author | paulson |
Thu, 06 Feb 2003 11:01:05 +0100 | |
changeset 13807 | a28a8fbc76d4 |
parent 13655 | 95b95cdb4704 |
child 14171 | 0cab06e3bbd0 |
permissions | -rw-r--r-- |
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(* Title: ZF/Constructible/L_axioms.thy |
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ID: $Id$ |
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Author: Lawrence C Paulson, Cambridge University Computer Laboratory |
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*) |
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header {* The ZF Axioms (Except Separation) in L *} |
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theory L_axioms = Formula + Relative + Reflection + MetaExists: |
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text {* The class L satisfies the premises of locale @{text M_trivial} *} |
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lemma transL: "[| y\<in>x; L(x) |] ==> L(y)" |
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apply (insert Transset_Lset) |
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apply (simp add: Transset_def L_def, blast) |
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done |
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lemma nonempty: "L(0)" |
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apply (simp add: L_def) |
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apply (blast intro: zero_in_Lset) |
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done |
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theorem upair_ax: "upair_ax(L)" |
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apply (simp add: upair_ax_def upair_def, clarify) |
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apply (rule_tac x="{x,y}" in rexI) |
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apply (simp_all add: doubleton_in_L) |
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done |
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theorem Union_ax: "Union_ax(L)" |
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apply (simp add: Union_ax_def big_union_def, clarify) |
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apply (rule_tac x="Union(x)" in rexI) |
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apply (simp_all add: Union_in_L, auto) |
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apply (blast intro: transL) |
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done |
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theorem power_ax: "power_ax(L)" |
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apply (simp add: power_ax_def powerset_def Relative.subset_def, clarify) |
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apply (rule_tac x="{y \<in> Pow(x). L(y)}" in rexI) |
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apply (simp_all add: LPow_in_L, auto) |
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apply (blast intro: transL) |
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done |
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text{*We don't actually need @{term L} to satisfy the foundation axiom.*} |
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theorem foundation_ax: "foundation_ax(L)" |
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apply (simp add: foundation_ax_def) |
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apply (rule rallI) |
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apply (cut_tac A=x in foundation) |
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apply (blast intro: transL) |
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done |
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subsection{*For L to satisfy Replacement *} |
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(*Can't move these to Formula unless the definition of univalent is moved |
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there too!*) |
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lemma LReplace_in_Lset: |
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"[|X \<in> Lset(i); univalent(L,X,Q); Ord(i)|] |
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==> \<exists>j. Ord(j) & Replace(X, %x y. Q(x,y) & L(y)) \<subseteq> Lset(j)" |
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apply (rule_tac x="\<Union>y \<in> Replace(X, %x y. Q(x,y) & L(y)). succ(lrank(y))" |
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in exI) |
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apply simp |
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apply clarify |
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apply (rule_tac a=x in UN_I) |
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apply (simp_all add: Replace_iff univalent_def) |
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apply (blast dest: transL L_I) |
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done |
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lemma LReplace_in_L: |
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"[|L(X); univalent(L,X,Q)|] |
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==> \<exists>Y. L(Y) & Replace(X, %x y. Q(x,y) & L(y)) \<subseteq> Y" |
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apply (drule L_D, clarify) |
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apply (drule LReplace_in_Lset, assumption+) |
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apply (blast intro: L_I Lset_in_Lset_succ) |
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done |
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theorem replacement: "replacement(L,P)" |
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apply (simp add: replacement_def, clarify) |
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apply (frule LReplace_in_L, assumption+, clarify) |
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apply (rule_tac x=Y in rexI) |
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apply (simp_all add: Replace_iff univalent_def, blast) |
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done |
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subsection{*Instantiating the locale @{text M_trivial}*} |
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text{*No instances of Separation yet.*} |
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lemma Lset_mono_le: "mono_le_subset(Lset)" |
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by (simp add: mono_le_subset_def le_imp_subset Lset_mono) |
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lemma Lset_cont: "cont_Ord(Lset)" |
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by (simp add: cont_Ord_def Limit_Lset_eq OUnion_def Limit_is_Ord) |
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lemmas L_nat = Ord_in_L [OF Ord_nat] |
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theorem M_trivial_L: "PROP M_trivial(L)" |
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apply (rule M_trivial.intro) |
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apply (erule (1) transL) |
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apply (rule upair_ax) |
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apply (rule Union_ax) |
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apply (rule power_ax) |
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apply (rule replacement) |
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apply (rule L_nat) |
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done |
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lemmas rall_abs = M_trivial.rall_abs [OF M_trivial_L] |
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and rex_abs = M_trivial.rex_abs [OF M_trivial_L] |
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and ball_iff_equiv = M_trivial.ball_iff_equiv [OF M_trivial_L] |
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and M_equalityI = M_trivial.M_equalityI [OF M_trivial_L] |
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and empty_abs = M_trivial.empty_abs [OF M_trivial_L] |
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and subset_abs = M_trivial.subset_abs [OF M_trivial_L] |
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and upair_abs = M_trivial.upair_abs [OF M_trivial_L] |
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and upair_in_M_iff = M_trivial.upair_in_M_iff [OF M_trivial_L] |
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and singleton_in_M_iff = M_trivial.singleton_in_M_iff [OF M_trivial_L] |
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and pair_abs = M_trivial.pair_abs [OF M_trivial_L] |
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and pair_in_M_iff = M_trivial.pair_in_M_iff [OF M_trivial_L] |
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and pair_components_in_M = M_trivial.pair_components_in_M [OF M_trivial_L] |
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and cartprod_abs = M_trivial.cartprod_abs [OF M_trivial_L] |
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and union_abs = M_trivial.union_abs [OF M_trivial_L] |
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and inter_abs = M_trivial.inter_abs [OF M_trivial_L] |
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and setdiff_abs = M_trivial.setdiff_abs [OF M_trivial_L] |
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and Union_abs = M_trivial.Union_abs [OF M_trivial_L] |
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and Union_closed = M_trivial.Union_closed [OF M_trivial_L] |
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and Un_closed = M_trivial.Un_closed [OF M_trivial_L] |
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and cons_closed = M_trivial.cons_closed [OF M_trivial_L] |
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and successor_abs = M_trivial.successor_abs [OF M_trivial_L] |
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and succ_in_M_iff = M_trivial.succ_in_M_iff [OF M_trivial_L] |
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and separation_closed = M_trivial.separation_closed [OF M_trivial_L] |
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and strong_replacementI = |
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M_trivial.strong_replacementI [OF M_trivial_L, rule_format] |
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and strong_replacement_closed = M_trivial.strong_replacement_closed [OF M_trivial_L] |
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and RepFun_closed = M_trivial.RepFun_closed [OF M_trivial_L] |
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and lam_closed = M_trivial.lam_closed [OF M_trivial_L] |
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and image_abs = M_trivial.image_abs [OF M_trivial_L] |
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and powerset_Pow = M_trivial.powerset_Pow [OF M_trivial_L] |
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and powerset_imp_subset_Pow = M_trivial.powerset_imp_subset_Pow [OF M_trivial_L] |
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and nat_into_M = M_trivial.nat_into_M [OF M_trivial_L] |
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and nat_case_closed = M_trivial.nat_case_closed [OF M_trivial_L] |
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and Inl_in_M_iff = M_trivial.Inl_in_M_iff [OF M_trivial_L] |
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and Inr_in_M_iff = M_trivial.Inr_in_M_iff [OF M_trivial_L] |
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and lt_closed = M_trivial.lt_closed [OF M_trivial_L] |
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and transitive_set_abs = M_trivial.transitive_set_abs [OF M_trivial_L] |
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and ordinal_abs = M_trivial.ordinal_abs [OF M_trivial_L] |
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and limit_ordinal_abs = M_trivial.limit_ordinal_abs [OF M_trivial_L] |
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and successor_ordinal_abs = M_trivial.successor_ordinal_abs [OF M_trivial_L] |
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and finite_ordinal_abs = M_trivial.finite_ordinal_abs [OF M_trivial_L] |
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and omega_abs = M_trivial.omega_abs [OF M_trivial_L] |
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and number1_abs = M_trivial.number1_abs [OF M_trivial_L] |
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and number2_abs = M_trivial.number2_abs [OF M_trivial_L] |
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and number3_abs = M_trivial.number3_abs [OF M_trivial_L] |
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declare rall_abs [simp] |
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declare rex_abs [simp] |
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declare empty_abs [simp] |
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declare subset_abs [simp] |
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declare upair_abs [simp] |
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declare upair_in_M_iff [iff] |
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declare singleton_in_M_iff [iff] |
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declare pair_abs [simp] |
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declare pair_in_M_iff [iff] |
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declare cartprod_abs [simp] |
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declare union_abs [simp] |
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declare inter_abs [simp] |
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declare setdiff_abs [simp] |
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declare Union_abs [simp] |
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declare Union_closed [intro, simp] |
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declare Un_closed [intro, simp] |
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declare cons_closed [intro, simp] |
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declare successor_abs [simp] |
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declare succ_in_M_iff [iff] |
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declare separation_closed [intro, simp] |
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declare strong_replacement_closed [intro, simp] |
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declare RepFun_closed [intro, simp] |
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declare lam_closed [intro, simp] |
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declare image_abs [simp] |
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declare nat_into_M [intro] |
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declare Inl_in_M_iff [iff] |
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declare Inr_in_M_iff [iff] |
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declare transitive_set_abs [simp] |
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declare ordinal_abs [simp] |
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declare limit_ordinal_abs [simp] |
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declare successor_ordinal_abs [simp] |
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declare finite_ordinal_abs [simp] |
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declare omega_abs [simp] |
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declare number1_abs [simp] |
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declare number2_abs [simp] |
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declare number3_abs [simp] |
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subsection{*Instantiation of the locale @{text reflection}*} |
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text{*instances of locale constants*} |
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constdefs |
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L_F0 :: "[i=>o,i] => i" |
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"L_F0(P,y) == \<mu>b. (\<exists>z. L(z) \<and> P(<y,z>)) --> (\<exists>z\<in>Lset(b). P(<y,z>))" |
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L_FF :: "[i=>o,i] => i" |
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"L_FF(P) == \<lambda>a. \<Union>y\<in>Lset(a). L_F0(P,y)" |
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L_ClEx :: "[i=>o,i] => o" |
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"L_ClEx(P) == \<lambda>a. Limit(a) \<and> normalize(L_FF(P),a) = a" |
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text{*We must use the meta-existential quantifier; otherwise the reflection |
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terms become enormous!*} |
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constdefs |
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L_Reflects :: "[i=>o,[i,i]=>o] => prop" ("(3REFLECTS/ [_,/ _])") |
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"REFLECTS[P,Q] == (??Cl. Closed_Unbounded(Cl) & |
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(\<forall>a. Cl(a) --> (\<forall>x \<in> Lset(a). P(x) <-> Q(a,x))))" |
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theorem Triv_reflection: |
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"REFLECTS[P, \<lambda>a x. P(x)]" |
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apply (simp add: L_Reflects_def) |
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apply (rule meta_exI) |
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apply (rule Closed_Unbounded_Ord) |
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done |
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theorem Not_reflection: |
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"REFLECTS[P,Q] ==> REFLECTS[\<lambda>x. ~P(x), \<lambda>a x. ~Q(a,x)]" |
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apply (unfold L_Reflects_def) |
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apply (erule meta_exE) |
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apply (rule_tac x=Cl in meta_exI, simp) |
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done |
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theorem And_reflection: |
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"[| REFLECTS[P,Q]; REFLECTS[P',Q'] |] |
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==> REFLECTS[\<lambda>x. P(x) \<and> P'(x), \<lambda>a x. Q(a,x) \<and> Q'(a,x)]" |
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apply (unfold L_Reflects_def) |
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apply (elim meta_exE) |
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apply (rule_tac x="\<lambda>a. Cl(a) \<and> Cla(a)" in meta_exI) |
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apply (simp add: Closed_Unbounded_Int, blast) |
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done |
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theorem Or_reflection: |
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"[| REFLECTS[P,Q]; REFLECTS[P',Q'] |] |
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==> REFLECTS[\<lambda>x. P(x) \<or> P'(x), \<lambda>a x. Q(a,x) \<or> Q'(a,x)]" |
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apply (unfold L_Reflects_def) |
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apply (elim meta_exE) |
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apply (rule_tac x="\<lambda>a. Cl(a) \<and> Cla(a)" in meta_exI) |
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apply (simp add: Closed_Unbounded_Int, blast) |
|
13314 | 239 |
done |
240 |
||
241 |
theorem Imp_reflection: |
|
13429 | 242 |
"[| REFLECTS[P,Q]; REFLECTS[P',Q'] |] |
13314 | 243 |
==> REFLECTS[\<lambda>x. P(x) --> P'(x), \<lambda>a x. Q(a,x) --> Q'(a,x)]" |
13429 | 244 |
apply (unfold L_Reflects_def) |
245 |
apply (elim meta_exE) |
|
246 |
apply (rule_tac x="\<lambda>a. Cl(a) \<and> Cla(a)" in meta_exI) |
|
247 |
apply (simp add: Closed_Unbounded_Int, blast) |
|
13314 | 248 |
done |
249 |
||
250 |
theorem Iff_reflection: |
|
13429 | 251 |
"[| REFLECTS[P,Q]; REFLECTS[P',Q'] |] |
13314 | 252 |
==> REFLECTS[\<lambda>x. P(x) <-> P'(x), \<lambda>a x. Q(a,x) <-> Q'(a,x)]" |
13429 | 253 |
apply (unfold L_Reflects_def) |
254 |
apply (elim meta_exE) |
|
255 |
apply (rule_tac x="\<lambda>a. Cl(a) \<and> Cla(a)" in meta_exI) |
|
256 |
apply (simp add: Closed_Unbounded_Int, blast) |
|
13314 | 257 |
done |
258 |
||
259 |
||
13434 | 260 |
lemma reflection_Lset: "reflection(Lset)" |
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by (blast intro: reflection.intro Lset_mono_le Lset_cont |
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Formula.Pair_in_LLimit)+ |
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263 |
|
13434 | 264 |
|
13314 | 265 |
theorem Ex_reflection: |
266 |
"REFLECTS[\<lambda>x. P(fst(x),snd(x)), \<lambda>a x. Q(a,fst(x),snd(x))] |
|
267 |
==> REFLECTS[\<lambda>x. \<exists>z. L(z) \<and> P(x,z), \<lambda>a x. \<exists>z\<in>Lset(a). Q(a,x,z)]" |
|
13429 | 268 |
apply (unfold L_Reflects_def L_ClEx_def L_FF_def L_F0_def L_def) |
269 |
apply (elim meta_exE) |
|
13314 | 270 |
apply (rule meta_exI) |
13434 | 271 |
apply (erule reflection.Ex_reflection [OF reflection_Lset]) |
13291 | 272 |
done |
273 |
||
13314 | 274 |
theorem All_reflection: |
275 |
"REFLECTS[\<lambda>x. P(fst(x),snd(x)), \<lambda>a x. Q(a,fst(x),snd(x))] |
|
13429 | 276 |
==> REFLECTS[\<lambda>x. \<forall>z. L(z) --> P(x,z), \<lambda>a x. \<forall>z\<in>Lset(a). Q(a,x,z)]" |
277 |
apply (unfold L_Reflects_def L_ClEx_def L_FF_def L_F0_def L_def) |
|
278 |
apply (elim meta_exE) |
|
13314 | 279 |
apply (rule meta_exI) |
13434 | 280 |
apply (erule reflection.All_reflection [OF reflection_Lset]) |
13291 | 281 |
done |
282 |
||
13314 | 283 |
theorem Rex_reflection: |
284 |
"REFLECTS[ \<lambda>x. P(fst(x),snd(x)), \<lambda>a x. Q(a,fst(x),snd(x))] |
|
285 |
==> REFLECTS[\<lambda>x. \<exists>z[L]. P(x,z), \<lambda>a x. \<exists>z\<in>Lset(a). Q(a,x,z)]" |
|
13429 | 286 |
apply (unfold rex_def) |
13314 | 287 |
apply (intro And_reflection Ex_reflection, assumption) |
288 |
done |
|
13291 | 289 |
|
13314 | 290 |
theorem Rall_reflection: |
291 |
"REFLECTS[\<lambda>x. P(fst(x),snd(x)), \<lambda>a x. Q(a,fst(x),snd(x))] |
|
13429 | 292 |
==> REFLECTS[\<lambda>x. \<forall>z[L]. P(x,z), \<lambda>a x. \<forall>z\<in>Lset(a). Q(a,x,z)]" |
293 |
apply (unfold rall_def) |
|
13314 | 294 |
apply (intro Imp_reflection All_reflection, assumption) |
295 |
done |
|
296 |
||
13440 | 297 |
text{*This version handles an alternative form of the bounded quantifier |
298 |
in the second argument of @{text REFLECTS}.*} |
|
299 |
theorem Rex_reflection': |
|
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"REFLECTS[\<lambda>x. P(fst(x),snd(x)), \<lambda>a x. Q(a,fst(x),snd(x))] |
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==> REFLECTS[\<lambda>x. \<exists>z[L]. P(x,z), \<lambda>a x. \<exists>z[##Lset(a)]. Q(a,x,z)]" |
13440 | 302 |
apply (unfold setclass_def rex_def) |
303 |
apply (erule Rex_reflection [unfolded rex_def Bex_def]) |
|
304 |
done |
|
305 |
||
306 |
text{*As above.*} |
|
307 |
theorem Rall_reflection': |
|
308 |
"REFLECTS[\<lambda>x. P(fst(x),snd(x)), \<lambda>a x. Q(a,fst(x),snd(x))] |
|
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==> REFLECTS[\<lambda>x. \<forall>z[L]. P(x,z), \<lambda>a x. \<forall>z[##Lset(a)]. Q(a,x,z)]" |
13440 | 310 |
apply (unfold setclass_def rall_def) |
311 |
apply (erule Rall_reflection [unfolded rall_def Ball_def]) |
|
312 |
done |
|
313 |
||
13429 | 314 |
lemmas FOL_reflections = |
13314 | 315 |
Triv_reflection Not_reflection And_reflection Or_reflection |
316 |
Imp_reflection Iff_reflection Ex_reflection All_reflection |
|
13440 | 317 |
Rex_reflection Rall_reflection Rex_reflection' Rall_reflection' |
13291 | 318 |
|
319 |
lemma ReflectsD: |
|
13429 | 320 |
"[|REFLECTS[P,Q]; Ord(i)|] |
13291 | 321 |
==> \<exists>j. i<j & (\<forall>x \<in> Lset(j). P(x) <-> Q(j,x))" |
13429 | 322 |
apply (unfold L_Reflects_def Closed_Unbounded_def) |
323 |
apply (elim meta_exE, clarify) |
|
324 |
apply (blast dest!: UnboundedD) |
|
13291 | 325 |
done |
326 |
||
327 |
lemma ReflectsE: |
|
13314 | 328 |
"[| REFLECTS[P,Q]; Ord(i); |
13291 | 329 |
!!j. [|i<j; \<forall>x \<in> Lset(j). P(x) <-> Q(j,x)|] ==> R |] |
330 |
==> R" |
|
13429 | 331 |
apply (drule ReflectsD, assumption, blast) |
13314 | 332 |
done |
13291 | 333 |
|
13428 | 334 |
lemma Collect_mem_eq: "{x\<in>A. x\<in>B} = A \<inter> B" |
13291 | 335 |
by blast |
336 |
||
337 |
||
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338 |
subsection{*Internalized Formulas for some Set-Theoretic Concepts*} |
13298 | 339 |
|
13306 | 340 |
subsubsection{*Some numbers to help write de Bruijn indices*} |
341 |
||
342 |
syntax |
|
343 |
"3" :: i ("3") |
|
344 |
"4" :: i ("4") |
|
345 |
"5" :: i ("5") |
|
346 |
"6" :: i ("6") |
|
347 |
"7" :: i ("7") |
|
348 |
"8" :: i ("8") |
|
349 |
"9" :: i ("9") |
|
350 |
||
351 |
translations |
|
352 |
"3" == "succ(2)" |
|
353 |
"4" == "succ(3)" |
|
354 |
"5" == "succ(4)" |
|
355 |
"6" == "succ(5)" |
|
356 |
"7" == "succ(6)" |
|
357 |
"8" == "succ(7)" |
|
358 |
"9" == "succ(8)" |
|
359 |
||
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360 |
|
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361 |
subsubsection{*The Empty Set, Internalized*} |
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362 |
|
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constdefs empty_fm :: "i=>i" |
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"empty_fm(x) == Forall(Neg(Member(0,succ(x))))" |
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365 |
|
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lemma empty_type [TC]: |
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"x \<in> nat ==> empty_fm(x) \<in> formula" |
13429 | 368 |
by (simp add: empty_fm_def) |
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|
369 |
|
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|
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lemma sats_empty_fm [simp]: |
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"[| x \<in> nat; env \<in> list(A)|] |
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==> sats(A, empty_fm(x), env) <-> empty(##A, nth(x,env))" |
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373 |
by (simp add: empty_fm_def empty_def) |
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|
374 |
|
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375 |
lemma empty_iff_sats: |
13429 | 376 |
"[| nth(i,env) = x; nth(j,env) = y; |
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i \<in> nat; env \<in> list(A)|] |
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==> empty(##A, x) <-> sats(A, empty_fm(i), env)" |
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379 |
by simp |
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|
380 |
|
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381 |
theorem empty_reflection: |
13429 | 382 |
"REFLECTS[\<lambda>x. empty(L,f(x)), |
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|
383 |
\<lambda>i x. empty(##Lset(i),f(x))]" |
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384 |
apply (simp only: empty_def) |
13429 | 385 |
apply (intro FOL_reflections) |
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|
386 |
done |
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|
387 |
|
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|
388 |
text{*Not used. But maybe useful?*} |
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|
389 |
lemma Transset_sats_empty_fm_eq_0: |
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|
390 |
"[| n \<in> nat; env \<in> list(A); Transset(A)|] |
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|
391 |
==> sats(A, empty_fm(n), env) <-> nth(n,env) = 0" |
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|
392 |
apply (simp add: empty_fm_def empty_def Transset_def, auto) |
13429 | 393 |
apply (case_tac "n < length(env)") |
394 |
apply (frule nth_type, assumption+, blast) |
|
395 |
apply (simp_all add: not_lt_iff_le nth_eq_0) |
|
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|
396 |
done |
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|
397 |
|
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|
398 |
|
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|
399 |
subsubsection{*Unordered Pairs, Internalized*} |
13298 | 400 |
|
401 |
constdefs upair_fm :: "[i,i,i]=>i" |
|
13429 | 402 |
"upair_fm(x,y,z) == |
403 |
And(Member(x,z), |
|
13298 | 404 |
And(Member(y,z), |
13429 | 405 |
Forall(Implies(Member(0,succ(z)), |
13298 | 406 |
Or(Equal(0,succ(x)), Equal(0,succ(y)))))))" |
407 |
||
408 |
lemma upair_type [TC]: |
|
409 |
"[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> upair_fm(x,y,z) \<in> formula" |
|
13429 | 410 |
by (simp add: upair_fm_def) |
13298 | 411 |
|
412 |
lemma sats_upair_fm [simp]: |
|
413 |
"[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|] |
|
13429 | 414 |
==> sats(A, upair_fm(x,y,z), env) <-> |
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|
415 |
upair(##A, nth(x,env), nth(y,env), nth(z,env))" |
13298 | 416 |
by (simp add: upair_fm_def upair_def) |
417 |
||
418 |
lemma upair_iff_sats: |
|
13429 | 419 |
"[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z; |
13298 | 420 |
i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|] |
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|
421 |
==> upair(##A, x, y, z) <-> sats(A, upair_fm(i,j,k), env)" |
13298 | 422 |
by (simp add: sats_upair_fm) |
423 |
||
424 |
text{*Useful? At least it refers to "real" unordered pairs*} |
|
425 |
lemma sats_upair_fm2 [simp]: |
|
426 |
"[| x \<in> nat; y \<in> nat; z < length(env); env \<in> list(A); Transset(A)|] |
|
13429 | 427 |
==> sats(A, upair_fm(x,y,z), env) <-> |
13298 | 428 |
nth(z,env) = {nth(x,env), nth(y,env)}" |
13429 | 429 |
apply (frule lt_length_in_nat, assumption) |
430 |
apply (simp add: upair_fm_def Transset_def, auto) |
|
431 |
apply (blast intro: nth_type) |
|
13298 | 432 |
done |
433 |
||
13314 | 434 |
theorem upair_reflection: |
13429 | 435 |
"REFLECTS[\<lambda>x. upair(L,f(x),g(x),h(x)), |
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|
436 |
\<lambda>i x. upair(##Lset(i),f(x),g(x),h(x))]" |
13314 | 437 |
apply (simp add: upair_def) |
13429 | 438 |
apply (intro FOL_reflections) |
13314 | 439 |
done |
13306 | 440 |
|
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441 |
subsubsection{*Ordered pairs, Internalized*} |
13298 | 442 |
|
443 |
constdefs pair_fm :: "[i,i,i]=>i" |
|
13429 | 444 |
"pair_fm(x,y,z) == |
13298 | 445 |
Exists(And(upair_fm(succ(x),succ(x),0), |
446 |
Exists(And(upair_fm(succ(succ(x)),succ(succ(y)),0), |
|
447 |
upair_fm(1,0,succ(succ(z)))))))" |
|
448 |
||
449 |
lemma pair_type [TC]: |
|
450 |
"[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> pair_fm(x,y,z) \<in> formula" |
|
13429 | 451 |
by (simp add: pair_fm_def) |
13298 | 452 |
|
453 |
lemma sats_pair_fm [simp]: |
|
454 |
"[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|] |
|
13429 | 455 |
==> sats(A, pair_fm(x,y,z), env) <-> |
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|
456 |
pair(##A, nth(x,env), nth(y,env), nth(z,env))" |
13298 | 457 |
by (simp add: pair_fm_def pair_def) |
458 |
||
459 |
lemma pair_iff_sats: |
|
13429 | 460 |
"[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z; |
13298 | 461 |
i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|] |
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|
462 |
==> pair(##A, x, y, z) <-> sats(A, pair_fm(i,j,k), env)" |
13298 | 463 |
by (simp add: sats_pair_fm) |
464 |
||
13314 | 465 |
theorem pair_reflection: |
13429 | 466 |
"REFLECTS[\<lambda>x. pair(L,f(x),g(x),h(x)), |
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|
467 |
\<lambda>i x. pair(##Lset(i),f(x),g(x),h(x))]" |
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|
468 |
apply (simp only: pair_def) |
13429 | 469 |
apply (intro FOL_reflections upair_reflection) |
13314 | 470 |
done |
13306 | 471 |
|
472 |
||
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473 |
subsubsection{*Binary Unions, Internalized*} |
13298 | 474 |
|
13306 | 475 |
constdefs union_fm :: "[i,i,i]=>i" |
13429 | 476 |
"union_fm(x,y,z) == |
13306 | 477 |
Forall(Iff(Member(0,succ(z)), |
478 |
Or(Member(0,succ(x)),Member(0,succ(y)))))" |
|
479 |
||
480 |
lemma union_type [TC]: |
|
481 |
"[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> union_fm(x,y,z) \<in> formula" |
|
13429 | 482 |
by (simp add: union_fm_def) |
13306 | 483 |
|
484 |
lemma sats_union_fm [simp]: |
|
485 |
"[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|] |
|
13429 | 486 |
==> sats(A, union_fm(x,y,z), env) <-> |
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|
487 |
union(##A, nth(x,env), nth(y,env), nth(z,env))" |
13306 | 488 |
by (simp add: union_fm_def union_def) |
489 |
||
490 |
lemma union_iff_sats: |
|
13429 | 491 |
"[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z; |
13306 | 492 |
i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|] |
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|
493 |
==> union(##A, x, y, z) <-> sats(A, union_fm(i,j,k), env)" |
13306 | 494 |
by (simp add: sats_union_fm) |
13298 | 495 |
|
13314 | 496 |
theorem union_reflection: |
13429 | 497 |
"REFLECTS[\<lambda>x. union(L,f(x),g(x),h(x)), |
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|
498 |
\<lambda>i x. union(##Lset(i),f(x),g(x),h(x))]" |
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|
499 |
apply (simp only: union_def) |
13429 | 500 |
apply (intro FOL_reflections) |
13314 | 501 |
done |
13306 | 502 |
|
13298 | 503 |
|
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|
504 |
subsubsection{*Set ``Cons,'' Internalized*} |
13306 | 505 |
|
506 |
constdefs cons_fm :: "[i,i,i]=>i" |
|
13429 | 507 |
"cons_fm(x,y,z) == |
13306 | 508 |
Exists(And(upair_fm(succ(x),succ(x),0), |
509 |
union_fm(0,succ(y),succ(z))))" |
|
13298 | 510 |
|
511 |
||
13306 | 512 |
lemma cons_type [TC]: |
513 |
"[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> cons_fm(x,y,z) \<in> formula" |
|
13429 | 514 |
by (simp add: cons_fm_def) |
13306 | 515 |
|
516 |
lemma sats_cons_fm [simp]: |
|
517 |
"[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|] |
|
13429 | 518 |
==> sats(A, cons_fm(x,y,z), env) <-> |
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|
519 |
is_cons(##A, nth(x,env), nth(y,env), nth(z,env))" |
13306 | 520 |
by (simp add: cons_fm_def is_cons_def) |
521 |
||
522 |
lemma cons_iff_sats: |
|
13429 | 523 |
"[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z; |
13306 | 524 |
i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|] |
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|
525 |
==> is_cons(##A, x, y, z) <-> sats(A, cons_fm(i,j,k), env)" |
13306 | 526 |
by simp |
527 |
||
13314 | 528 |
theorem cons_reflection: |
13429 | 529 |
"REFLECTS[\<lambda>x. is_cons(L,f(x),g(x),h(x)), |
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|
530 |
\<lambda>i x. is_cons(##Lset(i),f(x),g(x),h(x))]" |
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|
531 |
apply (simp only: is_cons_def) |
13429 | 532 |
apply (intro FOL_reflections upair_reflection union_reflection) |
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533 |
done |
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|
534 |
|
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535 |
|
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|
536 |
subsubsection{*Successor Function, Internalized*} |
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|
537 |
|
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538 |
constdefs succ_fm :: "[i,i]=>i" |
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539 |
"succ_fm(x,y) == cons_fm(x,x,y)" |
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|
540 |
|
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|
541 |
lemma succ_type [TC]: |
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|
542 |
"[| x \<in> nat; y \<in> nat |] ==> succ_fm(x,y) \<in> formula" |
13429 | 543 |
by (simp add: succ_fm_def) |
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|
544 |
|
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|
545 |
lemma sats_succ_fm [simp]: |
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|
546 |
"[| x \<in> nat; y \<in> nat; env \<in> list(A)|] |
13429 | 547 |
==> sats(A, succ_fm(x,y), env) <-> |
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|
548 |
successor(##A, nth(x,env), nth(y,env))" |
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|
549 |
by (simp add: succ_fm_def successor_def) |
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|
550 |
|
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|
551 |
lemma successor_iff_sats: |
13429 | 552 |
"[| nth(i,env) = x; nth(j,env) = y; |
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553 |
i \<in> nat; j \<in> nat; env \<in> list(A)|] |
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|
554 |
==> successor(##A, x, y) <-> sats(A, succ_fm(i,j), env)" |
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|
555 |
by simp |
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|
556 |
|
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|
557 |
theorem successor_reflection: |
13429 | 558 |
"REFLECTS[\<lambda>x. successor(L,f(x),g(x)), |
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|
559 |
\<lambda>i x. successor(##Lset(i),f(x),g(x))]" |
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Tidying up. New primitives is_iterates and is_iterates_fm.
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13651
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|
560 |
apply (simp only: successor_def) |
13429 | 561 |
apply (intro cons_reflection) |
13314 | 562 |
done |
13298 | 563 |
|
564 |
||
13363 | 565 |
subsubsection{*The Number 1, Internalized*} |
566 |
||
567 |
(* "number1(M,a) == (\<exists>x[M]. empty(M,x) & successor(M,x,a))" *) |
|
568 |
constdefs number1_fm :: "i=>i" |
|
569 |
"number1_fm(a) == Exists(And(empty_fm(0), succ_fm(0,succ(a))))" |
|
570 |
||
571 |
lemma number1_type [TC]: |
|
572 |
"x \<in> nat ==> number1_fm(x) \<in> formula" |
|
13429 | 573 |
by (simp add: number1_fm_def) |
13363 | 574 |
|
575 |
lemma sats_number1_fm [simp]: |
|
576 |
"[| x \<in> nat; env \<in> list(A)|] |
|
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diff
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|
577 |
==> sats(A, number1_fm(x), env) <-> number1(##A, nth(x,env))" |
13363 | 578 |
by (simp add: number1_fm_def number1_def) |
579 |
||
580 |
lemma number1_iff_sats: |
|
13429 | 581 |
"[| nth(i,env) = x; nth(j,env) = y; |
13363 | 582 |
i \<in> nat; env \<in> list(A)|] |
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|
583 |
==> number1(##A, x) <-> sats(A, number1_fm(i), env)" |
13363 | 584 |
by simp |
585 |
||
586 |
theorem number1_reflection: |
|
13429 | 587 |
"REFLECTS[\<lambda>x. number1(L,f(x)), |
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|
588 |
\<lambda>i x. number1(##Lset(i),f(x))]" |
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diff
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|
589 |
apply (simp only: number1_def) |
13363 | 590 |
apply (intro FOL_reflections empty_reflection successor_reflection) |
591 |
done |
|
592 |
||
593 |
||
13352 | 594 |
subsubsection{*Big Union, Internalized*} |
13306 | 595 |
|
13352 | 596 |
(* "big_union(M,A,z) == \<forall>x[M]. x \<in> z <-> (\<exists>y[M]. y\<in>A & x \<in> y)" *) |
597 |
constdefs big_union_fm :: "[i,i]=>i" |
|
13429 | 598 |
"big_union_fm(A,z) == |
13352 | 599 |
Forall(Iff(Member(0,succ(z)), |
600 |
Exists(And(Member(0,succ(succ(A))), Member(1,0)))))" |
|
13298 | 601 |
|
13352 | 602 |
lemma big_union_type [TC]: |
603 |
"[| x \<in> nat; y \<in> nat |] ==> big_union_fm(x,y) \<in> formula" |
|
13429 | 604 |
by (simp add: big_union_fm_def) |
13306 | 605 |
|
13352 | 606 |
lemma sats_big_union_fm [simp]: |
607 |
"[| x \<in> nat; y \<in> nat; env \<in> list(A)|] |
|
13429 | 608 |
==> sats(A, big_union_fm(x,y), env) <-> |
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|
609 |
big_union(##A, nth(x,env), nth(y,env))" |
13352 | 610 |
by (simp add: big_union_fm_def big_union_def) |
13306 | 611 |
|
13352 | 612 |
lemma big_union_iff_sats: |
13429 | 613 |
"[| nth(i,env) = x; nth(j,env) = y; |
13352 | 614 |
i \<in> nat; j \<in> nat; env \<in> list(A)|] |
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diff
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|
615 |
==> big_union(##A, x, y) <-> sats(A, big_union_fm(i,j), env)" |
13306 | 616 |
by simp |
617 |
||
13352 | 618 |
theorem big_union_reflection: |
13429 | 619 |
"REFLECTS[\<lambda>x. big_union(L,f(x),g(x)), |
13807
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|
620 |
\<lambda>i x. big_union(##Lset(i),f(x),g(x))]" |
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13651
diff
changeset
|
621 |
apply (simp only: big_union_def) |
13429 | 622 |
apply (intro FOL_reflections) |
13314 | 623 |
done |
13298 | 624 |
|
625 |
||
13306 | 626 |
subsubsection{*Variants of Satisfaction Definitions for Ordinals, etc.*} |
627 |
||
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|
628 |
text{*The @{text sats} theorems below are standard versions of the ones proved |
ac80e101306a
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13634
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|
629 |
in theory @{text Formula}. They relate elements of type @{term formula} to |
ac80e101306a
Cosmetic changes suggested by writing the paper. Deleted some
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parents:
13634
diff
changeset
|
630 |
relativized concepts such as @{term subset} or @{term ordinal} rather than to |
ac80e101306a
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parents:
13634
diff
changeset
|
631 |
real concepts such as @{term Ord}. Now that we have instantiated the locale |
ac80e101306a
Cosmetic changes suggested by writing the paper. Deleted some
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changeset
|
632 |
@{text M_trivial}, we no longer require the earlier versions.*} |
13306 | 633 |
|
634 |
lemma sats_subset_fm': |
|
635 |
"[|x \<in> nat; y \<in> nat; env \<in> list(A)|] |
|
13807
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|
636 |
==> sats(A, subset_fm(x,y), env) <-> subset(##A, nth(x,env), nth(y,env))" |
13429 | 637 |
by (simp add: subset_fm_def Relative.subset_def) |
13298 | 638 |
|
13314 | 639 |
theorem subset_reflection: |
13429 | 640 |
"REFLECTS[\<lambda>x. subset(L,f(x),g(x)), |
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diff
changeset
|
641 |
\<lambda>i x. subset(##Lset(i),f(x),g(x))]" |
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Tidying up. New primitives is_iterates and is_iterates_fm.
paulson
parents:
13651
diff
changeset
|
642 |
apply (simp only: Relative.subset_def) |
13429 | 643 |
apply (intro FOL_reflections) |
13314 | 644 |
done |
13306 | 645 |
|
646 |
lemma sats_transset_fm': |
|
647 |
"[|x \<in> nat; env \<in> list(A)|] |
|
13807
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diff
changeset
|
648 |
==> sats(A, transset_fm(x), env) <-> transitive_set(##A, nth(x,env))" |
13429 | 649 |
by (simp add: sats_subset_fm' transset_fm_def transitive_set_def) |
13298 | 650 |
|
13314 | 651 |
theorem transitive_set_reflection: |
652 |
"REFLECTS[\<lambda>x. transitive_set(L,f(x)), |
|
13807
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parents:
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diff
changeset
|
653 |
\<lambda>i x. transitive_set(##Lset(i),f(x))]" |
13655
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Tidying up. New primitives is_iterates and is_iterates_fm.
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parents:
13651
diff
changeset
|
654 |
apply (simp only: transitive_set_def) |
13429 | 655 |
apply (intro FOL_reflections subset_reflection) |
13314 | 656 |
done |
13306 | 657 |
|
658 |
lemma sats_ordinal_fm': |
|
659 |
"[|x \<in> nat; env \<in> list(A)|] |
|
13807
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13655
diff
changeset
|
660 |
==> sats(A, ordinal_fm(x), env) <-> ordinal(##A,nth(x,env))" |
13306 | 661 |
by (simp add: sats_transset_fm' ordinal_fm_def ordinal_def) |
662 |
||
663 |
lemma ordinal_iff_sats: |
|
664 |
"[| nth(i,env) = x; i \<in> nat; env \<in> list(A)|] |
|
13807
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diff
changeset
|
665 |
==> ordinal(##A, x) <-> sats(A, ordinal_fm(i), env)" |
13306 | 666 |
by (simp add: sats_ordinal_fm') |
667 |
||
13314 | 668 |
theorem ordinal_reflection: |
13807
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parents:
13655
diff
changeset
|
669 |
"REFLECTS[\<lambda>x. ordinal(L,f(x)), \<lambda>i x. ordinal(##Lset(i),f(x))]" |
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Tidying up. New primitives is_iterates and is_iterates_fm.
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changeset
|
670 |
apply (simp only: ordinal_def) |
13429 | 671 |
apply (intro FOL_reflections transitive_set_reflection) |
13314 | 672 |
done |
13298 | 673 |
|
674 |
||
13339
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diff
changeset
|
675 |
subsubsection{*Membership Relation, Internalized*} |
13298 | 676 |
|
13306 | 677 |
constdefs Memrel_fm :: "[i,i]=>i" |
13429 | 678 |
"Memrel_fm(A,r) == |
13306 | 679 |
Forall(Iff(Member(0,succ(r)), |
680 |
Exists(And(Member(0,succ(succ(A))), |
|
681 |
Exists(And(Member(0,succ(succ(succ(A)))), |
|
682 |
And(Member(1,0), |
|
683 |
pair_fm(1,0,2))))))))" |
|
684 |
||
685 |
lemma Memrel_type [TC]: |
|
686 |
"[| x \<in> nat; y \<in> nat |] ==> Memrel_fm(x,y) \<in> formula" |
|
13429 | 687 |
by (simp add: Memrel_fm_def) |
13298 | 688 |
|
13306 | 689 |
lemma sats_Memrel_fm [simp]: |
690 |
"[| x \<in> nat; y \<in> nat; env \<in> list(A)|] |
|
13429 | 691 |
==> sats(A, Memrel_fm(x,y), env) <-> |
13807
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paulson
parents:
13655
diff
changeset
|
692 |
membership(##A, nth(x,env), nth(y,env))" |
13306 | 693 |
by (simp add: Memrel_fm_def membership_def) |
13298 | 694 |
|
13306 | 695 |
lemma Memrel_iff_sats: |
13429 | 696 |
"[| nth(i,env) = x; nth(j,env) = y; |
13306 | 697 |
i \<in> nat; j \<in> nat; env \<in> list(A)|] |
13807
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13655
diff
changeset
|
698 |
==> membership(##A, x, y) <-> sats(A, Memrel_fm(i,j), env)" |
13306 | 699 |
by simp |
13304 | 700 |
|
13314 | 701 |
theorem membership_reflection: |
13429 | 702 |
"REFLECTS[\<lambda>x. membership(L,f(x),g(x)), |
13807
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parents:
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diff
changeset
|
703 |
\<lambda>i x. membership(##Lset(i),f(x),g(x))]" |
13655
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Tidying up. New primitives is_iterates and is_iterates_fm.
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13651
diff
changeset
|
704 |
apply (simp only: membership_def) |
13429 | 705 |
apply (intro FOL_reflections pair_reflection) |
13314 | 706 |
done |
13304 | 707 |
|
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changeset
|
708 |
subsubsection{*Predecessor Set, Internalized*} |
13304 | 709 |
|
13306 | 710 |
constdefs pred_set_fm :: "[i,i,i,i]=>i" |
13429 | 711 |
"pred_set_fm(A,x,r,B) == |
13306 | 712 |
Forall(Iff(Member(0,succ(B)), |
713 |
Exists(And(Member(0,succ(succ(r))), |
|
714 |
And(Member(1,succ(succ(A))), |
|
715 |
pair_fm(1,succ(succ(x)),0))))))" |
|
716 |
||
717 |
||
718 |
lemma pred_set_type [TC]: |
|
13429 | 719 |
"[| A \<in> nat; x \<in> nat; r \<in> nat; B \<in> nat |] |
13306 | 720 |
==> pred_set_fm(A,x,r,B) \<in> formula" |
13429 | 721 |
by (simp add: pred_set_fm_def) |
13304 | 722 |
|
13306 | 723 |
lemma sats_pred_set_fm [simp]: |
724 |
"[| U \<in> nat; x \<in> nat; r \<in> nat; B \<in> nat; env \<in> list(A)|] |
|
13429 | 725 |
==> sats(A, pred_set_fm(U,x,r,B), env) <-> |
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|
726 |
pred_set(##A, nth(U,env), nth(x,env), nth(r,env), nth(B,env))" |
13306 | 727 |
by (simp add: pred_set_fm_def pred_set_def) |
728 |
||
729 |
lemma pred_set_iff_sats: |
|
13429 | 730 |
"[| nth(i,env) = U; nth(j,env) = x; nth(k,env) = r; nth(l,env) = B; |
13306 | 731 |
i \<in> nat; j \<in> nat; k \<in> nat; l \<in> nat; env \<in> list(A)|] |
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changeset
|
732 |
==> pred_set(##A,U,x,r,B) <-> sats(A, pred_set_fm(i,j,k,l), env)" |
13306 | 733 |
by (simp add: sats_pred_set_fm) |
734 |
||
13314 | 735 |
theorem pred_set_reflection: |
13429 | 736 |
"REFLECTS[\<lambda>x. pred_set(L,f(x),g(x),h(x),b(x)), |
13807
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|
737 |
\<lambda>i x. pred_set(##Lset(i),f(x),g(x),h(x),b(x))]" |
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Tidying up. New primitives is_iterates and is_iterates_fm.
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changeset
|
738 |
apply (simp only: pred_set_def) |
13429 | 739 |
apply (intro FOL_reflections pair_reflection) |
13314 | 740 |
done |
13304 | 741 |
|
742 |
||
13298 | 743 |
|
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|
744 |
subsubsection{*Domain of a Relation, Internalized*} |
13306 | 745 |
|
13429 | 746 |
(* "is_domain(M,r,z) == |
747 |
\<forall>x[M]. (x \<in> z <-> (\<exists>w[M]. w\<in>r & (\<exists>y[M]. pair(M,x,y,w))))" *) |
|
13306 | 748 |
constdefs domain_fm :: "[i,i]=>i" |
13429 | 749 |
"domain_fm(r,z) == |
13306 | 750 |
Forall(Iff(Member(0,succ(z)), |
751 |
Exists(And(Member(0,succ(succ(r))), |
|
752 |
Exists(pair_fm(2,0,1))))))" |
|
753 |
||
754 |
lemma domain_type [TC]: |
|
755 |
"[| x \<in> nat; y \<in> nat |] ==> domain_fm(x,y) \<in> formula" |
|
13429 | 756 |
by (simp add: domain_fm_def) |
13306 | 757 |
|
758 |
lemma sats_domain_fm [simp]: |
|
759 |
"[| x \<in> nat; y \<in> nat; env \<in> list(A)|] |
|
13429 | 760 |
==> sats(A, domain_fm(x,y), env) <-> |
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|
761 |
is_domain(##A, nth(x,env), nth(y,env))" |
13306 | 762 |
by (simp add: domain_fm_def is_domain_def) |
763 |
||
764 |
lemma domain_iff_sats: |
|
13429 | 765 |
"[| nth(i,env) = x; nth(j,env) = y; |
13306 | 766 |
i \<in> nat; j \<in> nat; env \<in> list(A)|] |
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|
767 |
==> is_domain(##A, x, y) <-> sats(A, domain_fm(i,j), env)" |
13306 | 768 |
by simp |
769 |
||
13314 | 770 |
theorem domain_reflection: |
13429 | 771 |
"REFLECTS[\<lambda>x. is_domain(L,f(x),g(x)), |
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|
772 |
\<lambda>i x. is_domain(##Lset(i),f(x),g(x))]" |
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Tidying up. New primitives is_iterates and is_iterates_fm.
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parents:
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|
773 |
apply (simp only: is_domain_def) |
13429 | 774 |
apply (intro FOL_reflections pair_reflection) |
13314 | 775 |
done |
13306 | 776 |
|
777 |
||
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Fixed quantified variable name preservation for ball and bex (bounded quants)
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|
778 |
subsubsection{*Range of a Relation, Internalized*} |
13306 | 779 |
|
13429 | 780 |
(* "is_range(M,r,z) == |
781 |
\<forall>y[M]. (y \<in> z <-> (\<exists>w[M]. w\<in>r & (\<exists>x[M]. pair(M,x,y,w))))" *) |
|
13306 | 782 |
constdefs range_fm :: "[i,i]=>i" |
13429 | 783 |
"range_fm(r,z) == |
13306 | 784 |
Forall(Iff(Member(0,succ(z)), |
785 |
Exists(And(Member(0,succ(succ(r))), |
|
786 |
Exists(pair_fm(0,2,1))))))" |
|
787 |
||
788 |
lemma range_type [TC]: |
|
789 |
"[| x \<in> nat; y \<in> nat |] ==> range_fm(x,y) \<in> formula" |
|
13429 | 790 |
by (simp add: range_fm_def) |
13306 | 791 |
|
792 |
lemma sats_range_fm [simp]: |
|
793 |
"[| x \<in> nat; y \<in> nat; env \<in> list(A)|] |
|
13429 | 794 |
==> sats(A, range_fm(x,y), env) <-> |
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|
795 |
is_range(##A, nth(x,env), nth(y,env))" |
13306 | 796 |
by (simp add: range_fm_def is_range_def) |
797 |
||
798 |
lemma range_iff_sats: |
|
13429 | 799 |
"[| nth(i,env) = x; nth(j,env) = y; |
13306 | 800 |
i \<in> nat; j \<in> nat; env \<in> list(A)|] |
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|
801 |
==> is_range(##A, x, y) <-> sats(A, range_fm(i,j), env)" |
13306 | 802 |
by simp |
803 |
||
13314 | 804 |
theorem range_reflection: |
13429 | 805 |
"REFLECTS[\<lambda>x. is_range(L,f(x),g(x)), |
13807
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changeset
|
806 |
\<lambda>i x. is_range(##Lset(i),f(x),g(x))]" |
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95b95cdb4704
Tidying up. New primitives is_iterates and is_iterates_fm.
paulson
parents:
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diff
changeset
|
807 |
apply (simp only: is_range_def) |
13429 | 808 |
apply (intro FOL_reflections pair_reflection) |
13314 | 809 |
done |
13306 | 810 |
|
13429 | 811 |
|
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Fixed quantified variable name preservation for ball and bex (bounded quants)
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|
812 |
subsubsection{*Field of a Relation, Internalized*} |
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|
813 |
|
13429 | 814 |
(* "is_field(M,r,z) == |
815 |
\<exists>dr[M]. is_domain(M,r,dr) & |
|
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|
816 |
(\<exists>rr[M]. is_range(M,r,rr) & union(M,dr,rr,z))" *) |
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|
817 |
constdefs field_fm :: "[i,i]=>i" |
13429 | 818 |
"field_fm(r,z) == |
819 |
Exists(And(domain_fm(succ(r),0), |
|
820 |
Exists(And(range_fm(succ(succ(r)),0), |
|
13323
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|
821 |
union_fm(1,0,succ(succ(z)))))))" |
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|
822 |
|
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|
823 |
lemma field_type [TC]: |
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|
824 |
"[| x \<in> nat; y \<in> nat |] ==> field_fm(x,y) \<in> formula" |
13429 | 825 |
by (simp add: field_fm_def) |
13323
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|
826 |
|
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|
827 |
lemma sats_field_fm [simp]: |
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|
828 |
"[| x \<in> nat; y \<in> nat; env \<in> list(A)|] |
13429 | 829 |
==> sats(A, field_fm(x,y), env) <-> |
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|
830 |
is_field(##A, nth(x,env), nth(y,env))" |
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|
831 |
by (simp add: field_fm_def is_field_def) |
2c287f50c9f3
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diff
changeset
|
832 |
|
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|
833 |
lemma field_iff_sats: |
13429 | 834 |
"[| nth(i,env) = x; nth(j,env) = y; |
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|
835 |
i \<in> nat; j \<in> nat; env \<in> list(A)|] |
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paulson
parents:
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diff
changeset
|
836 |
==> is_field(##A, x, y) <-> sats(A, field_fm(i,j), env)" |
13323
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|
837 |
by simp |
2c287f50c9f3
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changeset
|
838 |
|
2c287f50c9f3
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|
839 |
theorem field_reflection: |
13429 | 840 |
"REFLECTS[\<lambda>x. is_field(L,f(x),g(x)), |
13807
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changed ** to ## to avoid conflict with new comment syntax
paulson
parents:
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diff
changeset
|
841 |
\<lambda>i x. is_field(##Lset(i),f(x),g(x))]" |
13655
95b95cdb4704
Tidying up. New primitives is_iterates and is_iterates_fm.
paulson
parents:
13651
diff
changeset
|
842 |
apply (simp only: is_field_def) |
13323
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More relativization, reflection and proofs of separation
paulson
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diff
changeset
|
843 |
apply (intro FOL_reflections domain_reflection range_reflection |
2c287f50c9f3
More relativization, reflection and proofs of separation
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parents:
13316
diff
changeset
|
844 |
union_reflection) |
2c287f50c9f3
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diff
changeset
|
845 |
done |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
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changeset
|
846 |
|
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More relativization, reflection and proofs of separation
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parents:
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diff
changeset
|
847 |
|
13339
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Fixed quantified variable name preservation for ball and bex (bounded quants)
paulson
parents:
13323
diff
changeset
|
848 |
subsubsection{*Image under a Relation, Internalized*} |
13306 | 849 |
|
13429 | 850 |
(* "image(M,r,A,z) == |
13306 | 851 |
\<forall>y[M]. (y \<in> z <-> (\<exists>w[M]. w\<in>r & (\<exists>x[M]. x\<in>A & pair(M,x,y,w))))" *) |
852 |
constdefs image_fm :: "[i,i,i]=>i" |
|
13429 | 853 |
"image_fm(r,A,z) == |
13306 | 854 |
Forall(Iff(Member(0,succ(z)), |
855 |
Exists(And(Member(0,succ(succ(r))), |
|
856 |
Exists(And(Member(0,succ(succ(succ(A)))), |
|
13429 | 857 |
pair_fm(0,2,1)))))))" |
13306 | 858 |
|
859 |
lemma image_type [TC]: |
|
860 |
"[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> image_fm(x,y,z) \<in> formula" |
|
13429 | 861 |
by (simp add: image_fm_def) |
13306 | 862 |
|
863 |
lemma sats_image_fm [simp]: |
|
864 |
"[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|] |
|
13429 | 865 |
==> sats(A, image_fm(x,y,z), env) <-> |
13807
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paulson
parents:
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diff
changeset
|
866 |
image(##A, nth(x,env), nth(y,env), nth(z,env))" |
13323
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More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
867 |
by (simp add: image_fm_def Relative.image_def) |
13306 | 868 |
|
869 |
lemma image_iff_sats: |
|
13429 | 870 |
"[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z; |
13306 | 871 |
i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|] |
13807
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paulson
parents:
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diff
changeset
|
872 |
==> image(##A, x, y, z) <-> sats(A, image_fm(i,j,k), env)" |
13306 | 873 |
by (simp add: sats_image_fm) |
874 |
||
13314 | 875 |
theorem image_reflection: |
13429 | 876 |
"REFLECTS[\<lambda>x. image(L,f(x),g(x),h(x)), |
13807
a28a8fbc76d4
changed ** to ## to avoid conflict with new comment syntax
paulson
parents:
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diff
changeset
|
877 |
\<lambda>i x. image(##Lset(i),f(x),g(x),h(x))]" |
13655
95b95cdb4704
Tidying up. New primitives is_iterates and is_iterates_fm.
paulson
parents:
13651
diff
changeset
|
878 |
apply (simp only: Relative.image_def) |
13429 | 879 |
apply (intro FOL_reflections pair_reflection) |
13314 | 880 |
done |
13306 | 881 |
|
882 |
||
13348 | 883 |
subsubsection{*Pre-Image under a Relation, Internalized*} |
884 |
||
13429 | 885 |
(* "pre_image(M,r,A,z) == |
886 |
\<forall>x[M]. x \<in> z <-> (\<exists>w[M]. w\<in>r & (\<exists>y[M]. y\<in>A & pair(M,x,y,w)))" *) |
|
13348 | 887 |
constdefs pre_image_fm :: "[i,i,i]=>i" |
13429 | 888 |
"pre_image_fm(r,A,z) == |
13348 | 889 |
Forall(Iff(Member(0,succ(z)), |
890 |
Exists(And(Member(0,succ(succ(r))), |
|
891 |
Exists(And(Member(0,succ(succ(succ(A)))), |
|
13429 | 892 |
pair_fm(2,0,1)))))))" |
13348 | 893 |
|
894 |
lemma pre_image_type [TC]: |
|
895 |
"[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> pre_image_fm(x,y,z) \<in> formula" |
|
13429 | 896 |
by (simp add: pre_image_fm_def) |
13348 | 897 |
|
898 |
lemma sats_pre_image_fm [simp]: |
|
899 |
"[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|] |
|
13429 | 900 |
==> sats(A, pre_image_fm(x,y,z), env) <-> |
13807
a28a8fbc76d4
changed ** to ## to avoid conflict with new comment syntax
paulson
parents:
13655
diff
changeset
|
901 |
pre_image(##A, nth(x,env), nth(y,env), nth(z,env))" |
13348 | 902 |
by (simp add: pre_image_fm_def Relative.pre_image_def) |
903 |
||
904 |
lemma pre_image_iff_sats: |
|
13429 | 905 |
"[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z; |
13348 | 906 |
i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|] |
13807
a28a8fbc76d4
changed ** to ## to avoid conflict with new comment syntax
paulson
parents:
13655
diff
changeset
|
907 |
==> pre_image(##A, x, y, z) <-> sats(A, pre_image_fm(i,j,k), env)" |
13348 | 908 |
by (simp add: sats_pre_image_fm) |
909 |
||
910 |
theorem pre_image_reflection: |
|
13429 | 911 |
"REFLECTS[\<lambda>x. pre_image(L,f(x),g(x),h(x)), |
13807
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changed ** to ## to avoid conflict with new comment syntax
paulson
parents:
13655
diff
changeset
|
912 |
\<lambda>i x. pre_image(##Lset(i),f(x),g(x),h(x))]" |
13655
95b95cdb4704
Tidying up. New primitives is_iterates and is_iterates_fm.
paulson
parents:
13651
diff
changeset
|
913 |
apply (simp only: Relative.pre_image_def) |
13429 | 914 |
apply (intro FOL_reflections pair_reflection) |
13348 | 915 |
done |
916 |
||
917 |
||
13352 | 918 |
subsubsection{*Function Application, Internalized*} |
919 |
||
13429 | 920 |
(* "fun_apply(M,f,x,y) == |
921 |
(\<exists>xs[M]. \<exists>fxs[M]. |
|
13352 | 922 |
upair(M,x,x,xs) & image(M,f,xs,fxs) & big_union(M,fxs,y))" *) |
923 |
constdefs fun_apply_fm :: "[i,i,i]=>i" |
|
13429 | 924 |
"fun_apply_fm(f,x,y) == |
13352 | 925 |
Exists(Exists(And(upair_fm(succ(succ(x)), succ(succ(x)), 1), |
13429 | 926 |
And(image_fm(succ(succ(f)), 1, 0), |
13352 | 927 |
big_union_fm(0,succ(succ(y)))))))" |
928 |
||
929 |
lemma fun_apply_type [TC]: |
|
930 |
"[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> fun_apply_fm(x,y,z) \<in> formula" |
|
13429 | 931 |
by (simp add: fun_apply_fm_def) |
13352 | 932 |
|
933 |
lemma sats_fun_apply_fm [simp]: |
|
934 |
"[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|] |
|
13429 | 935 |
==> sats(A, fun_apply_fm(x,y,z), env) <-> |
13807
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changed ** to ## to avoid conflict with new comment syntax
paulson
parents:
13655
diff
changeset
|
936 |
fun_apply(##A, nth(x,env), nth(y,env), nth(z,env))" |
13352 | 937 |
by (simp add: fun_apply_fm_def fun_apply_def) |
938 |
||
939 |
lemma fun_apply_iff_sats: |
|
13429 | 940 |
"[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z; |
13352 | 941 |
i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|] |
13807
a28a8fbc76d4
changed ** to ## to avoid conflict with new comment syntax
paulson
parents:
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diff
changeset
|
942 |
==> fun_apply(##A, x, y, z) <-> sats(A, fun_apply_fm(i,j,k), env)" |
13352 | 943 |
by simp |
944 |
||
945 |
theorem fun_apply_reflection: |
|
13429 | 946 |
"REFLECTS[\<lambda>x. fun_apply(L,f(x),g(x),h(x)), |
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|
947 |
\<lambda>i x. fun_apply(##Lset(i),f(x),g(x),h(x))]" |
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|
948 |
apply (simp only: fun_apply_def) |
13352 | 949 |
apply (intro FOL_reflections upair_reflection image_reflection |
13429 | 950 |
big_union_reflection) |
13352 | 951 |
done |
952 |
||
953 |
||
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|
954 |
subsubsection{*The Concept of Relation, Internalized*} |
13306 | 955 |
|
13429 | 956 |
(* "is_relation(M,r) == |
13306 | 957 |
(\<forall>z[M]. z\<in>r --> (\<exists>x[M]. \<exists>y[M]. pair(M,x,y,z)))" *) |
958 |
constdefs relation_fm :: "i=>i" |
|
13429 | 959 |
"relation_fm(r) == |
13306 | 960 |
Forall(Implies(Member(0,succ(r)), Exists(Exists(pair_fm(1,0,2)))))" |
961 |
||
962 |
lemma relation_type [TC]: |
|
963 |
"[| x \<in> nat |] ==> relation_fm(x) \<in> formula" |
|
13429 | 964 |
by (simp add: relation_fm_def) |
13306 | 965 |
|
966 |
lemma sats_relation_fm [simp]: |
|
967 |
"[| x \<in> nat; env \<in> list(A)|] |
|
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|
968 |
==> sats(A, relation_fm(x), env) <-> is_relation(##A, nth(x,env))" |
13306 | 969 |
by (simp add: relation_fm_def is_relation_def) |
970 |
||
971 |
lemma relation_iff_sats: |
|
13429 | 972 |
"[| nth(i,env) = x; nth(j,env) = y; |
13306 | 973 |
i \<in> nat; env \<in> list(A)|] |
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|
974 |
==> is_relation(##A, x) <-> sats(A, relation_fm(i), env)" |
13306 | 975 |
by simp |
976 |
||
13314 | 977 |
theorem is_relation_reflection: |
13429 | 978 |
"REFLECTS[\<lambda>x. is_relation(L,f(x)), |
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|
979 |
\<lambda>i x. is_relation(##Lset(i),f(x))]" |
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|
980 |
apply (simp only: is_relation_def) |
13429 | 981 |
apply (intro FOL_reflections pair_reflection) |
13314 | 982 |
done |
13306 | 983 |
|
984 |
||
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|
985 |
subsubsection{*The Concept of Function, Internalized*} |
13306 | 986 |
|
13429 | 987 |
(* "is_function(M,r) == |
988 |
\<forall>x[M]. \<forall>y[M]. \<forall>y'[M]. \<forall>p[M]. \<forall>p'[M]. |
|
13306 | 989 |
pair(M,x,y,p) --> pair(M,x,y',p') --> p\<in>r --> p'\<in>r --> y=y'" *) |
990 |
constdefs function_fm :: "i=>i" |
|
13429 | 991 |
"function_fm(r) == |
13306 | 992 |
Forall(Forall(Forall(Forall(Forall( |
993 |
Implies(pair_fm(4,3,1), |
|
994 |
Implies(pair_fm(4,2,0), |
|
995 |
Implies(Member(1,r#+5), |
|
996 |
Implies(Member(0,r#+5), Equal(3,2))))))))))" |
|
997 |
||
998 |
lemma function_type [TC]: |
|
999 |
"[| x \<in> nat |] ==> function_fm(x) \<in> formula" |
|
13429 | 1000 |
by (simp add: function_fm_def) |
13306 | 1001 |
|
1002 |
lemma sats_function_fm [simp]: |
|
1003 |
"[| x \<in> nat; env \<in> list(A)|] |
|
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|
1004 |
==> sats(A, function_fm(x), env) <-> is_function(##A, nth(x,env))" |
13306 | 1005 |
by (simp add: function_fm_def is_function_def) |
1006 |
||
13505 | 1007 |
lemma is_function_iff_sats: |
13429 | 1008 |
"[| nth(i,env) = x; nth(j,env) = y; |
13306 | 1009 |
i \<in> nat; env \<in> list(A)|] |
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|
1010 |
==> is_function(##A, x) <-> sats(A, function_fm(i), env)" |
13306 | 1011 |
by simp |
1012 |
||
13314 | 1013 |
theorem is_function_reflection: |
13429 | 1014 |
"REFLECTS[\<lambda>x. is_function(L,f(x)), |
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|
1015 |
\<lambda>i x. is_function(##Lset(i),f(x))]" |
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|
1016 |
apply (simp only: is_function_def) |
13429 | 1017 |
apply (intro FOL_reflections pair_reflection) |
13314 | 1018 |
done |
13298 | 1019 |
|
1020 |
||
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|
1021 |
subsubsection{*Typed Functions, Internalized*} |
13309 | 1022 |
|
13429 | 1023 |
(* "typed_function(M,A,B,r) == |
13309 | 1024 |
is_function(M,r) & is_relation(M,r) & is_domain(M,r,A) & |
1025 |
(\<forall>u[M]. u\<in>r --> (\<forall>x[M]. \<forall>y[M]. pair(M,x,y,u) --> y\<in>B))" *) |
|
1026 |
||
1027 |
constdefs typed_function_fm :: "[i,i,i]=>i" |
|
13429 | 1028 |
"typed_function_fm(A,B,r) == |
13309 | 1029 |
And(function_fm(r), |
1030 |
And(relation_fm(r), |
|
1031 |
And(domain_fm(r,A), |
|
1032 |
Forall(Implies(Member(0,succ(r)), |
|
1033 |
Forall(Forall(Implies(pair_fm(1,0,2),Member(0,B#+3)))))))))" |
|
1034 |
||
1035 |
lemma typed_function_type [TC]: |
|
1036 |
"[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> typed_function_fm(x,y,z) \<in> formula" |
|
13429 | 1037 |
by (simp add: typed_function_fm_def) |
13309 | 1038 |
|
1039 |
lemma sats_typed_function_fm [simp]: |
|
1040 |
"[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|] |
|
13429 | 1041 |
==> sats(A, typed_function_fm(x,y,z), env) <-> |
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|
1042 |
typed_function(##A, nth(x,env), nth(y,env), nth(z,env))" |
13309 | 1043 |
by (simp add: typed_function_fm_def typed_function_def) |
1044 |
||
1045 |
lemma typed_function_iff_sats: |
|
13429 | 1046 |
"[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z; |
13309 | 1047 |
i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|] |
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|
1048 |
==> typed_function(##A, x, y, z) <-> sats(A, typed_function_fm(i,j,k), env)" |
13309 | 1049 |
by simp |
1050 |
||
13429 | 1051 |
lemmas function_reflections = |
13363 | 1052 |
empty_reflection number1_reflection |
13429 | 1053 |
upair_reflection pair_reflection union_reflection |
1054 |
big_union_reflection cons_reflection successor_reflection |
|
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|
1055 |
fun_apply_reflection subset_reflection |
13429 | 1056 |
transitive_set_reflection membership_reflection |
1057 |
pred_set_reflection domain_reflection range_reflection field_reflection |
|
13348 | 1058 |
image_reflection pre_image_reflection |
13429 | 1059 |
is_relation_reflection is_function_reflection |
13309 | 1060 |
|
13429 | 1061 |
lemmas function_iff_sats = |
1062 |
empty_iff_sats number1_iff_sats |
|
1063 |
upair_iff_sats pair_iff_sats union_iff_sats |
|
13505 | 1064 |
big_union_iff_sats cons_iff_sats successor_iff_sats |
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|
1065 |
fun_apply_iff_sats Memrel_iff_sats |
13429 | 1066 |
pred_set_iff_sats domain_iff_sats range_iff_sats field_iff_sats |
1067 |
image_iff_sats pre_image_iff_sats |
|
13505 | 1068 |
relation_iff_sats is_function_iff_sats |
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|
1069 |
|
13309 | 1070 |
|
13314 | 1071 |
theorem typed_function_reflection: |
13429 | 1072 |
"REFLECTS[\<lambda>x. typed_function(L,f(x),g(x),h(x)), |
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|
1073 |
\<lambda>i x. typed_function(##Lset(i),f(x),g(x),h(x))]" |
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|
1074 |
apply (simp only: typed_function_def) |
13429 | 1075 |
apply (intro FOL_reflections function_reflections) |
13323
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|
1076 |
done |
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|
1077 |
|
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|
1078 |
|
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|
1079 |
subsubsection{*Composition of Relations, Internalized*} |
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|
1080 |
|
13429 | 1081 |
(* "composition(M,r,s,t) == |
1082 |
\<forall>p[M]. p \<in> t <-> |
|
1083 |
(\<exists>x[M]. \<exists>y[M]. \<exists>z[M]. \<exists>xy[M]. \<exists>yz[M]. |
|
1084 |
pair(M,x,z,p) & pair(M,x,y,xy) & pair(M,y,z,yz) & |
|
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|
1085 |
xy \<in> s & yz \<in> r)" *) |
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|
1086 |
constdefs composition_fm :: "[i,i,i]=>i" |
13429 | 1087 |
"composition_fm(r,s,t) == |
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|
1088 |
Forall(Iff(Member(0,succ(t)), |
13429 | 1089 |
Exists(Exists(Exists(Exists(Exists( |
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|
1090 |
And(pair_fm(4,2,5), |
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|
1091 |
And(pair_fm(4,3,1), |
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|
1092 |
And(pair_fm(3,2,0), |
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|
1093 |
And(Member(1,s#+6), Member(0,r#+6))))))))))))" |
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|
1094 |
|
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|
1095 |
lemma composition_type [TC]: |
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|
1096 |
"[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> composition_fm(x,y,z) \<in> formula" |
13429 | 1097 |
by (simp add: composition_fm_def) |
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|
1098 |
|
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|
1099 |
lemma sats_composition_fm [simp]: |
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|
1100 |
"[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|] |
13429 | 1101 |
==> sats(A, composition_fm(x,y,z), env) <-> |
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|
1102 |
composition(##A, nth(x,env), nth(y,env), nth(z,env))" |
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|
1103 |
by (simp add: composition_fm_def composition_def) |
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|
1104 |
|
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|
1105 |
lemma composition_iff_sats: |
13429 | 1106 |
"[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z; |
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|
1107 |
i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|] |
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|
1108 |
==> composition(##A, x, y, z) <-> sats(A, composition_fm(i,j,k), env)" |
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|
1109 |
by simp |
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|
1110 |
|
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|
1111 |
theorem composition_reflection: |
13429 | 1112 |
"REFLECTS[\<lambda>x. composition(L,f(x),g(x),h(x)), |
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|
1113 |
\<lambda>i x. composition(##Lset(i),f(x),g(x),h(x))]" |
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|
1114 |
apply (simp only: composition_def) |
13429 | 1115 |
apply (intro FOL_reflections pair_reflection) |
13314 | 1116 |
done |
1117 |
||
13309 | 1118 |
|
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|
1119 |
subsubsection{*Injections, Internalized*} |
13309 | 1120 |
|
13429 | 1121 |
(* "injection(M,A,B,f) == |
1122 |
typed_function(M,A,B,f) & |
|
1123 |
(\<forall>x[M]. \<forall>x'[M]. \<forall>y[M]. \<forall>p[M]. \<forall>p'[M]. |
|
13309 | 1124 |
pair(M,x,y,p) --> pair(M,x',y,p') --> p\<in>f --> p'\<in>f --> x=x')" *) |
1125 |
constdefs injection_fm :: "[i,i,i]=>i" |
|
13429 | 1126 |
"injection_fm(A,B,f) == |
13309 | 1127 |
And(typed_function_fm(A,B,f), |
1128 |
Forall(Forall(Forall(Forall(Forall( |
|
1129 |
Implies(pair_fm(4,2,1), |
|
1130 |
Implies(pair_fm(3,2,0), |
|
1131 |
Implies(Member(1,f#+5), |
|
1132 |
Implies(Member(0,f#+5), Equal(4,3)))))))))))" |
|
1133 |
||
1134 |
||
1135 |
lemma injection_type [TC]: |
|
1136 |
"[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> injection_fm(x,y,z) \<in> formula" |
|
13429 | 1137 |
by (simp add: injection_fm_def) |
13309 | 1138 |
|
1139 |
lemma sats_injection_fm [simp]: |
|
1140 |
"[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|] |
|
13429 | 1141 |
==> sats(A, injection_fm(x,y,z), env) <-> |
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|
1142 |
injection(##A, nth(x,env), nth(y,env), nth(z,env))" |
13309 | 1143 |
by (simp add: injection_fm_def injection_def) |
1144 |
||
1145 |
lemma injection_iff_sats: |
|
13429 | 1146 |
"[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z; |
13309 | 1147 |
i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|] |
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|
1148 |
==> injection(##A, x, y, z) <-> sats(A, injection_fm(i,j,k), env)" |
13309 | 1149 |
by simp |
1150 |
||
13314 | 1151 |
theorem injection_reflection: |
13429 | 1152 |
"REFLECTS[\<lambda>x. injection(L,f(x),g(x),h(x)), |
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|
1153 |
\<lambda>i x. injection(##Lset(i),f(x),g(x),h(x))]" |
13655
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changeset
|
1154 |
apply (simp only: injection_def) |
13429 | 1155 |
apply (intro FOL_reflections function_reflections typed_function_reflection) |
13314 | 1156 |
done |
13309 | 1157 |
|
1158 |
||
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|
1159 |
subsubsection{*Surjections, Internalized*} |
13309 | 1160 |
|
1161 |
(* surjection :: "[i=>o,i,i,i] => o" |
|
13429 | 1162 |
"surjection(M,A,B,f) == |
13309 | 1163 |
typed_function(M,A,B,f) & |
1164 |
(\<forall>y[M]. y\<in>B --> (\<exists>x[M]. x\<in>A & fun_apply(M,f,x,y)))" *) |
|
1165 |
constdefs surjection_fm :: "[i,i,i]=>i" |
|
13429 | 1166 |
"surjection_fm(A,B,f) == |
13309 | 1167 |
And(typed_function_fm(A,B,f), |
1168 |
Forall(Implies(Member(0,succ(B)), |
|
1169 |
Exists(And(Member(0,succ(succ(A))), |
|
1170 |
fun_apply_fm(succ(succ(f)),0,1))))))" |
|
1171 |
||
1172 |
lemma surjection_type [TC]: |
|
1173 |
"[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> surjection_fm(x,y,z) \<in> formula" |
|
13429 | 1174 |
by (simp add: surjection_fm_def) |
13309 | 1175 |
|
1176 |
lemma sats_surjection_fm [simp]: |
|
1177 |
"[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|] |
|
13429 | 1178 |
==> sats(A, surjection_fm(x,y,z), env) <-> |
13807
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parents:
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diff
changeset
|
1179 |
surjection(##A, nth(x,env), nth(y,env), nth(z,env))" |
13309 | 1180 |
by (simp add: surjection_fm_def surjection_def) |
1181 |
||
1182 |
lemma surjection_iff_sats: |
|
13429 | 1183 |
"[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z; |
13309 | 1184 |
i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|] |
13807
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parents:
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diff
changeset
|
1185 |
==> surjection(##A, x, y, z) <-> sats(A, surjection_fm(i,j,k), env)" |
13309 | 1186 |
by simp |
1187 |
||
13314 | 1188 |
theorem surjection_reflection: |
13429 | 1189 |
"REFLECTS[\<lambda>x. surjection(L,f(x),g(x),h(x)), |
13807
a28a8fbc76d4
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parents:
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diff
changeset
|
1190 |
\<lambda>i x. surjection(##Lset(i),f(x),g(x),h(x))]" |
13655
95b95cdb4704
Tidying up. New primitives is_iterates and is_iterates_fm.
paulson
parents:
13651
diff
changeset
|
1191 |
apply (simp only: surjection_def) |
13429 | 1192 |
apply (intro FOL_reflections function_reflections typed_function_reflection) |
13314 | 1193 |
done |
13309 | 1194 |
|
1195 |
||
1196 |
||
13339
0f89104dd377
Fixed quantified variable name preservation for ball and bex (bounded quants)
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parents:
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diff
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|
1197 |
subsubsection{*Bijections, Internalized*} |
13309 | 1198 |
|
1199 |
(* bijection :: "[i=>o,i,i,i] => o" |
|
1200 |
"bijection(M,A,B,f) == injection(M,A,B,f) & surjection(M,A,B,f)" *) |
|
1201 |
constdefs bijection_fm :: "[i,i,i]=>i" |
|
1202 |
"bijection_fm(A,B,f) == And(injection_fm(A,B,f), surjection_fm(A,B,f))" |
|
1203 |
||
1204 |
lemma bijection_type [TC]: |
|
1205 |
"[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> bijection_fm(x,y,z) \<in> formula" |
|
13429 | 1206 |
by (simp add: bijection_fm_def) |
13309 | 1207 |
|
1208 |
lemma sats_bijection_fm [simp]: |
|
1209 |
"[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|] |
|
13429 | 1210 |
==> sats(A, bijection_fm(x,y,z), env) <-> |
13807
a28a8fbc76d4
changed ** to ## to avoid conflict with new comment syntax
paulson
parents:
13655
diff
changeset
|
1211 |
bijection(##A, nth(x,env), nth(y,env), nth(z,env))" |
13309 | 1212 |
by (simp add: bijection_fm_def bijection_def) |
1213 |
||
1214 |
lemma bijection_iff_sats: |
|
13429 | 1215 |
"[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z; |
13309 | 1216 |
i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|] |
13807
a28a8fbc76d4
changed ** to ## to avoid conflict with new comment syntax
paulson
parents:
13655
diff
changeset
|
1217 |
==> bijection(##A, x, y, z) <-> sats(A, bijection_fm(i,j,k), env)" |
13309 | 1218 |
by simp |
1219 |
||
13314 | 1220 |
theorem bijection_reflection: |
13429 | 1221 |
"REFLECTS[\<lambda>x. bijection(L,f(x),g(x),h(x)), |
13807
a28a8fbc76d4
changed ** to ## to avoid conflict with new comment syntax
paulson
parents:
13655
diff
changeset
|
1222 |
\<lambda>i x. bijection(##Lset(i),f(x),g(x),h(x))]" |
13655
95b95cdb4704
Tidying up. New primitives is_iterates and is_iterates_fm.
paulson
parents:
13651
diff
changeset
|
1223 |
apply (simp only: bijection_def) |
13429 | 1224 |
apply (intro And_reflection injection_reflection surjection_reflection) |
13314 | 1225 |
done |
13309 | 1226 |
|
1227 |
||
13348 | 1228 |
subsubsection{*Restriction of a Relation, Internalized*} |
1229 |
||
1230 |
||
13429 | 1231 |
(* "restriction(M,r,A,z) == |
1232 |
\<forall>x[M]. x \<in> z <-> (x \<in> r & (\<exists>u[M]. u\<in>A & (\<exists>v[M]. pair(M,u,v,x))))" *) |
|
13348 | 1233 |
constdefs restriction_fm :: "[i,i,i]=>i" |
13429 | 1234 |
"restriction_fm(r,A,z) == |
13348 | 1235 |
Forall(Iff(Member(0,succ(z)), |
1236 |
And(Member(0,succ(r)), |
|
1237 |
Exists(And(Member(0,succ(succ(A))), |
|
1238 |
Exists(pair_fm(1,0,2)))))))" |
|
1239 |
||
1240 |
lemma restriction_type [TC]: |
|
1241 |
"[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> restriction_fm(x,y,z) \<in> formula" |
|
13429 | 1242 |
by (simp add: restriction_fm_def) |
13348 | 1243 |
|
1244 |
lemma sats_restriction_fm [simp]: |
|
1245 |
"[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|] |
|
13429 | 1246 |
==> sats(A, restriction_fm(x,y,z), env) <-> |
13807
a28a8fbc76d4
changed ** to ## to avoid conflict with new comment syntax
paulson
parents:
13655
diff
changeset
|
1247 |
restriction(##A, nth(x,env), nth(y,env), nth(z,env))" |
13348 | 1248 |
by (simp add: restriction_fm_def restriction_def) |
1249 |
||
1250 |
lemma restriction_iff_sats: |
|
13429 | 1251 |
"[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z; |
13348 | 1252 |
i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|] |
13807
a28a8fbc76d4
changed ** to ## to avoid conflict with new comment syntax
paulson
parents:
13655
diff
changeset
|
1253 |
==> restriction(##A, x, y, z) <-> sats(A, restriction_fm(i,j,k), env)" |
13348 | 1254 |
by simp |
1255 |
||
1256 |
theorem restriction_reflection: |
|
13429 | 1257 |
"REFLECTS[\<lambda>x. restriction(L,f(x),g(x),h(x)), |
13807
a28a8fbc76d4
changed ** to ## to avoid conflict with new comment syntax
paulson
parents:
13655
diff
changeset
|
1258 |
\<lambda>i x. restriction(##Lset(i),f(x),g(x),h(x))]" |
13655
95b95cdb4704
Tidying up. New primitives is_iterates and is_iterates_fm.
paulson
parents:
13651
diff
changeset
|
1259 |
apply (simp only: restriction_def) |
13429 | 1260 |
apply (intro FOL_reflections pair_reflection) |
13348 | 1261 |
done |
1262 |
||
13339
0f89104dd377
Fixed quantified variable name preservation for ball and bex (bounded quants)
paulson
parents:
13323
diff
changeset
|
1263 |
subsubsection{*Order-Isomorphisms, Internalized*} |
13309 | 1264 |
|
1265 |
(* order_isomorphism :: "[i=>o,i,i,i,i,i] => o" |
|
13429 | 1266 |
"order_isomorphism(M,A,r,B,s,f) == |
1267 |
bijection(M,A,B,f) & |
|
13309 | 1268 |
(\<forall>x[M]. x\<in>A --> (\<forall>y[M]. y\<in>A --> |
1269 |
(\<forall>p[M]. \<forall>fx[M]. \<forall>fy[M]. \<forall>q[M]. |
|
13429 | 1270 |
pair(M,x,y,p) --> fun_apply(M,f,x,fx) --> fun_apply(M,f,y,fy) --> |
13309 | 1271 |
pair(M,fx,fy,q) --> (p\<in>r <-> q\<in>s))))" |
1272 |
*) |
|
1273 |
||
1274 |
constdefs order_isomorphism_fm :: "[i,i,i,i,i]=>i" |
|
13429 | 1275 |
"order_isomorphism_fm(A,r,B,s,f) == |
1276 |
And(bijection_fm(A,B,f), |
|
13309 | 1277 |
Forall(Implies(Member(0,succ(A)), |
1278 |
Forall(Implies(Member(0,succ(succ(A))), |
|
1279 |
Forall(Forall(Forall(Forall( |
|
1280 |
Implies(pair_fm(5,4,3), |
|
1281 |
Implies(fun_apply_fm(f#+6,5,2), |
|
1282 |
Implies(fun_apply_fm(f#+6,4,1), |
|
13429 | 1283 |
Implies(pair_fm(2,1,0), |
13309 | 1284 |
Iff(Member(3,r#+6), Member(0,s#+6)))))))))))))))" |
1285 |
||
1286 |
lemma order_isomorphism_type [TC]: |
|
13429 | 1287 |
"[| A \<in> nat; r \<in> nat; B \<in> nat; s \<in> nat; f \<in> nat |] |
13309 | 1288 |
==> order_isomorphism_fm(A,r,B,s,f) \<in> formula" |
13429 | 1289 |
by (simp add: order_isomorphism_fm_def) |
13309 | 1290 |
|
1291 |
lemma sats_order_isomorphism_fm [simp]: |
|
1292 |
"[| U \<in> nat; r \<in> nat; B \<in> nat; s \<in> nat; f \<in> nat; env \<in> list(A)|] |
|
13429 | 1293 |
==> sats(A, order_isomorphism_fm(U,r,B,s,f), env) <-> |
13807
a28a8fbc76d4
changed ** to ## to avoid conflict with new comment syntax
paulson
parents:
13655
diff
changeset
|
1294 |
order_isomorphism(##A, nth(U,env), nth(r,env), nth(B,env), |
13309 | 1295 |
nth(s,env), nth(f,env))" |
1296 |
by (simp add: order_isomorphism_fm_def order_isomorphism_def) |
|
1297 |
||
1298 |
lemma order_isomorphism_iff_sats: |
|
13429 | 1299 |
"[| nth(i,env) = U; nth(j,env) = r; nth(k,env) = B; nth(j',env) = s; |
1300 |
nth(k',env) = f; |
|
13309 | 1301 |
i \<in> nat; j \<in> nat; k \<in> nat; j' \<in> nat; k' \<in> nat; env \<in> list(A)|] |
13807
a28a8fbc76d4
changed ** to ## to avoid conflict with new comment syntax
paulson
parents:
13655
diff
changeset
|
1302 |
==> order_isomorphism(##A,U,r,B,s,f) <-> |
13429 | 1303 |
sats(A, order_isomorphism_fm(i,j,k,j',k'), env)" |
13309 | 1304 |
by simp |
1305 |
||
13314 | 1306 |
theorem order_isomorphism_reflection: |
13429 | 1307 |
"REFLECTS[\<lambda>x. order_isomorphism(L,f(x),g(x),h(x),g'(x),h'(x)), |
13807
a28a8fbc76d4
changed ** to ## to avoid conflict with new comment syntax
paulson
parents:
13655
diff
changeset
|
1308 |
\<lambda>i x. order_isomorphism(##Lset(i),f(x),g(x),h(x),g'(x),h'(x))]" |
13655
95b95cdb4704
Tidying up. New primitives is_iterates and is_iterates_fm.
paulson
parents:
13651
diff
changeset
|
1309 |
apply (simp only: order_isomorphism_def) |
13429 | 1310 |
apply (intro FOL_reflections function_reflections bijection_reflection) |
13323
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
1311 |
done |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
1312 |
|
13339
0f89104dd377
Fixed quantified variable name preservation for ball and bex (bounded quants)
paulson
parents:
13323
diff
changeset
|
1313 |
subsubsection{*Limit Ordinals, Internalized*} |
13323
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
1314 |
|
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
1315 |
text{*A limit ordinal is a non-empty, successor-closed ordinal*} |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
1316 |
|
13429 | 1317 |
(* "limit_ordinal(M,a) == |
1318 |
ordinal(M,a) & ~ empty(M,a) & |
|
13323
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
1319 |
(\<forall>x[M]. x\<in>a --> (\<exists>y[M]. y\<in>a & successor(M,x,y)))" *) |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
1320 |
|
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
1321 |
constdefs limit_ordinal_fm :: "i=>i" |
13429 | 1322 |
"limit_ordinal_fm(x) == |
13323
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
1323 |
And(ordinal_fm(x), |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
1324 |
And(Neg(empty_fm(x)), |
13429 | 1325 |
Forall(Implies(Member(0,succ(x)), |
13323
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
1326 |
Exists(And(Member(0,succ(succ(x))), |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
1327 |
succ_fm(1,0)))))))" |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
1328 |
|
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
1329 |
lemma limit_ordinal_type [TC]: |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
1330 |
"x \<in> nat ==> limit_ordinal_fm(x) \<in> formula" |
13429 | 1331 |
by (simp add: limit_ordinal_fm_def) |
13323
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
1332 |
|
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
1333 |
lemma sats_limit_ordinal_fm [simp]: |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
1334 |
"[| x \<in> nat; env \<in> list(A)|] |
13807
a28a8fbc76d4
changed ** to ## to avoid conflict with new comment syntax
paulson
parents:
13655
diff
changeset
|
1335 |
==> sats(A, limit_ordinal_fm(x), env) <-> limit_ordinal(##A, nth(x,env))" |
13323
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
1336 |
by (simp add: limit_ordinal_fm_def limit_ordinal_def sats_ordinal_fm') |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
1337 |
|
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
1338 |
lemma limit_ordinal_iff_sats: |
13429 | 1339 |
"[| nth(i,env) = x; nth(j,env) = y; |
13323
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
1340 |
i \<in> nat; env \<in> list(A)|] |
13807
a28a8fbc76d4
changed ** to ## to avoid conflict with new comment syntax
paulson
parents:
13655
diff
changeset
|
1341 |
==> limit_ordinal(##A, x) <-> sats(A, limit_ordinal_fm(i), env)" |
13323
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
1342 |
by simp |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
1343 |
|
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
1344 |
theorem limit_ordinal_reflection: |
13429 | 1345 |
"REFLECTS[\<lambda>x. limit_ordinal(L,f(x)), |
13807
a28a8fbc76d4
changed ** to ## to avoid conflict with new comment syntax
paulson
parents:
13655
diff
changeset
|
1346 |
\<lambda>i x. limit_ordinal(##Lset(i),f(x))]" |
13655
95b95cdb4704
Tidying up. New primitives is_iterates and is_iterates_fm.
paulson
parents:
13651
diff
changeset
|
1347 |
apply (simp only: limit_ordinal_def) |
13429 | 1348 |
apply (intro FOL_reflections ordinal_reflection |
1349 |
empty_reflection successor_reflection) |
|
13314 | 1350 |
done |
13309 | 1351 |
|
13493
5aa68c051725
Lots of new results concerning recursive datatypes, towards absoluteness of
paulson
parents:
13440
diff
changeset
|
1352 |
subsubsection{*Finite Ordinals: The Predicate ``Is A Natural Number''*} |
5aa68c051725
Lots of new results concerning recursive datatypes, towards absoluteness of
paulson
parents:
13440
diff
changeset
|
1353 |
|
5aa68c051725
Lots of new results concerning recursive datatypes, towards absoluteness of
paulson
parents:
13440
diff
changeset
|
1354 |
(* "finite_ordinal(M,a) == |
5aa68c051725
Lots of new results concerning recursive datatypes, towards absoluteness of
paulson
parents:
13440
diff
changeset
|
1355 |
ordinal(M,a) & ~ limit_ordinal(M,a) & |
5aa68c051725
Lots of new results concerning recursive datatypes, towards absoluteness of
paulson
parents:
13440
diff
changeset
|
1356 |
(\<forall>x[M]. x\<in>a --> ~ limit_ordinal(M,x))" *) |
5aa68c051725
Lots of new results concerning recursive datatypes, towards absoluteness of
paulson
parents:
13440
diff
changeset
|
1357 |
constdefs finite_ordinal_fm :: "i=>i" |
5aa68c051725
Lots of new results concerning recursive datatypes, towards absoluteness of
paulson
parents:
13440
diff
changeset
|
1358 |
"finite_ordinal_fm(x) == |
5aa68c051725
Lots of new results concerning recursive datatypes, towards absoluteness of
paulson
parents:
13440
diff
changeset
|
1359 |
And(ordinal_fm(x), |
5aa68c051725
Lots of new results concerning recursive datatypes, towards absoluteness of
paulson
parents:
13440
diff
changeset
|
1360 |
And(Neg(limit_ordinal_fm(x)), |
5aa68c051725
Lots of new results concerning recursive datatypes, towards absoluteness of
paulson
parents:
13440
diff
changeset
|
1361 |
Forall(Implies(Member(0,succ(x)), |
5aa68c051725
Lots of new results concerning recursive datatypes, towards absoluteness of
paulson
parents:
13440
diff
changeset
|
1362 |
Neg(limit_ordinal_fm(0))))))" |
5aa68c051725
Lots of new results concerning recursive datatypes, towards absoluteness of
paulson
parents:
13440
diff
changeset
|
1363 |
|
5aa68c051725
Lots of new results concerning recursive datatypes, towards absoluteness of
paulson
parents:
13440
diff
changeset
|
1364 |
lemma finite_ordinal_type [TC]: |
5aa68c051725
Lots of new results concerning recursive datatypes, towards absoluteness of
paulson
parents:
13440
diff
changeset
|
1365 |
"x \<in> nat ==> finite_ordinal_fm(x) \<in> formula" |
5aa68c051725
Lots of new results concerning recursive datatypes, towards absoluteness of
paulson
parents:
13440
diff
changeset
|
1366 |
by (simp add: finite_ordinal_fm_def) |
5aa68c051725
Lots of new results concerning recursive datatypes, towards absoluteness of
paulson
parents:
13440
diff
changeset
|
1367 |
|
5aa68c051725
Lots of new results concerning recursive datatypes, towards absoluteness of
paulson
parents:
13440
diff
changeset
|
1368 |
lemma sats_finite_ordinal_fm [simp]: |
5aa68c051725
Lots of new results concerning recursive datatypes, towards absoluteness of
paulson
parents:
13440
diff
changeset
|
1369 |
"[| x \<in> nat; env \<in> list(A)|] |
13807
a28a8fbc76d4
changed ** to ## to avoid conflict with new comment syntax
paulson
parents:
13655
diff
changeset
|
1370 |
==> sats(A, finite_ordinal_fm(x), env) <-> finite_ordinal(##A, nth(x,env))" |
13493
5aa68c051725
Lots of new results concerning recursive datatypes, towards absoluteness of
paulson
parents:
13440
diff
changeset
|
1371 |
by (simp add: finite_ordinal_fm_def sats_ordinal_fm' finite_ordinal_def) |
5aa68c051725
Lots of new results concerning recursive datatypes, towards absoluteness of
paulson
parents:
13440
diff
changeset
|
1372 |
|
5aa68c051725
Lots of new results concerning recursive datatypes, towards absoluteness of
paulson
parents:
13440
diff
changeset
|
1373 |
lemma finite_ordinal_iff_sats: |
5aa68c051725
Lots of new results concerning recursive datatypes, towards absoluteness of
paulson
parents:
13440
diff
changeset
|
1374 |
"[| nth(i,env) = x; nth(j,env) = y; |
5aa68c051725
Lots of new results concerning recursive datatypes, towards absoluteness of
paulson
parents:
13440
diff
changeset
|
1375 |
i \<in> nat; env \<in> list(A)|] |
13807
a28a8fbc76d4
changed ** to ## to avoid conflict with new comment syntax
paulson
parents:
13655
diff
changeset
|
1376 |
==> finite_ordinal(##A, x) <-> sats(A, finite_ordinal_fm(i), env)" |
13493
5aa68c051725
Lots of new results concerning recursive datatypes, towards absoluteness of
paulson
parents:
13440
diff
changeset
|
1377 |
by simp |
5aa68c051725
Lots of new results concerning recursive datatypes, towards absoluteness of
paulson
parents:
13440
diff
changeset
|
1378 |
|
5aa68c051725
Lots of new results concerning recursive datatypes, towards absoluteness of
paulson
parents:
13440
diff
changeset
|
1379 |
theorem finite_ordinal_reflection: |
5aa68c051725
Lots of new results concerning recursive datatypes, towards absoluteness of
paulson
parents:
13440
diff
changeset
|
1380 |
"REFLECTS[\<lambda>x. finite_ordinal(L,f(x)), |
13807
a28a8fbc76d4
changed ** to ## to avoid conflict with new comment syntax
paulson
parents:
13655
diff
changeset
|
1381 |
\<lambda>i x. finite_ordinal(##Lset(i),f(x))]" |
13655
95b95cdb4704
Tidying up. New primitives is_iterates and is_iterates_fm.
paulson
parents:
13651
diff
changeset
|
1382 |
apply (simp only: finite_ordinal_def) |
13493
5aa68c051725
Lots of new results concerning recursive datatypes, towards absoluteness of
paulson
parents:
13440
diff
changeset
|
1383 |
apply (intro FOL_reflections ordinal_reflection limit_ordinal_reflection) |
5aa68c051725
Lots of new results concerning recursive datatypes, towards absoluteness of
paulson
parents:
13440
diff
changeset
|
1384 |
done |
5aa68c051725
Lots of new results concerning recursive datatypes, towards absoluteness of
paulson
parents:
13440
diff
changeset
|
1385 |
|
5aa68c051725
Lots of new results concerning recursive datatypes, towards absoluteness of
paulson
parents:
13440
diff
changeset
|
1386 |
|
13323
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
1387 |
subsubsection{*Omega: The Set of Natural Numbers*} |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
1388 |
|
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
1389 |
(* omega(M,a) == limit_ordinal(M,a) & (\<forall>x[M]. x\<in>a --> ~ limit_ordinal(M,x)) *) |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
1390 |
constdefs omega_fm :: "i=>i" |
13429 | 1391 |
"omega_fm(x) == |
13323
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
1392 |
And(limit_ordinal_fm(x), |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
1393 |
Forall(Implies(Member(0,succ(x)), |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
1394 |
Neg(limit_ordinal_fm(0)))))" |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
1395 |
|
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
1396 |
lemma omega_type [TC]: |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
1397 |
"x \<in> nat ==> omega_fm(x) \<in> formula" |
13429 | 1398 |
by (simp add: omega_fm_def) |
13323
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
1399 |
|
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
1400 |
lemma sats_omega_fm [simp]: |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
1401 |
"[| x \<in> nat; env \<in> list(A)|] |
13807
a28a8fbc76d4
changed ** to ## to avoid conflict with new comment syntax
paulson
parents:
13655
diff
changeset
|
1402 |
==> sats(A, omega_fm(x), env) <-> omega(##A, nth(x,env))" |
13323
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
1403 |
by (simp add: omega_fm_def omega_def) |
13316 | 1404 |
|
13323
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
1405 |
lemma omega_iff_sats: |
13429 | 1406 |
"[| nth(i,env) = x; nth(j,env) = y; |
13323
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
1407 |
i \<in> nat; env \<in> list(A)|] |
13807
a28a8fbc76d4
changed ** to ## to avoid conflict with new comment syntax
paulson
parents:
13655
diff
changeset
|
1408 |
==> omega(##A, x) <-> sats(A, omega_fm(i), env)" |
13323
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
1409 |
by simp |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
1410 |
|
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
1411 |
theorem omega_reflection: |
13429 | 1412 |
"REFLECTS[\<lambda>x. omega(L,f(x)), |
13807
a28a8fbc76d4
changed ** to ## to avoid conflict with new comment syntax
paulson
parents:
13655
diff
changeset
|
1413 |
\<lambda>i x. omega(##Lset(i),f(x))]" |
13655
95b95cdb4704
Tidying up. New primitives is_iterates and is_iterates_fm.
paulson
parents:
13651
diff
changeset
|
1414 |
apply (simp only: omega_def) |
13429 | 1415 |
apply (intro FOL_reflections limit_ordinal_reflection) |
13323
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
1416 |
done |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
1417 |
|
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
1418 |
|
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
1419 |
lemmas fun_plus_reflections = |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
1420 |
typed_function_reflection composition_reflection |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
1421 |
injection_reflection surjection_reflection |
13348 | 1422 |
bijection_reflection restriction_reflection |
13496
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
13493
diff
changeset
|
1423 |
order_isomorphism_reflection finite_ordinal_reflection |
13323
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
1424 |
ordinal_reflection limit_ordinal_reflection omega_reflection |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
1425 |
|
13429 | 1426 |
lemmas fun_plus_iff_sats = |
1427 |
typed_function_iff_sats composition_iff_sats |
|
1428 |
injection_iff_sats surjection_iff_sats |
|
1429 |
bijection_iff_sats restriction_iff_sats |
|
13496
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
13493
diff
changeset
|
1430 |
order_isomorphism_iff_sats finite_ordinal_iff_sats |
13323
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
1431 |
ordinal_iff_sats limit_ordinal_iff_sats omega_iff_sats |
13316 | 1432 |
|
13223 | 1433 |
end |