[487] | 1 | include "basics/types.ma". |
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| 2 | include "basics/logic.ma". |
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[1599] | 3 | include "basics/lists/list.ma". |
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[797] | 4 | include "common/PreIdentifiers.ma". |
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[1601] | 5 | include "basics/russell.ma". |
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[1640] | 6 | include "utilities/monad.ma". |
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[1882] | 7 | include "utilities/option.ma". |
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[2645] | 8 | include "common/ErrorMessages.ma". |
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[3] | 9 | |
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| 10 | (* * Error reporting and the error monad. *) |
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| 11 | |
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[797] | 12 | (* * * Representation of error messages. *) |
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| 13 | |
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| 14 | (* * Compile-time errors produce an error message, represented in Coq |
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[3] | 15 | as a list of either substrings or positive numbers encoding |
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| 16 | a source-level identifier (see module AST). *) |
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| 17 | |
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[797] | 18 | inductive errcode: Type[0] := |
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[2645] | 19 | | MSG: ErrorMessage → errcode |
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| 20 | | CTX: ∀tag:identifierTag. identifier tag → errcode. |
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[3] | 21 | |
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[797] | 22 | definition errmsg: Type[0] ≝ list errcode. |
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[3] | 23 | |
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[2645] | 24 | definition msg : ErrorMessage → errmsg ≝ λs. [MSG s]. |
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[797] | 25 | |
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[3] | 26 | (* * * The error monad *) |
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| 27 | |
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| 28 | (* * Compilation functions that can fail have return type [res A]. |
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| 29 | The return value is either [OK res] to indicate success, |
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| 30 | or [Error msg] to indicate failure. *) |
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| 31 | |
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[1882] | 32 | (* Paolo: not using except for backward compatbility |
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| 33 | (would break Error ? msg) *) |
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| 34 | |
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[487] | 35 | inductive res (A: Type[0]) : Type[0] ≝ |
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[3] | 36 | | OK: A → res A |
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[797] | 37 | | Error: errmsg → res A. |
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[3] | 38 | |
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| 39 | (*Implicit Arguments Error [A].*) |
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| 40 | |
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[1647] | 41 | definition Res ≝ MakeMonadProps |
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[1640] | 42 | (* the monad *) |
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| 43 | res |
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| 44 | (* return *) |
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| 45 | (λX.OK X) |
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| 46 | (* bind *) |
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| 47 | (λX,Y,m,f. match m with [ OK x ⇒ f x | Error msg ⇒ Error ? msg]) |
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[1882] | 48 | ????. |
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[1640] | 49 | // |
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[1647] | 50 | [(* bind_ret *) |
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| 51 | #X*normalize // |
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| 52 | |(* bind_bind *) |
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[1640] | 53 | #X#Y#Z*normalize // |
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[1882] | 54 | | normalize #X #Y * normalize /2/ |
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[1640] | 55 | ] |
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| 56 | qed. |
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[3] | 57 | |
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[1640] | 58 | include "hints_declaration.ma". |
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[1949] | 59 | alias symbol "hint_decl" (instance 1) = "hint_decl_Type1". |
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[1640] | 60 | unification hint 0 ≔ X; |
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[1882] | 61 | N ≟ max_def Res |
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[1640] | 62 | (*---------------*) ⊢ |
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[1882] | 63 | res X ≡ monad N X |
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[1640] | 64 | . |
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[3] | 65 | |
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[1640] | 66 | (*(* Dependent pair version. *) |
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[1316] | 67 | notation > "vbox('do' « ident v , ident p » ← e; break e')" with precedence 40 |
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[1608] | 68 | for @{ bind ?? ${e} (λ${fresh x}.match ${fresh x} with [ mk_Sig ${ident v} ${ident p} ⇒ ${e'} ]) }. |
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[1316] | 69 | |
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[1598] | 70 | lemma Prod_extensionality: |
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| 71 | ∀a, b: Type[0]. |
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| 72 | ∀p: a × b. |
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| 73 | p = 〈fst … p, snd … p〉. |
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| 74 | #a #b #p // |
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| 75 | qed. |
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| 76 | |
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[1351] | 77 | 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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| 78 | λA,B,C,P,e,f. |
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| 79 | match e with |
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[1601] | 80 | [ OK v ⇒ match v with [ mk_Sig v' p ⇒ f (fst … v') (snd … v') ? ] |
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[1351] | 81 | | Error msg ⇒ Error ? msg |
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| 82 | ]. |
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[1598] | 83 | <(Prod_extensionality A B v') |
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| 84 | assumption |
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| 85 | qed. |
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[1351] | 86 | |
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| 87 | 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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[1640] | 88 | interpretation "error monad sig Prod bind" 'sigbind2 e f = (sigbind2 ???? e f).*) |
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[1351] | 89 | |
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[1647] | 90 | lemma bind_inversion: |
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[487] | 91 | ∀A,B: Type[0]. ∀f: res A. ∀g: A → res B. ∀y: B. |
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[2453] | 92 | (! x ← f ; g x = return y) → |
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[1640] | 93 | ∃x. (f = return x) ∧ (g x = return y). |
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| 94 | #A #B #f #g #y cases f normalize |
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| 95 | [ #a #e %{a} /2 by conj/ |
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| 96 | | #m #H destruct (H) |
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[487] | 97 | ] qed. |
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[3] | 98 | |
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[487] | 99 | lemma bind2_inversion: |
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| 100 | ∀A,B,C: Type[0]. ∀f: res (A×B). ∀g: A → B → res C. ∀z: C. |
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[1640] | 101 | m_bind2 ???? f g = return z → |
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| 102 | ∃x. ∃y. f = return 〈x, y〉 ∧ g x y = return z. |
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| 103 | #A #B #C #f #g #z cases f normalize |
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| 104 | [ * #a #b normalize #e %{a} %{b} /2 by conj/ |
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| 105 | | #m #H destruct (H) |
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[487] | 106 | ] qed. |
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[3] | 107 | |
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[250] | 108 | (* |
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[1640] | 109 | Open Local Scope error_monad_scope.*) |
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[3] | 110 | |
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[636] | 111 | (* |
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| 112 | lemma mmap_inversion: |
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| 113 | ∀A, B: Type[0]. ∀f: A -> res B. ∀l: list A. ∀l': list B. |
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| 114 | mmap A B f l = OK ? l' → |
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[3] | 115 | list_forall2 (fun x y => f x = OK y) l l'. |
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| 116 | Proof. |
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| 117 | induction l; simpl; intros. |
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| 118 | inversion_clear H. constructor. |
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| 119 | destruct (bind_inversion _ _ H) as [hd' [P Q]]. |
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| 120 | destruct (bind_inversion _ _ Q) as [tl' [R S]]. |
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| 121 | inversion_clear S. |
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| 122 | constructor. auto. auto. |
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| 123 | Qed. |
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[1125] | 124 | *) |
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[3] | 125 | |
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[1355] | 126 | (* A monadic fold_lefti *) |
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| 127 | let rec mfold_left_i_aux (A: Type[0]) (B: Type[0]) |
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| 128 | (f: nat → A → B → res A) (x: res A) (i: nat) (l: list B) on l ≝ |
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| 129 | match l with |
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| 130 | [ nil ⇒ x |
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| 131 | | cons hd tl ⇒ |
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[1640] | 132 | ! x ← x ; |
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[1355] | 133 | mfold_left_i_aux A B f (f i x hd) (S i) tl |
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| 134 | ]. |
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| 135 | |
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| 136 | definition mfold_left_i ≝ λA,B,f,x. mfold_left_i_aux A B f x O. |
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| 137 | |
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| 138 | |
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[1125] | 139 | (* A monadic fold_left2 *) |
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| 140 | |
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| 141 | let rec mfold_left2 |
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| 142 | (A: Type[0]) (B: Type[0]) (C: Type[0]) (f: A → B → C → res A) (accu: res A) |
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| 143 | (left: list B) (right: list C) on left: res A ≝ |
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| 144 | match left with |
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| 145 | [ nil ⇒ |
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| 146 | match right with |
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| 147 | [ nil ⇒ accu |
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| 148 | | cons hd tl ⇒ Error ? (msg WrongLength) |
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| 149 | ] |
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| 150 | | cons hd tl ⇒ |
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| 151 | match right with |
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| 152 | [ nil ⇒ Error ? (msg WrongLength) |
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| 153 | | cons hd' tl' ⇒ |
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[1640] | 154 | ! accu ← accu; |
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[1125] | 155 | mfold_left2 … f (f accu hd hd') tl tl' |
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| 156 | ] |
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| 157 | ]. |
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| 158 | |
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| 159 | (* |
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[3] | 160 | (** * Reasoning over monadic computations *) |
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| 161 | |
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| 162 | (** The [monadInv H] tactic below simplifies hypotheses of the form |
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| 163 | << |
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| 164 | H: (do x <- a; b) = OK res |
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| 165 | >> |
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| 166 | By definition of the bind operation, both computations [a] and |
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| 167 | [b] must succeed for their composition to succeed. The tactic |
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| 168 | therefore generates the following hypotheses: |
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| 169 | |
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| 170 | x: ... |
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| 171 | H1: a = OK x |
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| 172 | H2: b x = OK res |
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| 173 | *) |
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| 174 | |
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| 175 | Ltac monadInv1 H := |
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| 176 | match type of H with |
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| 177 | | (OK _ = OK _) => |
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| 178 | inversion H; clear H; try subst |
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| 179 | | (Error _ = OK _) => |
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| 180 | discriminate |
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| 181 | | (bind ?F ?G = OK ?X) => |
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| 182 | let x := fresh "x" in ( |
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| 183 | let EQ1 := fresh "EQ" in ( |
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| 184 | let EQ2 := fresh "EQ" in ( |
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| 185 | destruct (bind_inversion F G H) as [x [EQ1 EQ2]]; |
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| 186 | clear H; |
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| 187 | try (monadInv1 EQ2)))) |
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| 188 | | (bind2 ?F ?G = OK ?X) => |
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| 189 | let x1 := fresh "x" in ( |
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| 190 | let x2 := fresh "x" in ( |
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| 191 | let EQ1 := fresh "EQ" in ( |
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| 192 | let EQ2 := fresh "EQ" in ( |
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| 193 | destruct (bind2_inversion F G H) as [x1 [x2 [EQ1 EQ2]]]; |
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| 194 | clear H; |
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| 195 | try (monadInv1 EQ2))))) |
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| 196 | | (mmap ?F ?L = OK ?M) => |
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| 197 | generalize (mmap_inversion F L H); intro |
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| 198 | end. |
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| 199 | |
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| 200 | Ltac monadInv H := |
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| 201 | match type of H with |
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| 202 | | (OK _ = OK _) => monadInv1 H |
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| 203 | | (Error _ = OK _) => monadInv1 H |
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| 204 | | (bind ?F ?G = OK ?X) => monadInv1 H |
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| 205 | | (bind2 ?F ?G = OK ?X) => monadInv1 H |
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| 206 | | (?F _ _ _ _ _ _ _ _ = OK _) => |
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| 207 | ((progress simpl in H) || unfold F in H); monadInv1 H |
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| 208 | | (?F _ _ _ _ _ _ _ = OK _) => |
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| 209 | ((progress simpl in H) || unfold F in H); monadInv1 H |
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| 210 | | (?F _ _ _ _ _ _ = OK _) => |
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| 211 | ((progress simpl in H) || unfold F in H); monadInv1 H |
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| 212 | | (?F _ _ _ _ _ = OK _) => |
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| 213 | ((progress simpl in H) || unfold F in H); monadInv1 H |
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| 214 | | (?F _ _ _ _ = OK _) => |
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| 215 | ((progress simpl in H) || unfold F in H); monadInv1 H |
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| 216 | | (?F _ _ _ = OK _) => |
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| 217 | ((progress simpl in H) || unfold F in H); monadInv1 H |
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| 218 | | (?F _ _ = OK _) => |
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| 219 | ((progress simpl in H) || unfold F in H); monadInv1 H |
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| 220 | | (?F _ = OK _) => |
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| 221 | ((progress simpl in H) || unfold F in H); monadInv1 H |
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| 222 | end. |
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| 223 | *) |
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[485] | 224 | |
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| 225 | |
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[797] | 226 | 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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| 227 | lemma opt_OK: ∀A,m,P,e. |
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[485] | 228 | (∀v. e = Some ? v → P v) → |
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[797] | 229 | match opt_to_res A m e with [ Error _ ⇒ True | OK v ⇒ P v ]. |
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| 230 | #A #m #P #e elim e; /2/; |
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[487] | 231 | qed. |
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[1214] | 232 | |
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[1954] | 233 | lemma opt_eq_from_res : ∀T,m,e,v. |
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| 234 | opt_to_res T m e = return v → |
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| 235 | e = Some T v. |
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| 236 | #T #m * [ #v #E normalize in E; destruct | #e' #v #E normalize in E; destruct % ] |
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| 237 | qed. |
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| 238 | |
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| 239 | coercion opt_eq_from_res : |
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| 240 | ∀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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| 241 | on _E:eq (res ?) ?? to eq (option ?) ??. |
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| 242 | |
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[1316] | 243 | (* A variation of bind and its notation that provides an equality proof for |
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| 244 | later use. *) |
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| 245 | |
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| 246 | definition bind_eq ≝ λA,B:Type[0]. λf: res A. λg: ∀a:A. f = OK ? a → res B. |
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| 247 | match f return λx. f = x → ? with |
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| 248 | [ OK x ⇒ g x |
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| 249 | | Error msg ⇒ λ_. Error ? msg |
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| 250 | ] (refl ? f). |
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| 251 | |
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[1647] | 252 | 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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[1316] | 253 | |
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[2439] | 254 | lemma bind_as_inversion: |
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| 255 | ∀A,B: Type[0]. ∀f: res A. ∀g: ∀a:A. f = OK A a → res B. ∀y: B. |
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| 256 | (do x as E ← f; g x E = return y) → |
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| 257 | ∃x. ∃E:f = return x. g x E = return y. |
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| 258 | #A #B #f cases f normalize |
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| 259 | [ #a #g #y #e %{a} /2/ |
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| 260 | | #m #g #y #H destruct (H) |
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| 261 | ] qed. |
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| 262 | |
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[1626] | 263 | 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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| 264 | match f return λx. f = x → ? with |
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| 265 | [ OK x ⇒ match x return λx. f = OK ? x → ? with [ mk_Prod a b ⇒ g a b ] |
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| 266 | | Error msg ⇒ λ_. Error ? msg |
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| 267 | ] (refl ? f). |
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| 268 | |
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[1647] | 269 | 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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[1626] | 270 | |
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[2496] | 271 | lemma bind2_eq_inversion: |
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| 272 | ∀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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| 273 | bind2_eq ??? f g = return z → |
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[2500] | 274 | ∃x. ∃y. ∃Eq. g x y Eq = return z. |
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[2496] | 275 | #A #B #C #f cases f |
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[2500] | 276 | [ * #a #b #g normalize #z #Heq %{a} %{b} %{(refl ? (OK ? 〈a,b〉))} @Heq |
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[2496] | 277 | | #errmsg #g #z normalize #Habsurd destruct (Habsurd) ] |
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[2500] | 278 | qed. |
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[2496] | 279 | |
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[1214] | 280 | definition res_to_opt : ∀A:Type[0]. res A → option A ≝ |
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[1316] | 281 | λA.λv. match v with [ Error _ ⇒ None ? | OK v ⇒ Some … v]. |
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| 282 | |
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[1647] | 283 | (* aliases for backward compatibility *) |
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| 284 | definition bind ≝ m_bind Res. |
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| 285 | definition bind2 ≝ m_bind2 Res. |
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| 286 | definition bind3 ≝ m_bind3 Res. |
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[1882] | 287 | definition mmap ≝ m_list_map Res. |
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| 288 | definition mmap_sigma ≝ m_list_map_sigma Res. |
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