1 InfDatatype = Datatype + Univ + Finite + Cardinal_AC |
1 (* Title: ZF/InfDatatype.ML |
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2 ID: $Id$ |
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3 Author: Lawrence C Paulson, Cambridge University Computer Laboratory |
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4 Copyright 1994 University of Cambridge |
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5 |
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6 Infinite-branching datatype definitions |
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7 *) |
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8 |
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9 theory InfDatatype = Datatype + Univ + Finite + Cardinal_AC: |
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10 |
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11 lemmas fun_Limit_VfromE = |
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12 Limit_VfromE [OF apply_funtype InfCard_csucc [THEN InfCard_is_Limit]] |
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13 |
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14 lemma fun_Vcsucc_lemma: |
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15 "[| f: D -> Vfrom(A,csucc(K)); |D| le K; InfCard(K) |] |
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16 ==> EX j. f: D -> Vfrom(A,j) & j < csucc(K)" |
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17 apply (rule_tac x = "UN d:D. LEAST i. f`d : Vfrom (A,i) " in exI) |
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18 apply (rule conjI) |
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19 apply (rule_tac [2] le_UN_Ord_lt_csucc) |
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20 apply (rule_tac [4] ballI, erule_tac [4] fun_Limit_VfromE) |
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21 apply (simp_all add: ) |
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22 prefer 2 apply (fast elim: Least_le [THEN lt_trans1] ltE) |
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23 apply (rule Pi_type) |
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24 apply (rename_tac [2] d) |
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25 apply (erule_tac [2] fun_Limit_VfromE, simp_all) |
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26 apply (subgoal_tac "f`d : Vfrom (A, LEAST i. f`d : Vfrom (A,i))") |
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27 apply (erule Vfrom_mono [OF subset_refl UN_upper, THEN subsetD]) |
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28 apply assumption |
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29 apply (fast elim: LeastI ltE) |
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30 done |
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31 |
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32 lemma subset_Vcsucc: |
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33 "[| D <= Vfrom(A,csucc(K)); |D| le K; InfCard(K) |] |
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34 ==> EX j. D <= Vfrom(A,j) & j < csucc(K)" |
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35 by (simp add: subset_iff_id fun_Vcsucc_lemma) |
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36 |
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37 (*Version for arbitrary index sets*) |
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38 lemma fun_Vcsucc: |
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39 "[| |D| le K; InfCard(K); D <= Vfrom(A,csucc(K)) |] ==> |
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40 D -> Vfrom(A,csucc(K)) <= Vfrom(A,csucc(K))" |
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41 apply (safe dest!: fun_Vcsucc_lemma subset_Vcsucc) |
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42 apply (rule Vfrom [THEN ssubst]) |
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43 apply (drule fun_is_rel) |
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44 (*This level includes the function, and is below csucc(K)*) |
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45 apply (rule_tac a1 = "succ (succ (j Un ja))" in UN_I [THEN UnI2]) |
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46 apply (blast intro: ltD InfCard_csucc InfCard_is_Limit Limit_has_succ |
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47 Un_least_lt); |
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48 apply (erule subset_trans [THEN PowI]) |
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49 apply (fast intro: Pair_in_Vfrom Vfrom_UnI1 Vfrom_UnI2) |
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50 done |
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51 |
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52 lemma fun_in_Vcsucc: |
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53 "[| f: D -> Vfrom(A, csucc(K)); |D| le K; InfCard(K); |
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54 D <= Vfrom(A,csucc(K)) |] |
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55 ==> f: Vfrom(A,csucc(K))" |
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56 by (blast intro: fun_Vcsucc [THEN subsetD]) |
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57 |
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58 (*Remove <= from the rule above*) |
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59 lemmas fun_in_Vcsucc' = fun_in_Vcsucc [OF _ _ _ subsetI] |
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60 |
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61 (** Version where K itself is the index set **) |
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62 |
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63 lemma Card_fun_Vcsucc: |
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64 "InfCard(K) ==> K -> Vfrom(A,csucc(K)) <= Vfrom(A,csucc(K))" |
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65 apply (frule InfCard_is_Card [THEN Card_is_Ord]) |
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66 apply (blast del: subsetI |
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67 intro: fun_Vcsucc Ord_cardinal_le i_subset_Vfrom |
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68 lt_csucc [THEN leI, THEN le_imp_subset, THEN subset_trans]) |
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69 done |
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70 |
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71 lemma Card_fun_in_Vcsucc: |
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72 "[| f: K -> Vfrom(A, csucc(K)); InfCard(K) |] ==> f: Vfrom(A,csucc(K))" |
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73 by (blast intro: Card_fun_Vcsucc [THEN subsetD]) |
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74 |
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75 lemma Limit_csucc: "InfCard(K) ==> Limit(csucc(K))" |
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76 by (erule InfCard_csucc [THEN InfCard_is_Limit]) |
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77 |
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78 lemmas Pair_in_Vcsucc = Pair_in_VLimit [OF _ _ Limit_csucc] |
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79 lemmas Inl_in_Vcsucc = Inl_in_VLimit [OF _ Limit_csucc] |
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80 lemmas Inr_in_Vcsucc = Inr_in_VLimit [OF _ Limit_csucc] |
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81 lemmas zero_in_Vcsucc = Limit_csucc [THEN zero_in_VLimit] |
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82 lemmas nat_into_Vcsucc = nat_into_VLimit [OF _ Limit_csucc] |
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83 |
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84 (*For handling Cardinals of the form (nat Un |X|) *) |
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85 |
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86 lemmas InfCard_nat_Un_cardinal = InfCard_Un [OF InfCard_nat Card_cardinal] |
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87 |
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88 lemmas le_nat_Un_cardinal = |
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89 Un_upper2_le [OF Ord_nat Card_cardinal [THEN Card_is_Ord]] |
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90 |
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91 lemmas UN_upper_cardinal = UN_upper [THEN subset_imp_lepoll, THEN lepoll_imp_Card_le] |
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92 |
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93 (*The new version of Data_Arg.intrs, declared in Datatype.ML*) |
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94 lemmas Data_Arg_intros = |
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95 SigmaI InlI InrI |
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96 Pair_in_univ Inl_in_univ Inr_in_univ |
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97 zero_in_univ A_into_univ nat_into_univ UnCI |
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98 |
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99 (*For most K-branching datatypes with domain Vfrom(A, csucc(K)) *) |
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100 lemmas inf_datatype_intros = |
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101 InfCard_nat InfCard_nat_Un_cardinal |
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102 Pair_in_Vcsucc Inl_in_Vcsucc Inr_in_Vcsucc |
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103 zero_in_Vcsucc A_into_Vfrom nat_into_Vcsucc |
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104 Card_fun_in_Vcsucc fun_in_Vcsucc' UN_I |
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105 |
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106 end |
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107 |