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From Coq Require Import ssreflect ssrfun ssrbool ZArith QArith Qcanon.
From HB Require Import structures.
Require Import classical.
Declare Scope hb_scope.
Delimit Scope hb_scope with G.
Local Open Scope classical_set_scope.
Local Open Scope hb_scope.
Module Stage10.
HB.mixin Record AddAG_of_TYPE A := {
zero : A;
add : A -> A -> A;
opp : A -> A;
addrA : associative add;
addrC : commutative add;
add0r : left_id zero add;
addNr : left_inverse zero opp add;
}.
HB.structure Definition AddAG := { A of AddAG_of_TYPE A }.
Notation "0" := zero : hb_scope.
Infix "+" := (@add _) : hb_scope.
Notation "- x" := (@opp _ x) : hb_scope.
Notation "x - y" := (x + - y) : hb_scope.
(* Theory *)
Section AddAGTheory.
Variable A : AddAG.type.
Implicit Type (x : A).
Lemma addr0 : right_id (@zero A) add.
Proof. by move=> x; rewrite addrC add0r. Qed.
Lemma addrN : right_inverse (@zero A) opp add.
Proof. by move=> x; rewrite addrC addNr. Qed.
Lemma subrr x : x - x = 0.
Proof. by rewrite addrN. Qed.
Lemma addrK : right_loop (@opp A) (@add A).
Proof. by move=> x y; rewrite -addrA subrr addr0. Qed.
Lemma addKr : left_loop (@opp A) (@add A).
Proof. by move=> x y; rewrite addrA addNr add0r. Qed.
Lemma addrNK : rev_right_loop (@opp A) (@add A).
Proof. by move=> y x; rewrite -addrA addNr addr0. Qed.
Lemma addNKr : rev_left_loop (@opp A) (@add A).
Proof. by move=> x y; rewrite addrA subrr add0r. Qed.
Lemma addrAC : right_commutative (@add A).
Proof. by move=> x y z; rewrite -!addrA [y + z]addrC. Qed.
Lemma addrCA : left_commutative (@add A).
Proof. by move=> x y z; rewrite !addrA [x + y]addrC. Qed.
Lemma addrACA : interchange (@add A) add.
Proof. by move=> x y z t; rewrite !addrA [x + y + z]addrAC. Qed.
Lemma opprK : involutive (@opp A).
Proof. by move=> x; apply: (can_inj (addrK (- x))); rewrite addNr addrN. Qed.
Lemma opprD x y : - (x + y) = - x - y.
Proof.
apply: (can_inj (addKr (x + y))).
by rewrite subrr addrACA !subrr addr0.
Qed.
Lemma opprB x y : - (x - y) = y - x.
Proof. by rewrite opprD opprK addrC. Qed.
End AddAGTheory.
HB.mixin Record Ring_of_AddAG A of AddAG A := {
one : A;
mul : A -> A -> A;
mulrA : associative mul;
mulr1 : left_id one mul;
mul1r : right_id one mul;
mulrDl : left_distributive mul add;
mulrDr : right_distributive mul add;
}.
HB.factory Record Ring_of_TYPE A := {
zero : A;
one : A;
add : A -> A -> A;
opp : A -> A;
mul : A -> A -> A;
addrA : associative add;
addrC : commutative add;
add0r : left_id zero add;
addNr : left_inverse zero opp add;
mulrA : associative mul;
mul1r : left_id one mul;
mulr1 : right_id one mul;
mulrDl : left_distributive mul add;
mulrDr : right_distributive mul add;
}.
HB.builders Context A (a : Ring_of_TYPE A).
HB.instance
Definition to_AddAG_of_TYPE := AddAG_of_TYPE.Build A
_ _ _ addrA addrC add0r addNr.
HB.instance
Definition to_Ring_of_AddAG :=
Ring_of_AddAG.Build A _ _ mulrA mul1r
mulr1 mulrDl mulrDr.
HB.end.
HB.structure Definition Ring := { A of Ring_of_TYPE A }.
Notation "1" := one : hb_scope.
Infix "*" := (@mul _) : hb_scope.
HB.mixin Record Topological T := {
open : (T -> Prop) -> Prop;
open_setT : open setT;
open_bigcup : forall {I} (D : set I) (F : I -> set T),
(forall i, D i -> open (F i)) -> open (\bigcup_(i in D) F i);
open_setI : forall X Y : set T, open X -> open Y -> open (setI X Y);
}.
HB.structure Definition TopologicalSpace := { A of Topological A }.
#[export] Hint Extern 0 (open setT) => now apply: open_setT : core.
HB.factory Record TopologicalBase T := {
open_base : set (set T);
open_base_covers : setT `<=` \bigcup_(X in open_base) X;
open_base_cup : forall X Y : set T, open_base X -> open_base Y ->
forall z, (X `&` Y) z -> exists2 Z, open_base Z & Z z /\ Z `<=` X `&` Y
}.
HB.builders Context T (a : TopologicalBase T).
Definition open_of : (T -> Prop) -> Prop :=
[set A | exists2 D, D `<=` open_base & A = \bigcup_(X in D) X].
Lemma open_of_setT : open_of setT. Proof.
exists open_base; rewrite // predeqE => x; split=> // _.
by apply: open_base_covers.
Qed.
Lemma open_of_bigcup {I} (D : set I) (F : I -> set T) :
(forall i, D i -> open_of (F i)) -> open_of (\bigcup_(i in D) F i).
Proof. Admitted.
Lemma open_of_cap X Y : open_of X -> open_of Y -> open_of (X `&` Y).
Proof. Admitted.
HB.instance
Definition to_Topological :=
Topological.Build T _ open_of_setT (@open_of_bigcup) open_of_cap.
HB.end.
Section ProductTopology.
Variables (T1 T2 : TopologicalSpace.type).
Definition prod_open_base :=
[set A | exists (A1 : set T1) (A2 : set T2),
open A1 /\ open A2 /\ A = setM A1 A2].
Lemma prod_open_base_covers : setT `<=` \bigcup_(X in prod_open_base) X.
Proof.
move=> X _; exists setT => //; exists setT, setT; do ?split.
- exact: open_setT.
- exact: open_setT.
- by rewrite predeqE.
Qed.
Lemma prod_open_base_setU X Y :
prod_open_base X -> prod_open_base Y ->
forall z, (X `&` Y) z ->
exists2 Z, prod_open_base Z & Z z /\ Z `<=` X `&` Y.
Proof.
move=> [A1 [A2 [A1open [A2open ->]]]] [B1 [B2 [B1open [B2open ->]]]].
move=> [z1 z2] [[/=Az1 Az2] [/= Bz1 Bz2]].
exists ((A1 `&` B1) `*` (A2 `&` B2)).
by eexists _, _; do ?[split; last first]; apply: open_setI.
by split => // [[x1 x2] [[/=Ax1 Bx1] [/=Ax2 Bx2]]].
Qed.
HB.instance Definition prod_topology :=
TopologicalBase.Build (T1 * T2)%type _ prod_open_base_covers prod_open_base_setU.
End ProductTopology.
Definition continuous {T T' : TopologicalSpace.type} (f : T -> T') :=
forall B : set T', open B -> open (f@^-1` B).
Definition continuous2 {T T' T'': TopologicalSpace.type}
(f : T -> T' -> T'') := continuous (fun xy => f xy.1 xy.2).
HB.mixin Record JoinTAddAG T of AddAG_of_TYPE T & Topological T := {
add_continuous : continuous2 (add : T -> T -> T);
opp_continuous : continuous (opp : T -> T)
}.
HB.structure Definition TAddAG := { A of Topological A & AddAG_of_TYPE A & JoinTAddAG A }.
(* Instance *)
HB.instance Definition Z_ring_axioms :=
Ring_of_TYPE.Build Z 0%Z 1%Z Z.add Z.opp Z.mul
Z.add_assoc Z.add_comm Z.add_0_l Z.add_opp_diag_l
Z.mul_assoc Z.mul_1_l Z.mul_1_r
Z.mul_add_distr_r Z.mul_add_distr_l.
Example test1 (m n : Z) : (m + n) - n + 0 = m.
Proof. by rewrite addrK addr0. Qed.
Import Qcanon.
Search _ Qc "plus" "opp".
Lemma Qcplus_opp_l q : - q + q = 0.
Proof. by rewrite Qcplus_comm Qcplus_opp_r. Qed.
HB.instance Definition Qc_ring_axioms :=
Ring_of_TYPE.Build Qc 0%Qc 1%Qc Qcplus Qcopp Qcmult
Qcplus_assoc Qcplus_comm Qcplus_0_l Qcplus_opp_l
Qcmult_assoc Qcmult_1_l Qcmult_1_r
Qcmult_plus_distr_l Qcmult_plus_distr_r.
Obligation Tactic := idtac.
Definition Qcopen_base : set (set Qc) :=
[set A | exists a b : Qc, forall z, A z <-> a < z /\ z < b].
Program Definition QcTopological := TopologicalBase.Build Qc Qcopen_base _ _.
Next Obligation.
move=> x _; exists [set y | x - 1 < y < x + 1].
by exists (x - 1), (x + 1).
split; rewrite Qclt_minus_iff.
by rewrite -[_ + _]/(x - (x - 1))%G opprB addrCA subrr.
by rewrite -[_ + _]/(x + 1 - x)%G addrAC subrr.
Qed.
Next Obligation.
move=> X Y [aX [bX Xeq]] [aY [bY Yeq]] z [/Xeq [aXz zbX] /Yeq [aYz zbY]].
Admitted.
HB.instance Definition _ : TopologicalBase Qc := QcTopological.
Program Definition QcJoinTAddAG := JoinTAddAG.Build Qc _ _.
Next Obligation. Admitted.
Next Obligation. Admitted.
HB.instance Definition _ : JoinTAddAG Qc := QcJoinTAddAG.
End Stage10.
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