) 
paulson@13298
   381
apply (blast intro: nth_type) 
paulson@13298
   382
done
paulson@13298
   383
paulson@13314
   384
theorem upair_reflection:
paulson@13314
   385
     "REFLECTS[\<lambda>x. upair(L,f(x),g(x),h(x)), 
paulson@13314
   386
               \<lambda>i x. upair(**Lset(i),f(x),g(x),h(x))]" 
paulson@13314
   387
apply (simp add: upair_def)
paulson@13314
   388
apply (intro FOL_reflection)  
paulson@13314
   389
done
paulson@13306
   390
paulson@13298
   391
subsubsection{*Ordered pairs*}
paulson@13298
   392
paulson@13298
   393
constdefs pair_fm :: "[i,i,i]=>i"
paulson@13298
   394
    "pair_fm(x,y,z) == 
paulson@13298
   395
       Exists(And(upair_fm(succ(x),succ(x),0),
paulson@13298
   396
              Exists(And(upair_fm(succ(succ(x)),succ(succ(y)),0),
paulson@13298
   397
                         upair_fm(1,0,succ(succ(z)))))))"
paulson@13298
   398
paulson@13298
   399
lemma pair_type [TC]:
paulson@13298
   400
     "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> pair_fm(x,y,z) \<in> formula"
paulson@13298
   401
by (simp add: pair_fm_def) 
paulson@13298
   402
paulson@13298
   403
lemma arity_pair_fm [simp]:
paulson@13298
   404
     "[| x \<in> nat; y \<in> nat; z \<in> nat |] 
paulson@13298
   405
      ==> arity(pair_fm(x,y,z)) = succ(x) \<union> succ(y) \<union> succ(z)"
paulson@13298
   406
by (simp add: pair_fm_def succ_Un_distrib [symmetric] Un_ac) 
paulson@13298
   407
paulson@13298
   408
lemma sats_pair_fm [simp]:
paulson@13298
   409
   "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
paulson@13298
   410
    ==> sats(A, pair_fm(x,y,z), env) <-> 
paulson@13298
   411
        pair(**A, nth(x,env), nth(y,env), nth(z,env))"
paulson@13298
   412
by (simp add: pair_fm_def pair_def)
paulson@13298
   413
paulson@13298
   414
lemma pair_iff_sats:
paulson@13298
   415
      "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z; 
paulson@13298
   416
          i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
paulson@13298
   417
       ==> pair(**A, x, y, z) <-> sats(A, pair_fm(i,j,k), env)"
paulson@13298
   418
by (simp add: sats_pair_fm)
paulson@13298
   419
paulson@13314
   420
theorem pair_reflection:
paulson@13314
   421
     "REFLECTS[\<lambda>x. pair(L,f(x),g(x),h(x)), 
paulson@13314
   422
               \<lambda>i x. pair(**Lset(i),f(x),g(x),h(x))]"
paulson@13314
   423
apply (simp only: pair_def setclass_simps)
paulson@13314
   424
apply (intro FOL_reflection upair_reflection)  
paulson@13314
   425
done
paulson@13306
   426
paulson@13306
   427
paulson@13306
   428
subsubsection{*Binary Unions*}
paulson@13298
   429
paulson@13306
   430
constdefs union_fm :: "[i,i,i]=>i"
paulson@13306
   431
    "union_fm(x,y,z) == 
paulson@13306
   432
       Forall(Iff(Member(0,succ(z)),
paulson@13306
   433
                  Or(Member(0,succ(x)),Member(0,succ(y)))))"
paulson@13306
   434
paulson@13306
   435
lemma union_type [TC]:
paulson@13306
   436
     "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> union_fm(x,y,z) \<in> formula"
paulson@13306
   437
by (simp add: union_fm_def) 
paulson@13306
   438
paulson@13306
   439
lemma arity_union_fm [simp]:
paulson@13306
   440
     "[| x \<in> nat; y \<in> nat; z \<in> nat |] 
paulson@13306
   441
      ==> arity(union_fm(x,y,z)) = succ(x) \<union> succ(y) \<union> succ(z)"
paulson@13306
   442
by (simp add: union_fm_def succ_Un_distrib [symmetric] Un_ac) 
paulson@13298
   443
paulson@13306
   444
lemma sats_union_fm [simp]:
paulson@13306
   445
   "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
paulson@13306
   446
    ==> sats(A, union_fm(x,y,z), env) <-> 
paulson@13306
   447
        union(**A, nth(x,env), nth(y,env), nth(z,env))"
paulson@13306
   448
by (simp add: union_fm_def union_def)
paulson@13306
   449
paulson@13306
   450
lemma union_iff_sats:
paulson@13306
   451
      "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z; 
paulson@13306
   452
          i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
paulson@13306
   453
       ==> union(**A, x, y, z) <-> sats(A, union_fm(i,j,k), env)"
paulson@13306
   454
by (simp add: sats_union_fm)
paulson@13298
   455
paulson@13314
   456
theorem union_reflection:
paulson@13314
   457
     "REFLECTS[\<lambda>x. union(L,f(x),g(x),h(x)), 
paulson@13314
   458
               \<lambda>i x. union(**Lset(i),f(x),g(x),h(x))]"
paulson@13314
   459
apply (simp only: union_def setclass_simps)
paulson@13314
   460
apply (intro FOL_reflection)  
paulson@13314
   461
done
paulson@13306
   462
paulson@13298
   463
paulson@13306
   464
subsubsection{*`Cons' for sets*}
paulson@13306
   465
paulson@13306
   466
constdefs cons_fm :: "[i,i,i]=>i"
paulson@13306
   467
    "cons_fm(x,y,z) == 
paulson@13306
   468
       Exists(And(upair_fm(succ(x),succ(x),0),
paulson@13306
   469
                  union_fm(0,succ(y),succ(z))))"
paulson@13298
   470
paulson@13298
   471
paulson@13306
   472
lemma cons_type [TC]:
paulson@13306
   473
     "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> cons_fm(x,y,z) \<in> formula"
paulson@13306
   474
by (simp add: cons_fm_def) 
paulson@13306
   475
paulson@13306
   476
lemma arity_cons_fm [simp]:
paulson@13306
   477
     "[| x \<in> nat; y \<in> nat; z \<in> nat |] 
paulson@13306
   478
      ==> arity(cons_fm(x,y,z)) = succ(x) \<union> succ(y) \<union> succ(z)"
paulson@13306
   479
by (simp add: cons_fm_def succ_Un_distrib [symmetric] Un_ac) 
paulson@13306
   480
paulson@13306
   481
lemma sats_cons_fm [simp]:
paulson@13306
   482
   "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
paulson@13306
   483
    ==> sats(A, cons_fm(x,y,z), env) <-> 
paulson@13306
   484
        is_cons(**A, nth(x,env), nth(y,env), nth(z,env))"
paulson@13306
   485
by (simp add: cons_fm_def is_cons_def)
paulson@13306
   486
paulson@13306
   487
lemma cons_iff_sats:
paulson@13306
   488
      "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z; 
paulson@13306
   489
          i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
paulson@13306
   490
       ==> is_cons(**A, x, y, z) <-> sats(A, cons_fm(i,j,k), env)"
paulson@13306
   491
by simp
paulson@13306
   492
paulson@13314
   493
theorem cons_reflection:
paulson@13314
   494
     "REFLECTS[\<lambda>x. is_cons(L,f(x),g(x),h(x)), 
paulson@13314
   495
               \<lambda>i x. is_cons(**Lset(i),f(x),g(x),h(x))]"
paulson@13314
   496
apply (simp only: is_cons_def setclass_simps)
paulson@13314
   497
apply (intro FOL_reflection upair_reflection union_reflection)  
paulson@13314
   498
done
paulson@13298
   499
paulson@13298
   500
paulson@13306
   501
subsubsection{*Function Applications*}
paulson@13306
   502
paulson@13306
   503
constdefs fun_apply_fm :: "[i,i,i]=>i"
paulson@13306
   504
    "fun_apply_fm(f,x,y) == 
paulson@13306
   505
       Forall(Iff(Exists(And(Member(0,succ(succ(f))),
paulson@13306
   506
                             pair_fm(succ(succ(x)), 1, 0))),
paulson@13306
   507
                  Equal(succ(y),0)))"
paulson@13298
   508
paulson@13306
   509
lemma fun_apply_type [TC]:
paulson@13306
   510
     "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> fun_apply_fm(x,y,z) \<in> formula"
paulson@13306
   511
by (simp add: fun_apply_fm_def) 
paulson@13306
   512
paulson@13306
   513
lemma arity_fun_apply_fm [simp]:
paulson@13306
   514
     "[| x \<in> nat; y \<in> nat; z \<in> nat |] 
paulson@13306
   515
      ==> arity(fun_apply_fm(x,y,z)) = succ(x) \<union> succ(y) \<union> succ(z)"
paulson@13306
   516
by (simp add: fun_apply_fm_def succ_Un_distrib [symmetric] Un_ac) 
paulson@13298
   517
paulson@13306
   518
lemma sats_fun_apply_fm [simp]:
paulson@13306
   519
   "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
paulson@13306
   520
    ==> sats(A, fun_apply_fm(x,y,z), env) <-> 
paulson@13306
   521
        fun_apply(**A, nth(x,env), nth(y,env), nth(z,env))"
paulson@13306
   522
by (simp add: fun_apply_fm_def fun_apply_def)
paulson@13306
   523
paulson@13306
   524
lemma fun_apply_iff_sats:
paulson@13306
   525
      "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z; 
paulson@13306
   526
          i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
paulson@13306
   527
       ==> fun_apply(**A, x, y, z) <-> sats(A, fun_apply_fm(i,j,k), env)"
paulson@13306
   528
by simp
paulson@13306
   529
paulson@13314
   530
theorem fun_apply_reflection:
paulson@13314
   531
     "REFLECTS[\<lambda>x. fun_apply(L,f(x),g(x),h(x)), 
paulson@13314
   532
               \<lambda>i x. fun_apply(**Lset(i),f(x),g(x),h(x))]" 
paulson@13314
   533
apply (simp only: fun_apply_def setclass_simps)
paulson@13314
   534
apply (intro FOL_reflection pair_reflection)  
paulson@13314
   535
done
paulson@13298
   536
paulson@13298
   537
paulson@13306
   538
subsubsection{*Variants of Satisfaction Definitions for Ordinals, etc.*}
paulson@13306
   539
paulson@13306
   540
text{*Differs from the one in Formula by using "ordinal" rather than "Ord"*}
paulson@13306
   541
paulson@13306
   542
paulson@13306
   543
lemma sats_subset_fm':
paulson@13306
   544
   "[|x \<in> nat; y \<in> nat; env \<in> list(A)|]
paulson@13306
   545
    ==> sats(A, subset_fm(x,y), env) <-> subset(**A, nth(x,env), nth(y,env))" 
paulson@13306
   546
by (simp add: subset_fm_def subset_def) 
paulson@13298
   547
paulson@13314
   548
theorem subset_reflection:
paulson@13314
   549
     "REFLECTS[\<lambda>x. subset(L,f(x),g(x)), 
paulson@13314
   550
               \<lambda>i x. subset(**Lset(i),f(x),g(x))]" 
paulson@13314
   551
apply (simp only: subset_def setclass_simps)
paulson@13314
   552
apply (intro FOL_reflection)  
paulson@13314
   553
done
paulson@13306
   554
paulson@13306
   555
lemma sats_transset_fm':
paulson@13306
   556
   "[|x \<in> nat; env \<in> list(A)|]
paulson@13306
   557
    ==> sats(A, transset_fm(x), env) <-> transitive_set(**A, nth(x,env))"
paulson@13306
   558
by (simp add: sats_subset_fm' transset_fm_def transitive_set_def) 
paulson@13298
   559
paulson@13314
   560
theorem transitive_set_reflection:
paulson@13314
   561
     "REFLECTS[\<lambda>x. transitive_set(L,f(x)),
paulson@13314
   562
               \<lambda>i x. transitive_set(**Lset(i),f(x))]"
paulson@13314
   563
apply (simp only: transitive_set_def setclass_simps)
paulson@13314
   564
apply (intro FOL_reflection subset_reflection)  
paulson@13314
   565
done
paulson@13306
   566
paulson@13306
   567
lemma sats_ordinal_fm':
paulson@13306
   568
   "[|x \<in> nat; env \<in> list(A)|]
paulson@13306
   569
    ==> sats(A, ordinal_fm(x), env) <-> ordinal(**A,nth(x,env))"
paulson@13306
   570
by (simp add: sats_transset_fm' ordinal_fm_def ordinal_def)
paulson@13306
   571
paulson@13306
   572
lemma ordinal_iff_sats:
paulson@13306
   573
      "[| nth(i,env) = x;  i \<in> nat; env \<in> list(A)|]
paulson@13306
   574
       ==> ordinal(**A, x) <-> sats(A, ordinal_fm(i), env)"
paulson@13306
   575
by (simp add: sats_ordinal_fm')
paulson@13306
   576
paulson@13314
   577
theorem ordinal_reflection:
paulson@13314
   578
     "REFLECTS[\<lambda>x. ordinal(L,f(x)), \<lambda>i x. ordinal(**Lset(i),f(x))]"
paulson@13314
   579
apply (simp only: ordinal_def setclass_simps)
paulson@13314
   580
apply (intro FOL_reflection transitive_set_reflection)  
paulson@13314
   581
done
paulson@13298
   582
paulson@13298
   583
paulson@13306
   584
subsubsection{*Membership Relation*}
paulson@13298
   585
paulson@13306
   586
constdefs Memrel_fm :: "[i,i]=>i"
paulson@13306
   587
    "Memrel_fm(A,r) == 
paulson@13306
   588
       Forall(Iff(Member(0,succ(r)),
paulson@13306
   589
                  Exists(And(Member(0,succ(succ(A))),
paulson@13306
   590
                             Exists(And(Member(0,succ(succ(succ(A)))),
paulson@13306
   591
                                        And(Member(1,0),
paulson@13306
   592
                                            pair_fm(1,0,2))))))))"
paulson@13306
   593
paulson@13306
   594
lemma Memrel_type [TC]:
paulson@13306
   595
     "[| x \<in> nat; y \<in> nat |] ==> Memrel_fm(x,y) \<in> formula"
paulson@13306
   596
by (simp add: Memrel_fm_def) 
paulson@13298
   597
paulson@13306
   598
lemma arity_Memrel_fm [simp]:
paulson@13306
   599
     "[| x \<in> nat; y \<in> nat |] 
paulson@13306
   600
      ==> arity(Memrel_fm(x,y)) = succ(x) \<union> succ(y)"
paulson@13306
   601
by (simp add: Memrel_fm_def succ_Un_distrib [symmetric] Un_ac) 
paulson@13306
   602
paulson@13306
   603
lemma sats_Memrel_fm [simp]:
paulson@13306
   604
   "[| x \<in> nat; y \<in> nat; env \<in> list(A)|]
paulson@13306
   605
    ==> sats(A, Memrel_fm(x,y), env) <-> 
paulson@13306
   606
        membership(**A, nth(x,env), nth(y,env))"
paulson@13306
   607
by (simp add: Memrel_fm_def membership_def)
paulson@13298
   608
paulson@13306
   609
lemma Memrel_iff_sats:
paulson@13306
   610
      "[| nth(i,env) = x; nth(j,env) = y; 
paulson@13306
   611
          i \<in> nat; j \<in> nat; env \<in> list(A)|]
paulson@13306
   612
       ==> membership(**A, x, y) <-> sats(A, Memrel_fm(i,j), env)"
paulson@13306
   613
by simp
paulson@13304
   614
paulson@13314
   615
theorem membership_reflection:
paulson@13314
   616
     "REFLECTS[\<lambda>x. membership(L,f(x),g(x)), 
paulson@13314
   617
               \<lambda>i x. membership(**Lset(i),f(x),g(x))]"
paulson@13314
   618
apply (simp only: membership_def setclass_simps)
paulson@13314
   619
apply (intro FOL_reflection pair_reflection)  
paulson@13314
   620
done
paulson@13304
   621
paulson@13306
   622
subsubsection{*Predecessor Set*}
paulson@13304
   623
paulson@13306
   624
constdefs pred_set_fm :: "[i,i,i,i]=>i"
paulson@13306
   625
    "pred_set_fm(A,x,r,B) == 
paulson@13306
   626
       Forall(Iff(Member(0,succ(B)),
paulson@13306
   627
                  Exists(And(Member(0,succ(succ(r))),
paulson@13306
   628
                             And(Member(1,succ(succ(A))),
paulson@13306
   629
                                 pair_fm(1,succ(succ(x)),0))))))"
paulson@13306
   630
paulson@13306
   631
paulson@13306
   632
lemma pred_set_type [TC]:
paulson@13306
   633
     "[| A \<in> nat; x \<in> nat; r \<in> nat; B \<in> nat |] 
paulson@13306
   634
      ==> pred_set_fm(A,x,r,B) \<in> formula"
paulson@13306
   635
by (simp add: pred_set_fm_def) 
paulson@13304
   636
paulson@13306
   637
lemma arity_pred_set_fm [simp]:
paulson@13306
   638
   "[| A \<in> nat; x \<in> nat; r \<in> nat; B \<in> nat |] 
paulson@13306
   639
    ==> arity(pred_set_fm(A,x,r,B)) = succ(A) \<union> succ(x) \<union> succ(r) \<union> succ(B)"
paulson@13306
   640
by (simp add: pred_set_fm_def succ_Un_distrib [symmetric] Un_ac) 
paulson@13306
   641
paulson@13306
   642
lemma sats_pred_set_fm [simp]:
paulson@13306
   643
   "[| U \<in> nat; x \<in> nat; r \<in> nat; B \<in> nat; env \<in> list(A)|]
paulson@13306
   644
    ==> sats(A, pred_set_fm(U,x,r,B), env) <-> 
paulson@13306
   645
        pred_set(**A, nth(U,env), nth(x,env), nth(r,env), nth(B,env))"
paulson@13306
   646
by (simp add: pred_set_fm_def pred_set_def)
paulson@13306
   647
paulson@13306
   648
lemma pred_set_iff_sats:
paulson@13306
   649
      "[| nth(i,env) = U; nth(j,env) = x; nth(k,env) = r; nth(l,env) = B; 
paulson@13306
   650
          i \<in> nat; j \<in> nat; k \<in> nat; l \<in> nat; env \<in> list(A)|]
paulson@13306
   651
       ==> pred_set(**A,U,x,r,B) <-> sats(A, pred_set_fm(i,j,k,l), env)"
paulson@13306
   652
by (simp add: sats_pred_set_fm)
paulson@13306
   653
paulson@13314
   654
theorem pred_set_reflection:
paulson@13314
   655
     "REFLECTS[\<lambda>x. pred_set(L,f(x),g(x),h(x),b(x)), 
paulson@13314
   656
               \<lambda>i x. pred_set(**Lset(i),f(x),g(x),h(x),b(x))]" 
paulson@13314
   657
apply (simp only: pred_set_def setclass_simps)
paulson@13314
   658
apply (intro FOL_reflection pair_reflection)  
paulson@13314
   659
done
paulson@13304
   660
paulson@13304
   661
paulson@13298
   662
paulson@13306
   663
subsubsection{*Domain*}
paulson@13306
   664
paulson@13306
   665
(* "is_domain(M,r,z) == 
paulson@13306
   666
	\<forall>x[M]. (x \<in> z <-> (\<exists>w[M]. w\<in>r & (\<exists>y[M]. pair(M,x,y,w))))" *)
paulson@13306
   667
constdefs domain_fm :: "[i,i]=>i"
paulson@13306
   668
    "domain_fm(r,z) == 
paulson@13306
   669
       Forall(Iff(Member(0,succ(z)),
paulson@13306
   670
                  Exists(And(Member(0,succ(succ(r))),
paulson@13306
   671
                             Exists(pair_fm(2,0,1))))))"
paulson@13306
   672
paulson@13306
   673
lemma domain_type [TC]:
paulson@13306
   674
     "[| x \<in> nat; y \<in> nat |] ==> domain_fm(x,y) \<in> formula"
paulson@13306
   675
by (simp add: domain_fm_def) 
paulson@13306
   676
paulson@13306
   677
lemma arity_domain_fm [simp]:
paulson@13306
   678
     "[| x \<in> nat; y \<in> nat |] 
paulson@13306
   679
      ==> arity(domain_fm(x,y)) = succ(x) \<union> succ(y)"
paulson@13306
   680
by (simp add: domain_fm_def succ_Un_distrib [symmetric] Un_ac) 
paulson@13306
   681
paulson@13306
   682
lemma sats_domain_fm [simp]:
paulson@13306
   683
   "[| x \<in> nat; y \<in> nat; env \<in> list(A)|]
paulson@13306
   684
    ==> sats(A, domain_fm(x,y), env) <-> 
paulson@13306
   685
        is_domain(**A, nth(x,env), nth(y,env))"
paulson@13306
   686
by (simp add: domain_fm_def is_domain_def)
paulson@13306
   687
paulson@13306
   688
lemma domain_iff_sats:
paulson@13306
   689
      "[| nth(i,env) = x; nth(j,env) = y; 
paulson@13306
   690
          i \<in> nat; j \<in> nat; env \<in> list(A)|]
paulson@13306
   691
       ==> is_domain(**A, x, y) <-> sats(A, domain_fm(i,j), env)"
paulson@13306
   692
by simp
paulson@13306
   693
paulson@13314
   694
theorem domain_reflection:
paulson@13314
   695
     "REFLECTS[\<lambda>x. is_domain(L,f(x),g(x)), 
paulson@13314
   696
               \<lambda>i x. is_domain(**Lset(i),f(x),g(x))]"
paulson@13314
   697
apply (simp only: is_domain_def setclass_simps)
paulson@13314
   698
apply (intro FOL_reflection pair_reflection)  
paulson@13314
   699
done
paulson@13306
   700
paulson@13306
   701
paulson@13306
   702
subsubsection{*Range*}
paulson@13306
   703
paulson@13306
   704
(* "is_range(M,r,z) == 
paulson@13306
   705
	\<forall>y[M]. (y \<in> z <-> (\<exists>w[M]. w\<in>r & (\<exists>x[M]. pair(M,x,y,w))))" *)
paulson@13306
   706
constdefs range_fm :: "[i,i]=>i"
paulson@13306
   707
    "range_fm(r,z) == 
paulson@13306
   708
       Forall(Iff(Member(0,succ(z)),
paulson@13306
   709
                  Exists(And(Member(0,succ(succ(r))),
paulson@13306
   710
                             Exists(pair_fm(0,2,1))))))"
paulson@13306
   711
paulson@13306
   712
lemma range_type [TC]:
paulson@13306
   713
     "[| x \<in> nat; y \<in> nat |] ==> range_fm(x,y) \<in> formula"
paulson@13306
   714
by (simp add: range_fm_def) 
paulson@13306
   715
paulson@13306
   716
lemma arity_range_fm [simp]:
paulson@13306
   717
     "[| x \<in> nat; y \<in> nat |] 
paulson@13306
   718
      ==> arity(range_fm(x,y)) = succ(x) \<union> succ(y)"
paulson@13306
   719
by (simp add: range_fm_def succ_Un_distrib [symmetric] Un_ac) 
paulson@13306
   720
paulson@13306
   721
lemma sats_range_fm [simp]:
paulson@13306
   722
   "[| x \<in> nat; y \<in> nat; env \<in> list(A)|]
paulson@13306
   723
    ==> sats(A, range_fm(x,y), env) <-> 
paulson@13306
   724
        is_range(**A, nth(x,env), nth(y,env))"
paulson@13306
   725
by (simp add: range_fm_def is_range_def)
paulson@13306
   726
paulson@13306
   727
lemma range_iff_sats:
paulson@13306
   728
      "[| nth(i,env) = x; nth(j,env) = y; 
paulson@13306
   729
          i \<in> nat; j \<in> nat; env \<in> list(A)|]
paulson@13306
   730
       ==> is_range(**A, x, y) <-> sats(A, range_fm(i,j), env)"
paulson@13306
   731
by simp
paulson@13306
   732
paulson@13314
   733
theorem range_reflection:
paulson@13314
   734
     "REFLECTS[\<lambda>x. is_range(L,f(x),g(x)), 
paulson@13314
   735
               \<lambda>i x. is_range(**Lset(i),f(x),g(x))]"
paulson@13314
   736
apply (simp only: is_range_def setclass_simps)
paulson@13314
   737
apply (intro FOL_reflection pair_reflection)  
paulson@13314
   738
done
paulson@13306
   739
paulson@13306
   740
 
paulson@13306
   741
subsubsection{*Image*}
paulson@13306
   742
paulson@13306
   743
(* "image(M,r,A,z) == 
paulson@13306
   744
        \<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
   745
constdefs image_fm :: "[i,i,i]=>i"
paulson@13306
   746
    "image_fm(r,A,z) == 
paulson@13306
   747
       Forall(Iff(Member(0,succ(z)),
paulson@13306
   748
                  Exists(And(Member(0,succ(succ(r))),
paulson@13306
   749
                             Exists(And(Member(0,succ(succ(succ(A)))),
paulson@13306
   750
	 			        pair_fm(0,2,1)))))))"
paulson@13306
   751
paulson@13306
   752
lemma image_type [TC]:
paulson@13306
   753
     "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> image_fm(x,y,z) \<in> formula"
paulson@13306
   754
by (simp add: image_fm_def) 
paulson@13306
   755
paulson@13306
   756
lemma arity_image_fm [simp]:
paulson@13306
   757
     "[| x \<in> nat; y \<in> nat; z \<in> nat |] 
paulson@13306
   758
      ==> arity(image_fm(x,y,z)) = succ(x) \<union> succ(y) \<union> succ(z)"
paulson@13306
   759
by (simp add: image_fm_def succ_Un_distrib [symmetric] Un_ac) 
paulson@13306
   760
paulson@13306
   761
lemma sats_image_fm [simp]:
paulson@13306
   762
   "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
paulson@13306
   763
    ==> sats(A, image_fm(x,y,z), env) <-> 
paulson@13306
   764
        image(**A, nth(x,env), nth(y,env), nth(z,env))"
paulson@13306
   765
by (simp add: image_fm_def image_def)
paulson@13306
   766
paulson@13306
   767
lemma image_iff_sats:
paulson@13306
   768
      "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z; 
paulson@13306
   769
          i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
paulson@13306
   770
       ==> image(**A, x, y, z) <-> sats(A, image_fm(i,j,k), env)"
paulson@13306
   771
by (simp add: sats_image_fm)
paulson@13306
   772
paulson@13314
   773
theorem image_reflection:
paulson@13314
   774
     "REFLECTS[\<lambda>x. image(L,f(x),g(x),h(x)), 
paulson@13314
   775
               \<lambda>i x. image(**Lset(i),f(x),g(x),h(x))]"
paulson@13314
   776
apply (simp only: image_def setclass_simps)
paulson@13314
   777
apply (intro FOL_reflection pair_reflection)  
paulson@13314
   778
done
paulson@13306
   779
paulson@13306
   780
paulson@13306
   781
subsubsection{*The Concept of Relation*}
paulson@13306
   782
paulson@13306
   783
(* "is_relation(M,r) == 
paulson@13306
   784
        (\<forall>z[M]. z\<in>r --> (\<exists>x[M]. \<exists>y[M]. pair(M,x,y,z)))" *)
paulson@13306
   785
constdefs relation_fm :: "i=>i"
paulson@13306
   786
    "relation_fm(r) == 
paulson@13306
   787
       Forall(Implies(Member(0,succ(r)), Exists(Exists(pair_fm(1,0,2)))))"
paulson@13306
   788
paulson@13306
   789
lemma relation_type [TC]:
paulson@13306
   790
     "[| x \<in> nat |] ==> relation_fm(x) \<in> formula"
paulson@13306
   791
by (simp add: relation_fm_def) 
paulson@13306
   792
paulson@13306
   793
lemma arity_relation_fm [simp]:
paulson@13306
   794
     "x \<in> nat ==> arity(relation_fm(x)) = succ(x)"
paulson@13306
   795
by (simp add: relation_fm_def succ_Un_distrib [symmetric] Un_ac) 
paulson@13306
   796
paulson@13306
   797
lemma sats_relation_fm [simp]:
paulson@13306
   798
   "[| x \<in> nat; env \<in> list(A)|]
paulson@13306
   799
    ==> sats(A, relation_fm(x), env) <-> is_relation(**A, nth(x,env))"
paulson@13306
   800
by (simp add: relation_fm_def is_relation_def)
paulson@13306
   801
paulson@13306
   802
lemma relation_iff_sats:
paulson@13306
   803
      "[| nth(i,env) = x; nth(j,env) = y; 
paulson@13306
   804
          i \<in> nat; env \<in> list(A)|]
paulson@13306
   805
       ==> is_relation(**A, x) <-> sats(A, relation_fm(i), env)"
paulson@13306
   806
by simp
paulson@13306
   807
paulson@13314
   808
theorem is_relation_reflection:
paulson@13314
   809
     "REFLECTS[\<lambda>x. is_relation(L,f(x)), 
paulson@13314
   810
               \<lambda>i x. is_relation(**Lset(i),f(x))]"
paulson@13314
   811
apply (simp only: is_relation_def setclass_simps)
paulson@13314
   812
apply (intro FOL_reflection pair_reflection)  
paulson@13314
   813
done
paulson@13306
   814
paulson@13306
   815
paulson@13306
   816
subsubsection{*The Concept of Function*}
paulson@13306
   817
paulson@13306
   818
(* "is_function(M,r) == 
paulson@13306
   819
	\<forall>x[M]. \<forall>y[M]. \<forall>y'[M]. \<forall>p[M]. \<forall>p'[M]. 
paulson@13306
   820
           pair(M,x,y,p) --> pair(M,x,y',p') --> p\<in>r --> p'\<in>r --> y=y'" *)
paulson@13306
   821
constdefs function_fm :: "i=>i"
paulson@13306
   822
    "function_fm(r) == 
paulson@13306
   823
       Forall(Forall(Forall(Forall(Forall(
paulson@13306
   824
         Implies(pair_fm(4,3,1),
paulson@13306
   825
                 Implies(pair_fm(4,2,0),
paulson@13306
   826
                         Implies(Member(1,r#+5),
paulson@13306
   827
                                 Implies(Member(0,r#+5), Equal(3,2))))))))))"
paulson@13306
   828
paulson@13306
   829
lemma function_type [TC]:
paulson@13306
   830
     "[| x \<in> nat |] ==> function_fm(x) \<in> formula"
paulson@13306
   831
by (simp add: function_fm_def) 
paulson@13306
   832
paulson@13306
   833
lemma arity_function_fm [simp]:
paulson@13306
   834
     "x \<in> nat ==> arity(function_fm(x)) = succ(x)"
paulson@13306
   835
by (simp add: function_fm_def succ_Un_distrib [symmetric] Un_ac) 
paulson@13306
   836
paulson@13306
   837
lemma sats_function_fm [simp]:
paulson@13306
   838
   "[| x \<in> nat; env \<in> list(A)|]
paulson@13306
   839
    ==> sats(A, function_fm(x), env) <-> is_function(**A, nth(x,env))"
paulson@13306
   840
by (simp add: function_fm_def is_function_def)
paulson@13306
   841
paulson@13306
   842
lemma function_iff_sats:
paulson@13306
   843
      "[| nth(i,env) = x; nth(j,env) = y; 
paulson@13306
   844
          i \<in> nat; env \<in> list(A)|]
paulson@13306
   845
       ==> is_function(**A, x) <-> sats(A, function_fm(i), env)"
paulson@13306
   846
by simp
paulson@13306
   847
paulson@13314
   848
theorem is_function_reflection:
paulson@13314
   849
     "REFLECTS[\<lambda>x. is_function(L,f(x)), 
paulson@13314
   850
               \<lambda>i x. is_function(**Lset(i),f(x))]"
paulson@13314
   851
apply (simp only: is_function_def setclass_simps)
paulson@13314
   852
apply (intro FOL_reflection pair_reflection)  
paulson@13314
   853
done
paulson@13298
   854
paulson@13298
   855
paulson@13309
   856
subsubsection{*Typed Functions*}
paulson@13309
   857
paulson@13309
   858
(* "typed_function(M,A,B,r) == 
paulson@13309
   859
        is_function(M,r) & is_relation(M,r) & is_domain(M,r,A) &
paulson@13309
   860
        (\<forall>u[M]. u\<in>r --> (\<forall>x[M]. \<forall>y[M]. pair(M,x,y,u) --> y\<in>B))" *)
paulson@13309
   861
paulson@13309
   862
constdefs typed_function_fm :: "[i,i,i]=>i"
paulson@13309
   863
    "typed_function_fm(A,B,r) == 
paulson@13309
   864
       And(function_fm(r),
paulson@13309
   865
         And(relation_fm(r),
paulson@13309
   866
           And(domain_fm(r,A),
paulson@13309
   867
             Forall(Implies(Member(0,succ(r)),
paulson@13309
   868
                  Forall(Forall(Implies(pair_fm(1,0,2),Member(0,B#+3)))))))))"
paulson@13309
   869
paulson@13309
   870
lemma typed_function_type [TC]:
paulson@13309
   871
     "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> typed_function_fm(x,y,z) \<in> formula"
paulson@13309
   872
by (simp add: typed_function_fm_def) 
paulson@13309
   873
paulson@13309
   874
lemma arity_typed_function_fm [simp]:
paulson@13309
   875
     "[| x \<in> nat; y \<in> nat; z \<in> nat |] 
paulson@13309
   876
      ==> arity(typed_function_fm(x,y,z)) = succ(x) \<union> succ(y) \<union> succ(z)"
paulson@13309
   877
by (simp add: typed_function_fm_def succ_Un_distrib [symmetric] Un_ac) 
paulson@13309
   878
paulson@13309
   879
lemma sats_typed_function_fm [simp]:
paulson@13309
   880
   "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
paulson@13309
   881
    ==> sats(A, typed_function_fm(x,y,z), env) <-> 
paulson@13309
   882
        typed_function(**A, nth(x,env), nth(y,env), nth(z,env))"
paulson@13309
   883
by (simp add: typed_function_fm_def typed_function_def)
paulson@13309
   884
paulson@13309
   885
lemma typed_function_iff_sats:
paulson@13309
   886
  "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z; 
paulson@13309
   887
      i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
paulson@13309
   888
   ==> typed_function(**A, x, y, z) <-> sats(A, typed_function_fm(i,j,k), env)"
paulson@13309
   889
by simp
paulson@13309
   890
paulson@13314
   891
lemmas function_reflection = 
paulson@13314
   892
        upair_reflection pair_reflection union_reflection
paulson@13314
   893
	cons_reflection fun_apply_reflection subset_reflection
paulson@13314
   894
	transitive_set_reflection ordinal_reflection membership_reflection
paulson@13314
   895
	pred_set_reflection domain_reflection range_reflection image_reflection
paulson@13314
   896
	is_relation_reflection is_function_reflection
paulson@13309
   897
paulson@13309
   898
paulson@13314
   899
theorem typed_function_reflection:
paulson@13314
   900
     "REFLECTS[\<lambda>x. typed_function(L,f(x),g(x),h(x)), 
paulson@13314
   901
               \<lambda>i x. typed_function(**Lset(i),f(x),g(x),h(x))]"
paulson@13314
   902
apply (simp only: typed_function_def setclass_simps)
paulson@13314
   903
apply (intro FOL_reflection function_reflection)  
paulson@13314
   904
done
paulson@13314
   905
paulson@13309
   906
paulson@13309
   907
subsubsection{*Injections*}
paulson@13309
   908
paulson@13309
   909
(* "injection(M,A,B,f) == 
paulson@13309
   910
	typed_function(M,A,B,f) &
paulson@13309
   911
        (\<forall>x[M]. \<forall>x'[M]. \<forall>y[M]. \<forall>p[M]. \<forall>p'[M]. 
paulson@13309
   912
          pair(M,x,y,p) --> pair(M,x',y,p') --> p\<in>f --> p'\<in>f --> x=x')" *)
paulson@13309
   913
constdefs injection_fm :: "[i,i,i]=>i"
paulson@13309
   914
 "injection_fm(A,B,f) == 
paulson@13309
   915
    And(typed_function_fm(A,B,f),
paulson@13309
   916
       Forall(Forall(Forall(Forall(Forall(
paulson@13309
   917
         Implies(pair_fm(4,2,1),
paulson@13309
   918
                 Implies(pair_fm(3,2,0),
paulson@13309
   919
                         Implies(Member(1,f#+5),
paulson@13309
   920
                                 Implies(Member(0,f#+5), Equal(4,3)))))))))))"
paulson@13309
   921
paulson@13309
   922
paulson@13309
   923
lemma injection_type [TC]:
paulson@13309
   924
     "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> injection_fm(x,y,z) \<in> formula"
paulson@13309
   925
by (simp add: injection_fm_def) 
paulson@13309
   926
paulson@13309
   927
lemma arity_injection_fm [simp]:
paulson@13309
   928
     "[| x \<in> nat; y \<in> nat; z \<in> nat |] 
paulson@13309
   929
      ==> arity(injection_fm(x,y,z)) = succ(x) \<union> succ(y) \<union> succ(z)"
paulson@13309
   930
by (simp add: injection_fm_def succ_Un_distrib [symmetric] Un_ac) 
paulson@13309
   931
paulson@13309
   932
lemma sats_injection_fm [simp]:
paulson@13309
   933
   "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
paulson@13309
   934
    ==> sats(A, injection_fm(x,y,z), env) <-> 
paulson@13309
   935
        injection(**A, nth(x,env), nth(y,env), nth(z,env))"
paulson@13309
   936
by (simp add: injection_fm_def injection_def)
paulson@13309
   937
paulson@13309
   938
lemma injection_iff_sats:
paulson@13309
   939
  "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z; 
paulson@13309
   940
      i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
paulson@13309
   941
   ==> injection(**A, x, y, z) <-> sats(A, injection_fm(i,j,k), env)"
paulson@13309
   942
by simp
paulson@13309
   943
paulson@13314
   944
theorem injection_reflection:
paulson@13314
   945
     "REFLECTS[\<lambda>x. injection(L,f(x),g(x),h(x)), 
paulson@13314
   946
               \<lambda>i x. injection(**Lset(i),f(x),g(x),h(x))]"
paulson@13314
   947
apply (simp only: injection_def setclass_simps)
paulson@13314
   948
apply (intro FOL_reflection function_reflection typed_function_reflection)  
paulson@13314
   949
done
paulson@13309
   950
paulson@13309
   951
paulson@13309
   952
subsubsection{*Surjections*}
paulson@13309
   953
paulson@13309
   954
(*  surjection :: "[i=>o,i,i,i] => o"
paulson@13309
   955
    "surjection(M,A,B,f) == 
paulson@13309
   956
        typed_function(M,A,B,f) &
paulson@13309
   957
        (\<forall>y[M]. y\<in>B --> (\<exists>x[M]. x\<in>A & fun_apply(M,f,x,y)))" *)
paulson@13309
   958
constdefs surjection_fm :: "[i,i,i]=>i"
paulson@13309
   959
 "surjection_fm(A,B,f) == 
paulson@13309
   960
    And(typed_function_fm(A,B,f),
paulson@13309
   961
       Forall(Implies(Member(0,succ(B)),
paulson@13309
   962
                      Exists(And(Member(0,succ(succ(A))),
paulson@13309
   963
                                 fun_apply_fm(succ(succ(f)),0,1))))))"
paulson@13309
   964
paulson@13309
   965
lemma surjection_type [TC]:
paulson@13309
   966
     "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> surjection_fm(x,y,z) \<in> formula"
paulson@13309
   967
by (simp add: surjection_fm_def) 
paulson@13309
   968
paulson@13309
   969
lemma arity_surjection_fm [simp]:
paulson@13309
   970
     "[| x \<in> nat; y \<in> nat; z \<in> nat |] 
paulson@13309
   971
      ==> arity(surjection_fm(x,y,z)) = succ(x) \<union> succ(y) \<union> succ(z)"
paulson@13309
   972
by (simp add: surjection_fm_def succ_Un_distrib [symmetric] Un_ac) 
paulson@13309
   973
paulson@13309
   974
lemma sats_surjection_fm [simp]:
paulson@13309
   975
   "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
paulson@13309
   976
    ==> sats(A, surjection_fm(x,y,z), env) <-> 
paulson@13309
   977
        surjection(**A, nth(x,env), nth(y,env), nth(z,env))"
paulson@13309
   978
by (simp add: surjection_fm_def surjection_def)
paulson@13309
   979
paulson@13309
   980
lemma surjection_iff_sats:
paulson@13309
   981
  "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z; 
paulson@13309
   982
      i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
paulson@13309
   983
   ==> surjection(**A, x, y, z) <-> sats(A, surjection_fm(i,j,k), env)"
paulson@13309
   984
by simp
paulson@13309
   985
paulson@13314
   986
theorem surjection_reflection:
paulson@13314
   987
     "REFLECTS[\<lambda>x. surjection(L,f(x),g(x),h(x)), 
paulson@13314
   988
               \<lambda>i x. surjection(**Lset(i),f(x),g(x),h(x))]"
paulson@13314
   989
apply (simp only: surjection_def setclass_simps)
paulson@13314
   990
apply (intro FOL_reflection function_reflection typed_function_reflection)  
paulson@13314
   991
done
paulson@13309
   992
paulson@13309
   993
paulson@13309
   994
paulson@13309
   995
subsubsection{*Bijections*}
paulson@13309
   996
paulson@13309
   997
(*   bijection :: "[i=>o,i,i,i] => o"
paulson@13309
   998
    "bijection(M,A,B,f) == injection(M,A,B,f) & surjection(M,A,B,f)" *)
paulson@13309
   999
constdefs bijection_fm :: "[i,i,i]=>i"
paulson@13309
  1000
 "bijection_fm(A,B,f) == And(injection_fm(A,B,f), surjection_fm(A,B,f))"
paulson@13309
  1001
paulson@13309
  1002
lemma bijection_type [TC]:
paulson@13309
  1003
     "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> bijection_fm(x,y,z) \<in> formula"
paulson@13309
  1004
by (simp add: bijection_fm_def) 
paulson@13309
  1005
paulson@13309
  1006
lemma arity_bijection_fm [simp]:
paulson@13309
  1007
     "[| x \<in> nat; y \<in> nat; z \<in> nat |] 
paulson@13309
  1008
      ==> arity(bijection_fm(x,y,z)) = succ(x) \<union> succ(y) \<union> succ(z)"
paulson@13309
  1009
by (simp add: bijection_fm_def succ_Un_distrib [symmetric] Un_ac) 
paulson@13309
  1010
paulson@13309
  1011
lemma sats_bijection_fm [simp]:
paulson@13309
  1012
   "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
paulson@13309
  1013
    ==> sats(A, bijection_fm(x,y,z), env) <-> 
paulson@13309
  1014
        bijection(**A, nth(x,env), nth(y,env), nth(z,env))"
paulson@13309
  1015
by (simp add: bijection_fm_def bijection_def)
paulson@13309
  1016
paulson@13309
  1017
lemma bijection_iff_sats:
paulson@13309
  1018
  "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z; 
paulson@13309
  1019
      i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
paulson@13309
  1020
   ==> bijection(**A, x, y, z) <-> sats(A, bijection_fm(i,j,k), env)"
paulson@13309
  1021
by simp
paulson@13309
  1022
paulson@13314
  1023
theorem bijection_reflection:
paulson@13314
  1024
     "REFLECTS[\<lambda>x. bijection(L,f(x),g(x),h(x)), 
paulson@13314
  1025
               \<lambda>i x. bijection(**Lset(i),f(x),g(x),h(x))]"
paulson@13314
  1026
apply (simp only: bijection_def setclass_simps)
paulson@13314
  1027
apply (intro And_reflection injection_reflection surjection_reflection)  
paulson@13314
  1028
done
paulson@13309
  1029
paulson@13309
  1030
paulson@13309
  1031
subsubsection{*Order-Isomorphisms*}
paulson@13309
  1032
paulson@13309
  1033
(*  order_isomorphism :: "[i=>o,i,i,i,i,i] => o"
paulson@13309
  1034
   "order_isomorphism(M,A,r,B,s,f) == 
paulson@13309
  1035
        bijection(M,A,B,f) & 
paulson@13309
  1036
        (\<forall>x[M]. x\<in>A --> (\<forall>y[M]. y\<in>A -->
paulson@13309
  1037
          (\<forall>p[M]. \<forall>fx[M]. \<forall>fy[M]. \<forall>q[M].
paulson@13309
  1038
            pair(M,x,y,p) --> fun_apply(M,f,x,fx) --> fun_apply(M,f,y,fy) --> 
paulson@13309
  1039
            pair(M,fx,fy,q) --> (p\<in>r <-> q\<in>s))))"
paulson@13309
  1040
  *)
paulson@13309
  1041
paulson@13309
  1042
constdefs order_isomorphism_fm :: "[i,i,i,i,i]=>i"
paulson@13309
  1043
 "order_isomorphism_fm(A,r,B,s,f) == 
paulson@13309
  1044
   And(bijection_fm(A,B,f), 
paulson@13309
  1045
     Forall(Implies(Member(0,succ(A)),
paulson@13309
  1046
       Forall(Implies(Member(0,succ(succ(A))),
paulson@13309
  1047
         Forall(Forall(Forall(Forall(
paulson@13309
  1048
           Implies(pair_fm(5,4,3),
paulson@13309
  1049
             Implies(fun_apply_fm(f#+6,5,2),
paulson@13309
  1050
               Implies(fun_apply_fm(f#+6,4,1),
paulson@13309
  1051
                 Implies(pair_fm(2,1,0), 
paulson@13309
  1052
                   Iff(Member(3,r#+6), Member(0,s#+6)))))))))))))))"
paulson@13309
  1053
paulson@13309
  1054
lemma order_isomorphism_type [TC]:
paulson@13309
  1055
     "[| A \<in> nat; r \<in> nat; B \<in> nat; s \<in> nat; f \<in> nat |]  
paulson@13309
  1056
      ==> order_isomorphism_fm(A,r,B,s,f) \<in> formula"
paulson@13309
  1057
by (simp add: order_isomorphism_fm_def) 
paulson@13309
  1058
paulson@13309
  1059
lemma arity_order_isomorphism_fm [simp]:
paulson@13309
  1060
     "[| A \<in> nat; r \<in> nat; B \<in> nat; s \<in> nat; f \<in> nat |] 
paulson@13309
  1061
      ==> arity(order_isomorphism_fm(A,r,B,s,f)) = 
paulson@13309
  1062
          succ(A) \<union> succ(r) \<union> succ(B) \<union> succ(s) \<union> succ(f)" 
paulson@13309
  1063
by (simp add: order_isomorphism_fm_def succ_Un_distrib [symmetric] Un_ac) 
paulson@13309
  1064
paulson@13309
  1065
lemma sats_order_isomorphism_fm [simp]:
paulson@13309
  1066
   "[| U \<in> nat; r \<in> nat; B \<in> nat; s \<in> nat; f \<in> nat; env \<in> list(A)|]
paulson@13309
  1067
    ==> sats(A, order_isomorphism_fm(U,r,B,s,f), env) <-> 
paulson@13309
  1068
        order_isomorphism(**A, nth(U,env), nth(r,env), nth(B,env), 
paulson@13309
  1069
                               nth(s,env), nth(f,env))"
paulson@13309
  1070
by (simp add: order_isomorphism_fm_def order_isomorphism_def)
paulson@13309
  1071
paulson@13309
  1072
lemma order_isomorphism_iff_sats:
paulson@13309
  1073
  "[| nth(i,env) = U; nth(j,env) = r; nth(k,env) = B; nth(j',env) = s; 
paulson@13309
  1074
      nth(k',env) = f; 
paulson@13309
  1075
      i \<in> nat; j \<in> nat; k \<in> nat; j' \<in> nat; k' \<in> nat; env \<in> list(A)|]
paulson@13309
  1076
   ==> order_isomorphism(**A,U,r,B,s,f) <-> 
paulson@13309
  1077
       sats(A, order_isomorphism_fm(i,j,k,j',k'), env)" 
paulson@13309
  1078
by simp
paulson@13309
  1079
paulson@13314
  1080
theorem order_isomorphism_reflection:
paulson@13314
  1081
     "REFLECTS[\<lambda>x. order_isomorphism(L,f(x),g(x),h(x),g'(x),h'(x)), 
paulson@13314
  1082
               \<lambda>i x. order_isomorphism(**Lset(i),f(x),g(x),h(x),g'(x),h'(x))]"
paulson@13314
  1083
apply (simp only: order_isomorphism_def setclass_simps)
paulson@13314
  1084
apply (intro FOL_reflection function_reflection bijection_reflection)  
paulson@13314
  1085
done
paulson@13309
  1086
paulson@13223
  1087
end