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header {* The class L satisfies the axioms of ZF*}
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theory L_axioms = Formula + Relative + Reflection:
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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 [rule_format]
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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_Reflects :: "[i=>o,i=>o,[i,i]=>o] => o"
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"L_Reflects(Cl,P,Q) == 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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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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theorem Triv_reflection [intro]:
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"L_Reflects(Ord, P, \<lambda>a x. P(x))"
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by (simp add: L_Reflects_def)
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theorem Not_reflection [intro]:
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"L_Reflects(Cl,P,Q) ==> L_Reflects(Cl, \<lambda>x. ~P(x), \<lambda>a x. ~Q(a,x))"
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by (simp add: L_Reflects_def)
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theorem And_reflection [intro]:
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"[| L_Reflects(Cl,P,Q); L_Reflects(C',P',Q') |]
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==> L_Reflects(\<lambda>a. Cl(a) \<and> C'(a), \<lambda>x. P(x) \<and> P'(x),
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\<lambda>a x. Q(a,x) \<and> Q'(a,x))"
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by (simp add: L_Reflects_def Closed_Unbounded_Int, blast)
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theorem Or_reflection [intro]:
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"[| L_Reflects(Cl,P,Q); L_Reflects(C',P',Q') |]
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==> L_Reflects(\<lambda>a. Cl(a) \<and> C'(a), \<lambda>x. P(x) \<or> P'(x),
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\<lambda>a x. Q(a,x) \<or> Q'(a,x))"
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by (simp add: L_Reflects_def Closed_Unbounded_Int, blast)
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theorem Imp_reflection [intro]:
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"[| L_Reflects(Cl,P,Q); L_Reflects(C',P',Q') |]
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==> L_Reflects(\<lambda>a. Cl(a) \<and> C'(a),
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\<lambda>x. P(x) --> P'(x),
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\<lambda>a x. Q(a,x) --> Q'(a,x))"
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by (simp add: L_Reflects_def Closed_Unbounded_Int, blast)
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theorem Iff_reflection [intro]:
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"[| L_Reflects(Cl,P,Q); L_Reflects(C',P',Q') |]
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==> L_Reflects(\<lambda>a. Cl(a) \<and> C'(a),
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\<lambda>x. P(x) <-> P'(x),
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\<lambda>a x. Q(a,x) <-> Q'(a,x))"
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by (simp add: L_Reflects_def Closed_Unbounded_Int, blast)
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theorem Ex_reflection [intro]:
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"L_Reflects(Cl, \<lambda>x. P(fst(x),snd(x)), \<lambda>a x. Q(a,fst(x),snd(x)))
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==> L_Reflects(\<lambda>a. Cl(a) \<and> L_ClEx(\<lambda>x. P(fst(x),snd(x)), a),
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\<lambda>x. \<exists>z. L(z) \<and> P(x,z),
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\<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 (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 [intro]:
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"L_Reflects(Cl, \<lambda>x. P(fst(x),snd(x)), \<lambda>a x. Q(a,fst(x),snd(x)))
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==> L_Reflects(\<lambda>a. Cl(a) \<and> L_ClEx(\<lambda>x. ~P(fst(x),snd(x)), a),
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\<lambda>x. \<forall>z. L(z) --> P(x,z),
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\<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 (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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theorem Rex_reflection [intro]:
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"L_Reflects(Cl, \<lambda>x. P(fst(x),snd(x)), \<lambda>a x. Q(a,fst(x),snd(x)))
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==> L_Reflects(\<lambda>a. Cl(a) \<and> L_ClEx(\<lambda>x. P(fst(x),snd(x)), a),
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\<lambda>x. \<exists>z[L]. P(x,z),
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\<lambda>a x. \<exists>z\<in>Lset(a). Q(a,x,z))"
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by (unfold rex_def, blast)
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theorem Rall_reflection [intro]:
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"L_Reflects(Cl, \<lambda>x. P(fst(x),snd(x)), \<lambda>a x. Q(a,fst(x),snd(x)))
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==> L_Reflects(\<lambda>a. Cl(a) \<and> L_ClEx(\<lambda>x. ~P(fst(x),snd(x)), a),
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\<lambda>x. \<forall>z[L]. P(x,z),
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\<lambda>a x. \<forall>z\<in>Lset(a). Q(a,x,z))"
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by (unfold rall_def, blast)
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lemma ReflectsD:
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"[|L_Reflects(Cl,P,Q); Ord(i)|]
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==> \<exists>j. i<j & (\<forall>x \<in> Lset(j). P(x) <-> Q(j,x))"
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apply (simp add: L_Reflects_def Closed_Unbounded_def, clarify)
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apply (blast dest!: UnboundedD)
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done
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lemma ReflectsE:
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"[| L_Reflects(Cl,P,Q); Ord(i);
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!!j. [|i<j; \<forall>x \<in> Lset(j). P(x) <-> Q(j,x)|] ==> R |]
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==> R"
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by (blast dest!: ReflectsD)
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lemma Collect_mem_eq: "{x\<in>A. x\<in>B} = A \<inter> B";
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by blast
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subsection{*Internalized formulas for some relativized ones*}
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subsubsection{*Unordered pairs*}
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constdefs upair_fm :: "[i,i,i]=>i"
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"upair_fm(x,y,z) ==
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297 |
And(Member(x,z),
|
|
298 |
And(Member(y,z),
|
|
299 |
Forall(Implies(Member(0,succ(z)),
|
|
300 |
Or(Equal(0,succ(x)), Equal(0,succ(y)))))))"
|
|
301 |
|
|
302 |
lemma upair_type [TC]:
|
|
303 |
"[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> upair_fm(x,y,z) \<in> formula"
|
|
304 |
by (simp add: upair_fm_def)
|
|
305 |
|
|
306 |
lemma arity_upair_fm [simp]:
|
|
307 |
"[| x \<in> nat; y \<in> nat; z \<in> nat |]
|
|
308 |
==> arity(upair_fm(x,y,z)) = succ(x) \<union> succ(y) \<union> succ(z)"
|
|
309 |
by (simp add: upair_fm_def succ_Un_distrib [symmetric] Un_ac)
|
|
310 |
|
|
311 |
lemma sats_upair_fm [simp]:
|
|
312 |
"[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
|
|
313 |
==> sats(A, upair_fm(x,y,z), env) <->
|
|
314 |
upair(**A, nth(x,env), nth(y,env), nth(z,env))"
|
|
315 |
by (simp add: upair_fm_def upair_def)
|
|
316 |
|
|
317 |
lemma upair_iff_sats:
|
|
318 |
"[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z;
|
|
319 |
i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
|
|
320 |
==> upair(**A, x, y, z) <-> sats(A, upair_fm(i,j,k), env)"
|
|
321 |
by (simp add: sats_upair_fm)
|
|
322 |
|
|
323 |
text{*Useful? At least it refers to "real" unordered pairs*}
|
|
324 |
lemma sats_upair_fm2 [simp]:
|
|
325 |
"[| x \<in> nat; y \<in> nat; z < length(env); env \<in> list(A); Transset(A)|]
|
|
326 |
==> sats(A, upair_fm(x,y,z), env) <->
|
|
327 |
nth(z,env) = {nth(x,env), nth(y,env)}"
|
|
328 |
apply (frule lt_length_in_nat, assumption)
|
|
329 |
apply (simp add: upair_fm_def Transset_def, auto)
|
|
330 |
apply (blast intro: nth_type)
|
|
331 |
done
|
|
332 |
|
|
333 |
subsubsection{*Ordered pairs*}
|
|
334 |
|
|
335 |
constdefs pair_fm :: "[i,i,i]=>i"
|
|
336 |
"pair_fm(x,y,z) ==
|
|
337 |
Exists(And(upair_fm(succ(x),succ(x),0),
|
|
338 |
Exists(And(upair_fm(succ(succ(x)),succ(succ(y)),0),
|
|
339 |
upair_fm(1,0,succ(succ(z)))))))"
|
|
340 |
|
|
341 |
lemma pair_type [TC]:
|
|
342 |
"[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> pair_fm(x,y,z) \<in> formula"
|
|
343 |
by (simp add: pair_fm_def)
|
|
344 |
|
|
345 |
lemma arity_pair_fm [simp]:
|
|
346 |
"[| x \<in> nat; y \<in> nat; z \<in> nat |]
|
|
347 |
==> arity(pair_fm(x,y,z)) = succ(x) \<union> succ(y) \<union> succ(z)"
|
|
348 |
by (simp add: pair_fm_def succ_Un_distrib [symmetric] Un_ac)
|
|
349 |
|
|
350 |
lemma sats_pair_fm [simp]:
|
|
351 |
"[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
|
|
352 |
==> sats(A, pair_fm(x,y,z), env) <->
|
|
353 |
pair(**A, nth(x,env), nth(y,env), nth(z,env))"
|
|
354 |
by (simp add: pair_fm_def pair_def)
|
|
355 |
|
|
356 |
lemma pair_iff_sats:
|
|
357 |
"[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z;
|
|
358 |
i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
|
|
359 |
==> pair(**A, x, y, z) <-> sats(A, pair_fm(i,j,k), env)"
|
|
360 |
by (simp add: sats_pair_fm)
|
|
361 |
|
|
362 |
|
|
363 |
|
|
364 |
subsection{*Proving instances of Separation using Reflection!*}
|
|
365 |
|
|
366 |
text{*Helps us solve for de Bruijn indices!*}
|
|
367 |
lemma nth_ConsI: "[|nth(n,l) = x; n \<in> nat|] ==> nth(succ(n), Cons(a,l)) = x"
|
|
368 |
by simp
|
|
369 |
|
|
370 |
|
|
371 |
lemma Collect_conj_in_DPow:
|
|
372 |
"[| {x\<in>A. P(x)} \<in> DPow(A); {x\<in>A. Q(x)} \<in> DPow(A) |]
|
|
373 |
==> {x\<in>A. P(x) & Q(x)} \<in> DPow(A)"
|
|
374 |
by (simp add: Int_in_DPow Collect_Int_Collect_eq [symmetric])
|
|
375 |
|
|
376 |
lemma Collect_conj_in_DPow_Lset:
|
|
377 |
"[|z \<in> Lset(j); {x \<in> Lset(j). P(x)} \<in> DPow(Lset(j))|]
|
|
378 |
==> {x \<in> Lset(j). x \<in> z & P(x)} \<in> DPow(Lset(j))"
|
|
379 |
apply (frule mem_Lset_imp_subset_Lset)
|
|
380 |
apply (simp add: Collect_conj_in_DPow Collect_mem_eq
|
|
381 |
subset_Int_iff2 elem_subset_in_DPow)
|
|
382 |
done
|
|
383 |
|
|
384 |
lemma separation_CollectI:
|
|
385 |
"(\<And>z. L(z) ==> L({x \<in> z . P(x)})) ==> separation(L, \<lambda>x. P(x))"
|
|
386 |
apply (unfold separation_def, clarify)
|
|
387 |
apply (rule_tac x="{x\<in>z. P(x)}" in rexI)
|
|
388 |
apply simp_all
|
|
389 |
done
|
|
390 |
|
|
391 |
text{*Reduces the original comprehension to the reflected one*}
|
|
392 |
lemma reflection_imp_L_separation:
|
|
393 |
"[| \<forall>x\<in>Lset(j). P(x) <-> Q(x);
|
|
394 |
{x \<in> Lset(j) . Q(x)} \<in> DPow(Lset(j));
|
|
395 |
Ord(j); z \<in> Lset(j)|] ==> L({x \<in> z . P(x)})"
|
|
396 |
apply (rule_tac i = "succ(j)" in L_I)
|
|
397 |
prefer 2 apply simp
|
|
398 |
apply (subgoal_tac "{x \<in> z. P(x)} = {x \<in> Lset(j). x \<in> z & (Q(x))}")
|
|
399 |
prefer 2
|
|
400 |
apply (blast dest: mem_Lset_imp_subset_Lset)
|
|
401 |
apply (simp add: Lset_succ Collect_conj_in_DPow_Lset)
|
|
402 |
done
|
|
403 |
|
|
404 |
|
|
405 |
subsubsection{*Separation for Intersection*}
|
|
406 |
|
|
407 |
lemma Inter_Reflects:
|
|
408 |
"L_Reflects(?Cl, \<lambda>x. \<forall>y[L]. y\<in>A --> x \<in> y,
|
|
409 |
\<lambda>i x. \<forall>y\<in>Lset(i). y\<in>A --> x \<in> y)"
|
|
410 |
by fast
|
|
411 |
|
|
412 |
lemma Inter_separation:
|
|
413 |
"L(A) ==> separation(L, \<lambda>x. \<forall>y[L]. y\<in>A --> x\<in>y)"
|
|
414 |
apply (rule separation_CollectI)
|
|
415 |
apply (rule_tac A="{A,z}" in subset_LsetE, blast )
|
|
416 |
apply (rule ReflectsE [OF Inter_Reflects], assumption)
|
|
417 |
apply (drule subset_Lset_ltD, assumption)
|
|
418 |
apply (erule reflection_imp_L_separation)
|
|
419 |
apply (simp_all add: lt_Ord2, clarify)
|
|
420 |
apply (rule DPowI2)
|
|
421 |
apply (rule ball_iff_sats)
|
|
422 |
apply (rule imp_iff_sats)
|
|
423 |
apply (rule_tac [2] i=1 and j=0 and env="[y,x,A]" in mem_iff_sats)
|
|
424 |
apply (rule_tac i=0 and j=2 in mem_iff_sats)
|
|
425 |
apply (simp_all add: succ_Un_distrib [symmetric])
|
|
426 |
done
|
|
427 |
|
|
428 |
subsubsection{*Separation for Cartesian Product*}
|
|
429 |
|
|
430 |
text{*The @{text simplified} attribute tidies up the reflecting class.*}
|
|
431 |
theorem upair_reflection [simplified,intro]:
|
|
432 |
"L_Reflects(?Cl, \<lambda>x. upair(L,f(x),g(x),h(x)),
|
|
433 |
\<lambda>i x. upair(**Lset(i),f(x),g(x),h(x)))"
|
|
434 |
by (simp add: upair_def, fast)
|
|
435 |
|
|
436 |
theorem pair_reflection [simplified,intro]:
|
|
437 |
"L_Reflects(?Cl, \<lambda>x. pair(L,f(x),g(x),h(x)),
|
|
438 |
\<lambda>i x. pair(**Lset(i),f(x),g(x),h(x)))"
|
|
439 |
by (simp only: pair_def rex_setclass_is_bex, fast)
|
|
440 |
|
|
441 |
lemma cartprod_Reflects [simplified]:
|
|
442 |
"L_Reflects(?Cl, \<lambda>z. \<exists>x[L]. x\<in>A & (\<exists>y[L]. y\<in>B & pair(L,x,y,z)),
|
|
443 |
\<lambda>i z. \<exists>x\<in>Lset(i). x\<in>A & (\<exists>y\<in>Lset(i). y\<in>B &
|
|
444 |
pair(**Lset(i),x,y,z)))"
|
|
445 |
by fast
|
|
446 |
|
|
447 |
lemma cartprod_separation:
|
|
448 |
"[| L(A); L(B) |]
|
|
449 |
==> separation(L, \<lambda>z. \<exists>x[L]. x\<in>A & (\<exists>y[L]. y\<in>B & pair(L,x,y,z)))"
|
|
450 |
apply (rule separation_CollectI)
|
|
451 |
apply (rule_tac A="{A,B,z}" in subset_LsetE, blast )
|
|
452 |
apply (rule ReflectsE [OF cartprod_Reflects], assumption)
|
|
453 |
apply (drule subset_Lset_ltD, assumption)
|
|
454 |
apply (erule reflection_imp_L_separation)
|
|
455 |
apply (simp_all add: lt_Ord2, clarify)
|
|
456 |
apply (rule DPowI2)
|
|
457 |
apply (rename_tac u)
|
|
458 |
apply (rule bex_iff_sats)
|
|
459 |
apply (rule conj_iff_sats)
|
|
460 |
apply (rule_tac i=0 and j=2 and env="[x,u,A,B]" in mem_iff_sats, simp_all)
|
|
461 |
apply (rule bex_iff_sats)
|
|
462 |
apply (rule conj_iff_sats)
|
|
463 |
apply (rule mem_iff_sats)
|
|
464 |
apply (blast intro: nth_0 nth_ConsI)
|
|
465 |
apply (blast intro: nth_0 nth_ConsI, simp_all)
|
|
466 |
apply (rule_tac i=1 and j=0 and k=2 in pair_iff_sats)
|
|
467 |
apply (simp_all add: succ_Un_distrib [symmetric])
|
|
468 |
done
|
|
469 |
|
|
470 |
subsubsection{*Separation for Image*}
|
|
471 |
|
|
472 |
text{*No @{text simplified} here: it simplifies the occurrence of
|
|
473 |
the predicate @{term pair}!*}
|
|
474 |
lemma image_Reflects:
|
|
475 |
"L_Reflects(?Cl, \<lambda>y. \<exists>p[L]. p\<in>r & (\<exists>x[L]. x\<in>A & pair(L,x,y,p)),
|
|
476 |
\<lambda>i y. \<exists>p\<in>Lset(i). p\<in>r & (\<exists>x\<in>Lset(i). x\<in>A & pair(**Lset(i),x,y,p)))"
|
|
477 |
by fast
|
|
478 |
|
|
479 |
|
|
480 |
lemma image_separation:
|
|
481 |
"[| L(A); L(r) |]
|
|
482 |
==> separation(L, \<lambda>y. \<exists>p[L]. p\<in>r & (\<exists>x[L]. x\<in>A & pair(L,x,y,p)))"
|
|
483 |
apply (rule separation_CollectI)
|
|
484 |
apply (rule_tac A="{A,r,z}" in subset_LsetE, blast )
|
|
485 |
apply (rule ReflectsE [OF image_Reflects], assumption)
|
|
486 |
apply (drule subset_Lset_ltD, assumption)
|
|
487 |
apply (erule reflection_imp_L_separation)
|
|
488 |
apply (simp_all add: lt_Ord2, clarify)
|
|
489 |
apply (rule DPowI2)
|
|
490 |
apply (rule bex_iff_sats)
|
|
491 |
apply (rule conj_iff_sats)
|
|
492 |
apply (rule_tac env="[p,y,A,r]" in mem_iff_sats)
|
|
493 |
apply (blast intro: nth_0 nth_ConsI)
|
|
494 |
apply (blast intro: nth_0 nth_ConsI, simp_all)
|
|
495 |
apply (rule bex_iff_sats)
|
|
496 |
apply (rule conj_iff_sats)
|
|
497 |
apply (rule mem_iff_sats)
|
|
498 |
apply (blast intro: nth_0 nth_ConsI)
|
|
499 |
apply (blast intro: nth_0 nth_ConsI, simp_all)
|
|
500 |
apply (rule pair_iff_sats)
|
|
501 |
apply (blast intro: nth_0 nth_ConsI)
|
|
502 |
apply (blast intro: nth_0 nth_ConsI)
|
|
503 |
apply (blast intro: nth_0 nth_ConsI)
|
|
504 |
apply (simp_all add: succ_Un_distrib [symmetric])
|
|
505 |
done
|
|
506 |
|
|
507 |
|
|
508 |
subsubsection{*Separation for Converse*}
|
|
509 |
|
|
510 |
lemma converse_Reflects:
|
|
511 |
"L_Reflects(?Cl,
|
|
512 |
\<lambda>z. \<exists>p[L]. p\<in>r & (\<exists>x[L]. \<exists>y[L]. pair(L,x,y,p) & pair(L,y,x,z)),
|
|
513 |
\<lambda>i z. \<exists>p\<in>Lset(i). p\<in>r & (\<exists>x\<in>Lset(i). \<exists>y\<in>Lset(i).
|
|
514 |
pair(**Lset(i),x,y,p) & pair(**Lset(i),y,x,z)))"
|
|
515 |
by fast
|
|
516 |
|
|
517 |
lemma converse_separation:
|
|
518 |
"L(r) ==> separation(L,
|
|
519 |
\<lambda>z. \<exists>p[L]. p\<in>r & (\<exists>x[L]. \<exists>y[L]. pair(L,x,y,p) & pair(L,y,x,z)))"
|
|
520 |
apply (rule separation_CollectI)
|
|
521 |
apply (rule_tac A="{r,z}" in subset_LsetE, blast )
|
|
522 |
apply (rule ReflectsE [OF converse_Reflects], assumption)
|
|
523 |
apply (drule subset_Lset_ltD, assumption)
|
|
524 |
apply (erule reflection_imp_L_separation)
|
|
525 |
apply (simp_all add: lt_Ord2, clarify)
|
|
526 |
apply (rule DPowI2)
|
|
527 |
apply (rename_tac u)
|
|
528 |
apply (rule bex_iff_sats)
|
|
529 |
apply (rule conj_iff_sats)
|
|
530 |
apply (rule_tac i=0 and j="2" and env="[p,u,r]" in mem_iff_sats, simp_all)
|
|
531 |
apply (rule bex_iff_sats)
|
|
532 |
apply (rule bex_iff_sats)
|
|
533 |
apply (rule conj_iff_sats)
|
|
534 |
apply (rule_tac i=1 and j=0 and k=2 in pair_iff_sats, simp_all)
|
|
535 |
apply (rule pair_iff_sats)
|
|
536 |
apply (blast intro: nth_0 nth_ConsI)
|
|
537 |
apply (blast intro: nth_0 nth_ConsI)
|
|
538 |
apply (blast intro: nth_0 nth_ConsI)
|
|
539 |
apply (simp_all add: succ_Un_distrib [symmetric])
|
|
540 |
done
|
|
541 |
|
|
542 |
|
|
543 |
subsubsection{*Separation for Restriction*}
|
|
544 |
|
|
545 |
lemma restrict_Reflects:
|
|
546 |
"L_Reflects(?Cl, \<lambda>z. \<exists>x[L]. x\<in>A & (\<exists>y[L]. pair(L,x,y,z)),
|
|
547 |
\<lambda>i z. \<exists>x\<in>Lset(i). x\<in>A & (\<exists>y\<in>Lset(i). pair(**Lset(i),x,y,z)))"
|
|
548 |
by fast
|
|
549 |
|
|
550 |
lemma restrict_separation:
|
|
551 |
"L(A) ==> separation(L, \<lambda>z. \<exists>x[L]. x\<in>A & (\<exists>y[L]. pair(L,x,y,z)))"
|
|
552 |
apply (rule separation_CollectI)
|
|
553 |
apply (rule_tac A="{A,z}" in subset_LsetE, blast )
|
|
554 |
apply (rule ReflectsE [OF restrict_Reflects], assumption)
|
|
555 |
apply (drule subset_Lset_ltD, assumption)
|
|
556 |
apply (erule reflection_imp_L_separation)
|
|
557 |
apply (simp_all add: lt_Ord2, clarify)
|
|
558 |
apply (rule DPowI2)
|
|
559 |
apply (rename_tac u)
|
|
560 |
apply (rule bex_iff_sats)
|
|
561 |
apply (rule conj_iff_sats)
|
|
562 |
apply (rule_tac i=0 and j="2" and env="[x,u,A]" in mem_iff_sats, simp_all)
|
|
563 |
apply (rule bex_iff_sats)
|
|
564 |
apply (rule_tac i=1 and j=0 and k=2 in pair_iff_sats)
|
|
565 |
apply (simp_all add: succ_Un_distrib [symmetric])
|
|
566 |
done
|
|
567 |
|
|
568 |
|
|
569 |
subsubsection{*Separation for Composition*}
|
|
570 |
|
|
571 |
lemma comp_Reflects:
|
|
572 |
"L_Reflects(?Cl, \<lambda>xz. \<exists>x[L]. \<exists>y[L]. \<exists>z[L]. \<exists>xy[L]. \<exists>yz[L].
|
|
573 |
pair(L,x,z,xz) & pair(L,x,y,xy) & pair(L,y,z,yz) &
|
|
574 |
xy\<in>s & yz\<in>r,
|
|
575 |
\<lambda>i xz. \<exists>x\<in>Lset(i). \<exists>y\<in>Lset(i). \<exists>z\<in>Lset(i). \<exists>xy\<in>Lset(i). \<exists>yz\<in>Lset(i).
|
|
576 |
pair(**Lset(i),x,z,xz) & pair(**Lset(i),x,y,xy) &
|
|
577 |
pair(**Lset(i),y,z,yz) & xy\<in>s & yz\<in>r)"
|
|
578 |
by fast
|
|
579 |
|
|
580 |
lemma comp_separation:
|
|
581 |
"[| L(r); L(s) |]
|
|
582 |
==> separation(L, \<lambda>xz. \<exists>x[L]. \<exists>y[L]. \<exists>z[L]. \<exists>xy[L]. \<exists>yz[L].
|
|
583 |
pair(L,x,z,xz) & pair(L,x,y,xy) & pair(L,y,z,yz) &
|
|
584 |
xy\<in>s & yz\<in>r)"
|
|
585 |
apply (rule separation_CollectI)
|
|
586 |
apply (rule_tac A="{r,s,z}" in subset_LsetE, blast )
|
|
587 |
apply (rule ReflectsE [OF comp_Reflects], assumption)
|
|
588 |
apply (drule subset_Lset_ltD, assumption)
|
|
589 |
apply (erule reflection_imp_L_separation)
|
|
590 |
apply (simp_all add: lt_Ord2, clarify)
|
|
591 |
apply (rule DPowI2)
|
|
592 |
apply (rename_tac u)
|
|
593 |
apply (rule bex_iff_sats)+
|
|
594 |
apply (rename_tac x y z)
|
|
595 |
apply (rule conj_iff_sats)
|
|
596 |
apply (rule_tac env="[z,y,x,u,r,s]" in pair_iff_sats)
|
|
597 |
apply (blast intro: nth_0 nth_ConsI)
|
|
598 |
apply (blast intro: nth_0 nth_ConsI)
|
|
599 |
apply (blast intro: nth_0 nth_ConsI, simp_all)
|
|
600 |
apply (rule bex_iff_sats)
|
|
601 |
apply (rule conj_iff_sats)
|
|
602 |
apply (rule pair_iff_sats)
|
|
603 |
apply (blast intro: nth_0 nth_ConsI)
|
|
604 |
apply (blast intro: nth_0 nth_ConsI)
|
|
605 |
apply (blast intro: nth_0 nth_ConsI, simp_all)
|
|
606 |
apply (rule bex_iff_sats)
|
|
607 |
apply (rule conj_iff_sats)
|
|
608 |
apply (rule pair_iff_sats)
|
|
609 |
apply (blast intro: nth_0 nth_ConsI)
|
|
610 |
apply (blast intro: nth_0 nth_ConsI)
|
|
611 |
apply (blast intro: nth_0 nth_ConsI, simp_all)
|
|
612 |
apply (rule conj_iff_sats)
|
|
613 |
apply (rule mem_iff_sats)
|
|
614 |
apply (blast intro: nth_0 nth_ConsI)
|
|
615 |
apply (blast intro: nth_0 nth_ConsI, simp)
|
|
616 |
apply (rule mem_iff_sats)
|
|
617 |
apply (blast intro: nth_0 nth_ConsI)
|
|
618 |
apply (blast intro: nth_0 nth_ConsI)
|
|
619 |
apply (simp_all add: succ_Un_distrib [symmetric])
|
|
620 |
done
|
|
621 |
|
13304
|
622 |
subsubsection{*Separation for Predecessors in an Order*}
|
|
623 |
|
|
624 |
lemma pred_Reflects:
|
|
625 |
"L_Reflects(?Cl, \<lambda>y. \<exists>p[L]. p\<in>r & pair(L,y,x,p),
|
|
626 |
\<lambda>i y. \<exists>p \<in> Lset(i). p\<in>r & pair(**Lset(i),y,x,p))"
|
|
627 |
by fast
|
|
628 |
|
|
629 |
lemma pred_separation:
|
|
630 |
"[| L(r); L(x) |] ==> separation(L, \<lambda>y. \<exists>p[L]. p\<in>r & pair(L,y,x,p))"
|
|
631 |
apply (rule separation_CollectI)
|
|
632 |
apply (rule_tac A="{r,x,z}" in subset_LsetE, blast )
|
|
633 |
apply (rule ReflectsE [OF pred_Reflects], assumption)
|
|
634 |
apply (drule subset_Lset_ltD, assumption)
|
|
635 |
apply (erule reflection_imp_L_separation)
|
|
636 |
apply (simp_all add: lt_Ord2, clarify)
|
|
637 |
apply (rule DPowI2)
|
|
638 |
apply (rename_tac u)
|
|
639 |
apply (rule bex_iff_sats)
|
|
640 |
apply (rule conj_iff_sats)
|
|
641 |
apply (rule_tac env = "[p,u,r,x]" in mem_iff_sats)
|
|
642 |
apply (blast intro: nth_0 nth_ConsI)
|
|
643 |
apply (blast intro: nth_0 nth_ConsI, simp)
|
|
644 |
apply (rule pair_iff_sats)
|
|
645 |
apply (blast intro: nth_0 nth_ConsI)
|
|
646 |
apply (blast intro: nth_0 nth_ConsI)
|
|
647 |
apply (blast intro: nth_0 nth_ConsI, simp_all)
|
|
648 |
apply (simp_all add: succ_Un_distrib [symmetric])
|
|
649 |
done
|
|
650 |
|
|
651 |
|
|
652 |
subsubsection{*Separation for the Membership Relation*}
|
|
653 |
|
|
654 |
lemma Memrel_Reflects:
|
|
655 |
"L_Reflects(?Cl, \<lambda>z. \<exists>x[L]. \<exists>y[L]. pair(L,x,y,z) & x \<in> y,
|
|
656 |
\<lambda>i z. \<exists>x \<in> Lset(i). \<exists>y \<in> Lset(i). pair(**Lset(i),x,y,z) & x \<in> y)"
|
|
657 |
by fast
|
|
658 |
|
|
659 |
lemma Memrel_separation:
|
|
660 |
"separation(L, \<lambda>z. \<exists>x[L]. \<exists>y[L]. pair(L,x,y,z) & x \<in> y)"
|
|
661 |
apply (rule separation_CollectI)
|
|
662 |
apply (rule_tac A="{z}" in subset_LsetE, blast )
|
|
663 |
apply (rule ReflectsE [OF Memrel_Reflects], assumption)
|
|
664 |
apply (drule subset_Lset_ltD, assumption)
|
|
665 |
apply (erule reflection_imp_L_separation)
|
|
666 |
apply (simp_all add: lt_Ord2)
|
|
667 |
apply (rule DPowI2)
|
|
668 |
apply (rename_tac u)
|
|
669 |
apply (rule bex_iff_sats)+
|
|
670 |
apply (rule conj_iff_sats)
|
|
671 |
apply (rule_tac env = "[y,x,u]" in pair_iff_sats)
|
|
672 |
apply (blast intro: nth_0 nth_ConsI)
|
|
673 |
apply (blast intro: nth_0 nth_ConsI)
|
|
674 |
apply (blast intro: nth_0 nth_ConsI, simp_all)
|
|
675 |
apply (rule mem_iff_sats)
|
|
676 |
apply (blast intro: nth_0 nth_ConsI)
|
|
677 |
apply (blast intro: nth_0 nth_ConsI)
|
|
678 |
apply (simp_all add: succ_Un_distrib [symmetric])
|
|
679 |
done
|
|
680 |
|
|
681 |
|
13298
|
682 |
|
|
683 |
|
|
684 |
|
13223
|
685 |
end
|