1 (* Title: HOL/Tools/inductive_package.ML |
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2 Author: Lawrence C Paulson, Cambridge University Computer Laboratory |
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3 Author: Stefan Berghofer and Markus Wenzel, TU Muenchen |
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4 |
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5 (Co)Inductive Definition module for HOL. |
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6 |
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7 Features: |
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8 * least or greatest fixedpoints |
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9 * mutually recursive definitions |
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10 * definitions involving arbitrary monotone operators |
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11 * automatically proves introduction and elimination rules |
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12 |
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13 Introduction rules have the form |
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14 [| M Pj ti, ..., Q x, ... |] ==> Pk t |
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15 where M is some monotone operator (usually the identity) |
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16 Q x is any side condition on the free variables |
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17 ti, t are any terms |
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18 Pj, Pk are two of the predicates being defined in mutual recursion |
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19 *) |
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20 |
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21 signature BASIC_INDUCTIVE_PACKAGE = |
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22 sig |
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23 type inductive_result |
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24 val morph_result: morphism -> inductive_result -> inductive_result |
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25 type inductive_info |
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26 val the_inductive: Proof.context -> string -> inductive_info |
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27 val print_inductives: Proof.context -> unit |
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28 val mono_add: attribute |
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29 val mono_del: attribute |
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30 val get_monos: Proof.context -> thm list |
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31 val mk_cases: Proof.context -> term -> thm |
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32 val inductive_forall_name: string |
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33 val inductive_forall_def: thm |
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34 val rulify: thm -> thm |
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35 val inductive_cases: (Attrib.binding * string list) list -> local_theory -> |
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36 thm list list * local_theory |
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37 val inductive_cases_i: (Attrib.binding * term list) list -> local_theory -> |
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38 thm list list * local_theory |
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39 type inductive_flags |
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40 val add_inductive_i: |
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41 inductive_flags -> ((binding * typ) * mixfix) list -> |
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42 (string * typ) list -> (Attrib.binding * term) list -> thm list -> local_theory -> |
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43 inductive_result * local_theory |
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44 val add_inductive: bool -> bool -> |
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45 (binding * string option * mixfix) list -> |
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46 (binding * string option * mixfix) list -> |
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47 (Attrib.binding * string) list -> |
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48 (Facts.ref * Attrib.src list) list -> |
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49 bool -> local_theory -> inductive_result * local_theory |
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50 val add_inductive_global: string -> inductive_flags -> |
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51 ((binding * typ) * mixfix) list -> (string * typ) list -> (Attrib.binding * term) list -> |
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52 thm list -> theory -> inductive_result * theory |
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53 val arities_of: thm -> (string * int) list |
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54 val params_of: thm -> term list |
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55 val partition_rules: thm -> thm list -> (string * thm list) list |
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56 val partition_rules': thm -> (thm * 'a) list -> (string * (thm * 'a) list) list |
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57 val unpartition_rules: thm list -> (string * 'a list) list -> 'a list |
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58 val infer_intro_vars: thm -> int -> thm list -> term list list |
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59 val setup: theory -> theory |
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60 end; |
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61 |
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62 signature INDUCTIVE_PACKAGE = |
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63 sig |
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64 include BASIC_INDUCTIVE_PACKAGE |
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65 type add_ind_def |
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66 val declare_rules: string -> binding -> bool -> bool -> string list -> |
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67 thm list -> binding list -> Attrib.src list list -> (thm * string list) list -> |
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68 thm -> local_theory -> thm list * thm list * thm * local_theory |
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69 val add_ind_def: add_ind_def |
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70 val gen_add_inductive_i: add_ind_def -> inductive_flags -> |
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71 ((binding * typ) * mixfix) list -> (string * typ) list -> (Attrib.binding * term) list -> |
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72 thm list -> local_theory -> inductive_result * local_theory |
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73 val gen_add_inductive: add_ind_def -> bool -> bool -> |
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74 (binding * string option * mixfix) list -> |
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75 (binding * string option * mixfix) list -> |
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76 (Attrib.binding * string) list -> (Facts.ref * Attrib.src list) list -> |
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77 bool -> local_theory -> inductive_result * local_theory |
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78 val gen_ind_decl: add_ind_def -> bool -> |
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79 OuterParse.token list -> (bool -> local_theory -> local_theory) * OuterParse.token list |
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80 end; |
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81 |
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82 structure InductivePackage: INDUCTIVE_PACKAGE = |
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83 struct |
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84 |
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85 |
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86 (** theory context references **) |
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87 |
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88 val inductive_forall_name = "HOL.induct_forall"; |
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89 val inductive_forall_def = thm "induct_forall_def"; |
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90 val inductive_conj_name = "HOL.induct_conj"; |
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91 val inductive_conj_def = thm "induct_conj_def"; |
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92 val inductive_conj = thms "induct_conj"; |
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93 val inductive_atomize = thms "induct_atomize"; |
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94 val inductive_rulify = thms "induct_rulify"; |
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95 val inductive_rulify_fallback = thms "induct_rulify_fallback"; |
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96 |
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97 val notTrueE = TrueI RSN (2, notE); |
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98 val notFalseI = Seq.hd (atac 1 notI); |
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99 val simp_thms' = map (fn s => mk_meta_eq (the (find_first |
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100 (equal (OldGoals.read_prop @{theory HOL} s) o prop_of) simp_thms))) |
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101 ["(~True) = False", "(~False) = True", |
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102 "(True --> ?P) = ?P", "(False --> ?P) = True", |
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103 "(?P & True) = ?P", "(True & ?P) = ?P"]; |
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104 |
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105 |
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106 |
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107 (** context data **) |
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108 |
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109 type inductive_result = |
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110 {preds: term list, elims: thm list, raw_induct: thm, |
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111 induct: thm, intrs: thm list}; |
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112 |
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113 fun morph_result phi {preds, elims, raw_induct: thm, induct, intrs} = |
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114 let |
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115 val term = Morphism.term phi; |
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116 val thm = Morphism.thm phi; |
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117 val fact = Morphism.fact phi; |
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118 in |
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119 {preds = map term preds, elims = fact elims, raw_induct = thm raw_induct, |
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120 induct = thm induct, intrs = fact intrs} |
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121 end; |
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122 |
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123 type inductive_info = |
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124 {names: string list, coind: bool} * inductive_result; |
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125 |
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126 structure InductiveData = GenericDataFun |
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127 ( |
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128 type T = inductive_info Symtab.table * thm list; |
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129 val empty = (Symtab.empty, []); |
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130 val extend = I; |
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131 fun merge _ ((tab1, monos1), (tab2, monos2)) = |
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132 (Symtab.merge (K true) (tab1, tab2), Thm.merge_thms (monos1, monos2)); |
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133 ); |
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134 |
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135 val get_inductives = InductiveData.get o Context.Proof; |
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136 |
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137 fun print_inductives ctxt = |
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138 let |
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139 val (tab, monos) = get_inductives ctxt; |
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140 val space = Consts.space_of (ProofContext.consts_of ctxt); |
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141 in |
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142 [Pretty.strs ("(co)inductives:" :: map #1 (NameSpace.extern_table (space, tab))), |
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143 Pretty.big_list "monotonicity rules:" (map (ProofContext.pretty_thm ctxt) monos)] |
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144 |> Pretty.chunks |> Pretty.writeln |
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145 end; |
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146 |
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147 |
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148 (* get and put data *) |
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149 |
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150 fun the_inductive ctxt name = |
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151 (case Symtab.lookup (#1 (get_inductives ctxt)) name of |
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152 NONE => error ("Unknown (co)inductive predicate " ^ quote name) |
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153 | SOME info => info); |
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154 |
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155 fun put_inductives names info = InductiveData.map |
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156 (apfst (fold (fn name => Symtab.update (name, info)) names)); |
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157 |
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158 |
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159 |
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160 (** monotonicity rules **) |
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161 |
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162 val get_monos = #2 o get_inductives; |
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163 val map_monos = InductiveData.map o apsnd; |
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164 |
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165 fun mk_mono thm = |
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166 let |
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167 val concl = concl_of thm; |
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168 fun eq2mono thm' = [thm' RS (thm' RS eq_to_mono)] @ |
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169 (case concl of |
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170 (_ $ (_ $ (Const ("Not", _) $ _) $ _)) => [] |
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171 | _ => [thm' RS (thm' RS eq_to_mono2)]); |
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172 fun dest_less_concl thm = dest_less_concl (thm RS le_funD) |
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173 handle THM _ => thm RS le_boolD |
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174 in |
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175 case concl of |
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176 Const ("==", _) $ _ $ _ => eq2mono (thm RS meta_eq_to_obj_eq) |
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177 | _ $ (Const ("op =", _) $ _ $ _) => eq2mono thm |
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178 | _ $ (Const ("HOL.ord_class.less_eq", _) $ _ $ _) => |
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179 [dest_less_concl (Seq.hd (REPEAT (FIRSTGOAL |
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180 (resolve_tac [le_funI, le_boolI'])) thm))] |
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181 | _ => [thm] |
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182 end handle THM _ => error ("Bad monotonicity theorem:\n" ^ Display.string_of_thm thm); |
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183 |
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184 val mono_add = Thm.declaration_attribute (map_monos o fold Thm.add_thm o mk_mono); |
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185 val mono_del = Thm.declaration_attribute (map_monos o fold Thm.del_thm o mk_mono); |
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186 |
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187 |
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188 |
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189 (** misc utilities **) |
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190 |
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191 fun message quiet_mode s = if quiet_mode then () else writeln s; |
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192 fun clean_message quiet_mode s = if ! quick_and_dirty then () else message quiet_mode s; |
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193 |
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194 fun coind_prefix true = "co" |
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195 | coind_prefix false = ""; |
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196 |
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197 fun log (b:int) m n = if m >= n then 0 else 1 + log b (b * m) n; |
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198 |
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199 fun make_bool_args f g [] i = [] |
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200 | make_bool_args f g (x :: xs) i = |
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201 (if i mod 2 = 0 then f x else g x) :: make_bool_args f g xs (i div 2); |
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202 |
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203 fun make_bool_args' xs = |
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204 make_bool_args (K HOLogic.false_const) (K HOLogic.true_const) xs; |
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205 |
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206 fun find_arg T x [] = sys_error "find_arg" |
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207 | find_arg T x ((p as (_, (SOME _, _))) :: ps) = |
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208 apsnd (cons p) (find_arg T x ps) |
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209 | find_arg T x ((p as (U, (NONE, y))) :: ps) = |
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210 if (T: typ) = U then (y, (U, (SOME x, y)) :: ps) |
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211 else apsnd (cons p) (find_arg T x ps); |
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212 |
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213 fun make_args Ts xs = |
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214 map (fn (T, (NONE, ())) => Const (@{const_name undefined}, T) | (_, (SOME t, ())) => t) |
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215 (fold (fn (t, T) => snd o find_arg T t) xs (map (rpair (NONE, ())) Ts)); |
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216 |
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217 fun make_args' Ts xs Us = |
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218 fst (fold_map (fn T => find_arg T ()) Us (Ts ~~ map (pair NONE) xs)); |
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219 |
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220 fun dest_predicate cs params t = |
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221 let |
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222 val k = length params; |
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223 val (c, ts) = strip_comb t; |
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224 val (xs, ys) = chop k ts; |
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225 val i = find_index_eq c cs; |
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226 in |
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227 if xs = params andalso i >= 0 then |
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228 SOME (c, i, ys, chop (length ys) |
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229 (List.drop (binder_types (fastype_of c), k))) |
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230 else NONE |
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231 end; |
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232 |
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233 fun mk_names a 0 = [] |
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234 | mk_names a 1 = [a] |
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235 | mk_names a n = map (fn i => a ^ string_of_int i) (1 upto n); |
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236 |
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237 |
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238 |
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239 (** process rules **) |
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240 |
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241 local |
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242 |
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243 fun err_in_rule ctxt name t msg = |
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244 error (cat_lines ["Ill-formed introduction rule " ^ quote name, |
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245 Syntax.string_of_term ctxt t, msg]); |
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246 |
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247 fun err_in_prem ctxt name t p msg = |
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248 error (cat_lines ["Ill-formed premise", Syntax.string_of_term ctxt p, |
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249 "in introduction rule " ^ quote name, Syntax.string_of_term ctxt t, msg]); |
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250 |
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251 val bad_concl = "Conclusion of introduction rule must be an inductive predicate"; |
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252 |
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253 val bad_ind_occ = "Inductive predicate occurs in argument of inductive predicate"; |
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254 |
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255 val bad_app = "Inductive predicate must be applied to parameter(s) "; |
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256 |
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257 fun atomize_term thy = MetaSimplifier.rewrite_term thy inductive_atomize []; |
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258 |
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259 in |
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260 |
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261 fun check_rule ctxt cs params ((binding, att), rule) = |
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262 let |
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263 val err_name = Binding.str_of binding; |
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264 val params' = Term.variant_frees rule (Logic.strip_params rule); |
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265 val frees = rev (map Free params'); |
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266 val concl = subst_bounds (frees, Logic.strip_assums_concl rule); |
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267 val prems = map (curry subst_bounds frees) (Logic.strip_assums_hyp rule); |
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268 val rule' = Logic.list_implies (prems, concl); |
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269 val aprems = map (atomize_term (ProofContext.theory_of ctxt)) prems; |
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270 val arule = list_all_free (params', Logic.list_implies (aprems, concl)); |
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271 |
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272 fun check_ind err t = case dest_predicate cs params t of |
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273 NONE => err (bad_app ^ |
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274 commas (map (Syntax.string_of_term ctxt) params)) |
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275 | SOME (_, _, ys, _) => |
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276 if exists (fn c => exists (fn t => Logic.occs (c, t)) ys) cs |
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277 then err bad_ind_occ else (); |
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278 |
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279 fun check_prem' prem t = |
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280 if head_of t mem cs then |
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281 check_ind (err_in_prem ctxt err_name rule prem) t |
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282 else (case t of |
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283 Abs (_, _, t) => check_prem' prem t |
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284 | t $ u => (check_prem' prem t; check_prem' prem u) |
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285 | _ => ()); |
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286 |
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287 fun check_prem (prem, aprem) = |
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288 if can HOLogic.dest_Trueprop aprem then check_prem' prem prem |
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289 else err_in_prem ctxt err_name rule prem "Non-atomic premise"; |
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290 in |
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291 (case concl of |
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292 Const ("Trueprop", _) $ t => |
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293 if head_of t mem cs then |
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294 (check_ind (err_in_rule ctxt err_name rule') t; |
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295 List.app check_prem (prems ~~ aprems)) |
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296 else err_in_rule ctxt err_name rule' bad_concl |
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297 | _ => err_in_rule ctxt err_name rule' bad_concl); |
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298 ((binding, att), arule) |
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299 end; |
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300 |
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301 val rulify = |
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302 hol_simplify inductive_conj |
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303 #> hol_simplify inductive_rulify |
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304 #> hol_simplify inductive_rulify_fallback |
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305 #> Simplifier.norm_hhf; |
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306 |
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307 end; |
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308 |
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309 |
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310 |
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311 (** proofs for (co)inductive predicates **) |
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312 |
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313 (* prove monotonicity *) |
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314 |
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315 fun prove_mono quiet_mode skip_mono fork_mono predT fp_fun monos ctxt = |
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316 (message (quiet_mode orelse skip_mono andalso !quick_and_dirty orelse fork_mono) |
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317 " Proving monotonicity ..."; |
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318 (if skip_mono then SkipProof.prove else if fork_mono then Goal.prove_future else Goal.prove) ctxt |
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319 [] [] |
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320 (HOLogic.mk_Trueprop |
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321 (Const (@{const_name Orderings.mono}, (predT --> predT) --> HOLogic.boolT) $ fp_fun)) |
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322 (fn _ => EVERY [rtac @{thm monoI} 1, |
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323 REPEAT (resolve_tac [le_funI, le_boolI'] 1), |
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324 REPEAT (FIRST |
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325 [atac 1, |
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326 resolve_tac (List.concat (map mk_mono monos) @ get_monos ctxt) 1, |
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327 etac le_funE 1, dtac le_boolD 1])])); |
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328 |
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329 |
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330 (* prove introduction rules *) |
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331 |
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332 fun prove_intrs quiet_mode coind mono fp_def k params intr_ts rec_preds_defs ctxt = |
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333 let |
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334 val _ = clean_message quiet_mode " Proving the introduction rules ..."; |
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335 |
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336 val unfold = funpow k (fn th => th RS fun_cong) |
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337 (mono RS (fp_def RS |
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338 (if coind then def_gfp_unfold else def_lfp_unfold))); |
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339 |
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340 fun select_disj 1 1 = [] |
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341 | select_disj _ 1 = [rtac disjI1] |
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342 | select_disj n i = (rtac disjI2)::(select_disj (n - 1) (i - 1)); |
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343 |
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344 val rules = [refl, TrueI, notFalseI, exI, conjI]; |
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345 |
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346 val intrs = map_index (fn (i, intr) => rulify |
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347 (SkipProof.prove ctxt (map (fst o dest_Free) params) [] intr (fn _ => EVERY |
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348 [rewrite_goals_tac rec_preds_defs, |
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349 rtac (unfold RS iffD2) 1, |
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350 EVERY1 (select_disj (length intr_ts) (i + 1)), |
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351 (*Not ares_tac, since refl must be tried before any equality assumptions; |
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352 backtracking may occur if the premises have extra variables!*) |
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353 DEPTH_SOLVE_1 (resolve_tac rules 1 APPEND assume_tac 1)]))) intr_ts |
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354 |
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355 in (intrs, unfold) end; |
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356 |
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357 |
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358 (* prove elimination rules *) |
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359 |
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360 fun prove_elims quiet_mode cs params intr_ts intr_names unfold rec_preds_defs ctxt = |
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361 let |
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362 val _ = clean_message quiet_mode " Proving the elimination rules ..."; |
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363 |
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364 val ([pname], ctxt') = ctxt |> |
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365 Variable.add_fixes (map (fst o dest_Free) params) |> snd |> |
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366 Variable.variant_fixes ["P"]; |
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367 val P = HOLogic.mk_Trueprop (Free (pname, HOLogic.boolT)); |
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368 |
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369 fun dest_intr r = |
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370 (the (dest_predicate cs params (HOLogic.dest_Trueprop (Logic.strip_assums_concl r))), |
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371 Logic.strip_assums_hyp r, Logic.strip_params r); |
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372 |
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373 val intrs = map dest_intr intr_ts ~~ intr_names; |
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374 |
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375 val rules1 = [disjE, exE, FalseE]; |
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376 val rules2 = [conjE, FalseE, notTrueE]; |
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377 |
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378 fun prove_elim c = |
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379 let |
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380 val Ts = List.drop (binder_types (fastype_of c), length params); |
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381 val (anames, ctxt'') = Variable.variant_fixes (mk_names "a" (length Ts)) ctxt'; |
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382 val frees = map Free (anames ~~ Ts); |
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383 |
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384 fun mk_elim_prem ((_, _, us, _), ts, params') = |
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385 list_all (params', |
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386 Logic.list_implies (map (HOLogic.mk_Trueprop o HOLogic.mk_eq) |
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387 (frees ~~ us) @ ts, P)); |
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388 val c_intrs = (List.filter (equal c o #1 o #1 o #1) intrs); |
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389 val prems = HOLogic.mk_Trueprop (list_comb (c, params @ frees)) :: |
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390 map mk_elim_prem (map #1 c_intrs) |
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391 in |
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392 (SkipProof.prove ctxt'' [] prems P |
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393 (fn {prems, ...} => EVERY |
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394 [cut_facts_tac [hd prems] 1, |
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395 rewrite_goals_tac rec_preds_defs, |
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396 dtac (unfold RS iffD1) 1, |
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397 REPEAT (FIRSTGOAL (eresolve_tac rules1)), |
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398 REPEAT (FIRSTGOAL (eresolve_tac rules2)), |
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399 EVERY (map (fn prem => |
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400 DEPTH_SOLVE_1 (ares_tac [rewrite_rule rec_preds_defs prem, conjI] 1)) (tl prems))]) |
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401 |> rulify |
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402 |> singleton (ProofContext.export ctxt'' ctxt), |
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403 map #2 c_intrs) |
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404 end |
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405 |
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406 in map prove_elim cs end; |
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407 |
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408 |
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409 (* derivation of simplified elimination rules *) |
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410 |
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411 local |
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412 |
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413 (*delete needless equality assumptions*) |
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414 val refl_thin = Goal.prove_global @{theory HOL} [] [] @{prop "!!P. a = a ==> P ==> P"} |
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415 (fn _ => assume_tac 1); |
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416 val elim_rls = [asm_rl, FalseE, refl_thin, conjE, exE]; |
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417 val elim_tac = REPEAT o Tactic.eresolve_tac elim_rls; |
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418 |
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419 fun simp_case_tac ss i = |
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420 EVERY' [elim_tac, asm_full_simp_tac ss, elim_tac, REPEAT o bound_hyp_subst_tac] i; |
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421 |
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422 in |
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423 |
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424 fun mk_cases ctxt prop = |
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425 let |
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426 val thy = ProofContext.theory_of ctxt; |
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427 val ss = Simplifier.local_simpset_of ctxt; |
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428 |
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429 fun err msg = |
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430 error (Pretty.string_of (Pretty.block |
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431 [Pretty.str msg, Pretty.fbrk, Syntax.pretty_term ctxt prop])); |
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432 |
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433 val elims = Induct.find_casesP ctxt prop; |
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434 |
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435 val cprop = Thm.cterm_of thy prop; |
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436 val tac = ALLGOALS (simp_case_tac ss) THEN prune_params_tac; |
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437 fun mk_elim rl = |
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438 Thm.implies_intr cprop (Tactic.rule_by_tactic tac (Thm.assume cprop RS rl)) |
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439 |> singleton (Variable.export (Variable.auto_fixes prop ctxt) ctxt); |
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440 in |
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441 (case get_first (try mk_elim) elims of |
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442 SOME r => r |
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443 | NONE => err "Proposition not an inductive predicate:") |
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444 end; |
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445 |
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446 end; |
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447 |
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448 |
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449 (* inductive_cases *) |
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450 |
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451 fun gen_inductive_cases prep_att prep_prop args lthy = |
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452 let |
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453 val thy = ProofContext.theory_of lthy; |
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454 val facts = args |> map (fn ((a, atts), props) => |
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455 ((a, map (prep_att thy) atts), |
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456 map (Thm.no_attributes o single o mk_cases lthy o prep_prop lthy) props)); |
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457 in lthy |> LocalTheory.notes Thm.generatedK facts |>> map snd end; |
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458 |
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459 val inductive_cases = gen_inductive_cases Attrib.intern_src Syntax.read_prop; |
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460 val inductive_cases_i = gen_inductive_cases (K I) Syntax.check_prop; |
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461 |
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462 |
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463 val ind_cases_setup = |
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464 Method.setup @{binding ind_cases} |
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465 (Scan.lift (Scan.repeat1 Args.name_source -- |
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466 Scan.optional (Args.$$$ "for" |-- Scan.repeat1 Args.name) []) >> |
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467 (fn (raw_props, fixes) => fn ctxt => |
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468 let |
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469 val (_, ctxt') = Variable.add_fixes fixes ctxt; |
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470 val props = Syntax.read_props ctxt' raw_props; |
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471 val ctxt'' = fold Variable.declare_term props ctxt'; |
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472 val rules = ProofContext.export ctxt'' ctxt (map (mk_cases ctxt'') props) |
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473 in Method.erule 0 rules end)) |
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474 "dynamic case analysis on predicates"; |
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475 |
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476 |
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477 (* prove induction rule *) |
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478 |
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479 fun prove_indrule quiet_mode cs argTs bs xs rec_const params intr_ts mono |
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480 fp_def rec_preds_defs ctxt = |
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481 let |
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482 val _ = clean_message quiet_mode " Proving the induction rule ..."; |
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483 val thy = ProofContext.theory_of ctxt; |
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484 |
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485 (* predicates for induction rule *) |
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486 |
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487 val (pnames, ctxt') = ctxt |> |
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488 Variable.add_fixes (map (fst o dest_Free) params) |> snd |> |
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489 Variable.variant_fixes (mk_names "P" (length cs)); |
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490 val preds = map Free (pnames ~~ |
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491 map (fn c => List.drop (binder_types (fastype_of c), length params) ---> |
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492 HOLogic.boolT) cs); |
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493 |
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494 (* transform an introduction rule into a premise for induction rule *) |
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495 |
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496 fun mk_ind_prem r = |
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497 let |
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498 fun subst s = (case dest_predicate cs params s of |
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499 SOME (_, i, ys, (_, Ts)) => |
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500 let |
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501 val k = length Ts; |
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502 val bs = map Bound (k - 1 downto 0); |
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503 val P = list_comb (List.nth (preds, i), |
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504 map (incr_boundvars k) ys @ bs); |
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505 val Q = list_abs (mk_names "x" k ~~ Ts, |
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506 HOLogic.mk_binop inductive_conj_name |
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507 (list_comb (incr_boundvars k s, bs), P)) |
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508 in (Q, case Ts of [] => SOME (s, P) | _ => NONE) end |
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509 | NONE => (case s of |
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510 (t $ u) => (fst (subst t) $ fst (subst u), NONE) |
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511 | (Abs (a, T, t)) => (Abs (a, T, fst (subst t)), NONE) |
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512 | _ => (s, NONE))); |
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513 |
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514 fun mk_prem (s, prems) = (case subst s of |
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515 (_, SOME (t, u)) => t :: u :: prems |
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516 | (t, _) => t :: prems); |
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517 |
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518 val SOME (_, i, ys, _) = dest_predicate cs params |
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519 (HOLogic.dest_Trueprop (Logic.strip_assums_concl r)) |
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520 |
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521 in list_all_free (Logic.strip_params r, |
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522 Logic.list_implies (map HOLogic.mk_Trueprop (List.foldr mk_prem |
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523 [] (map HOLogic.dest_Trueprop (Logic.strip_assums_hyp r))), |
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524 HOLogic.mk_Trueprop (list_comb (List.nth (preds, i), ys)))) |
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525 end; |
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526 |
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527 val ind_prems = map mk_ind_prem intr_ts; |
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528 |
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529 |
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530 (* make conclusions for induction rules *) |
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531 |
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532 val Tss = map (binder_types o fastype_of) preds; |
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533 val (xnames, ctxt'') = |
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534 Variable.variant_fixes (mk_names "x" (length (flat Tss))) ctxt'; |
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535 val mutual_ind_concl = HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj |
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536 (map (fn (((xnames, Ts), c), P) => |
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537 let val frees = map Free (xnames ~~ Ts) |
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538 in HOLogic.mk_imp |
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539 (list_comb (c, params @ frees), list_comb (P, frees)) |
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540 end) (unflat Tss xnames ~~ Tss ~~ cs ~~ preds))); |
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541 |
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542 |
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543 (* make predicate for instantiation of abstract induction rule *) |
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544 |
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545 val ind_pred = fold_rev lambda (bs @ xs) (foldr1 HOLogic.mk_conj |
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546 (map_index (fn (i, P) => List.foldr HOLogic.mk_imp |
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547 (list_comb (P, make_args' argTs xs (binder_types (fastype_of P)))) |
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548 (make_bool_args HOLogic.mk_not I bs i)) preds)); |
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549 |
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550 val ind_concl = HOLogic.mk_Trueprop |
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551 (HOLogic.mk_binrel "HOL.ord_class.less_eq" (rec_const, ind_pred)); |
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552 |
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553 val raw_fp_induct = (mono RS (fp_def RS def_lfp_induct)); |
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554 |
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555 val induct = SkipProof.prove ctxt'' [] ind_prems ind_concl |
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556 (fn {prems, ...} => EVERY |
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557 [rewrite_goals_tac [inductive_conj_def], |
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558 DETERM (rtac raw_fp_induct 1), |
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559 REPEAT (resolve_tac [le_funI, le_boolI] 1), |
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560 rewrite_goals_tac (inf_fun_eq :: inf_bool_eq :: simp_thms'), |
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561 (*This disjE separates out the introduction rules*) |
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562 REPEAT (FIRSTGOAL (eresolve_tac [disjE, exE, FalseE])), |
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563 (*Now break down the individual cases. No disjE here in case |
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564 some premise involves disjunction.*) |
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565 REPEAT (FIRSTGOAL (etac conjE ORELSE' bound_hyp_subst_tac)), |
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566 REPEAT (FIRSTGOAL |
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567 (resolve_tac [conjI, impI] ORELSE' (etac notE THEN' atac))), |
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568 EVERY (map (fn prem => DEPTH_SOLVE_1 (ares_tac [rewrite_rule |
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569 (inductive_conj_def :: rec_preds_defs @ simp_thms') prem, |
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570 conjI, refl] 1)) prems)]); |
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571 |
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572 val lemma = SkipProof.prove ctxt'' [] [] |
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573 (Logic.mk_implies (ind_concl, mutual_ind_concl)) (fn _ => EVERY |
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574 [rewrite_goals_tac rec_preds_defs, |
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575 REPEAT (EVERY |
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576 [REPEAT (resolve_tac [conjI, impI] 1), |
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577 REPEAT (eresolve_tac [le_funE, le_boolE] 1), |
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578 atac 1, |
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579 rewrite_goals_tac simp_thms', |
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580 atac 1])]) |
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581 |
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582 in singleton (ProofContext.export ctxt'' ctxt) (induct RS lemma) end; |
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583 |
|
584 |
|
585 |
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586 (** specification of (co)inductive predicates **) |
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587 |
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588 fun mk_ind_def quiet_mode skip_mono fork_mono alt_name coind cs intr_ts monos params cnames_syn ctxt = |
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589 let |
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590 val fp_name = if coind then @{const_name Inductive.gfp} else @{const_name Inductive.lfp}; |
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591 |
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592 val argTs = fold (fn c => fn Ts => Ts @ |
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593 (List.drop (binder_types (fastype_of c), length params) \\ Ts)) cs []; |
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594 val k = log 2 1 (length cs); |
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595 val predT = replicate k HOLogic.boolT ---> argTs ---> HOLogic.boolT; |
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596 val p :: xs = map Free (Variable.variant_frees ctxt intr_ts |
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597 (("p", predT) :: (mk_names "x" (length argTs) ~~ argTs))); |
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598 val bs = map Free (Variable.variant_frees ctxt (p :: xs @ intr_ts) |
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599 (map (rpair HOLogic.boolT) (mk_names "b" k))); |
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600 |
|
601 fun subst t = (case dest_predicate cs params t of |
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602 SOME (_, i, ts, (Ts, Us)) => |
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603 let |
|
604 val l = length Us; |
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605 val zs = map Bound (l - 1 downto 0) |
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606 in |
|
607 list_abs (map (pair "z") Us, list_comb (p, |
|
608 make_bool_args' bs i @ make_args argTs |
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609 ((map (incr_boundvars l) ts ~~ Ts) @ (zs ~~ Us)))) |
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610 end |
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611 | NONE => (case t of |
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612 t1 $ t2 => subst t1 $ subst t2 |
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613 | Abs (x, T, u) => Abs (x, T, subst u) |
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614 | _ => t)); |
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615 |
|
616 (* transform an introduction rule into a conjunction *) |
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617 (* [| p_i t; ... |] ==> p_j u *) |
|
618 (* is transformed into *) |
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619 (* b_j & x_j = u & p b_j t & ... *) |
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620 |
|
621 fun transform_rule r = |
|
622 let |
|
623 val SOME (_, i, ts, (Ts, _)) = dest_predicate cs params |
|
624 (HOLogic.dest_Trueprop (Logic.strip_assums_concl r)); |
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625 val ps = make_bool_args HOLogic.mk_not I bs i @ |
|
626 map HOLogic.mk_eq (make_args' argTs xs Ts ~~ ts) @ |
|
627 map (subst o HOLogic.dest_Trueprop) |
|
628 (Logic.strip_assums_hyp r) |
|
629 in List.foldr (fn ((x, T), P) => HOLogic.exists_const T $ (Abs (x, T, P))) |
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630 (if null ps then HOLogic.true_const else foldr1 HOLogic.mk_conj ps) |
|
631 (Logic.strip_params r) |
|
632 end |
|
633 |
|
634 (* make a disjunction of all introduction rules *) |
|
635 |
|
636 val fp_fun = fold_rev lambda (p :: bs @ xs) |
|
637 (if null intr_ts then HOLogic.false_const |
|
638 else foldr1 HOLogic.mk_disj (map transform_rule intr_ts)); |
|
639 |
|
640 (* add definiton of recursive predicates to theory *) |
|
641 |
|
642 val rec_name = |
|
643 if Binding.is_empty alt_name then |
|
644 Binding.name (space_implode "_" (map (Binding.name_of o fst) cnames_syn)) |
|
645 else alt_name; |
|
646 |
|
647 val ((rec_const, (_, fp_def)), ctxt') = ctxt |> |
|
648 LocalTheory.define Thm.internalK |
|
649 ((rec_name, case cnames_syn of [(_, syn)] => syn | _ => NoSyn), |
|
650 (Attrib.empty_binding, fold_rev lambda params |
|
651 (Const (fp_name, (predT --> predT) --> predT) $ fp_fun))); |
|
652 val fp_def' = Simplifier.rewrite (HOL_basic_ss addsimps [fp_def]) |
|
653 (cterm_of (ProofContext.theory_of ctxt') (list_comb (rec_const, params))); |
|
654 val specs = if length cs < 2 then [] else |
|
655 map_index (fn (i, (name_mx, c)) => |
|
656 let |
|
657 val Ts = List.drop (binder_types (fastype_of c), length params); |
|
658 val xs = map Free (Variable.variant_frees ctxt intr_ts |
|
659 (mk_names "x" (length Ts) ~~ Ts)) |
|
660 in |
|
661 (name_mx, (Attrib.empty_binding, fold_rev lambda (params @ xs) |
|
662 (list_comb (rec_const, params @ make_bool_args' bs i @ |
|
663 make_args argTs (xs ~~ Ts))))) |
|
664 end) (cnames_syn ~~ cs); |
|
665 val (consts_defs, ctxt'') = fold_map (LocalTheory.define Thm.internalK) specs ctxt'; |
|
666 val preds = (case cs of [_] => [rec_const] | _ => map #1 consts_defs); |
|
667 |
|
668 val mono = prove_mono quiet_mode skip_mono fork_mono predT fp_fun monos ctxt''; |
|
669 val ((_, [mono']), ctxt''') = |
|
670 LocalTheory.note Thm.internalK (Attrib.empty_binding, [mono]) ctxt''; |
|
671 |
|
672 in (ctxt''', rec_name, mono', fp_def', map (#2 o #2) consts_defs, |
|
673 list_comb (rec_const, params), preds, argTs, bs, xs) |
|
674 end; |
|
675 |
|
676 fun declare_rules kind rec_binding coind no_ind cnames intrs intr_bindings intr_atts |
|
677 elims raw_induct ctxt = |
|
678 let |
|
679 val rec_name = Binding.name_of rec_binding; |
|
680 val rec_qualified = Binding.qualify false rec_name; |
|
681 val intr_names = map Binding.name_of intr_bindings; |
|
682 val ind_case_names = RuleCases.case_names intr_names; |
|
683 val induct = |
|
684 if coind then |
|
685 (raw_induct, [RuleCases.case_names [rec_name], |
|
686 RuleCases.case_conclusion (rec_name, intr_names), |
|
687 RuleCases.consumes 1, Induct.coinduct_pred (hd cnames)]) |
|
688 else if no_ind orelse length cnames > 1 then |
|
689 (raw_induct, [ind_case_names, RuleCases.consumes 0]) |
|
690 else (raw_induct RSN (2, rev_mp), [ind_case_names, RuleCases.consumes 1]); |
|
691 |
|
692 val (intrs', ctxt1) = |
|
693 ctxt |> |
|
694 LocalTheory.notes kind |
|
695 (map rec_qualified intr_bindings ~~ intr_atts ~~ map (fn th => [([th], |
|
696 [Attrib.internal (K (ContextRules.intro_query NONE)), |
|
697 Attrib.internal (K Nitpick_Ind_Intro_Thms.add)])]) intrs) |>> |
|
698 map (hd o snd); |
|
699 val (((_, elims'), (_, [induct'])), ctxt2) = |
|
700 ctxt1 |> |
|
701 LocalTheory.note kind ((rec_qualified (Binding.name "intros"), []), intrs') ||>> |
|
702 fold_map (fn (name, (elim, cases)) => |
|
703 LocalTheory.note kind ((Binding.qualified_name (Long_Name.qualify (Long_Name.base_name name) "cases"), |
|
704 [Attrib.internal (K (RuleCases.case_names cases)), |
|
705 Attrib.internal (K (RuleCases.consumes 1)), |
|
706 Attrib.internal (K (Induct.cases_pred name)), |
|
707 Attrib.internal (K (ContextRules.elim_query NONE))]), [elim]) #> |
|
708 apfst (hd o snd)) (if null elims then [] else cnames ~~ elims) ||>> |
|
709 LocalTheory.note kind |
|
710 ((rec_qualified (Binding.name (coind_prefix coind ^ "induct")), |
|
711 map (Attrib.internal o K) (#2 induct)), [rulify (#1 induct)]); |
|
712 |
|
713 val ctxt3 = if no_ind orelse coind then ctxt2 else |
|
714 let val inducts = cnames ~~ ProjectRule.projects ctxt2 (1 upto length cnames) induct' |
|
715 in |
|
716 ctxt2 |> |
|
717 LocalTheory.notes kind [((rec_qualified (Binding.name "inducts"), []), |
|
718 inducts |> map (fn (name, th) => ([th], |
|
719 [Attrib.internal (K ind_case_names), |
|
720 Attrib.internal (K (RuleCases.consumes 1)), |
|
721 Attrib.internal (K (Induct.induct_pred name))])))] |> snd |
|
722 end |
|
723 in (intrs', elims', induct', ctxt3) end; |
|
724 |
|
725 type inductive_flags = |
|
726 {quiet_mode: bool, verbose: bool, kind: string, alt_name: binding, |
|
727 coind: bool, no_elim: bool, no_ind: bool, skip_mono: bool, fork_mono: bool} |
|
728 |
|
729 type add_ind_def = |
|
730 inductive_flags -> |
|
731 term list -> (Attrib.binding * term) list -> thm list -> |
|
732 term list -> (binding * mixfix) list -> |
|
733 local_theory -> inductive_result * local_theory |
|
734 |
|
735 fun add_ind_def {quiet_mode, verbose, kind, alt_name, coind, no_elim, no_ind, skip_mono, fork_mono} |
|
736 cs intros monos params cnames_syn ctxt = |
|
737 let |
|
738 val _ = null cnames_syn andalso error "No inductive predicates given"; |
|
739 val names = map (Binding.name_of o fst) cnames_syn; |
|
740 val _ = message (quiet_mode andalso not verbose) |
|
741 ("Proofs for " ^ coind_prefix coind ^ "inductive predicate(s) " ^ commas_quote names); |
|
742 |
|
743 val cnames = map (LocalTheory.full_name ctxt o #1) cnames_syn; (* FIXME *) |
|
744 val ((intr_names, intr_atts), intr_ts) = |
|
745 apfst split_list (split_list (map (check_rule ctxt cs params) intros)); |
|
746 |
|
747 val (ctxt1, rec_name, mono, fp_def, rec_preds_defs, rec_const, preds, |
|
748 argTs, bs, xs) = mk_ind_def quiet_mode skip_mono fork_mono alt_name coind cs intr_ts |
|
749 monos params cnames_syn ctxt; |
|
750 |
|
751 val (intrs, unfold) = prove_intrs quiet_mode coind mono fp_def (length bs + length xs) |
|
752 params intr_ts rec_preds_defs ctxt1; |
|
753 val elims = if no_elim then [] else |
|
754 prove_elims quiet_mode cs params intr_ts (map Binding.name_of intr_names) |
|
755 unfold rec_preds_defs ctxt1; |
|
756 val raw_induct = zero_var_indexes |
|
757 (if no_ind then Drule.asm_rl else |
|
758 if coind then |
|
759 singleton (ProofContext.export |
|
760 (snd (Variable.add_fixes (map (fst o dest_Free) params) ctxt1)) ctxt1) |
|
761 (rotate_prems ~1 (ObjectLogic.rulify |
|
762 (fold_rule rec_preds_defs |
|
763 (rewrite_rule [le_fun_def, le_bool_def, sup_fun_eq, sup_bool_eq] |
|
764 (mono RS (fp_def RS def_coinduct)))))) |
|
765 else |
|
766 prove_indrule quiet_mode cs argTs bs xs rec_const params intr_ts mono fp_def |
|
767 rec_preds_defs ctxt1); |
|
768 |
|
769 val (intrs', elims', induct, ctxt2) = declare_rules kind rec_name coind no_ind |
|
770 cnames intrs intr_names intr_atts elims raw_induct ctxt1; |
|
771 |
|
772 val result = |
|
773 {preds = preds, |
|
774 intrs = intrs', |
|
775 elims = elims', |
|
776 raw_induct = rulify raw_induct, |
|
777 induct = induct}; |
|
778 |
|
779 val ctxt3 = ctxt2 |
|
780 |> LocalTheory.declaration (fn phi => |
|
781 let val result' = morph_result phi result; |
|
782 in put_inductives cnames (*global names!?*) ({names = cnames, coind = coind}, result') end); |
|
783 in (result, ctxt3) end; |
|
784 |
|
785 |
|
786 (* external interfaces *) |
|
787 |
|
788 fun gen_add_inductive_i mk_def |
|
789 (flags as {quiet_mode, verbose, kind, alt_name, coind, no_elim, no_ind, skip_mono, fork_mono}) |
|
790 cnames_syn pnames spec monos lthy = |
|
791 let |
|
792 val thy = ProofContext.theory_of lthy; |
|
793 val _ = Theory.requires thy "Inductive" (coind_prefix coind ^ "inductive definitions"); |
|
794 |
|
795 |
|
796 (* abbrevs *) |
|
797 |
|
798 val (_, ctxt1) = Variable.add_fixes (map (Binding.name_of o fst o fst) cnames_syn) lthy; |
|
799 |
|
800 fun get_abbrev ((name, atts), t) = |
|
801 if can (Logic.strip_assums_concl #> Logic.dest_equals) t then |
|
802 let |
|
803 val _ = Binding.is_empty name andalso null atts orelse |
|
804 error "Abbreviations may not have names or attributes"; |
|
805 val ((x, T), rhs) = LocalDefs.abs_def (snd (LocalDefs.cert_def ctxt1 t)); |
|
806 val var = |
|
807 (case find_first (fn ((c, _), _) => Binding.name_of c = x) cnames_syn of |
|
808 NONE => error ("Undeclared head of abbreviation " ^ quote x) |
|
809 | SOME ((b, T'), mx) => |
|
810 if T <> T' then error ("Bad type specification for abbreviation " ^ quote x) |
|
811 else (b, mx)); |
|
812 in SOME (var, rhs) end |
|
813 else NONE; |
|
814 |
|
815 val abbrevs = map_filter get_abbrev spec; |
|
816 val bs = map (Binding.name_of o fst o fst) abbrevs; |
|
817 |
|
818 |
|
819 (* predicates *) |
|
820 |
|
821 val pre_intros = filter_out (is_some o get_abbrev) spec; |
|
822 val cnames_syn' = filter_out (member (op =) bs o Binding.name_of o fst o fst) cnames_syn; |
|
823 val cs = map (Free o apfst Binding.name_of o fst) cnames_syn'; |
|
824 val ps = map Free pnames; |
|
825 |
|
826 val (_, ctxt2) = lthy |> Variable.add_fixes (map (Binding.name_of o fst o fst) cnames_syn'); |
|
827 val _ = map (fn abbr => LocalDefs.fixed_abbrev abbr ctxt2) abbrevs; |
|
828 val ctxt3 = ctxt2 |> fold (snd oo LocalDefs.fixed_abbrev) abbrevs; |
|
829 val expand = Assumption.export_term ctxt3 lthy #> ProofContext.cert_term lthy; |
|
830 |
|
831 fun close_rule r = list_all_free (rev (fold_aterms |
|
832 (fn t as Free (v as (s, _)) => |
|
833 if Variable.is_fixed ctxt1 s orelse |
|
834 member (op =) ps t then I else insert (op =) v |
|
835 | _ => I) r []), r); |
|
836 |
|
837 val intros = map (apsnd (Syntax.check_term lthy #> close_rule #> expand)) pre_intros; |
|
838 val preds = map (fn ((c, _), mx) => (c, mx)) cnames_syn'; |
|
839 in |
|
840 lthy |
|
841 |> mk_def flags cs intros monos ps preds |
|
842 ||> fold (snd oo LocalTheory.abbrev Syntax.mode_default) abbrevs |
|
843 end; |
|
844 |
|
845 fun gen_add_inductive mk_def verbose coind cnames_syn pnames_syn intro_srcs raw_monos int lthy = |
|
846 let |
|
847 val ((vars, intrs), _) = lthy |
|
848 |> ProofContext.set_mode ProofContext.mode_abbrev |
|
849 |> Specification.read_spec (cnames_syn @ pnames_syn) intro_srcs; |
|
850 val (cs, ps) = chop (length cnames_syn) vars; |
|
851 val monos = Attrib.eval_thms lthy raw_monos; |
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852 val flags = {quiet_mode = false, verbose = verbose, kind = Thm.generatedK, |
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853 alt_name = Binding.empty, coind = coind, no_elim = false, no_ind = false, |
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854 skip_mono = false, fork_mono = not int}; |
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855 in |
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856 lthy |
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857 |> LocalTheory.set_group (serial_string ()) |
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858 |> gen_add_inductive_i mk_def flags cs (map (apfst Binding.name_of o fst) ps) intrs monos |
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859 end; |
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860 |
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861 val add_inductive_i = gen_add_inductive_i add_ind_def; |
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862 val add_inductive = gen_add_inductive add_ind_def; |
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863 |
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864 fun add_inductive_global group flags cnames_syn pnames pre_intros monos thy = |
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865 let |
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866 val name = Sign.full_name thy (fst (fst (hd cnames_syn))); |
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867 val ctxt' = thy |
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868 |> TheoryTarget.init NONE |
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869 |> LocalTheory.set_group group |
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870 |> add_inductive_i flags cnames_syn pnames pre_intros monos |> snd |
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871 |> LocalTheory.exit; |
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872 val info = #2 (the_inductive ctxt' name); |
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873 in (info, ProofContext.theory_of ctxt') end; |
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874 |
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875 |
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876 (* read off arities of inductive predicates from raw induction rule *) |
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877 fun arities_of induct = |
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878 map (fn (_ $ t $ u) => |
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879 (fst (dest_Const (head_of t)), length (snd (strip_comb u)))) |
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880 (HOLogic.dest_conj (HOLogic.dest_Trueprop (concl_of induct))); |
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881 |
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882 (* read off parameters of inductive predicate from raw induction rule *) |
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883 fun params_of induct = |
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884 let |
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885 val (_ $ t $ u :: _) = |
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886 HOLogic.dest_conj (HOLogic.dest_Trueprop (concl_of induct)); |
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887 val (_, ts) = strip_comb t; |
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888 val (_, us) = strip_comb u |
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889 in |
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890 List.take (ts, length ts - length us) |
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891 end; |
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892 |
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893 val pname_of_intr = |
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894 concl_of #> HOLogic.dest_Trueprop #> head_of #> dest_Const #> fst; |
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895 |
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896 (* partition introduction rules according to predicate name *) |
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897 fun gen_partition_rules f induct intros = |
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898 fold_rev (fn r => AList.map_entry op = (pname_of_intr (f r)) (cons r)) intros |
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899 (map (rpair [] o fst) (arities_of induct)); |
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900 |
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901 val partition_rules = gen_partition_rules I; |
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902 fun partition_rules' induct = gen_partition_rules fst induct; |
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903 |
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904 fun unpartition_rules intros xs = |
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905 fold_map (fn r => AList.map_entry_yield op = (pname_of_intr r) |
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906 (fn x :: xs => (x, xs)) #>> the) intros xs |> fst; |
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907 |
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908 (* infer order of variables in intro rules from order of quantifiers in elim rule *) |
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909 fun infer_intro_vars elim arity intros = |
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910 let |
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911 val thy = theory_of_thm elim; |
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912 val _ :: cases = prems_of elim; |
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913 val used = map (fst o fst) (Term.add_vars (prop_of elim) []); |
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914 fun mtch (t, u) = |
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915 let |
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916 val params = Logic.strip_params t; |
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917 val vars = map (Var o apfst (rpair 0)) |
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918 (Name.variant_list used (map fst params) ~~ map snd params); |
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919 val ts = map (curry subst_bounds (rev vars)) |
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920 (List.drop (Logic.strip_assums_hyp t, arity)); |
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921 val us = Logic.strip_imp_prems u; |
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922 val tab = fold (Pattern.first_order_match thy) (ts ~~ us) |
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923 (Vartab.empty, Vartab.empty); |
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924 in |
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925 map (Envir.subst_vars tab) vars |
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926 end |
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927 in |
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928 map (mtch o apsnd prop_of) (cases ~~ intros) |
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929 end; |
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930 |
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931 |
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932 |
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933 (** package setup **) |
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934 |
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935 (* setup theory *) |
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936 |
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937 val setup = |
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938 ind_cases_setup #> |
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939 Attrib.setup @{binding mono} (Attrib.add_del mono_add mono_del) |
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940 "declaration of monotonicity rule"; |
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941 |
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942 |
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943 (* outer syntax *) |
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944 |
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945 local structure P = OuterParse and K = OuterKeyword in |
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946 |
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947 val _ = OuterKeyword.keyword "monos"; |
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948 |
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949 fun gen_ind_decl mk_def coind = |
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950 P.fixes -- P.for_fixes -- |
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951 Scan.optional SpecParse.where_alt_specs [] -- |
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952 Scan.optional (P.$$$ "monos" |-- P.!!! SpecParse.xthms1) [] |
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953 >> (fn (((preds, params), specs), monos) => |
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954 (snd oo gen_add_inductive mk_def true coind preds params specs monos)); |
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955 |
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956 val ind_decl = gen_ind_decl add_ind_def; |
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957 |
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958 val _ = OuterSyntax.local_theory' "inductive" "define inductive predicates" K.thy_decl (ind_decl false); |
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959 val _ = OuterSyntax.local_theory' "coinductive" "define coinductive predicates" K.thy_decl (ind_decl true); |
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960 |
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961 val _ = |
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962 OuterSyntax.local_theory "inductive_cases" |
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963 "create simplified instances of elimination rules (improper)" K.thy_script |
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964 (P.and_list1 SpecParse.specs >> (snd oo inductive_cases)); |
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965 |
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966 end; |
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967 |
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968 end; |
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