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Require Import Coq.Numbers.BinNums.
Require Import Coq.Classes.Morphisms.
Require Import Crypto.Spec.WeierstrassCurve.
Require Import Crypto.Algebra Crypto.Algebra.Field.
Require Import Crypto.Util.Decidable Crypto.Util.Tactics.
Require Import BinPos.
Module W.
Section W.
Context {F Feq Fzero Fone Fopp Fadd Fsub Fmul Finv Fdiv} {a b:F}
{field:@Algebra.field F Feq Fzero Fone Fopp Fadd Fsub Fmul Finv Fdiv}
{char_gt_3:@Ring.char_gt F Feq Fzero Fone Fopp Fadd Fsub Fmul (BinNat.N.succ_pos (BinNat.N.two))}
{char_gt_11:@Ring.char_gt F Feq Fzero Fone Fopp Fadd Fsub Fmul 11%positive} (* FIXME: we shouldn't need this *)
{Feq_dec:DecidableRel Feq}.
Local Infix "=" := Feq : type_scope. Local Notation "a <> b" := (not (a = b)) : type_scope.
Local Notation "0" := Fzero. Local Notation "1" := Fone.
Local Infix "+" := Fadd. Local Infix "*" := Fmul.
Local Infix "-" := Fsub. Local Infix "/" := Fdiv.
Local Notation eq := (@W.eq F Feq Fadd Fmul a b).
Local Notation point := (@W.point F Feq Fadd Fmul a b).
Local Notation "0" := Fzero. Local Notation "1" := Fone.
Local Notation "2" := (1+1). Local Notation "3" := (1+2). Local Notation "4" := (1+3).
Local Notation "8" := (1+(1+(1+(1+4)))). Local Notation "12" := (1+(1+(1+(1+8)))).
Local Notation "16" := (1+(1+(1+(1+12)))). Local Notation "20" := (1+(1+(1+(1+16)))).
Local Notation "24" := (1+(1+(1+(1+20)))). Local Notation "27" := (1+(1+(1+24))).
Local Notation "x ^ 2" := (x*x) (at level 30). Local Notation "x ^ 3" := (x*x^2) (at level 30).
Context {discriminant_nonzero:4*a^3 + 27*b^2 <> 0}.
Lemma same_x_same_y
(xA yA : F)
(A : yA ^ 2 = xA ^ 3 + a * xA + b)
(xB yB : F)
(B : yB ^ 2 = xB ^ 3 + a * xB + b)
(Hx: xA = xB)
(Hy:yB <> Fopp yA)
: yB = yA.
Proof. fsatz. Qed.
Definition is_redundant {T} (x:T) := x.
Global Arguments is_redundant : simpl never.
Ltac clear_marked_redundant :=
repeat match goal with
[H:?P, Hr:is_redundant ?P |- _] => clear H Hr
end.
Ltac t_step :=
match goal with
| _ => exact _
| _ => intro
| [ A : ?yA ^ 2 = ?xA ^ 3 + a * ?xA + b,
B : ?yB ^ 2 = ?xB ^ 3 + a * ?xB + b,
Hx: ?xA = ?xB,
Hy: ?yB <> Fopp ?yA
|- _] => unique pose proof (same_x_same_y _ _ A _ _ B Hx Hy)
| |- Equivalence _ => split
| |- abelian_group => split | |- group => split | |- monoid => split
| |- is_associative => split | |- is_commutative => split
| |- is_left_inverse => split | |- is_right_inverse => split
| |- is_left_identity => split | |- is_right_identity => split
| p:point |- _ => destruct p
| _ => progress destruct_head' sum
| _ => progress destruct_head' prod
| _ => progress destruct_head' unit
| _ => progress destruct_head' and
| |- context[?P] =>
unique pose proof (proj2_sig P);
unique pose proof (proj2_sig P:(is_redundant _))
| _ => progress cbv [eq W.eq W.add W.coordinates proj1_sig] in *
| _ => progress simpl in *
| _ => progress break_match
| |- _ /\ _ => split | |- _ <-> _ => split
| _ => abstract (trivial || contradiction || clear_marked_redundant; fsatz)
end.
Ltac t := repeat t_step.
Program Definition inv (P:point) : point
:= exist
_
(match W.coordinates P return _ with
| inl (x1, y1) => inl (x1, Fopp y1)
| _ => P
end)
_.
Next Obligation. t. Qed.
Global Instance commutative_group : abelian_group(eq:=W.eq)(op:=W.add)(id:=W.zero)(inv:=inv).
Proof. Time t. Time Qed.
End W.
End W.
(*
Times for individual subgoal of the associativity proof:
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Finished transaction in 268.648 secs (268.609u,0.096s) (successful)
*)
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