(** 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 *)
Proof fun _ P x0 xs H
=> let H0
:  forall x, In x (x0 :: xs) -> P x
:= proj1 (Forall_forall P (x0 :: xs)) H in
let H1
:  forall x, In x xs -> P x
:= fun x H2
=> H0 x (or_intror (x0 = x) H2) in
proj2 (Forall_forall P xs) H1.
Arguments Forall_tail {A} {P} x0 xs.

(** Accepts two predicates, [P] and [Q], and a list, [xs], and proves that, if [P -> Q], and there exists an element in [xs] for which [P] is true, then there exists an element in [xs] for which [Q] is true. *)
Theorem Exists_impl
:  forall (A : Type) (P Q : A -> Prop),
(forall x : A, P x -> Q x) ->
forall xs : list A,
Exists P xs ->
Exists Q xs.