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(*
* Vericert: Verified high-level synthesis.
* Copyright (C) 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/>.
*)
From vericert Require Import Monad.
From Coq Require Import Lists.List.
Module Option <: Monad.
Definition mon := option.
Definition ret {A: Type} (x: A) := Some x.
Definition default {T : Type} (x : T) (u : option T) : T :=
match u with
| Some y => y
| _ => x
end.
Definition map {S : Type} {T : Type} (f : S -> T) (u : option S) : option T :=
match u with
| Some y => Some (f y)
| _ => None
end.
Definition liftA2 {T : Type} (f : T -> T -> T) (a : option T) (b : option T) : option T :=
match a with
| Some x => map (f x) b
| _ => None
end.
Definition bind {A B : Type} (g : A -> option B) (f : option A) : option B :=
match f with
| Some a => g a
| _ => None
end.
Definition bind2 {A B C : Type} (g : A -> B -> option C) (f : mon (A * B)) : option C :=
match f with
| Some (a, b) => g a b
| _ => None
end.
Definition join {A : Type} (a : option (option A)) : option A :=
match a with
| None => None
| Some a' => a'
end.
#[global] Instance option_ret : MRet option := @ret.
#[global] Instance option_bind : MBind option := @bind.
#[global] Instance option_join : MJoin option := @join.
#[global] Instance option_map : FMap option := @map.
#[global] Instance option_omap : OMap option := @bind.
End Option.
Module OptionExtra.
Module Export OE := MonadExtra(Option).
Lemma mfold_left_Some :
forall A B f x n y,
@mfold_left A B f x n = Some y ->
exists n', n = Some n'.
Proof. induction x; intros; destruct n; eauto. Qed.
End OptionExtra.
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