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Diffstat (limited to 'src/common/NonEmpty.v')
-rw-r--r-- | src/common/NonEmpty.v | 110 |
1 files changed, 110 insertions, 0 deletions
diff --git a/src/common/NonEmpty.v b/src/common/NonEmpty.v new file mode 100644 index 0000000..9cb7650 --- /dev/null +++ b/src/common/NonEmpty.v @@ -0,0 +1,110 @@ +(* + * Vericert: Verified high-level synthesis. + * Copyright (C) 2021-2022 Yann Herklotz <yann@yannherklotz.com> + * + * This program is free software: you can redistribute it and/or modify + * it under the terms of the GNU General Public License as published by + * the Free Software Foundation, either version 3 of the License, or + * (at your option) any later version. + * + * This program is distributed in the hope that it will be useful, + * but WITHOUT ANY WARRANTY; without even the implied warranty of + * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the + * GNU General Public License for more details. + * + * You should have received a copy of the GNU General Public License + * along with this program. If not, see <https://www.gnu.org/licenses/>. + *) + +Require Import Coq.Lists.List. + +Inductive non_empty (A: Type) := +| singleton : A -> non_empty A +| cons : A -> non_empty A -> non_empty A +. + +Arguments singleton [A]. +Arguments cons [A]. + +Declare Scope non_empty_scope. +Delimit Scope non_empty_scope with non_empty. + +Module NonEmptyNotation. + Infix "::|" := cons (at level 60, right associativity) : non_empty_scope. +End NonEmptyNotation. +Import NonEmptyNotation. + +#[local] Open Scope non_empty_scope. + +Fixpoint map {A B} (f: A -> B) (l: non_empty A): non_empty B := + match l with + | singleton a => singleton (f a) + | a ::| b => f a ::| map f b + end. + +Fixpoint to_list {A} (l: non_empty A): list A := + match l with + | singleton a => a::nil + | a ::| b => a :: to_list b + end. + +Fixpoint app {A} (l1 l2: non_empty A) := + match l1 with + | singleton a => a ::| l2 + | a ::| b => a ::| app b l2 + end. + +Fixpoint non_empty_prod {A B} (l: non_empty A) (l': non_empty B) := + match l with + | singleton a => map (fun x => (a, x)) l' + | a ::| b => app (map (fun x => (a, x)) l') (non_empty_prod b l') + end. + +Fixpoint of_list {A} (l: list A): option (non_empty A) := + match l with + | a::b => + match of_list b with + | Some b' => Some (a ::| b') + | _ => None + end + | nil => None + end. + +Fixpoint replace {A} (f: A -> A -> bool) (a b: A) (l: non_empty A) := + match l with + | a' ::| l' => if f a a' then b ::| replace f a b l' else a' ::| replace f a b l' + | singleton a' => if f a a' then singleton b else singleton a' + end. + +Inductive In {A: Type} (x: A) : non_empty A -> Prop := +| In_cons : forall a b, x = a \/ In x b -> In x (a ::| b) +| In_single : In x (singleton x). + +Ltac inv X := inversion X; clear X; subst. + +Lemma in_dec: + forall A (a: A) (l: non_empty A), + (forall a b: A, {a = b} + {a <> b}) -> + {In a l} + {~ In a l}. +Proof. + induction l; intros. + { specialize (X a a0). inv X. + left. constructor. + right. unfold not. intros. apply H. inv H0. auto. } + { pose proof X as X2. + specialize (X a a0). inv X. + left. constructor; tauto. + apply IHl in X2. inv X2. + left. constructor; tauto. + right. unfold not in *; intros. apply H0. inv H1. now inv H3. } +Qed. + +Fixpoint filter {A: Type} (f: A -> bool) (l: non_empty A) : option (non_empty A) := + match l with + | singleton a => + if f a then Some (singleton a) else None + | a ::| b => + if f a then + match filter f b with Some x => Some (a ::| x) | None => Some (singleton a) end + else filter f b + end. |