1 (* Title: HOL/Nominal/nominal.ML |
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2 Author: Stefan Berghofer and Christian Urban, TU Muenchen |
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3 |
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4 Nominal datatype package for Isabelle/HOL. |
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5 *) |
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6 |
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7 signature NOMINAL = |
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8 sig |
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9 val add_nominal_datatype : Datatype.config -> string list -> |
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10 (string list * bstring * mixfix * |
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11 (bstring * string list * mixfix) list) list -> theory -> theory |
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12 type descr |
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13 type nominal_datatype_info |
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14 val get_nominal_datatypes : theory -> nominal_datatype_info Symtab.table |
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15 val get_nominal_datatype : theory -> string -> nominal_datatype_info option |
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16 val mk_perm: typ list -> term -> term -> term |
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17 val perm_of_pair: term * term -> term |
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18 val mk_not_sym: thm list -> thm list |
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19 val perm_simproc: simproc |
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20 val fresh_const: typ -> typ -> term |
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21 val fresh_star_const: typ -> typ -> term |
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22 end |
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23 |
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24 structure Nominal : NOMINAL = |
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25 struct |
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26 |
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27 val finite_emptyI = thm "finite.emptyI"; |
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28 val finite_Diff = thm "finite_Diff"; |
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29 val finite_Un = thm "finite_Un"; |
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30 val Un_iff = thm "Un_iff"; |
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31 val In0_eq = thm "In0_eq"; |
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32 val In1_eq = thm "In1_eq"; |
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33 val In0_not_In1 = thm "In0_not_In1"; |
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34 val In1_not_In0 = thm "In1_not_In0"; |
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35 val Un_assoc = thm "Un_assoc"; |
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36 val Collect_disj_eq = thm "Collect_disj_eq"; |
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37 val empty_def = thm "empty_def"; |
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38 val empty_iff = thm "empty_iff"; |
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39 |
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40 open DatatypeAux; |
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41 open NominalAtoms; |
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42 |
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43 (** FIXME: Datatype should export this function **) |
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44 |
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45 local |
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46 |
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47 fun dt_recs (DtTFree _) = [] |
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48 | dt_recs (DtType (_, dts)) = List.concat (map dt_recs dts) |
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49 | dt_recs (DtRec i) = [i]; |
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50 |
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51 fun dt_cases (descr: descr) (_, args, constrs) = |
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52 let |
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53 fun the_bname i = Long_Name.base_name (#1 (valOf (AList.lookup (op =) descr i))); |
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54 val bnames = map the_bname (distinct op = (List.concat (map dt_recs args))); |
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55 in map (fn (c, _) => space_implode "_" (Long_Name.base_name c :: bnames)) constrs end; |
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56 |
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57 |
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58 fun induct_cases descr = |
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59 DatatypeProp.indexify_names (List.concat (map (dt_cases descr) (map #2 descr))); |
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60 |
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61 fun exhaust_cases descr i = dt_cases descr (valOf (AList.lookup (op =) descr i)); |
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62 |
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63 in |
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64 |
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65 fun mk_case_names_induct descr = RuleCases.case_names (induct_cases descr); |
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66 |
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67 fun mk_case_names_exhausts descr new = |
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68 map (RuleCases.case_names o exhaust_cases descr o #1) |
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69 (List.filter (fn ((_, (name, _, _))) => name mem_string new) descr); |
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70 |
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71 end; |
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72 |
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73 (* theory data *) |
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74 |
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75 type descr = (int * (string * dtyp list * (string * (dtyp list * dtyp) list) list)) list; |
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76 |
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77 type nominal_datatype_info = |
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78 {index : int, |
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79 descr : descr, |
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80 sorts : (string * sort) list, |
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81 rec_names : string list, |
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82 rec_rewrites : thm list, |
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83 induction : thm, |
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84 distinct : thm list, |
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85 inject : thm list}; |
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86 |
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87 structure NominalDatatypesData = TheoryDataFun |
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88 ( |
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89 type T = nominal_datatype_info Symtab.table; |
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90 val empty = Symtab.empty; |
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91 val copy = I; |
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92 val extend = I; |
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93 fun merge _ tabs : T = Symtab.merge (K true) tabs; |
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94 ); |
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95 |
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96 val get_nominal_datatypes = NominalDatatypesData.get; |
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97 val put_nominal_datatypes = NominalDatatypesData.put; |
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98 val map_nominal_datatypes = NominalDatatypesData.map; |
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99 val get_nominal_datatype = Symtab.lookup o get_nominal_datatypes; |
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100 |
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101 |
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102 (**** make datatype info ****) |
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103 |
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104 fun make_dt_info descr sorts induct reccomb_names rec_thms |
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105 (((i, (_, (tname, _, _))), distinct), inject) = |
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106 (tname, |
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107 {index = i, |
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108 descr = descr, |
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109 sorts = sorts, |
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110 rec_names = reccomb_names, |
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111 rec_rewrites = rec_thms, |
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112 induction = induct, |
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113 distinct = distinct, |
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114 inject = inject}); |
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115 |
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116 (*******************************) |
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117 |
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118 val (_ $ (_ $ (_ $ (distinct_f $ _) $ _))) = hd (prems_of distinct_lemma); |
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119 |
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120 |
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121 (** simplification procedure for sorting permutations **) |
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122 |
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123 val dj_cp = thm "dj_cp"; |
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124 |
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125 fun dest_permT (Type ("fun", [Type ("List.list", [Type ("*", [T, _])]), |
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126 Type ("fun", [_, U])])) = (T, U); |
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127 |
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128 fun permTs_of (Const ("Nominal.perm", T) $ t $ u) = fst (dest_permT T) :: permTs_of u |
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129 | permTs_of _ = []; |
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130 |
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131 fun perm_simproc' thy ss (Const ("Nominal.perm", T) $ t $ (u as Const ("Nominal.perm", U) $ r $ s)) = |
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132 let |
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133 val (aT as Type (a, []), S) = dest_permT T; |
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134 val (bT as Type (b, []), _) = dest_permT U |
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135 in if aT mem permTs_of u andalso aT <> bT then |
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136 let |
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137 val cp = cp_inst_of thy a b; |
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138 val dj = dj_thm_of thy b a; |
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139 val dj_cp' = [cp, dj] MRS dj_cp; |
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140 val cert = SOME o cterm_of thy |
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141 in |
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142 SOME (mk_meta_eq (Drule.instantiate' [SOME (ctyp_of thy S)] |
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143 [cert t, cert r, cert s] dj_cp')) |
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144 end |
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145 else NONE |
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146 end |
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147 | perm_simproc' thy ss _ = NONE; |
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148 |
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149 val perm_simproc = |
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150 Simplifier.simproc (the_context ()) "perm_simp" ["pi1 \<bullet> (pi2 \<bullet> x)"] perm_simproc'; |
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151 |
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152 val meta_spec = thm "meta_spec"; |
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153 |
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154 fun projections rule = |
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155 ProjectRule.projections (ProofContext.init (Thm.theory_of_thm rule)) rule |
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156 |> map (standard #> RuleCases.save rule); |
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157 |
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158 val supp_prod = thm "supp_prod"; |
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159 val fresh_prod = thm "fresh_prod"; |
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160 val supports_fresh = thm "supports_fresh"; |
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161 val supports_def = thm "Nominal.supports_def"; |
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162 val fresh_def = thm "fresh_def"; |
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163 val supp_def = thm "supp_def"; |
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164 val rev_simps = thms "rev.simps"; |
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165 val app_simps = thms "append.simps"; |
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166 val at_fin_set_supp = thm "at_fin_set_supp"; |
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167 val at_fin_set_fresh = thm "at_fin_set_fresh"; |
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168 val abs_fun_eq1 = thm "abs_fun_eq1"; |
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169 |
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170 val collect_simp = rewrite_rule [mk_meta_eq mem_Collect_eq]; |
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171 |
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172 fun mk_perm Ts t u = |
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173 let |
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174 val T = fastype_of1 (Ts, t); |
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175 val U = fastype_of1 (Ts, u) |
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176 in Const ("Nominal.perm", T --> U --> U) $ t $ u end; |
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177 |
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178 fun perm_of_pair (x, y) = |
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179 let |
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180 val T = fastype_of x; |
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181 val pT = mk_permT T |
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182 in Const ("List.list.Cons", HOLogic.mk_prodT (T, T) --> pT --> pT) $ |
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183 HOLogic.mk_prod (x, y) $ Const ("List.list.Nil", pT) |
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184 end; |
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185 |
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186 fun mk_not_sym ths = maps (fn th => case prop_of th of |
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187 _ $ (Const ("Not", _) $ (Const ("op =", _) $ _ $ _)) => [th, th RS not_sym] |
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188 | _ => [th]) ths; |
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189 |
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190 fun fresh_const T U = Const ("Nominal.fresh", T --> U --> HOLogic.boolT); |
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191 fun fresh_star_const T U = |
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192 Const ("Nominal.fresh_star", HOLogic.mk_setT T --> U --> HOLogic.boolT); |
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193 |
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194 fun gen_add_nominal_datatype prep_typ config new_type_names dts thy = |
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195 let |
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196 (* this theory is used just for parsing *) |
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197 |
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198 val tmp_thy = thy |> |
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199 Theory.copy |> |
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200 Sign.add_types (map (fn (tvs, tname, mx, _) => |
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201 (Binding.name tname, length tvs, mx)) dts); |
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202 |
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203 val atoms = atoms_of thy; |
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204 |
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205 fun prep_constr ((constrs, sorts), (cname, cargs, mx)) = |
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206 let val (cargs', sorts') = Library.foldl (prep_typ tmp_thy) (([], sorts), cargs) |
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207 in (constrs @ [(cname, cargs', mx)], sorts') end |
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208 |
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209 fun prep_dt_spec ((dts, sorts), (tvs, tname, mx, constrs)) = |
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210 let val (constrs', sorts') = Library.foldl prep_constr (([], sorts), constrs) |
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211 in (dts @ [(tvs, tname, mx, constrs')], sorts') end |
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212 |
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213 val (dts', sorts) = Library.foldl prep_dt_spec (([], []), dts); |
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214 val tyvars = map (map (fn s => |
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215 (s, the (AList.lookup (op =) sorts s))) o #1) dts'; |
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216 |
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217 fun inter_sort thy S S' = Type.inter_sort (Sign.tsig_of thy) (S, S'); |
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218 fun augment_sort_typ thy S = |
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219 let val S = Sign.certify_sort thy S |
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220 in map_type_tfree (fn (s, S') => TFree (s, |
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221 if member (op = o apsnd fst) sorts s then inter_sort thy S S' else S')) |
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222 end; |
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223 fun augment_sort thy S = map_types (augment_sort_typ thy S); |
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224 |
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225 val types_syntax = map (fn (tvs, tname, mx, constrs) => (tname, mx)) dts'; |
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226 val constr_syntax = map (fn (tvs, tname, mx, constrs) => |
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227 map (fn (cname, cargs, mx) => (cname, mx)) constrs) dts'; |
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228 |
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229 val ps = map (fn (_, n, _, _) => |
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230 (Sign.full_bname tmp_thy n, Sign.full_bname tmp_thy (n ^ "_Rep"))) dts; |
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231 val rps = map Library.swap ps; |
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232 |
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233 fun replace_types (Type ("Nominal.ABS", [T, U])) = |
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234 Type ("fun", [T, Type ("Nominal.noption", [replace_types U])]) |
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235 | replace_types (Type (s, Ts)) = |
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236 Type (getOpt (AList.lookup op = ps s, s), map replace_types Ts) |
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237 | replace_types T = T; |
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238 |
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239 val dts'' = map (fn (tvs, tname, mx, constrs) => (tvs, Binding.name (tname ^ "_Rep"), NoSyn, |
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240 map (fn (cname, cargs, mx) => (Binding.name (cname ^ "_Rep"), |
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241 map replace_types cargs, NoSyn)) constrs)) dts'; |
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242 |
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243 val new_type_names' = map (fn n => n ^ "_Rep") new_type_names; |
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244 |
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245 val (full_new_type_names',thy1) = |
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246 Datatype.add_datatype config new_type_names' dts'' thy; |
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247 |
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248 val {descr, induction, ...} = |
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249 Datatype.the_info thy1 (hd full_new_type_names'); |
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250 fun nth_dtyp i = typ_of_dtyp descr sorts (DtRec i); |
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251 |
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252 val big_name = space_implode "_" new_type_names; |
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253 |
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254 |
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255 (**** define permutation functions ****) |
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256 |
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257 val permT = mk_permT (TFree ("'x", HOLogic.typeS)); |
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258 val pi = Free ("pi", permT); |
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259 val perm_types = map (fn (i, _) => |
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260 let val T = nth_dtyp i |
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261 in permT --> T --> T end) descr; |
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262 val perm_names' = DatatypeProp.indexify_names (map (fn (i, _) => |
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263 "perm_" ^ name_of_typ (nth_dtyp i)) descr); |
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264 val perm_names = replicate (length new_type_names) "Nominal.perm" @ |
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265 map (Sign.full_bname thy1) (List.drop (perm_names', length new_type_names)); |
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266 val perm_names_types = perm_names ~~ perm_types; |
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267 val perm_names_types' = perm_names' ~~ perm_types; |
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268 |
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269 val perm_eqs = maps (fn (i, (_, _, constrs)) => |
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270 let val T = nth_dtyp i |
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271 in map (fn (cname, dts) => |
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272 let |
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273 val Ts = map (typ_of_dtyp descr sorts) dts; |
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274 val names = Name.variant_list ["pi"] (DatatypeProp.make_tnames Ts); |
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275 val args = map Free (names ~~ Ts); |
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276 val c = Const (cname, Ts ---> T); |
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277 fun perm_arg (dt, x) = |
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278 let val T = type_of x |
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279 in if is_rec_type dt then |
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280 let val (Us, _) = strip_type T |
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281 in list_abs (map (pair "x") Us, |
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282 Free (nth perm_names_types' (body_index dt)) $ pi $ |
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283 list_comb (x, map (fn (i, U) => |
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284 Const ("Nominal.perm", permT --> U --> U) $ |
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285 (Const ("List.rev", permT --> permT) $ pi) $ |
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286 Bound i) ((length Us - 1 downto 0) ~~ Us))) |
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287 end |
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288 else Const ("Nominal.perm", permT --> T --> T) $ pi $ x |
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289 end; |
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290 in |
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291 (Attrib.empty_binding, HOLogic.mk_Trueprop (HOLogic.mk_eq |
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292 (Free (nth perm_names_types' i) $ |
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293 Free ("pi", mk_permT (TFree ("'x", HOLogic.typeS))) $ |
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294 list_comb (c, args), |
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295 list_comb (c, map perm_arg (dts ~~ args))))) |
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296 end) constrs |
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297 end) descr; |
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298 |
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299 val (perm_simps, thy2) = |
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300 Primrec.add_primrec_overloaded |
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301 (map (fn (s, sT) => (s, sT, false)) |
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302 (List.take (perm_names' ~~ perm_names_types, length new_type_names))) |
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303 (map (fn s => (Binding.name s, NONE, NoSyn)) perm_names') perm_eqs thy1; |
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304 |
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305 (**** prove that permutation functions introduced by unfolding are ****) |
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306 (**** equivalent to already existing permutation functions ****) |
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307 |
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308 val _ = warning ("length descr: " ^ string_of_int (length descr)); |
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309 val _ = warning ("length new_type_names: " ^ string_of_int (length new_type_names)); |
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310 |
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311 val perm_indnames = DatatypeProp.make_tnames (map body_type perm_types); |
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312 val perm_fun_def = PureThy.get_thm thy2 "perm_fun_def"; |
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313 |
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314 val unfolded_perm_eq_thms = |
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315 if length descr = length new_type_names then [] |
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316 else map standard (List.drop (split_conj_thm |
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317 (Goal.prove_global thy2 [] [] |
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318 (HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj |
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319 (map (fn (c as (s, T), x) => |
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320 let val [T1, T2] = binder_types T |
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321 in HOLogic.mk_eq (Const c $ pi $ Free (x, T2), |
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322 Const ("Nominal.perm", T) $ pi $ Free (x, T2)) |
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323 end) |
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324 (perm_names_types ~~ perm_indnames)))) |
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325 (fn _ => EVERY [indtac induction perm_indnames 1, |
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326 ALLGOALS (asm_full_simp_tac |
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327 (simpset_of thy2 addsimps [perm_fun_def]))])), |
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328 length new_type_names)); |
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329 |
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330 (**** prove [] \<bullet> t = t ****) |
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331 |
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332 val _ = warning "perm_empty_thms"; |
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333 |
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334 val perm_empty_thms = List.concat (map (fn a => |
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335 let val permT = mk_permT (Type (a, [])) |
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336 in map standard (List.take (split_conj_thm |
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337 (Goal.prove_global thy2 [] [] |
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338 (augment_sort thy2 [pt_class_of thy2 a] |
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339 (HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj |
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340 (map (fn ((s, T), x) => HOLogic.mk_eq |
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341 (Const (s, permT --> T --> T) $ |
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342 Const ("List.list.Nil", permT) $ Free (x, T), |
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343 Free (x, T))) |
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344 (perm_names ~~ |
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345 map body_type perm_types ~~ perm_indnames))))) |
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346 (fn _ => EVERY [indtac induction perm_indnames 1, |
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347 ALLGOALS (asm_full_simp_tac (simpset_of thy2))])), |
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348 length new_type_names)) |
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349 end) |
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350 atoms); |
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351 |
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352 (**** prove (pi1 @ pi2) \<bullet> t = pi1 \<bullet> (pi2 \<bullet> t) ****) |
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353 |
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354 val _ = warning "perm_append_thms"; |
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355 |
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356 (*FIXME: these should be looked up statically*) |
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357 val at_pt_inst = PureThy.get_thm thy2 "at_pt_inst"; |
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358 val pt2 = PureThy.get_thm thy2 "pt2"; |
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359 |
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360 val perm_append_thms = List.concat (map (fn a => |
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361 let |
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362 val permT = mk_permT (Type (a, [])); |
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363 val pi1 = Free ("pi1", permT); |
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364 val pi2 = Free ("pi2", permT); |
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365 val pt_inst = pt_inst_of thy2 a; |
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366 val pt2' = pt_inst RS pt2; |
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367 val pt2_ax = PureThy.get_thm thy2 (Long_Name.map_base_name (fn s => "pt_" ^ s ^ "2") a); |
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368 in List.take (map standard (split_conj_thm |
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369 (Goal.prove_global thy2 [] [] |
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370 (augment_sort thy2 [pt_class_of thy2 a] |
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371 (HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj |
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372 (map (fn ((s, T), x) => |
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373 let val perm = Const (s, permT --> T --> T) |
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374 in HOLogic.mk_eq |
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375 (perm $ (Const ("List.append", permT --> permT --> permT) $ |
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376 pi1 $ pi2) $ Free (x, T), |
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377 perm $ pi1 $ (perm $ pi2 $ Free (x, T))) |
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378 end) |
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379 (perm_names ~~ |
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380 map body_type perm_types ~~ perm_indnames))))) |
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381 (fn _ => EVERY [indtac induction perm_indnames 1, |
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382 ALLGOALS (asm_full_simp_tac (simpset_of thy2 addsimps [pt2', pt2_ax]))]))), |
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383 length new_type_names) |
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384 end) atoms); |
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385 |
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386 (**** prove pi1 ~ pi2 ==> pi1 \<bullet> t = pi2 \<bullet> t ****) |
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387 |
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388 val _ = warning "perm_eq_thms"; |
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389 |
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390 val pt3 = PureThy.get_thm thy2 "pt3"; |
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391 val pt3_rev = PureThy.get_thm thy2 "pt3_rev"; |
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392 |
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393 val perm_eq_thms = List.concat (map (fn a => |
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394 let |
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395 val permT = mk_permT (Type (a, [])); |
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396 val pi1 = Free ("pi1", permT); |
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397 val pi2 = Free ("pi2", permT); |
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398 val at_inst = at_inst_of thy2 a; |
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399 val pt_inst = pt_inst_of thy2 a; |
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400 val pt3' = pt_inst RS pt3; |
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401 val pt3_rev' = at_inst RS (pt_inst RS pt3_rev); |
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402 val pt3_ax = PureThy.get_thm thy2 (Long_Name.map_base_name (fn s => "pt_" ^ s ^ "3") a); |
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403 in List.take (map standard (split_conj_thm |
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404 (Goal.prove_global thy2 [] [] |
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405 (augment_sort thy2 [pt_class_of thy2 a] (Logic.mk_implies |
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406 (HOLogic.mk_Trueprop (Const ("Nominal.prm_eq", |
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407 permT --> permT --> HOLogic.boolT) $ pi1 $ pi2), |
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408 HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj |
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409 (map (fn ((s, T), x) => |
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410 let val perm = Const (s, permT --> T --> T) |
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411 in HOLogic.mk_eq |
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412 (perm $ pi1 $ Free (x, T), |
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413 perm $ pi2 $ Free (x, T)) |
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414 end) |
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415 (perm_names ~~ |
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416 map body_type perm_types ~~ perm_indnames)))))) |
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417 (fn _ => EVERY [indtac induction perm_indnames 1, |
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418 ALLGOALS (asm_full_simp_tac (simpset_of thy2 addsimps [pt3', pt3_rev', pt3_ax]))]))), |
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419 length new_type_names) |
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420 end) atoms); |
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421 |
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422 (**** prove pi1 \<bullet> (pi2 \<bullet> t) = (pi1 \<bullet> pi2) \<bullet> (pi1 \<bullet> t) ****) |
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423 |
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424 val cp1 = PureThy.get_thm thy2 "cp1"; |
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425 val dj_cp = PureThy.get_thm thy2 "dj_cp"; |
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426 val pt_perm_compose = PureThy.get_thm thy2 "pt_perm_compose"; |
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427 val pt_perm_compose_rev = PureThy.get_thm thy2 "pt_perm_compose_rev"; |
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428 val dj_perm_perm_forget = PureThy.get_thm thy2 "dj_perm_perm_forget"; |
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429 |
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430 fun composition_instance name1 name2 thy = |
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431 let |
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432 val cp_class = cp_class_of thy name1 name2; |
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433 val pt_class = |
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434 if name1 = name2 then [pt_class_of thy name1] |
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435 else []; |
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436 val permT1 = mk_permT (Type (name1, [])); |
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437 val permT2 = mk_permT (Type (name2, [])); |
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438 val Ts = map body_type perm_types; |
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439 val cp_inst = cp_inst_of thy name1 name2; |
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440 val simps = simpset_of thy addsimps (perm_fun_def :: |
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441 (if name1 <> name2 then |
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442 let val dj = dj_thm_of thy name2 name1 |
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443 in [dj RS (cp_inst RS dj_cp), dj RS dj_perm_perm_forget] end |
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444 else |
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445 let |
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446 val at_inst = at_inst_of thy name1; |
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447 val pt_inst = pt_inst_of thy name1; |
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448 in |
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449 [cp_inst RS cp1 RS sym, |
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450 at_inst RS (pt_inst RS pt_perm_compose) RS sym, |
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451 at_inst RS (pt_inst RS pt_perm_compose_rev) RS sym] |
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452 end)) |
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453 val sort = Sign.certify_sort thy (cp_class :: pt_class); |
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454 val thms = split_conj_thm (Goal.prove_global thy [] [] |
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455 (augment_sort thy sort |
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456 (HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj |
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457 (map (fn ((s, T), x) => |
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458 let |
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459 val pi1 = Free ("pi1", permT1); |
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460 val pi2 = Free ("pi2", permT2); |
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461 val perm1 = Const (s, permT1 --> T --> T); |
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462 val perm2 = Const (s, permT2 --> T --> T); |
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463 val perm3 = Const ("Nominal.perm", permT1 --> permT2 --> permT2) |
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464 in HOLogic.mk_eq |
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465 (perm1 $ pi1 $ (perm2 $ pi2 $ Free (x, T)), |
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466 perm2 $ (perm3 $ pi1 $ pi2) $ (perm1 $ pi1 $ Free (x, T))) |
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467 end) |
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468 (perm_names ~~ Ts ~~ perm_indnames))))) |
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469 (fn _ => EVERY [indtac induction perm_indnames 1, |
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470 ALLGOALS (asm_full_simp_tac simps)])) |
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471 in |
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472 fold (fn (s, tvs) => fn thy => AxClass.prove_arity |
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473 (s, map (inter_sort thy sort o snd) tvs, [cp_class]) |
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474 (Class.intro_classes_tac [] THEN ALLGOALS (resolve_tac thms)) thy) |
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475 (full_new_type_names' ~~ tyvars) thy |
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476 end; |
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477 |
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478 val (perm_thmss,thy3) = thy2 |> |
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479 fold (fn name1 => fold (composition_instance name1) atoms) atoms |> |
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480 fold (fn atom => fn thy => |
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481 let val pt_name = pt_class_of thy atom |
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482 in |
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483 fold (fn (s, tvs) => fn thy => AxClass.prove_arity |
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484 (s, map (inter_sort thy [pt_name] o snd) tvs, [pt_name]) |
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485 (EVERY |
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486 [Class.intro_classes_tac [], |
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487 resolve_tac perm_empty_thms 1, |
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488 resolve_tac perm_append_thms 1, |
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489 resolve_tac perm_eq_thms 1, assume_tac 1]) thy) |
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490 (full_new_type_names' ~~ tyvars) thy |
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491 end) atoms |> |
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492 PureThy.add_thmss |
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493 [((Binding.name (space_implode "_" new_type_names ^ "_unfolded_perm_eq"), |
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494 unfolded_perm_eq_thms), [Simplifier.simp_add]), |
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495 ((Binding.name (space_implode "_" new_type_names ^ "_perm_empty"), |
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496 perm_empty_thms), [Simplifier.simp_add]), |
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497 ((Binding.name (space_implode "_" new_type_names ^ "_perm_append"), |
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498 perm_append_thms), [Simplifier.simp_add]), |
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499 ((Binding.name (space_implode "_" new_type_names ^ "_perm_eq"), |
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500 perm_eq_thms), [Simplifier.simp_add])]; |
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501 |
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502 (**** Define representing sets ****) |
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503 |
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504 val _ = warning "representing sets"; |
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505 |
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506 val rep_set_names = DatatypeProp.indexify_names |
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507 (map (fn (i, _) => name_of_typ (nth_dtyp i) ^ "_set") descr); |
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508 val big_rep_name = |
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509 space_implode "_" (DatatypeProp.indexify_names (List.mapPartial |
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510 (fn (i, ("Nominal.noption", _, _)) => NONE |
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511 | (i, _) => SOME (name_of_typ (nth_dtyp i))) descr)) ^ "_set"; |
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512 val _ = warning ("big_rep_name: " ^ big_rep_name); |
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513 |
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514 fun strip_option (dtf as DtType ("fun", [dt, DtRec i])) = |
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515 (case AList.lookup op = descr i of |
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516 SOME ("Nominal.noption", _, [(_, [dt']), _]) => |
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517 apfst (cons dt) (strip_option dt') |
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518 | _ => ([], dtf)) |
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519 | strip_option (DtType ("fun", [dt, DtType ("Nominal.noption", [dt'])])) = |
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520 apfst (cons dt) (strip_option dt') |
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521 | strip_option dt = ([], dt); |
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522 |
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523 val dt_atomTs = distinct op = (map (typ_of_dtyp descr sorts) |
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524 (List.concat (map (fn (_, (_, _, cs)) => List.concat |
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525 (map (List.concat o map (fst o strip_option) o snd) cs)) descr))); |
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526 val dt_atoms = map (fst o dest_Type) dt_atomTs; |
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527 |
|
528 fun make_intr s T (cname, cargs) = |
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529 let |
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530 fun mk_prem (dt, (j, j', prems, ts)) = |
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531 let |
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532 val (dts, dt') = strip_option dt; |
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533 val (dts', dt'') = strip_dtyp dt'; |
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534 val Ts = map (typ_of_dtyp descr sorts) dts; |
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535 val Us = map (typ_of_dtyp descr sorts) dts'; |
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536 val T = typ_of_dtyp descr sorts dt''; |
|
537 val free = mk_Free "x" (Us ---> T) j; |
|
538 val free' = app_bnds free (length Us); |
|
539 fun mk_abs_fun (T, (i, t)) = |
|
540 let val U = fastype_of t |
|
541 in (i + 1, Const ("Nominal.abs_fun", [T, U, T] ---> |
|
542 Type ("Nominal.noption", [U])) $ mk_Free "y" T i $ t) |
|
543 end |
|
544 in (j + 1, j' + length Ts, |
|
545 case dt'' of |
|
546 DtRec k => list_all (map (pair "x") Us, |
|
547 HOLogic.mk_Trueprop (Free (List.nth (rep_set_names, k), |
|
548 T --> HOLogic.boolT) $ free')) :: prems |
|
549 | _ => prems, |
|
550 snd (List.foldr mk_abs_fun (j', free) Ts) :: ts) |
|
551 end; |
|
552 |
|
553 val (_, _, prems, ts) = List.foldr mk_prem (1, 1, [], []) cargs; |
|
554 val concl = HOLogic.mk_Trueprop (Free (s, T --> HOLogic.boolT) $ |
|
555 list_comb (Const (cname, map fastype_of ts ---> T), ts)) |
|
556 in Logic.list_implies (prems, concl) |
|
557 end; |
|
558 |
|
559 val (intr_ts, (rep_set_names', recTs')) = |
|
560 apfst List.concat (apsnd ListPair.unzip (ListPair.unzip (List.mapPartial |
|
561 (fn ((_, ("Nominal.noption", _, _)), _) => NONE |
|
562 | ((i, (_, _, constrs)), rep_set_name) => |
|
563 let val T = nth_dtyp i |
|
564 in SOME (map (make_intr rep_set_name T) constrs, |
|
565 (rep_set_name, T)) |
|
566 end) |
|
567 (descr ~~ rep_set_names)))); |
|
568 val rep_set_names'' = map (Sign.full_bname thy3) rep_set_names'; |
|
569 |
|
570 val ({raw_induct = rep_induct, intrs = rep_intrs, ...}, thy4) = |
|
571 Inductive.add_inductive_global (serial_string ()) |
|
572 {quiet_mode = false, verbose = false, kind = Thm.internalK, |
|
573 alt_name = Binding.name big_rep_name, coind = false, no_elim = true, no_ind = false, |
|
574 skip_mono = true, fork_mono = false} |
|
575 (map (fn (s, T) => ((Binding.name s, T --> HOLogic.boolT), NoSyn)) |
|
576 (rep_set_names' ~~ recTs')) |
|
577 [] (map (fn x => (Attrib.empty_binding, x)) intr_ts) [] thy3; |
|
578 |
|
579 (**** Prove that representing set is closed under permutation ****) |
|
580 |
|
581 val _ = warning "proving closure under permutation..."; |
|
582 |
|
583 val abs_perm = PureThy.get_thms thy4 "abs_perm"; |
|
584 |
|
585 val perm_indnames' = List.mapPartial |
|
586 (fn (x, (_, ("Nominal.noption", _, _))) => NONE | (x, _) => SOME x) |
|
587 (perm_indnames ~~ descr); |
|
588 |
|
589 fun mk_perm_closed name = map (fn th => standard (th RS mp)) |
|
590 (List.take (split_conj_thm (Goal.prove_global thy4 [] [] |
|
591 (augment_sort thy4 |
|
592 (pt_class_of thy4 name :: map (cp_class_of thy4 name) (dt_atoms \ name)) |
|
593 (HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj (map |
|
594 (fn ((s, T), x) => |
|
595 let |
|
596 val S = Const (s, T --> HOLogic.boolT); |
|
597 val permT = mk_permT (Type (name, [])) |
|
598 in HOLogic.mk_imp (S $ Free (x, T), |
|
599 S $ (Const ("Nominal.perm", permT --> T --> T) $ |
|
600 Free ("pi", permT) $ Free (x, T))) |
|
601 end) (rep_set_names'' ~~ recTs' ~~ perm_indnames'))))) |
|
602 (fn _ => EVERY |
|
603 [indtac rep_induct [] 1, |
|
604 ALLGOALS (simp_tac (simpset_of thy4 addsimps |
|
605 (symmetric perm_fun_def :: abs_perm))), |
|
606 ALLGOALS (resolve_tac rep_intrs THEN_ALL_NEW assume_tac)])), |
|
607 length new_type_names)); |
|
608 |
|
609 val perm_closed_thmss = map mk_perm_closed atoms; |
|
610 |
|
611 (**** typedef ****) |
|
612 |
|
613 val _ = warning "defining type..."; |
|
614 |
|
615 val (typedefs, thy6) = |
|
616 thy4 |
|
617 |> fold_map (fn ((((name, mx), tvs), (cname, U)), name') => fn thy => |
|
618 Typedef.add_typedef false (SOME (Binding.name name')) |
|
619 (Binding.name name, map fst tvs, mx) |
|
620 (Const ("Collect", (U --> HOLogic.boolT) --> HOLogic.mk_setT U) $ |
|
621 Const (cname, U --> HOLogic.boolT)) NONE |
|
622 (rtac exI 1 THEN rtac CollectI 1 THEN |
|
623 QUIET_BREADTH_FIRST (has_fewer_prems 1) |
|
624 (resolve_tac rep_intrs 1)) thy |> (fn ((_, r), thy) => |
|
625 let |
|
626 val permT = mk_permT |
|
627 (TFree (Name.variant (map fst tvs) "'a", HOLogic.typeS)); |
|
628 val pi = Free ("pi", permT); |
|
629 val T = Type (Sign.intern_type thy name, map TFree tvs); |
|
630 in apfst (pair r o hd) |
|
631 (PureThy.add_defs_unchecked true [((Binding.name ("prm_" ^ name ^ "_def"), Logic.mk_equals |
|
632 (Const ("Nominal.perm", permT --> T --> T) $ pi $ Free ("x", T), |
|
633 Const (Sign.intern_const thy ("Abs_" ^ name), U --> T) $ |
|
634 (Const ("Nominal.perm", permT --> U --> U) $ pi $ |
|
635 (Const (Sign.intern_const thy ("Rep_" ^ name), T --> U) $ |
|
636 Free ("x", T))))), [])] thy) |
|
637 end)) |
|
638 (types_syntax ~~ tyvars ~~ |
|
639 List.take (rep_set_names'' ~~ recTs', length new_type_names) ~~ |
|
640 new_type_names); |
|
641 |
|
642 val perm_defs = map snd typedefs; |
|
643 val Abs_inverse_thms = map (collect_simp o #Abs_inverse o fst) typedefs; |
|
644 val Rep_inverse_thms = map (#Rep_inverse o fst) typedefs; |
|
645 val Rep_thms = map (collect_simp o #Rep o fst) typedefs; |
|
646 |
|
647 |
|
648 (** prove that new types are in class pt_<name> **) |
|
649 |
|
650 val _ = warning "prove that new types are in class pt_<name> ..."; |
|
651 |
|
652 fun pt_instance (atom, perm_closed_thms) = |
|
653 fold (fn ((((((Abs_inverse, Rep_inverse), Rep), |
|
654 perm_def), name), tvs), perm_closed) => fn thy => |
|
655 let |
|
656 val pt_class = pt_class_of thy atom; |
|
657 val sort = Sign.certify_sort thy |
|
658 (pt_class :: map (cp_class_of thy atom) (dt_atoms \ atom)) |
|
659 in AxClass.prove_arity |
|
660 (Sign.intern_type thy name, |
|
661 map (inter_sort thy sort o snd) tvs, [pt_class]) |
|
662 (EVERY [Class.intro_classes_tac [], |
|
663 rewrite_goals_tac [perm_def], |
|
664 asm_full_simp_tac (simpset_of thy addsimps [Rep_inverse]) 1, |
|
665 asm_full_simp_tac (simpset_of thy addsimps |
|
666 [Rep RS perm_closed RS Abs_inverse]) 1, |
|
667 asm_full_simp_tac (HOL_basic_ss addsimps [PureThy.get_thm thy |
|
668 ("pt_" ^ Long_Name.base_name atom ^ "3")]) 1]) thy |
|
669 end) |
|
670 (Abs_inverse_thms ~~ Rep_inverse_thms ~~ Rep_thms ~~ perm_defs ~~ |
|
671 new_type_names ~~ tyvars ~~ perm_closed_thms); |
|
672 |
|
673 |
|
674 (** prove that new types are in class cp_<name1>_<name2> **) |
|
675 |
|
676 val _ = warning "prove that new types are in class cp_<name1>_<name2> ..."; |
|
677 |
|
678 fun cp_instance (atom1, perm_closed_thms1) (atom2, perm_closed_thms2) thy = |
|
679 let |
|
680 val cp_class = cp_class_of thy atom1 atom2; |
|
681 val sort = Sign.certify_sort thy |
|
682 (pt_class_of thy atom1 :: map (cp_class_of thy atom1) (dt_atoms \ atom1) @ |
|
683 (if atom1 = atom2 then [cp_class_of thy atom1 atom1] else |
|
684 pt_class_of thy atom2 :: map (cp_class_of thy atom2) (dt_atoms \ atom2))); |
|
685 val cp1' = cp_inst_of thy atom1 atom2 RS cp1 |
|
686 in fold (fn ((((((Abs_inverse, Rep), |
|
687 perm_def), name), tvs), perm_closed1), perm_closed2) => fn thy => |
|
688 AxClass.prove_arity |
|
689 (Sign.intern_type thy name, |
|
690 map (inter_sort thy sort o snd) tvs, [cp_class]) |
|
691 (EVERY [Class.intro_classes_tac [], |
|
692 rewrite_goals_tac [perm_def], |
|
693 asm_full_simp_tac (simpset_of thy addsimps |
|
694 ((Rep RS perm_closed1 RS Abs_inverse) :: |
|
695 (if atom1 = atom2 then [] |
|
696 else [Rep RS perm_closed2 RS Abs_inverse]))) 1, |
|
697 cong_tac 1, |
|
698 rtac refl 1, |
|
699 rtac cp1' 1]) thy) |
|
700 (Abs_inverse_thms ~~ Rep_thms ~~ perm_defs ~~ new_type_names ~~ |
|
701 tyvars ~~ perm_closed_thms1 ~~ perm_closed_thms2) thy |
|
702 end; |
|
703 |
|
704 val thy7 = fold (fn x => fn thy => thy |> |
|
705 pt_instance x |> |
|
706 fold (cp_instance x) (atoms ~~ perm_closed_thmss)) |
|
707 (atoms ~~ perm_closed_thmss) thy6; |
|
708 |
|
709 (**** constructors ****) |
|
710 |
|
711 fun mk_abs_fun (x, t) = |
|
712 let |
|
713 val T = fastype_of x; |
|
714 val U = fastype_of t |
|
715 in |
|
716 Const ("Nominal.abs_fun", T --> U --> T --> |
|
717 Type ("Nominal.noption", [U])) $ x $ t |
|
718 end; |
|
719 |
|
720 val (ty_idxs, _) = List.foldl |
|
721 (fn ((i, ("Nominal.noption", _, _)), p) => p |
|
722 | ((i, _), (ty_idxs, j)) => (ty_idxs @ [(i, j)], j + 1)) ([], 0) descr; |
|
723 |
|
724 fun reindex (DtType (s, dts)) = DtType (s, map reindex dts) |
|
725 | reindex (DtRec i) = DtRec (the (AList.lookup op = ty_idxs i)) |
|
726 | reindex dt = dt; |
|
727 |
|
728 fun strip_suffix i s = implode (List.take (explode s, size s - i)); |
|
729 |
|
730 (** strips the "_Rep" in type names *) |
|
731 fun strip_nth_name i s = |
|
732 let val xs = Long_Name.explode s; |
|
733 in Long_Name.implode (Library.nth_map (length xs - i) (strip_suffix 4) xs) end; |
|
734 |
|
735 val (descr'', ndescr) = ListPair.unzip (map_filter |
|
736 (fn (i, ("Nominal.noption", _, _)) => NONE |
|
737 | (i, (s, dts, constrs)) => |
|
738 let |
|
739 val SOME index = AList.lookup op = ty_idxs i; |
|
740 val (constrs2, constrs1) = |
|
741 map_split (fn (cname, cargs) => |
|
742 apsnd (pair (strip_nth_name 2 (strip_nth_name 1 cname))) |
|
743 (fold_map (fn dt => fn dts => |
|
744 let val (dts', dt') = strip_option dt |
|
745 in ((length dts, length dts'), dts @ dts' @ [reindex dt']) end) |
|
746 cargs [])) constrs |
|
747 in SOME ((index, (strip_nth_name 1 s, map reindex dts, constrs1)), |
|
748 (index, constrs2)) |
|
749 end) descr); |
|
750 |
|
751 val (descr1, descr2) = chop (length new_type_names) descr''; |
|
752 val descr' = [descr1, descr2]; |
|
753 |
|
754 fun partition_cargs idxs xs = map (fn (i, j) => |
|
755 (List.take (List.drop (xs, i), j), List.nth (xs, i + j))) idxs; |
|
756 |
|
757 val pdescr = map (fn ((i, (s, dts, constrs)), (_, idxss)) => (i, (s, dts, |
|
758 map (fn ((cname, cargs), idxs) => (cname, partition_cargs idxs cargs)) |
|
759 (constrs ~~ idxss)))) (descr'' ~~ ndescr); |
|
760 |
|
761 fun nth_dtyp' i = typ_of_dtyp descr'' sorts (DtRec i); |
|
762 |
|
763 val rep_names = map (fn s => |
|
764 Sign.intern_const thy7 ("Rep_" ^ s)) new_type_names; |
|
765 val abs_names = map (fn s => |
|
766 Sign.intern_const thy7 ("Abs_" ^ s)) new_type_names; |
|
767 |
|
768 val recTs = get_rec_types descr'' sorts; |
|
769 val newTs' = Library.take (length new_type_names, recTs'); |
|
770 val newTs = Library.take (length new_type_names, recTs); |
|
771 |
|
772 val full_new_type_names = map (Sign.full_bname thy) new_type_names; |
|
773 |
|
774 fun make_constr_def tname T T' ((thy, defs, eqns), |
|
775 (((cname_rep, _), (cname, cargs)), (cname', mx))) = |
|
776 let |
|
777 fun constr_arg ((dts, dt), (j, l_args, r_args)) = |
|
778 let |
|
779 val xs = map (fn (dt, i) => mk_Free "x" (typ_of_dtyp descr'' sorts dt) i) |
|
780 (dts ~~ (j upto j + length dts - 1)) |
|
781 val x = mk_Free "x" (typ_of_dtyp descr'' sorts dt) (j + length dts) |
|
782 in |
|
783 (j + length dts + 1, |
|
784 xs @ x :: l_args, |
|
785 List.foldr mk_abs_fun |
|
786 (case dt of |
|
787 DtRec k => if k < length new_type_names then |
|
788 Const (List.nth (rep_names, k), typ_of_dtyp descr'' sorts dt --> |
|
789 typ_of_dtyp descr sorts dt) $ x |
|
790 else error "nested recursion not (yet) supported" |
|
791 | _ => x) xs :: r_args) |
|
792 end |
|
793 |
|
794 val (_, l_args, r_args) = List.foldr constr_arg (1, [], []) cargs; |
|
795 val abs_name = Sign.intern_const thy ("Abs_" ^ tname); |
|
796 val rep_name = Sign.intern_const thy ("Rep_" ^ tname); |
|
797 val constrT = map fastype_of l_args ---> T; |
|
798 val lhs = list_comb (Const (cname, constrT), l_args); |
|
799 val rhs = list_comb (Const (cname_rep, map fastype_of r_args ---> T'), r_args); |
|
800 val def = Logic.mk_equals (lhs, Const (abs_name, T' --> T) $ rhs); |
|
801 val eqn = HOLogic.mk_Trueprop (HOLogic.mk_eq |
|
802 (Const (rep_name, T --> T') $ lhs, rhs)); |
|
803 val def_name = (Long_Name.base_name cname) ^ "_def"; |
|
804 val ([def_thm], thy') = thy |> |
|
805 Sign.add_consts_i [(Binding.name cname', constrT, mx)] |> |
|
806 (PureThy.add_defs false o map Thm.no_attributes) [(Binding.name def_name, def)] |
|
807 in (thy', defs @ [def_thm], eqns @ [eqn]) end; |
|
808 |
|
809 fun dt_constr_defs ((thy, defs, eqns, dist_lemmas), ((((((_, (_, _, constrs)), |
|
810 (_, (_, _, constrs'))), tname), T), T'), constr_syntax)) = |
|
811 let |
|
812 val rep_const = cterm_of thy |
|
813 (Const (Sign.intern_const thy ("Rep_" ^ tname), T --> T')); |
|
814 val dist = standard (cterm_instantiate [(cterm_of thy distinct_f, rep_const)] distinct_lemma); |
|
815 val (thy', defs', eqns') = Library.foldl (make_constr_def tname T T') |
|
816 ((Sign.add_path tname thy, defs, []), constrs ~~ constrs' ~~ constr_syntax) |
|
817 in |
|
818 (parent_path (#flat_names config) thy', defs', eqns @ [eqns'], dist_lemmas @ [dist]) |
|
819 end; |
|
820 |
|
821 val (thy8, constr_defs, constr_rep_eqns, dist_lemmas) = Library.foldl dt_constr_defs |
|
822 ((thy7, [], [], []), List.take (descr, length new_type_names) ~~ |
|
823 List.take (pdescr, length new_type_names) ~~ |
|
824 new_type_names ~~ newTs ~~ newTs' ~~ constr_syntax); |
|
825 |
|
826 val abs_inject_thms = map (collect_simp o #Abs_inject o fst) typedefs |
|
827 val rep_inject_thms = map (#Rep_inject o fst) typedefs |
|
828 |
|
829 (* prove theorem Rep_i (Constr_j ...) = Constr'_j ... *) |
|
830 |
|
831 fun prove_constr_rep_thm eqn = |
|
832 let |
|
833 val inj_thms = map (fn r => r RS iffD1) abs_inject_thms; |
|
834 val rewrites = constr_defs @ map mk_meta_eq Rep_inverse_thms |
|
835 in Goal.prove_global thy8 [] [] eqn (fn _ => EVERY |
|
836 [resolve_tac inj_thms 1, |
|
837 rewrite_goals_tac rewrites, |
|
838 rtac refl 3, |
|
839 resolve_tac rep_intrs 2, |
|
840 REPEAT (resolve_tac Rep_thms 1)]) |
|
841 end; |
|
842 |
|
843 val constr_rep_thmss = map (map prove_constr_rep_thm) constr_rep_eqns; |
|
844 |
|
845 (* prove theorem pi \<bullet> Rep_i x = Rep_i (pi \<bullet> x) *) |
|
846 |
|
847 fun prove_perm_rep_perm (atom, perm_closed_thms) = map (fn th => |
|
848 let |
|
849 val _ $ (_ $ (Rep $ x)) = Logic.unvarify (prop_of th); |
|
850 val Type ("fun", [T, U]) = fastype_of Rep; |
|
851 val permT = mk_permT (Type (atom, [])); |
|
852 val pi = Free ("pi", permT); |
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853 in |
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854 Goal.prove_global thy8 [] [] |
|
855 (augment_sort thy8 |
|
856 (pt_class_of thy8 atom :: map (cp_class_of thy8 atom) (dt_atoms \ atom)) |
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857 (HOLogic.mk_Trueprop (HOLogic.mk_eq |
|
858 (Const ("Nominal.perm", permT --> U --> U) $ pi $ (Rep $ x), |
|
859 Rep $ (Const ("Nominal.perm", permT --> T --> T) $ pi $ x))))) |
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860 (fn _ => simp_tac (HOL_basic_ss addsimps (perm_defs @ Abs_inverse_thms @ |
|
861 perm_closed_thms @ Rep_thms)) 1) |
|
862 end) Rep_thms; |
|
863 |
|
864 val perm_rep_perm_thms = List.concat (map prove_perm_rep_perm |
|
865 (atoms ~~ perm_closed_thmss)); |
|
866 |
|
867 (* prove distinctness theorems *) |
|
868 |
|
869 val distinct_props = DatatypeProp.make_distincts descr' sorts; |
|
870 val dist_rewrites = map2 (fn rep_thms => fn dist_lemma => |
|
871 dist_lemma :: rep_thms @ [In0_eq, In1_eq, In0_not_In1, In1_not_In0]) |
|
872 constr_rep_thmss dist_lemmas; |
|
873 |
|
874 fun prove_distinct_thms _ (_, []) = [] |
|
875 | prove_distinct_thms (p as (rep_thms, dist_lemma)) (k, t :: ts) = |
|
876 let |
|
877 val dist_thm = Goal.prove_global thy8 [] [] t (fn _ => |
|
878 simp_tac (simpset_of thy8 addsimps (dist_lemma :: rep_thms)) 1) |
|
879 in dist_thm :: standard (dist_thm RS not_sym) :: |
|
880 prove_distinct_thms p (k, ts) |
|
881 end; |
|
882 |
|
883 val distinct_thms = map2 prove_distinct_thms |
|
884 (constr_rep_thmss ~~ dist_lemmas) distinct_props; |
|
885 |
|
886 (** prove equations for permutation functions **) |
|
887 |
|
888 val perm_simps' = map (fn (((i, (_, _, constrs)), tname), constr_rep_thms) => |
|
889 let val T = nth_dtyp' i |
|
890 in List.concat (map (fn (atom, perm_closed_thms) => |
|
891 map (fn ((cname, dts), constr_rep_thm) => |
|
892 let |
|
893 val cname = Sign.intern_const thy8 |
|
894 (Long_Name.append tname (Long_Name.base_name cname)); |
|
895 val permT = mk_permT (Type (atom, [])); |
|
896 val pi = Free ("pi", permT); |
|
897 |
|
898 fun perm t = |
|
899 let val T = fastype_of t |
|
900 in Const ("Nominal.perm", permT --> T --> T) $ pi $ t end; |
|
901 |
|
902 fun constr_arg ((dts, dt), (j, l_args, r_args)) = |
|
903 let |
|
904 val Ts = map (typ_of_dtyp descr'' sorts) dts; |
|
905 val xs = map (fn (T, i) => mk_Free "x" T i) |
|
906 (Ts ~~ (j upto j + length dts - 1)) |
|
907 val x = mk_Free "x" (typ_of_dtyp descr'' sorts dt) (j + length dts) |
|
908 in |
|
909 (j + length dts + 1, |
|
910 xs @ x :: l_args, |
|
911 map perm (xs @ [x]) @ r_args) |
|
912 end |
|
913 |
|
914 val (_, l_args, r_args) = List.foldr constr_arg (1, [], []) dts; |
|
915 val c = Const (cname, map fastype_of l_args ---> T) |
|
916 in |
|
917 Goal.prove_global thy8 [] [] |
|
918 (augment_sort thy8 |
|
919 (pt_class_of thy8 atom :: map (cp_class_of thy8 atom) (dt_atoms \ atom)) |
|
920 (HOLogic.mk_Trueprop (HOLogic.mk_eq |
|
921 (perm (list_comb (c, l_args)), list_comb (c, r_args))))) |
|
922 (fn _ => EVERY |
|
923 [simp_tac (simpset_of thy8 addsimps (constr_rep_thm :: perm_defs)) 1, |
|
924 simp_tac (HOL_basic_ss addsimps (Rep_thms @ Abs_inverse_thms @ |
|
925 constr_defs @ perm_closed_thms)) 1, |
|
926 TRY (simp_tac (HOL_basic_ss addsimps |
|
927 (symmetric perm_fun_def :: abs_perm)) 1), |
|
928 TRY (simp_tac (HOL_basic_ss addsimps |
|
929 (perm_fun_def :: perm_defs @ Rep_thms @ Abs_inverse_thms @ |
|
930 perm_closed_thms)) 1)]) |
|
931 end) (constrs ~~ constr_rep_thms)) (atoms ~~ perm_closed_thmss)) |
|
932 end) (List.take (pdescr, length new_type_names) ~~ new_type_names ~~ constr_rep_thmss); |
|
933 |
|
934 (** prove injectivity of constructors **) |
|
935 |
|
936 val rep_inject_thms' = map (fn th => th RS sym) rep_inject_thms; |
|
937 val alpha = PureThy.get_thms thy8 "alpha"; |
|
938 val abs_fresh = PureThy.get_thms thy8 "abs_fresh"; |
|
939 |
|
940 val pt_cp_sort = |
|
941 map (pt_class_of thy8) dt_atoms @ |
|
942 maps (fn s => map (cp_class_of thy8 s) (dt_atoms \ s)) dt_atoms; |
|
943 |
|
944 val inject_thms = map (fn (((i, (_, _, constrs)), tname), constr_rep_thms) => |
|
945 let val T = nth_dtyp' i |
|
946 in List.mapPartial (fn ((cname, dts), constr_rep_thm) => |
|
947 if null dts then NONE else SOME |
|
948 let |
|
949 val cname = Sign.intern_const thy8 |
|
950 (Long_Name.append tname (Long_Name.base_name cname)); |
|
951 |
|
952 fun make_inj ((dts, dt), (j, args1, args2, eqs)) = |
|
953 let |
|
954 val Ts_idx = map (typ_of_dtyp descr'' sorts) dts ~~ (j upto j + length dts - 1); |
|
955 val xs = map (fn (T, i) => mk_Free "x" T i) Ts_idx; |
|
956 val ys = map (fn (T, i) => mk_Free "y" T i) Ts_idx; |
|
957 val x = mk_Free "x" (typ_of_dtyp descr'' sorts dt) (j + length dts); |
|
958 val y = mk_Free "y" (typ_of_dtyp descr'' sorts dt) (j + length dts) |
|
959 in |
|
960 (j + length dts + 1, |
|
961 xs @ (x :: args1), ys @ (y :: args2), |
|
962 HOLogic.mk_eq |
|
963 (List.foldr mk_abs_fun x xs, List.foldr mk_abs_fun y ys) :: eqs) |
|
964 end; |
|
965 |
|
966 val (_, args1, args2, eqs) = List.foldr make_inj (1, [], [], []) dts; |
|
967 val Ts = map fastype_of args1; |
|
968 val c = Const (cname, Ts ---> T) |
|
969 in |
|
970 Goal.prove_global thy8 [] [] |
|
971 (augment_sort thy8 pt_cp_sort |
|
972 (HOLogic.mk_Trueprop (HOLogic.mk_eq |
|
973 (HOLogic.mk_eq (list_comb (c, args1), list_comb (c, args2)), |
|
974 foldr1 HOLogic.mk_conj eqs)))) |
|
975 (fn _ => EVERY |
|
976 [asm_full_simp_tac (simpset_of thy8 addsimps (constr_rep_thm :: |
|
977 rep_inject_thms')) 1, |
|
978 TRY (asm_full_simp_tac (HOL_basic_ss addsimps (fresh_def :: supp_def :: |
|
979 alpha @ abs_perm @ abs_fresh @ rep_inject_thms @ |
|
980 perm_rep_perm_thms)) 1)]) |
|
981 end) (constrs ~~ constr_rep_thms) |
|
982 end) (List.take (pdescr, length new_type_names) ~~ new_type_names ~~ constr_rep_thmss); |
|
983 |
|
984 (** equations for support and freshness **) |
|
985 |
|
986 val (supp_thms, fresh_thms) = ListPair.unzip (map ListPair.unzip |
|
987 (map (fn ((((i, (_, _, constrs)), tname), inject_thms'), perm_thms') => |
|
988 let val T = nth_dtyp' i |
|
989 in List.concat (map (fn (cname, dts) => map (fn atom => |
|
990 let |
|
991 val cname = Sign.intern_const thy8 |
|
992 (Long_Name.append tname (Long_Name.base_name cname)); |
|
993 val atomT = Type (atom, []); |
|
994 |
|
995 fun process_constr ((dts, dt), (j, args1, args2)) = |
|
996 let |
|
997 val Ts_idx = map (typ_of_dtyp descr'' sorts) dts ~~ (j upto j + length dts - 1); |
|
998 val xs = map (fn (T, i) => mk_Free "x" T i) Ts_idx; |
|
999 val x = mk_Free "x" (typ_of_dtyp descr'' sorts dt) (j + length dts) |
|
1000 in |
|
1001 (j + length dts + 1, |
|
1002 xs @ (x :: args1), List.foldr mk_abs_fun x xs :: args2) |
|
1003 end; |
|
1004 |
|
1005 val (_, args1, args2) = List.foldr process_constr (1, [], []) dts; |
|
1006 val Ts = map fastype_of args1; |
|
1007 val c = list_comb (Const (cname, Ts ---> T), args1); |
|
1008 fun supp t = |
|
1009 Const ("Nominal.supp", fastype_of t --> HOLogic.mk_setT atomT) $ t; |
|
1010 fun fresh t = fresh_const atomT (fastype_of t) $ Free ("a", atomT) $ t; |
|
1011 val supp_thm = Goal.prove_global thy8 [] [] |
|
1012 (augment_sort thy8 pt_cp_sort |
|
1013 (HOLogic.mk_Trueprop (HOLogic.mk_eq |
|
1014 (supp c, |
|
1015 if null dts then HOLogic.mk_set atomT [] |
|
1016 else foldr1 (HOLogic.mk_binop @{const_name Un}) (map supp args2))))) |
|
1017 (fn _ => |
|
1018 simp_tac (HOL_basic_ss addsimps (supp_def :: |
|
1019 Un_assoc :: de_Morgan_conj :: Collect_disj_eq :: finite_Un :: |
|
1020 symmetric empty_def :: finite_emptyI :: simp_thms @ |
|
1021 abs_perm @ abs_fresh @ inject_thms' @ perm_thms')) 1) |
|
1022 in |
|
1023 (supp_thm, |
|
1024 Goal.prove_global thy8 [] [] (augment_sort thy8 pt_cp_sort |
|
1025 (HOLogic.mk_Trueprop (HOLogic.mk_eq |
|
1026 (fresh c, |
|
1027 if null dts then HOLogic.true_const |
|
1028 else foldr1 HOLogic.mk_conj (map fresh args2))))) |
|
1029 (fn _ => |
|
1030 simp_tac (HOL_ss addsimps [Un_iff, empty_iff, fresh_def, supp_thm]) 1)) |
|
1031 end) atoms) constrs) |
|
1032 end) (List.take (pdescr, length new_type_names) ~~ new_type_names ~~ inject_thms ~~ perm_simps'))); |
|
1033 |
|
1034 (**** weak induction theorem ****) |
|
1035 |
|
1036 fun mk_indrule_lemma ((prems, concls), (((i, _), T), U)) = |
|
1037 let |
|
1038 val Rep_t = Const (List.nth (rep_names, i), T --> U) $ |
|
1039 mk_Free "x" T i; |
|
1040 |
|
1041 val Abs_t = Const (List.nth (abs_names, i), U --> T) |
|
1042 |
|
1043 in (prems @ [HOLogic.imp $ |
|
1044 (Const (List.nth (rep_set_names'', i), U --> HOLogic.boolT) $ Rep_t) $ |
|
1045 (mk_Free "P" (T --> HOLogic.boolT) (i + 1) $ (Abs_t $ Rep_t))], |
|
1046 concls @ [mk_Free "P" (T --> HOLogic.boolT) (i + 1) $ mk_Free "x" T i]) |
|
1047 end; |
|
1048 |
|
1049 val (indrule_lemma_prems, indrule_lemma_concls) = |
|
1050 Library.foldl mk_indrule_lemma (([], []), (descr'' ~~ recTs ~~ recTs')); |
|
1051 |
|
1052 val indrule_lemma = Goal.prove_global thy8 [] [] |
|
1053 (Logic.mk_implies |
|
1054 (HOLogic.mk_Trueprop (mk_conj indrule_lemma_prems), |
|
1055 HOLogic.mk_Trueprop (mk_conj indrule_lemma_concls))) (fn _ => EVERY |
|
1056 [REPEAT (etac conjE 1), |
|
1057 REPEAT (EVERY |
|
1058 [TRY (rtac conjI 1), full_simp_tac (HOL_basic_ss addsimps Rep_inverse_thms) 1, |
|
1059 etac mp 1, resolve_tac Rep_thms 1])]); |
|
1060 |
|
1061 val Ps = map head_of (HOLogic.dest_conj (HOLogic.dest_Trueprop (concl_of indrule_lemma))); |
|
1062 val frees = if length Ps = 1 then [Free ("P", snd (dest_Var (hd Ps)))] else |
|
1063 map (Free o apfst fst o dest_Var) Ps; |
|
1064 val indrule_lemma' = cterm_instantiate |
|
1065 (map (cterm_of thy8) Ps ~~ map (cterm_of thy8) frees) indrule_lemma; |
|
1066 |
|
1067 val Abs_inverse_thms' = map (fn r => r RS subst) Abs_inverse_thms; |
|
1068 |
|
1069 val dt_induct_prop = DatatypeProp.make_ind descr' sorts; |
|
1070 val dt_induct = Goal.prove_global thy8 [] |
|
1071 (Logic.strip_imp_prems dt_induct_prop) (Logic.strip_imp_concl dt_induct_prop) |
|
1072 (fn {prems, ...} => EVERY |
|
1073 [rtac indrule_lemma' 1, |
|
1074 (indtac rep_induct [] THEN_ALL_NEW ObjectLogic.atomize_prems_tac) 1, |
|
1075 EVERY (map (fn (prem, r) => (EVERY |
|
1076 [REPEAT (eresolve_tac Abs_inverse_thms' 1), |
|
1077 simp_tac (HOL_basic_ss addsimps [symmetric r]) 1, |
|
1078 DEPTH_SOLVE_1 (ares_tac [prem] 1 ORELSE etac allE 1)])) |
|
1079 (prems ~~ constr_defs))]); |
|
1080 |
|
1081 val case_names_induct = mk_case_names_induct descr''; |
|
1082 |
|
1083 (**** prove that new datatypes have finite support ****) |
|
1084 |
|
1085 val _ = warning "proving finite support for the new datatype"; |
|
1086 |
|
1087 val indnames = DatatypeProp.make_tnames recTs; |
|
1088 |
|
1089 val abs_supp = PureThy.get_thms thy8 "abs_supp"; |
|
1090 val supp_atm = PureThy.get_thms thy8 "supp_atm"; |
|
1091 |
|
1092 val finite_supp_thms = map (fn atom => |
|
1093 let val atomT = Type (atom, []) |
|
1094 in map standard (List.take |
|
1095 (split_conj_thm (Goal.prove_global thy8 [] [] |
|
1096 (augment_sort thy8 (fs_class_of thy8 atom :: pt_cp_sort) |
|
1097 (HOLogic.mk_Trueprop |
|
1098 (foldr1 HOLogic.mk_conj (map (fn (s, T) => |
|
1099 Const ("Finite_Set.finite", HOLogic.mk_setT atomT --> HOLogic.boolT) $ |
|
1100 (Const ("Nominal.supp", T --> HOLogic.mk_setT atomT) $ Free (s, T))) |
|
1101 (indnames ~~ recTs))))) |
|
1102 (fn _ => indtac dt_induct indnames 1 THEN |
|
1103 ALLGOALS (asm_full_simp_tac (simpset_of thy8 addsimps |
|
1104 (abs_supp @ supp_atm @ |
|
1105 PureThy.get_thms thy8 ("fs_" ^ Long_Name.base_name atom ^ "1") @ |
|
1106 List.concat supp_thms))))), |
|
1107 length new_type_names)) |
|
1108 end) atoms; |
|
1109 |
|
1110 val simp_atts = replicate (length new_type_names) [Simplifier.simp_add]; |
|
1111 |
|
1112 (* Function to add both the simp and eqvt attributes *) |
|
1113 (* These two attributes are duplicated on all the types in the mutual nominal datatypes *) |
|
1114 |
|
1115 val simp_eqvt_atts = replicate (length new_type_names) [Simplifier.simp_add, NominalThmDecls.eqvt_add]; |
|
1116 |
|
1117 val (_, thy9) = thy8 |> |
|
1118 Sign.add_path big_name |> |
|
1119 PureThy.add_thms [((Binding.name "induct", dt_induct), [case_names_induct])] ||>> |
|
1120 PureThy.add_thmss [((Binding.name "inducts", projections dt_induct), [case_names_induct])] ||> |
|
1121 Sign.parent_path ||>> |
|
1122 DatatypeAux.store_thmss_atts "distinct" new_type_names simp_atts distinct_thms ||>> |
|
1123 DatatypeAux.store_thmss "constr_rep" new_type_names constr_rep_thmss ||>> |
|
1124 DatatypeAux.store_thmss_atts "perm" new_type_names simp_eqvt_atts perm_simps' ||>> |
|
1125 DatatypeAux.store_thmss "inject" new_type_names inject_thms ||>> |
|
1126 DatatypeAux.store_thmss "supp" new_type_names supp_thms ||>> |
|
1127 DatatypeAux.store_thmss_atts "fresh" new_type_names simp_atts fresh_thms ||> |
|
1128 fold (fn (atom, ths) => fn thy => |
|
1129 let |
|
1130 val class = fs_class_of thy atom; |
|
1131 val sort = Sign.certify_sort thy (class :: pt_cp_sort) |
|
1132 in fold (fn Type (s, Ts) => AxClass.prove_arity |
|
1133 (s, map (inter_sort thy sort o snd o dest_TFree) Ts, [class]) |
|
1134 (Class.intro_classes_tac [] THEN resolve_tac ths 1)) newTs thy |
|
1135 end) (atoms ~~ finite_supp_thms); |
|
1136 |
|
1137 (**** strong induction theorem ****) |
|
1138 |
|
1139 val pnames = if length descr'' = 1 then ["P"] |
|
1140 else map (fn i => "P" ^ string_of_int i) (1 upto length descr''); |
|
1141 val ind_sort = if null dt_atomTs then HOLogic.typeS |
|
1142 else Sign.certify_sort thy9 (map (fs_class_of thy9) dt_atoms); |
|
1143 val fsT = TFree ("'n", ind_sort); |
|
1144 val fsT' = TFree ("'n", HOLogic.typeS); |
|
1145 |
|
1146 val fresh_fs = map (fn (s, T) => (T, Free (s, fsT' --> HOLogic.mk_setT T))) |
|
1147 (DatatypeProp.indexify_names (replicate (length dt_atomTs) "f") ~~ dt_atomTs); |
|
1148 |
|
1149 fun make_pred fsT i T = |
|
1150 Free (List.nth (pnames, i), fsT --> T --> HOLogic.boolT); |
|
1151 |
|
1152 fun mk_fresh1 xs [] = [] |
|
1153 | mk_fresh1 xs ((y as (_, T)) :: ys) = map (fn x => HOLogic.mk_Trueprop |
|
1154 (HOLogic.mk_not (HOLogic.mk_eq (Free y, Free x)))) |
|
1155 (filter (fn (_, U) => T = U) (rev xs)) @ |
|
1156 mk_fresh1 (y :: xs) ys; |
|
1157 |
|
1158 fun mk_fresh2 xss [] = [] |
|
1159 | mk_fresh2 xss ((p as (ys, _)) :: yss) = List.concat (map (fn y as (_, T) => |
|
1160 map (fn (_, x as (_, U)) => HOLogic.mk_Trueprop |
|
1161 (fresh_const T U $ Free y $ Free x)) (rev xss @ yss)) ys) @ |
|
1162 mk_fresh2 (p :: xss) yss; |
|
1163 |
|
1164 fun make_ind_prem fsT f k T ((cname, cargs), idxs) = |
|
1165 let |
|
1166 val recs = List.filter is_rec_type cargs; |
|
1167 val Ts = map (typ_of_dtyp descr'' sorts) cargs; |
|
1168 val recTs' = map (typ_of_dtyp descr'' sorts) recs; |
|
1169 val tnames = Name.variant_list pnames (DatatypeProp.make_tnames Ts); |
|
1170 val rec_tnames = map fst (List.filter (is_rec_type o snd) (tnames ~~ cargs)); |
|
1171 val frees = tnames ~~ Ts; |
|
1172 val frees' = partition_cargs idxs frees; |
|
1173 val z = (Name.variant tnames "z", fsT); |
|
1174 |
|
1175 fun mk_prem ((dt, s), T) = |
|
1176 let |
|
1177 val (Us, U) = strip_type T; |
|
1178 val l = length Us |
|
1179 in list_all (z :: map (pair "x") Us, HOLogic.mk_Trueprop |
|
1180 (make_pred fsT (body_index dt) U $ Bound l $ app_bnds (Free (s, T)) l)) |
|
1181 end; |
|
1182 |
|
1183 val prems = map mk_prem (recs ~~ rec_tnames ~~ recTs'); |
|
1184 val prems' = map (fn p as (_, T) => HOLogic.mk_Trueprop |
|
1185 (f T (Free p) (Free z))) (List.concat (map fst frees')) @ |
|
1186 mk_fresh1 [] (List.concat (map fst frees')) @ |
|
1187 mk_fresh2 [] frees' |
|
1188 |
|
1189 in list_all_free (frees @ [z], Logic.list_implies (prems' @ prems, |
|
1190 HOLogic.mk_Trueprop (make_pred fsT k T $ Free z $ |
|
1191 list_comb (Const (cname, Ts ---> T), map Free frees)))) |
|
1192 end; |
|
1193 |
|
1194 val ind_prems = List.concat (map (fn (((i, (_, _, constrs)), (_, idxss)), T) => |
|
1195 map (make_ind_prem fsT (fn T => fn t => fn u => |
|
1196 fresh_const T fsT $ t $ u) i T) |
|
1197 (constrs ~~ idxss)) (descr'' ~~ ndescr ~~ recTs)); |
|
1198 val tnames = DatatypeProp.make_tnames recTs; |
|
1199 val zs = Name.variant_list tnames (replicate (length descr'') "z"); |
|
1200 val ind_concl = HOLogic.mk_Trueprop (foldr1 (HOLogic.mk_binop "op &") |
|
1201 (map (fn ((((i, _), T), tname), z) => |
|
1202 make_pred fsT i T $ Free (z, fsT) $ Free (tname, T)) |
|
1203 (descr'' ~~ recTs ~~ tnames ~~ zs))); |
|
1204 val induct = Logic.list_implies (ind_prems, ind_concl); |
|
1205 |
|
1206 val ind_prems' = |
|
1207 map (fn (_, f as Free (_, T)) => list_all_free ([("x", fsT')], |
|
1208 HOLogic.mk_Trueprop (Const ("Finite_Set.finite", |
|
1209 (snd (split_last (binder_types T)) --> HOLogic.boolT) --> |
|
1210 HOLogic.boolT) $ (f $ Free ("x", fsT'))))) fresh_fs @ |
|
1211 List.concat (map (fn (((i, (_, _, constrs)), (_, idxss)), T) => |
|
1212 map (make_ind_prem fsT' (fn T => fn t => fn u => HOLogic.Not $ |
|
1213 HOLogic.mk_mem (t, the (AList.lookup op = fresh_fs T) $ u)) i T) |
|
1214 (constrs ~~ idxss)) (descr'' ~~ ndescr ~~ recTs)); |
|
1215 val ind_concl' = HOLogic.mk_Trueprop (foldr1 (HOLogic.mk_binop "op &") |
|
1216 (map (fn ((((i, _), T), tname), z) => |
|
1217 make_pred fsT' i T $ Free (z, fsT') $ Free (tname, T)) |
|
1218 (descr'' ~~ recTs ~~ tnames ~~ zs))); |
|
1219 val induct' = Logic.list_implies (ind_prems', ind_concl'); |
|
1220 |
|
1221 val aux_ind_vars = |
|
1222 (DatatypeProp.indexify_names (replicate (length dt_atomTs) "pi") ~~ |
|
1223 map mk_permT dt_atomTs) @ [("z", fsT')]; |
|
1224 val aux_ind_Ts = rev (map snd aux_ind_vars); |
|
1225 val aux_ind_concl = HOLogic.mk_Trueprop (foldr1 (HOLogic.mk_binop "op &") |
|
1226 (map (fn (((i, _), T), tname) => |
|
1227 HOLogic.list_all (aux_ind_vars, make_pred fsT' i T $ Bound 0 $ |
|
1228 fold_rev (mk_perm aux_ind_Ts) (map Bound (length dt_atomTs downto 1)) |
|
1229 (Free (tname, T)))) |
|
1230 (descr'' ~~ recTs ~~ tnames))); |
|
1231 |
|
1232 val fin_set_supp = map (fn s => |
|
1233 at_inst_of thy9 s RS at_fin_set_supp) dt_atoms; |
|
1234 val fin_set_fresh = map (fn s => |
|
1235 at_inst_of thy9 s RS at_fin_set_fresh) dt_atoms; |
|
1236 val pt1_atoms = map (fn Type (s, _) => |
|
1237 PureThy.get_thm thy9 ("pt_" ^ Long_Name.base_name s ^ "1")) dt_atomTs; |
|
1238 val pt2_atoms = map (fn Type (s, _) => |
|
1239 PureThy.get_thm thy9 ("pt_" ^ Long_Name.base_name s ^ "2") RS sym) dt_atomTs; |
|
1240 val exists_fresh' = PureThy.get_thms thy9 "exists_fresh'"; |
|
1241 val fs_atoms = PureThy.get_thms thy9 "fin_supp"; |
|
1242 val abs_supp = PureThy.get_thms thy9 "abs_supp"; |
|
1243 val perm_fresh_fresh = PureThy.get_thms thy9 "perm_fresh_fresh"; |
|
1244 val calc_atm = PureThy.get_thms thy9 "calc_atm"; |
|
1245 val fresh_atm = PureThy.get_thms thy9 "fresh_atm"; |
|
1246 val fresh_left = PureThy.get_thms thy9 "fresh_left"; |
|
1247 val perm_swap = PureThy.get_thms thy9 "perm_swap"; |
|
1248 |
|
1249 fun obtain_fresh_name' ths ts T (freshs1, freshs2, ctxt) = |
|
1250 let |
|
1251 val p = foldr1 HOLogic.mk_prod (ts @ freshs1); |
|
1252 val ex = Goal.prove ctxt [] [] (HOLogic.mk_Trueprop |
|
1253 (HOLogic.exists_const T $ Abs ("x", T, |
|
1254 fresh_const T (fastype_of p) $ |
|
1255 Bound 0 $ p))) |
|
1256 (fn _ => EVERY |
|
1257 [resolve_tac exists_fresh' 1, |
|
1258 simp_tac (HOL_ss addsimps (supp_prod :: finite_Un :: fs_atoms @ |
|
1259 fin_set_supp @ ths)) 1]); |
|
1260 val (([cx], ths), ctxt') = Obtain.result |
|
1261 (fn _ => EVERY |
|
1262 [etac exE 1, |
|
1263 full_simp_tac (HOL_ss addsimps (fresh_prod :: fresh_atm)) 1, |
|
1264 REPEAT (etac conjE 1)]) |
|
1265 [ex] ctxt |
|
1266 in (freshs1 @ [term_of cx], freshs2 @ ths, ctxt') end; |
|
1267 |
|
1268 fun fresh_fresh_inst thy a b = |
|
1269 let |
|
1270 val T = fastype_of a; |
|
1271 val SOME th = find_first (fn th => case prop_of th of |
|
1272 _ $ (_ $ (Const (_, Type (_, [U, _])) $ _ $ _)) $ _ => U = T |
|
1273 | _ => false) perm_fresh_fresh |
|
1274 in |
|
1275 Drule.instantiate' [] |
|
1276 [SOME (cterm_of thy a), NONE, SOME (cterm_of thy b)] th |
|
1277 end; |
|
1278 |
|
1279 val fs_cp_sort = |
|
1280 map (fs_class_of thy9) dt_atoms @ |
|
1281 maps (fn s => map (cp_class_of thy9 s) (dt_atoms \ s)) dt_atoms; |
|
1282 |
|
1283 (********************************************************************** |
|
1284 The subgoals occurring in the proof of induct_aux have the |
|
1285 following parameters: |
|
1286 |
|
1287 x_1 ... x_k p_1 ... p_m z |
|
1288 |
|
1289 where |
|
1290 |
|
1291 x_i : constructor arguments (introduced by weak induction rule) |
|
1292 p_i : permutations (one for each atom type in the data type) |
|
1293 z : freshness context |
|
1294 ***********************************************************************) |
|
1295 |
|
1296 val _ = warning "proving strong induction theorem ..."; |
|
1297 |
|
1298 val induct_aux = Goal.prove_global thy9 [] |
|
1299 (map (augment_sort thy9 fs_cp_sort) ind_prems') |
|
1300 (augment_sort thy9 fs_cp_sort ind_concl') (fn {prems, context} => |
|
1301 let |
|
1302 val (prems1, prems2) = chop (length dt_atomTs) prems; |
|
1303 val ind_ss2 = HOL_ss addsimps |
|
1304 finite_Diff :: abs_fresh @ abs_supp @ fs_atoms; |
|
1305 val ind_ss1 = ind_ss2 addsimps fresh_left @ calc_atm @ |
|
1306 fresh_atm @ rev_simps @ app_simps; |
|
1307 val ind_ss3 = HOL_ss addsimps abs_fun_eq1 :: |
|
1308 abs_perm @ calc_atm @ perm_swap; |
|
1309 val ind_ss4 = HOL_basic_ss addsimps fresh_left @ prems1 @ |
|
1310 fin_set_fresh @ calc_atm; |
|
1311 val ind_ss5 = HOL_basic_ss addsimps pt1_atoms; |
|
1312 val ind_ss6 = HOL_basic_ss addsimps flat perm_simps'; |
|
1313 val th = Goal.prove context [] [] |
|
1314 (augment_sort thy9 fs_cp_sort aux_ind_concl) |
|
1315 (fn {context = context1, ...} => |
|
1316 EVERY (indtac dt_induct tnames 1 :: |
|
1317 maps (fn ((_, (_, _, constrs)), (_, constrs')) => |
|
1318 map (fn ((cname, cargs), is) => |
|
1319 REPEAT (rtac allI 1) THEN |
|
1320 SUBPROOF (fn {prems = iprems, params, concl, |
|
1321 context = context2, ...} => |
|
1322 let |
|
1323 val concl' = term_of concl; |
|
1324 val _ $ (_ $ _ $ u) = concl'; |
|
1325 val U = fastype_of u; |
|
1326 val (xs, params') = |
|
1327 chop (length cargs) (map term_of params); |
|
1328 val Ts = map fastype_of xs; |
|
1329 val cnstr = Const (cname, Ts ---> U); |
|
1330 val (pis, z) = split_last params'; |
|
1331 val mk_pi = fold_rev (mk_perm []) pis; |
|
1332 val xs' = partition_cargs is xs; |
|
1333 val xs'' = map (fn (ts, u) => (map mk_pi ts, mk_pi u)) xs'; |
|
1334 val ts = maps (fn (ts, u) => ts @ [u]) xs''; |
|
1335 val (freshs1, freshs2, context3) = fold (fn t => |
|
1336 let val T = fastype_of t |
|
1337 in obtain_fresh_name' prems1 |
|
1338 (the (AList.lookup op = fresh_fs T) $ z :: ts) T |
|
1339 end) (maps fst xs') ([], [], context2); |
|
1340 val freshs1' = unflat (map fst xs') freshs1; |
|
1341 val freshs2' = map (Simplifier.simplify ind_ss4) |
|
1342 (mk_not_sym freshs2); |
|
1343 val ind_ss1' = ind_ss1 addsimps freshs2'; |
|
1344 val ind_ss3' = ind_ss3 addsimps freshs2'; |
|
1345 val rename_eq = |
|
1346 if forall (null o fst) xs' then [] |
|
1347 else [Goal.prove context3 [] [] |
|
1348 (HOLogic.mk_Trueprop (HOLogic.mk_eq |
|
1349 (list_comb (cnstr, ts), |
|
1350 list_comb (cnstr, maps (fn ((bs, t), cs) => |
|
1351 cs @ [fold_rev (mk_perm []) (map perm_of_pair |
|
1352 (bs ~~ cs)) t]) (xs'' ~~ freshs1'))))) |
|
1353 (fn _ => EVERY |
|
1354 (simp_tac (HOL_ss addsimps flat inject_thms) 1 :: |
|
1355 REPEAT (FIRSTGOAL (rtac conjI)) :: |
|
1356 maps (fn ((bs, t), cs) => |
|
1357 if null bs then [] |
|
1358 else rtac sym 1 :: maps (fn (b, c) => |
|
1359 [rtac trans 1, rtac sym 1, |
|
1360 rtac (fresh_fresh_inst thy9 b c) 1, |
|
1361 simp_tac ind_ss1' 1, |
|
1362 simp_tac ind_ss2 1, |
|
1363 simp_tac ind_ss3' 1]) (bs ~~ cs)) |
|
1364 (xs'' ~~ freshs1')))]; |
|
1365 val th = Goal.prove context3 [] [] concl' (fn _ => EVERY |
|
1366 [simp_tac (ind_ss6 addsimps rename_eq) 1, |
|
1367 cut_facts_tac iprems 1, |
|
1368 (resolve_tac prems THEN_ALL_NEW |
|
1369 SUBGOAL (fn (t, i) => case Logic.strip_assums_concl t of |
|
1370 _ $ (Const ("Nominal.fresh", _) $ _ $ _) => |
|
1371 simp_tac ind_ss1' i |
|
1372 | _ $ (Const ("Not", _) $ _) => |
|
1373 resolve_tac freshs2' i |
|
1374 | _ => asm_simp_tac (HOL_basic_ss addsimps |
|
1375 pt2_atoms addsimprocs [perm_simproc]) i)) 1]) |
|
1376 val final = ProofContext.export context3 context2 [th] |
|
1377 in |
|
1378 resolve_tac final 1 |
|
1379 end) context1 1) (constrs ~~ constrs')) (descr'' ~~ ndescr))) |
|
1380 in |
|
1381 EVERY |
|
1382 [cut_facts_tac [th] 1, |
|
1383 REPEAT (eresolve_tac [conjE, @{thm allE_Nil}] 1), |
|
1384 REPEAT (etac allE 1), |
|
1385 REPEAT (TRY (rtac conjI 1) THEN asm_full_simp_tac ind_ss5 1)] |
|
1386 end); |
|
1387 |
|
1388 val induct_aux' = Thm.instantiate ([], |
|
1389 map (fn (s, v as Var (_, T)) => |
|
1390 (cterm_of thy9 v, cterm_of thy9 (Free (s, T)))) |
|
1391 (pnames ~~ map head_of (HOLogic.dest_conj |
|
1392 (HOLogic.dest_Trueprop (concl_of induct_aux)))) @ |
|
1393 map (fn (_, f) => |
|
1394 let val f' = Logic.varify f |
|
1395 in (cterm_of thy9 f', |
|
1396 cterm_of thy9 (Const ("Nominal.supp", fastype_of f'))) |
|
1397 end) fresh_fs) induct_aux; |
|
1398 |
|
1399 val induct = Goal.prove_global thy9 [] |
|
1400 (map (augment_sort thy9 fs_cp_sort) ind_prems) |
|
1401 (augment_sort thy9 fs_cp_sort ind_concl) |
|
1402 (fn {prems, ...} => EVERY |
|
1403 [rtac induct_aux' 1, |
|
1404 REPEAT (resolve_tac fs_atoms 1), |
|
1405 REPEAT ((resolve_tac prems THEN_ALL_NEW |
|
1406 (etac meta_spec ORELSE' full_simp_tac (HOL_basic_ss addsimps [fresh_def]))) 1)]) |
|
1407 |
|
1408 val (_, thy10) = thy9 |> |
|
1409 Sign.add_path big_name |> |
|
1410 PureThy.add_thms [((Binding.name "strong_induct'", induct_aux), [])] ||>> |
|
1411 PureThy.add_thms [((Binding.name "strong_induct", induct), [case_names_induct])] ||>> |
|
1412 PureThy.add_thmss [((Binding.name "strong_inducts", projections induct), [case_names_induct])]; |
|
1413 |
|
1414 (**** recursion combinator ****) |
|
1415 |
|
1416 val _ = warning "defining recursion combinator ..."; |
|
1417 |
|
1418 val used = List.foldr OldTerm.add_typ_tfree_names [] recTs; |
|
1419 |
|
1420 val (rec_result_Ts', rec_fn_Ts') = DatatypeProp.make_primrec_Ts descr' sorts used; |
|
1421 |
|
1422 val rec_sort = if null dt_atomTs then HOLogic.typeS else |
|
1423 Sign.certify_sort thy10 pt_cp_sort; |
|
1424 |
|
1425 val rec_result_Ts = map (fn TFree (s, _) => TFree (s, rec_sort)) rec_result_Ts'; |
|
1426 val rec_fn_Ts = map (typ_subst_atomic (rec_result_Ts' ~~ rec_result_Ts)) rec_fn_Ts'; |
|
1427 |
|
1428 val rec_set_Ts = map (fn (T1, T2) => |
|
1429 rec_fn_Ts @ [T1, T2] ---> HOLogic.boolT) (recTs ~~ rec_result_Ts); |
|
1430 |
|
1431 val big_rec_name = big_name ^ "_rec_set"; |
|
1432 val rec_set_names' = |
|
1433 if length descr'' = 1 then [big_rec_name] else |
|
1434 map ((curry (op ^) (big_rec_name ^ "_")) o string_of_int) |
|
1435 (1 upto (length descr'')); |
|
1436 val rec_set_names = map (Sign.full_bname thy10) rec_set_names'; |
|
1437 |
|
1438 val rec_fns = map (uncurry (mk_Free "f")) |
|
1439 (rec_fn_Ts ~~ (1 upto (length rec_fn_Ts))); |
|
1440 val rec_sets' = map (fn c => list_comb (Free c, rec_fns)) |
|
1441 (rec_set_names' ~~ rec_set_Ts); |
|
1442 val rec_sets = map (fn c => list_comb (Const c, rec_fns)) |
|
1443 (rec_set_names ~~ rec_set_Ts); |
|
1444 |
|
1445 (* introduction rules for graph of recursion function *) |
|
1446 |
|
1447 val rec_preds = map (fn (a, T) => |
|
1448 Free (a, T --> HOLogic.boolT)) (pnames ~~ rec_result_Ts); |
|
1449 |
|
1450 fun mk_fresh3 rs [] = [] |
|
1451 | mk_fresh3 rs ((p as (ys, z)) :: yss) = List.concat (map (fn y as (_, T) => |
|
1452 List.mapPartial (fn ((_, (_, x)), r as (_, U)) => if z = x then NONE |
|
1453 else SOME (HOLogic.mk_Trueprop |
|
1454 (fresh_const T U $ Free y $ Free r))) rs) ys) @ |
|
1455 mk_fresh3 rs yss; |
|
1456 |
|
1457 (* FIXME: avoid collisions with other variable names? *) |
|
1458 val rec_ctxt = Free ("z", fsT'); |
|
1459 |
|
1460 fun make_rec_intr T p rec_set ((rec_intr_ts, rec_prems, rec_prems', |
|
1461 rec_eq_prems, l), ((cname, cargs), idxs)) = |
|
1462 let |
|
1463 val Ts = map (typ_of_dtyp descr'' sorts) cargs; |
|
1464 val frees = map (fn i => "x" ^ string_of_int i) (1 upto length Ts) ~~ Ts; |
|
1465 val frees' = partition_cargs idxs frees; |
|
1466 val binders = List.concat (map fst frees'); |
|
1467 val atomTs = distinct op = (maps (map snd o fst) frees'); |
|
1468 val recs = List.mapPartial |
|
1469 (fn ((_, DtRec i), p) => SOME (i, p) | _ => NONE) |
|
1470 (partition_cargs idxs cargs ~~ frees'); |
|
1471 val frees'' = map (fn i => "y" ^ string_of_int i) (1 upto length recs) ~~ |
|
1472 map (fn (i, _) => List.nth (rec_result_Ts, i)) recs; |
|
1473 val prems1 = map (fn ((i, (_, x)), y) => HOLogic.mk_Trueprop |
|
1474 (List.nth (rec_sets', i) $ Free x $ Free y)) (recs ~~ frees''); |
|
1475 val prems2 = |
|
1476 map (fn f => map (fn p as (_, T) => HOLogic.mk_Trueprop |
|
1477 (fresh_const T (fastype_of f) $ Free p $ f)) binders) rec_fns; |
|
1478 val prems3 = mk_fresh1 [] binders @ mk_fresh2 [] frees'; |
|
1479 val prems4 = map (fn ((i, _), y) => |
|
1480 HOLogic.mk_Trueprop (List.nth (rec_preds, i) $ Free y)) (recs ~~ frees''); |
|
1481 val prems5 = mk_fresh3 (recs ~~ frees'') frees'; |
|
1482 val prems6 = maps (fn aT => map (fn y as (_, T) => HOLogic.mk_Trueprop |
|
1483 (Const ("Finite_Set.finite", HOLogic.mk_setT aT --> HOLogic.boolT) $ |
|
1484 (Const ("Nominal.supp", T --> HOLogic.mk_setT aT) $ Free y))) |
|
1485 frees'') atomTs; |
|
1486 val prems7 = map (fn x as (_, T) => HOLogic.mk_Trueprop |
|
1487 (fresh_const T fsT' $ Free x $ rec_ctxt)) binders; |
|
1488 val result = list_comb (List.nth (rec_fns, l), map Free (frees @ frees'')); |
|
1489 val result_freshs = map (fn p as (_, T) => |
|
1490 fresh_const T (fastype_of result) $ Free p $ result) binders; |
|
1491 val P = HOLogic.mk_Trueprop (p $ result) |
|
1492 in |
|
1493 (rec_intr_ts @ [Logic.list_implies (List.concat prems2 @ prems3 @ prems1, |
|
1494 HOLogic.mk_Trueprop (rec_set $ |
|
1495 list_comb (Const (cname, Ts ---> T), map Free frees) $ result))], |
|
1496 rec_prems @ [list_all_free (frees @ frees'', Logic.list_implies (prems4, P))], |
|
1497 rec_prems' @ map (fn fr => list_all_free (frees @ frees'', |
|
1498 Logic.list_implies (List.nth (prems2, l) @ prems3 @ prems5 @ prems7 @ prems6 @ [P], |
|
1499 HOLogic.mk_Trueprop fr))) result_freshs, |
|
1500 rec_eq_prems @ [List.concat prems2 @ prems3], |
|
1501 l + 1) |
|
1502 end; |
|
1503 |
|
1504 val (rec_intr_ts, rec_prems, rec_prems', rec_eq_prems, _) = |
|
1505 Library.foldl (fn (x, ((((d, d'), T), p), rec_set)) => |
|
1506 Library.foldl (make_rec_intr T p rec_set) (x, #3 (snd d) ~~ snd d')) |
|
1507 (([], [], [], [], 0), descr'' ~~ ndescr ~~ recTs ~~ rec_preds ~~ rec_sets'); |
|
1508 |
|
1509 val ({intrs = rec_intrs, elims = rec_elims, raw_induct = rec_induct, ...}, thy11) = |
|
1510 thy10 |> |
|
1511 Inductive.add_inductive_global (serial_string ()) |
|
1512 {quiet_mode = #quiet config, verbose = false, kind = Thm.internalK, |
|
1513 alt_name = Binding.name big_rec_name, coind = false, no_elim = false, no_ind = false, |
|
1514 skip_mono = true, fork_mono = false} |
|
1515 (map (fn (s, T) => ((Binding.name s, T), NoSyn)) (rec_set_names' ~~ rec_set_Ts)) |
|
1516 (map dest_Free rec_fns) |
|
1517 (map (fn x => (Attrib.empty_binding, x)) rec_intr_ts) [] ||> |
|
1518 PureThy.hide_fact true (Long_Name.append (Sign.full_bname thy10 big_rec_name) "induct"); |
|
1519 |
|
1520 (** equivariance **) |
|
1521 |
|
1522 val fresh_bij = PureThy.get_thms thy11 "fresh_bij"; |
|
1523 val perm_bij = PureThy.get_thms thy11 "perm_bij"; |
|
1524 |
|
1525 val (rec_equiv_thms, rec_equiv_thms') = ListPair.unzip (map (fn aT => |
|
1526 let |
|
1527 val permT = mk_permT aT; |
|
1528 val pi = Free ("pi", permT); |
|
1529 val rec_fns_pi = map (mk_perm [] pi o uncurry (mk_Free "f")) |
|
1530 (rec_fn_Ts ~~ (1 upto (length rec_fn_Ts))); |
|
1531 val rec_sets_pi = map (fn c => list_comb (Const c, rec_fns_pi)) |
|
1532 (rec_set_names ~~ rec_set_Ts); |
|
1533 val ps = map (fn ((((T, U), R), R'), i) => |
|
1534 let |
|
1535 val x = Free ("x" ^ string_of_int i, T); |
|
1536 val y = Free ("y" ^ string_of_int i, U) |
|
1537 in |
|
1538 (R $ x $ y, R' $ mk_perm [] pi x $ mk_perm [] pi y) |
|
1539 end) (recTs ~~ rec_result_Ts ~~ rec_sets ~~ rec_sets_pi ~~ (1 upto length recTs)); |
|
1540 val ths = map (fn th => standard (th RS mp)) (split_conj_thm |
|
1541 (Goal.prove_global thy11 [] [] |
|
1542 (augment_sort thy1 pt_cp_sort |
|
1543 (HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj (map HOLogic.mk_imp ps)))) |
|
1544 (fn _ => rtac rec_induct 1 THEN REPEAT |
|
1545 (simp_tac (Simplifier.theory_context thy11 HOL_basic_ss |
|
1546 addsimps flat perm_simps' |
|
1547 addsimprocs [NominalPermeq.perm_simproc_app]) 1 THEN |
|
1548 (resolve_tac rec_intrs THEN_ALL_NEW |
|
1549 asm_simp_tac (HOL_ss addsimps (fresh_bij @ perm_bij))) 1)))) |
|
1550 val ths' = map (fn ((P, Q), th) => |
|
1551 Goal.prove_global thy11 [] [] |
|
1552 (augment_sort thy1 pt_cp_sort |
|
1553 (Logic.mk_implies (HOLogic.mk_Trueprop Q, HOLogic.mk_Trueprop P))) |
|
1554 (fn _ => dtac (Thm.instantiate ([], |
|
1555 [(cterm_of thy11 (Var (("pi", 0), permT)), |
|
1556 cterm_of thy11 (Const ("List.rev", permT --> permT) $ pi))]) th) 1 THEN |
|
1557 NominalPermeq.perm_simp_tac HOL_ss 1)) (ps ~~ ths) |
|
1558 in (ths, ths') end) dt_atomTs); |
|
1559 |
|
1560 (** finite support **) |
|
1561 |
|
1562 val rec_fin_supp_thms = map (fn aT => |
|
1563 let |
|
1564 val name = Long_Name.base_name (fst (dest_Type aT)); |
|
1565 val fs_name = PureThy.get_thm thy11 ("fs_" ^ name ^ "1"); |
|
1566 val aset = HOLogic.mk_setT aT; |
|
1567 val finite = Const ("Finite_Set.finite", aset --> HOLogic.boolT); |
|
1568 val fins = map (fn (f, T) => HOLogic.mk_Trueprop |
|
1569 (finite $ (Const ("Nominal.supp", T --> aset) $ f))) |
|
1570 (rec_fns ~~ rec_fn_Ts) |
|
1571 in |
|
1572 map (fn th => standard (th RS mp)) (split_conj_thm |
|
1573 (Goal.prove_global thy11 [] |
|
1574 (map (augment_sort thy11 fs_cp_sort) fins) |
|
1575 (augment_sort thy11 fs_cp_sort |
|
1576 (HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj |
|
1577 (map (fn (((T, U), R), i) => |
|
1578 let |
|
1579 val x = Free ("x" ^ string_of_int i, T); |
|
1580 val y = Free ("y" ^ string_of_int i, U) |
|
1581 in |
|
1582 HOLogic.mk_imp (R $ x $ y, |
|
1583 finite $ (Const ("Nominal.supp", U --> aset) $ y)) |
|
1584 end) (recTs ~~ rec_result_Ts ~~ rec_sets ~~ |
|
1585 (1 upto length recTs)))))) |
|
1586 (fn {prems = fins, ...} => |
|
1587 (rtac rec_induct THEN_ALL_NEW cut_facts_tac fins) 1 THEN REPEAT |
|
1588 (NominalPermeq.finite_guess_tac (HOL_ss addsimps [fs_name]) 1)))) |
|
1589 end) dt_atomTs; |
|
1590 |
|
1591 (** freshness **) |
|
1592 |
|
1593 val finite_premss = map (fn aT => |
|
1594 map (fn (f, T) => HOLogic.mk_Trueprop |
|
1595 (Const ("Finite_Set.finite", HOLogic.mk_setT aT --> HOLogic.boolT) $ |
|
1596 (Const ("Nominal.supp", T --> HOLogic.mk_setT aT) $ f))) |
|
1597 (rec_fns ~~ rec_fn_Ts)) dt_atomTs; |
|
1598 |
|
1599 val rec_fns' = map (augment_sort thy11 fs_cp_sort) rec_fns; |
|
1600 |
|
1601 val rec_fresh_thms = map (fn ((aT, eqvt_ths), finite_prems) => |
|
1602 let |
|
1603 val name = Long_Name.base_name (fst (dest_Type aT)); |
|
1604 val fs_name = PureThy.get_thm thy11 ("fs_" ^ name ^ "1"); |
|
1605 val a = Free ("a", aT); |
|
1606 val freshs = map (fn (f, fT) => HOLogic.mk_Trueprop |
|
1607 (fresh_const aT fT $ a $ f)) (rec_fns ~~ rec_fn_Ts) |
|
1608 in |
|
1609 map (fn (((T, U), R), eqvt_th) => |
|
1610 let |
|
1611 val x = Free ("x", augment_sort_typ thy11 fs_cp_sort T); |
|
1612 val y = Free ("y", U); |
|
1613 val y' = Free ("y'", U) |
|
1614 in |
|
1615 standard (Goal.prove (ProofContext.init thy11) [] |
|
1616 (map (augment_sort thy11 fs_cp_sort) |
|
1617 (finite_prems @ |
|
1618 [HOLogic.mk_Trueprop (R $ x $ y), |
|
1619 HOLogic.mk_Trueprop (HOLogic.mk_all ("y'", U, |
|
1620 HOLogic.mk_imp (R $ x $ y', HOLogic.mk_eq (y', y)))), |
|
1621 HOLogic.mk_Trueprop (fresh_const aT T $ a $ x)] @ |
|
1622 freshs)) |
|
1623 (HOLogic.mk_Trueprop (fresh_const aT U $ a $ y)) |
|
1624 (fn {prems, context} => |
|
1625 let |
|
1626 val (finite_prems, rec_prem :: unique_prem :: |
|
1627 fresh_prems) = chop (length finite_prems) prems; |
|
1628 val unique_prem' = unique_prem RS spec RS mp; |
|
1629 val unique = [unique_prem', unique_prem' RS sym] MRS trans; |
|
1630 val _ $ (_ $ (_ $ S $ _)) $ _ = prop_of supports_fresh; |
|
1631 val tuple = foldr1 HOLogic.mk_prod (x :: rec_fns') |
|
1632 in EVERY |
|
1633 [rtac (Drule.cterm_instantiate |
|
1634 [(cterm_of thy11 S, |
|
1635 cterm_of thy11 (Const ("Nominal.supp", |
|
1636 fastype_of tuple --> HOLogic.mk_setT aT) $ tuple))] |
|
1637 supports_fresh) 1, |
|
1638 simp_tac (HOL_basic_ss addsimps |
|
1639 [supports_def, symmetric fresh_def, fresh_prod]) 1, |
|
1640 REPEAT_DETERM (resolve_tac [allI, impI] 1), |
|
1641 REPEAT_DETERM (etac conjE 1), |
|
1642 rtac unique 1, |
|
1643 SUBPROOF (fn {prems = prems', params = [a, b], ...} => EVERY |
|
1644 [cut_facts_tac [rec_prem] 1, |
|
1645 rtac (Thm.instantiate ([], |
|
1646 [(cterm_of thy11 (Var (("pi", 0), mk_permT aT)), |
|
1647 cterm_of thy11 (perm_of_pair (term_of a, term_of b)))]) eqvt_th) 1, |
|
1648 asm_simp_tac (HOL_ss addsimps |
|
1649 (prems' @ perm_swap @ perm_fresh_fresh)) 1]) context 1, |
|
1650 rtac rec_prem 1, |
|
1651 simp_tac (HOL_ss addsimps (fs_name :: |
|
1652 supp_prod :: finite_Un :: finite_prems)) 1, |
|
1653 simp_tac (HOL_ss addsimps (symmetric fresh_def :: |
|
1654 fresh_prod :: fresh_prems)) 1] |
|
1655 end)) |
|
1656 end) (recTs ~~ rec_result_Ts ~~ rec_sets ~~ eqvt_ths) |
|
1657 end) (dt_atomTs ~~ rec_equiv_thms' ~~ finite_premss); |
|
1658 |
|
1659 (** uniqueness **) |
|
1660 |
|
1661 val fun_tuple = foldr1 HOLogic.mk_prod (rec_ctxt :: rec_fns); |
|
1662 val fun_tupleT = fastype_of fun_tuple; |
|
1663 val rec_unique_frees = |
|
1664 DatatypeProp.indexify_names (replicate (length recTs) "x") ~~ recTs; |
|
1665 val rec_unique_frees'' = map (fn (s, T) => (s ^ "'", T)) rec_unique_frees; |
|
1666 val rec_unique_frees' = |
|
1667 DatatypeProp.indexify_names (replicate (length recTs) "y") ~~ rec_result_Ts; |
|
1668 val rec_unique_concls = map (fn ((x, U), R) => |
|
1669 Const ("Ex1", (U --> HOLogic.boolT) --> HOLogic.boolT) $ |
|
1670 Abs ("y", U, R $ Free x $ Bound 0)) |
|
1671 (rec_unique_frees ~~ rec_result_Ts ~~ rec_sets); |
|
1672 |
|
1673 val induct_aux_rec = Drule.cterm_instantiate |
|
1674 (map (pairself (cterm_of thy11) o apsnd (augment_sort thy11 fs_cp_sort)) |
|
1675 (map (fn (aT, f) => (Logic.varify f, Abs ("z", HOLogic.unitT, |
|
1676 Const ("Nominal.supp", fun_tupleT --> HOLogic.mk_setT aT) $ fun_tuple))) |
|
1677 fresh_fs @ |
|
1678 map (fn (((P, T), (x, U)), Q) => |
|
1679 (Var ((P, 0), Logic.varifyT (fsT' --> T --> HOLogic.boolT)), |
|
1680 Abs ("z", HOLogic.unitT, absfree (x, U, Q)))) |
|
1681 (pnames ~~ recTs ~~ rec_unique_frees ~~ rec_unique_concls) @ |
|
1682 map (fn (s, T) => (Var ((s, 0), Logic.varifyT T), Free (s, T))) |
|
1683 rec_unique_frees)) induct_aux; |
|
1684 |
|
1685 fun obtain_fresh_name vs ths rec_fin_supp T (freshs1, freshs2, ctxt) = |
|
1686 let |
|
1687 val p = foldr1 HOLogic.mk_prod (vs @ freshs1); |
|
1688 val ex = Goal.prove ctxt [] [] (HOLogic.mk_Trueprop |
|
1689 (HOLogic.exists_const T $ Abs ("x", T, |
|
1690 fresh_const T (fastype_of p) $ Bound 0 $ p))) |
|
1691 (fn _ => EVERY |
|
1692 [cut_facts_tac ths 1, |
|
1693 REPEAT_DETERM (dresolve_tac (the (AList.lookup op = rec_fin_supp T)) 1), |
|
1694 resolve_tac exists_fresh' 1, |
|
1695 asm_simp_tac (HOL_ss addsimps (supp_prod :: finite_Un :: fs_atoms)) 1]); |
|
1696 val (([cx], ths), ctxt') = Obtain.result |
|
1697 (fn _ => EVERY |
|
1698 [etac exE 1, |
|
1699 full_simp_tac (HOL_ss addsimps (fresh_prod :: fresh_atm)) 1, |
|
1700 REPEAT (etac conjE 1)]) |
|
1701 [ex] ctxt |
|
1702 in (freshs1 @ [term_of cx], freshs2 @ ths, ctxt') end; |
|
1703 |
|
1704 val finite_ctxt_prems = map (fn aT => |
|
1705 HOLogic.mk_Trueprop |
|
1706 (Const ("Finite_Set.finite", HOLogic.mk_setT aT --> HOLogic.boolT) $ |
|
1707 (Const ("Nominal.supp", fsT' --> HOLogic.mk_setT aT) $ rec_ctxt))) dt_atomTs; |
|
1708 |
|
1709 val rec_unique_thms = split_conj_thm (Goal.prove |
|
1710 (ProofContext.init thy11) (map fst rec_unique_frees) |
|
1711 (map (augment_sort thy11 fs_cp_sort) |
|
1712 (List.concat finite_premss @ finite_ctxt_prems @ rec_prems @ rec_prems')) |
|
1713 (augment_sort thy11 fs_cp_sort |
|
1714 (HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj rec_unique_concls))) |
|
1715 (fn {prems, context} => |
|
1716 let |
|
1717 val k = length rec_fns; |
|
1718 val (finite_thss, ths1) = fold_map (fn T => fn xs => |
|
1719 apfst (pair T) (chop k xs)) dt_atomTs prems; |
|
1720 val (finite_ctxt_ths, ths2) = chop (length dt_atomTs) ths1; |
|
1721 val (P_ind_ths, fcbs) = chop k ths2; |
|
1722 val P_ths = map (fn th => th RS mp) (split_conj_thm |
|
1723 (Goal.prove context |
|
1724 (map fst (rec_unique_frees'' @ rec_unique_frees')) [] |
|
1725 (augment_sort thy11 fs_cp_sort |
|
1726 (HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj |
|
1727 (map (fn (((x, y), S), P) => HOLogic.mk_imp |
|
1728 (S $ Free x $ Free y, P $ (Free y))) |
|
1729 (rec_unique_frees'' ~~ rec_unique_frees' ~~ |
|
1730 rec_sets ~~ rec_preds))))) |
|
1731 (fn _ => |
|
1732 rtac rec_induct 1 THEN |
|
1733 REPEAT ((resolve_tac P_ind_ths THEN_ALL_NEW assume_tac) 1)))); |
|
1734 val rec_fin_supp_thms' = map |
|
1735 (fn (ths, (T, fin_ths)) => (T, map (curry op MRS fin_ths) ths)) |
|
1736 (rec_fin_supp_thms ~~ finite_thss); |
|
1737 in EVERY |
|
1738 ([rtac induct_aux_rec 1] @ |
|
1739 maps (fn ((_, finite_ths), finite_th) => |
|
1740 [cut_facts_tac (finite_th :: finite_ths) 1, |
|
1741 asm_simp_tac (HOL_ss addsimps [supp_prod, finite_Un]) 1]) |
|
1742 (finite_thss ~~ finite_ctxt_ths) @ |
|
1743 maps (fn ((_, idxss), elim) => maps (fn idxs => |
|
1744 [full_simp_tac (HOL_ss addsimps [symmetric fresh_def, supp_prod, Un_iff]) 1, |
|
1745 REPEAT_DETERM (eresolve_tac [conjE, ex1E] 1), |
|
1746 rtac ex1I 1, |
|
1747 (resolve_tac rec_intrs THEN_ALL_NEW atac) 1, |
|
1748 rotate_tac ~1 1, |
|
1749 ((DETERM o etac elim) THEN_ALL_NEW full_simp_tac |
|
1750 (HOL_ss addsimps List.concat distinct_thms)) 1] @ |
|
1751 (if null idxs then [] else [hyp_subst_tac 1, |
|
1752 SUBPROOF (fn {asms, concl, prems = prems', params, context = context', ...} => |
|
1753 let |
|
1754 val SOME prem = find_first (can (HOLogic.dest_eq o |
|
1755 HOLogic.dest_Trueprop o prop_of)) prems'; |
|
1756 val _ $ (_ $ lhs $ rhs) = prop_of prem; |
|
1757 val _ $ (_ $ lhs' $ rhs') = term_of concl; |
|
1758 val rT = fastype_of lhs'; |
|
1759 val (c, cargsl) = strip_comb lhs; |
|
1760 val cargsl' = partition_cargs idxs cargsl; |
|
1761 val boundsl = List.concat (map fst cargsl'); |
|
1762 val (_, cargsr) = strip_comb rhs; |
|
1763 val cargsr' = partition_cargs idxs cargsr; |
|
1764 val boundsr = List.concat (map fst cargsr'); |
|
1765 val (params1, _ :: params2) = |
|
1766 chop (length params div 2) (map term_of params); |
|
1767 val params' = params1 @ params2; |
|
1768 val rec_prems = filter (fn th => case prop_of th of |
|
1769 _ $ p => (case head_of p of |
|
1770 Const (s, _) => s mem rec_set_names |
|
1771 | _ => false) |
|
1772 | _ => false) prems'; |
|
1773 val fresh_prems = filter (fn th => case prop_of th of |
|
1774 _ $ (Const ("Nominal.fresh", _) $ _ $ _) => true |
|
1775 | _ $ (Const ("Not", _) $ _) => true |
|
1776 | _ => false) prems'; |
|
1777 val Ts = map fastype_of boundsl; |
|
1778 |
|
1779 val _ = warning "step 1: obtaining fresh names"; |
|
1780 val (freshs1, freshs2, context'') = fold |
|
1781 (obtain_fresh_name (rec_ctxt :: rec_fns' @ params') |
|
1782 (List.concat (map snd finite_thss) @ |
|
1783 finite_ctxt_ths @ rec_prems) |
|
1784 rec_fin_supp_thms') |
|
1785 Ts ([], [], context'); |
|
1786 val pi1 = map perm_of_pair (boundsl ~~ freshs1); |
|
1787 val rpi1 = rev pi1; |
|
1788 val pi2 = map perm_of_pair (boundsr ~~ freshs1); |
|
1789 val rpi2 = rev pi2; |
|
1790 |
|
1791 val fresh_prems' = mk_not_sym fresh_prems; |
|
1792 val freshs2' = mk_not_sym freshs2; |
|
1793 |
|
1794 (** as, bs, cs # K as ts, K bs us **) |
|
1795 val _ = warning "step 2: as, bs, cs # K as ts, K bs us"; |
|
1796 val prove_fresh_ss = HOL_ss addsimps |
|
1797 (finite_Diff :: List.concat fresh_thms @ |
|
1798 fs_atoms @ abs_fresh @ abs_supp @ fresh_atm); |
|
1799 (* FIXME: avoid asm_full_simp_tac ? *) |
|
1800 fun prove_fresh ths y x = Goal.prove context'' [] [] |
|
1801 (HOLogic.mk_Trueprop (fresh_const |
|
1802 (fastype_of x) (fastype_of y) $ x $ y)) |
|
1803 (fn _ => cut_facts_tac ths 1 THEN asm_full_simp_tac prove_fresh_ss 1); |
|
1804 val constr_fresh_thms = |
|
1805 map (prove_fresh fresh_prems lhs) boundsl @ |
|
1806 map (prove_fresh fresh_prems rhs) boundsr @ |
|
1807 map (prove_fresh freshs2 lhs) freshs1 @ |
|
1808 map (prove_fresh freshs2 rhs) freshs1; |
|
1809 |
|
1810 (** pi1 o (K as ts) = pi2 o (K bs us) **) |
|
1811 val _ = warning "step 3: pi1 o (K as ts) = pi2 o (K bs us)"; |
|
1812 val pi1_pi2_eq = Goal.prove context'' [] [] |
|
1813 (HOLogic.mk_Trueprop (HOLogic.mk_eq |
|
1814 (fold_rev (mk_perm []) pi1 lhs, fold_rev (mk_perm []) pi2 rhs))) |
|
1815 (fn _ => EVERY |
|
1816 [cut_facts_tac constr_fresh_thms 1, |
|
1817 asm_simp_tac (HOL_basic_ss addsimps perm_fresh_fresh) 1, |
|
1818 rtac prem 1]); |
|
1819 |
|
1820 (** pi1 o ts = pi2 o us **) |
|
1821 val _ = warning "step 4: pi1 o ts = pi2 o us"; |
|
1822 val pi1_pi2_eqs = map (fn (t, u) => |
|
1823 Goal.prove context'' [] [] |
|
1824 (HOLogic.mk_Trueprop (HOLogic.mk_eq |
|
1825 (fold_rev (mk_perm []) pi1 t, fold_rev (mk_perm []) pi2 u))) |
|
1826 (fn _ => EVERY |
|
1827 [cut_facts_tac [pi1_pi2_eq] 1, |
|
1828 asm_full_simp_tac (HOL_ss addsimps |
|
1829 (calc_atm @ List.concat perm_simps' @ |
|
1830 fresh_prems' @ freshs2' @ abs_perm @ |
|
1831 alpha @ List.concat inject_thms)) 1])) |
|
1832 (map snd cargsl' ~~ map snd cargsr'); |
|
1833 |
|
1834 (** pi1^-1 o pi2 o us = ts **) |
|
1835 val _ = warning "step 5: pi1^-1 o pi2 o us = ts"; |
|
1836 val rpi1_pi2_eqs = map (fn ((t, u), eq) => |
|
1837 Goal.prove context'' [] [] |
|
1838 (HOLogic.mk_Trueprop (HOLogic.mk_eq |
|
1839 (fold_rev (mk_perm []) (rpi1 @ pi2) u, t))) |
|
1840 (fn _ => simp_tac (HOL_ss addsimps |
|
1841 ((eq RS sym) :: perm_swap)) 1)) |
|
1842 (map snd cargsl' ~~ map snd cargsr' ~~ pi1_pi2_eqs); |
|
1843 |
|
1844 val (rec_prems1, rec_prems2) = |
|
1845 chop (length rec_prems div 2) rec_prems; |
|
1846 |
|
1847 (** (ts, pi1^-1 o pi2 o vs) in rec_set **) |
|
1848 val _ = warning "step 6: (ts, pi1^-1 o pi2 o vs) in rec_set"; |
|
1849 val rec_prems' = map (fn th => |
|
1850 let |
|
1851 val _ $ (S $ x $ y) = prop_of th; |
|
1852 val Const (s, _) = head_of S; |
|
1853 val k = find_index (equal s) rec_set_names; |
|
1854 val pi = rpi1 @ pi2; |
|
1855 fun mk_pi z = fold_rev (mk_perm []) pi z; |
|
1856 fun eqvt_tac p = |
|
1857 let |
|
1858 val U as Type (_, [Type (_, [T, _])]) = fastype_of p; |
|
1859 val l = find_index (equal T) dt_atomTs; |
|
1860 val th = List.nth (List.nth (rec_equiv_thms', l), k); |
|
1861 val th' = Thm.instantiate ([], |
|
1862 [(cterm_of thy11 (Var (("pi", 0), U)), |
|
1863 cterm_of thy11 p)]) th; |
|
1864 in rtac th' 1 end; |
|
1865 val th' = Goal.prove context'' [] [] |
|
1866 (HOLogic.mk_Trueprop (S $ mk_pi x $ mk_pi y)) |
|
1867 (fn _ => EVERY |
|
1868 (map eqvt_tac pi @ |
|
1869 [simp_tac (HOL_ss addsimps (fresh_prems' @ freshs2' @ |
|
1870 perm_swap @ perm_fresh_fresh)) 1, |
|
1871 rtac th 1])) |
|
1872 in |
|
1873 Simplifier.simplify |
|
1874 (HOL_basic_ss addsimps rpi1_pi2_eqs) th' |
|
1875 end) rec_prems2; |
|
1876 |
|
1877 val ihs = filter (fn th => case prop_of th of |
|
1878 _ $ (Const ("All", _) $ _) => true | _ => false) prems'; |
|
1879 |
|
1880 (** pi1 o rs = pi2 o vs , rs = pi1^-1 o pi2 o vs **) |
|
1881 val _ = warning "step 7: pi1 o rs = pi2 o vs , rs = pi1^-1 o pi2 o vs"; |
|
1882 val rec_eqns = map (fn (th, ih) => |
|
1883 let |
|
1884 val th' = th RS (ih RS spec RS mp) RS sym; |
|
1885 val _ $ (_ $ lhs $ rhs) = prop_of th'; |
|
1886 fun strip_perm (_ $ _ $ t) = strip_perm t |
|
1887 | strip_perm t = t; |
|
1888 in |
|
1889 Goal.prove context'' [] [] |
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1890 (HOLogic.mk_Trueprop (HOLogic.mk_eq |
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1891 (fold_rev (mk_perm []) pi1 lhs, |
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1892 fold_rev (mk_perm []) pi2 (strip_perm rhs)))) |
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1893 (fn _ => simp_tac (HOL_basic_ss addsimps |
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1894 (th' :: perm_swap)) 1) |
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1895 end) (rec_prems' ~~ ihs); |
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1896 |
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1897 (** as # rs **) |
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1898 val _ = warning "step 8: as # rs"; |
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1899 val rec_freshs = List.concat |
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1900 (map (fn (rec_prem, ih) => |
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1901 let |
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1902 val _ $ (S $ x $ (y as Free (_, T))) = |
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1903 prop_of rec_prem; |
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1904 val k = find_index (equal S) rec_sets; |
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1905 val atoms = List.concat (List.mapPartial (fn (bs, z) => |
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1906 if z = x then NONE else SOME bs) cargsl') |
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1907 in |
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1908 map (fn a as Free (_, aT) => |
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1909 let val l = find_index (equal aT) dt_atomTs; |
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1910 in |
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1911 Goal.prove context'' [] [] |
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1912 (HOLogic.mk_Trueprop (fresh_const aT T $ a $ y)) |
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1913 (fn _ => EVERY |
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1914 (rtac (List.nth (List.nth (rec_fresh_thms, l), k)) 1 :: |
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1915 map (fn th => rtac th 1) |
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1916 (snd (List.nth (finite_thss, l))) @ |
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1917 [rtac rec_prem 1, rtac ih 1, |
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1918 REPEAT_DETERM (resolve_tac fresh_prems 1)])) |
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1919 end) atoms |
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1920 end) (rec_prems1 ~~ ihs)); |
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1921 |
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1922 (** as # fK as ts rs , bs # fK bs us vs **) |
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1923 val _ = warning "step 9: as # fK as ts rs , bs # fK bs us vs"; |
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1924 fun prove_fresh_result (a as Free (_, aT)) = |
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1925 Goal.prove context'' [] [] |
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1926 (HOLogic.mk_Trueprop (fresh_const aT rT $ a $ rhs')) |
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1927 (fn _ => EVERY |
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1928 [resolve_tac fcbs 1, |
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1929 REPEAT_DETERM (resolve_tac |
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1930 (fresh_prems @ rec_freshs) 1), |
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1931 REPEAT_DETERM (resolve_tac (maps snd rec_fin_supp_thms') 1 |
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1932 THEN resolve_tac rec_prems 1), |
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1933 resolve_tac P_ind_ths 1, |
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1934 REPEAT_DETERM (resolve_tac (P_ths @ rec_prems) 1)]); |
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1935 |
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1936 val fresh_results'' = map prove_fresh_result boundsl; |
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1937 |
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1938 fun prove_fresh_result'' ((a as Free (_, aT), b), th) = |
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1939 let val th' = Goal.prove context'' [] [] |
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1940 (HOLogic.mk_Trueprop (fresh_const aT rT $ |
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1941 fold_rev (mk_perm []) (rpi2 @ pi1) a $ |
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1942 fold_rev (mk_perm []) (rpi2 @ pi1) rhs')) |
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1943 (fn _ => simp_tac (HOL_ss addsimps fresh_bij) 1 THEN |
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1944 rtac th 1) |
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1945 in |
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1946 Goal.prove context'' [] [] |
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1947 (HOLogic.mk_Trueprop (fresh_const aT rT $ b $ lhs')) |
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1948 (fn _ => EVERY |
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1949 [cut_facts_tac [th'] 1, |
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1950 full_simp_tac (Simplifier.theory_context thy11 HOL_ss |
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1951 addsimps rec_eqns @ pi1_pi2_eqs @ perm_swap |
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1952 addsimprocs [NominalPermeq.perm_simproc_app]) 1, |
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1953 full_simp_tac (HOL_ss addsimps (calc_atm @ |
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1954 fresh_prems' @ freshs2' @ perm_fresh_fresh)) 1]) |
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1955 end; |
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1956 |
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1957 val fresh_results = fresh_results'' @ map prove_fresh_result'' |
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1958 (boundsl ~~ boundsr ~~ fresh_results''); |
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1959 |
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1960 (** cs # fK as ts rs , cs # fK bs us vs **) |
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1961 val _ = warning "step 10: cs # fK as ts rs , cs # fK bs us vs"; |
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1962 fun prove_fresh_result' recs t (a as Free (_, aT)) = |
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1963 Goal.prove context'' [] [] |
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1964 (HOLogic.mk_Trueprop (fresh_const aT rT $ a $ t)) |
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1965 (fn _ => EVERY |
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1966 [cut_facts_tac recs 1, |
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1967 REPEAT_DETERM (dresolve_tac |
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1968 (the (AList.lookup op = rec_fin_supp_thms' aT)) 1), |
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1969 NominalPermeq.fresh_guess_tac |
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1970 (HOL_ss addsimps (freshs2 @ |
|
1971 fs_atoms @ fresh_atm @ |
|
1972 List.concat (map snd finite_thss))) 1]); |
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1973 |
|
1974 val fresh_results' = |
|
1975 map (prove_fresh_result' rec_prems1 rhs') freshs1 @ |
|
1976 map (prove_fresh_result' rec_prems2 lhs') freshs1; |
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1977 |
|
1978 (** pi1 o (fK as ts rs) = pi2 o (fK bs us vs) **) |
|
1979 val _ = warning "step 11: pi1 o (fK as ts rs) = pi2 o (fK bs us vs)"; |
|
1980 val pi1_pi2_result = Goal.prove context'' [] [] |
|
1981 (HOLogic.mk_Trueprop (HOLogic.mk_eq |
|
1982 (fold_rev (mk_perm []) pi1 rhs', fold_rev (mk_perm []) pi2 lhs'))) |
|
1983 (fn _ => simp_tac (Simplifier.context context'' HOL_ss |
|
1984 addsimps pi1_pi2_eqs @ rec_eqns |
|
1985 addsimprocs [NominalPermeq.perm_simproc_app]) 1 THEN |
|
1986 TRY (simp_tac (HOL_ss addsimps |
|
1987 (fresh_prems' @ freshs2' @ calc_atm @ perm_fresh_fresh)) 1)); |
|
1988 |
|
1989 val _ = warning "final result"; |
|
1990 val final = Goal.prove context'' [] [] (term_of concl) |
|
1991 (fn _ => cut_facts_tac [pi1_pi2_result RS sym] 1 THEN |
|
1992 full_simp_tac (HOL_basic_ss addsimps perm_fresh_fresh @ |
|
1993 fresh_results @ fresh_results') 1); |
|
1994 val final' = ProofContext.export context'' context' [final]; |
|
1995 val _ = warning "finished!" |
|
1996 in |
|
1997 resolve_tac final' 1 |
|
1998 end) context 1])) idxss) (ndescr ~~ rec_elims)) |
|
1999 end)); |
|
2000 |
|
2001 val rec_total_thms = map (fn r => r RS theI') rec_unique_thms; |
|
2002 |
|
2003 (* define primrec combinators *) |
|
2004 |
|
2005 val big_reccomb_name = (space_implode "_" new_type_names) ^ "_rec"; |
|
2006 val reccomb_names = map (Sign.full_bname thy11) |
|
2007 (if length descr'' = 1 then [big_reccomb_name] else |
|
2008 (map ((curry (op ^) (big_reccomb_name ^ "_")) o string_of_int) |
|
2009 (1 upto (length descr'')))); |
|
2010 val reccombs = map (fn ((name, T), T') => list_comb |
|
2011 (Const (name, rec_fn_Ts @ [T] ---> T'), rec_fns)) |
|
2012 (reccomb_names ~~ recTs ~~ rec_result_Ts); |
|
2013 |
|
2014 val (reccomb_defs, thy12) = |
|
2015 thy11 |
|
2016 |> Sign.add_consts_i (map (fn ((name, T), T') => |
|
2017 (Binding.name (Long_Name.base_name name), rec_fn_Ts @ [T] ---> T', NoSyn)) |
|
2018 (reccomb_names ~~ recTs ~~ rec_result_Ts)) |
|
2019 |> (PureThy.add_defs false o map Thm.no_attributes) (map (fn ((((name, comb), set), T), T') => |
|
2020 (Binding.name (Long_Name.base_name name ^ "_def"), Logic.mk_equals (comb, absfree ("x", T, |
|
2021 Const ("The", (T' --> HOLogic.boolT) --> T') $ absfree ("y", T', |
|
2022 set $ Free ("x", T) $ Free ("y", T')))))) |
|
2023 (reccomb_names ~~ reccombs ~~ rec_sets ~~ recTs ~~ rec_result_Ts)); |
|
2024 |
|
2025 (* prove characteristic equations for primrec combinators *) |
|
2026 |
|
2027 val rec_thms = map (fn (prems, concl) => |
|
2028 let |
|
2029 val _ $ (_ $ (_ $ x) $ _) = concl; |
|
2030 val (_, cargs) = strip_comb x; |
|
2031 val ps = map (fn (x as Free (_, T), i) => |
|
2032 (Free ("x" ^ string_of_int i, T), x)) (cargs ~~ (1 upto length cargs)); |
|
2033 val concl' = subst_atomic_types (rec_result_Ts' ~~ rec_result_Ts) concl; |
|
2034 val prems' = List.concat finite_premss @ finite_ctxt_prems @ |
|
2035 rec_prems @ rec_prems' @ map (subst_atomic ps) prems; |
|
2036 fun solve rules prems = resolve_tac rules THEN_ALL_NEW |
|
2037 (resolve_tac prems THEN_ALL_NEW atac) |
|
2038 in |
|
2039 Goal.prove_global thy12 [] |
|
2040 (map (augment_sort thy12 fs_cp_sort) prems') |
|
2041 (augment_sort thy12 fs_cp_sort concl') |
|
2042 (fn {prems, ...} => EVERY |
|
2043 [rewrite_goals_tac reccomb_defs, |
|
2044 rtac the1_equality 1, |
|
2045 solve rec_unique_thms prems 1, |
|
2046 resolve_tac rec_intrs 1, |
|
2047 REPEAT (solve (prems @ rec_total_thms) prems 1)]) |
|
2048 end) (rec_eq_prems ~~ |
|
2049 DatatypeProp.make_primrecs new_type_names descr' sorts thy12); |
|
2050 |
|
2051 val dt_infos = map (make_dt_info pdescr sorts induct reccomb_names rec_thms) |
|
2052 ((0 upto length descr1 - 1) ~~ descr1 ~~ distinct_thms ~~ inject_thms); |
|
2053 |
|
2054 (* FIXME: theorems are stored in database for testing only *) |
|
2055 val (_, thy13) = thy12 |> |
|
2056 PureThy.add_thmss |
|
2057 [((Binding.name "rec_equiv", List.concat rec_equiv_thms), []), |
|
2058 ((Binding.name "rec_equiv'", List.concat rec_equiv_thms'), []), |
|
2059 ((Binding.name "rec_fin_supp", List.concat rec_fin_supp_thms), []), |
|
2060 ((Binding.name "rec_fresh", List.concat rec_fresh_thms), []), |
|
2061 ((Binding.name "rec_unique", map standard rec_unique_thms), []), |
|
2062 ((Binding.name "recs", rec_thms), [])] ||> |
|
2063 Sign.parent_path ||> |
|
2064 map_nominal_datatypes (fold Symtab.update dt_infos); |
|
2065 |
|
2066 in |
|
2067 thy13 |
|
2068 end; |
|
2069 |
|
2070 val add_nominal_datatype = gen_add_nominal_datatype Datatype.read_typ; |
|
2071 |
|
2072 |
|
2073 (* FIXME: The following stuff should be exported by Datatype *) |
|
2074 |
|
2075 local structure P = OuterParse and K = OuterKeyword in |
|
2076 |
|
2077 val datatype_decl = |
|
2078 Scan.option (P.$$$ "(" |-- P.name --| P.$$$ ")") -- P.type_args -- P.name -- P.opt_infix -- |
|
2079 (P.$$$ "=" |-- P.enum1 "|" (P.name -- Scan.repeat P.typ -- P.opt_mixfix)); |
|
2080 |
|
2081 fun mk_datatype args = |
|
2082 let |
|
2083 val names = map (fn ((((NONE, _), t), _), _) => t | ((((SOME t, _), _), _), _) => t) args; |
|
2084 val specs = map (fn ((((_, vs), t), mx), cons) => |
|
2085 (vs, t, mx, map (fn ((x, y), z) => (x, y, z)) cons)) args; |
|
2086 in add_nominal_datatype Datatype.default_config names specs end; |
|
2087 |
|
2088 val _ = |
|
2089 OuterSyntax.command "nominal_datatype" "define inductive datatypes" K.thy_decl |
|
2090 (P.and_list1 datatype_decl >> (Toplevel.theory o mk_datatype)); |
|
2091 |
|
2092 end; |
|
2093 |
|
2094 end |
|