(* (c) Copyright 2006-2016 Microsoft Corporation and Inria.                  *)
(* Distributed under the terms of CeCILL-B.                                  *)
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(* This file defines a datatype for rational numbers and equips it with a     *)
(* structure of archimedean, real field, with int and nat declared as closed  *)
(* subrings.                                                                  *)
(*          rat == the type of rational number, with single constructor Rat   *)
(*         n%:Q == explicit cast from int to rat, ie. the specialization to   *)
(*                 rationals of the generic ring morphism n%:~R               *)
(*       numq r == numerator of (r : rat)                                     *)
(*       denq r == denominator of (r : rat)                                   *)
(* x \is a Qint == x is an element of rat whose denominator is equal to 1     *)
(* x \is a Qnat == x is a non-negative element of rat whose denominator       *)
(*                 is equal to 1                                              *)
(*       ratr r == generic embedding of (r : rat) into an arbitrary unitring. *)
(* [rat x // y] == smart constructor for rationals, definitionally equal      *)
(*                 to x / y for concrete values, intended for printing only   *)
(*                 of normal forms. The parsable notation is for debugging.   *)
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Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
Reserved Notation "[ 'rat' x // y ]" (format "[ 'rat'  x  //  y ]", at level 0).
Reserved Notation "n %:Q" (at level 2, left associativity, format "n %:Q").
Local Open Scope ring_scope.
Local Notation sgr := Num.sg.
Record rat : Set := Rat {
valq : (int * int);
_ : (0 < valq.2) && coprime `|valq.1| `|valq.2|
}.
Bind Scope ring_scope with rat.
Delimit Scope rat_scope with Q.
Definition ratz (n : int) := @Rat (n, 1) (coprimen1 _).
(* Coercion ratz (n : int) := @Rat (n, 1) (coprimen1 _). *)
Canonical rat_subType := Eval hnf in [subType for valq].
Definition rat_eqMixin := [eqMixin of rat by <:].
Canonical rat_eqType := EqType rat rat_eqMixin.
Definition rat_choiceMixin := [choiceMixin of rat by <:].
Canonical rat_choiceType := ChoiceType rat rat_choiceMixin.
Definition rat_countMixin := [countMixin of rat by <:].
Canonical rat_countType := CountType rat rat_countMixin.
Canonical rat_subCountType := [subCountType of rat].
Definition numq x := nosimpl ((valq x).1).
Definition denq x := nosimpl ((valq x).2).
Fact fracq_subproof x (y := fracq_opt_subdef x) :
(0 < y.2) && (coprime `|y.1| `|y.2|).
(* We use a match expression in order to "lock" the definition of fracq.   *)
(* Indeed, the kernel will try to reduce a fracq only when applied to      *)
(* a term which has "enough" constructors: i.e. it reduces to a pair of    *)
(* a Posz or Negz on the first component, and a Posz of 0 or S, or a Negz  *)
(* on the second component. See issue #698.                                *)
(* Additionally, we use fracq_opt_subdef to precompute the normal form     *)
(* before we use fracq_subproof in order to make sure the proof will be    *)
(* independent from the input of fracq. This ensure reflexivity of any     *)
(* computation involving rationals as long as all operators use fracq.     *)
(* As a consequence val (fracq x) = fracq_opt_subdef (fracq_opt_subdef x)) *)
Definition fracq '((n', d')) : rat :=
match d', n' with
| Posz 0 as d, _ as n => Rat (fracq_subproof (1, 0))
| _ as d, Posz _ as n | _ as d, _ as n =>
Rat (fracq_subproof (fracq_opt_subdef (n, d)))
end.
Arguments fracq : simpl never.
Fact valqK x : fracq (valq x) = x.
Definition scalq '(n, d) := sgr d * (gcdn `|n| `|d|)%:Z.
Definition zeroq := fracq (0, 1).
Definition oneq := fracq (1, 1).
Fact frac0q x : fracq (0, x) = zeroq.
Fact fracq0  x : fracq (x, 0) = zeroq. Proof. exact/eqP. Qed.
Variant fracq_spec (x : int * int) : int * int -> rat -> Type :=
| FracqSpecN of x.2 = 0 : fracq_spec x (x.1, 0) zeroq
| FracqSpecP k fx of k != 0 : fracq_spec x (k * numq fx, k * denq fx) fx.
Fact fracqP x : fracq_spec x x (fracq x).

Lemma absz_denq x : `|denq x|%N = denq x :> int.