1 | include "basics/types.ma". |
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2 | include "basics/logic.ma". |
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3 | include "basics/lists/list.ma". |
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4 | include "common/PreIdentifiers.ma". |
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5 | include "basics/russell.ma". |
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6 | include "utilities/monad.ma". |
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7 | include "utilities/option.ma". |
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8 | |
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9 | (* * Error reporting and the error monad. *) |
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10 | |
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11 | (* * * Representation of error messages. *) |
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12 | |
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13 | (* * Compile-time errors produce an error message, represented in Coq |
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14 | as a list of either substrings or positive numbers encoding |
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15 | a source-level identifier (see module AST). *) |
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16 | |
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17 | inductive errcode: Type[0] := |
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18 | | MSG: String → errcode |
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19 | | CTX: ∀tag:String. identifier tag → errcode. |
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20 | |
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21 | definition errmsg: Type[0] ≝ list errcode. |
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22 | |
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23 | definition msg : String → errmsg ≝ λs. [MSG s]. |
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24 | |
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25 | (* * * The error monad *) |
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26 | |
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27 | (* * Compilation functions that can fail have return type [res A]. |
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28 | The return value is either [OK res] to indicate success, |
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29 | or [Error msg] to indicate failure. *) |
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30 | |
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31 | (* Paolo: not using except for backward compatbility |
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32 | (would break Error ? msg) *) |
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33 | |
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34 | inductive res (A: Type[0]) : Type[0] ≝ |
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35 | | OK: A → res A |
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36 | | Error: errmsg → res A. |
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37 | |
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38 | (*Implicit Arguments Error [A].*) |
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39 | |
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40 | definition Res ≝ MakeMonadProps |
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41 | (* the monad *) |
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42 | res |
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43 | (* return *) |
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44 | (λX.OK X) |
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45 | (* bind *) |
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46 | (λX,Y,m,f. match m with [ OK x ⇒ f x | Error msg ⇒ Error ? msg]) |
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47 | ????. |
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48 | // |
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49 | [(* bind_ret *) |
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50 | #X*normalize // |
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51 | |(* bind_bind *) |
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52 | #X#Y#Z*normalize // |
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53 | | normalize #X #Y * normalize /2/ |
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54 | ] |
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55 | qed. |
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56 | |
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57 | include "hints_declaration.ma". |
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58 | alias symbol "hint_decl" (instance 1) = "hint_decl_Type1". |
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59 | unification hint 0 ≔ X; |
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60 | N ≟ max_def Res |
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61 | (*---------------*) ⊢ |
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62 | res X ≡ monad N X |
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63 | . |
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64 | |
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65 | (*(* Dependent pair version. *) |
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66 | notation > "vbox('do' « ident v , ident p » ← e; break e')" with precedence 40 |
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67 | for @{ bind ?? ${e} (λ${fresh x}.match ${fresh x} with [ mk_Sig ${ident v} ${ident p} ⇒ ${e'} ]) }. |
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68 | |
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69 | lemma Prod_extensionality: |
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70 | ∀a, b: Type[0]. |
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71 | ∀p: a × b. |
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72 | p = 〈fst … p, snd … p〉. |
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73 | #a #b #p // |
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74 | qed. |
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75 | |
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76 | definition sigbind2 : ∀A,B,C:Type[0]. ∀P:A×B → Prop. res (Σx:A×B.P x) → (∀a,b. P 〈a,b〉 → res C) → res C ≝ |
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77 | λA,B,C,P,e,f. |
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78 | match e with |
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79 | [ OK v ⇒ match v with [ mk_Sig v' p ⇒ f (fst … v') (snd … v') ? ] |
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80 | | Error msg ⇒ Error ? msg |
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81 | ]. |
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82 | <(Prod_extensionality A B v') |
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83 | assumption |
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84 | qed. |
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85 | |
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86 | notation > "vbox('do' «ident v1, ident v2, ident H» ← e; break e')" with precedence 40 for @{'sigbind2 ${e} (λ${ident v1}.λ${ident v2}.λ${ident H}.${e'})}. |
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87 | interpretation "error monad sig Prod bind" 'sigbind2 e f = (sigbind2 ???? e f).*) |
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88 | |
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89 | lemma bind_inversion: |
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90 | ∀A,B: Type[0]. ∀f: res A. ∀g: A → res B. ∀y: B. |
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91 | (! x ← f ; g x = return y) → |
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92 | ∃x. (f = return x) ∧ (g x = return y). |
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93 | #A #B #f #g #y cases f normalize |
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94 | [ #a #e %{a} /2 by conj/ |
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95 | | #m #H destruct (H) |
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96 | ] qed. |
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97 | |
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98 | lemma bind2_inversion: |
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99 | ∀A,B,C: Type[0]. ∀f: res (A×B). ∀g: A → B → res C. ∀z: C. |
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100 | m_bind2 ???? f g = return z → |
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101 | ∃x. ∃y. f = return 〈x, y〉 ∧ g x y = return z. |
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102 | #A #B #C #f #g #z cases f normalize |
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103 | [ * #a #b normalize #e %{a} %{b} /2 by conj/ |
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104 | | #m #H destruct (H) |
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105 | ] qed. |
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106 | |
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107 | (* |
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108 | Open Local Scope error_monad_scope.*) |
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109 | |
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110 | (* |
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111 | lemma mmap_inversion: |
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112 | ∀A, B: Type[0]. ∀f: A -> res B. ∀l: list A. ∀l': list B. |
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113 | mmap A B f l = OK ? l' → |
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114 | list_forall2 (fun x y => f x = OK y) l l'. |
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115 | Proof. |
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116 | induction l; simpl; intros. |
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117 | inversion_clear H. constructor. |
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118 | destruct (bind_inversion _ _ H) as [hd' [P Q]]. |
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119 | destruct (bind_inversion _ _ Q) as [tl' [R S]]. |
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120 | inversion_clear S. |
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121 | constructor. auto. auto. |
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122 | Qed. |
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123 | *) |
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124 | |
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125 | (* A monadic fold_lefti *) |
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126 | let rec mfold_left_i_aux (A: Type[0]) (B: Type[0]) |
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127 | (f: nat → A → B → res A) (x: res A) (i: nat) (l: list B) on l ≝ |
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128 | match l with |
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129 | [ nil ⇒ x |
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130 | | cons hd tl ⇒ |
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131 | ! x ← x ; |
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132 | mfold_left_i_aux A B f (f i x hd) (S i) tl |
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133 | ]. |
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134 | |
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135 | definition mfold_left_i ≝ λA,B,f,x. mfold_left_i_aux A B f x O. |
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136 | |
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137 | |
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138 | (* A monadic fold_left2 *) |
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139 | |
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140 | axiom WrongLength: String. |
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141 | |
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142 | let rec mfold_left2 |
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143 | (A: Type[0]) (B: Type[0]) (C: Type[0]) (f: A → B → C → res A) (accu: res A) |
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144 | (left: list B) (right: list C) on left: res A ≝ |
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145 | match left with |
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146 | [ nil ⇒ |
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147 | match right with |
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148 | [ nil ⇒ accu |
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149 | | cons hd tl ⇒ Error ? (msg WrongLength) |
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150 | ] |
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151 | | cons hd tl ⇒ |
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152 | match right with |
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153 | [ nil ⇒ Error ? (msg WrongLength) |
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154 | | cons hd' tl' ⇒ |
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155 | ! accu ← accu; |
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156 | mfold_left2 … f (f accu hd hd') tl tl' |
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157 | ] |
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158 | ]. |
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159 | |
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160 | (* |
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161 | (** * Reasoning over monadic computations *) |
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162 | |
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163 | (** The [monadInv H] tactic below simplifies hypotheses of the form |
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164 | << |
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165 | H: (do x <- a; b) = OK res |
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166 | >> |
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167 | By definition of the bind operation, both computations [a] and |
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168 | [b] must succeed for their composition to succeed. The tactic |
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169 | therefore generates the following hypotheses: |
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170 | |
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171 | x: ... |
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172 | H1: a = OK x |
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173 | H2: b x = OK res |
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174 | *) |
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175 | |
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176 | Ltac monadInv1 H := |
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177 | match type of H with |
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178 | | (OK _ = OK _) => |
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179 | inversion H; clear H; try subst |
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180 | | (Error _ = OK _) => |
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181 | discriminate |
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182 | | (bind ?F ?G = OK ?X) => |
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183 | let x := fresh "x" in ( |
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184 | let EQ1 := fresh "EQ" in ( |
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185 | let EQ2 := fresh "EQ" in ( |
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186 | destruct (bind_inversion F G H) as [x [EQ1 EQ2]]; |
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187 | clear H; |
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188 | try (monadInv1 EQ2)))) |
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189 | | (bind2 ?F ?G = OK ?X) => |
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190 | let x1 := fresh "x" in ( |
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191 | let x2 := fresh "x" in ( |
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192 | let EQ1 := fresh "EQ" in ( |
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193 | let EQ2 := fresh "EQ" in ( |
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194 | destruct (bind2_inversion F G H) as [x1 [x2 [EQ1 EQ2]]]; |
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195 | clear H; |
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196 | try (monadInv1 EQ2))))) |
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197 | | (mmap ?F ?L = OK ?M) => |
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198 | generalize (mmap_inversion F L H); intro |
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199 | end. |
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200 | |
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201 | Ltac monadInv H := |
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202 | match type of H with |
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203 | | (OK _ = OK _) => monadInv1 H |
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204 | | (Error _ = OK _) => monadInv1 H |
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205 | | (bind ?F ?G = OK ?X) => monadInv1 H |
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206 | | (bind2 ?F ?G = OK ?X) => monadInv1 H |
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207 | | (?F _ _ _ _ _ _ _ _ = OK _) => |
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208 | ((progress simpl in H) || unfold F in H); monadInv1 H |
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209 | | (?F _ _ _ _ _ _ _ = OK _) => |
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210 | ((progress simpl in H) || unfold F in H); monadInv1 H |
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211 | | (?F _ _ _ _ _ _ = OK _) => |
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212 | ((progress simpl in H) || unfold F in H); monadInv1 H |
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213 | | (?F _ _ _ _ _ = OK _) => |
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214 | ((progress simpl in H) || unfold F in H); monadInv1 H |
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215 | | (?F _ _ _ _ = OK _) => |
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216 | ((progress simpl in H) || unfold F in H); monadInv1 H |
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217 | | (?F _ _ _ = OK _) => |
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218 | ((progress simpl in H) || unfold F in H); monadInv1 H |
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219 | | (?F _ _ = OK _) => |
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220 | ((progress simpl in H) || unfold F in H); monadInv1 H |
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221 | | (?F _ = OK _) => |
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222 | ((progress simpl in H) || unfold F in H); monadInv1 H |
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223 | end. |
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224 | *) |
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225 | |
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226 | |
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227 | definition opt_to_res ≝ λA.λmsg.λv:option A. match v with [ None ⇒ Error A msg | Some v ⇒ OK A v ]. |
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228 | lemma opt_OK: ∀A,m,P,e. |
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229 | (∀v. e = Some ? v → P v) → |
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230 | match opt_to_res A m e with [ Error _ ⇒ True | OK v ⇒ P v ]. |
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231 | #A #m #P #e elim e; /2/; |
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232 | qed. |
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233 | |
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234 | lemma opt_eq_from_res : ∀T,m,e,v. |
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235 | opt_to_res T m e = return v → |
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236 | e = Some T v. |
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237 | #T #m * [ #v #E normalize in E; destruct | #e' #v #E normalize in E; destruct % ] |
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238 | qed. |
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239 | |
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240 | coercion opt_eq_from_res : |
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241 | ∀T,m,e,v. ∀E:opt_to_res T m e = return v. e = Some T v ≝ opt_eq_from_res |
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242 | on _E:eq (res ?) ?? to eq (option ?) ??. |
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243 | |
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244 | (* A variation of bind and its notation that provides an equality proof for |
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245 | later use. *) |
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246 | |
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247 | definition bind_eq ≝ λA,B:Type[0]. λf: res A. λg: ∀a:A. f = OK ? a → res B. |
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248 | match f return λx. f = x → ? with |
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249 | [ OK x ⇒ g x |
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250 | | Error msg ⇒ λ_. Error ? msg |
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251 | ] (refl ? f). |
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252 | |
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253 | notation > "vbox('do' ident v 'as' ident E ← e; break e')" with precedence 48 for @{ bind_eq ?? ${e} (λ${ident v}.λ${ident E}.${e'})}. |
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254 | |
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255 | lemma bind_as_inversion: |
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256 | ∀A,B: Type[0]. ∀f: res A. ∀g: ∀a:A. f = OK A a → res B. ∀y: B. |
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257 | (do x as E ← f; g x E = return y) → |
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258 | ∃x. ∃E:f = return x. g x E = return y. |
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259 | #A #B #f cases f normalize |
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260 | [ #a #g #y #e %{a} /2/ |
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261 | | #m #g #y #H destruct (H) |
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262 | ] qed. |
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263 | |
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264 | definition bind2_eq ≝ λA,B,C:Type[0]. λf: res (A×B). λg: ∀a:A.∀b:B. f = OK ? 〈a,b〉 → res C. |
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265 | match f return λx. f = x → ? with |
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266 | [ OK x ⇒ match x return λx. f = OK ? x → ? with [ mk_Prod a b ⇒ g a b ] |
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267 | | Error msg ⇒ λ_. Error ? msg |
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268 | ] (refl ? f). |
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269 | |
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270 | notation > "vbox('do' 〈ident v1, ident v2〉 'as' ident E ← e; break e')" with precedence 48 for @{ bind2_eq ??? ${e} (λ${ident v1}.λ${ident v2}.λ${ident E}.${e'})}. |
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271 | |
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272 | lemma bind2_eq_inversion: |
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273 | ∀A,B,C: Type[0]. ∀f: res (A×B). ∀g: ∀a:A.∀b:B. f = OK ? 〈a,b〉 → res C. ∀z. |
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274 | bind2_eq ??? f g = return z → |
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275 | ∃x. ∃y. ∃Eq. g x y Eq = return z. |
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276 | #A #B #C #f cases f |
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277 | [ * #a #b #g normalize #z #Heq %{a} %{b} %{(refl ? (OK ? 〈a,b〉))} @Heq |
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278 | | #errmsg #g #z normalize #Habsurd destruct (Habsurd) ] |
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279 | qed. |
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280 | |
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281 | definition res_to_opt : ∀A:Type[0]. res A → option A ≝ |
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282 | λA.λv. match v with [ Error _ ⇒ None ? | OK v ⇒ Some … v]. |
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283 | |
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284 | (* aliases for backward compatibility *) |
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285 | definition bind ≝ m_bind Res. |
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286 | definition bind2 ≝ m_bind2 Res. |
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287 | definition bind3 ≝ m_bind3 Res. |
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288 | definition mmap ≝ m_list_map Res. |
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289 | definition mmap_sigma ≝ m_list_map_sigma Res. |
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