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(************************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2010 *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(************************************************************************)
(* non commutative rings *)
Require Import Setoid.
Require Import BinPos.
Require Import BinNat.
Require Export Morphisms Setoid Bool.
Require Export Algebra_syntax.
Set Implicit Arguments.
Class Ring (R:Type) := {
ring0: R; ring1: R;
ring_plus: R->R->R; ring_mult: R->R->R;
ring_sub: R->R->R; ring_opp: R->R;
ring_eq : R -> R -> Prop;
ring_setoid: Equivalence ring_eq;
ring_plus_comp: Proper (ring_eq==>ring_eq==>ring_eq) ring_plus;
ring_mult_comp: Proper (ring_eq==>ring_eq==>ring_eq) ring_mult;
ring_sub_comp: Proper (ring_eq==>ring_eq==>ring_eq) ring_sub;
ring_opp_comp: Proper (ring_eq==>ring_eq) ring_opp;
ring_add_0_l : forall x, ring_eq (ring_plus ring0 x) x;
ring_add_comm : forall x y, ring_eq (ring_plus x y) (ring_plus y x);
ring_add_assoc : forall x y z, ring_eq (ring_plus x (ring_plus y z))
(ring_plus (ring_plus x y) z);
ring_mul_1_l : forall x, ring_eq (ring_mult ring1 x) x;
ring_mul_1_r : forall x, ring_eq (ring_mult x ring1) x;
ring_mul_assoc : forall x y z, ring_eq (ring_mult x (ring_mult y z))
(ring_mult (ring_mult x y) z);
ring_distr_l : forall x y z, ring_eq (ring_mult (ring_plus x y) z)
(ring_plus (ring_mult x z) (ring_mult y z));
ring_distr_r : forall x y z, ring_eq (ring_mult z (ring_plus x y))
(ring_plus (ring_mult z x) (ring_mult z y));
ring_sub_def : forall x y, ring_eq (ring_sub x y) (ring_plus x (ring_opp y));
ring_opp_def : forall x, ring_eq (ring_plus x (ring_opp x)) ring0
}.
Instance zero_ring (R:Type)(Rr:Ring R) : Zero R := {zero := ring0}.
Instance one_ring(R:Type)(Rr:Ring R) : One R := {one := ring1}.
Instance addition_ring(R:Type)(Rr:Ring R) : Addition R :=
{addition x y := ring_plus x y}.
Instance multiplication_ring(R:Type)(Rr:Ring R) : Multiplication:=
{multiplication x y := ring_mult x y}.
Instance subtraction_ring(R:Type)(Rr:Ring R) : Subtraction R :=
{subtraction x y := ring_sub x y}.
Instance opposite_ring(R:Type)(Rr:Ring R) : Opposite R :=
{opposite x := ring_opp x}.
Instance equality_ring(R:Type)(Rr:Ring R) : Equality :=
{equality x y := ring_eq x y}.
Existing Instance ring_setoid.
Existing Instance ring_plus_comp.
Existing Instance ring_mult_comp.
Existing Instance ring_sub_comp.
Existing Instance ring_opp_comp.
(** Interpretation morphisms definition*)
Class Ring_morphism (C R:Type)`{Cr:Ring C} `{Rr:Ring R}:= {
ring_morphism_fun: C -> R;
ring_morphism0 : ring_morphism_fun 0 == 0;
ring_morphism1 : ring_morphism_fun 1 == 1;
ring_morphism_add : forall x y, ring_morphism_fun (x + y)
== ring_morphism_fun x + ring_morphism_fun y;
ring_morphism_sub : forall x y, ring_morphism_fun (x - y)
== ring_morphism_fun x - ring_morphism_fun y;
ring_morphism_mul : forall x y, ring_morphism_fun (x * y)
== ring_morphism_fun x * ring_morphism_fun y;
ring_morphism_opp : forall x, ring_morphism_fun (-x)
== -(ring_morphism_fun x);
ring_morphism_eq : forall x y, x == y
-> ring_morphism_fun x == ring_morphism_fun y}.
Instance bracket_ring (C R:Type)`{Cr:Ring C} `{Rr:Ring R}
`{phi:@Ring_morphism C R Cr Rr}
: Bracket C R :=
{bracket x := ring_morphism_fun x}.
(* Tactics for rings *)
Lemma ring_syntax1:forall (A:Type)(Ar:Ring A), (@ring_eq _ Ar) = equality.
intros. symmetry. simpl; reflexivity. Qed.
Lemma ring_syntax2:forall (A:Type)(Ar:Ring A), (@ring_plus _ Ar) = addition.
intros. symmetry. simpl; reflexivity. Qed.
Lemma ring_syntax3:forall (A:Type)(Ar:Ring A), (@ring_mult _ Ar) = multiplication.
intros. symmetry. simpl; reflexivity. Qed.
Lemma ring_syntax4:forall (A:Type)(Ar:Ring A), (@ring_sub _ Ar) = subtraction.
intros. symmetry. simpl; reflexivity. Qed.
Lemma ring_syntax5:forall (A:Type)(Ar:Ring A), (@ring_opp _ Ar) = opposite.
intros. symmetry. simpl; reflexivity. Qed.
Lemma ring_syntax6:forall (A:Type)(Ar:Ring A), (@ring0 _ Ar) = zero.
intros. symmetry. simpl; reflexivity. Qed.
Lemma ring_syntax7:forall (A:Type)(Ar:Ring A), (@ring1 _ Ar) = one.
intros. symmetry. simpl; reflexivity. Qed.
Lemma ring_syntax8:forall (A:Type)(Ar:Ring A)(B:Type)(Br:Ring B)
(pM:@Ring_morphism A B Ar Br), (@ring_morphism_fun _ _ _ _ pM) = bracket.
intros. symmetry. simpl; reflexivity. Qed.
Ltac set_ring_notations :=
repeat (rewrite ring_syntax1);
repeat (rewrite ring_syntax2);
repeat (rewrite ring_syntax3);
repeat (rewrite ring_syntax4);
repeat (rewrite ring_syntax5);
repeat (rewrite ring_syntax6);
repeat (rewrite ring_syntax7);
repeat (rewrite ring_syntax8).
Ltac unset_ring_notations :=
unfold equality, equality_ring, addition, addition_ring,
multiplication, multiplication_ring, subtraction, subtraction_ring,
opposite, opposite_ring, one, one_ring, zero, zero_ring,
bracket, bracket_ring.
Ltac ring_simpl := simpl; set_ring_notations.
Ltac ring_rewrite H:=
generalize H;
let h := fresh "H" in
unset_ring_notations; intro h;
rewrite h; clear h;
set_ring_notations.
Ltac ring_rewrite_rev H:=
generalize H;
let h := fresh "H" in
unset_ring_notations; intro h;
rewrite <- h; clear h;
set_ring_notations.
Ltac rrefl := unset_ring_notations; reflexivity.
Section Ring.
Variable R: Type.
Variable Rr: Ring R.
(* Powers *)
Fixpoint pow_pos (x:R) (i:positive) {struct i}: R :=
match i with
| xH => x
| xO i => let p := pow_pos x i in p * p
| xI i => let p := pow_pos x i in x * (p * p)
end.
Add Setoid R ring_eq ring_setoid as R_set_Power.
Add Morphism ring_mult : rmul_ext_Power. exact ring_mult_comp. Qed.
Lemma pow_pos_comm : forall x j, x * pow_pos x j == pow_pos x j * x.
induction j; ring_simpl.
ring_rewrite_rev ring_mul_assoc. ring_rewrite_rev ring_mul_assoc.
ring_rewrite_rev IHj. ring_rewrite (ring_mul_assoc (pow_pos x j) x (pow_pos x j)).
ring_rewrite_rev IHj. ring_rewrite_rev ring_mul_assoc. rrefl.
ring_rewrite_rev ring_mul_assoc. ring_rewrite_rev IHj.
ring_rewrite ring_mul_assoc. ring_rewrite IHj.
ring_rewrite_rev ring_mul_assoc. ring_rewrite IHj. rrefl. rrefl.
Qed.
Lemma pow_pos_Psucc : forall x j, pow_pos x (Psucc j) == x * pow_pos x j.
Proof.
induction j; ring_simpl.
ring_rewrite IHj.
ring_rewrite_rev (ring_mul_assoc x (pow_pos x j) (x * pow_pos x j)).
ring_rewrite (ring_mul_assoc (pow_pos x j) x (pow_pos x j)).
ring_rewrite_rev pow_pos_comm. unset_ring_notations.
rewrite <- ring_mul_assoc. reflexivity.
rrefl. rrefl.
Qed.
Lemma pow_pos_Pplus : forall x i j, pow_pos x (i + j) == pow_pos x i * pow_pos x j.
Proof.
intro x;induction i;intros.
rewrite xI_succ_xO;rewrite Pplus_one_succ_r.
rewrite <- Pplus_diag;repeat rewrite <- Pplus_assoc.
repeat ring_rewrite IHi.
rewrite Pplus_comm;rewrite <- Pplus_one_succ_r;
ring_rewrite pow_pos_Psucc.
ring_simpl;repeat ring_rewrite ring_mul_assoc. rrefl.
rewrite <- Pplus_diag;repeat rewrite <- Pplus_assoc.
repeat ring_rewrite IHi. ring_rewrite ring_mul_assoc. rrefl.
rewrite Pplus_comm;rewrite <- Pplus_one_succ_r;ring_rewrite pow_pos_Psucc.
simpl. reflexivity.
Qed.
Definition pow_N (x:R) (p:N) :=
match p with
| N0 => 1
| Npos p => pow_pos x p
end.
Definition id_phi_N (x:N) : N := x.
Lemma pow_N_pow_N : forall x n, pow_N x (id_phi_N n) == pow_N x n.
Proof.
intros; rrefl.
Qed.
End Ring.
Section Ring2.
Variable R: Type.
Variable Rr: Ring R.
(** Identity is a morphism *)
Definition IDphi (x:R) := x.
Lemma IDmorph : @Ring_morphism R R Rr Rr.
Proof.
apply (Build_Ring_morphism Rr Rr IDphi);intros;unfold IDphi; try rrefl. trivial.
Qed.
Ltac ring_replace a b :=
unset_ring_notations; setoid_replace a with b; set_ring_notations.
(** rings are almost rings*)
Lemma ring_mul_0_l : forall x, 0 * x == 0.
Proof.
intro x. ring_replace (0*x) ((0+1)*x + -x).
ring_rewrite ring_add_0_l. ring_rewrite ring_mul_1_l .
ring_rewrite ring_opp_def ;rrefl.
ring_rewrite ring_distr_l ;ring_rewrite ring_mul_1_l .
ring_rewrite_rev ring_add_assoc ; ring_rewrite ring_opp_def .
ring_rewrite ring_add_comm ; ring_rewrite ring_add_0_l ;rrefl.
Qed.
Lemma ring_mul_0_r : forall x, x * 0 == 0.
Proof.
intro x; ring_replace (x*0) (x*(0+1) + -x).
ring_rewrite ring_add_0_l ; ring_rewrite ring_mul_1_r .
ring_rewrite ring_opp_def ;rrefl.
ring_rewrite ring_distr_r ;ring_rewrite ring_mul_1_r .
ring_rewrite_rev ring_add_assoc ; ring_rewrite ring_opp_def .
ring_rewrite ring_add_comm ; ring_rewrite ring_add_0_l ;rrefl.
Qed.
Lemma ring_opp_mul_l : forall x y, -(x * y) == -x * y.
Proof.
intros x y;ring_rewrite_rev (ring_add_0_l (- x * y)).
ring_rewrite ring_add_comm .
ring_rewrite_rev (ring_opp_def (x*y)).
ring_rewrite ring_add_assoc .
ring_rewrite_rev ring_distr_l.
ring_rewrite (ring_add_comm (-x));ring_rewrite ring_opp_def .
ring_rewrite ring_mul_0_l;ring_rewrite ring_add_0_l ;rrefl.
Qed.
Lemma ring_opp_mul_r : forall x y, -(x * y) == x * -y.
Proof.
intros x y;ring_rewrite_rev (ring_add_0_l (x * - y)).
ring_rewrite ring_add_comm .
ring_rewrite_rev (ring_opp_def (x*y)).
ring_rewrite ring_add_assoc .
ring_rewrite_rev ring_distr_r .
ring_rewrite (ring_add_comm (-y));ring_rewrite ring_opp_def .
ring_rewrite ring_mul_0_r;ring_rewrite ring_add_0_l ;rrefl.
Qed.
Lemma ring_opp_add : forall x y, -(x + y) == -x + -y.
Proof.
intros x y;ring_rewrite_rev (ring_add_0_l (-(x+y))).
ring_rewrite_rev (ring_opp_def x).
ring_rewrite_rev (ring_add_0_l (x + - x + - (x + y))).
ring_rewrite_rev (ring_opp_def y).
ring_rewrite (ring_add_comm x).
ring_rewrite (ring_add_comm y).
ring_rewrite_rev (ring_add_assoc (-y)).
ring_rewrite_rev (ring_add_assoc (- x)).
ring_rewrite (ring_add_assoc y).
ring_rewrite (ring_add_comm y).
ring_rewrite_rev (ring_add_assoc (- x)).
ring_rewrite (ring_add_assoc y).
ring_rewrite (ring_add_comm y);ring_rewrite ring_opp_def .
ring_rewrite (ring_add_comm (-x) 0);ring_rewrite ring_add_0_l .
ring_rewrite ring_add_comm; rrefl.
Qed.
Lemma ring_opp_opp : forall x, - -x == x.
Proof.
intros x; ring_rewrite_rev (ring_add_0_l (- -x)).
ring_rewrite_rev (ring_opp_def x).
ring_rewrite_rev ring_add_assoc ; ring_rewrite ring_opp_def .
ring_rewrite (ring_add_comm x); ring_rewrite ring_add_0_l . rrefl.
Qed.
Lemma ring_sub_ext :
forall x1 x2, x1 == x2 -> forall y1 y2, y1 == y2 -> x1 - y1 == x2 - y2.
Proof.
intros.
ring_replace (x1 - y1) (x1 + -y1).
ring_replace (x2 - y2) (x2 + -y2).
ring_rewrite H;ring_rewrite H0;rrefl.
ring_rewrite ring_sub_def. rrefl.
ring_rewrite ring_sub_def. rrefl.
Qed.
Ltac mring_rewrite :=
repeat first
[ ring_rewrite ring_add_0_l
| ring_rewrite_rev (ring_add_comm 0)
| ring_rewrite ring_mul_1_l
| ring_rewrite ring_mul_0_l
| ring_rewrite ring_distr_l
| rrefl
].
Lemma ring_add_0_r : forall x, (x + 0) == x.
Proof. intros; mring_rewrite. Qed.
Lemma ring_add_assoc1 : forall x y z, (x + y) + z == (y + z) + x.
Proof.
intros;ring_rewrite_rev (ring_add_assoc x).
ring_rewrite (ring_add_comm x);rrefl.
Qed.
Lemma ring_add_assoc2 : forall x y z, (y + x) + z == (y + z) + x.
Proof.
intros; repeat ring_rewrite_rev ring_add_assoc.
ring_rewrite (ring_add_comm x); rrefl.
Qed.
Lemma ring_opp_zero : -0 == 0.
Proof.
ring_rewrite_rev (ring_mul_0_r 0). ring_rewrite ring_opp_mul_l.
repeat ring_rewrite ring_mul_0_r. rrefl.
Qed.
End Ring2.
(** Some simplification tactics*)
Ltac gen_reflexivity := rrefl.
Ltac gen_ring_rewrite :=
repeat first
[ rrefl
| progress ring_rewrite ring_opp_zero
| ring_rewrite ring_add_0_l
| ring_rewrite ring_add_0_r
| ring_rewrite ring_mul_1_l
| ring_rewrite ring_mul_1_r
| ring_rewrite ring_mul_0_l
| ring_rewrite ring_mul_0_r
| ring_rewrite ring_distr_l
| ring_rewrite ring_distr_r
| ring_rewrite ring_add_assoc
| ring_rewrite ring_mul_assoc
| progress ring_rewrite ring_opp_add
| progress ring_rewrite ring_sub_def
| progress ring_rewrite_rev ring_opp_mul_l
| progress ring_rewrite_rev ring_opp_mul_r ].
Ltac gen_add_push x :=
set_ring_notations;
repeat (match goal with
| |- context [(?y + x) + ?z] =>
progress ring_rewrite (@ring_add_assoc2 _ _ x y z)
| |- context [(x + ?y) + ?z] =>
progress ring_rewrite (@ring_add_assoc1 _ _ x y z)
end).
|
