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(*
* 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/>.
*)
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).
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.
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