1 | (* *********************************************************************) |
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2 | (* *) |
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3 | (* The Compcert verified compiler *) |
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4 | (* *) |
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5 | (* Xavier Leroy, INRIA Paris-Rocquencourt *) |
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6 | (* *) |
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7 | (* Copyright Institut National de Recherche en Informatique et en *) |
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8 | (* Automatique. All rights reserved. This file is distributed *) |
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9 | (* under the terms of the GNU General Public License as published by *) |
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10 | (* the Free Software Foundation, either version 2 of the License, or *) |
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11 | (* (at your option) any later version. This file is also distributed *) |
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12 | (* under the terms of the INRIA Non-Commercial License Agreement. *) |
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13 | (* *) |
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14 | (* *********************************************************************) |
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15 | |
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16 | include "basics/types.ma". |
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17 | include "basics/logic.ma". |
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18 | include "basics/list.ma". |
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19 | |
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20 | (* * Error reporting and the error monad. *) |
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21 | (* |
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22 | (** * Representation of error messages. *) |
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23 | |
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24 | (** Compile-time errors produce an error message, represented in Coq |
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25 | as a list of either substrings or positive numbers encoding |
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26 | a source-level identifier (see module AST). *) |
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27 | |
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28 | Inductive errcode: Type := |
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29 | | MSG: string -> errcode |
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30 | | CTX: positive -> errcode. |
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31 | |
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32 | Definition errmsg: Type := list errcode. |
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33 | |
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34 | Definition msg (s: string) : errmsg := MSG s :: nil. |
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35 | *) |
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36 | (* * * The error monad *) |
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37 | |
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38 | (* * Compilation functions that can fail have return type [res A]. |
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39 | The return value is either [OK res] to indicate success, |
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40 | or [Error msg] to indicate failure. *) |
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41 | |
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42 | inductive res (A: Type[0]) : Type[0] ≝ |
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43 | | OK: A → res A |
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44 | | Error: (* FIXME errmsg →*) res A. |
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45 | |
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46 | (*Implicit Arguments Error [A].*) |
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47 | |
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48 | (* * To automate the propagation of errors, we use a monadic style |
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49 | with the following [bind] operation. *) |
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50 | |
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51 | definition bind ≝ λA,B:Type[0]. λf: res A. λg: A → res B. |
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52 | match f with |
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53 | [ OK x ⇒ g x |
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54 | | Error (*msg*) ⇒ Error ? (*msg*) |
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55 | ]. |
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56 | |
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57 | definition bind2 ≝ λA,B,C: Type[0]. λf: res (A × B). λg: A → B → res C. |
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58 | match f with |
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59 | [ OK v ⇒ match v with [ pair x y ⇒ g x y ] |
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60 | | Error (*msg*) => Error ? (*msg*) |
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61 | ]. |
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62 | |
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63 | definition bind3 ≝ λA,B,C,D: Type[0]. λf: res (A × B × C). λg: A → B → C → res D. |
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64 | match f with |
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65 | [ OK v ⇒ match v with [ pair xy z ⇒ match xy with [ pair x y ⇒ g x y z ] ] |
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66 | | Error (*msg*) => Error ? (*msg*) |
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67 | ]. |
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68 | |
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69 | (* Not sure what level to use *) |
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70 | notation > "vbox('do' ident v ← e; break e')" with precedence 40 for @{'bind ${e} (λ${ident v}.${e'})}. |
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71 | notation > "vbox('do' ident v : ty ← e; break e')" with precedence 40 for @{'bind ${e} (λ${ident v} : ${ty}.${e'})}. |
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72 | notation < "vbox('do' \nbsp ident v ← e; break e')" with precedence 40 for @{'bind ${e} (λ${ident v}.${e'})}. |
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73 | notation < "vbox('do' \nbsp ident v : ty ← e; break e')" with precedence 40 for @{'bind ${e} (λ${ident v} : ${ty}.${e'})}. |
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74 | interpretation "error monad bind" 'bind e f = (bind ?? e f). |
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75 | notation > "vbox('do' 〈ident v1, ident v2〉 ← e; break e')" with precedence 40 for @{'bind2 ${e} (λ${ident v1}.λ${ident v2}.${e'})}. |
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76 | notation > "vbox('do' 〈ident v1 : ty1, ident v2 : ty2〉 ← e; break e')" with precedence 40 for @{'bind2 ${e} (λ${ident v1} : ${ty1}.λ${ident v2} : ${ty2}.${e'})}. |
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77 | notation < "vbox('do' \nbsp 〈ident v1, ident v2〉 ← e; break e')" with precedence 40 for @{'bind2 ${e} (λ${ident v1}.λ${ident v2}.${e'})}. |
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78 | notation < "vbox('do' \nbsp 〈ident v1 : ty1, ident v2 : ty2〉 ← e; break e')" with precedence 40 for @{'bind2 ${e} (λ${ident v1} : ${ty1}.λ${ident v2} : ${ty2}.${e'})}. |
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79 | interpretation "error monad Prod bind" 'bind2 e f = (bind2 ??? e f). |
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80 | notation > "vbox('do' 〈ident v1, ident v2, ident v3〉 ← e; break e')" with precedence 40 for @{'bind3 ${e} (λ${ident v1}.λ${ident v2}.λ${ident v3}.${e'})}. |
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81 | notation > "vbox('do' 〈ident v1 : ty1, ident v2 : ty2, ident v3 : ty3〉 ← e; break e')" with precedence 40 for @{'bind3 ${e} (λ${ident v1} : ${ty1}.λ${ident v2} : ${ty2}.λ${ident v3} : ${ty3}.${e'})}. |
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82 | notation < "vbox('do' \nbsp 〈ident v1, ident v2, ident v3〉 ← e; break e')" with precedence 40 for @{'bind3 ${e} (λ${ident v1}.λ${ident v2}.λ${ident v3}.${e'})}. |
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83 | notation < "vbox('do' \nbsp 〈ident v1 : ty1, ident v2 : ty2, ident v3 : ty3〉 ← e; break e')" with precedence 40 for @{'bind3 ${e} (λ${ident v1} : ${ty1}.λ${ident v2} : ${ty2}.λ${ident v3} : ${ty3}.${e'})}. |
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84 | interpretation "error monad triple bind" 'bind3 e f = (bind3 ???? e f). |
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85 | (*interpretation "error monad ret" 'ret e = (ret ? e). |
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86 | notation "'ret' e" non associative with precedence 45 for @{'ret ${e}}.*) |
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87 | |
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88 | (* |
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89 | (** The [do] notation, inspired by Haskell's, keeps the code readable. *) |
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90 | |
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91 | Notation "'do' X <- A ; B" := (bind A (fun X => B)) |
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92 | (at level 200, X ident, A at level 100, B at level 200) |
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93 | : error_monad_scope. |
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94 | |
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95 | Notation "'do' ( X , Y ) <- A ; B" := (bind2 A (fun X Y => B)) |
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96 | (at level 200, X ident, Y ident, A at level 100, B at level 200) |
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97 | : error_monad_scope. |
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98 | *) |
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99 | lemma bind_inversion: |
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100 | ∀A,B: Type[0]. ∀f: res A. ∀g: A → res B. ∀y: B. |
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101 | bind ?? f g = OK ? y → |
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102 | ∃x. f = OK ? x ∧ g x = OK ? y. |
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103 | #A #B #f #g #y cases f; |
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104 | [ #a #e %{a} whd in e:(??%?); /2/; |
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105 | | #H whd in H:(??%?); destruct (H); |
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106 | ] qed. |
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107 | |
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108 | lemma bind2_inversion: |
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109 | ∀A,B,C: Type[0]. ∀f: res (A×B). ∀g: A → B → res C. ∀z: C. |
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110 | bind2 ??? f g = OK ? z → |
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111 | ∃x. ∃y. f = OK ? 〈x, y〉 ∧ g x y = OK ? z. |
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112 | #A #B #C #f #g #z cases f; |
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113 | [ #ab cases ab; #a #b #e %{a} %{b} whd in e:(??%?); /2/; |
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114 | | #H whd in H:(??%?); destruct |
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115 | ] qed. |
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116 | |
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117 | (* |
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118 | Open Local Scope error_monad_scope. |
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119 | |
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120 | (** This is the familiar monadic map iterator. *) |
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121 | *) |
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122 | |
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123 | let rec mmap (A, B: Type[0]) (f: A → res B) (l: list A) on l : res (list B) ≝ |
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124 | match l with |
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125 | [ nil ⇒ OK ? [] |
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126 | | cons hd tl ⇒ do hd' ← f hd; do tl' ← mmap ?? f tl; OK ? (hd'::tl') |
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127 | ]. |
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128 | |
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129 | (* |
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130 | lemma mmap_inversion: |
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131 | ∀A, B: Type[0]. ∀f: A -> res B. ∀l: list A. ∀l': list B. |
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132 | mmap A B f l = OK ? l' → |
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133 | list_forall2 (fun x y => f x = OK y) l l'. |
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134 | Proof. |
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135 | induction l; simpl; intros. |
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136 | inversion_clear H. constructor. |
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137 | destruct (bind_inversion _ _ H) as [hd' [P Q]]. |
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138 | destruct (bind_inversion _ _ Q) as [tl' [R S]]. |
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139 | inversion_clear S. |
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140 | constructor. auto. auto. |
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141 | Qed. |
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142 | |
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143 | (** * Reasoning over monadic computations *) |
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144 | |
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145 | (** The [monadInv H] tactic below simplifies hypotheses of the form |
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146 | << |
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147 | H: (do x <- a; b) = OK res |
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148 | >> |
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149 | By definition of the bind operation, both computations [a] and |
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150 | [b] must succeed for their composition to succeed. The tactic |
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151 | therefore generates the following hypotheses: |
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152 | |
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153 | x: ... |
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154 | H1: a = OK x |
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155 | H2: b x = OK res |
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156 | *) |
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157 | |
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158 | Ltac monadInv1 H := |
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159 | match type of H with |
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160 | | (OK _ = OK _) => |
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161 | inversion H; clear H; try subst |
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162 | | (Error _ = OK _) => |
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163 | discriminate |
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164 | | (bind ?F ?G = OK ?X) => |
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165 | let x := fresh "x" in ( |
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166 | let EQ1 := fresh "EQ" in ( |
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167 | let EQ2 := fresh "EQ" in ( |
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168 | destruct (bind_inversion F G H) as [x [EQ1 EQ2]]; |
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169 | clear H; |
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170 | try (monadInv1 EQ2)))) |
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171 | | (bind2 ?F ?G = OK ?X) => |
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172 | let x1 := fresh "x" in ( |
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173 | let x2 := fresh "x" in ( |
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174 | let EQ1 := fresh "EQ" in ( |
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175 | let EQ2 := fresh "EQ" in ( |
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176 | destruct (bind2_inversion F G H) as [x1 [x2 [EQ1 EQ2]]]; |
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177 | clear H; |
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178 | try (monadInv1 EQ2))))) |
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179 | | (mmap ?F ?L = OK ?M) => |
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180 | generalize (mmap_inversion F L H); intro |
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181 | end. |
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182 | |
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183 | Ltac monadInv H := |
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184 | match type of H with |
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185 | | (OK _ = OK _) => monadInv1 H |
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186 | | (Error _ = OK _) => monadInv1 H |
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187 | | (bind ?F ?G = OK ?X) => monadInv1 H |
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188 | | (bind2 ?F ?G = OK ?X) => monadInv1 H |
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189 | | (?F _ _ _ _ _ _ _ _ = OK _) => |
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190 | ((progress simpl in H) || unfold F in H); monadInv1 H |
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191 | | (?F _ _ _ _ _ _ _ = OK _) => |
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192 | ((progress simpl in H) || unfold F in H); monadInv1 H |
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193 | | (?F _ _ _ _ _ _ = OK _) => |
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194 | ((progress simpl in H) || unfold F in H); monadInv1 H |
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195 | | (?F _ _ _ _ _ = OK _) => |
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196 | ((progress simpl in H) || unfold F in H); monadInv1 H |
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197 | | (?F _ _ _ _ = OK _) => |
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198 | ((progress simpl in H) || unfold F in H); monadInv1 H |
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199 | | (?F _ _ _ = OK _) => |
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200 | ((progress simpl in H) || unfold F in H); monadInv1 H |
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201 | | (?F _ _ = OK _) => |
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202 | ((progress simpl in H) || unfold F in H); monadInv1 H |
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203 | | (?F _ = OK _) => |
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204 | ((progress simpl in H) || unfold F in H); monadInv1 H |
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205 | end. |
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206 | *) |
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207 | |
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208 | |
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209 | definition opt_to_res ≝ λA.λv:option A. match v with [ None ⇒ Error A | Some v ⇒ OK A v ]. |
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210 | lemma opt_OK: ∀A,P,e. |
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211 | (∀v. e = Some ? v → P v) → |
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212 | match opt_to_res A e with [ Error ⇒ True | OK v ⇒ P v ]. |
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213 | #A #P #e elim e; /2/; |
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214 | qed. |
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