# Library mathcomp.field.algC

(* (c) Copyright 2006-2016 Microsoft Corporation and Inria.
Distributed under the terms of CeCILL-B.                                  *)

From HB Require Import structures.
From mathcomp Require Import ssreflect ssrbool ssrfun ssrnat eqtype seq choice.
From mathcomp Require Import div fintype path bigop finset prime order ssralg.
From mathcomp Require Import poly polydiv mxpoly generic_quotient countalg.
From mathcomp Require Import ssrnum closed_field ssrint rat intdiv.
From mathcomp Require Import algebraics_fundamentals.

This file provides an axiomatic construction of the algebraic numbers. The construction only assumes the existence of an algebraically closed filed with an automorphism of order 2; this amounts to the purely algebraic contents of the Fundamenta Theorem of Algebra. algC == the closed, countable field of algebraic numbers. algCeq, algCring, ..., algCnumField == structures for algC. The ssrnum interfaces are implemented for algC as follows: x <= y <=> (y - x) is a nonnegative real x < y <=> (y - x) is a (strictly) positive real |z| == the complex norm of z, i.e., sqrtC (z * z^* ). Creal == the subset of real numbers (:= Num.real for algC). 'i == the imaginary number (:= sqrtC (-1)). 'Re z == the real component of z. 'Im z == the imaginary component of z. z^* == the complex conjugate of z (:= conjC z). sqrtC z == a nonnegative square root of z, i.e., 0 <= sqrt x if 0 <= x. n.-root z == more generally, for n > 0, an nth root of z, chosen with a minimal non-negative argument for n > 1 (i.e., with a maximal real part subject to a nonnegative imaginary part). Note that n.-root (-1) is a primitive 2nth root of unity, an thus not equal to -1 for n odd > 1 (this will be shown in file cyclotomic.v). In addition, we provide: Crat == the subset of rational numbers. Cint == the subset of integers. Cnat == the subset of natural integers. getCrat z == some a : rat such that ratr a = z, provided z \in Crat. floorC z == for z \in Creal, an m : int s.t. m%:~R <= z < (m + 1)%:~R. truncC z == for z >= 0, an n : nat s.t. n%:R <= z < n.+1%:R, else 0%N. minCpoly z == the minimal (monic) polynomial over Crat with root z. algC_invaut nu == an inverse of nu : {rmorphism algC -> algC}. (x %| y)%C <=> y is an integer (Cint) multiple of x; if x or y are (x %| y)%Cx of type nat or int they are coerced to algC here. The (x %| y)%Cx display form is a workaround for design limitations of the Coq Notation facilities. (x == y % [mod z])%C <=> x and y differ by an integer (Cint) multiple of z; as above, arguments of type nat or int are cast to algC. (x != y % [mod z])%C <=> x and y do not differ by an integer multiple of z. Note that in file algnum we give an alternative definition of divisibility based on algebraic integers, overloading the notation in the %A scope.

Set Implicit Arguments.

Declare Scope C_scope.
Declare Scope C_core_scope.
Declare Scope C_expanded_scope.

Import Order.TTheory GRing.Theory Num.Theory.
Local Open Scope ring_scope.

Lemma nz2: 2 != 0 :> L.

Lemma mul2I: injective (fun z : Lz *+ 2).

Definition sqrt x : L :=
sval (sig_eqW (@solve_monicpoly _ 2%N (nth 0 [:: x]) isT)).

Lemma sqrtK x: sqrt x ^+ 2 = x.

Lemma sqrtE x y: y ^+ 2 = x {b : bool | y = (-1) ^+ b × sqrt x}.

Definition i := sqrt (- 1).

Lemma sqrMi x: (i × x) ^+ 2 = - x ^+ 2.

Lemma iJ : conj i = - i.

Definition norm x := sqrt x × conj (sqrt x).

Lemma normK x : norm x ^+ 2 = x × conj x.

Lemma normE x y : y ^+ 2 = x norm x = y × conj y.

Lemma norm_eq0 x : norm x = 0 x = 0.

Lemma normM x y : norm (x × y) = norm x × norm y.

Lemma normN x : norm (- x) = norm x.

Definition le x y := norm (y - x) == y - x.
Definition lt x y := (y != x) && le x y.

Lemma posE x: le 0 x = (norm x == x).

Lemma leB x y: le x y = le 0 (y - x).

Lemma posP x : reflect ( y, x = y × conj y) (le 0 x).

Lemma posJ x : le 0 x conj x = x.

Lemma pos_linear x y : le 0 x le 0 y le x y || le y x.

Lemma sposDl x y : lt 0 x le 0 y lt 0 (x + y).

Lemma sposD x y : lt 0 x lt 0 y lt 0 (x + y).

Lemma normD x y : le (norm (x + y)) (norm x + norm y).

Module Algebraics.

Module Type Specification.

Parameter type : Type.

Parameter conjMixin : Num.ClosedField type.

Parameter isCountable : Countable type.

Axiom algebraic : integralRange (@ratr (Num.ClosedField.Pack conjMixin)).

End Specification.

Module Implementation : Specification.

Definition L := tag Fundamental_Theorem_of_Algebraics.

Definition conjL : {rmorphism L L} :=
s2val (tagged Fundamental_Theorem_of_Algebraics).

Fact conjL_K : involutive conjL.

Fact conjL_nt : ¬ conjL =1 id.

Definition L' : Type := eta L.

Notation cfType := [the Num.ClosedField.type of L'].

Definition QtoL := [rmorphism of @ratr cfType].

Notation pQtoL := (map_poly QtoL).

Definition rootQtoL p_j :=
if p_j.1 == 0 then 0 else
(sval (closed_field_poly_normal (pQtoL p_j.1)))_p_j.2.

Definition eq_root p_j q_k := rootQtoL p_j == rootQtoL q_k.

Fact eq_root_is_equiv : equiv_class_of eq_root.
Canonical eq_root_equiv := EquivRelPack eq_root_is_equiv.

Definition type : Type := {eq_quot eq_root}%qT.

Definition CtoL (u : type) := rootQtoL (repr u).

Fact CtoL_inj : injective CtoL.

Fact CtoL_P u : integralOver QtoL (CtoL u).

Fact LtoC_subproof z : integralOver QtoL z {u | CtoL u = z}.

Definition LtoC z Az := sval (@LtoC_subproof z Az).
Fact LtoC_K z Az : CtoL (@LtoC z Az) = z.

Fact CtoL_K u : LtoC (CtoL_P u) = u.

Definition zero := LtoC (integral0 _).
Definition opp u := LtoC (integral_opp (CtoL_P u)).

Definition one := LtoC (integral1 _).
Definition mul u v := LtoC (integral_mul (CtoL_P u) (CtoL_P v)).
Definition inv u := LtoC (integral_inv (CtoL_P u)).

Fact mulA : associative mul.

Fact mulC : commutative mul.

Fact mul1 : left_id one mul.

Fact mulD : left_distributive mul +%R.

Fact one_nz : one != 0 :> type.

Fact CtoL_is_multiplicative : multiplicative CtoL.

Fact mulVf u : u != 0 inv u × u = 1.

Fact inv0 : inv 0 = 0.

Fact closedFieldAxiom : GRing.closed_field_axiom [the ringType of type].

Fact conj_subproof u : integralOver QtoL (conjL (CtoL u)).

Fact conj_is_multiplicative : multiplicative (fun uLtoC (conj_subproof u)).

Definition conj : {rmorphism type type} :=
GRing.RMorphism.Pack
(GRing.RMorphism.Class
(GRing.isMultiplicative.Build _ _ _ conj_is_multiplicative)).

Lemma conjK : involutive conj.

Fact conj_nt : ¬ conj =1 id.

Lemma normK u : |u| ^+ 2 = u × conj u.

Definition conjMixin := Num.ClosedField.on type.

Lemma algebraic : integralRange (@ratr [the unitRingType of type]).

Definition isCountable := Countable.on type.

End Implementation.

Definition divisor := Implementation.type.

#[export] HB.instance Definition _ := Implementation.conjMixin.
#[export] HB.instance Definition _ := Implementation.isCountable.

Module Internals.

Import Implementation.

Fact algCi_subproof : {i : algC | i ^+ 2 = -1}.

Variant getCrat_spec : Type := GetCrat_spec CtoQ of cancel QtoC CtoQ.

Fact getCrat_subproof : getCrat_spec.

Fact floorC_subproof x : {m | x \is Creal ZtoC m x < ZtoC (m + 1)}.

Fact minCpoly_subproof (x : algC) :
{p | p \is monic & q, root (pQtoC q) x = (p %| q)%R}.

Definition algC_divisor (x : algC) := x : divisor.
Definition int_divisor m := m%:~R : divisor.
Definition nat_divisor n := n%:R : divisor.

End Internals.

Module Import Exports.

Import Implementation Internals.

Notation algC := type.
Delimit Scope C_scope with C.
Delimit Scope C_core_scope with Cc.
Delimit Scope C_expanded_scope with Cx.
Open Scope C_core_scope.
Notation algCeq := [the eqType of type].
Notation algCzmod := [the zmodType of type].
Notation algCring := [the ringType of type].
Notation algCuring := [the unitRingType of type].
Notation algCnum := [the numDomainType of type].
Notation algCfield := [the fieldType of type].
Notation algCnumField := [the numFieldType of type].
Notation algCnumClosedField := [the numClosedFieldType of type].

Notation Creal := (@Num.Def.Rreal algCnum).

Definition getCrat := let: GetCrat_spec CtoQ _ := getCrat_subproof in CtoQ.
Definition Crat : {pred algC} := fun xratr (getCrat x) == x.

Definition floorC x := sval (floorC_subproof x).
Definition Cint : {pred algC} := fun x(floorC x)%:~R == x.

Definition truncC x := if x 0 then |floorC x|%N else 0%N.
Definition Cnat : {pred algC} := fun x(truncC x)%:R == x.

Definition minCpoly x : {poly algC} :=
let: exist2 p _ _ := minCpoly_subproof x in map_poly ratr p.

Coercion nat_divisor : nat >-> divisor.
Coercion int_divisor : int >-> divisor.
Coercion algC_divisor : algC >-> divisor.

Lemma nCdivE (p : nat) : p = p%:R :> divisor.
Lemma zCdivE (p : int) : p = p%:~R :> divisor.
Definition CdivE := (nCdivE, zCdivE).

Definition dvdC (x : divisor) : {pred algC} :=
fun yif x == 0 then y == 0 else y / x \in Cint.
Notation "x %| y" := (y \in dvdC x) : C_expanded_scope.
Notation "x %| y" := (@in_mem divisor y (mem (dvdC x))) : C_scope.

Definition eqCmod (e x y : divisor) := (e %| x - y)%C.

Notation "x == y %[mod e ]" := (eqCmod e x y) : C_scope.
Notation "x != y %[mod e ]" := (~~ (x == y %[mod e])%C) : C_scope.

End Exports.

Module HBExports. End HBExports.

End Algebraics.

Export Algebraics.Exports.

Export Algebraics.HBExports.

Section AlgebraicsTheory.

Implicit Types (x y z : algC) (n : nat) (m : int) (b : bool).
Import Algebraics.Internals.

Let intr_inj_ZtoC := (intr_inj : injective ZtoC).
#[local] Hint Resolve intr_inj_ZtoC : core.

Specialization of a few basic ssrnum order lemmas.

Definition eqC_nat n p : (n%:R == p%:R :> algC) = (n == p) := eqr_nat _ n p.
Definition leC_nat n p : (n%:R p%:R :> algC) = (n p)%N := ler_nat _ n p.
Definition ltC_nat n p : (n%:R < p%:R :> algC) = (n < p)%N := ltr_nat _ n p.
Definition Cchar : [char algC] =i pred0 := @char_num _.

This can be used in the converse direction to evaluate assertions over manifest rationals, such as 3^-1 + 7%:%^-1 < 2%:%^-1 :> algC. Missing norm and integer exponent, due to gaps in ssrint and rat.
Conjugation and norm.
Real number subset.

Lemma Creal0 : 0 \is Creal.
Lemma Creal1 : 1 \is Creal.
Trivial cannot resolve a general real0 hint.
Hint Resolve Creal0 Creal1 : core.

Lemma algCrect x : x = 'Re x + 'i × 'Im x.

Lemma algCreal_Re x : 'Re x \is Creal.

Lemma algCreal_Im x : 'Im x \is Creal.
Hint Resolve algCreal_Re algCreal_Im : core.

Integer subset. Not relying on the undocumented interval library, for now.

Lemma floorC_itv x : x \is Creal (floorC x)%:~R x < (floorC x + 1)%:~R.

Lemma floorC_def x m : m%:~R x < (m + 1)%:~R floorC x = m.

Lemma intCK : cancel intr floorC.

Lemma floorCK : {in Cint, cancel floorC intr}.

Lemma floorC0 : floorC 0 = 0.
Lemma floorC1 : floorC 1 = 1.
Hint Resolve floorC0 floorC1 : core.

Lemma floorCpK (p : {poly algC}) :
p \is a polyOver Cint map_poly intr (map_poly floorC p) = p.

Lemma floorCpP (p : {poly algC}) :
p \is a polyOver Cint {q | p = map_poly intr q}.

Lemma Cint_int m : m%:~R \in Cint.

Lemma CintP x : reflect ( m, x = m%:~R) (x \in Cint).

Lemma floorCD : {in Cint & Creal, {morph floorC : x y / x + y}}.

Lemma floorCN : {in Cint, {morph floorC : x / - x}}.

Lemma floorCM : {in Cint &, {morph floorC : x y / x × y}}.

Lemma floorCX n : {in Cint, {morph floorC : x / x ^+ n}}.

Lemma rpred_Cint (S : subringClosed algC) x : x \in Cint x \in S.

Lemma Cint0 : 0 \in Cint.
Lemma Cint1 : 1 \in Cint.
Hint Resolve Cint0 Cint1 : core.

Fact Cint_subring : subring_closed Cint.

Lemma Creal_Cint : {subset Cint Creal}.

Lemma conj_Cint x : x \in Cint x^* = x.

Lemma Cint_normK x : x \in Cint |x| ^+ 2 = x ^+ 2.

Lemma CintEsign x : x \in Cint x = (-1) ^+ (x < 0)%C × |x|.

Natural integer subset.

Lemma truncC_itv x : 0 x (truncC x)%:R x < (truncC x).+1%:R.

Lemma truncC_def x n : n%:R x < n.+1%:R truncC x = n.

Lemma natCK n : truncC n%:R = n.

Lemma CnatP x : reflect ( n, x = n%:R) (x \in Cnat).

Lemma truncCK : {in Cnat, cancel truncC (GRing.natmul 1)}.

Lemma truncC_gt0 x : (0 < truncC x)%N = (1 x).

Lemma truncC0Pn x : reflect (truncC x = 0%N) (~~ (1 x)).

Lemma truncC0 : truncC 0 = 0%N.
Lemma truncC1 : truncC 1 = 1%N.

Lemma truncCD :
{in Cnat & Num.nneg, {morph truncC : x y / x + y >-> (x + y)%N}}.

Lemma truncCM : {in Cnat &, {morph truncC : x y / x × y >-> (x × y)%N}}.

Lemma truncCX n : {in Cnat, {morph truncC : x / x ^+ n >-> (x ^ n)%N}}.

Lemma rpred_Cnat (S : semiringClosed algC) x : x \in Cnat x \in S.

Lemma Cnat_nat n : n%:R \in Cnat.
Lemma Cnat0 : 0 \in Cnat.
Lemma Cnat1 : 1 \in Cnat.
Hint Resolve Cnat_nat Cnat0 Cnat1 : core.

Fact Cnat_semiring : semiring_closed Cnat.

Lemma Cnat_ge0 x : x \in Cnat 0 x.

Lemma Cnat_gt0 x : x \in Cnat (0 < x) = (x != 0).

Lemma conj_Cnat x : x \in Cnat x^* = x.

Lemma norm_Cnat x : x \in Cnat |x| = x.

Lemma Creal_Cnat : {subset Cnat Creal}.

Lemma Cnat_sum_eq1 (I : finType) (P : pred I) (F : I algC) :
( i, P i F i \in Cnat) \sum_(i | P i) F i = 1
{i : I | [/\ P i, F i = 1 & j, j != i P j F j = 0]}.

Lemma Cnat_mul_eq1 x y :
x \in Cnat y \in Cnat (x × y == 1) = (x == 1) && (y == 1).

Lemma Cnat_prod_eq1 (I : finType) (P : pred I) (F : I algC) :
( i, P i F i \in Cnat) \prod_(i | P i) F i = 1
i, P i F i = 1.

Relating Cint and Cnat.
Integer divisibility.

Lemma dvdCP x y : reflect (exists2 z, z \in Cint & y = z × x) (x %| y)%C.

Lemma dvdCP_nat x y : 0 x 0 y (x %| y)%C {n | y = n%:R × x}.

Lemma dvdC0 x : (x %| 0)%C.

Lemma dvd0C x : (0 %| x)%C = (x == 0).

Lemma dvdC_mull x y z : y \in Cint (x %| z)%C (x %| y × z)%C.

Lemma dvdC_mulr x y z : y \in Cint (x %| z)%C (x %| z × y)%C.

Lemma dvdC_mul2r x y z : y != 0 (x × y %| z × y)%C = (x %| z)%C.

Lemma dvdC_mul2l x y z : y != 0 (y × x %| y × z)%C = (x %| z)%C.

Lemma dvdC_trans x y z : (x %| y)%C (y %| z)%C (x %| z)%C.

Lemma dvdC_refl x : (x %| x)%C.
Hint Resolve dvdC_refl : core.

Lemma dvdC_zmod x : zmod_closed (dvdC x).

Lemma dvdC_nat (p n : nat) : (p %| n)%C = (p %| n)%N.

Lemma dvdC_int (p : nat) x : x \in Cint (p %| x)%C = (p %| |floorC x|)%N.

Elementary modular arithmetic.

Lemma eqCmod_refl e x : (x == x %[mod e])%C.

Lemma eqCmodm0 e : (e == 0 %[mod e])%C.
Hint Resolve eqCmod_refl eqCmodm0 : core.

Lemma eqCmod0 e x : (x == 0 %[mod e])%C = (e %| x)%C.

Lemma eqCmod_sym e x y : ((x == y %[mod e]) = (y == x %[mod e]))%C.

Lemma eqCmod_trans e y x z :
(x == y %[mod e] y == z %[mod e] x == z %[mod e])%C.

Lemma eqCmod_transl e x y z :
(x == y %[mod e])%C (x == z %[mod e])%C = (y == z %[mod e])%C.

Lemma eqCmod_transr e x y z :
(x == y %[mod e])%C (z == x %[mod e])%C = (z == y %[mod e])%C.

Lemma eqCmodN e x y : (- x == y %[mod e])%C = (x == - y %[mod e])%C.

Lemma eqCmodDr e x y z : (y + x == z + x %[mod e])%C = (y == z %[mod e])%C.

Lemma eqCmodDl e x y z : (x + y == x + z %[mod e])%C = (y == z %[mod e])%C.

Lemma eqCmodD e x1 x2 y1 y2 :
(x1 == x2 %[mod e] y1 == y2 %[mod e] x1 + y1 == x2 + y2 %[mod e])%C.

Lemma eqCmod_nat (e m n : nat) : (m == n %[mod e])%C = (m == n %[mod e]).

Lemma eqCmod0_nat (e m : nat) : (m == 0 %[mod e])%C = (e %| m)%N.

Lemma eqCmodMr e :
{in Cint, z x y, x == y %[mod e] x × z == y × z %[mod e]}%C.

Lemma eqCmodMl e :
{in Cint, z x y, x == y %[mod e] z × x == z × y %[mod e]}%C.

Lemma eqCmodMl0 e : {in Cint, x, x × e == 0 %[mod e]}%C.

Lemma eqCmodMr0 e : {in Cint, x, e × x == 0 %[mod e]}%C.

Lemma eqCmod_addl_mul e : {in Cint, x y, x × e + y == y %[mod e]}%C.

Lemma eqCmodM e : {in Cint & Cint, x1 y2 x2 y1,
x1 == x2 %[mod e] y1 == y2 %[mod e] x1 × y1 == x2 × y2 %[mod e]}%C.

Rational number subset.

Lemma ratCK : cancel QtoC CtoQ.

Lemma getCratK : {in Crat, cancel CtoQ QtoC}.

Lemma Crat_rat (a : rat) : QtoC a \in Crat.

Lemma CratP x : reflect ( a, x = QtoC a) (x \in Crat).

Lemma Crat0 : 0 \in Crat.
Lemma Crat1 : 1 \in Crat.
Hint Resolve Crat0 Crat1 : core.

Fact Crat_divring_closed : divring_closed Crat.

Lemma rpred_Crat (S : divringClosed algC) : {subset Crat S}.

Lemma conj_Crat z : z \in Crat z^* = z.

Lemma Creal_Crat : {subset Crat Creal}.

Lemma Cint_rat a : (QtoC a \in Cint) = (a \in Qint).

Lemma minCpolyP x :
{p | minCpoly x = pQtoC p p \is monic
& q, root (pQtoC q) x = (p %| q)%R}.

Lemma minCpoly_monic x : minCpoly x \is monic.

Lemma minCpoly_eq0 x : (minCpoly x == 0) = false.

Lemma root_minCpoly x : root (minCpoly x) x.

Lemma size_minCpoly x : (1 < size (minCpoly x))%N.

Basic properties of automorphisms.
Section AutC.

Implicit Type nu : {rmorphism algC algC}.

Lemma aut_Cnat nu : {in Cnat, nu =1 id}.

Lemma aut_Cint nu : {in Cint, nu =1 id}.

Lemma aut_Crat nu : {in Crat, nu =1 id}.

Lemma Cnat_aut nu x : (nu x \in Cnat) = (x \in Cnat).

Lemma Cint_aut nu x : (nu x \in Cint) = (x \in Cint).

Lemma Crat_aut nu x : (nu x \in Crat) = (x \in Crat).

Lemma algC_invaut_subproof nu x : {y | nu y = x}.
Definition algC_invaut nu x := sval (algC_invaut_subproof nu x).

Lemma algC_invautK nu : cancel (algC_invaut nu) nu.

Lemma algC_autK nu : cancel nu (algC_invaut nu).

Fact algC_invaut_is_rmorphism nu : multiplicative (algC_invaut nu).

Lemma minCpoly_aut nu x : minCpoly (nu x) = minCpoly x.

End AutC.

Section AutLmodC.

Variables (U V : lmodType algC) (f : {additive U V}).

Lemma raddfZ_Cnat a u : a \in Cnat f (a *: u) = a *: f u.

Lemma raddfZ_Cint a u : a \in Cint f (a *: u) = a *: f u.

End AutLmodC.

Section PredCmod.

Variable V : lmodType algC.

Lemma rpredZ_Cnat (S : addrClosed V) : {in Cnat & S, z u, z *: u \in S}.

Lemma rpredZ_Cint (S : zmodClosed V) : {in Cint & S, z u, z *: u \in S}.

End PredCmod.

End AlgebraicsTheory.
#[global] Hint Resolve Creal0 Creal1 Cnat_nat Cnat0 Cnat1 Cint0 Cint1 : core.
#[global] Hint Resolve floorC0 Crat0 Crat1 : core.
#[global] Hint Resolve dvdC0 dvdC_refl eqCmod_refl eqCmodm0 : core.