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Require Import Crypto.Algebra.Field.
Require Import Crypto.Spec.MontgomeryCurve Crypto.Curves.Montgomery.Affine.
Require Import Crypto.Spec.WeierstrassCurve Crypto.Curves.Weierstrass.Affine.
Require Import Crypto.Curves.Weierstrass.AffineProofs.
Require Import Crypto.Curves.Montgomery.AffineProofs.
Require Import Coq.Classes.RelationClasses.
Module M.
Section MontgomeryCurve.
Import BinNat.
Context {F Feq Fzero Fone Fopp Fadd Fsub Fmul Finv Fdiv}
{field:@Algebra.Hierarchy.field F Feq Fzero Fone Fopp Fadd Fsub Fmul Finv Fdiv}
{Feq_dec:Decidable.DecidableRel Feq}.
Local Infix "=" := Feq : type_scope. Local Notation "a <> b" := (not (a = b)) : type_scope.
Local Infix "+" := Fadd. Local Infix "*" := Fmul.
Local Infix "-" := Fsub. Local Infix "/" := Fdiv.
Local Notation "- x" := (Fopp x).
Local Notation "x ^ 2" := (x*x) (at level 30).
Local Notation "0" := Fzero.
Local Notation "1" := Fone.
Local Notation "4" := (1+1+1+1).
Global Instance MontgomeryWeierstrassIsomorphism
{a b: F}
(b_nonzero : b <> 0)
(discriminant_nonzero: a^2 - 4 <> 0)
{char_ge_3:@Ring.char_ge F Feq Fzero Fone Fopp Fadd Fsub Fmul 3}
{char_ge_12:@Ring.char_ge F Feq Fzero Fone Fopp Fadd Fsub Fmul 12}
{char_ge_28:@Ring.char_ge F Feq Fzero Fone Fopp Fadd Fsub Fmul 28} (* XXX: this is a workaround for nsatz assuming arbitrary characteristic *)
:
@Group.isomorphic_commutative_groups
(@W.point F Feq Fadd Fmul _ _)
W.eq
(@W.add F Feq Fzero Fone Fopp Fadd Fsub Fmul Finv Fdiv field _ char_ge_3 _ _)
W.zero
(@W.opp F Feq Fzero Fone Fopp Fadd Fsub Fmul Finv Fdiv _ _ field _)
(@M.point F Feq Fadd Fmul a b)
M.eq
(M.add(char_ge_3:=char_ge_3)(b_nonzero:=b_nonzero))
M.zero
(M.opp(b_nonzero:=b_nonzero))
(M.of_Weierstrass(Haw:=reflexivity _)(Hbw:=reflexivity _)(b_nonzero:=b_nonzero))
(M.to_Weierstrass(Haw:=reflexivity _)(Hbw:=reflexivity _)(b_nonzero:=b_nonzero)).
Proof.
eapply @AffineProofs.M.MontgomeryWeierstrassIsomorphism; try assumption; cbv [id]; fsatz.
Qed.
End MontgomeryCurve.
End M.
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