(** This module defines functions and notations shared by all of the modules in this package.  Copyright (C) 2018 Larry D. Lee Jr. <llee454@gmail.com>  This program is free software: you can redistribute it and/or modify it under the terms of the GNU Lesser 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 Lesser General Public License for more details.  You should have received a copy of the GNU Lesser General Public License along with this program. If not, see <https://www.gnu.org/licenses/>. *)
(** The following notations are introduced here to simplify sequences of algebraic rewrites which would otherwise be expressed as long sequences of eq_ind*. *)
Notation "A || B @ X 'by' E"
:= (eq_ind_r (fun X => B) A E) (at level 40, left associativity).
Notation "A || B @ X 'by' <- H"
:= (eq_ind_r (fun X => B) A (eq_sym H)) (at level 40, left associativity).
(** The following notation can be used to define equality assertions. These are like unittests in that they check that a given expression reduces to a given value. *)
Notation "A =:= B"
:= (eq_refl A : A = B) (at level 90).
(** * II. Auxiliary Theorems *)

(* Specifications: *)
(** Accepts a predicate, [P], and a list, [x0 :: xs], and proves that if [P] is true for every element in [x0 :: xs], then [P] is true for every element in [xs]. *)
Theorem Forall_tail
:  forall (A : Type) (P : A -> Prop) (x0 : A) (xs : list A), Forall P (x0 :: xs) -> Forall P xs.