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Require Import BinInt BinNat ZArith Znumtheory.
Require Export Coq.Classes.Morphisms Setoid.
Require Export Ring_theory Field_theory Field_tac.
Require Export Crypto.Galois.GaloisField.
Set Implicit Arguments.
Unset Strict Implicits.
Module GaloisFieldTheory (M: Modulus).
Module Field := GaloisField M.
Import M Field.
(* Notations *)
Infix "{+}" := GFplus (at level 60, right associativity).
Infix "{-}" := GFminus (at level 60, right associativity).
Infix "{*}" := GFmult (at level 50, left associativity).
Infix "{/}" := GFdiv (at level 50, left associativity).
Infix "{^}" := GFexp (at level 40, left associativity).
Notation "{1}" := GFone.
Notation "{0}" := GFzero.
(* Basic Properties *)
(* Fails iff the input term does some arithmetic with mod'd values. *)
Ltac notFancy E :=
match E with
| - (_ mod _) => idtac
| context[_ mod _] => fail 1
| _ => idtac
end.
Lemma Zplus_neg : forall n m, n + -m = n - m.
Proof.
auto.
Qed.
(* Remove redundant [mod] operations from the conclusion. *)
Ltac demod :=
repeat match goal with
| [ |- context[(?x mod _ + _) mod _] ] =>
notFancy x; rewrite (Zplus_mod_idemp_l x)
| [ |- context[(_ + ?y mod _) mod _] ] =>
notFancy y; rewrite (Zplus_mod_idemp_r y)
| [ |- context[(?x mod _ - _) mod _] ] =>
notFancy x; rewrite (Zminus_mod_idemp_l x)
| [ |- context[(_ - ?y mod _) mod _] ] =>
notFancy y; rewrite (Zminus_mod_idemp_r _ y)
| [ |- context[(?x mod _ * _) mod _] ] =>
notFancy x; rewrite (Zmult_mod_idemp_l x)
| [ |- context[(_ * (?y mod _)) mod _] ] =>
notFancy y; rewrite (Zmult_mod_idemp_r y)
| _ => rewrite Zplus_neg in * || rewrite Z.sub_diag in *
end.
(* General big hammer for proving Galois arithmetic facts *)
Ltac galois := intros; apply gf_eq; simpl;
repeat match goal with
| [ x : GF |- _ ] => destruct x
| [ H : ex _ |- _ ] => destruct H; subst
end; simpl in *; autorewrite with core;
intuition; demod; try solve [ f_equal; intuition ].
Lemma modmatch_eta : forall n,
match n with
| 0 => 0
| Z.pos y' => Z.pos y'
| Z.neg y' => Z.neg y'
end = n.
Proof.
destruct n; auto.
Qed.
Hint Rewrite Zmod_mod modmatch_eta.
Theorem GFplus_0_l: forall x : GF, {0} {+} x = x.
Proof.
galois.
Qed.
Theorem GFplus_commutative: forall x y : GF, x {+} y = y {+} x.
Proof.
galois.
Qed.
Theorem GFplus_associative: forall x y z : GF, x {+} (y {+} z) = (x {+} y) {+} z.
Proof.
galois.
Qed.
Theorem GFmult_1_l: forall x : GF, {1} {*} x = x.
Proof.
galois.
Qed.
Theorem GFmult_commutative: forall x y : GF, x {*} y = y {*} x.
Proof.
galois.
Qed.
Theorem GFmult_associative: forall x y z : GF, x {*} (y {*} z) = (x {*} y) {*} z.
Proof.
galois.
Qed.
Theorem GFdistributive: forall x y z : GF, (x {+} y) {*} z = x {*} z {+} y {*} z.
Proof.
galois.
Qed.
Theorem GFminus_opp: forall x y : GF, x {-} y = x {+} GFopp y.
Proof.
galois.
Qed.
Theorem GFopp_inverse: forall x : GF, x {+} GFopp x = {0}.
Proof.
galois.
Qed.
(* Ring Theory*)
Ltac compat := repeat intro; subst; trivial.
Instance GFplus_compat : Proper (eq==>eq==>eq) GFplus.
Proof.
compat.
Qed.
Instance GFminus_compat : Proper (eq==>eq==>eq) GFminus.
Proof.
compat.
Qed.
Instance GFmult_compat : Proper (eq==>eq==>eq) GFmult.
Proof.
compat.
Qed.
Instance GFopp_compat : Proper (eq==>eq) GFopp.
Proof.
compat.
Qed.
Definition GFring_theory : ring_theory GFzero GFone GFplus GFmult GFminus GFopp eq.
Proof.
constructor; galois.
Qed.
Add Ring GFring : GFring_theory.
(* Power theory *)
Lemma GFpower_theory : power_theory GFone GFmult eq id GFexp.
Proof.
constructor. intros.
induction n; simpl; intuition.
(* Prove the doubling case, which we'll use alot *)
assert (forall q r0, GFexp' r0 q{*}GFexp' r0 q = GFexp' (r0{*}r0) q).
induction q.
intros. assert (Q := (IHq (r0{*}r0))).
unfold GFexp'; simpl; fold GFexp'.
rewrite <- Q. ring.
intros. assert (Q := (IHq (r0{*}r0))).
unfold GFexp'; simpl; fold GFexp'.
rewrite <- Q. ring.
intros; simpl; ring.
(* Take it case-by-case *)
unfold GFexp; induction p.
(* Odd case *)
unfold GFexp'; simpl; fold GFexp'; fold pow_pos.
rewrite <- (H p r).
rewrite <- IHp. trivial.
(* Even case *)
unfold GFexp'; simpl; fold GFexp'; fold pow_pos.
rewrite <- (H p r).
rewrite <- IHp. trivial.
(* Base case *)
simpl; intuition.
Qed.
(* Field Theory*)
Instance GFinv_compat : Proper (eq==>eq) GFinv.
Proof.
compat.
Qed.
Instance GFdiv_compat : Proper (eq==>eq==>eq) GFdiv.
Proof.
compat.
Qed.
Definition GFfield_theory : field_theory GFzero GFone GFplus GFmult GFminus GFopp GFdiv GFinv eq.
Proof.
constructor; simpl; intuition.
exact GFring_theory.
unfold GFone, GFzero in H; apply exist_eq in H; simpl in H; intuition.
(* Prove multiplicative inverses *)
unfold GFinv.
destruct totient; simpl. destruct i.
replace (p{^}(N.pred x){*}p) with (p{^}x); simpl; intuition.
apply N.gt_lt in H0; apply N.lt_neq in H0.
replace x with (N.succ (N.pred x)).
induction (N.pred x).
simpl. ring.
rewrite N.pred_succ.
(* Just the N.succ case *)
generalize p. induction p0; simpl in *; intuition.
rewrite (IHp0 (p1{*}p1)). ring.
ring.
(* cleanup *)
rewrite N.succ_pred; intuition.
Defined.
Add Field GFfield : GFfield_theory.
End GaloisFieldTheory.
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