author | paulson |
Wed, 24 Jul 2002 17:59:12 +0200 | |
changeset 13418 | 7c0ba9dba978 |
parent 13385 | 31df66ca0780 |
child 13428 | 99e52e78eb65 |
permissions | -rw-r--r-- |
13339
0f89104dd377
Fixed quantified variable name preservation for ball and bex (bounded quants)
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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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Fixed quantified variable name preservation for ball and bex (bounded quants)
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text {* The class L satisfies the premises of locale @{text M_triv_axioms} *} |
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lemma transL: "[| y\<in>x; L(x) |] ==> L(y)" |
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apply (insert Transset_Lset) |
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apply (simp add: Transset_def L_def, blast) |
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done |
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lemma nonempty: "L(0)" |
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apply (simp add: L_def) |
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apply (blast intro: zero_in_Lset) |
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done |
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lemma upair_ax: "upair_ax(L)" |
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apply (simp add: upair_ax_def upair_def, clarify) |
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apply (rule_tac x="{x,y}" in rexI) |
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apply (simp_all add: doubleton_in_L) |
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done |
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lemma Union_ax: "Union_ax(L)" |
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apply (simp add: Union_ax_def big_union_def, clarify) |
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apply (rule_tac x="Union(x)" in rexI) |
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apply (simp_all add: Union_in_L, auto) |
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apply (blast intro: transL) |
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done |
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lemma power_ax: "power_ax(L)" |
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apply (simp add: power_ax_def powerset_def Relative.subset_def, clarify) |
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apply (rule_tac x="{y \<in> Pow(x). L(y)}" in rexI) |
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apply (simp_all add: LPow_in_L, auto) |
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apply (blast intro: transL) |
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done |
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subsubsection{*For L to satisfy Replacement *} |
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(*Can't move these to Formula unless the definition of univalent is moved |
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there too!*) |
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lemma LReplace_in_Lset: |
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"[|X \<in> Lset(i); univalent(L,X,Q); Ord(i)|] |
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==> \<exists>j. Ord(j) & Replace(X, %x y. Q(x,y) & L(y)) \<subseteq> Lset(j)" |
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apply (rule_tac x="\<Union>y \<in> Replace(X, %x y. Q(x,y) & L(y)). succ(lrank(y))" |
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in exI) |
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apply simp |
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apply clarify |
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apply (rule_tac a=x in UN_I) |
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apply (simp_all add: Replace_iff univalent_def) |
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apply (blast dest: transL L_I) |
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done |
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lemma LReplace_in_L: |
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"[|L(X); univalent(L,X,Q)|] |
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==> \<exists>Y. L(Y) & Replace(X, %x y. Q(x,y) & L(y)) \<subseteq> Y" |
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apply (drule L_D, clarify) |
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apply (drule LReplace_in_Lset, assumption+) |
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apply (blast intro: L_I Lset_in_Lset_succ) |
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done |
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lemma replacement: "replacement(L,P)" |
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apply (simp add: replacement_def, clarify) |
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apply (frule LReplace_in_L, assumption+, clarify) |
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apply (rule_tac x=Y in rexI) |
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apply (simp_all add: Replace_iff univalent_def, blast) |
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done |
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subsection{*Instantiating the locale @{text M_triv_axioms}*} |
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text{*No instances of Separation yet.*} |
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lemma Lset_mono_le: "mono_le_subset(Lset)" |
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by (simp add: mono_le_subset_def le_imp_subset Lset_mono) |
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lemma Lset_cont: "cont_Ord(Lset)" |
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by (simp add: cont_Ord_def Limit_Lset_eq OUnion_def Limit_is_Ord) |
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lemmas Pair_in_Lset = Formula.Pair_in_LLimit; |
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lemmas L_nat = Ord_in_L [OF Ord_nat]; |
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ML |
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{* |
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val transL = thm "transL"; |
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val nonempty = thm "nonempty"; |
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val upair_ax = thm "upair_ax"; |
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val Union_ax = thm "Union_ax"; |
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val power_ax = thm "power_ax"; |
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val replacement = thm "replacement"; |
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val L_nat = thm "L_nat"; |
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fun kill_flex_triv_prems st = Seq.hd ((REPEAT_FIRST assume_tac) st); |
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fun triv_axioms_L th = |
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kill_flex_triv_prems |
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([transL, nonempty, upair_ax, Union_ax, power_ax, replacement, L_nat] |
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MRS (inst "M" "L" th)); |
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bind_thm ("rall_abs", triv_axioms_L (thm "M_triv_axioms.rall_abs")); |
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bind_thm ("rex_abs", triv_axioms_L (thm "M_triv_axioms.rex_abs")); |
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bind_thm ("ball_iff_equiv", triv_axioms_L (thm "M_triv_axioms.ball_iff_equiv")); |
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bind_thm ("M_equalityI", triv_axioms_L (thm "M_triv_axioms.M_equalityI")); |
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bind_thm ("empty_abs", triv_axioms_L (thm "M_triv_axioms.empty_abs")); |
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bind_thm ("subset_abs", triv_axioms_L (thm "M_triv_axioms.subset_abs")); |
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bind_thm ("upair_abs", triv_axioms_L (thm "M_triv_axioms.upair_abs")); |
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bind_thm ("upair_in_M_iff", triv_axioms_L (thm "M_triv_axioms.upair_in_M_iff")); |
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bind_thm ("singleton_in_M_iff", triv_axioms_L (thm "M_triv_axioms.singleton_in_M_iff")); |
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bind_thm ("pair_abs", triv_axioms_L (thm "M_triv_axioms.pair_abs")); |
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bind_thm ("pair_in_M_iff", triv_axioms_L (thm "M_triv_axioms.pair_in_M_iff")); |
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bind_thm ("pair_components_in_M", triv_axioms_L (thm "M_triv_axioms.pair_components_in_M")); |
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bind_thm ("cartprod_abs", triv_axioms_L (thm "M_triv_axioms.cartprod_abs")); |
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bind_thm ("union_abs", triv_axioms_L (thm "M_triv_axioms.union_abs")); |
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bind_thm ("inter_abs", triv_axioms_L (thm "M_triv_axioms.inter_abs")); |
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bind_thm ("setdiff_abs", triv_axioms_L (thm "M_triv_axioms.setdiff_abs")); |
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bind_thm ("Union_abs", triv_axioms_L (thm "M_triv_axioms.Union_abs")); |
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bind_thm ("Union_closed", triv_axioms_L (thm "M_triv_axioms.Union_closed")); |
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bind_thm ("Un_closed", triv_axioms_L (thm "M_triv_axioms.Un_closed")); |
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bind_thm ("cons_closed", triv_axioms_L (thm "M_triv_axioms.cons_closed")); |
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bind_thm ("successor_abs", triv_axioms_L (thm "M_triv_axioms.successor_abs")); |
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bind_thm ("succ_in_M_iff", triv_axioms_L (thm "M_triv_axioms.succ_in_M_iff")); |
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bind_thm ("separation_closed", triv_axioms_L (thm "M_triv_axioms.separation_closed")); |
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bind_thm ("strong_replacementI", triv_axioms_L (thm "M_triv_axioms.strong_replacementI")); |
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bind_thm ("strong_replacement_closed", triv_axioms_L (thm "M_triv_axioms.strong_replacement_closed")); |
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bind_thm ("RepFun_closed", triv_axioms_L (thm "M_triv_axioms.RepFun_closed")); |
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bind_thm ("lam_closed", triv_axioms_L (thm "M_triv_axioms.lam_closed")); |
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bind_thm ("image_abs", triv_axioms_L (thm "M_triv_axioms.image_abs")); |
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bind_thm ("powerset_Pow", triv_axioms_L (thm "M_triv_axioms.powerset_Pow")); |
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bind_thm ("powerset_imp_subset_Pow", triv_axioms_L (thm "M_triv_axioms.powerset_imp_subset_Pow")); |
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bind_thm ("nat_into_M", triv_axioms_L (thm "M_triv_axioms.nat_into_M")); |
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bind_thm ("nat_case_closed", triv_axioms_L (thm "M_triv_axioms.nat_case_closed")); |
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bind_thm ("Inl_in_M_iff", triv_axioms_L (thm "M_triv_axioms.Inl_in_M_iff")); |
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bind_thm ("Inr_in_M_iff", triv_axioms_L (thm "M_triv_axioms.Inr_in_M_iff")); |
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bind_thm ("lt_closed", triv_axioms_L (thm "M_triv_axioms.lt_closed")); |
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bind_thm ("transitive_set_abs", triv_axioms_L (thm "M_triv_axioms.transitive_set_abs")); |
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bind_thm ("ordinal_abs", triv_axioms_L (thm "M_triv_axioms.ordinal_abs")); |
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bind_thm ("limit_ordinal_abs", triv_axioms_L (thm "M_triv_axioms.limit_ordinal_abs")); |
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bind_thm ("successor_ordinal_abs", triv_axioms_L (thm "M_triv_axioms.successor_ordinal_abs")); |
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bind_thm ("finite_ordinal_abs", triv_axioms_L (thm "M_triv_axioms.finite_ordinal_abs")); |
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bind_thm ("omega_abs", triv_axioms_L (thm "M_triv_axioms.omega_abs")); |
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bind_thm ("number1_abs", triv_axioms_L (thm "M_triv_axioms.number1_abs")); |
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bind_thm ("number1_abs", triv_axioms_L (thm "M_triv_axioms.number1_abs")); |
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bind_thm ("number3_abs", triv_axioms_L (thm "M_triv_axioms.number3_abs")); |
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*} |
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declare rall_abs [simp] |
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declare rex_abs [simp] |
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declare empty_abs [simp] |
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declare subset_abs [simp] |
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declare upair_abs [simp] |
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declare upair_in_M_iff [iff] |
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declare singleton_in_M_iff [iff] |
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declare pair_abs [simp] |
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declare pair_in_M_iff [iff] |
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declare cartprod_abs [simp] |
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declare union_abs [simp] |
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declare inter_abs [simp] |
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declare setdiff_abs [simp] |
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declare Union_abs [simp] |
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declare Union_closed [intro,simp] |
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declare Un_closed [intro,simp] |
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declare cons_closed [intro,simp] |
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declare successor_abs [simp] |
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declare succ_in_M_iff [iff] |
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declare separation_closed [intro,simp] |
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declare strong_replacementI |
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declare strong_replacement_closed [intro,simp] |
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declare RepFun_closed [intro,simp] |
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declare lam_closed [intro,simp] |
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declare image_abs [simp] |
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declare nat_into_M [intro] |
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declare Inl_in_M_iff [iff] |
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declare Inr_in_M_iff [iff] |
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declare transitive_set_abs [simp] |
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declare ordinal_abs [simp] |
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declare limit_ordinal_abs [simp] |
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declare successor_ordinal_abs [simp] |
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declare finite_ordinal_abs [simp] |
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declare omega_abs [simp] |
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declare number1_abs [simp] |
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declare number1_abs [simp] |
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declare number3_abs [simp] |
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subsection{*Instantiation of the locale @{text reflection}*} |
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text{*instances of locale constants*} |
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constdefs |
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L_F0 :: "[i=>o,i] => i" |
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"L_F0(P,y) == \<mu>b. (\<exists>z. L(z) \<and> P(<y,z>)) --> (\<exists>z\<in>Lset(b). P(<y,z>))" |
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L_FF :: "[i=>o,i] => i" |
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"L_FF(P) == \<lambda>a. \<Union>y\<in>Lset(a). L_F0(P,y)" |
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L_ClEx :: "[i=>o,i] => o" |
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"L_ClEx(P) == \<lambda>a. Limit(a) \<and> normalize(L_FF(P),a) = a" |
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text{*We must use the meta-existential quantifier; otherwise the reflection |
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terms become enormous!*} |
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constdefs |
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L_Reflects :: "[i=>o,[i,i]=>o] => prop" ("(3REFLECTS/ [_,/ _])") |
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"REFLECTS[P,Q] == (??Cl. Closed_Unbounded(Cl) & |
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(\<forall>a. Cl(a) --> (\<forall>x \<in> Lset(a). P(x) <-> Q(a,x))))" |
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theorem Triv_reflection: |
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"REFLECTS[P, \<lambda>a x. P(x)]" |
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apply (simp add: L_Reflects_def) |
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apply (rule meta_exI) |
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apply (rule Closed_Unbounded_Ord) |
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done |
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theorem Not_reflection: |
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"REFLECTS[P,Q] ==> REFLECTS[\<lambda>x. ~P(x), \<lambda>a x. ~Q(a,x)]" |
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apply (unfold L_Reflects_def) |
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apply (erule meta_exE) |
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apply (rule_tac x=Cl in meta_exI, simp) |
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done |
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theorem And_reflection: |
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"[| REFLECTS[P,Q]; REFLECTS[P',Q'] |] |
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==> REFLECTS[\<lambda>x. P(x) \<and> P'(x), \<lambda>a x. Q(a,x) \<and> Q'(a,x)]" |
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apply (unfold L_Reflects_def) |
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apply (elim meta_exE) |
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apply (rule_tac x="\<lambda>a. Cl(a) \<and> Cla(a)" in meta_exI) |
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apply (simp add: Closed_Unbounded_Int, blast) |
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done |
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theorem Or_reflection: |
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"[| REFLECTS[P,Q]; REFLECTS[P',Q'] |] |
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==> REFLECTS[\<lambda>x. P(x) \<or> P'(x), \<lambda>a x. Q(a,x) \<or> Q'(a,x)]" |
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apply (unfold L_Reflects_def) |
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apply (elim meta_exE) |
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apply (rule_tac x="\<lambda>a. Cl(a) \<and> Cla(a)" in meta_exI) |
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apply (simp add: Closed_Unbounded_Int, blast) |
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done |
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theorem Imp_reflection: |
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"[| REFLECTS[P,Q]; REFLECTS[P',Q'] |] |
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==> REFLECTS[\<lambda>x. P(x) --> P'(x), \<lambda>a x. Q(a,x) --> Q'(a,x)]" |
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apply (unfold L_Reflects_def) |
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apply (elim meta_exE) |
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apply (rule_tac x="\<lambda>a. Cl(a) \<and> Cla(a)" in meta_exI) |
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apply (simp add: Closed_Unbounded_Int, blast) |
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done |
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theorem Iff_reflection: |
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"[| REFLECTS[P,Q]; REFLECTS[P',Q'] |] |
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==> REFLECTS[\<lambda>x. P(x) <-> P'(x), \<lambda>a x. Q(a,x) <-> Q'(a,x)]" |
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apply (unfold L_Reflects_def) |
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apply (elim meta_exE) |
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apply (rule_tac x="\<lambda>a. Cl(a) \<and> Cla(a)" in meta_exI) |
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apply (simp add: Closed_Unbounded_Int, blast) |
|
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done |
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theorem Ex_reflection: |
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"REFLECTS[\<lambda>x. P(fst(x),snd(x)), \<lambda>a x. Q(a,fst(x),snd(x))] |
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==> REFLECTS[\<lambda>x. \<exists>z. L(z) \<and> P(x,z), \<lambda>a x. \<exists>z\<in>Lset(a). Q(a,x,z)]" |
|
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apply (unfold L_Reflects_def L_ClEx_def L_FF_def L_F0_def L_def) |
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apply (elim meta_exE) |
262 |
apply (rule meta_exI) |
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apply (rule reflection.Ex_reflection [OF Lset_mono_le Lset_cont Pair_in_Lset], |
264 |
assumption+) |
|
265 |
done |
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||
13314 | 267 |
theorem All_reflection: |
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"REFLECTS[\<lambda>x. P(fst(x),snd(x)), \<lambda>a x. Q(a,fst(x),snd(x))] |
|
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==> REFLECTS[\<lambda>x. \<forall>z. L(z) --> P(x,z), \<lambda>a x. \<forall>z\<in>Lset(a). Q(a,x,z)]" |
|
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apply (unfold L_Reflects_def L_ClEx_def L_FF_def L_F0_def L_def) |
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apply (elim meta_exE) |
272 |
apply (rule meta_exI) |
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apply (rule reflection.All_reflection [OF Lset_mono_le Lset_cont Pair_in_Lset], |
274 |
assumption+) |
|
275 |
done |
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||
13314 | 277 |
theorem Rex_reflection: |
278 |
"REFLECTS[ \<lambda>x. P(fst(x),snd(x)), \<lambda>a x. Q(a,fst(x),snd(x))] |
|
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==> REFLECTS[\<lambda>x. \<exists>z[L]. P(x,z), \<lambda>a x. \<exists>z\<in>Lset(a). Q(a,x,z)]" |
|
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apply (unfold rex_def) |
|
281 |
apply (intro And_reflection Ex_reflection, assumption) |
|
282 |
done |
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|
13314 | 284 |
theorem Rall_reflection: |
285 |
"REFLECTS[\<lambda>x. P(fst(x),snd(x)), \<lambda>a x. Q(a,fst(x),snd(x))] |
|
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==> REFLECTS[\<lambda>x. \<forall>z[L]. P(x,z), \<lambda>a x. \<forall>z\<in>Lset(a). Q(a,x,z)]" |
|
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apply (unfold rall_def) |
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apply (intro Imp_reflection All_reflection, assumption) |
|
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done |
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More relativization, reflection and proofs of separation
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291 |
lemmas FOL_reflections = |
13314 | 292 |
Triv_reflection Not_reflection And_reflection Or_reflection |
293 |
Imp_reflection Iff_reflection Ex_reflection All_reflection |
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Rex_reflection Rall_reflection |
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13291 | 295 |
|
296 |
lemma ReflectsD: |
|
13314 | 297 |
"[|REFLECTS[P,Q]; Ord(i)|] |
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==> \<exists>j. i<j & (\<forall>x \<in> Lset(j). P(x) <-> Q(j,x))" |
13314 | 299 |
apply (unfold L_Reflects_def Closed_Unbounded_def) |
300 |
apply (elim meta_exE, clarify) |
|
13291 | 301 |
apply (blast dest!: UnboundedD) |
302 |
done |
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303 |
||
304 |
lemma ReflectsE: |
|
13314 | 305 |
"[| REFLECTS[P,Q]; Ord(i); |
13291 | 306 |
!!j. [|i<j; \<forall>x \<in> Lset(j). P(x) <-> Q(j,x)|] ==> R |] |
307 |
==> R" |
|
13316 | 308 |
apply (drule ReflectsD, assumption, blast) |
13314 | 309 |
done |
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|
311 |
lemma Collect_mem_eq: "{x\<in>A. x\<in>B} = A \<inter> B"; |
|
312 |
by blast |
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13339
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Fixed quantified variable name preservation for ball and bex (bounded quants)
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|
315 |
subsection{*Internalized Formulas for some Set-Theoretic Concepts*} |
13298 | 316 |
|
13306 | 317 |
lemmas setclass_simps = rall_setclass_is_ball rex_setclass_is_bex |
318 |
||
319 |
subsubsection{*Some numbers to help write de Bruijn indices*} |
|
320 |
||
321 |
syntax |
|
322 |
"3" :: i ("3") |
|
323 |
"4" :: i ("4") |
|
324 |
"5" :: i ("5") |
|
325 |
"6" :: i ("6") |
|
326 |
"7" :: i ("7") |
|
327 |
"8" :: i ("8") |
|
328 |
"9" :: i ("9") |
|
329 |
||
330 |
translations |
|
331 |
"3" == "succ(2)" |
|
332 |
"4" == "succ(3)" |
|
333 |
"5" == "succ(4)" |
|
334 |
"6" == "succ(5)" |
|
335 |
"7" == "succ(6)" |
|
336 |
"8" == "succ(7)" |
|
337 |
"9" == "succ(8)" |
|
338 |
||
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339 |
|
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340 |
subsubsection{*The Empty Set, Internalized*} |
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|
341 |
|
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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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|
344 |
|
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345 |
lemma empty_type [TC]: |
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346 |
"x \<in> nat ==> empty_fm(x) \<in> formula" |
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347 |
by (simp add: empty_fm_def) |
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|
348 |
|
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349 |
lemma arity_empty_fm [simp]: |
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350 |
"x \<in> nat ==> arity(empty_fm(x)) = succ(x)" |
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by (simp add: empty_fm_def succ_Un_distrib [symmetric] Un_ac) |
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|
352 |
|
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|
353 |
lemma sats_empty_fm [simp]: |
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|
354 |
"[| x \<in> nat; env \<in> list(A)|] |
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355 |
==> sats(A, empty_fm(x), env) <-> empty(**A, nth(x,env))" |
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|
356 |
by (simp add: empty_fm_def empty_def) |
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|
357 |
|
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|
358 |
lemma empty_iff_sats: |
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|
359 |
"[| nth(i,env) = x; nth(j,env) = y; |
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i \<in> nat; env \<in> list(A)|] |
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|
361 |
==> empty(**A, x) <-> sats(A, empty_fm(i), env)" |
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|
362 |
by simp |
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|
363 |
|
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|
364 |
theorem empty_reflection: |
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|
365 |
"REFLECTS[\<lambda>x. empty(L,f(x)), |
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|
366 |
\<lambda>i x. empty(**Lset(i),f(x))]" |
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|
367 |
apply (simp only: empty_def setclass_simps) |
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|
368 |
apply (intro FOL_reflections) |
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|
369 |
done |
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|
370 |
|
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|
371 |
text{*Not used. But maybe useful?*} |
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|
372 |
lemma Transset_sats_empty_fm_eq_0: |
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|
373 |
"[| n \<in> nat; env \<in> list(A); Transset(A)|] |
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|
374 |
==> sats(A, empty_fm(n), env) <-> nth(n,env) = 0" |
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|
375 |
apply (simp add: empty_fm_def empty_def Transset_def, auto) |
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|
376 |
apply (case_tac "n < length(env)") |
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|
377 |
apply (frule nth_type, assumption+, blast) |
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|
378 |
apply (simp_all add: not_lt_iff_le nth_eq_0) |
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|
379 |
done |
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|
380 |
|
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381 |
|
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|
382 |
subsubsection{*Unordered Pairs, Internalized*} |
13298 | 383 |
|
384 |
constdefs upair_fm :: "[i,i,i]=>i" |
|
385 |
"upair_fm(x,y,z) == |
|
386 |
And(Member(x,z), |
|
387 |
And(Member(y,z), |
|
388 |
Forall(Implies(Member(0,succ(z)), |
|
389 |
Or(Equal(0,succ(x)), Equal(0,succ(y)))))))" |
|
390 |
||
391 |
lemma upair_type [TC]: |
|
392 |
"[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> upair_fm(x,y,z) \<in> formula" |
|
393 |
by (simp add: upair_fm_def) |
|
394 |
||
395 |
lemma arity_upair_fm [simp]: |
|
396 |
"[| x \<in> nat; y \<in> nat; z \<in> nat |] |
|
397 |
==> arity(upair_fm(x,y,z)) = succ(x) \<union> succ(y) \<union> succ(z)" |
|
398 |
by (simp add: upair_fm_def succ_Un_distrib [symmetric] Un_ac) |
|
399 |
||
400 |
lemma sats_upair_fm [simp]: |
|
401 |
"[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|] |
|
402 |
==> sats(A, upair_fm(x,y,z), env) <-> |
|
403 |
upair(**A, nth(x,env), nth(y,env), nth(z,env))" |
|
404 |
by (simp add: upair_fm_def upair_def) |
|
405 |
||
406 |
lemma upair_iff_sats: |
|
407 |
"[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z; |
|
408 |
i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|] |
|
409 |
==> upair(**A, x, y, z) <-> sats(A, upair_fm(i,j,k), env)" |
|
410 |
by (simp add: sats_upair_fm) |
|
411 |
||
412 |
text{*Useful? At least it refers to "real" unordered pairs*} |
|
413 |
lemma sats_upair_fm2 [simp]: |
|
414 |
"[| x \<in> nat; y \<in> nat; z < length(env); env \<in> list(A); Transset(A)|] |
|
415 |
==> sats(A, upair_fm(x,y,z), env) <-> |
|
416 |
nth(z,env) = {nth(x,env), nth(y,env)}" |
|
417 |
apply (frule lt_length_in_nat, assumption) |
|
418 |
apply (simp add: upair_fm_def Transset_def, auto) |
|
419 |
apply (blast intro: nth_type) |
|
420 |
done |
|
421 |
||
13314 | 422 |
theorem upair_reflection: |
423 |
"REFLECTS[\<lambda>x. upair(L,f(x),g(x),h(x)), |
|
424 |
\<lambda>i x. upair(**Lset(i),f(x),g(x),h(x))]" |
|
425 |
apply (simp add: upair_def) |
|
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426 |
apply (intro FOL_reflections) |
13314 | 427 |
done |
13306 | 428 |
|
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|
429 |
subsubsection{*Ordered pairs, Internalized*} |
13298 | 430 |
|
431 |
constdefs pair_fm :: "[i,i,i]=>i" |
|
432 |
"pair_fm(x,y,z) == |
|
433 |
Exists(And(upair_fm(succ(x),succ(x),0), |
|
434 |
Exists(And(upair_fm(succ(succ(x)),succ(succ(y)),0), |
|
435 |
upair_fm(1,0,succ(succ(z)))))))" |
|
436 |
||
437 |
lemma pair_type [TC]: |
|
438 |
"[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> pair_fm(x,y,z) \<in> formula" |
|
439 |
by (simp add: pair_fm_def) |
|
440 |
||
441 |
lemma arity_pair_fm [simp]: |
|
442 |
"[| x \<in> nat; y \<in> nat; z \<in> nat |] |
|
443 |
==> arity(pair_fm(x,y,z)) = succ(x) \<union> succ(y) \<union> succ(z)" |
|
444 |
by (simp add: pair_fm_def succ_Un_distrib [symmetric] Un_ac) |
|
445 |
||
446 |
lemma sats_pair_fm [simp]: |
|
447 |
"[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|] |
|
448 |
==> sats(A, pair_fm(x,y,z), env) <-> |
|
449 |
pair(**A, nth(x,env), nth(y,env), nth(z,env))" |
|
450 |
by (simp add: pair_fm_def pair_def) |
|
451 |
||
452 |
lemma pair_iff_sats: |
|
453 |
"[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z; |
|
454 |
i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|] |
|
455 |
==> pair(**A, x, y, z) <-> sats(A, pair_fm(i,j,k), env)" |
|
456 |
by (simp add: sats_pair_fm) |
|
457 |
||
13314 | 458 |
theorem pair_reflection: |
459 |
"REFLECTS[\<lambda>x. pair(L,f(x),g(x),h(x)), |
|
460 |
\<lambda>i x. pair(**Lset(i),f(x),g(x),h(x))]" |
|
461 |
apply (simp only: pair_def setclass_simps) |
|
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462 |
apply (intro FOL_reflections upair_reflection) |
13314 | 463 |
done |
13306 | 464 |
|
465 |
||
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|
466 |
subsubsection{*Binary Unions, Internalized*} |
13298 | 467 |
|
13306 | 468 |
constdefs union_fm :: "[i,i,i]=>i" |
469 |
"union_fm(x,y,z) == |
|
470 |
Forall(Iff(Member(0,succ(z)), |
|
471 |
Or(Member(0,succ(x)),Member(0,succ(y)))))" |
|
472 |
||
473 |
lemma union_type [TC]: |
|
474 |
"[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> union_fm(x,y,z) \<in> formula" |
|
475 |
by (simp add: union_fm_def) |
|
476 |
||
477 |
lemma arity_union_fm [simp]: |
|
478 |
"[| x \<in> nat; y \<in> nat; z \<in> nat |] |
|
479 |
==> arity(union_fm(x,y,z)) = succ(x) \<union> succ(y) \<union> succ(z)" |
|
480 |
by (simp add: union_fm_def succ_Un_distrib [symmetric] Un_ac) |
|
13298 | 481 |
|
13306 | 482 |
lemma sats_union_fm [simp]: |
483 |
"[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|] |
|
484 |
==> sats(A, union_fm(x,y,z), env) <-> |
|
485 |
union(**A, nth(x,env), nth(y,env), nth(z,env))" |
|
486 |
by (simp add: union_fm_def union_def) |
|
487 |
||
488 |
lemma union_iff_sats: |
|
489 |
"[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z; |
|
490 |
i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|] |
|
491 |
==> union(**A, x, y, z) <-> sats(A, union_fm(i,j,k), env)" |
|
492 |
by (simp add: sats_union_fm) |
|
13298 | 493 |
|
13314 | 494 |
theorem union_reflection: |
495 |
"REFLECTS[\<lambda>x. union(L,f(x),g(x),h(x)), |
|
496 |
\<lambda>i x. union(**Lset(i),f(x),g(x),h(x))]" |
|
497 |
apply (simp only: union_def setclass_simps) |
|
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|
498 |
apply (intro FOL_reflections) |
13314 | 499 |
done |
13306 | 500 |
|
13298 | 501 |
|
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|
502 |
subsubsection{*Set ``Cons,'' Internalized*} |
13306 | 503 |
|
504 |
constdefs cons_fm :: "[i,i,i]=>i" |
|
505 |
"cons_fm(x,y,z) == |
|
506 |
Exists(And(upair_fm(succ(x),succ(x),0), |
|
507 |
union_fm(0,succ(y),succ(z))))" |
|
13298 | 508 |
|
509 |
||
13306 | 510 |
lemma cons_type [TC]: |
511 |
"[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> cons_fm(x,y,z) \<in> formula" |
|
512 |
by (simp add: cons_fm_def) |
|
513 |
||
514 |
lemma arity_cons_fm [simp]: |
|
515 |
"[| x \<in> nat; y \<in> nat; z \<in> nat |] |
|
516 |
==> arity(cons_fm(x,y,z)) = succ(x) \<union> succ(y) \<union> succ(z)" |
|
517 |
by (simp add: cons_fm_def succ_Un_distrib [symmetric] Un_ac) |
|
518 |
||
519 |
lemma sats_cons_fm [simp]: |
|
520 |
"[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|] |
|
521 |
==> sats(A, cons_fm(x,y,z), env) <-> |
|
522 |
is_cons(**A, nth(x,env), nth(y,env), nth(z,env))" |
|
523 |
by (simp add: cons_fm_def is_cons_def) |
|
524 |
||
525 |
lemma cons_iff_sats: |
|
526 |
"[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z; |
|
527 |
i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|] |
|
528 |
==> is_cons(**A, x, y, z) <-> sats(A, cons_fm(i,j,k), env)" |
|
529 |
by simp |
|
530 |
||
13314 | 531 |
theorem cons_reflection: |
532 |
"REFLECTS[\<lambda>x. is_cons(L,f(x),g(x),h(x)), |
|
533 |
\<lambda>i x. is_cons(**Lset(i),f(x),g(x),h(x))]" |
|
534 |
apply (simp only: is_cons_def setclass_simps) |
|
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|
535 |
apply (intro FOL_reflections upair_reflection union_reflection) |
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|
536 |
done |
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|
537 |
|
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|
538 |
|
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|
539 |
subsubsection{*Successor Function, Internalized*} |
13323
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|
540 |
|
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|
541 |
constdefs succ_fm :: "[i,i]=>i" |
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|
542 |
"succ_fm(x,y) == cons_fm(x,x,y)" |
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|
543 |
|
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|
544 |
lemma succ_type [TC]: |
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|
545 |
"[| x \<in> nat; y \<in> nat |] ==> succ_fm(x,y) \<in> formula" |
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|
546 |
by (simp add: succ_fm_def) |
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|
547 |
|
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|
548 |
lemma arity_succ_fm [simp]: |
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|
549 |
"[| x \<in> nat; y \<in> nat |] |
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|
550 |
==> arity(succ_fm(x,y)) = succ(x) \<union> succ(y)" |
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|
551 |
by (simp add: succ_fm_def) |
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|
552 |
|
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|
553 |
lemma sats_succ_fm [simp]: |
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|
554 |
"[| x \<in> nat; y \<in> nat; env \<in> list(A)|] |
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|
555 |
==> sats(A, succ_fm(x,y), env) <-> |
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|
556 |
successor(**A, nth(x,env), nth(y,env))" |
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changeset
|
557 |
by (simp add: succ_fm_def successor_def) |
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|
558 |
|
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|
559 |
lemma successor_iff_sats: |
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|
560 |
"[| nth(i,env) = x; nth(j,env) = y; |
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|
561 |
i \<in> nat; j \<in> nat; env \<in> list(A)|] |
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|
562 |
==> successor(**A, x, y) <-> sats(A, succ_fm(i,j), env)" |
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|
563 |
by simp |
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changeset
|
564 |
|
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|
565 |
theorem successor_reflection: |
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changeset
|
566 |
"REFLECTS[\<lambda>x. successor(L,f(x),g(x)), |
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|
567 |
\<lambda>i x. successor(**Lset(i),f(x),g(x))]" |
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changeset
|
568 |
apply (simp only: successor_def setclass_simps) |
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|
569 |
apply (intro cons_reflection) |
13314 | 570 |
done |
13298 | 571 |
|
572 |
||
13363 | 573 |
subsubsection{*The Number 1, Internalized*} |
574 |
||
575 |
(* "number1(M,a) == (\<exists>x[M]. empty(M,x) & successor(M,x,a))" *) |
|
576 |
constdefs number1_fm :: "i=>i" |
|
577 |
"number1_fm(a) == Exists(And(empty_fm(0), succ_fm(0,succ(a))))" |
|
578 |
||
579 |
lemma number1_type [TC]: |
|
580 |
"x \<in> nat ==> number1_fm(x) \<in> formula" |
|
581 |
by (simp add: number1_fm_def) |
|
582 |
||
583 |
lemma arity_number1_fm [simp]: |
|
584 |
"x \<in> nat ==> arity(number1_fm(x)) = succ(x)" |
|
585 |
by (simp add: number1_fm_def succ_Un_distrib [symmetric] Un_ac) |
|
586 |
||
587 |
lemma sats_number1_fm [simp]: |
|
588 |
"[| x \<in> nat; env \<in> list(A)|] |
|
589 |
==> sats(A, number1_fm(x), env) <-> number1(**A, nth(x,env))" |
|
590 |
by (simp add: number1_fm_def number1_def) |
|
591 |
||
592 |
lemma number1_iff_sats: |
|
593 |
"[| nth(i,env) = x; nth(j,env) = y; |
|
594 |
i \<in> nat; env \<in> list(A)|] |
|
595 |
==> number1(**A, x) <-> sats(A, number1_fm(i), env)" |
|
596 |
by simp |
|
597 |
||
598 |
theorem number1_reflection: |
|
599 |
"REFLECTS[\<lambda>x. number1(L,f(x)), |
|
600 |
\<lambda>i x. number1(**Lset(i),f(x))]" |
|
601 |
apply (simp only: number1_def setclass_simps) |
|
602 |
apply (intro FOL_reflections empty_reflection successor_reflection) |
|
603 |
done |
|
604 |
||
605 |
||
13352 | 606 |
subsubsection{*Big Union, Internalized*} |
13306 | 607 |
|
13352 | 608 |
(* "big_union(M,A,z) == \<forall>x[M]. x \<in> z <-> (\<exists>y[M]. y\<in>A & x \<in> y)" *) |
609 |
constdefs big_union_fm :: "[i,i]=>i" |
|
610 |
"big_union_fm(A,z) == |
|
611 |
Forall(Iff(Member(0,succ(z)), |
|
612 |
Exists(And(Member(0,succ(succ(A))), Member(1,0)))))" |
|
13298 | 613 |
|
13352 | 614 |
lemma big_union_type [TC]: |
615 |
"[| x \<in> nat; y \<in> nat |] ==> big_union_fm(x,y) \<in> formula" |
|
616 |
by (simp add: big_union_fm_def) |
|
13306 | 617 |
|
13352 | 618 |
lemma arity_big_union_fm [simp]: |
619 |
"[| x \<in> nat; y \<in> nat |] |
|
620 |
==> arity(big_union_fm(x,y)) = succ(x) \<union> succ(y)" |
|
621 |
by (simp add: big_union_fm_def succ_Un_distrib [symmetric] Un_ac) |
|
13298 | 622 |
|
13352 | 623 |
lemma sats_big_union_fm [simp]: |
624 |
"[| x \<in> nat; y \<in> nat; env \<in> list(A)|] |
|
625 |
==> sats(A, big_union_fm(x,y), env) <-> |
|
626 |
big_union(**A, nth(x,env), nth(y,env))" |
|
627 |
by (simp add: big_union_fm_def big_union_def) |
|
13306 | 628 |
|
13352 | 629 |
lemma big_union_iff_sats: |
630 |
"[| nth(i,env) = x; nth(j,env) = y; |
|
631 |
i \<in> nat; j \<in> nat; env \<in> list(A)|] |
|
632 |
==> big_union(**A, x, y) <-> sats(A, big_union_fm(i,j), env)" |
|
13306 | 633 |
by simp |
634 |
||
13352 | 635 |
theorem big_union_reflection: |
636 |
"REFLECTS[\<lambda>x. big_union(L,f(x),g(x)), |
|
637 |
\<lambda>i x. big_union(**Lset(i),f(x),g(x))]" |
|
638 |
apply (simp only: big_union_def setclass_simps) |
|
639 |
apply (intro FOL_reflections) |
|
13314 | 640 |
done |
13298 | 641 |
|
642 |
||
13306 | 643 |
subsubsection{*Variants of Satisfaction Definitions for Ordinals, etc.*} |
644 |
||
645 |
text{*Differs from the one in Formula by using "ordinal" rather than "Ord"*} |
|
646 |
||
647 |
||
648 |
lemma sats_subset_fm': |
|
649 |
"[|x \<in> nat; y \<in> nat; env \<in> list(A)|] |
|
650 |
==> sats(A, subset_fm(x,y), env) <-> subset(**A, nth(x,env), nth(y,env))" |
|
13323
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|
651 |
by (simp add: subset_fm_def Relative.subset_def) |
13298 | 652 |
|
13314 | 653 |
theorem subset_reflection: |
654 |
"REFLECTS[\<lambda>x. subset(L,f(x),g(x)), |
|
655 |
\<lambda>i x. subset(**Lset(i),f(x),g(x))]" |
|
13323
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More relativization, reflection and proofs of separation
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|
656 |
apply (simp only: Relative.subset_def setclass_simps) |
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More relativization, reflection and proofs of separation
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diff
changeset
|
657 |
apply (intro FOL_reflections) |
13314 | 658 |
done |
13306 | 659 |
|
660 |
lemma sats_transset_fm': |
|
661 |
"[|x \<in> nat; env \<in> list(A)|] |
|
662 |
==> sats(A, transset_fm(x), env) <-> transitive_set(**A, nth(x,env))" |
|
663 |
by (simp add: sats_subset_fm' transset_fm_def transitive_set_def) |
|
13298 | 664 |
|
13314 | 665 |
theorem transitive_set_reflection: |
666 |
"REFLECTS[\<lambda>x. transitive_set(L,f(x)), |
|
667 |
\<lambda>i x. transitive_set(**Lset(i),f(x))]" |
|
668 |
apply (simp only: transitive_set_def setclass_simps) |
|
13323
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More relativization, reflection and proofs of separation
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changeset
|
669 |
apply (intro FOL_reflections subset_reflection) |
13314 | 670 |
done |
13306 | 671 |
|
672 |
lemma sats_ordinal_fm': |
|
673 |
"[|x \<in> nat; env \<in> list(A)|] |
|
674 |
==> sats(A, ordinal_fm(x), env) <-> ordinal(**A,nth(x,env))" |
|
675 |
by (simp add: sats_transset_fm' ordinal_fm_def ordinal_def) |
|
676 |
||
677 |
lemma ordinal_iff_sats: |
|
678 |
"[| nth(i,env) = x; i \<in> nat; env \<in> list(A)|] |
|
679 |
==> ordinal(**A, x) <-> sats(A, ordinal_fm(i), env)" |
|
680 |
by (simp add: sats_ordinal_fm') |
|
681 |
||
13314 | 682 |
theorem ordinal_reflection: |
683 |
"REFLECTS[\<lambda>x. ordinal(L,f(x)), \<lambda>i x. ordinal(**Lset(i),f(x))]" |
|
684 |
apply (simp only: ordinal_def setclass_simps) |
|
13323
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More relativization, reflection and proofs of separation
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|
685 |
apply (intro FOL_reflections transitive_set_reflection) |
13314 | 686 |
done |
13298 | 687 |
|
688 |
||
13339
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Fixed quantified variable name preservation for ball and bex (bounded quants)
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13323
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changeset
|
689 |
subsubsection{*Membership Relation, Internalized*} |
13298 | 690 |
|
13306 | 691 |
constdefs Memrel_fm :: "[i,i]=>i" |
692 |
"Memrel_fm(A,r) == |
|
693 |
Forall(Iff(Member(0,succ(r)), |
|
694 |
Exists(And(Member(0,succ(succ(A))), |
|
695 |
Exists(And(Member(0,succ(succ(succ(A)))), |
|
696 |
And(Member(1,0), |
|
697 |
pair_fm(1,0,2))))))))" |
|
698 |
||
699 |
lemma Memrel_type [TC]: |
|
700 |
"[| x \<in> nat; y \<in> nat |] ==> Memrel_fm(x,y) \<in> formula" |
|
701 |
by (simp add: Memrel_fm_def) |
|
13298 | 702 |
|
13306 | 703 |
lemma arity_Memrel_fm [simp]: |
704 |
"[| x \<in> nat; y \<in> nat |] |
|
705 |
==> arity(Memrel_fm(x,y)) = succ(x) \<union> succ(y)" |
|
706 |
by (simp add: Memrel_fm_def succ_Un_distrib [symmetric] Un_ac) |
|
707 |
||
708 |
lemma sats_Memrel_fm [simp]: |
|
709 |
"[| x \<in> nat; y \<in> nat; env \<in> list(A)|] |
|
710 |
==> sats(A, Memrel_fm(x,y), env) <-> |
|
711 |
membership(**A, nth(x,env), nth(y,env))" |
|
712 |
by (simp add: Memrel_fm_def membership_def) |
|
13298 | 713 |
|
13306 | 714 |
lemma Memrel_iff_sats: |
715 |
"[| nth(i,env) = x; nth(j,env) = y; |
|
716 |
i \<in> nat; j \<in> nat; env \<in> list(A)|] |
|
717 |
==> membership(**A, x, y) <-> sats(A, Memrel_fm(i,j), env)" |
|
718 |
by simp |
|
13304 | 719 |
|
13314 | 720 |
theorem membership_reflection: |
721 |
"REFLECTS[\<lambda>x. membership(L,f(x),g(x)), |
|
722 |
\<lambda>i x. membership(**Lset(i),f(x),g(x))]" |
|
723 |
apply (simp only: membership_def setclass_simps) |
|
13323
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
724 |
apply (intro FOL_reflections pair_reflection) |
13314 | 725 |
done |
13304 | 726 |
|
13339
0f89104dd377
Fixed quantified variable name preservation for ball and bex (bounded quants)
paulson
parents:
13323
diff
changeset
|
727 |
subsubsection{*Predecessor Set, Internalized*} |
13304 | 728 |
|
13306 | 729 |
constdefs pred_set_fm :: "[i,i,i,i]=>i" |
730 |
"pred_set_fm(A,x,r,B) == |
|
731 |
Forall(Iff(Member(0,succ(B)), |
|
732 |
Exists(And(Member(0,succ(succ(r))), |
|
733 |
And(Member(1,succ(succ(A))), |
|
734 |
pair_fm(1,succ(succ(x)),0))))))" |
|
735 |
||
736 |
||
737 |
lemma pred_set_type [TC]: |
|
738 |
"[| A \<in> nat; x \<in> nat; r \<in> nat; B \<in> nat |] |
|
739 |
==> pred_set_fm(A,x,r,B) \<in> formula" |
|
740 |
by (simp add: pred_set_fm_def) |
|
13304 | 741 |
|
13306 | 742 |
lemma arity_pred_set_fm [simp]: |
743 |
"[| A \<in> nat; x \<in> nat; r \<in> nat; B \<in> nat |] |
|
744 |
==> arity(pred_set_fm(A,x,r,B)) = succ(A) \<union> succ(x) \<union> succ(r) \<union> succ(B)" |
|
745 |
by (simp add: pred_set_fm_def succ_Un_distrib [symmetric] Un_ac) |
|
746 |
||
747 |
lemma sats_pred_set_fm [simp]: |
|
748 |
"[| U \<in> nat; x \<in> nat; r \<in> nat; B \<in> nat; env \<in> list(A)|] |
|
749 |
==> sats(A, pred_set_fm(U,x,r,B), env) <-> |
|
750 |
pred_set(**A, nth(U,env), nth(x,env), nth(r,env), nth(B,env))" |
|
751 |
by (simp add: pred_set_fm_def pred_set_def) |
|
752 |
||
753 |
lemma pred_set_iff_sats: |
|
754 |
"[| nth(i,env) = U; nth(j,env) = x; nth(k,env) = r; nth(l,env) = B; |
|
755 |
i \<in> nat; j \<in> nat; k \<in> nat; l \<in> nat; env \<in> list(A)|] |
|
756 |
==> pred_set(**A,U,x,r,B) <-> sats(A, pred_set_fm(i,j,k,l), env)" |
|
757 |
by (simp add: sats_pred_set_fm) |
|
758 |
||
13314 | 759 |
theorem pred_set_reflection: |
760 |
"REFLECTS[\<lambda>x. pred_set(L,f(x),g(x),h(x),b(x)), |
|
761 |
\<lambda>i x. pred_set(**Lset(i),f(x),g(x),h(x),b(x))]" |
|
762 |
apply (simp only: pred_set_def setclass_simps) |
|
13323
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
763 |
apply (intro FOL_reflections pair_reflection) |
13314 | 764 |
done |
13304 | 765 |
|
766 |
||
13298 | 767 |
|
13339
0f89104dd377
Fixed quantified variable name preservation for ball and bex (bounded quants)
paulson
parents:
13323
diff
changeset
|
768 |
subsubsection{*Domain of a Relation, Internalized*} |
13306 | 769 |
|
770 |
(* "is_domain(M,r,z) == |
|
771 |
\<forall>x[M]. (x \<in> z <-> (\<exists>w[M]. w\<in>r & (\<exists>y[M]. pair(M,x,y,w))))" *) |
|
772 |
constdefs domain_fm :: "[i,i]=>i" |
|
773 |
"domain_fm(r,z) == |
|
774 |
Forall(Iff(Member(0,succ(z)), |
|
775 |
Exists(And(Member(0,succ(succ(r))), |
|
776 |
Exists(pair_fm(2,0,1))))))" |
|
777 |
||
778 |
lemma domain_type [TC]: |
|
779 |
"[| x \<in> nat; y \<in> nat |] ==> domain_fm(x,y) \<in> formula" |
|
780 |
by (simp add: domain_fm_def) |
|
781 |
||
782 |
lemma arity_domain_fm [simp]: |
|
783 |
"[| x \<in> nat; y \<in> nat |] |
|
784 |
==> arity(domain_fm(x,y)) = succ(x) \<union> succ(y)" |
|
785 |
by (simp add: domain_fm_def succ_Un_distrib [symmetric] Un_ac) |
|
786 |
||
787 |
lemma sats_domain_fm [simp]: |
|
788 |
"[| x \<in> nat; y \<in> nat; env \<in> list(A)|] |
|
789 |
==> sats(A, domain_fm(x,y), env) <-> |
|
790 |
is_domain(**A, nth(x,env), nth(y,env))" |
|
791 |
by (simp add: domain_fm_def is_domain_def) |
|
792 |
||
793 |
lemma domain_iff_sats: |
|
794 |
"[| nth(i,env) = x; nth(j,env) = y; |
|
795 |
i \<in> nat; j \<in> nat; env \<in> list(A)|] |
|
796 |
==> is_domain(**A, x, y) <-> sats(A, domain_fm(i,j), env)" |
|
797 |
by simp |
|
798 |
||
13314 | 799 |
theorem domain_reflection: |
800 |
"REFLECTS[\<lambda>x. is_domain(L,f(x),g(x)), |
|
801 |
\<lambda>i x. is_domain(**Lset(i),f(x),g(x))]" |
|
802 |
apply (simp only: is_domain_def setclass_simps) |
|
13323
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
803 |
apply (intro FOL_reflections pair_reflection) |
13314 | 804 |
done |
13306 | 805 |
|
806 |
||
13339
0f89104dd377
Fixed quantified variable name preservation for ball and bex (bounded quants)
paulson
parents:
13323
diff
changeset
|
807 |
subsubsection{*Range of a Relation, Internalized*} |
13306 | 808 |
|
809 |
(* "is_range(M,r,z) == |
|
810 |
\<forall>y[M]. (y \<in> z <-> (\<exists>w[M]. w\<in>r & (\<exists>x[M]. pair(M,x,y,w))))" *) |
|
811 |
constdefs range_fm :: "[i,i]=>i" |
|
812 |
"range_fm(r,z) == |
|
813 |
Forall(Iff(Member(0,succ(z)), |
|
814 |
Exists(And(Member(0,succ(succ(r))), |
|
815 |
Exists(pair_fm(0,2,1))))))" |
|
816 |
||
817 |
lemma range_type [TC]: |
|
818 |
"[| x \<in> nat; y \<in> nat |] ==> range_fm(x,y) \<in> formula" |
|
819 |
by (simp add: range_fm_def) |
|
820 |
||
821 |
lemma arity_range_fm [simp]: |
|
822 |
"[| x \<in> nat; y \<in> nat |] |
|
823 |
==> arity(range_fm(x,y)) = succ(x) \<union> succ(y)" |
|
824 |
by (simp add: range_fm_def succ_Un_distrib [symmetric] Un_ac) |
|
825 |
||
826 |
lemma sats_range_fm [simp]: |
|
827 |
"[| x \<in> nat; y \<in> nat; env \<in> list(A)|] |
|
828 |
==> sats(A, range_fm(x,y), env) <-> |
|
829 |
is_range(**A, nth(x,env), nth(y,env))" |
|
830 |
by (simp add: range_fm_def is_range_def) |
|
831 |
||
832 |
lemma range_iff_sats: |
|
833 |
"[| nth(i,env) = x; nth(j,env) = y; |
|
834 |
i \<in> nat; j \<in> nat; env \<in> list(A)|] |
|
835 |
==> is_range(**A, x, y) <-> sats(A, range_fm(i,j), env)" |
|
836 |
by simp |
|
837 |
||
13314 | 838 |
theorem range_reflection: |
839 |
"REFLECTS[\<lambda>x. is_range(L,f(x),g(x)), |
|
840 |
\<lambda>i x. is_range(**Lset(i),f(x),g(x))]" |
|
841 |
apply (simp only: is_range_def setclass_simps) |
|
13323
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
842 |
apply (intro FOL_reflections pair_reflection) |
13314 | 843 |
done |
13306 | 844 |
|
845 |
||
13339
0f89104dd377
Fixed quantified variable name preservation for ball and bex (bounded quants)
paulson
parents:
13323
diff
changeset
|
846 |
subsubsection{*Field of a Relation, Internalized*} |
13323
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
847 |
|
2c287f50c9f3
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parents:
13316
diff
changeset
|
848 |
(* "is_field(M,r,z) == |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
849 |
\<exists>dr[M]. is_domain(M,r,dr) & |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
850 |
(\<exists>rr[M]. is_range(M,r,rr) & union(M,dr,rr,z))" *) |
2c287f50c9f3
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paulson
parents:
13316
diff
changeset
|
851 |
constdefs field_fm :: "[i,i]=>i" |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
852 |
"field_fm(r,z) == |
2c287f50c9f3
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paulson
parents:
13316
diff
changeset
|
853 |
Exists(And(domain_fm(succ(r),0), |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
854 |
Exists(And(range_fm(succ(succ(r)),0), |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
855 |
union_fm(1,0,succ(succ(z)))))))" |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
856 |
|
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
857 |
lemma field_type [TC]: |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
858 |
"[| x \<in> nat; y \<in> nat |] ==> field_fm(x,y) \<in> formula" |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
859 |
by (simp add: field_fm_def) |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
860 |
|
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
861 |
lemma arity_field_fm [simp]: |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
862 |
"[| x \<in> nat; y \<in> nat |] |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
863 |
==> arity(field_fm(x,y)) = succ(x) \<union> succ(y)" |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
864 |
by (simp add: field_fm_def succ_Un_distrib [symmetric] Un_ac) |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
865 |
|
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
866 |
lemma sats_field_fm [simp]: |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
867 |
"[| x \<in> nat; y \<in> nat; env \<in> list(A)|] |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
868 |
==> sats(A, field_fm(x,y), env) <-> |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
869 |
is_field(**A, nth(x,env), nth(y,env))" |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
870 |
by (simp add: field_fm_def is_field_def) |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
871 |
|
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
872 |
lemma field_iff_sats: |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
873 |
"[| nth(i,env) = x; nth(j,env) = y; |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
874 |
i \<in> nat; j \<in> nat; env \<in> list(A)|] |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
875 |
==> is_field(**A, x, y) <-> sats(A, field_fm(i,j), env)" |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
876 |
by simp |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
877 |
|
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
878 |
theorem field_reflection: |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
879 |
"REFLECTS[\<lambda>x. is_field(L,f(x),g(x)), |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
880 |
\<lambda>i x. is_field(**Lset(i),f(x),g(x))]" |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
881 |
apply (simp only: is_field_def setclass_simps) |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
882 |
apply (intro FOL_reflections domain_reflection range_reflection |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
883 |
union_reflection) |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
884 |
done |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
885 |
|
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
886 |
|
13339
0f89104dd377
Fixed quantified variable name preservation for ball and bex (bounded quants)
paulson
parents:
13323
diff
changeset
|
887 |
subsubsection{*Image under a Relation, Internalized*} |
13306 | 888 |
|
889 |
(* "image(M,r,A,z) == |
|
890 |
\<forall>y[M]. (y \<in> z <-> (\<exists>w[M]. w\<in>r & (\<exists>x[M]. x\<in>A & pair(M,x,y,w))))" *) |
|
891 |
constdefs image_fm :: "[i,i,i]=>i" |
|
892 |
"image_fm(r,A,z) == |
|
893 |
Forall(Iff(Member(0,succ(z)), |
|
894 |
Exists(And(Member(0,succ(succ(r))), |
|
895 |
Exists(And(Member(0,succ(succ(succ(A)))), |
|
896 |
pair_fm(0,2,1)))))))" |
|
897 |
||
898 |
lemma image_type [TC]: |
|
899 |
"[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> image_fm(x,y,z) \<in> formula" |
|
900 |
by (simp add: image_fm_def) |
|
901 |
||
902 |
lemma arity_image_fm [simp]: |
|
903 |
"[| x \<in> nat; y \<in> nat; z \<in> nat |] |
|
904 |
==> arity(image_fm(x,y,z)) = succ(x) \<union> succ(y) \<union> succ(z)" |
|
905 |
by (simp add: image_fm_def succ_Un_distrib [symmetric] Un_ac) |
|
906 |
||
907 |
lemma sats_image_fm [simp]: |
|
908 |
"[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|] |
|
909 |
==> sats(A, image_fm(x,y,z), env) <-> |
|
910 |
image(**A, nth(x,env), nth(y,env), nth(z,env))" |
|
13323
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
911 |
by (simp add: image_fm_def Relative.image_def) |
13306 | 912 |
|
913 |
lemma image_iff_sats: |
|
914 |
"[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z; |
|
915 |
i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|] |
|
916 |
==> image(**A, x, y, z) <-> sats(A, image_fm(i,j,k), env)" |
|
917 |
by (simp add: sats_image_fm) |
|
918 |
||
13314 | 919 |
theorem image_reflection: |
920 |
"REFLECTS[\<lambda>x. image(L,f(x),g(x),h(x)), |
|
921 |
\<lambda>i x. image(**Lset(i),f(x),g(x),h(x))]" |
|
13323
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
922 |
apply (simp only: Relative.image_def setclass_simps) |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
923 |
apply (intro FOL_reflections pair_reflection) |
13314 | 924 |
done |
13306 | 925 |
|
926 |
||
13348 | 927 |
subsubsection{*Pre-Image under a Relation, Internalized*} |
928 |
||
929 |
(* "pre_image(M,r,A,z) == |
|
930 |
\<forall>x[M]. x \<in> z <-> (\<exists>w[M]. w\<in>r & (\<exists>y[M]. y\<in>A & pair(M,x,y,w)))" *) |
|
931 |
constdefs pre_image_fm :: "[i,i,i]=>i" |
|
932 |
"pre_image_fm(r,A,z) == |
|
933 |
Forall(Iff(Member(0,succ(z)), |
|
934 |
Exists(And(Member(0,succ(succ(r))), |
|
935 |
Exists(And(Member(0,succ(succ(succ(A)))), |
|
936 |
pair_fm(2,0,1)))))))" |
|
937 |
||
938 |
lemma pre_image_type [TC]: |
|
939 |
"[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> pre_image_fm(x,y,z) \<in> formula" |
|
940 |
by (simp add: pre_image_fm_def) |
|
941 |
||
942 |
lemma arity_pre_image_fm [simp]: |
|
943 |
"[| x \<in> nat; y \<in> nat; z \<in> nat |] |
|
944 |
==> arity(pre_image_fm(x,y,z)) = succ(x) \<union> succ(y) \<union> succ(z)" |
|
945 |
by (simp add: pre_image_fm_def succ_Un_distrib [symmetric] Un_ac) |
|
946 |
||
947 |
lemma sats_pre_image_fm [simp]: |
|
948 |
"[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|] |
|
949 |
==> sats(A, pre_image_fm(x,y,z), env) <-> |
|
950 |
pre_image(**A, nth(x,env), nth(y,env), nth(z,env))" |
|
951 |
by (simp add: pre_image_fm_def Relative.pre_image_def) |
|
952 |
||
953 |
lemma pre_image_iff_sats: |
|
954 |
"[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z; |
|
955 |
i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|] |
|
956 |
==> pre_image(**A, x, y, z) <-> sats(A, pre_image_fm(i,j,k), env)" |
|
957 |
by (simp add: sats_pre_image_fm) |
|
958 |
||
959 |
theorem pre_image_reflection: |
|
960 |
"REFLECTS[\<lambda>x. pre_image(L,f(x),g(x),h(x)), |
|
961 |
\<lambda>i x. pre_image(**Lset(i),f(x),g(x),h(x))]" |
|
962 |
apply (simp only: Relative.pre_image_def setclass_simps) |
|
963 |
apply (intro FOL_reflections pair_reflection) |
|
964 |
done |
|
965 |
||
966 |
||
13352 | 967 |
subsubsection{*Function Application, Internalized*} |
968 |
||
969 |
(* "fun_apply(M,f,x,y) == |
|
970 |
(\<exists>xs[M]. \<exists>fxs[M]. |
|
971 |
upair(M,x,x,xs) & image(M,f,xs,fxs) & big_union(M,fxs,y))" *) |
|
972 |
constdefs fun_apply_fm :: "[i,i,i]=>i" |
|
973 |
"fun_apply_fm(f,x,y) == |
|
974 |
Exists(Exists(And(upair_fm(succ(succ(x)), succ(succ(x)), 1), |
|
975 |
And(image_fm(succ(succ(f)), 1, 0), |
|
976 |
big_union_fm(0,succ(succ(y)))))))" |
|
977 |
||
978 |
lemma fun_apply_type [TC]: |
|
979 |
"[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> fun_apply_fm(x,y,z) \<in> formula" |
|
980 |
by (simp add: fun_apply_fm_def) |
|
981 |
||
982 |
lemma arity_fun_apply_fm [simp]: |
|
983 |
"[| x \<in> nat; y \<in> nat; z \<in> nat |] |
|
984 |
==> arity(fun_apply_fm(x,y,z)) = succ(x) \<union> succ(y) \<union> succ(z)" |
|
985 |
by (simp add: fun_apply_fm_def succ_Un_distrib [symmetric] Un_ac) |
|
986 |
||
987 |
lemma sats_fun_apply_fm [simp]: |
|
988 |
"[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|] |
|
989 |
==> sats(A, fun_apply_fm(x,y,z), env) <-> |
|
990 |
fun_apply(**A, nth(x,env), nth(y,env), nth(z,env))" |
|
991 |
by (simp add: fun_apply_fm_def fun_apply_def) |
|
992 |
||
993 |
lemma fun_apply_iff_sats: |
|
994 |
"[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z; |
|
995 |
i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|] |
|
996 |
==> fun_apply(**A, x, y, z) <-> sats(A, fun_apply_fm(i,j,k), env)" |
|
997 |
by simp |
|
998 |
||
999 |
theorem fun_apply_reflection: |
|
1000 |
"REFLECTS[\<lambda>x. fun_apply(L,f(x),g(x),h(x)), |
|
1001 |
\<lambda>i x. fun_apply(**Lset(i),f(x),g(x),h(x))]" |
|
1002 |
apply (simp only: fun_apply_def setclass_simps) |
|
1003 |
apply (intro FOL_reflections upair_reflection image_reflection |
|
1004 |
big_union_reflection) |
|
1005 |
done |
|
1006 |
||
1007 |
||
13339
0f89104dd377
Fixed quantified variable name preservation for ball and bex (bounded quants)
paulson
parents:
13323
diff
changeset
|
1008 |
subsubsection{*The Concept of Relation, Internalized*} |
13306 | 1009 |
|
1010 |
(* "is_relation(M,r) == |
|
1011 |
(\<forall>z[M]. z\<in>r --> (\<exists>x[M]. \<exists>y[M]. pair(M,x,y,z)))" *) |
|
1012 |
constdefs relation_fm :: "i=>i" |
|
1013 |
"relation_fm(r) == |
|
1014 |
Forall(Implies(Member(0,succ(r)), Exists(Exists(pair_fm(1,0,2)))))" |
|
1015 |
||
1016 |
lemma relation_type [TC]: |
|
1017 |
"[| x \<in> nat |] ==> relation_fm(x) \<in> formula" |
|
1018 |
by (simp add: relation_fm_def) |
|
1019 |
||
1020 |
lemma arity_relation_fm [simp]: |
|
1021 |
"x \<in> nat ==> arity(relation_fm(x)) = succ(x)" |
|
1022 |
by (simp add: relation_fm_def succ_Un_distrib [symmetric] Un_ac) |
|
1023 |
||
1024 |
lemma sats_relation_fm [simp]: |
|
1025 |
"[| x \<in> nat; env \<in> list(A)|] |
|
1026 |
==> sats(A, relation_fm(x), env) <-> is_relation(**A, nth(x,env))" |
|
1027 |
by (simp add: relation_fm_def is_relation_def) |
|
1028 |
||
1029 |
lemma relation_iff_sats: |
|
1030 |
"[| nth(i,env) = x; nth(j,env) = y; |
|
1031 |
i \<in> nat; env \<in> list(A)|] |
|
1032 |
==> is_relation(**A, x) <-> sats(A, relation_fm(i), env)" |
|
1033 |
by simp |
|
1034 |
||
13314 | 1035 |
theorem is_relation_reflection: |
1036 |
"REFLECTS[\<lambda>x. is_relation(L,f(x)), |
|
1037 |
\<lambda>i x. is_relation(**Lset(i),f(x))]" |
|
1038 |
apply (simp only: is_relation_def setclass_simps) |
|
13323
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
1039 |
apply (intro FOL_reflections pair_reflection) |
13314 | 1040 |
done |
13306 | 1041 |
|
1042 |
||
13339
0f89104dd377
Fixed quantified variable name preservation for ball and bex (bounded quants)
paulson
parents:
13323
diff
changeset
|
1043 |
subsubsection{*The Concept of Function, Internalized*} |
13306 | 1044 |
|
1045 |
(* "is_function(M,r) == |
|
1046 |
\<forall>x[M]. \<forall>y[M]. \<forall>y'[M]. \<forall>p[M]. \<forall>p'[M]. |
|
1047 |
pair(M,x,y,p) --> pair(M,x,y',p') --> p\<in>r --> p'\<in>r --> y=y'" *) |
|
1048 |
constdefs function_fm :: "i=>i" |
|
1049 |
"function_fm(r) == |
|
1050 |
Forall(Forall(Forall(Forall(Forall( |
|
1051 |
Implies(pair_fm(4,3,1), |
|
1052 |
Implies(pair_fm(4,2,0), |
|
1053 |
Implies(Member(1,r#+5), |
|
1054 |
Implies(Member(0,r#+5), Equal(3,2))))))))))" |
|
1055 |
||
1056 |
lemma function_type [TC]: |
|
1057 |
"[| x \<in> nat |] ==> function_fm(x) \<in> formula" |
|
1058 |
by (simp add: function_fm_def) |
|
1059 |
||
1060 |
lemma arity_function_fm [simp]: |
|
1061 |
"x \<in> nat ==> arity(function_fm(x)) = succ(x)" |
|
1062 |
by (simp add: function_fm_def succ_Un_distrib [symmetric] Un_ac) |
|
1063 |
||
1064 |
lemma sats_function_fm [simp]: |
|
1065 |
"[| x \<in> nat; env \<in> list(A)|] |
|
1066 |
==> sats(A, function_fm(x), env) <-> is_function(**A, nth(x,env))" |
|
1067 |
by (simp add: function_fm_def is_function_def) |
|
1068 |
||
1069 |
lemma function_iff_sats: |
|
1070 |
"[| nth(i,env) = x; nth(j,env) = y; |
|
1071 |
i \<in> nat; env \<in> list(A)|] |
|
1072 |
==> is_function(**A, x) <-> sats(A, function_fm(i), env)" |
|
1073 |
by simp |
|
1074 |
||
13314 | 1075 |
theorem is_function_reflection: |
1076 |
"REFLECTS[\<lambda>x. is_function(L,f(x)), |
|
1077 |
\<lambda>i x. is_function(**Lset(i),f(x))]" |
|
1078 |
apply (simp only: is_function_def setclass_simps) |
|
13323
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
1079 |
apply (intro FOL_reflections pair_reflection) |
13314 | 1080 |
done |
13298 | 1081 |
|
1082 |
||
13339
0f89104dd377
Fixed quantified variable name preservation for ball and bex (bounded quants)
paulson
parents:
13323
diff
changeset
|
1083 |
subsubsection{*Typed Functions, Internalized*} |
13309 | 1084 |
|
1085 |
(* "typed_function(M,A,B,r) == |
|
1086 |
is_function(M,r) & is_relation(M,r) & is_domain(M,r,A) & |
|
1087 |
(\<forall>u[M]. u\<in>r --> (\<forall>x[M]. \<forall>y[M]. pair(M,x,y,u) --> y\<in>B))" *) |
|
1088 |
||
1089 |
constdefs typed_function_fm :: "[i,i,i]=>i" |
|
1090 |
"typed_function_fm(A,B,r) == |
|
1091 |
And(function_fm(r), |
|
1092 |
And(relation_fm(r), |
|
1093 |
And(domain_fm(r,A), |
|
1094 |
Forall(Implies(Member(0,succ(r)), |
|
1095 |
Forall(Forall(Implies(pair_fm(1,0,2),Member(0,B#+3)))))))))" |
|
1096 |
||
1097 |
lemma typed_function_type [TC]: |
|
1098 |
"[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> typed_function_fm(x,y,z) \<in> formula" |
|
1099 |
by (simp add: typed_function_fm_def) |
|
1100 |
||
1101 |
lemma arity_typed_function_fm [simp]: |
|
1102 |
"[| x \<in> nat; y \<in> nat; z \<in> nat |] |
|
1103 |
==> arity(typed_function_fm(x,y,z)) = succ(x) \<union> succ(y) \<union> succ(z)" |
|
1104 |
by (simp add: typed_function_fm_def succ_Un_distrib [symmetric] Un_ac) |
|
1105 |
||
1106 |
lemma sats_typed_function_fm [simp]: |
|
1107 |
"[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|] |
|
1108 |
==> sats(A, typed_function_fm(x,y,z), env) <-> |
|
1109 |
typed_function(**A, nth(x,env), nth(y,env), nth(z,env))" |
|
1110 |
by (simp add: typed_function_fm_def typed_function_def) |
|
1111 |
||
1112 |
lemma typed_function_iff_sats: |
|
1113 |
"[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z; |
|
1114 |
i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|] |
|
1115 |
==> typed_function(**A, x, y, z) <-> sats(A, typed_function_fm(i,j,k), env)" |
|
1116 |
by simp |
|
1117 |
||
13323
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
1118 |
lemmas function_reflections = |
13363 | 1119 |
empty_reflection number1_reflection |
1120 |
upair_reflection pair_reflection union_reflection |
|
13352 | 1121 |
big_union_reflection cons_reflection successor_reflection |
13323
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
1122 |
fun_apply_reflection subset_reflection |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
1123 |
transitive_set_reflection membership_reflection |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
1124 |
pred_set_reflection domain_reflection range_reflection field_reflection |
13348 | 1125 |
image_reflection pre_image_reflection |
13314 | 1126 |
is_relation_reflection is_function_reflection |
13309 | 1127 |
|
13323
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
1128 |
lemmas function_iff_sats = |
13363 | 1129 |
empty_iff_sats number1_iff_sats |
1130 |
upair_iff_sats pair_iff_sats union_iff_sats |
|
13323
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
1131 |
cons_iff_sats successor_iff_sats |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
1132 |
fun_apply_iff_sats Memrel_iff_sats |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
1133 |
pred_set_iff_sats domain_iff_sats range_iff_sats field_iff_sats |
13348 | 1134 |
image_iff_sats pre_image_iff_sats |
13323
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
1135 |
relation_iff_sats function_iff_sats |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
1136 |
|
13309 | 1137 |
|
13314 | 1138 |
theorem typed_function_reflection: |
1139 |
"REFLECTS[\<lambda>x. typed_function(L,f(x),g(x),h(x)), |
|
1140 |
\<lambda>i x. typed_function(**Lset(i),f(x),g(x),h(x))]" |
|
1141 |
apply (simp only: typed_function_def setclass_simps) |
|
13323
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
1142 |
apply (intro FOL_reflections function_reflections) |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
1143 |
done |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
1144 |
|
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
1145 |
|
13339
0f89104dd377
Fixed quantified variable name preservation for ball and bex (bounded quants)
paulson
parents:
13323
diff
changeset
|
1146 |
subsubsection{*Composition of Relations, Internalized*} |
13323
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
1147 |
|
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
1148 |
(* "composition(M,r,s,t) == |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
1149 |
\<forall>p[M]. p \<in> t <-> |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
1150 |
(\<exists>x[M]. \<exists>y[M]. \<exists>z[M]. \<exists>xy[M]. \<exists>yz[M]. |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
1151 |
pair(M,x,z,p) & pair(M,x,y,xy) & pair(M,y,z,yz) & |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
1152 |
xy \<in> s & yz \<in> r)" *) |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
1153 |
constdefs composition_fm :: "[i,i,i]=>i" |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
1154 |
"composition_fm(r,s,t) == |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
1155 |
Forall(Iff(Member(0,succ(t)), |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
1156 |
Exists(Exists(Exists(Exists(Exists( |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
1157 |
And(pair_fm(4,2,5), |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
1158 |
And(pair_fm(4,3,1), |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
1159 |
And(pair_fm(3,2,0), |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
1160 |
And(Member(1,s#+6), Member(0,r#+6))))))))))))" |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
1161 |
|
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
1162 |
lemma composition_type [TC]: |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
1163 |
"[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> composition_fm(x,y,z) \<in> formula" |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
1164 |
by (simp add: composition_fm_def) |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
1165 |
|
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
1166 |
lemma arity_composition_fm [simp]: |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
1167 |
"[| x \<in> nat; y \<in> nat; z \<in> nat |] |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
1168 |
==> arity(composition_fm(x,y,z)) = succ(x) \<union> succ(y) \<union> succ(z)" |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
1169 |
by (simp add: composition_fm_def succ_Un_distrib [symmetric] Un_ac) |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
1170 |
|
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
1171 |
lemma sats_composition_fm [simp]: |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
1172 |
"[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|] |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
1173 |
==> sats(A, composition_fm(x,y,z), env) <-> |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
1174 |
composition(**A, nth(x,env), nth(y,env), nth(z,env))" |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
1175 |
by (simp add: composition_fm_def composition_def) |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
1176 |
|
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
1177 |
lemma composition_iff_sats: |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
1178 |
"[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z; |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
1179 |
i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|] |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
1180 |
==> composition(**A, x, y, z) <-> sats(A, composition_fm(i,j,k), env)" |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
1181 |
by simp |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
1182 |
|
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
1183 |
theorem composition_reflection: |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
1184 |
"REFLECTS[\<lambda>x. composition(L,f(x),g(x),h(x)), |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
1185 |
\<lambda>i x. composition(**Lset(i),f(x),g(x),h(x))]" |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
1186 |
apply (simp only: composition_def setclass_simps) |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
1187 |
apply (intro FOL_reflections pair_reflection) |
13314 | 1188 |
done |
1189 |
||
13309 | 1190 |
|
13339
0f89104dd377
Fixed quantified variable name preservation for ball and bex (bounded quants)
paulson
parents:
13323
diff
changeset
|
1191 |
subsubsection{*Injections, Internalized*} |
13309 | 1192 |
|
1193 |
(* "injection(M,A,B,f) == |
|
1194 |
typed_function(M,A,B,f) & |
|
1195 |
(\<forall>x[M]. \<forall>x'[M]. \<forall>y[M]. \<forall>p[M]. \<forall>p'[M]. |
|
1196 |
pair(M,x,y,p) --> pair(M,x',y,p') --> p\<in>f --> p'\<in>f --> x=x')" *) |
|
1197 |
constdefs injection_fm :: "[i,i,i]=>i" |
|
1198 |
"injection_fm(A,B,f) == |
|
1199 |
And(typed_function_fm(A,B,f), |
|
1200 |
Forall(Forall(Forall(Forall(Forall( |
|
1201 |
Implies(pair_fm(4,2,1), |
|
1202 |
Implies(pair_fm(3,2,0), |
|
1203 |
Implies(Member(1,f#+5), |
|
1204 |
Implies(Member(0,f#+5), Equal(4,3)))))))))))" |
|
1205 |
||
1206 |
||
1207 |
lemma injection_type [TC]: |
|
1208 |
"[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> injection_fm(x,y,z) \<in> formula" |
|
1209 |
by (simp add: injection_fm_def) |
|
1210 |
||
1211 |
lemma arity_injection_fm [simp]: |
|
1212 |
"[| x \<in> nat; y \<in> nat; z \<in> nat |] |
|
1213 |
==> arity(injection_fm(x,y,z)) = succ(x) \<union> succ(y) \<union> succ(z)" |
|
1214 |
by (simp add: injection_fm_def succ_Un_distrib [symmetric] Un_ac) |
|
1215 |
||
1216 |
lemma sats_injection_fm [simp]: |
|
1217 |
"[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|] |
|
1218 |
==> sats(A, injection_fm(x,y,z), env) <-> |
|
1219 |
injection(**A, nth(x,env), nth(y,env), nth(z,env))" |
|
1220 |
by (simp add: injection_fm_def injection_def) |
|
1221 |
||
1222 |
lemma injection_iff_sats: |
|
1223 |
"[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z; |
|
1224 |
i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|] |
|
1225 |
==> injection(**A, x, y, z) <-> sats(A, injection_fm(i,j,k), env)" |
|
1226 |
by simp |
|
1227 |
||
13314 | 1228 |
theorem injection_reflection: |
1229 |
"REFLECTS[\<lambda>x. injection(L,f(x),g(x),h(x)), |
|
1230 |
\<lambda>i x. injection(**Lset(i),f(x),g(x),h(x))]" |
|
1231 |
apply (simp only: injection_def setclass_simps) |
|
13323
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
1232 |
apply (intro FOL_reflections function_reflections typed_function_reflection) |
13314 | 1233 |
done |
13309 | 1234 |
|
1235 |
||
13339
0f89104dd377
Fixed quantified variable name preservation for ball and bex (bounded quants)
paulson
parents:
13323
diff
changeset
|
1236 |
subsubsection{*Surjections, Internalized*} |
13309 | 1237 |
|
1238 |
(* surjection :: "[i=>o,i,i,i] => o" |
|
1239 |
"surjection(M,A,B,f) == |
|
1240 |
typed_function(M,A,B,f) & |
|
1241 |
(\<forall>y[M]. y\<in>B --> (\<exists>x[M]. x\<in>A & fun_apply(M,f,x,y)))" *) |
|
1242 |
constdefs surjection_fm :: "[i,i,i]=>i" |
|
1243 |
"surjection_fm(A,B,f) == |
|
1244 |
And(typed_function_fm(A,B,f), |
|
1245 |
Forall(Implies(Member(0,succ(B)), |
|
1246 |
Exists(And(Member(0,succ(succ(A))), |
|
1247 |
fun_apply_fm(succ(succ(f)),0,1))))))" |
|
1248 |
||
1249 |
lemma surjection_type [TC]: |
|
1250 |
"[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> surjection_fm(x,y,z) \<in> formula" |
|
1251 |
by (simp add: surjection_fm_def) |
|
1252 |
||
1253 |
lemma arity_surjection_fm [simp]: |
|
1254 |
"[| x \<in> nat; y \<in> nat; z \<in> nat |] |
|
1255 |
==> arity(surjection_fm(x,y,z)) = succ(x) \<union> succ(y) \<union> succ(z)" |
|
1256 |
by (simp add: surjection_fm_def succ_Un_distrib [symmetric] Un_ac) |
|
1257 |
||
1258 |
lemma sats_surjection_fm [simp]: |
|
1259 |
"[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|] |
|
1260 |
==> sats(A, surjection_fm(x,y,z), env) <-> |
|
1261 |
surjection(**A, nth(x,env), nth(y,env), nth(z,env))" |
|
1262 |
by (simp add: surjection_fm_def surjection_def) |
|
1263 |
||
1264 |
lemma surjection_iff_sats: |
|
1265 |
"[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z; |
|
1266 |
i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|] |
|
1267 |
==> surjection(**A, x, y, z) <-> sats(A, surjection_fm(i,j,k), env)" |
|
1268 |
by simp |
|
1269 |
||
13314 | 1270 |
theorem surjection_reflection: |
1271 |
"REFLECTS[\<lambda>x. surjection(L,f(x),g(x),h(x)), |
|
1272 |
\<lambda>i x. surjection(**Lset(i),f(x),g(x),h(x))]" |
|
1273 |
apply (simp only: surjection_def setclass_simps) |
|
13323
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
1274 |
apply (intro FOL_reflections function_reflections typed_function_reflection) |
13314 | 1275 |
done |
13309 | 1276 |
|
1277 |
||
1278 |
||
13339
0f89104dd377
Fixed quantified variable name preservation for ball and bex (bounded quants)
paulson
parents:
13323
diff
changeset
|
1279 |
subsubsection{*Bijections, Internalized*} |
13309 | 1280 |
|
1281 |
(* bijection :: "[i=>o,i,i,i] => o" |
|
1282 |
"bijection(M,A,B,f) == injection(M,A,B,f) & surjection(M,A,B,f)" *) |
|
1283 |
constdefs bijection_fm :: "[i,i,i]=>i" |
|
1284 |
"bijection_fm(A,B,f) == And(injection_fm(A,B,f), surjection_fm(A,B,f))" |
|
1285 |
||
1286 |
lemma bijection_type [TC]: |
|
1287 |
"[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> bijection_fm(x,y,z) \<in> formula" |
|
1288 |
by (simp add: bijection_fm_def) |
|
1289 |
||
1290 |
lemma arity_bijection_fm [simp]: |
|
1291 |
"[| x \<in> nat; y \<in> nat; z \<in> nat |] |
|
1292 |
==> arity(bijection_fm(x,y,z)) = succ(x) \<union> succ(y) \<union> succ(z)" |
|
1293 |
by (simp add: bijection_fm_def succ_Un_distrib [symmetric] Un_ac) |
|
1294 |
||
1295 |
lemma sats_bijection_fm [simp]: |
|
1296 |
"[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|] |
|
1297 |
==> sats(A, bijection_fm(x,y,z), env) <-> |
|
1298 |
bijection(**A, nth(x,env), nth(y,env), nth(z,env))" |
|
1299 |
by (simp add: bijection_fm_def bijection_def) |
|
1300 |
||
1301 |
lemma bijection_iff_sats: |
|
1302 |
"[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z; |
|
1303 |
i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|] |
|
1304 |
==> bijection(**A, x, y, z) <-> sats(A, bijection_fm(i,j,k), env)" |
|
1305 |
by simp |
|
1306 |
||
13314 | 1307 |
theorem bijection_reflection: |
1308 |
"REFLECTS[\<lambda>x. bijection(L,f(x),g(x),h(x)), |
|
1309 |
\<lambda>i x. bijection(**Lset(i),f(x),g(x),h(x))]" |
|
1310 |
apply (simp only: bijection_def setclass_simps) |
|
1311 |
apply (intro And_reflection injection_reflection surjection_reflection) |
|
1312 |
done |
|
13309 | 1313 |
|
1314 |
||
13348 | 1315 |
subsubsection{*Restriction of a Relation, Internalized*} |
1316 |
||
1317 |
||
1318 |
(* "restriction(M,r,A,z) == |
|
1319 |
\<forall>x[M]. x \<in> z <-> (x \<in> r & (\<exists>u[M]. u\<in>A & (\<exists>v[M]. pair(M,u,v,x))))" *) |
|
1320 |
constdefs restriction_fm :: "[i,i,i]=>i" |
|
1321 |
"restriction_fm(r,A,z) == |
|
1322 |
Forall(Iff(Member(0,succ(z)), |
|
1323 |
And(Member(0,succ(r)), |
|
1324 |
Exists(And(Member(0,succ(succ(A))), |
|
1325 |
Exists(pair_fm(1,0,2)))))))" |
|
1326 |
||
1327 |
lemma restriction_type [TC]: |
|
1328 |
"[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> restriction_fm(x,y,z) \<in> formula" |
|
1329 |
by (simp add: restriction_fm_def) |
|
1330 |
||
1331 |
lemma arity_restriction_fm [simp]: |
|
1332 |
"[| x \<in> nat; y \<in> nat; z \<in> nat |] |
|
1333 |
==> arity(restriction_fm(x,y,z)) = succ(x) \<union> succ(y) \<union> succ(z)" |
|
1334 |
by (simp add: restriction_fm_def succ_Un_distrib [symmetric] Un_ac) |
|
1335 |
||
1336 |
lemma sats_restriction_fm [simp]: |
|
1337 |
"[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|] |
|
1338 |
==> sats(A, restriction_fm(x,y,z), env) <-> |
|
1339 |
restriction(**A, nth(x,env), nth(y,env), nth(z,env))" |
|
1340 |
by (simp add: restriction_fm_def restriction_def) |
|
1341 |
||
1342 |
lemma restriction_iff_sats: |
|
1343 |
"[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z; |
|
1344 |
i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|] |
|
1345 |
==> restriction(**A, x, y, z) <-> sats(A, restriction_fm(i,j,k), env)" |
|
1346 |
by simp |
|
1347 |
||
1348 |
theorem restriction_reflection: |
|
1349 |
"REFLECTS[\<lambda>x. restriction(L,f(x),g(x),h(x)), |
|
1350 |
\<lambda>i x. restriction(**Lset(i),f(x),g(x),h(x))]" |
|
1351 |
apply (simp only: restriction_def setclass_simps) |
|
1352 |
apply (intro FOL_reflections pair_reflection) |
|
1353 |
done |
|
1354 |
||
13339
0f89104dd377
Fixed quantified variable name preservation for ball and bex (bounded quants)
paulson
parents:
13323
diff
changeset
|
1355 |
subsubsection{*Order-Isomorphisms, Internalized*} |
13309 | 1356 |
|
1357 |
(* order_isomorphism :: "[i=>o,i,i,i,i,i] => o" |
|
1358 |
"order_isomorphism(M,A,r,B,s,f) == |
|
1359 |
bijection(M,A,B,f) & |
|
1360 |
(\<forall>x[M]. x\<in>A --> (\<forall>y[M]. y\<in>A --> |
|
1361 |
(\<forall>p[M]. \<forall>fx[M]. \<forall>fy[M]. \<forall>q[M]. |
|
1362 |
pair(M,x,y,p) --> fun_apply(M,f,x,fx) --> fun_apply(M,f,y,fy) --> |
|
1363 |
pair(M,fx,fy,q) --> (p\<in>r <-> q\<in>s))))" |
|
1364 |
*) |
|
1365 |
||
1366 |
constdefs order_isomorphism_fm :: "[i,i,i,i,i]=>i" |
|
1367 |
"order_isomorphism_fm(A,r,B,s,f) == |
|
1368 |
And(bijection_fm(A,B,f), |
|
1369 |
Forall(Implies(Member(0,succ(A)), |
|
1370 |
Forall(Implies(Member(0,succ(succ(A))), |
|
1371 |
Forall(Forall(Forall(Forall( |
|
1372 |
Implies(pair_fm(5,4,3), |
|
1373 |
Implies(fun_apply_fm(f#+6,5,2), |
|
1374 |
Implies(fun_apply_fm(f#+6,4,1), |
|
1375 |
Implies(pair_fm(2,1,0), |
|
1376 |
Iff(Member(3,r#+6), Member(0,s#+6)))))))))))))))" |
|
1377 |
||
1378 |
lemma order_isomorphism_type [TC]: |
|
1379 |
"[| A \<in> nat; r \<in> nat; B \<in> nat; s \<in> nat; f \<in> nat |] |
|
1380 |
==> order_isomorphism_fm(A,r,B,s,f) \<in> formula" |
|
1381 |
by (simp add: order_isomorphism_fm_def) |
|
1382 |
||
1383 |
lemma arity_order_isomorphism_fm [simp]: |
|
1384 |
"[| A \<in> nat; r \<in> nat; B \<in> nat; s \<in> nat; f \<in> nat |] |
|
1385 |
==> arity(order_isomorphism_fm(A,r,B,s,f)) = |
|
1386 |
succ(A) \<union> succ(r) \<union> succ(B) \<union> succ(s) \<union> succ(f)" |
|
1387 |
by (simp add: order_isomorphism_fm_def succ_Un_distrib [symmetric] Un_ac) |
|
1388 |
||
1389 |
lemma sats_order_isomorphism_fm [simp]: |
|
1390 |
"[| U \<in> nat; r \<in> nat; B \<in> nat; s \<in> nat; f \<in> nat; env \<in> list(A)|] |
|
1391 |
==> sats(A, order_isomorphism_fm(U,r,B,s,f), env) <-> |
|
1392 |
order_isomorphism(**A, nth(U,env), nth(r,env), nth(B,env), |
|
1393 |
nth(s,env), nth(f,env))" |
|
1394 |
by (simp add: order_isomorphism_fm_def order_isomorphism_def) |
|
1395 |
||
1396 |
lemma order_isomorphism_iff_sats: |
|
1397 |
"[| nth(i,env) = U; nth(j,env) = r; nth(k,env) = B; nth(j',env) = s; |
|
1398 |
nth(k',env) = f; |
|
1399 |
i \<in> nat; j \<in> nat; k \<in> nat; j' \<in> nat; k' \<in> nat; env \<in> list(A)|] |
|
1400 |
==> order_isomorphism(**A,U,r,B,s,f) <-> |
|
1401 |
sats(A, order_isomorphism_fm(i,j,k,j',k'), env)" |
|
1402 |
by simp |
|
1403 |
||
13314 | 1404 |
theorem order_isomorphism_reflection: |
1405 |
"REFLECTS[\<lambda>x. order_isomorphism(L,f(x),g(x),h(x),g'(x),h'(x)), |
|
1406 |
\<lambda>i x. order_isomorphism(**Lset(i),f(x),g(x),h(x),g'(x),h'(x))]" |
|
1407 |
apply (simp only: order_isomorphism_def setclass_simps) |
|
13323
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
1408 |
apply (intro FOL_reflections function_reflections bijection_reflection) |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
1409 |
done |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
1410 |
|
13339
0f89104dd377
Fixed quantified variable name preservation for ball and bex (bounded quants)
paulson
parents:
13323
diff
changeset
|
1411 |
subsubsection{*Limit Ordinals, Internalized*} |
13323
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
1412 |
|
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
1413 |
text{*A limit ordinal is a non-empty, successor-closed ordinal*} |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
1414 |
|
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
1415 |
(* "limit_ordinal(M,a) == |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
1416 |
ordinal(M,a) & ~ empty(M,a) & |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
1417 |
(\<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
|
1418 |
|
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
1419 |
constdefs limit_ordinal_fm :: "i=>i" |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
1420 |
"limit_ordinal_fm(x) == |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
1421 |
And(ordinal_fm(x), |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
1422 |
And(Neg(empty_fm(x)), |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
1423 |
Forall(Implies(Member(0,succ(x)), |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
1424 |
Exists(And(Member(0,succ(succ(x))), |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
1425 |
succ_fm(1,0)))))))" |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
1426 |
|
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
1427 |
lemma limit_ordinal_type [TC]: |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
1428 |
"x \<in> nat ==> limit_ordinal_fm(x) \<in> formula" |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
1429 |
by (simp add: limit_ordinal_fm_def) |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
1430 |
|
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
1431 |
lemma arity_limit_ordinal_fm [simp]: |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
1432 |
"x \<in> nat ==> arity(limit_ordinal_fm(x)) = succ(x)" |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
1433 |
by (simp add: limit_ordinal_fm_def succ_Un_distrib [symmetric] Un_ac) |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
1434 |
|
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
1435 |
lemma sats_limit_ordinal_fm [simp]: |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
1436 |
"[| x \<in> nat; env \<in> list(A)|] |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
1437 |
==> sats(A, limit_ordinal_fm(x), env) <-> limit_ordinal(**A, nth(x,env))" |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
1438 |
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
|
1439 |
|
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
1440 |
lemma limit_ordinal_iff_sats: |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
1441 |
"[| nth(i,env) = x; nth(j,env) = y; |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
1442 |
i \<in> nat; env \<in> list(A)|] |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
1443 |
==> limit_ordinal(**A, x) <-> sats(A, limit_ordinal_fm(i), env)" |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
1444 |
by simp |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
1445 |
|
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
1446 |
theorem limit_ordinal_reflection: |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
1447 |
"REFLECTS[\<lambda>x. limit_ordinal(L,f(x)), |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
1448 |
\<lambda>i x. limit_ordinal(**Lset(i),f(x))]" |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
1449 |
apply (simp only: limit_ordinal_def setclass_simps) |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
1450 |
apply (intro FOL_reflections ordinal_reflection |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
1451 |
empty_reflection successor_reflection) |
13314 | 1452 |
done |
13309 | 1453 |
|
13323
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
1454 |
subsubsection{*Omega: The Set of Natural Numbers*} |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
1455 |
|
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
1456 |
(* 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
|
1457 |
constdefs omega_fm :: "i=>i" |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
1458 |
"omega_fm(x) == |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
1459 |
And(limit_ordinal_fm(x), |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
1460 |
Forall(Implies(Member(0,succ(x)), |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
1461 |
Neg(limit_ordinal_fm(0)))))" |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
1462 |
|
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
1463 |
lemma omega_type [TC]: |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
1464 |
"x \<in> nat ==> omega_fm(x) \<in> formula" |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
1465 |
by (simp add: omega_fm_def) |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
1466 |
|
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
1467 |
lemma arity_omega_fm [simp]: |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
1468 |
"x \<in> nat ==> arity(omega_fm(x)) = succ(x)" |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
1469 |
by (simp add: omega_fm_def succ_Un_distrib [symmetric] Un_ac) |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
1470 |
|
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
1471 |
lemma sats_omega_fm [simp]: |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
1472 |
"[| x \<in> nat; env \<in> list(A)|] |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
1473 |
==> sats(A, omega_fm(x), env) <-> omega(**A, nth(x,env))" |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
1474 |
by (simp add: omega_fm_def omega_def) |
13316 | 1475 |
|
13323
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
1476 |
lemma omega_iff_sats: |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
1477 |
"[| nth(i,env) = x; nth(j,env) = y; |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
1478 |
i \<in> nat; env \<in> list(A)|] |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
1479 |
==> omega(**A, x) <-> sats(A, omega_fm(i), env)" |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
1480 |
by simp |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
1481 |
|
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
1482 |
theorem omega_reflection: |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
1483 |
"REFLECTS[\<lambda>x. omega(L,f(x)), |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
1484 |
\<lambda>i x. omega(**Lset(i),f(x))]" |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
1485 |
apply (simp only: omega_def setclass_simps) |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
1486 |
apply (intro FOL_reflections limit_ordinal_reflection) |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
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diff
changeset
|
1487 |
done |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
1488 |
|
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
1489 |
|
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
1490 |
lemmas fun_plus_reflections = |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
1491 |
typed_function_reflection composition_reflection |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
1492 |
injection_reflection surjection_reflection |
13348 | 1493 |
bijection_reflection restriction_reflection |
1494 |
order_isomorphism_reflection |
|
13323
2c287f50c9f3
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paulson
parents:
13316
diff
changeset
|
1495 |
ordinal_reflection limit_ordinal_reflection omega_reflection |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
1496 |
|
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
1497 |
lemmas fun_plus_iff_sats = |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
1498 |
typed_function_iff_sats composition_iff_sats |
13348 | 1499 |
injection_iff_sats surjection_iff_sats |
1500 |
bijection_iff_sats restriction_iff_sats |
|
13316 | 1501 |
order_isomorphism_iff_sats |
13323
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13316
diff
changeset
|
1502 |
ordinal_iff_sats limit_ordinal_iff_sats omega_iff_sats |
13316 | 1503 |
|
13223 | 1504 |
end |