option.v 6.4 KB
 Robbert Krebbers committed Aug 29, 2012 1 2 3 4 5 6 7 8 ``````(* Copyright (c) 2012, Robbert Krebbers. *) (* This file is distributed under the terms of the BSD license. *) (** This file collects general purpose definitions and theorems on the option data type that are not in the Coq standard library. *) Require Export base tactics decidable orders. (** * General definitions and theorems *) (** Basic properties about equality. *) `````` Robbert Krebbers committed Jun 11, 2012 9 10 11 12 13 14 ``````Lemma None_ne_Some `(a : A) : None ≠ Some a. Proof. congruence. Qed. Lemma Some_ne_None `(a : A) : Some a ≠ None. Proof. congruence. Qed. Lemma eq_None_ne_Some `(x : option A) a : x = None → x ≠ Some a. Proof. congruence. Qed. `````` Robbert Krebbers committed Aug 21, 2012 15 ``````Instance Some_inj {A} : Injective (=) (=) (@Some A). `````` Robbert Krebbers committed Jun 11, 2012 16 17 ``````Proof. congruence. Qed. `````` Robbert Krebbers committed Aug 29, 2012 18 19 ``````(** The non dependent elimination principle on the option type. *) Definition option_case {A B} (f : A → B) (b : B) (x : option A) : B := `````` Robbert Krebbers committed Jun 11, 2012 20 21 22 23 24 `````` match x with | None => b | Some a => f a end. `````` Robbert Krebbers committed Aug 29, 2012 25 26 27 ``````(** The [maybe] function allows us to get the value out of the option type by specifying a default value. *) Definition maybe {A} (a : A) (x : option A) : A := `````` Robbert Krebbers committed Aug 21, 2012 28 29 `````` match x with | None => a `````` Robbert Krebbers committed Aug 29, 2012 30 `````` | Some b => b `````` Robbert Krebbers committed Aug 21, 2012 31 32 `````` end. `````` Robbert Krebbers committed Aug 29, 2012 33 34 ``````(** An alternative, but equivalent, definition of equality on the option data type. This theorem is useful to prove that two options are the same. *) `````` Robbert Krebbers committed Jun 11, 2012 35 36 37 38 ``````Lemma option_eq {A} (x y : option A) : x = y ↔ ∀ a, x = Some a ↔ y = Some a. Proof. split. `````` Robbert Krebbers committed Aug 29, 2012 39 40 41 42 43 44 `````` { intros. now subst. } intros E. destruct x, y. + now apply E. + symmetry. now apply E. + now apply E. + easy. `````` Robbert Krebbers committed Jun 11, 2012 45 46 ``````Qed. `````` Robbert Krebbers committed Aug 29, 2012 47 48 49 ``````(** We define [is_Some] as a [sig] instead of a [sigT] as extraction of witnesses can be derived (see [is_Some_sigT] below). *) Definition is_Some `(x : option A) : Prop := ∃ a, x = Some a. `````` Robbert Krebbers committed Jun 11, 2012 50 51 ``````Hint Extern 10 (is_Some _) => solve [eexists; eauto]. `````` Robbert Krebbers committed Aug 21, 2012 52 ``````Ltac simplify_is_Some := repeat intro; repeat `````` Robbert Krebbers committed Jun 11, 2012 53 `````` match goal with `````` Robbert Krebbers committed Aug 29, 2012 54 `````` | _ => progress simplify_equality `````` Robbert Krebbers committed Jun 11, 2012 55 56 `````` | H : is_Some _ |- _ => destruct H as [??] | |- is_Some _ => eauto `````` Robbert Krebbers committed Aug 21, 2012 57 `````` end. `````` Robbert Krebbers committed Jun 11, 2012 58 59 60 61 62 63 `````` Lemma Some_is_Some `(a : A) : is_Some (Some a). Proof. simplify_is_Some. Qed. Lemma None_not_is_Some {A} : ¬is_Some (@None A). Proof. simplify_is_Some. Qed. `````` Robbert Krebbers committed Aug 29, 2012 64 ``````Definition is_Some_sigT `(x : option A) : is_Some x → { a | x = Some a } := `````` Robbert Krebbers committed Jun 11, 2012 65 66 67 68 `````` match x with | None => False_rect _ ∘ ex_ind None_ne_Some | Some a => λ _, a↾eq_refl end. `````` Robbert Krebbers committed Aug 29, 2012 69 ``````Lemma eq_Some_is_Some `(x : option A) a : x = Some a → is_Some x. `````` Robbert Krebbers committed Jun 11, 2012 70 71 72 73 74 ``````Proof. simplify_is_Some. Qed. Lemma eq_None_not_Some `(x : option A) : x = None ↔ ¬is_Some x. Proof. destruct x; simpl; firstorder congruence. Qed. `````` Robbert Krebbers committed Aug 29, 2012 75 ``````Lemma make_eq_Some {A} (x : option A) a : `````` Robbert Krebbers committed Jun 11, 2012 76 77 78 `````` is_Some x → (∀ b, x = Some b → b = a) → x = Some a. Proof. intros [??] H. subst. f_equal. auto. Qed. `````` Robbert Krebbers committed Aug 29, 2012 79 80 81 ``````(** Equality on [option] is decidable. *) Instance option_eq_dec `{dec : ∀ x y : A, Decision (x = y)} (x y : option A) : Decision (x = y) := `````` Robbert Krebbers committed Jun 11, 2012 82 83 84 85 86 87 `````` match x with | Some a => match y with | Some b => match dec a b with | left H => left (f_equal _ H) `````` Robbert Krebbers committed Aug 21, 2012 88 `````` | right H => right (H ∘ injective Some _ _) `````` Robbert Krebbers committed Jun 11, 2012 89 90 91 92 93 94 95 96 97 98 `````` end | None => right (Some_ne_None _) end | None => match y with | Some _ => right (None_ne_Some _) | None => left eq_refl end end. `````` Robbert Krebbers committed Aug 29, 2012 99 ``````(** * Monadic operations *) `````` Robbert Krebbers committed Jun 11, 2012 100 101 102 103 104 105 106 107 108 109 110 111 112 ``````Instance option_ret: MRet option := @Some. Instance option_bind: MBind option := λ A B f x, match x with | Some a => f a | None => None end. Instance option_join: MJoin option := λ A x, match x with | Some x => x | None => None end. Instance option_fmap: FMap option := @option_map. `````` Robbert Krebbers committed Aug 21, 2012 113 114 ``````Lemma option_fmap_is_Some {A B} (f : A → B) (x : option A) : is_Some x ↔ is_Some (f <\$> x). `````` Robbert Krebbers committed Jun 11, 2012 115 ``````Proof. destruct x; split; intros [??]; subst; compute; eauto; discriminate. Qed. `````` Robbert Krebbers committed Aug 21, 2012 116 117 ``````Lemma option_fmap_is_None {A B} (f : A → B) (x : option A) : x = None ↔ f <\$> x = None. `````` Robbert Krebbers committed Jun 11, 2012 118 119 ``````Proof. unfold fmap, option_fmap. destruct x; simpl; split; congruence. Qed. `````` Robbert Krebbers committed Aug 29, 2012 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 ``````Ltac simplify_option_bind := repeat match goal with | |- context C [mbind (M:=option) ?f None] => let X := (context C [ None ]) in change X | H : context C [mbind (M:=option) ?f None] |- _ => let X := (context C [ None ]) in change X in H | |- context C [mbind (M:=option) ?f (Some ?a)] => let X := (context C [ f a ]) in let X' := eval simpl in X in change X' | H : context C [mbind (M:=option) ?f (Some ?a)] |- _ => let X := context C [ f a ] in let X' := eval simpl in X in change X' in H | _ => progress simplify_equality | H : mbind (M:=option) ?f ?o = ?x |- _ => destruct o eqn:? | H : context [ ?o = _ ] |- mbind (M:=option) ?f ?o = ?x => erewrite H by eauto end. (** * Union, intersection and difference *) `````` Robbert Krebbers committed Jun 11, 2012 140 141 142 143 144 145 146 ``````Instance option_union: UnionWith option := λ A f x y, match x, y with | Some a, Some b => Some (f a b) | Some a, None => Some a | None, Some b => Some b | None, None => None end. `````` Robbert Krebbers committed Aug 21, 2012 147 ``````Instance option_intersection: IntersectionWith option := λ A f x y, `````` Robbert Krebbers committed Jun 11, 2012 148 149 150 151 `````` match x, y with | Some a, Some b => Some (f a b) | _, _ => None end. `````` Robbert Krebbers committed Aug 21, 2012 152 153 154 155 156 157 ``````Instance option_difference: DifferenceWith option := λ A f x y, match x, y with | Some a, Some b => f a b | Some a, None => Some a | None, _ => None end. `````` Robbert Krebbers committed Jun 11, 2012 158 `````` `````` Robbert Krebbers committed Aug 21, 2012 159 ``````Section option_union_intersection. `````` Robbert Krebbers committed Jun 11, 2012 160 161 162 163 164 165 166 `````` Context {A} (f : A → A → A). Global Instance: LeftId (=) None (union_with f). Proof. now intros [?|]. Qed. Global Instance: RightId (=) None (union_with f). Proof. now intros [?|]. Qed. Global Instance: Commutative (=) f → Commutative (=) (union_with f). `````` Robbert Krebbers committed Aug 29, 2012 167 168 169 170 `````` Proof. intros ? [?|] [?|]; compute; try reflexivity. now rewrite (commutative f). Qed. `````` Robbert Krebbers committed Jun 11, 2012 171 `````` Global Instance: Associative (=) f → Associative (=) (union_with f). `````` Robbert Krebbers committed Aug 29, 2012 172 173 174 175 `````` Proof. intros ? [?|] [?|] [?|]; compute; try reflexivity. now rewrite (associative f). Qed. `````` Robbert Krebbers committed Jun 11, 2012 176 `````` Global Instance: Idempotent (=) f → Idempotent (=) (union_with f). `````` Robbert Krebbers committed Aug 29, 2012 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 `````` Proof. intros ? [?|]; compute; try reflexivity. now rewrite (idempotent f). Qed. Global Instance: Commutative (=) f → Commutative (=) (intersection_with f). Proof. intros ? [?|] [?|]; compute; try reflexivity. now rewrite (commutative f). Qed. Global Instance: Associative (=) f → Associative (=) (intersection_with f). Proof. intros ? [?|] [?|] [?|]; compute; try reflexivity. now rewrite (associative f). Qed. Global Instance: Idempotent (=) f → Idempotent (=) (intersection_with f). Proof. intros ? [?|]; compute; try reflexivity. now rewrite (idempotent f). Qed. `````` Robbert Krebbers committed Aug 21, 2012 197 198 199 200 201 202 203 204 ``````End option_union_intersection. Section option_difference. Context {A} (f : A → A → option A). Global Instance: RightId (=) None (difference_with f). Proof. now intros [?|]. Qed. End option_difference.``````