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-Require Import Coq.Classes.Morphisms. Require Coq.Setoids.Setoid.
-Require Import Crypto.Algebra.Field.
-Require Import Crypto.Util.Tactics.DestructHead.
-Require Import Crypto.Util.Tactics.BreakMatch.
-Require Import Crypto.Util.Notations.
-Require Import Crypto.Util.Decidable.
-Import BinNums.
-
-Local Open Scope core_scope.
-
-Section Pre.
- 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}
- {char_ge_3:@Ring.char_ge F Feq Fzero Fone Fopp Fadd Fsub Fmul (BinNat.N.succ_pos (BinNat.N.two))}
- {eq_dec: DecidableRel Feq}.
- Local Infix "=" := Feq. Local Notation "a <> b" := (not (a = b)).
- 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 "- x" := (Fopp x).
- Local Notation "x ^ 2" := (x*x). Local Notation "x ^ 3" := (x*x^2).
- Local Notation "'∞'" := unit : type_scope.
- Local Notation "'∞'" := (inr tt) : core_scope.
- Local Notation "2" := (1+1). Local Notation "3" := (1+2).
- Local Notation "( x , y )" := (inl (pair x y)).
-
- Context {a:F}.
- Context {b:F}.
-
- (* the canonical definitions are in Spec *)
- Let onCurve (P:F*F + ∞) := match P with
- | (x, y) => y^2 = x^3 + a*x + b
- | ∞ => True
- end.
- Let add (P1' P2':F*F + ∞) : F*F + ∞ :=
- match P1', P2' return _ with
- | (x1, y1), (x2, y2) =>
- if dec (x1 = x2)
- then
- if dec (y2 = -y1)
- then ∞
- else let k := (3*x1^2+a)/(2*y1) in
- let x3 := k^2-x1-x1 in
- let y3 := k*(x1-x3)-y1 in
- (x3, y3)
- else let k := (y2-y1)/(x2-x1) in
- let x3 := k^2-x1-x2 in
- let y3 := k*(x1-x3)-y1 in
- (x3, y3)
- | ∞, ∞ => ∞
- | ∞, _ => P2'
- | _, ∞ => P1'
- end.
-
- Lemma add_onCurve P1 P2 (_:onCurve P1) (_:onCurve P2) :
- onCurve (add P1 P2).
- Proof using a b char_ge_3 eq_dec field.
- destruct_head' sum; destruct_head' prod;
- cbv [onCurve add] in *; break_match; trivial; [|]; fsatz.
- Qed.
-End Pre.