(* (c) Copyright 2006-2016 Microsoft Corporation and Inria.                  *)
(* Distributed under the terms of CeCILL-B.                                  *)
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(* This file develops a basic theory of signed integers, defining:            *)
(*         int == the type of signed integers, with two constructors Posz for *)
(*                non-negative integers and Negz for negative integers. It    *)
(*                supports the realDomainType interface (and its parents).    *)
(*        n%:Z == explicit cast from nat to int (:= Posz n); displayed as n.  *)
(*                However (Posz m = Posz n) is displayed as (m = n :> int)    *)
(*                (and so are ==, != and <>)                                  *)
(*                Lemma NegzE : turns (Negz n) into - n.+1%:Z.                *)
(*      x *~ m == m times x, with m : int;                                    *)
(*                convertible to x *+ n if m is Posz n                        *)
(*                convertible to x *- n.+1 if m is Negz n.                    *)
(*       m%:~R == the image of m : int in a generic ring (:= 1 *~ m).         *)
(*       x ^ m == x to the m, with m : int;                                   *)
(*                convertible to x ^+ n if m is Posz n                        *)
(*                convertible to x ^- n.+1 if m is Negz n.                    *)
(*       sgz x == sign of x : R,                                              *)
(*                equals (0 : int) if and only x == 0,                        *)
(*                equals (1 : int) if x is positive                           *)
(*                and (-1 : int) otherwise.                                   *)
(*      `|m|%N == the n : nat such that `|m|%R = n%:Z, for m : int.           *)
(*  `|m - n|%N == the distance between m and n; the '-' is specialized to     *)
(*                the int type, so m and n can be either of type nat or int   *)
(*                thanks to the Posz coercion; m and n are however parsed in  *)
(*                the %N scope. The IntDist submodule provides this notation  *)
(*                and the corresponding theory independently of the rest of   *)
(*                of the int and ssralg libraries (and notations).            *)
(* Warning: due to the declaration of Posz as a coercion, two terms might be  *)
(* displayed the same while not being convertible, for instance:              *)
(* (Posz (x - y)) and (Posz x) - (Posz y) for x, y : nat.                     *)
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Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
Declare Scope int_scope.
Declare Scope distn_scope.
Declare Scope rat_scope.
Reserved Notation "n %:Z" (at level 2, left associativity, format "n %:Z").
Reserved Notation "n = m :> 'int'"
(at level 70, m at next level, format "n  =  m  :>  'int'", only printing).
Reserved Notation "n == m :> 'int'"
(at level 70, m at next level, format "n  ==  m  :>  'int'", only printing).
Reserved Notation "n != m :> 'int'"
(at level 70, m at next level, format "n  !=  m  :>  'int'", only printing).
Reserved Notation "n <> m :> 'int'"
(at level 70, m at next level, format "n  <>  m  :>  'int'", only printing).
Delimit Scope int_scope with Z.
Local Open Scope int_scope.
(* Defining int *)
Variant int : Set := Posz of nat | Negz of nat.
(* This must be deferred to module DistInt to work around the design flaws of *)
(* the Coq module system.                                                     *)
(* Coercion Posz : nat >-> int. *)
Notation "n %:Z" := (Posz n) (only parsing) : int_scope.
Notation "n %:Z" := (Posz n) (only parsing) : ring_scope.
Notation "n = m :> 'int'"  := (Posz n = Posz m) (only printing)  : ring_scope.
Notation "n == m :> 'int'" := (Posz n == Posz m) (only printing) : ring_scope.
Notation "n != m :> 'int'" := (Posz n != Posz m) (only printing) : ring_scope.
Notation "n <> m :> 'int'" := (Posz n <> Posz m) (only printing) : ring_scope.
Definition natsum_of_int (m : int) : nat + nat :=
match m with Posz p => inl _ p | Negz n => inr _ n end.
Definition int_of_natsum (m : nat + nat) :=
match m with inl p => Posz p | inr n => Negz n end.
Definition int_rec := int_rect.
Definition int_ind := int_rect.
Variant int_spec (x : int) : int -> Type :=
| ZintNull of x = 0 : int_spec x 0
| ZintPos n of x = n.+1 : int_spec x n.+1
| ZintNeg n of x = - (n.+1)%:Z : int_spec x (- n.+1).
Definition int_rec := int_rect.
Definition int_ind := int_rect.
Variant int_spec (x : int) : int -> Type :=
| ZintNull : int_spec x 0
| ZintPos n : int_spec x n.+1
| ZintNeg n : int_spec x (- (n.+1)%:Z).

Lemma intP x : int_spec x x.