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Require Import Coq.Numbers.BinNums.
Require Import Coq.Classes.Morphisms.
Require Import Crypto.Spec.WeierstrassCurve Crypto.Curves.Weierstrass.Affine.
Require Import Crypto.Algebra.Field Crypto.Algebra.Hierarchy.
Require Import Crypto.Util.Decidable Crypto.Util.Tactics.DestructHead Crypto.Util.Tactics.BreakMatch.
Require Import Coq.PArith.BinPos.
Module W.
Section W.
Context {F Feq Fzero Fone Fopp Fadd Fsub Fmul Finv Fdiv} {a b:F}
{field:@Algebra.Hierarchy.field F Feq Fzero Fone Fopp Fadd Fsub Fmul Finv Fdiv}
{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 "-" := Fsub. Local Infix "*" := Fmul.
Local Notation "4" := (1+1+1+1). Local Notation "27" := (4*4 + 4+4 +1+1+1).
Local Ltac Algebra_split :=
repeat match goal with
| |- Proper _ _ => cbv [Proper respectful]; intros
| |- Equivalence _ => split; [intros ? | intros ??? | intros ????? ]
| |- monoid => split
| |- group => split
| |- commutative_group => split
| |- is_associative => split; intros ???
| |- is_commutative => split; intros ??
| |- is_left_inverse => split; intros ?
| |- is_right_inverse => split; intros ?
| |- is_left_identity => split; intros ?
| |- is_right_identity => split; intros ?
end.
Global Instance commutative_group
char_ge_3
{char_ge_12:@Ring.char_ge F Feq Fzero Fone Fopp Fadd Fsub Fmul 12%positive} (* FIXME: shouldn't need we need 4, not 12? *)
{discriminant_nonzero:id(4*a*a*a + 27*b*b <> 0)}
: commutative_group(eq:=W.eq(a:=a)(b:=b))(op:=W.add(char_ge_3:=char_ge_3))(id:=W.zero)(inv:=W.opp).
Proof using Type.
Time
cbv [W.opp W.eq W.zero W.add W.coordinates proj1_sig];
repeat match goal with
| _ => progress Algebra_split
| H: _ /\ _ |- _ => destruct H
| |- _ /\ _ => split
| _ => progress break_match
| _ => progress break_match_hyps
end; try contradiction; trivial.
Time par: abstract (fsatz || (cbv [id] in *; fsatz)).
Time Qed.
End W.
End W.
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