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1 (* Title: HOL/Tools/Old_Datatype/old_datatype.ML |
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2 Author: Stefan Berghofer, TU Muenchen |
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3 |
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4 Datatype package: definitional introduction of datatypes |
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5 with proof of characteristic theorems: injectivity / distinctness |
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6 of constructors and induction. Main interface to datatypes |
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7 after full bootstrap of datatype package. |
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8 *) |
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9 |
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10 signature OLD_DATATYPE = |
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11 sig |
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12 val distinct_lemma: thm |
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13 type spec = |
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14 (binding * (string * sort) list * mixfix) * |
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15 (binding * typ list * mixfix) list |
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16 type spec_cmd = |
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17 (binding * (string * string option) list * mixfix) * |
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18 (binding * string list * mixfix) list |
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19 val read_specs: spec_cmd list -> theory -> spec list * Proof.context |
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20 val check_specs: spec list -> theory -> spec list * Proof.context |
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21 val add_datatype: Old_Datatype_Aux.config -> spec list -> theory -> string list * theory |
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22 val add_datatype_cmd: Old_Datatype_Aux.config -> spec_cmd list -> theory -> string list * theory |
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23 val spec_cmd: spec_cmd parser |
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24 end; |
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25 |
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26 structure Old_Datatype : OLD_DATATYPE = |
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27 struct |
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28 |
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29 (** auxiliary **) |
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30 |
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31 val distinct_lemma = @{lemma "f x \<noteq> f y ==> x \<noteq> y" by iprover}; |
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32 val (_ $ (_ $ (_ $ (distinct_f $ _) $ _))) = hd (prems_of distinct_lemma); |
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33 |
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34 fun exh_thm_of (dt_info : Old_Datatype_Aux.info Symtab.table) tname = |
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35 #exhaust (the (Symtab.lookup dt_info tname)); |
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36 |
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37 val In0_inject = @{thm In0_inject}; |
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38 val In1_inject = @{thm In1_inject}; |
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39 val Scons_inject = @{thm Scons_inject}; |
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40 val Leaf_inject = @{thm Leaf_inject}; |
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41 val In0_eq = @{thm In0_eq}; |
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42 val In1_eq = @{thm In1_eq}; |
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43 val In0_not_In1 = @{thm In0_not_In1}; |
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44 val In1_not_In0 = @{thm In1_not_In0}; |
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45 val Lim_inject = @{thm Lim_inject}; |
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46 val Inl_inject = @{thm Inl_inject}; |
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47 val Inr_inject = @{thm Inr_inject}; |
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48 val Suml_inject = @{thm Suml_inject}; |
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49 val Sumr_inject = @{thm Sumr_inject}; |
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50 |
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51 val datatype_injI = |
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52 @{lemma "(!!x. ALL y. f x = f y --> x = y) ==> inj f" by (simp add: inj_on_def)}; |
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53 |
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54 |
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55 (** proof of characteristic theorems **) |
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56 |
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57 fun representation_proofs (config : Old_Datatype_Aux.config) |
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58 (dt_info : Old_Datatype_Aux.info Symtab.table) descr types_syntax constr_syntax case_names_induct |
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59 thy = |
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60 let |
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61 val descr' = flat descr; |
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62 val new_type_names = map (Binding.name_of o fst) types_syntax; |
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63 val big_name = space_implode "_" new_type_names; |
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64 val thy1 = Sign.add_path big_name thy; |
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65 val big_rec_name = "rep_set_" ^ big_name; |
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66 val rep_set_names' = |
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67 if length descr' = 1 then [big_rec_name] |
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68 else map (prefix (big_rec_name ^ "_") o string_of_int) (1 upto length descr'); |
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69 val rep_set_names = map (Sign.full_bname thy1) rep_set_names'; |
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70 |
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71 val tyvars = map (fn (_, (_, Ts, _)) => map Old_Datatype_Aux.dest_DtTFree Ts) (hd descr); |
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72 val leafTs' = Old_Datatype_Aux.get_nonrec_types descr'; |
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73 val branchTs = Old_Datatype_Aux.get_branching_types descr'; |
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74 val branchT = |
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75 if null branchTs then HOLogic.unitT |
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76 else Balanced_Tree.make (fn (T, U) => Type (@{type_name Sum_Type.sum}, [T, U])) branchTs; |
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77 val arities = remove (op =) 0 (Old_Datatype_Aux.get_arities descr'); |
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78 val unneeded_vars = |
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79 subtract (op =) (fold Term.add_tfreesT (leafTs' @ branchTs) []) (hd tyvars); |
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80 val leafTs = leafTs' @ map TFree unneeded_vars; |
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81 val recTs = Old_Datatype_Aux.get_rec_types descr'; |
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82 val (newTs, oldTs) = chop (length (hd descr)) recTs; |
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83 val sumT = |
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84 if null leafTs then HOLogic.unitT |
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85 else Balanced_Tree.make (fn (T, U) => Type (@{type_name Sum_Type.sum}, [T, U])) leafTs; |
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86 val Univ_elT = HOLogic.mk_setT (Type (@{type_name Old_Datatype.node}, [sumT, branchT])); |
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87 val UnivT = HOLogic.mk_setT Univ_elT; |
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88 val UnivT' = Univ_elT --> HOLogic.boolT; |
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89 val Collect = Const (@{const_name Collect}, UnivT' --> UnivT); |
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90 |
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91 val In0 = Const (@{const_name Old_Datatype.In0}, Univ_elT --> Univ_elT); |
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92 val In1 = Const (@{const_name Old_Datatype.In1}, Univ_elT --> Univ_elT); |
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93 val Leaf = Const (@{const_name Old_Datatype.Leaf}, sumT --> Univ_elT); |
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94 val Lim = Const (@{const_name Old_Datatype.Lim}, (branchT --> Univ_elT) --> Univ_elT); |
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95 |
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96 (* make injections needed for embedding types in leaves *) |
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97 |
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98 fun mk_inj T' x = |
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99 let |
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100 fun mk_inj' T n i = |
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101 if n = 1 then x |
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102 else |
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103 let |
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104 val n2 = n div 2; |
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105 val Type (_, [T1, T2]) = T; |
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106 in |
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107 if i <= n2 |
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108 then Const (@{const_name Inl}, T1 --> T) $ mk_inj' T1 n2 i |
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109 else Const (@{const_name Inr}, T2 --> T) $ mk_inj' T2 (n - n2) (i - n2) |
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110 end; |
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111 in mk_inj' sumT (length leafTs) (1 + find_index (fn T'' => T'' = T') leafTs) end; |
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112 |
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113 (* make injections for constructors *) |
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114 |
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115 fun mk_univ_inj ts = Balanced_Tree.access |
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116 {left = fn t => In0 $ t, |
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117 right = fn t => In1 $ t, |
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118 init = |
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119 if ts = [] then Const (@{const_name undefined}, Univ_elT) |
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120 else foldr1 (HOLogic.mk_binop @{const_name Old_Datatype.Scons}) ts}; |
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121 |
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122 (* function spaces *) |
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123 |
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124 fun mk_fun_inj T' x = |
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125 let |
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126 fun mk_inj T n i = |
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127 if n = 1 then x |
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128 else |
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129 let |
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130 val n2 = n div 2; |
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131 val Type (_, [T1, T2]) = T; |
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132 fun mkT U = (U --> Univ_elT) --> T --> Univ_elT; |
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133 in |
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134 if i <= n2 then Const (@{const_name Sum_Type.Suml}, mkT T1) $ mk_inj T1 n2 i |
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135 else Const (@{const_name Sum_Type.Sumr}, mkT T2) $ mk_inj T2 (n - n2) (i - n2) |
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136 end; |
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137 in mk_inj branchT (length branchTs) (1 + find_index (fn T'' => T'' = T') branchTs) end; |
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138 |
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139 fun mk_lim t Ts = fold_rev (fn T => fn t => Lim $ mk_fun_inj T (Abs ("x", T, t))) Ts t; |
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140 |
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141 (************** generate introduction rules for representing set **********) |
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142 |
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143 val _ = Old_Datatype_Aux.message config "Constructing representing sets ..."; |
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144 |
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145 (* make introduction rule for a single constructor *) |
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146 |
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147 fun make_intr s n (i, (_, cargs)) = |
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148 let |
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149 fun mk_prem dt (j, prems, ts) = |
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150 (case Old_Datatype_Aux.strip_dtyp dt of |
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151 (dts, Old_Datatype_Aux.DtRec k) => |
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152 let |
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153 val Ts = map (Old_Datatype_Aux.typ_of_dtyp descr') dts; |
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154 val free_t = |
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155 Old_Datatype_Aux.app_bnds (Old_Datatype_Aux.mk_Free "x" (Ts ---> Univ_elT) j) |
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156 (length Ts) |
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157 in |
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158 (j + 1, Logic.list_all (map (pair "x") Ts, |
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159 HOLogic.mk_Trueprop |
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160 (Free (nth rep_set_names' k, UnivT') $ free_t)) :: prems, |
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161 mk_lim free_t Ts :: ts) |
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162 end |
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163 | _ => |
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164 let val T = Old_Datatype_Aux.typ_of_dtyp descr' dt |
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165 in (j + 1, prems, (Leaf $ mk_inj T (Old_Datatype_Aux.mk_Free "x" T j)) :: ts) end); |
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166 |
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167 val (_, prems, ts) = fold_rev mk_prem cargs (1, [], []); |
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168 val concl = HOLogic.mk_Trueprop (Free (s, UnivT') $ mk_univ_inj ts n i); |
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169 in Logic.list_implies (prems, concl) end; |
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170 |
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171 val intr_ts = maps (fn ((_, (_, _, constrs)), rep_set_name) => |
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172 map (make_intr rep_set_name (length constrs)) |
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173 ((1 upto length constrs) ~~ constrs)) (descr' ~~ rep_set_names'); |
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174 |
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175 val ({raw_induct = rep_induct, intrs = rep_intrs, ...}, thy2) = |
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176 thy1 |
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177 |> Sign.map_naming Name_Space.conceal |
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178 |> Inductive.add_inductive_global |
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179 {quiet_mode = #quiet config, verbose = false, alt_name = Binding.name big_rec_name, |
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180 coind = false, no_elim = true, no_ind = false, skip_mono = true} |
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181 (map (fn s => ((Binding.name s, UnivT'), NoSyn)) rep_set_names') [] |
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182 (map (fn x => (Attrib.empty_binding, x)) intr_ts) [] |
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183 ||> Sign.restore_naming thy1; |
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184 |
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185 (********************************* typedef ********************************) |
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186 |
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187 val (typedefs, thy3) = thy2 |
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188 |> Sign.parent_path |
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189 |> fold_map |
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190 (fn (((name, mx), tvs), c) => |
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191 Typedef.add_typedef_global (name, tvs, mx) |
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192 (Collect $ Const (c, UnivT')) NONE |
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193 (rtac exI 1 THEN rtac CollectI 1 THEN |
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194 QUIET_BREADTH_FIRST (has_fewer_prems 1) |
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195 (resolve_tac rep_intrs 1))) |
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196 (types_syntax ~~ tyvars ~~ take (length newTs) rep_set_names) |
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197 ||> Sign.add_path big_name; |
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198 |
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199 (*********************** definition of constructors ***********************) |
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200 |
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201 val big_rep_name = big_name ^ "_Rep_"; |
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202 val rep_names' = map (fn i => big_rep_name ^ string_of_int i) (1 upto length (flat (tl descr))); |
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203 val all_rep_names = |
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204 map (#Rep_name o #1 o #2) typedefs @ |
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205 map (Sign.full_bname thy3) rep_names'; |
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206 |
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207 (* isomorphism declarations *) |
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208 |
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209 val iso_decls = map (fn (T, s) => (Binding.name s, T --> Univ_elT, NoSyn)) |
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210 (oldTs ~~ rep_names'); |
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211 |
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212 (* constructor definitions *) |
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213 |
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214 fun make_constr_def (typedef: Typedef.info) T n |
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215 ((cname, cargs), (cname', mx)) (thy, defs, eqns, i) = |
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216 let |
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217 fun constr_arg dt (j, l_args, r_args) = |
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218 let |
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219 val T = Old_Datatype_Aux.typ_of_dtyp descr' dt; |
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220 val free_t = Old_Datatype_Aux.mk_Free "x" T j; |
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221 in |
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222 (case (Old_Datatype_Aux.strip_dtyp dt, strip_type T) of |
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223 ((_, Old_Datatype_Aux.DtRec m), (Us, U)) => |
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224 (j + 1, free_t :: l_args, mk_lim |
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225 (Const (nth all_rep_names m, U --> Univ_elT) $ |
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226 Old_Datatype_Aux.app_bnds free_t (length Us)) Us :: r_args) |
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227 | _ => (j + 1, free_t :: l_args, (Leaf $ mk_inj T free_t) :: r_args)) |
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228 end; |
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229 |
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230 val (_, l_args, r_args) = fold_rev constr_arg cargs (1, [], []); |
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231 val constrT = map (Old_Datatype_Aux.typ_of_dtyp descr') cargs ---> T; |
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232 val ({Abs_name, Rep_name, ...}, _) = typedef; |
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233 val lhs = list_comb (Const (cname, constrT), l_args); |
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234 val rhs = mk_univ_inj r_args n i; |
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235 val def = Logic.mk_equals (lhs, Const (Abs_name, Univ_elT --> T) $ rhs); |
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236 val def_name = Thm.def_name (Long_Name.base_name cname); |
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237 val eqn = |
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238 HOLogic.mk_Trueprop (HOLogic.mk_eq (Const (Rep_name, T --> Univ_elT) $ lhs, rhs)); |
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239 val ([def_thm], thy') = |
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240 thy |
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241 |> Sign.add_consts [(cname', constrT, mx)] |
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242 |> (Global_Theory.add_defs false o map Thm.no_attributes) [(Binding.name def_name, def)]; |
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243 |
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244 in (thy', defs @ [def_thm], eqns @ [eqn], i + 1) end; |
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245 |
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246 (* constructor definitions for datatype *) |
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247 |
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248 fun dt_constr_defs (((((_, (_, _, constrs)), tname), typedef: Typedef.info), T), constr_syntax) |
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249 (thy, defs, eqns, rep_congs, dist_lemmas) = |
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250 let |
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251 val _ $ (_ $ (cong_f $ _) $ _) = concl_of arg_cong; |
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252 val rep_const = cterm_of thy (Const (#Rep_name (#1 typedef), T --> Univ_elT)); |
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253 val cong' = cterm_instantiate [(cterm_of thy cong_f, rep_const)] arg_cong; |
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254 val dist = cterm_instantiate [(cterm_of thy distinct_f, rep_const)] distinct_lemma; |
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255 val (thy', defs', eqns', _) = |
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256 fold (make_constr_def typedef T (length constrs)) |
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257 (constrs ~~ constr_syntax) (Sign.add_path tname thy, defs, [], 1); |
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258 in |
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259 (Sign.parent_path thy', defs', eqns @ [eqns'], |
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260 rep_congs @ [cong'], dist_lemmas @ [dist]) |
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261 end; |
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262 |
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263 val (thy4, constr_defs, constr_rep_eqns, rep_congs, dist_lemmas) = |
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264 fold dt_constr_defs |
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265 (hd descr ~~ new_type_names ~~ map #2 typedefs ~~ newTs ~~ constr_syntax) |
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266 (thy3 |> Sign.add_consts iso_decls |> Sign.parent_path, [], [], [], []); |
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267 |
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268 |
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269 (*********** isomorphisms for new types (introduced by typedef) ***********) |
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270 |
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271 val _ = Old_Datatype_Aux.message config "Proving isomorphism properties ..."; |
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272 |
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273 val collect_simp = rewrite_rule (Proof_Context.init_global thy4) [mk_meta_eq mem_Collect_eq]; |
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274 |
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275 val newT_iso_axms = typedefs |> map (fn (_, (_, {Abs_inverse, Rep_inverse, Rep, ...})) => |
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276 (collect_simp Abs_inverse, Rep_inverse, collect_simp Rep)); |
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277 |
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278 val newT_iso_inj_thms = typedefs |> map (fn (_, (_, {Abs_inject, Rep_inject, ...})) => |
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279 (collect_simp Abs_inject RS iffD1, Rep_inject RS iffD1)); |
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280 |
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281 (********* isomorphisms between existing types and "unfolded" types *******) |
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282 |
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283 (*---------------------------------------------------------------------*) |
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284 (* isomorphisms are defined using primrec-combinators: *) |
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285 (* generate appropriate functions for instantiating primrec-combinator *) |
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286 (* *) |
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287 (* e.g. Rep_dt_i = list_rec ... (%h t y. In1 (Scons (Leaf h) y)) *) |
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288 (* *) |
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289 (* also generate characteristic equations for isomorphisms *) |
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290 (* *) |
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291 (* e.g. Rep_dt_i (cons h t) = In1 (Scons (Rep_dt_j h) (Rep_dt_i t)) *) |
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292 (*---------------------------------------------------------------------*) |
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293 |
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294 fun make_iso_def k ks n (cname, cargs) (fs, eqns, i) = |
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295 let |
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296 val argTs = map (Old_Datatype_Aux.typ_of_dtyp descr') cargs; |
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297 val T = nth recTs k; |
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298 val rep_const = Const (nth all_rep_names k, T --> Univ_elT); |
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299 val constr = Const (cname, argTs ---> T); |
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300 |
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301 fun process_arg ks' dt (i2, i2', ts, Ts) = |
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302 let |
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303 val T' = Old_Datatype_Aux.typ_of_dtyp descr' dt; |
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304 val (Us, U) = strip_type T' |
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305 in |
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306 (case Old_Datatype_Aux.strip_dtyp dt of |
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307 (_, Old_Datatype_Aux.DtRec j) => |
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308 if member (op =) ks' j then |
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309 (i2 + 1, i2' + 1, ts @ [mk_lim (Old_Datatype_Aux.app_bnds |
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310 (Old_Datatype_Aux.mk_Free "y" (Us ---> Univ_elT) i2') (length Us)) Us], |
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311 Ts @ [Us ---> Univ_elT]) |
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312 else |
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313 (i2 + 1, i2', ts @ [mk_lim |
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314 (Const (nth all_rep_names j, U --> Univ_elT) $ |
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315 Old_Datatype_Aux.app_bnds |
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316 (Old_Datatype_Aux.mk_Free "x" T' i2) (length Us)) Us], Ts) |
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317 | _ => (i2 + 1, i2', ts @ [Leaf $ mk_inj T' (Old_Datatype_Aux.mk_Free "x" T' i2)], Ts)) |
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318 end; |
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319 |
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320 val (i2, i2', ts, Ts) = fold (process_arg ks) cargs (1, 1, [], []); |
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321 val xs = map (uncurry (Old_Datatype_Aux.mk_Free "x")) (argTs ~~ (1 upto (i2 - 1))); |
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322 val ys = map (uncurry (Old_Datatype_Aux.mk_Free "y")) (Ts ~~ (1 upto (i2' - 1))); |
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323 val f = fold_rev lambda (xs @ ys) (mk_univ_inj ts n i); |
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324 |
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325 val (_, _, ts', _) = fold (process_arg []) cargs (1, 1, [], []); |
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326 val eqn = HOLogic.mk_Trueprop (HOLogic.mk_eq |
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327 (rep_const $ list_comb (constr, xs), mk_univ_inj ts' n i)) |
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328 |
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329 in (fs @ [f], eqns @ [eqn], i + 1) end; |
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330 |
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331 (* define isomorphisms for all mutually recursive datatypes in list ds *) |
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332 |
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333 fun make_iso_defs ds (thy, char_thms) = |
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334 let |
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335 val ks = map fst ds; |
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336 val (_, (tname, _, _)) = hd ds; |
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337 val {rec_rewrites, rec_names, ...} = the (Symtab.lookup dt_info tname); |
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338 |
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339 fun process_dt (k, (_, _, constrs)) (fs, eqns, isos) = |
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340 let |
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341 val (fs', eqns', _) = fold (make_iso_def k ks (length constrs)) constrs (fs, eqns, 1); |
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342 val iso = (nth recTs k, nth all_rep_names k); |
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343 in (fs', eqns', isos @ [iso]) end; |
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344 |
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345 val (fs, eqns, isos) = fold process_dt ds ([], [], []); |
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346 val fTs = map fastype_of fs; |
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347 val defs = |
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348 map (fn (rec_name, (T, iso_name)) => |
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349 (Binding.name (Thm.def_name (Long_Name.base_name iso_name)), |
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350 Logic.mk_equals (Const (iso_name, T --> Univ_elT), |
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351 list_comb (Const (rec_name, fTs @ [T] ---> Univ_elT), fs)))) (rec_names ~~ isos); |
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352 val (def_thms, thy') = |
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353 (Global_Theory.add_defs false o map Thm.no_attributes) defs thy; |
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354 |
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355 (* prove characteristic equations *) |
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356 |
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357 val rewrites = def_thms @ map mk_meta_eq rec_rewrites; |
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358 val char_thms' = |
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359 map (fn eqn => Goal.prove_sorry_global thy' [] [] eqn |
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360 (fn {context = ctxt, ...} => EVERY [rewrite_goals_tac ctxt rewrites, rtac refl 1])) eqns; |
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361 |
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362 in (thy', char_thms' @ char_thms) end; |
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363 |
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364 val (thy5, iso_char_thms) = |
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365 fold_rev make_iso_defs (tl descr) (Sign.add_path big_name thy4, []); |
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366 |
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367 (* prove isomorphism properties *) |
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368 |
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369 fun mk_funs_inv thy thm = |
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370 let |
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371 val prop = Thm.prop_of thm; |
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372 val _ $ (_ $ ((S as Const (_, Type (_, [U, _]))) $ _ )) $ |
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373 (_ $ (_ $ (r $ (a $ _)) $ _)) = Type.legacy_freeze prop; |
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374 val used = Term.add_tfree_names a []; |
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375 |
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376 fun mk_thm i = |
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377 let |
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378 val Ts = map (TFree o rpair @{sort type}) (Name.variant_list used (replicate i "'t")); |
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379 val f = Free ("f", Ts ---> U); |
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380 in |
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381 Goal.prove_sorry_global thy [] [] |
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382 (Logic.mk_implies |
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383 (HOLogic.mk_Trueprop (HOLogic.list_all |
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384 (map (pair "x") Ts, S $ Old_Datatype_Aux.app_bnds f i)), |
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385 HOLogic.mk_Trueprop (HOLogic.mk_eq (fold_rev (Term.abs o pair "x") Ts |
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386 (r $ (a $ Old_Datatype_Aux.app_bnds f i)), f)))) |
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387 (fn _ => EVERY [REPEAT_DETERM_N i (rtac @{thm ext} 1), |
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388 REPEAT (etac allE 1), rtac thm 1, atac 1]) |
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389 end |
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390 in map (fn r => r RS subst) (thm :: map mk_thm arities) end; |
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391 |
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392 (* prove inj Rep_dt_i and Rep_dt_i x : rep_set_dt_i *) |
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393 |
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394 val fun_congs = |
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395 map (fn T => make_elim (Drule.instantiate' [SOME (ctyp_of thy5 T)] [] fun_cong)) branchTs; |
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396 |
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397 fun prove_iso_thms ds (inj_thms, elem_thms) = |
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398 let |
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399 val (_, (tname, _, _)) = hd ds; |
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400 val induct = #induct (the (Symtab.lookup dt_info tname)); |
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401 |
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402 fun mk_ind_concl (i, _) = |
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403 let |
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404 val T = nth recTs i; |
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405 val Rep_t = Const (nth all_rep_names i, T --> Univ_elT); |
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406 val rep_set_name = nth rep_set_names i; |
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407 val concl1 = |
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408 HOLogic.all_const T $ Abs ("y", T, HOLogic.imp $ |
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409 HOLogic.mk_eq (Rep_t $ Old_Datatype_Aux.mk_Free "x" T i, Rep_t $ Bound 0) $ |
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410 HOLogic.mk_eq (Old_Datatype_Aux.mk_Free "x" T i, Bound 0)); |
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411 val concl2 = Const (rep_set_name, UnivT') $ (Rep_t $ Old_Datatype_Aux.mk_Free "x" T i); |
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412 in (concl1, concl2) end; |
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413 |
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414 val (ind_concl1, ind_concl2) = split_list (map mk_ind_concl ds); |
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415 |
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416 val rewrites = map mk_meta_eq iso_char_thms; |
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417 val inj_thms' = map snd newT_iso_inj_thms @ map (fn r => r RS @{thm injD}) inj_thms; |
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418 |
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419 val inj_thm = |
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420 Goal.prove_sorry_global thy5 [] [] |
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421 (HOLogic.mk_Trueprop (Old_Datatype_Aux.mk_conj ind_concl1)) |
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422 (fn {context = ctxt, ...} => EVERY |
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423 [(Old_Datatype_Aux.ind_tac induct [] THEN_ALL_NEW Object_Logic.atomize_prems_tac ctxt) 1, |
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424 REPEAT (EVERY |
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425 [rtac allI 1, rtac impI 1, |
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426 Old_Datatype_Aux.exh_tac (exh_thm_of dt_info) 1, |
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427 REPEAT (EVERY |
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428 [hyp_subst_tac ctxt 1, |
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429 rewrite_goals_tac ctxt rewrites, |
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430 REPEAT (dresolve_tac [In0_inject, In1_inject] 1), |
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431 (eresolve_tac [In0_not_In1 RS notE, In1_not_In0 RS notE] 1) |
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432 ORELSE (EVERY |
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433 [REPEAT (eresolve_tac (Scons_inject :: |
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434 map make_elim [Leaf_inject, Inl_inject, Inr_inject]) 1), |
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435 REPEAT (cong_tac 1), rtac refl 1, |
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436 REPEAT (atac 1 ORELSE (EVERY |
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437 [REPEAT (rtac @{thm ext} 1), |
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438 REPEAT (eresolve_tac (mp :: allE :: |
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439 map make_elim (Suml_inject :: Sumr_inject :: |
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440 Lim_inject :: inj_thms') @ fun_congs) 1), |
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441 atac 1]))])])])]); |
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442 |
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443 val inj_thms'' = map (fn r => r RS datatype_injI) (Old_Datatype_Aux.split_conj_thm inj_thm); |
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444 |
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445 val elem_thm = |
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446 Goal.prove_sorry_global thy5 [] [] |
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447 (HOLogic.mk_Trueprop (Old_Datatype_Aux.mk_conj ind_concl2)) |
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448 (fn {context = ctxt, ...} => |
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449 EVERY [ |
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450 (Old_Datatype_Aux.ind_tac induct [] THEN_ALL_NEW Object_Logic.atomize_prems_tac ctxt) 1, |
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451 rewrite_goals_tac ctxt rewrites, |
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452 REPEAT ((resolve_tac rep_intrs THEN_ALL_NEW |
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453 ((REPEAT o etac allE) THEN' ares_tac elem_thms)) 1)]); |
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454 |
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455 in (inj_thms'' @ inj_thms, elem_thms @ Old_Datatype_Aux.split_conj_thm elem_thm) end; |
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456 |
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457 val (iso_inj_thms_unfolded, iso_elem_thms) = |
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458 fold_rev prove_iso_thms (tl descr) ([], map #3 newT_iso_axms); |
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459 val iso_inj_thms = |
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460 map snd newT_iso_inj_thms @ map (fn r => r RS @{thm injD}) iso_inj_thms_unfolded; |
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461 |
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462 (* prove rep_set_dt_i x --> x : range Rep_dt_i *) |
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463 |
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464 fun mk_iso_t (((set_name, iso_name), i), T) = |
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465 let val isoT = T --> Univ_elT in |
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466 HOLogic.imp $ |
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467 (Const (set_name, UnivT') $ Old_Datatype_Aux.mk_Free "x" Univ_elT i) $ |
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468 (if i < length newTs then @{term True} |
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469 else HOLogic.mk_mem (Old_Datatype_Aux.mk_Free "x" Univ_elT i, |
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470 Const (@{const_name image}, isoT --> HOLogic.mk_setT T --> UnivT) $ |
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471 Const (iso_name, isoT) $ Const (@{const_abbrev UNIV}, HOLogic.mk_setT T))) |
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472 end; |
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473 |
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474 val iso_t = HOLogic.mk_Trueprop (Old_Datatype_Aux.mk_conj (map mk_iso_t |
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475 (rep_set_names ~~ all_rep_names ~~ (0 upto (length descr' - 1)) ~~ recTs))); |
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476 |
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477 (* all the theorems are proved by one single simultaneous induction *) |
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478 |
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479 val range_eqs = map (fn r => mk_meta_eq (r RS @{thm range_ex1_eq})) iso_inj_thms_unfolded; |
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480 |
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481 val iso_thms = |
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482 if length descr = 1 then [] |
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483 else |
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484 drop (length newTs) (Old_Datatype_Aux.split_conj_thm |
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485 (Goal.prove_sorry_global thy5 [] [] iso_t (fn {context = ctxt, ...} => EVERY |
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486 [(Old_Datatype_Aux.ind_tac rep_induct [] THEN_ALL_NEW |
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487 Object_Logic.atomize_prems_tac ctxt) 1, |
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488 REPEAT (rtac TrueI 1), |
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489 rewrite_goals_tac ctxt (mk_meta_eq @{thm choice_eq} :: |
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490 Thm.symmetric (mk_meta_eq @{thm fun_eq_iff}) :: range_eqs), |
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491 rewrite_goals_tac ctxt (map Thm.symmetric range_eqs), |
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492 REPEAT (EVERY |
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493 [REPEAT (eresolve_tac ([rangeE, @{thm ex1_implies_ex} RS exE] @ |
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494 maps (mk_funs_inv thy5 o #1) newT_iso_axms) 1), |
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495 TRY (hyp_subst_tac ctxt 1), |
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496 rtac (sym RS range_eqI) 1, |
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497 resolve_tac iso_char_thms 1])]))); |
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498 |
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499 val Abs_inverse_thms' = |
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500 map #1 newT_iso_axms @ |
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501 map2 (fn r_inj => fn r => @{thm f_the_inv_into_f} OF [r_inj, r RS mp]) |
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502 iso_inj_thms_unfolded iso_thms; |
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503 |
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504 val Abs_inverse_thms = maps (mk_funs_inv thy5) Abs_inverse_thms'; |
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505 |
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506 (******************* freeness theorems for constructors *******************) |
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507 |
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508 val _ = Old_Datatype_Aux.message config "Proving freeness of constructors ..."; |
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509 |
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510 (* prove theorem Rep_i (Constr_j ...) = Inj_j ... *) |
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511 |
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512 fun prove_constr_rep_thm eqn = |
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513 let |
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514 val inj_thms = map fst newT_iso_inj_thms; |
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515 val rewrites = @{thm o_def} :: constr_defs @ map (mk_meta_eq o #2) newT_iso_axms; |
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516 in |
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517 Goal.prove_sorry_global thy5 [] [] eqn |
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518 (fn {context = ctxt, ...} => EVERY |
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519 [resolve_tac inj_thms 1, |
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520 rewrite_goals_tac ctxt rewrites, |
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521 rtac refl 3, |
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522 resolve_tac rep_intrs 2, |
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523 REPEAT (resolve_tac iso_elem_thms 1)]) |
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524 end; |
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525 |
|
526 (*--------------------------------------------------------------*) |
|
527 (* constr_rep_thms and rep_congs are used to prove distinctness *) |
|
528 (* of constructors. *) |
|
529 (*--------------------------------------------------------------*) |
|
530 |
|
531 val constr_rep_thms = map (map prove_constr_rep_thm) constr_rep_eqns; |
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532 |
|
533 val dist_rewrites = |
|
534 map (fn (rep_thms, dist_lemma) => |
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535 dist_lemma :: (rep_thms @ [In0_eq, In1_eq, In0_not_In1, In1_not_In0])) |
|
536 (constr_rep_thms ~~ dist_lemmas); |
|
537 |
|
538 fun prove_distinct_thms dist_rewrites' = |
|
539 let |
|
540 fun prove [] = [] |
|
541 | prove (t :: ts) = |
|
542 let |
|
543 val dist_thm = Goal.prove_sorry_global thy5 [] [] t (fn {context = ctxt, ...} => |
|
544 EVERY [simp_tac (put_simpset HOL_ss ctxt addsimps dist_rewrites') 1]) |
|
545 in dist_thm :: Drule.zero_var_indexes (dist_thm RS not_sym) :: prove ts end; |
|
546 in prove end; |
|
547 |
|
548 val distinct_thms = |
|
549 map2 (prove_distinct_thms) dist_rewrites (Old_Datatype_Prop.make_distincts descr); |
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550 |
|
551 (* prove injectivity of constructors *) |
|
552 |
|
553 fun prove_constr_inj_thm rep_thms t = |
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554 let |
|
555 val inj_thms = Scons_inject :: |
|
556 map make_elim |
|
557 (iso_inj_thms @ |
|
558 [In0_inject, In1_inject, Leaf_inject, Inl_inject, Inr_inject, |
|
559 Lim_inject, Suml_inject, Sumr_inject]) |
|
560 in |
|
561 Goal.prove_sorry_global thy5 [] [] t |
|
562 (fn {context = ctxt, ...} => EVERY |
|
563 [rtac iffI 1, |
|
564 REPEAT (etac conjE 2), hyp_subst_tac ctxt 2, rtac refl 2, |
|
565 dresolve_tac rep_congs 1, dtac @{thm box_equals} 1, |
|
566 REPEAT (resolve_tac rep_thms 1), |
|
567 REPEAT (eresolve_tac inj_thms 1), |
|
568 REPEAT (ares_tac [conjI] 1 ORELSE (EVERY [REPEAT (rtac @{thm ext} 1), |
|
569 REPEAT (eresolve_tac (make_elim fun_cong :: inj_thms) 1), |
|
570 atac 1]))]) |
|
571 end; |
|
572 |
|
573 val constr_inject = |
|
574 map (fn (ts, thms) => map (prove_constr_inj_thm thms) ts) |
|
575 (Old_Datatype_Prop.make_injs descr ~~ constr_rep_thms); |
|
576 |
|
577 val ((constr_inject', distinct_thms'), thy6) = |
|
578 thy5 |
|
579 |> Sign.parent_path |
|
580 |> Old_Datatype_Aux.store_thmss "inject" new_type_names constr_inject |
|
581 ||>> Old_Datatype_Aux.store_thmss "distinct" new_type_names distinct_thms; |
|
582 |
|
583 (*************************** induction theorem ****************************) |
|
584 |
|
585 val _ = Old_Datatype_Aux.message config "Proving induction rule for datatypes ..."; |
|
586 |
|
587 val Rep_inverse_thms = |
|
588 map (fn (_, iso, _) => iso RS subst) newT_iso_axms @ |
|
589 map (fn r => r RS @{thm the_inv_f_f} RS subst) iso_inj_thms_unfolded; |
|
590 val Rep_inverse_thms' = map (fn r => r RS @{thm the_inv_f_f}) iso_inj_thms_unfolded; |
|
591 |
|
592 fun mk_indrule_lemma (i, _) T = |
|
593 let |
|
594 val Rep_t = Const (nth all_rep_names i, T --> Univ_elT) $ Old_Datatype_Aux.mk_Free "x" T i; |
|
595 val Abs_t = |
|
596 if i < length newTs then |
|
597 Const (#Abs_name (#1 (#2 (nth typedefs i))), Univ_elT --> T) |
|
598 else |
|
599 Const (@{const_name the_inv_into}, |
|
600 [HOLogic.mk_setT T, T --> Univ_elT, Univ_elT] ---> T) $ |
|
601 HOLogic.mk_UNIV T $ Const (nth all_rep_names i, T --> Univ_elT); |
|
602 val prem = |
|
603 HOLogic.imp $ |
|
604 (Const (nth rep_set_names i, UnivT') $ Rep_t) $ |
|
605 (Old_Datatype_Aux.mk_Free "P" (T --> HOLogic.boolT) (i + 1) $ (Abs_t $ Rep_t)); |
|
606 val concl = |
|
607 Old_Datatype_Aux.mk_Free "P" (T --> HOLogic.boolT) (i + 1) $ |
|
608 Old_Datatype_Aux.mk_Free "x" T i; |
|
609 in (prem, concl) end; |
|
610 |
|
611 val (indrule_lemma_prems, indrule_lemma_concls) = |
|
612 split_list (map2 mk_indrule_lemma descr' recTs); |
|
613 |
|
614 val cert = cterm_of thy6; |
|
615 |
|
616 val indrule_lemma = |
|
617 Goal.prove_sorry_global thy6 [] [] |
|
618 (Logic.mk_implies |
|
619 (HOLogic.mk_Trueprop (Old_Datatype_Aux.mk_conj indrule_lemma_prems), |
|
620 HOLogic.mk_Trueprop (Old_Datatype_Aux.mk_conj indrule_lemma_concls))) |
|
621 (fn _ => |
|
622 EVERY |
|
623 [REPEAT (etac conjE 1), |
|
624 REPEAT (EVERY |
|
625 [TRY (rtac conjI 1), resolve_tac Rep_inverse_thms 1, |
|
626 etac mp 1, resolve_tac iso_elem_thms 1])]); |
|
627 |
|
628 val Ps = map head_of (HOLogic.dest_conj (HOLogic.dest_Trueprop (concl_of indrule_lemma))); |
|
629 val frees = |
|
630 if length Ps = 1 then [Free ("P", snd (dest_Var (hd Ps)))] |
|
631 else map (Free o apfst fst o dest_Var) Ps; |
|
632 val indrule_lemma' = cterm_instantiate (map cert Ps ~~ map cert frees) indrule_lemma; |
|
633 |
|
634 val dt_induct_prop = Old_Datatype_Prop.make_ind descr; |
|
635 val dt_induct = |
|
636 Goal.prove_sorry_global thy6 [] |
|
637 (Logic.strip_imp_prems dt_induct_prop) |
|
638 (Logic.strip_imp_concl dt_induct_prop) |
|
639 (fn {context = ctxt, prems, ...} => |
|
640 EVERY |
|
641 [rtac indrule_lemma' 1, |
|
642 (Old_Datatype_Aux.ind_tac rep_induct [] THEN_ALL_NEW |
|
643 Object_Logic.atomize_prems_tac ctxt) 1, |
|
644 EVERY (map (fn (prem, r) => (EVERY |
|
645 [REPEAT (eresolve_tac Abs_inverse_thms 1), |
|
646 simp_tac (put_simpset HOL_basic_ss ctxt |
|
647 addsimps (Thm.symmetric r :: Rep_inverse_thms')) 1, |
|
648 DEPTH_SOLVE_1 (ares_tac [prem] 1 ORELSE etac allE 1)])) |
|
649 (prems ~~ (constr_defs @ map mk_meta_eq iso_char_thms)))]); |
|
650 |
|
651 val ([(_, [dt_induct'])], thy7) = |
|
652 thy6 |
|
653 |> Global_Theory.note_thmss "" |
|
654 [((Binding.qualify true big_name (Binding.name "induct"), [case_names_induct]), |
|
655 [([dt_induct], [])])]; |
|
656 in |
|
657 ((constr_inject', distinct_thms', dt_induct'), thy7) |
|
658 end; |
|
659 |
|
660 |
|
661 |
|
662 (** datatype definition **) |
|
663 |
|
664 (* specifications *) |
|
665 |
|
666 type spec = (binding * (string * sort) list * mixfix) * (binding * typ list * mixfix) list; |
|
667 |
|
668 type spec_cmd = |
|
669 (binding * (string * string option) list * mixfix) * (binding * string list * mixfix) list; |
|
670 |
|
671 local |
|
672 |
|
673 fun parse_spec ctxt ((b, args, mx), constrs) = |
|
674 ((b, map (apsnd (Typedecl.read_constraint ctxt)) args, mx), |
|
675 constrs |> map (fn (c, Ts, mx') => (c, map (Syntax.parse_typ ctxt) Ts, mx'))); |
|
676 |
|
677 fun check_specs ctxt (specs: spec list) = |
|
678 let |
|
679 fun prep_spec ((tname, args, mx), constrs) tys = |
|
680 let |
|
681 val (args', tys1) = chop (length args) tys; |
|
682 val (constrs', tys3) = (constrs, tys1) |-> fold_map (fn (cname, cargs, mx') => fn tys2 => |
|
683 let val (cargs', tys3) = chop (length cargs) tys2; |
|
684 in ((cname, cargs', mx'), tys3) end); |
|
685 in (((tname, map dest_TFree args', mx), constrs'), tys3) end; |
|
686 |
|
687 val all_tys = |
|
688 specs |> maps (fn ((_, args, _), cs) => map TFree args @ maps #2 cs) |
|
689 |> Syntax.check_typs ctxt; |
|
690 |
|
691 in #1 (fold_map prep_spec specs all_tys) end; |
|
692 |
|
693 fun prep_specs parse raw_specs thy = |
|
694 let |
|
695 val ctxt = thy |
|
696 |> Sign.add_types_global (map (fn ((b, args, mx), _) => (b, length args, mx)) raw_specs) |
|
697 |> Proof_Context.init_global |
|
698 |> fold (fn ((_, args, _), _) => fold (fn (a, _) => |
|
699 Variable.declare_typ (TFree (a, dummyS))) args) raw_specs; |
|
700 val specs = check_specs ctxt (map (parse ctxt) raw_specs); |
|
701 in (specs, ctxt) end; |
|
702 |
|
703 in |
|
704 |
|
705 val read_specs = prep_specs parse_spec; |
|
706 val check_specs = prep_specs (K I); |
|
707 |
|
708 end; |
|
709 |
|
710 |
|
711 (* main commands *) |
|
712 |
|
713 fun gen_add_datatype prep_specs config raw_specs thy = |
|
714 let |
|
715 val _ = Theory.requires thy (Context.theory_name @{theory}) "datatype definitions"; |
|
716 |
|
717 val (dts, spec_ctxt) = prep_specs raw_specs thy; |
|
718 val ((_, tyvars, _), _) :: _ = dts; |
|
719 val string_of_tyvar = Syntax.string_of_typ spec_ctxt o TFree; |
|
720 |
|
721 val (new_dts, types_syntax) = dts |> map (fn ((tname, tvs, mx), _) => |
|
722 let val full_tname = Sign.full_name thy tname in |
|
723 (case duplicates (op =) tvs of |
|
724 [] => |
|
725 if eq_set (op =) (tyvars, tvs) then ((full_tname, tvs), (tname, mx)) |
|
726 else error "Mutually recursive datatypes must have same type parameters" |
|
727 | dups => |
|
728 error ("Duplicate parameter(s) for datatype " ^ Binding.print tname ^ |
|
729 " : " ^ commas (map string_of_tyvar dups))) |
|
730 end) |> split_list; |
|
731 val dt_names = map fst new_dts; |
|
732 |
|
733 val _ = |
|
734 (case duplicates (op =) (map fst new_dts) of |
|
735 [] => () |
|
736 | dups => error ("Duplicate datatypes: " ^ commas_quote dups)); |
|
737 |
|
738 fun prep_dt_spec ((tname, tvs, _), constrs) (dts', constr_syntax, i) = |
|
739 let |
|
740 fun prep_constr (cname, cargs, mx) (constrs, constr_syntax') = |
|
741 let |
|
742 val _ = |
|
743 (case subtract (op =) tvs (fold Term.add_tfreesT cargs []) of |
|
744 [] => () |
|
745 | vs => error ("Extra type variables on rhs: " ^ commas (map string_of_tyvar vs))); |
|
746 val c = Sign.full_name_path thy (Binding.name_of tname) cname; |
|
747 in |
|
748 (constrs @ [(c, map (Old_Datatype_Aux.dtyp_of_typ new_dts) cargs)], |
|
749 constr_syntax' @ [(cname, mx)]) |
|
750 end handle ERROR msg => |
|
751 cat_error msg ("The error above occurred in constructor " ^ Binding.print cname ^ |
|
752 " of datatype " ^ Binding.print tname); |
|
753 |
|
754 val (constrs', constr_syntax') = fold prep_constr constrs ([], []); |
|
755 in |
|
756 (case duplicates (op =) (map fst constrs') of |
|
757 [] => |
|
758 (dts' @ [(i, (Sign.full_name thy tname, map Old_Datatype_Aux.DtTFree tvs, constrs'))], |
|
759 constr_syntax @ [constr_syntax'], i + 1) |
|
760 | dups => |
|
761 error ("Duplicate constructors " ^ commas_quote dups ^ |
|
762 " in datatype " ^ Binding.print tname)) |
|
763 end; |
|
764 |
|
765 val (dts', constr_syntax, i) = fold prep_dt_spec dts ([], [], 0); |
|
766 |
|
767 val dt_info = Old_Datatype_Data.get_all thy; |
|
768 val (descr, _) = Old_Datatype_Aux.unfold_datatypes spec_ctxt dts' dt_info dts' i; |
|
769 val _ = |
|
770 Old_Datatype_Aux.check_nonempty descr |
|
771 handle (exn as Old_Datatype_Aux.Datatype_Empty s) => |
|
772 if #strict config then error ("Nonemptiness check failed for datatype " ^ quote s) |
|
773 else reraise exn; |
|
774 |
|
775 val _ = |
|
776 Old_Datatype_Aux.message config |
|
777 ("Constructing datatype(s) " ^ commas_quote (map (Binding.name_of o #1 o #1) dts)); |
|
778 in |
|
779 thy |
|
780 |> representation_proofs config dt_info descr types_syntax constr_syntax |
|
781 (Old_Datatype_Data.mk_case_names_induct (flat descr)) |
|
782 |-> (fn (inject, distinct, induct) => |
|
783 Old_Rep_Datatype.derive_datatype_props config dt_names descr induct inject distinct) |
|
784 end; |
|
785 |
|
786 val add_datatype = gen_add_datatype check_specs; |
|
787 val add_datatype_cmd = gen_add_datatype read_specs; |
|
788 |
|
789 |
|
790 (* outer syntax *) |
|
791 |
|
792 val spec_cmd = |
|
793 Parse.type_args_constrained -- Parse.binding -- Parse.opt_mixfix -- |
|
794 (@{keyword "="} |-- Parse.enum1 "|" (Parse.binding -- Scan.repeat Parse.typ -- Parse.opt_mixfix)) |
|
795 >> (fn (((vs, t), mx), cons) => ((t, vs, mx), map Parse.triple1 cons)); |
|
796 |
|
797 val _ = |
|
798 Outer_Syntax.command @{command_spec "datatype"} "define inductive datatypes" |
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799 (Parse.and_list1 spec_cmd |
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800 >> (Toplevel.theory o (snd oo add_datatype_cmd Old_Datatype_Aux.default_config))); |
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801 |
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802 end; |