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base.v
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base.v
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(**
This module defines functions and notations shared by all of the
modules in this package.
Copyright (C) 2018 Larry D. Lee Jr. <[email protected]>
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).
(**
Proves that every boolean value is either true
or false.
*)
Definition bool_dec0
: forall b : bool, {b = true}+{b = false}
:= bool_rect
(fun b => {b = true}+{b = false})
(left (true = false) (eq_refl true))
(right (false = true) (eq_refl false)).