The Coq Standard Library

# Library Coq.Numbers.Rational.SpecViaQ.QSig

Require Import QArith Qpower Qminmax Orders RelationPairs GenericMinMax.
Open Scope Q_scope.

# QSig

Interface of a rich structure about rational numbers. Specifications are written via translation to Q.
Module Type QType.
Parameter t : Type.
Parameter to_Q : t -> Q.
Definition eq x y := [x] == [y].
Definition lt x y := [x] < [y].
Definition le x y := [x] <= [y].
Parameter of_Q : Q -> t.
Parameter spec_of_Q: forall x, to_Q (of_Q x) == x.
Parameter red : t -> t.
Parameter compare : t -> t -> comparison.
Parameter eq_bool : t -> t -> bool.
Parameter max : t -> t -> t.
Parameter min : t -> t -> t.
Parameter zero : t.
Parameter one : t.
Parameter minus_one : t.
Parameter add : t -> t -> t.
Parameter sub : t -> t -> t.
Parameter opp : t -> t.
Parameter mul : t -> t -> t.
Parameter square : t -> t.
Parameter inv : t -> t.
Parameter div : t -> t -> t.
Parameter power : t -> Z -> t.
Parameter spec_red : forall x, [red x] == [x].
Parameter strong_spec_red : forall x, [red x] = Qred [x].
Parameter spec_compare : forall x y, compare x y = ([x] ?= [y]).
Parameter spec_eq_bool : forall x y, eq_bool x y = Qeq_bool [x] [y].
Parameter spec_max : forall x y, [max x y] == Qmax [x] [y].
Parameter spec_min : forall x y, [min x y] == Qmin [x] [y].
Parameter spec_0: [zero] == 0.
Parameter spec_1: [one] == 1.
Parameter spec_m1: [minus_one] == -(1).
Parameter spec_add: forall x y, [add x y] == [x] + [y].
Parameter spec_sub: forall x y, [sub x y] == [x] - [y].
Parameter spec_opp: forall x, [opp x] == - [x].
Parameter spec_mul: forall x y, [mul x y] == [x] * [y].
Parameter spec_square: forall x, [square x] == [x] ^ 2.
Parameter spec_inv : forall x, [inv x] == / [x].
Parameter spec_div: forall x y, [div x y] == [x] / [y].
Parameter spec_power: forall x z, [power x z] == [x] ^ z.
End QType.
NB: several of the above functions come with ..._norm variants that expect reduced arguments and return reduced results.
TODO : also speak of specifications via Qcanon ...
Module Type QType_Notation (Import Q : QType).
Notation "[ x ]" := (to_Q x).
Infix "==" := eq (at level 70).
Notation "x != y" := (~x==y) (at level 70).
Infix "<=" := le.
Infix "<" := lt.
Notation "0" := zero.
Notation "1" := one.
Infix "+" := add.
Infix "-" := sub.
Infix "*" := mul.
Notation "- x" := (opp x).
Infix "/" := div.
Notation "/ x" := (inv x).
Infix "^" := power.
End QType_Notation.
Module Type QType' := QType <+ QType_Notation.
Module QProperties (Import Q : QType').
Conversion to Q
Hint Rewrite
spec_red spec_compare spec_eq_bool spec_min spec_max
spec_add spec_sub spec_opp spec_mul spec_square spec_inv spec_div
spec_power : qsimpl.
Ltac qify := unfold eq, lt, le in *; autorewrite with qsimpl;
try rewrite spec_0 in *; try rewrite spec_1 in *; try rewrite spec_m1 in *.
NB: do not add spec_0 in the autorewrite database. Otherwise, after instantiation in BigQ, this lemma become convertible to 0=0, and autorewrite loops. Idem for spec_1 and spec_m1
Morphisms
Ltac solve_wd1 := intros x x' Hx; qify; now rewrite Hx.
Ltac solve_wd2 := intros x x' Hx y y' Hy; qify; now rewrite Hx, Hy.
Instance : Measure to_Q.
Instance eq_equiv : Equivalence eq.
Program Instance lt_wd : Proper (eq==>eq==>iff) lt.
Program Instance le_wd : Proper (eq==>eq==>iff) le.
Program Instance red_wd : Proper (eq==>eq) red.
Program Instance compare_wd : Proper (eq==>eq==>Logic.eq) compare.
Program Instance eq_bool_wd : Proper (eq==>eq==>Logic.eq) eq_bool.
Program Instance min_wd : Proper (eq==>eq==>eq) min.
Program Instance max_wd : Proper (eq==>eq==>eq) max.
Program Instance add_wd : Proper (eq==>eq==>eq) add.
Program Instance sub_wd : Proper (eq==>eq==>eq) sub.
Program Instance opp_wd : Proper (eq==>eq) opp.
Program Instance mul_wd : Proper (eq==>eq==>eq) mul.
Program Instance square_wd : Proper (eq==>eq) square.
Program Instance inv_wd : Proper (eq==>eq) inv.
Program Instance div_wd : Proper (eq==>eq==>eq) div.
Program Instance power_wd : Proper (eq==>Logic.eq==>eq) power.
Let's implement HasCompare
Lemma compare_spec : forall x y, CompareSpec (x==y) (x<y) (y<x) (compare x y).
Let's implement TotalOrder
Definition lt_compat := lt_wd.
Instance lt_strorder : StrictOrder lt.
Lemma le_lteq : forall x y, x<=y <-> x<y \/ x==y.
Lemma lt_total : forall x y, x<y \/ x==y \/ y<x.
Let's implement HasEqBool
Definition eqb := eq_bool.
Lemma eqb_eq : forall x y, eq_bool x y = true <-> x == y.
Lemma eqb_correct : forall x y, eq_bool x y = true -> x == y.
Lemma eqb_complete : forall x y, x == y -> eq_bool x y = true.
Let's implement HasMinMax
Lemma max_l : forall x y, y<=x -> max x y == x.
Lemma max_r : forall x y, x<=y -> max x y == y.
Lemma min_l : forall x y, x<=y -> min x y == x.
Lemma min_r : forall x y, y<=x -> min x y == y.
Q is a ring
Lemma add_0_l : forall x, 0+x == x.
Lemma add_comm : forall x y, x+y == y+x.
Lemma add_assoc : forall x y z, x+(y+z) == x+y+z.
Lemma mul_1_l : forall x, 1*x == x.
Lemma mul_comm : forall x y, x*y == y*x.
Lemma mul_assoc : forall x y z, x*(y*z) == x*y*z.
Lemma mul_add_distr_r : forall x y z, (x+y)*z == x*z + y*z.
Lemma sub_add_opp : forall x y, x-y == x+(-y).
Lemma add_opp_diag_r : forall x, x+(-x) == 0.
Q is a field
Lemma neq_1_0 : 1!=0.
Lemma div_mul_inv : forall x y, x/y == x*(/y).
Lemma mul_inv_diag_l : forall x, x!=0 -> /x * x == 1.
End QProperties.
Module QTypeExt (Q : QType)
<: QType <: TotalOrder <: HasCompare Q <: HasMinMax Q <: HasEqBool Q
:= Q <+ QProperties.