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(************************************************************************)
(* * The Coq Proof Assistant / The Coq Development Team *)
(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *)
(* <O___,, * (see CREDITS file for the list of authors) *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(* * (see LICENSE file for the text of the license) *)
(************************************************************************)
Require Import Bool NZAxioms NZMulOrder NZParity NZPow NZDiv NZLog.
(** Axiomatization of some bitwise operations *)
Module Type Bits (Import A : Typ).
Parameter Inline testbit : t -> t -> bool.
Parameters Inline shiftl shiftr land lor ldiff lxor : t -> t -> t.
Parameter Inline div2 : t -> t.
End Bits.
Module Type BitsNotation (Import A : Typ)(Import B : Bits A).
Notation "a .[ n ]" := (testbit a n) (at level 5, format "a .[ n ]").
Infix ">>" := shiftr (at level 30, no associativity).
Infix "<<" := shiftl (at level 30, no associativity).
End BitsNotation.
Module Type Bits' (A:Typ) := Bits A <+ BitsNotation A.
Module Type NZBitsSpec
(Import A : NZOrdAxiomsSig')(Import B : Bits' A).
Declare Instance testbit_wd : Proper (eq==>eq==>Logic.eq) testbit.
Axiom testbit_odd_0 : forall a, (2*a+1).[0] = true.
Axiom testbit_even_0 : forall a, (2*a).[0] = false.
Axiom testbit_odd_succ : forall a n, 0<=n -> (2*a+1).[S n] = a.[n].
Axiom testbit_even_succ : forall a n, 0<=n -> (2*a).[S n] = a.[n].
Axiom testbit_neg_r : forall a n, n<0 -> a.[n] = false.
Axiom shiftr_spec : forall a n m, 0<=m -> (a >> n).[m] = a.[m+n].
Axiom shiftl_spec_high : forall a n m, 0<=m -> n<=m -> (a << n).[m] = a.[m-n].
Axiom shiftl_spec_low : forall a n m, m<n -> (a << n).[m] = false.
Axiom land_spec : forall a b n, (land a b).[n] = a.[n] && b.[n].
Axiom lor_spec : forall a b n, (lor a b).[n] = a.[n] || b.[n].
Axiom ldiff_spec : forall a b n, (ldiff a b).[n] = a.[n] && negb b.[n].
Axiom lxor_spec : forall a b n, (lxor a b).[n] = xorb a.[n] b.[n].
Axiom div2_spec : forall a, div2 a == a >> 1.
End NZBitsSpec.
Module Type NZBits (A:NZOrdAxiomsSig) := Bits A <+ NZBitsSpec A.
Module Type NZBits' (A:NZOrdAxiomsSig) := Bits' A <+ NZBitsSpec A.
(** In the functor of properties will also be defined:
- [setbit : t -> t -> t ] defined as [lor a (1<<n)].
- [clearbit : t -> t -> t ] defined as [ldiff a (1<<n)].
- [ones : t -> t], the number with [n] initial true bits,
corresponding to [2^n - 1].
- a logical complement [lnot]. For integer numbers it will
be a [t->t], doing a swap of all bits, while on natural
numbers, it will be a bounded complement [t->t->t], swapping
only the first [n] bits.
*)
(** For the moment, no shared properties about NZ here,
since properties and proofs for N and Z are quite different *)
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