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https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/EllipticCurve/Model.lean | Model.c4_zero_iff_b2_zero_of_char_three | [428, 1] | [436, 11] | norm_num | K : Type u
inst✝¹ : CommRing K
inst✝ : IsDomain K
e : Model K
h : ringChar K = 3
hchar' : ↑(ringChar K) = 3
hchar'' : 3 = 0
⊢ 24 = 3 * 8 | no goals |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/EllipticCurve/Model.lean | Model.a3_zero_of_a1_zero_of_disc_zero_of_char_two | [439, 1] | [451, 11] | have hchar' : (ringChar K : K) = 2 := by simp [h] | K : Type u
inst✝¹ : CommRing K
inst✝ : IsDomain K
e : Model K
h : ringChar K = 2
hdisc : discr e = 0
ha1 : e.a1 = 0
⊢ e.a3 = 0 | K : Type u
inst✝¹ : CommRing K
inst✝ : IsDomain K
e : Model K
h : ringChar K = 2
hdisc : discr e = 0
ha1 : e.a1 = 0
hchar' : ↑(ringChar K) = 2
⊢ e.a3 = 0 |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/EllipticCurve/Model.lean | Model.a3_zero_of_a1_zero_of_disc_zero_of_char_two | [439, 1] | [451, 11] | have hchar'' : (2 : K) = 0 := by simp [← hchar', ringChar.Nat.cast_ringChar] | K : Type u
inst✝¹ : CommRing K
inst✝ : IsDomain K
e : Model K
h : ringChar K = 2
hdisc : discr e = 0
ha1 : e.a1 = 0
hchar' : ↑(ringChar K) = 2
⊢ e.a3 = 0 | K : Type u
inst✝¹ : CommRing K
inst✝ : IsDomain K
e : Model K
h : ringChar K = 2
hdisc : discr e = 0
ha1 : e.a1 = 0
hchar' : ↑(ringChar K) = 2
hchar'' : 2 = 0
⊢ e.a3 = 0 |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/EllipticCurve/Model.lean | Model.a3_zero_of_a1_zero_of_disc_zero_of_char_two | [439, 1] | [451, 11] | rw [discr, b2, b4, b6, b8, ha1,
show (8 : K) = 2 * 4 by norm_num, show (4 : K) = 2 * 2 by norm_num, show (27 : K) = 2 * 13 + 1 by norm_num, hchar''] at hdisc | K : Type u
inst✝¹ : CommRing K
inst✝ : IsDomain K
e : Model K
h : ringChar K = 2
hdisc : discr e = 0
ha1 : e.a1 = 0
hchar' : ↑(ringChar K) = 2
hchar'' : 2 = 0
⊢ e.a3 = 0 | K : Type u
inst✝¹ : CommRing K
inst✝ : IsDomain K
e : Model K
h : ringChar K = 2
hdisc :
-(0 * 0 + 0 * 0 * e.a2) * (0 * 0 + 0 * 0 * e.a2) *
(0 * 0 * e.a6 - 0 * e.a3 * e.a4 + 0 * 0 * e.a2 * e.a6 + e.a2 * e.a3 * e.a3 - e.a4 * e.a4) -
0 * (0 * 0) * (0 * e.a3 + 0 * e.a4) ^ 3 -
(0 * 13 + 1) * (e.a3 * e.a3 + 0 * 0 * e.a6) * (e.a3 * e.a3 + 0 * 0 * e.a6) +
9 * (0 * 0 + 0 * 0 * e.a2) * (0 * e.a3 + 0 * e.a4) * (e.a3 * e.a3 + 0 * 0 * e.a6) =
0
ha1 : e.a1 = 0
hchar' : ↑(ringChar K) = 2
hchar'' : 2 = 0
⊢ e.a3 = 0 |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/EllipticCurve/Model.lean | Model.a3_zero_of_a1_zero_of_disc_zero_of_char_two | [439, 1] | [451, 11] | simp only [mul_zero, zero_mul, add_zero, neg_zero, sub_self, zero_add, one_mul, zero_sub, neg_eq_zero]
at hdisc | K : Type u
inst✝¹ : CommRing K
inst✝ : IsDomain K
e : Model K
h : ringChar K = 2
hdisc :
-(0 * 0 + 0 * 0 * e.a2) * (0 * 0 + 0 * 0 * e.a2) *
(0 * 0 * e.a6 - 0 * e.a3 * e.a4 + 0 * 0 * e.a2 * e.a6 + e.a2 * e.a3 * e.a3 - e.a4 * e.a4) -
0 * (0 * 0) * (0 * e.a3 + 0 * e.a4) ^ 3 -
(0 * 13 + 1) * (e.a3 * e.a3 + 0 * 0 * e.a6) * (e.a3 * e.a3 + 0 * 0 * e.a6) +
9 * (0 * 0 + 0 * 0 * e.a2) * (0 * e.a3 + 0 * e.a4) * (e.a3 * e.a3 + 0 * 0 * e.a6) =
0
ha1 : e.a1 = 0
hchar' : ↑(ringChar K) = 2
hchar'' : 2 = 0
⊢ e.a3 = 0 | K : Type u
inst✝¹ : CommRing K
inst✝ : IsDomain K
e : Model K
h : ringChar K = 2
ha1 : e.a1 = 0
hchar' : ↑(ringChar K) = 2
hchar'' : 2 = 0
hdisc : e.a3 * e.a3 * (e.a3 * e.a3) = 0
⊢ e.a3 = 0 |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/EllipticCurve/Model.lean | Model.a3_zero_of_a1_zero_of_disc_zero_of_char_two | [439, 1] | [451, 11] | rw [← pow_two, ← pow_two, ← pow_mul] at hdisc | K : Type u
inst✝¹ : CommRing K
inst✝ : IsDomain K
e : Model K
h : ringChar K = 2
ha1 : e.a1 = 0
hchar' : ↑(ringChar K) = 2
hchar'' : 2 = 0
hdisc : e.a3 * e.a3 * (e.a3 * e.a3) = 0
⊢ e.a3 = 0 | K : Type u
inst✝¹ : CommRing K
inst✝ : IsDomain K
e : Model K
h : ringChar K = 2
ha1 : e.a1 = 0
hchar' : ↑(ringChar K) = 2
hchar'' : 2 = 0
hdisc : e.a3 ^ (2 * 2) = 0
⊢ e.a3 = 0 |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/EllipticCurve/Model.lean | Model.a3_zero_of_a1_zero_of_disc_zero_of_char_two | [439, 1] | [451, 11] | rwa [pow_eq_zero_iff] at hdisc | K : Type u
inst✝¹ : CommRing K
inst✝ : IsDomain K
e : Model K
h : ringChar K = 2
ha1 : e.a1 = 0
hchar' : ↑(ringChar K) = 2
hchar'' : 2 = 0
hdisc : e.a3 ^ (2 * 2) = 0
⊢ e.a3 = 0 | K : Type u
inst✝¹ : CommRing K
inst✝ : IsDomain K
e : Model K
h : ringChar K = 2
ha1 : e.a1 = 0
hchar' : ↑(ringChar K) = 2
hchar'' : 2 = 0
hdisc : e.a3 ^ (2 * 2) = 0
⊢ 0 < 2 * 2 |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/EllipticCurve/Model.lean | Model.a3_zero_of_a1_zero_of_disc_zero_of_char_two | [439, 1] | [451, 11] | norm_num | K : Type u
inst✝¹ : CommRing K
inst✝ : IsDomain K
e : Model K
h : ringChar K = 2
ha1 : e.a1 = 0
hchar' : ↑(ringChar K) = 2
hchar'' : 2 = 0
hdisc : e.a3 ^ (2 * 2) = 0
⊢ 0 < 2 * 2 | no goals |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/EllipticCurve/Model.lean | Model.a3_zero_of_a1_zero_of_disc_zero_of_char_two | [439, 1] | [451, 11] | simp [h] | K : Type u
inst✝¹ : CommRing K
inst✝ : IsDomain K
e : Model K
h : ringChar K = 2
hdisc : discr e = 0
ha1 : e.a1 = 0
⊢ ↑(ringChar K) = 2 | no goals |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/EllipticCurve/Model.lean | Model.a3_zero_of_a1_zero_of_disc_zero_of_char_two | [439, 1] | [451, 11] | simp [← hchar', ringChar.Nat.cast_ringChar] | K : Type u
inst✝¹ : CommRing K
inst✝ : IsDomain K
e : Model K
h : ringChar K = 2
hdisc : discr e = 0
ha1 : e.a1 = 0
hchar' : ↑(ringChar K) = 2
⊢ 2 = 0 | no goals |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/EllipticCurve/Model.lean | Model.a3_zero_of_a1_zero_of_disc_zero_of_char_two | [439, 1] | [451, 11] | norm_num | K : Type u
inst✝¹ : CommRing K
inst✝ : IsDomain K
e : Model K
h : ringChar K = 2
hdisc :
-(0 * 0 + 4 * e.a2) * (0 * 0 + 4 * e.a2) *
(0 * 0 * e.a6 - 0 * e.a3 * e.a4 + 4 * e.a2 * e.a6 + e.a2 * e.a3 * e.a3 - e.a4 * e.a4) -
8 * (0 * e.a3 + 2 * e.a4) ^ 3 -
27 * (e.a3 * e.a3 + 4 * e.a6) * (e.a3 * e.a3 + 4 * e.a6) +
9 * (0 * 0 + 4 * e.a2) * (0 * e.a3 + 2 * e.a4) * (e.a3 * e.a3 + 4 * e.a6) =
0
ha1 : e.a1 = 0
hchar' : ↑(ringChar K) = 2
hchar'' : 2 = 0
⊢ 8 = 2 * 4 | no goals |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/EllipticCurve/Model.lean | Model.a3_zero_of_a1_zero_of_disc_zero_of_char_two | [439, 1] | [451, 11] | norm_num | K : Type u
inst✝¹ : CommRing K
inst✝ : IsDomain K
e : Model K
h : ringChar K = 2
hdisc :
-(0 * 0 + 4 * e.a2) * (0 * 0 + 4 * e.a2) *
(0 * 0 * e.a6 - 0 * e.a3 * e.a4 + 4 * e.a2 * e.a6 + e.a2 * e.a3 * e.a3 - e.a4 * e.a4) -
2 * 4 * (0 * e.a3 + 2 * e.a4) ^ 3 -
27 * (e.a3 * e.a3 + 4 * e.a6) * (e.a3 * e.a3 + 4 * e.a6) +
9 * (0 * 0 + 4 * e.a2) * (0 * e.a3 + 2 * e.a4) * (e.a3 * e.a3 + 4 * e.a6) =
0
ha1 : e.a1 = 0
hchar' : ↑(ringChar K) = 2
hchar'' : 2 = 0
⊢ 4 = 2 * 2 | no goals |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/EllipticCurve/Model.lean | Model.a3_zero_of_a1_zero_of_disc_zero_of_char_two | [439, 1] | [451, 11] | norm_num | K : Type u
inst✝¹ : CommRing K
inst✝ : IsDomain K
e : Model K
h : ringChar K = 2
hdisc :
-(0 * 0 + 2 * 2 * e.a2) * (0 * 0 + 2 * 2 * e.a2) *
(0 * 0 * e.a6 - 0 * e.a3 * e.a4 + 2 * 2 * e.a2 * e.a6 + e.a2 * e.a3 * e.a3 - e.a4 * e.a4) -
2 * (2 * 2) * (0 * e.a3 + 2 * e.a4) ^ 3 -
27 * (e.a3 * e.a3 + 2 * 2 * e.a6) * (e.a3 * e.a3 + 2 * 2 * e.a6) +
9 * (0 * 0 + 2 * 2 * e.a2) * (0 * e.a3 + 2 * e.a4) * (e.a3 * e.a3 + 2 * 2 * e.a6) =
0
ha1 : e.a1 = 0
hchar' : ↑(ringChar K) = 2
hchar'' : 2 = 0
⊢ 27 = 2 * 13 + 1 | no goals |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/EllipticCurve/Model.lean | Model.b4_zero_of_b2_zero_of_disc_zero_of_char_three | [454, 1] | [464, 20] | have hchar' : (ringChar K : K) = 3 := by simp [h] | K : Type u
inst✝¹ : CommRing K
inst✝ : IsDomain K
e : Model K
h : ringChar K = 3
hdisc : discr e = 0
hb2 : b2 e = 0
⊢ b4 e = 0 | K : Type u
inst✝¹ : CommRing K
inst✝ : IsDomain K
e : Model K
h : ringChar K = 3
hdisc : discr e = 0
hb2 : b2 e = 0
hchar' : ↑(ringChar K) = 3
⊢ b4 e = 0 |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/EllipticCurve/Model.lean | Model.b4_zero_of_b2_zero_of_disc_zero_of_char_three | [454, 1] | [464, 20] | have hchar'' : (3 : K) = 0 := by simp [← hchar', ringChar.Nat.cast_ringChar] | K : Type u
inst✝¹ : CommRing K
inst✝ : IsDomain K
e : Model K
h : ringChar K = 3
hdisc : discr e = 0
hb2 : b2 e = 0
hchar' : ↑(ringChar K) = 3
⊢ b4 e = 0 | K : Type u
inst✝¹ : CommRing K
inst✝ : IsDomain K
e : Model K
h : ringChar K = 3
hdisc : discr e = 0
hb2 : b2 e = 0
hchar' : ↑(ringChar K) = 3
hchar'' : 3 = 0
⊢ b4 e = 0 |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/EllipticCurve/Model.lean | Model.b4_zero_of_b2_zero_of_disc_zero_of_char_three | [454, 1] | [464, 20] | rw [discr, hb2,
show (27 : K) = 3 * 9 by norm_num,
show (8 : K) = 3 * 3 - 1 by norm_num,
hchar''] at hdisc | K : Type u
inst✝¹ : CommRing K
inst✝ : IsDomain K
e : Model K
h : ringChar K = 3
hdisc : discr e = 0
hb2 : b2 e = 0
hchar' : ↑(ringChar K) = 3
hchar'' : 3 = 0
⊢ b4 e = 0 | K : Type u
inst✝¹ : CommRing K
inst✝ : IsDomain K
e : Model K
h : ringChar K = 3
hdisc : -0 * 0 * b8 e - (0 * 0 - 1) * b4 e ^ 3 - 0 * 9 * b6 e * b6 e + 9 * 0 * b4 e * b6 e = 0
hb2 : b2 e = 0
hchar' : ↑(ringChar K) = 3
hchar'' : 3 = 0
⊢ b4 e = 0 |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/EllipticCurve/Model.lean | Model.b4_zero_of_b2_zero_of_disc_zero_of_char_three | [454, 1] | [464, 20] | simpa using hdisc | K : Type u
inst✝¹ : CommRing K
inst✝ : IsDomain K
e : Model K
h : ringChar K = 3
hdisc : -0 * 0 * b8 e - (0 * 0 - 1) * b4 e ^ 3 - 0 * 9 * b6 e * b6 e + 9 * 0 * b4 e * b6 e = 0
hb2 : b2 e = 0
hchar' : ↑(ringChar K) = 3
hchar'' : 3 = 0
⊢ b4 e = 0 | no goals |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/EllipticCurve/Model.lean | Model.b4_zero_of_b2_zero_of_disc_zero_of_char_three | [454, 1] | [464, 20] | simp [h] | K : Type u
inst✝¹ : CommRing K
inst✝ : IsDomain K
e : Model K
h : ringChar K = 3
hdisc : discr e = 0
hb2 : b2 e = 0
⊢ ↑(ringChar K) = 3 | no goals |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/EllipticCurve/Model.lean | Model.b4_zero_of_b2_zero_of_disc_zero_of_char_three | [454, 1] | [464, 20] | simp [← hchar', ringChar.Nat.cast_ringChar] | K : Type u
inst✝¹ : CommRing K
inst✝ : IsDomain K
e : Model K
h : ringChar K = 3
hdisc : discr e = 0
hb2 : b2 e = 0
hchar' : ↑(ringChar K) = 3
⊢ 3 = 0 | no goals |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/EllipticCurve/Model.lean | Model.b4_zero_of_b2_zero_of_disc_zero_of_char_three | [454, 1] | [464, 20] | norm_num | K : Type u
inst✝¹ : CommRing K
inst✝ : IsDomain K
e : Model K
h : ringChar K = 3
hdisc : -0 * 0 * b8 e - 8 * b4 e ^ 3 - 27 * b6 e * b6 e + 9 * 0 * b4 e * b6 e = 0
hb2 : b2 e = 0
hchar' : ↑(ringChar K) = 3
hchar'' : 3 = 0
⊢ 27 = 3 * 9 | no goals |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/EllipticCurve/Model.lean | Model.b4_zero_of_b2_zero_of_disc_zero_of_char_three | [454, 1] | [464, 20] | norm_num | K : Type u
inst✝¹ : CommRing K
inst✝ : IsDomain K
e : Model K
h : ringChar K = 3
hdisc : -0 * 0 * b8 e - 8 * b4 e ^ 3 - 3 * 9 * b6 e * b6 e + 9 * 0 * b4 e * b6 e = 0
hb2 : b2 e = 0
hchar' : ↑(ringChar K) = 3
hchar'' : 3 = 0
⊢ 8 = 3 * 3 - 1 | no goals |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/EllipticCurve/Model.lean | Model.Field.ringChar_eq_of_Prime | [488, 1] | [496, 15] | rw [← Nat.cast_eq_ofNat, ringChar.spec] at hn | K : Type u
inst✝¹ : Field K
n : ℕ
inst✝ : Nat.AtLeastTwo n
hn : OfNat.ofNat n = 0
hnp : Nat.Prime n
⊢ ringChar K = n | K : Type u
inst✝¹ : Field K
n : ℕ
inst✝ : Nat.AtLeastTwo n
hn : ringChar K ∣ n
hnp : Nat.Prime n
⊢ ringChar K = n |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/EllipticCurve/Model.lean | Model.Field.ringChar_eq_of_Prime | [488, 1] | [496, 15] | cases (Nat.dvd_prime hnp).mp hn with
| inl h =>
simpa [h] using CharP.char_is_prime_or_zero K (ringChar K)
| inr h =>
assumption | K : Type u
inst✝¹ : Field K
n : ℕ
inst✝ : Nat.AtLeastTwo n
hn : ringChar K ∣ n
hnp : Nat.Prime n
⊢ ringChar K = n | no goals |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/EllipticCurve/Model.lean | Model.Field.ringChar_eq_of_Prime | [488, 1] | [496, 15] | simpa [h] using CharP.char_is_prime_or_zero K (ringChar K) | case inl
K : Type u
inst✝¹ : Field K
n : ℕ
inst✝ : Nat.AtLeastTwo n
hn : ringChar K ∣ n
hnp : Nat.Prime n
h : ringChar K = 1
⊢ ringChar K = n | no goals |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/EllipticCurve/Model.lean | Model.Field.ringChar_eq_of_Prime | [488, 1] | [496, 15] | assumption | case inr
K : Type u
inst✝¹ : Field K
n : ℕ
inst✝ : Nat.AtLeastTwo n
hn : ringChar K ∣ n
hnp : Nat.Prime n
h : ringChar K = n
⊢ ringChar K = n | no goals |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/EllipticCurve/Model.lean | Model.Field.is_singular_point_singular_point | [509, 1] | [694, 11] | rw [singular_point] | K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
⊢ is_singular_point e (singular_point e) | K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
⊢ is_singular_point e
(if c4 e = 0 then
match ringChar K with
| 2 => (pth_root e.a4, pth_root (e.a2 * e.a4 + e.a6))
| 3 => (pth_root (-e.a3 ^ 2 - e.a6), e.a1 * pth_root (-e.a3 ^ 2 - e.a6) + e.a3)
| x => (-b2 e / 12, -(-e.a1 * b2 e / 12 + e.a3) / 2)
else ((18 * b6 e - b2 e * b4 e) / c4 e, (b2 e * b5 e + 3 * b7 e) / c4 e)) |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/EllipticCurve/Model.lean | Model.Field.is_singular_point_singular_point | [509, 1] | [694, 11] | split_ifs with hc4 | K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
⊢ is_singular_point e
(if c4 e = 0 then
match ringChar K with
| 2 => (pth_root e.a4, pth_root (e.a2 * e.a4 + e.a6))
| 3 => (pth_root (-e.a3 ^ 2 - e.a6), e.a1 * pth_root (-e.a3 ^ 2 - e.a6) + e.a3)
| x => (-b2 e / 12, -(-e.a1 * b2 e / 12 + e.a3) / 2)
else ((18 * b6 e - b2 e * b4 e) / c4 e, (b2 e * b5 e + 3 * b7 e) / c4 e)) | case pos
K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : c4 e = 0
⊢ is_singular_point e
(match ringChar K with
| 2 => (pth_root e.a4, pth_root (e.a2 * e.a4 + e.a6))
| 3 => (pth_root (-e.a3 ^ 2 - e.a6), e.a1 * pth_root (-e.a3 ^ 2 - e.a6) + e.a3)
| x => (-b2 e / 12, -(-e.a1 * b2 e / 12 + e.a3) / 2))
case neg
K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : ¬c4 e = 0
⊢ is_singular_point e ((18 * b6 e - b2 e * b4 e) / c4 e, (b2 e * b5 e + 3 * b7 e) / c4 e) |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/EllipticCurve/Model.lean | Model.Field.is_singular_point_singular_point | [509, 1] | [694, 11] | . have hc6 : c6 e = 0 := by
simpa [h, hc4, pow_succ, mul_eq_zero] using discr_identity e
split
. rw [is_singular_point]
have hchar : ringChar K = 2 := by assumption
have hchar' : (ringChar K : K) = 2 := by simp [hchar]
have hchar'' : (2 : K) = 0 := by simp [← hchar', ringChar.Nat.cast_ringChar]
have hcharne : ringChar K ≠ 0 := by simp [hchar]
have ha1 : e.a1 = 0 := by simpa [c4_zero_iff_a1_zero_of_char_two e hchar] using hc4
have ha3 : e.a3 = 0 := a3_zero_of_a1_zero_of_disc_zero_of_char_two e hchar h ha1
refine ⟨?_, ?_, ?_⟩
. rw [weierstrass]
simp only [ha1, ha3, mul_zero, zero_add, zero_mul, add_zero]
rw [show 3 = 2 + 1 by norm_num]
rw [pow_succ _ 2]
rw [← hchar, pth_root_pow_char hcharne]
rw [pth_root_pow_char hcharne]
ring_nf
simp [hchar'']
. rw [dweierstrass_dx]
simp only [ha1, zero_mul, hchar'', add_zero, zero_sub, neg_add_rev]
rw [← hchar, pth_root_pow_char hcharne, ← sub_eq_add_neg]
simp only [add_sub_add_right_eq_sub, sub_eq_iff_eq_add, zero_add]
rw [show (3 : K) = 2 * 2 - 1 by norm_num]
rw [hchar'']
simp [← neg_mul_eq_neg_mul]
. simp [dweierstrass_dy, ha1, ha3, hchar'']
. rw [is_singular_point]
have hchar : ringChar K = 3 := by assumption
have hcharne : ringChar K ≠ 0 := by simp [hchar]
have hchar' : (ringChar K : K) = 3 := by simp [hchar]
have hchar'' : (3 : K) = 0 := by simp [← hchar', ringChar.Nat.cast_ringChar]
have hb2 : e.b2 = 0 := by simpa [c4_zero_iff_b2_zero_of_char_three e hchar] using hc4
have hb4 : e.b4 = 0 := b4_zero_of_b2_zero_of_disc_zero_of_char_three e hchar h hb2
rw [b2] at hb2 rw [b4] at hb4
refine ⟨?_, ?_, ?_⟩
. rw [weierstrass]
rw [← hchar, pth_root_pow_char hcharne]
simp only
rw [show
(e.a1 * pth_root (-(e.a3 ^ 2) - e.a6) + e.a3) ^ 2 +
e.a1 * pth_root (-(e.a3 ^ 2) - e.a6) * (e.a1 * pth_root (-(e.a3 ^ 2) - e.a6) + e.a3) +
e.a3 * (e.a1 * pth_root (-(e.a3 ^ 2) - e.a6) + e.a3) -
(-(e.a3 ^ 2) - e.a6 + e.a2 * pth_root (-(e.a3 ^ 2) - e.a6) ^ 2 + e.a4 * pth_root (-(e.a3 ^ 2) - e.a6) + e.a6) =
(2 * e.a1 ^ 2 - e.a2) * pth_root (-(e.a3 ^ 2) - e.a6) ^ 2 +
(4 * e.a1 * e.a3 - e.a4) * pth_root (-(e.a3 ^ 2) - e.a6) +
3 * e.a3 ^ 2
by ring]
rw [show 2 * e.a1 ^ 2 - e.a2 = 0 from ?_]
rw [show 4 * e.a1 * e.a3 - e.a4 = 0 from ?_]
simp only [zero_mul, add_zero, hchar'']
. rw [show (2 : K) = -1 by rw [← add_zero (-1), ← hchar'']; norm_num] at hb4
rw [show (4 : K) = 1 by rw [← add_zero 1, ← hchar'']; norm_num]
simp only [neg_mul, one_mul] at hb4
simp [sub_eq_add_neg, hb4]
. linear_combination (norm := ring_nf <;> simp [hchar'']) -hb2
. rw [dweierstrass_dx]
rw [hchar'', zero_mul, zero_add]
simp only
rw [show e.a1 * (e.a1 * pth_root (-(e.a3 ^ 2) - e.a6) + e.a3) - (2 * e.a2 * pth_root (-(e.a3 ^ 2) - e.a6) + e.a4) =
(e.a1 * e.a1 - 2 * e.a2) * pth_root (-(e.a3 ^ 2) - e.a6) + (e.a1 * e.a3 - e.a4)
by ring]
rw [show (2 : K) = -1 by rw [← add_zero (-1), ← hchar'']; norm_num] at hb4
rw [show (4 : K) = -2 by rw [← add_zero (-2), ← zero_mul (2 : K), ← hchar'']; norm_num] at hb2
simp only [neg_mul, one_mul, ← sub_eq_add_neg] at hb4 hb2
rw [hb4, hb2, zero_mul, zero_add]
. rw [dweierstrass_dy]
simp only
rw [show 2 * (e.a1 * pth_root (-(e.a3 ^ 2) - e.a6) + e.a3)
+ e.a1 * pth_root (- (e.a3 ^ 2) - e.a6) + e.a3 = 3 * ((e.a1 * pth_root (-(e.a3 ^ 2) - e.a6)) + e.a3) by ring]
rw [hchar'', zero_mul]
. rename_i hn2 hn3
rw [is_singular_point]
have h2 : (2 : K) ≠ 0 := fun hh => hn2 (ringChar_eq_of_Prime hh (by norm_num))
have h3 : (3 : K) ≠ 0 := fun hh => hn3 (ringChar_eq_of_Prime hh (by norm_num))
have h12 : (12 : K) ≠ 0 := by
rw [show (12 : K) = 2 * 2 * 3 by norm_num]
repeat' apply mul_ne_zero
all_goals assumption
refine ⟨?_, ?_, ?_⟩
. apply nzero_mul_left_cancel (12 ^ 3) _ _ (pow_ne_zero _ h12)
simp only [weierstrass, div_eq_mul_inv, mul_zero]
rw [show
12 ^ 3 * ((-(-e.a1 * b2 e * 12⁻¹ + e.a3) * 2⁻¹) ^ 2 +
e.a1 * (-b2 e * 12⁻¹) * (-(-e.a1 * b2 e * 12⁻¹ + e.a3) * 2⁻¹) +
e.a3 * (-(-e.a1 * b2 e * 12⁻¹ + e.a3) * 2⁻¹) -
((-b2 e * 12⁻¹) ^ 3 + e.a2 * (-b2 e * 12⁻¹) ^ 2 + e.a4 * (-b2 e * 12⁻¹) + e.a6)) =
3*(-(-e.a1 * b2 e * (12 * 12⁻¹) + 12 * e.a3) * (2 * 2⁻¹)) ^ 2 +
e.a1 * (-b2 e * (12 * 12⁻¹)) * (-(-e.a1 * b2 e * (12 * 12⁻¹) + 12 * e.a3) * (6 * (2 * 2⁻¹))) +
12 * e.a3 * (-(-e.a1 * b2 e * (12 * 12⁻¹) + 12 * e.a3) * (6 * (2 * 2⁻¹))) -
((-b2 e * (12 * 12⁻¹)) ^ 3 + 12 * e.a2 * (-b2 e * (12 * 12⁻¹)) ^ 2 + 12 ^ 2 * e.a4 * (-b2 e * (12 * 12⁻¹)) + 12 ^ 3 * e.a6) by ring]
simp only [mul_inv_cancel h2, mul_inv_cancel h12, one_mul, mul_one]
rw [← mul_zero (2 : K), ← hc6]
simp only [c6, b2, b4, b6]
ring
. apply nzero_mul_left_cancel (12 ^ 2) _ _ (pow_ne_zero _ h12)
simp only [dweierstrass_dx, div_eq_mul_inv, mul_zero]
rw [show
12 ^ 2 * (e.a1 * (-(-e.a1 * b2 e * 12⁻¹ + e.a3) * 2⁻¹) - (3 * (-b2 e * 12⁻¹) ^ 2 + 2 * e.a2 * (-b2 e * 12⁻¹) + e.a4))
=
e.a1 * (-(-e.a1 * b2 e * (12 * 12⁻¹) + 12* e.a3) * 6 * (2 * 2⁻¹)) - (3 * (-b2 e * (12 * 12⁻¹)) ^ 2 + 24 * e.a2 * (-b2 e * (12 * 12⁻¹)) + 144 * e.a4)
by ring]
simp only [mul_inv_cancel h2, mul_inv_cancel h12, one_mul, mul_one]
rw [← mul_zero (3 : K), ← hc4]
simp only [c4, c6, b2, b4, b6]
ring
. apply nzero_mul_left_cancel 12 _ _ h12
simp only [dweierstrass_dy, div_eq_mul_inv, mul_zero]
rw [show
12 * (2 * (-(-e.a1 * b2 e * 12⁻¹ + e.a3) * 2⁻¹) + e.a1 * (-b2 e * 12⁻¹) + e.a3)
=
(-(-e.a1 * b2 e * (12 * 12⁻¹) + 12 * e.a3) * (2 * 2⁻¹)) + e.a1 * (-b2 e * (12 * 12⁻¹)) + 12 * e.a3
by ring]
simp only [mul_inv_cancel h2, mul_inv_cancel h12, one_mul, mul_one]
simp only [c6, b2, b4, b6]
ring | case pos
K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : c4 e = 0
⊢ is_singular_point e
(match ringChar K with
| 2 => (pth_root e.a4, pth_root (e.a2 * e.a4 + e.a6))
| 3 => (pth_root (-e.a3 ^ 2 - e.a6), e.a1 * pth_root (-e.a3 ^ 2 - e.a6) + e.a3)
| x => (-b2 e / 12, -(-e.a1 * b2 e / 12 + e.a3) / 2))
case neg
K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : ¬c4 e = 0
⊢ is_singular_point e ((18 * b6 e - b2 e * b4 e) / c4 e, (b2 e * b5 e + 3 * b7 e) / c4 e) | case neg
K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : ¬c4 e = 0
⊢ is_singular_point e ((18 * b6 e - b2 e * b4 e) / c4 e, (b2 e * b5 e + 3 * b7 e) / c4 e) |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/EllipticCurve/Model.lean | Model.Field.is_singular_point_singular_point | [509, 1] | [694, 11] | . rw [is_singular_point]
refine ⟨?_, ?_, ?_⟩
. rw [weierstrass]
apply nzero_mul_left_cancel (e.c4 ^ 3) _ _ (pow_ne_zero _ hc4)
rw [mul_zero]
simp only [div_eq_mul_inv]
rw [show c4 e ^ 3 * (((b2 e * b5 e + 3 * b7 e) * (c4 e)⁻¹) ^ 2 +
e.a1 * ((18 * b6 e - b2 e * b4 e) * (c4 e)⁻¹) * ((b2 e * b5 e + 3 * b7 e) * (c4 e)⁻¹) +
e.a3 * ((b2 e * b5 e + 3 * b7 e) * (c4 e)⁻¹) -
(((18 * b6 e - b2 e * b4 e) * (c4 e)⁻¹) ^ 3 + e.a2 * ((18 * b6 e - b2 e * b4 e) * (c4 e)⁻¹) ^ 2 +
e.a4 * ((18 * b6 e - b2 e * b4 e) * (c4 e)⁻¹) + e.a6)) =
(c4 e * (c4 e)⁻¹ * c4 e * (c4 e)⁻¹ * c4 e * ((b2 e * b5 e + 3 * b7 e)) ^ 2 +
c4 e * (c4 e)⁻¹ * c4 e * (c4 e)⁻¹ * c4 e * e.a1 * ((18 * b6 e - b2 e * b4 e)) * ((b2 e * b5 e + 3 * b7 e)) +
c4 e * (c4 e)⁻¹ * c4 e * c4 e * e.a3 * ((b2 e * b5 e + 3 * b7 e)) -
(c4 e * (c4 e)⁻¹ * c4 e * (c4 e)⁻¹ * c4 e * (c4 e)⁻¹ * ((18 * b6 e - b2 e * b4 e)) ^ 3 +
c4 e * (c4 e)⁻¹ * c4 e * (c4 e)⁻¹ * c4 e * e.a2 * ((18 * b6 e - b2 e * b4 e)) ^ 2 +
c4 e * (c4 e)⁻¹ * c4 e * c4 e * e.a4 * ((18 * b6 e - b2 e * b4 e)) +
c4 e * c4 e * c4 e * e.a6)) by ring]
simp only [mul_inv_cancel hc4, one_mul]
rw [b5, b7, c4, b2, b4, b6]
rw [← mul_zero (e.a1^6 + 12*e.a1^4*e.a2 + 48*e.a1^2*e.a2^2 - 36*e.a1^3*e.a3 + 64*e.a2^3
- 144*e.a1*e.a2*e.a3 - 72*e.a1^2*e.a4 + 216*e.a3^2 - 288*e.a2*e.a4 + 864*e.a6),
← h, discr_eq_neg_singular]
ring
. rw [dweierstrass_dx]
apply nzero_mul_left_cancel (e.c4 ^ 2) _ _ (pow_ne_zero _ hc4)
rw [mul_zero, pow_two]
simp only [div_eq_mul_inv]
rw [show c4 e * c4 e *
(e.a1 * ((b2 e * b5 e + 3 * b7 e) * (c4 e)⁻¹) -
(3 * ((18 * b6 e - b2 e * b4 e) * (c4 e)⁻¹) ^ 2
+ 2 * e.a2 * ((18 * b6 e - b2 e * b4 e) * (c4 e)⁻¹) + e.a4)) =
c4 e * (c4 e)⁻¹ * c4 e * (e.a1 * (b2 e * b5 e + 3 * b7 e)
- 2 * e.a2 * ((18 * b6 e - b2 e * b4 e)))
- c4 e * (c4 e)⁻¹ * c4 e * (c4 e)⁻¹ *
3 * (18 * b6 e - b2 e * b4 e) ^ 2 - e.a4 * c4 e * c4 e
by ring]
simp only [mul_inv_cancel hc4, one_mul]
rw [b5, b7, c4, b2, b4, b6]
rw [← mul_zero (36 : K), ← h, discr_eq_neg_singular]
ring
. rw [dweierstrass_dy]
apply nzero_mul_left_cancel e.c4 _ _ hc4
simp only [div_eq_mul_inv, mul_zero]
rw [show c4 e * (2 * ((b2 e * b5 e + 3 * b7 e) * (c4 e)⁻¹)
+ e.a1 * ((18 * b6 e - b2 e * b4 e) * (c4 e)⁻¹) + e.a3) =
c4 e * (c4 e)⁻¹ * (2 * (b2 e * b5 e + 3 * b7 e)
+ e.a1 * ((18 * b6 e - b2 e * b4 e))) + c4 e * e.a3 by ring]
simp only [mul_inv_cancel hc4, one_mul]
rw [b5, b7, c4, b2, b4, b6]
ring | case neg
K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : ¬c4 e = 0
⊢ is_singular_point e ((18 * b6 e - b2 e * b4 e) / c4 e, (b2 e * b5 e + 3 * b7 e) / c4 e) | no goals |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/EllipticCurve/Model.lean | Model.Field.is_singular_point_singular_point | [509, 1] | [694, 11] | have hc6 : c6 e = 0 := by
simpa [h, hc4, pow_succ, mul_eq_zero] using discr_identity e | case pos
K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : c4 e = 0
⊢ is_singular_point e
(match ringChar K with
| 2 => (pth_root e.a4, pth_root (e.a2 * e.a4 + e.a6))
| 3 => (pth_root (-e.a3 ^ 2 - e.a6), e.a1 * pth_root (-e.a3 ^ 2 - e.a6) + e.a3)
| x => (-b2 e / 12, -(-e.a1 * b2 e / 12 + e.a3) / 2)) | case pos
K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : c4 e = 0
hc6 : c6 e = 0
⊢ is_singular_point e
(match ringChar K with
| 2 => (pth_root e.a4, pth_root (e.a2 * e.a4 + e.a6))
| 3 => (pth_root (-e.a3 ^ 2 - e.a6), e.a1 * pth_root (-e.a3 ^ 2 - e.a6) + e.a3)
| x => (-b2 e / 12, -(-e.a1 * b2 e / 12 + e.a3) / 2)) |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/EllipticCurve/Model.lean | Model.Field.is_singular_point_singular_point | [509, 1] | [694, 11] | split | case pos
K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : c4 e = 0
hc6 : c6 e = 0
⊢ is_singular_point e
(match ringChar K with
| 2 => (pth_root e.a4, pth_root (e.a2 * e.a4 + e.a6))
| 3 => (pth_root (-e.a3 ^ 2 - e.a6), e.a1 * pth_root (-e.a3 ^ 2 - e.a6) + e.a3)
| x => (-b2 e / 12, -(-e.a1 * b2 e / 12 + e.a3) / 2)) | case pos.h_1
K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : c4 e = 0
hc6 : c6 e = 0
x✝ : ℕ
heq✝ : ringChar K = 2
⊢ is_singular_point e (pth_root e.a4, pth_root (e.a2 * e.a4 + e.a6))
case pos.h_2
K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : c4 e = 0
hc6 : c6 e = 0
x✝ : ℕ
heq✝ : ringChar K = 3
⊢ is_singular_point e (pth_root (-e.a3 ^ 2 - e.a6), e.a1 * pth_root (-e.a3 ^ 2 - e.a6) + e.a3)
case pos.h_3
K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : c4 e = 0
hc6 : c6 e = 0
x✝² : ℕ
x✝¹ : ringChar K = 2 → False
x✝ : ringChar K = 3 → False
⊢ is_singular_point e (-b2 e / 12, -(-e.a1 * b2 e / 12 + e.a3) / 2) |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/EllipticCurve/Model.lean | Model.Field.is_singular_point_singular_point | [509, 1] | [694, 11] | . rw [is_singular_point]
have hchar : ringChar K = 2 := by assumption
have hchar' : (ringChar K : K) = 2 := by simp [hchar]
have hchar'' : (2 : K) = 0 := by simp [← hchar', ringChar.Nat.cast_ringChar]
have hcharne : ringChar K ≠ 0 := by simp [hchar]
have ha1 : e.a1 = 0 := by simpa [c4_zero_iff_a1_zero_of_char_two e hchar] using hc4
have ha3 : e.a3 = 0 := a3_zero_of_a1_zero_of_disc_zero_of_char_two e hchar h ha1
refine ⟨?_, ?_, ?_⟩
. rw [weierstrass]
simp only [ha1, ha3, mul_zero, zero_add, zero_mul, add_zero]
rw [show 3 = 2 + 1 by norm_num]
rw [pow_succ _ 2]
rw [← hchar, pth_root_pow_char hcharne]
rw [pth_root_pow_char hcharne]
ring_nf
simp [hchar'']
. rw [dweierstrass_dx]
simp only [ha1, zero_mul, hchar'', add_zero, zero_sub, neg_add_rev]
rw [← hchar, pth_root_pow_char hcharne, ← sub_eq_add_neg]
simp only [add_sub_add_right_eq_sub, sub_eq_iff_eq_add, zero_add]
rw [show (3 : K) = 2 * 2 - 1 by norm_num]
rw [hchar'']
simp [← neg_mul_eq_neg_mul]
. simp [dweierstrass_dy, ha1, ha3, hchar''] | case pos.h_1
K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : c4 e = 0
hc6 : c6 e = 0
x✝ : ℕ
heq✝ : ringChar K = 2
⊢ is_singular_point e (pth_root e.a4, pth_root (e.a2 * e.a4 + e.a6))
case pos.h_2
K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : c4 e = 0
hc6 : c6 e = 0
x✝ : ℕ
heq✝ : ringChar K = 3
⊢ is_singular_point e (pth_root (-e.a3 ^ 2 - e.a6), e.a1 * pth_root (-e.a3 ^ 2 - e.a6) + e.a3)
case pos.h_3
K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : c4 e = 0
hc6 : c6 e = 0
x✝² : ℕ
x✝¹ : ringChar K = 2 → False
x✝ : ringChar K = 3 → False
⊢ is_singular_point e (-b2 e / 12, -(-e.a1 * b2 e / 12 + e.a3) / 2) | case pos.h_2
K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : c4 e = 0
hc6 : c6 e = 0
x✝ : ℕ
heq✝ : ringChar K = 3
⊢ is_singular_point e (pth_root (-e.a3 ^ 2 - e.a6), e.a1 * pth_root (-e.a3 ^ 2 - e.a6) + e.a3)
case pos.h_3
K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : c4 e = 0
hc6 : c6 e = 0
x✝² : ℕ
x✝¹ : ringChar K = 2 → False
x✝ : ringChar K = 3 → False
⊢ is_singular_point e (-b2 e / 12, -(-e.a1 * b2 e / 12 + e.a3) / 2) |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/EllipticCurve/Model.lean | Model.Field.is_singular_point_singular_point | [509, 1] | [694, 11] | . rw [is_singular_point]
have hchar : ringChar K = 3 := by assumption
have hcharne : ringChar K ≠ 0 := by simp [hchar]
have hchar' : (ringChar K : K) = 3 := by simp [hchar]
have hchar'' : (3 : K) = 0 := by simp [← hchar', ringChar.Nat.cast_ringChar]
have hb2 : e.b2 = 0 := by simpa [c4_zero_iff_b2_zero_of_char_three e hchar] using hc4
have hb4 : e.b4 = 0 := b4_zero_of_b2_zero_of_disc_zero_of_char_three e hchar h hb2
rw [b2] at hb2 rw [b4] at hb4
refine ⟨?_, ?_, ?_⟩
. rw [weierstrass]
rw [← hchar, pth_root_pow_char hcharne]
simp only
rw [show
(e.a1 * pth_root (-(e.a3 ^ 2) - e.a6) + e.a3) ^ 2 +
e.a1 * pth_root (-(e.a3 ^ 2) - e.a6) * (e.a1 * pth_root (-(e.a3 ^ 2) - e.a6) + e.a3) +
e.a3 * (e.a1 * pth_root (-(e.a3 ^ 2) - e.a6) + e.a3) -
(-(e.a3 ^ 2) - e.a6 + e.a2 * pth_root (-(e.a3 ^ 2) - e.a6) ^ 2 + e.a4 * pth_root (-(e.a3 ^ 2) - e.a6) + e.a6) =
(2 * e.a1 ^ 2 - e.a2) * pth_root (-(e.a3 ^ 2) - e.a6) ^ 2 +
(4 * e.a1 * e.a3 - e.a4) * pth_root (-(e.a3 ^ 2) - e.a6) +
3 * e.a3 ^ 2
by ring]
rw [show 2 * e.a1 ^ 2 - e.a2 = 0 from ?_]
rw [show 4 * e.a1 * e.a3 - e.a4 = 0 from ?_]
simp only [zero_mul, add_zero, hchar'']
. rw [show (2 : K) = -1 by rw [← add_zero (-1), ← hchar'']; norm_num] at hb4
rw [show (4 : K) = 1 by rw [← add_zero 1, ← hchar'']; norm_num]
simp only [neg_mul, one_mul] at hb4
simp [sub_eq_add_neg, hb4]
. linear_combination (norm := ring_nf <;> simp [hchar'']) -hb2
. rw [dweierstrass_dx]
rw [hchar'', zero_mul, zero_add]
simp only
rw [show e.a1 * (e.a1 * pth_root (-(e.a3 ^ 2) - e.a6) + e.a3) - (2 * e.a2 * pth_root (-(e.a3 ^ 2) - e.a6) + e.a4) =
(e.a1 * e.a1 - 2 * e.a2) * pth_root (-(e.a3 ^ 2) - e.a6) + (e.a1 * e.a3 - e.a4)
by ring]
rw [show (2 : K) = -1 by rw [← add_zero (-1), ← hchar'']; norm_num] at hb4
rw [show (4 : K) = -2 by rw [← add_zero (-2), ← zero_mul (2 : K), ← hchar'']; norm_num] at hb2
simp only [neg_mul, one_mul, ← sub_eq_add_neg] at hb4 hb2
rw [hb4, hb2, zero_mul, zero_add]
. rw [dweierstrass_dy]
simp only
rw [show 2 * (e.a1 * pth_root (-(e.a3 ^ 2) - e.a6) + e.a3)
+ e.a1 * pth_root (- (e.a3 ^ 2) - e.a6) + e.a3 = 3 * ((e.a1 * pth_root (-(e.a3 ^ 2) - e.a6)) + e.a3) by ring]
rw [hchar'', zero_mul] | case pos.h_2
K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : c4 e = 0
hc6 : c6 e = 0
x✝ : ℕ
heq✝ : ringChar K = 3
⊢ is_singular_point e (pth_root (-e.a3 ^ 2 - e.a6), e.a1 * pth_root (-e.a3 ^ 2 - e.a6) + e.a3)
case pos.h_3
K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : c4 e = 0
hc6 : c6 e = 0
x✝² : ℕ
x✝¹ : ringChar K = 2 → False
x✝ : ringChar K = 3 → False
⊢ is_singular_point e (-b2 e / 12, -(-e.a1 * b2 e / 12 + e.a3) / 2) | case pos.h_3
K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : c4 e = 0
hc6 : c6 e = 0
x✝² : ℕ
x✝¹ : ringChar K = 2 → False
x✝ : ringChar K = 3 → False
⊢ is_singular_point e (-b2 e / 12, -(-e.a1 * b2 e / 12 + e.a3) / 2) |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/EllipticCurve/Model.lean | Model.Field.is_singular_point_singular_point | [509, 1] | [694, 11] | . rename_i hn2 hn3
rw [is_singular_point]
have h2 : (2 : K) ≠ 0 := fun hh => hn2 (ringChar_eq_of_Prime hh (by norm_num))
have h3 : (3 : K) ≠ 0 := fun hh => hn3 (ringChar_eq_of_Prime hh (by norm_num))
have h12 : (12 : K) ≠ 0 := by
rw [show (12 : K) = 2 * 2 * 3 by norm_num]
repeat' apply mul_ne_zero
all_goals assumption
refine ⟨?_, ?_, ?_⟩
. apply nzero_mul_left_cancel (12 ^ 3) _ _ (pow_ne_zero _ h12)
simp only [weierstrass, div_eq_mul_inv, mul_zero]
rw [show
12 ^ 3 * ((-(-e.a1 * b2 e * 12⁻¹ + e.a3) * 2⁻¹) ^ 2 +
e.a1 * (-b2 e * 12⁻¹) * (-(-e.a1 * b2 e * 12⁻¹ + e.a3) * 2⁻¹) +
e.a3 * (-(-e.a1 * b2 e * 12⁻¹ + e.a3) * 2⁻¹) -
((-b2 e * 12⁻¹) ^ 3 + e.a2 * (-b2 e * 12⁻¹) ^ 2 + e.a4 * (-b2 e * 12⁻¹) + e.a6)) =
3*(-(-e.a1 * b2 e * (12 * 12⁻¹) + 12 * e.a3) * (2 * 2⁻¹)) ^ 2 +
e.a1 * (-b2 e * (12 * 12⁻¹)) * (-(-e.a1 * b2 e * (12 * 12⁻¹) + 12 * e.a3) * (6 * (2 * 2⁻¹))) +
12 * e.a3 * (-(-e.a1 * b2 e * (12 * 12⁻¹) + 12 * e.a3) * (6 * (2 * 2⁻¹))) -
((-b2 e * (12 * 12⁻¹)) ^ 3 + 12 * e.a2 * (-b2 e * (12 * 12⁻¹)) ^ 2 + 12 ^ 2 * e.a4 * (-b2 e * (12 * 12⁻¹)) + 12 ^ 3 * e.a6) by ring]
simp only [mul_inv_cancel h2, mul_inv_cancel h12, one_mul, mul_one]
rw [← mul_zero (2 : K), ← hc6]
simp only [c6, b2, b4, b6]
ring
. apply nzero_mul_left_cancel (12 ^ 2) _ _ (pow_ne_zero _ h12)
simp only [dweierstrass_dx, div_eq_mul_inv, mul_zero]
rw [show
12 ^ 2 * (e.a1 * (-(-e.a1 * b2 e * 12⁻¹ + e.a3) * 2⁻¹) - (3 * (-b2 e * 12⁻¹) ^ 2 + 2 * e.a2 * (-b2 e * 12⁻¹) + e.a4))
=
e.a1 * (-(-e.a1 * b2 e * (12 * 12⁻¹) + 12* e.a3) * 6 * (2 * 2⁻¹)) - (3 * (-b2 e * (12 * 12⁻¹)) ^ 2 + 24 * e.a2 * (-b2 e * (12 * 12⁻¹)) + 144 * e.a4)
by ring]
simp only [mul_inv_cancel h2, mul_inv_cancel h12, one_mul, mul_one]
rw [← mul_zero (3 : K), ← hc4]
simp only [c4, c6, b2, b4, b6]
ring
. apply nzero_mul_left_cancel 12 _ _ h12
simp only [dweierstrass_dy, div_eq_mul_inv, mul_zero]
rw [show
12 * (2 * (-(-e.a1 * b2 e * 12⁻¹ + e.a3) * 2⁻¹) + e.a1 * (-b2 e * 12⁻¹) + e.a3)
=
(-(-e.a1 * b2 e * (12 * 12⁻¹) + 12 * e.a3) * (2 * 2⁻¹)) + e.a1 * (-b2 e * (12 * 12⁻¹)) + 12 * e.a3
by ring]
simp only [mul_inv_cancel h2, mul_inv_cancel h12, one_mul, mul_one]
simp only [c6, b2, b4, b6]
ring | case pos.h_3
K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : c4 e = 0
hc6 : c6 e = 0
x✝² : ℕ
x✝¹ : ringChar K = 2 → False
x✝ : ringChar K = 3 → False
⊢ is_singular_point e (-b2 e / 12, -(-e.a1 * b2 e / 12 + e.a3) / 2) | no goals |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/EllipticCurve/Model.lean | Model.Field.is_singular_point_singular_point | [509, 1] | [694, 11] | simpa [h, hc4, pow_succ, mul_eq_zero] using discr_identity e | K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : c4 e = 0
⊢ c6 e = 0 | no goals |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/EllipticCurve/Model.lean | Model.Field.is_singular_point_singular_point | [509, 1] | [694, 11] | rw [is_singular_point] | case pos.h_1
K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : c4 e = 0
hc6 : c6 e = 0
x✝ : ℕ
heq✝ : ringChar K = 2
⊢ is_singular_point e (pth_root e.a4, pth_root (e.a2 * e.a4 + e.a6)) | case pos.h_1
K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : c4 e = 0
hc6 : c6 e = 0
x✝ : ℕ
heq✝ : ringChar K = 2
⊢ weierstrass e (pth_root e.a4, pth_root (e.a2 * e.a4 + e.a6)) = 0 ∧
dweierstrass_dx e (pth_root e.a4, pth_root (e.a2 * e.a4 + e.a6)) = 0 ∧
dweierstrass_dy e (pth_root e.a4, pth_root (e.a2 * e.a4 + e.a6)) = 0 |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/EllipticCurve/Model.lean | Model.Field.is_singular_point_singular_point | [509, 1] | [694, 11] | have hchar : ringChar K = 2 := by assumption | case pos.h_1
K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : c4 e = 0
hc6 : c6 e = 0
x✝ : ℕ
heq✝ : ringChar K = 2
⊢ weierstrass e (pth_root e.a4, pth_root (e.a2 * e.a4 + e.a6)) = 0 ∧
dweierstrass_dx e (pth_root e.a4, pth_root (e.a2 * e.a4 + e.a6)) = 0 ∧
dweierstrass_dy e (pth_root e.a4, pth_root (e.a2 * e.a4 + e.a6)) = 0 | case pos.h_1
K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : c4 e = 0
hc6 : c6 e = 0
x✝ : ℕ
heq✝ : ringChar K = 2
hchar : ringChar K = 2
⊢ weierstrass e (pth_root e.a4, pth_root (e.a2 * e.a4 + e.a6)) = 0 ∧
dweierstrass_dx e (pth_root e.a4, pth_root (e.a2 * e.a4 + e.a6)) = 0 ∧
dweierstrass_dy e (pth_root e.a4, pth_root (e.a2 * e.a4 + e.a6)) = 0 |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/EllipticCurve/Model.lean | Model.Field.is_singular_point_singular_point | [509, 1] | [694, 11] | have hchar' : (ringChar K : K) = 2 := by simp [hchar] | case pos.h_1
K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : c4 e = 0
hc6 : c6 e = 0
x✝ : ℕ
heq✝ : ringChar K = 2
hchar : ringChar K = 2
⊢ weierstrass e (pth_root e.a4, pth_root (e.a2 * e.a4 + e.a6)) = 0 ∧
dweierstrass_dx e (pth_root e.a4, pth_root (e.a2 * e.a4 + e.a6)) = 0 ∧
dweierstrass_dy e (pth_root e.a4, pth_root (e.a2 * e.a4 + e.a6)) = 0 | case pos.h_1
K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : c4 e = 0
hc6 : c6 e = 0
x✝ : ℕ
heq✝ : ringChar K = 2
hchar : ringChar K = 2
hchar' : ↑(ringChar K) = 2
⊢ weierstrass e (pth_root e.a4, pth_root (e.a2 * e.a4 + e.a6)) = 0 ∧
dweierstrass_dx e (pth_root e.a4, pth_root (e.a2 * e.a4 + e.a6)) = 0 ∧
dweierstrass_dy e (pth_root e.a4, pth_root (e.a2 * e.a4 + e.a6)) = 0 |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/EllipticCurve/Model.lean | Model.Field.is_singular_point_singular_point | [509, 1] | [694, 11] | have hchar'' : (2 : K) = 0 := by simp [← hchar', ringChar.Nat.cast_ringChar] | case pos.h_1
K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : c4 e = 0
hc6 : c6 e = 0
x✝ : ℕ
heq✝ : ringChar K = 2
hchar : ringChar K = 2
hchar' : ↑(ringChar K) = 2
⊢ weierstrass e (pth_root e.a4, pth_root (e.a2 * e.a4 + e.a6)) = 0 ∧
dweierstrass_dx e (pth_root e.a4, pth_root (e.a2 * e.a4 + e.a6)) = 0 ∧
dweierstrass_dy e (pth_root e.a4, pth_root (e.a2 * e.a4 + e.a6)) = 0 | case pos.h_1
K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : c4 e = 0
hc6 : c6 e = 0
x✝ : ℕ
heq✝ : ringChar K = 2
hchar : ringChar K = 2
hchar' : ↑(ringChar K) = 2
hchar'' : 2 = 0
⊢ weierstrass e (pth_root e.a4, pth_root (e.a2 * e.a4 + e.a6)) = 0 ∧
dweierstrass_dx e (pth_root e.a4, pth_root (e.a2 * e.a4 + e.a6)) = 0 ∧
dweierstrass_dy e (pth_root e.a4, pth_root (e.a2 * e.a4 + e.a6)) = 0 |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/EllipticCurve/Model.lean | Model.Field.is_singular_point_singular_point | [509, 1] | [694, 11] | have hcharne : ringChar K ≠ 0 := by simp [hchar] | case pos.h_1
K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : c4 e = 0
hc6 : c6 e = 0
x✝ : ℕ
heq✝ : ringChar K = 2
hchar : ringChar K = 2
hchar' : ↑(ringChar K) = 2
hchar'' : 2 = 0
⊢ weierstrass e (pth_root e.a4, pth_root (e.a2 * e.a4 + e.a6)) = 0 ∧
dweierstrass_dx e (pth_root e.a4, pth_root (e.a2 * e.a4 + e.a6)) = 0 ∧
dweierstrass_dy e (pth_root e.a4, pth_root (e.a2 * e.a4 + e.a6)) = 0 | case pos.h_1
K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : c4 e = 0
hc6 : c6 e = 0
x✝ : ℕ
heq✝ : ringChar K = 2
hchar : ringChar K = 2
hchar' : ↑(ringChar K) = 2
hchar'' : 2 = 0
hcharne : ringChar K ≠ 0
⊢ weierstrass e (pth_root e.a4, pth_root (e.a2 * e.a4 + e.a6)) = 0 ∧
dweierstrass_dx e (pth_root e.a4, pth_root (e.a2 * e.a4 + e.a6)) = 0 ∧
dweierstrass_dy e (pth_root e.a4, pth_root (e.a2 * e.a4 + e.a6)) = 0 |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/EllipticCurve/Model.lean | Model.Field.is_singular_point_singular_point | [509, 1] | [694, 11] | have ha1 : e.a1 = 0 := by simpa [c4_zero_iff_a1_zero_of_char_two e hchar] using hc4 | case pos.h_1
K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : c4 e = 0
hc6 : c6 e = 0
x✝ : ℕ
heq✝ : ringChar K = 2
hchar : ringChar K = 2
hchar' : ↑(ringChar K) = 2
hchar'' : 2 = 0
hcharne : ringChar K ≠ 0
⊢ weierstrass e (pth_root e.a4, pth_root (e.a2 * e.a4 + e.a6)) = 0 ∧
dweierstrass_dx e (pth_root e.a4, pth_root (e.a2 * e.a4 + e.a6)) = 0 ∧
dweierstrass_dy e (pth_root e.a4, pth_root (e.a2 * e.a4 + e.a6)) = 0 | case pos.h_1
K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : c4 e = 0
hc6 : c6 e = 0
x✝ : ℕ
heq✝ : ringChar K = 2
hchar : ringChar K = 2
hchar' : ↑(ringChar K) = 2
hchar'' : 2 = 0
hcharne : ringChar K ≠ 0
ha1 : e.a1 = 0
⊢ weierstrass e (pth_root e.a4, pth_root (e.a2 * e.a4 + e.a6)) = 0 ∧
dweierstrass_dx e (pth_root e.a4, pth_root (e.a2 * e.a4 + e.a6)) = 0 ∧
dweierstrass_dy e (pth_root e.a4, pth_root (e.a2 * e.a4 + e.a6)) = 0 |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/EllipticCurve/Model.lean | Model.Field.is_singular_point_singular_point | [509, 1] | [694, 11] | have ha3 : e.a3 = 0 := a3_zero_of_a1_zero_of_disc_zero_of_char_two e hchar h ha1 | case pos.h_1
K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : c4 e = 0
hc6 : c6 e = 0
x✝ : ℕ
heq✝ : ringChar K = 2
hchar : ringChar K = 2
hchar' : ↑(ringChar K) = 2
hchar'' : 2 = 0
hcharne : ringChar K ≠ 0
ha1 : e.a1 = 0
⊢ weierstrass e (pth_root e.a4, pth_root (e.a2 * e.a4 + e.a6)) = 0 ∧
dweierstrass_dx e (pth_root e.a4, pth_root (e.a2 * e.a4 + e.a6)) = 0 ∧
dweierstrass_dy e (pth_root e.a4, pth_root (e.a2 * e.a4 + e.a6)) = 0 | case pos.h_1
K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : c4 e = 0
hc6 : c6 e = 0
x✝ : ℕ
heq✝ : ringChar K = 2
hchar : ringChar K = 2
hchar' : ↑(ringChar K) = 2
hchar'' : 2 = 0
hcharne : ringChar K ≠ 0
ha1 : e.a1 = 0
ha3 : e.a3 = 0
⊢ weierstrass e (pth_root e.a4, pth_root (e.a2 * e.a4 + e.a6)) = 0 ∧
dweierstrass_dx e (pth_root e.a4, pth_root (e.a2 * e.a4 + e.a6)) = 0 ∧
dweierstrass_dy e (pth_root e.a4, pth_root (e.a2 * e.a4 + e.a6)) = 0 |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/EllipticCurve/Model.lean | Model.Field.is_singular_point_singular_point | [509, 1] | [694, 11] | refine ⟨?_, ?_, ?_⟩ | case pos.h_1
K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : c4 e = 0
hc6 : c6 e = 0
x✝ : ℕ
heq✝ : ringChar K = 2
hchar : ringChar K = 2
hchar' : ↑(ringChar K) = 2
hchar'' : 2 = 0
hcharne : ringChar K ≠ 0
ha1 : e.a1 = 0
ha3 : e.a3 = 0
⊢ weierstrass e (pth_root e.a4, pth_root (e.a2 * e.a4 + e.a6)) = 0 ∧
dweierstrass_dx e (pth_root e.a4, pth_root (e.a2 * e.a4 + e.a6)) = 0 ∧
dweierstrass_dy e (pth_root e.a4, pth_root (e.a2 * e.a4 + e.a6)) = 0 | case pos.h_1.refine_1
K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : c4 e = 0
hc6 : c6 e = 0
x✝ : ℕ
heq✝ : ringChar K = 2
hchar : ringChar K = 2
hchar' : ↑(ringChar K) = 2
hchar'' : 2 = 0
hcharne : ringChar K ≠ 0
ha1 : e.a1 = 0
ha3 : e.a3 = 0
⊢ weierstrass e (pth_root e.a4, pth_root (e.a2 * e.a4 + e.a6)) = 0
case pos.h_1.refine_2
K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : c4 e = 0
hc6 : c6 e = 0
x✝ : ℕ
heq✝ : ringChar K = 2
hchar : ringChar K = 2
hchar' : ↑(ringChar K) = 2
hchar'' : 2 = 0
hcharne : ringChar K ≠ 0
ha1 : e.a1 = 0
ha3 : e.a3 = 0
⊢ dweierstrass_dx e (pth_root e.a4, pth_root (e.a2 * e.a4 + e.a6)) = 0
case pos.h_1.refine_3
K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : c4 e = 0
hc6 : c6 e = 0
x✝ : ℕ
heq✝ : ringChar K = 2
hchar : ringChar K = 2
hchar' : ↑(ringChar K) = 2
hchar'' : 2 = 0
hcharne : ringChar K ≠ 0
ha1 : e.a1 = 0
ha3 : e.a3 = 0
⊢ dweierstrass_dy e (pth_root e.a4, pth_root (e.a2 * e.a4 + e.a6)) = 0 |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/EllipticCurve/Model.lean | Model.Field.is_singular_point_singular_point | [509, 1] | [694, 11] | . rw [weierstrass]
simp only [ha1, ha3, mul_zero, zero_add, zero_mul, add_zero]
rw [show 3 = 2 + 1 by norm_num]
rw [pow_succ _ 2]
rw [← hchar, pth_root_pow_char hcharne]
rw [pth_root_pow_char hcharne]
ring_nf
simp [hchar''] | case pos.h_1.refine_1
K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : c4 e = 0
hc6 : c6 e = 0
x✝ : ℕ
heq✝ : ringChar K = 2
hchar : ringChar K = 2
hchar' : ↑(ringChar K) = 2
hchar'' : 2 = 0
hcharne : ringChar K ≠ 0
ha1 : e.a1 = 0
ha3 : e.a3 = 0
⊢ weierstrass e (pth_root e.a4, pth_root (e.a2 * e.a4 + e.a6)) = 0
case pos.h_1.refine_2
K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : c4 e = 0
hc6 : c6 e = 0
x✝ : ℕ
heq✝ : ringChar K = 2
hchar : ringChar K = 2
hchar' : ↑(ringChar K) = 2
hchar'' : 2 = 0
hcharne : ringChar K ≠ 0
ha1 : e.a1 = 0
ha3 : e.a3 = 0
⊢ dweierstrass_dx e (pth_root e.a4, pth_root (e.a2 * e.a4 + e.a6)) = 0
case pos.h_1.refine_3
K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : c4 e = 0
hc6 : c6 e = 0
x✝ : ℕ
heq✝ : ringChar K = 2
hchar : ringChar K = 2
hchar' : ↑(ringChar K) = 2
hchar'' : 2 = 0
hcharne : ringChar K ≠ 0
ha1 : e.a1 = 0
ha3 : e.a3 = 0
⊢ dweierstrass_dy e (pth_root e.a4, pth_root (e.a2 * e.a4 + e.a6)) = 0 | case pos.h_1.refine_2
K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : c4 e = 0
hc6 : c6 e = 0
x✝ : ℕ
heq✝ : ringChar K = 2
hchar : ringChar K = 2
hchar' : ↑(ringChar K) = 2
hchar'' : 2 = 0
hcharne : ringChar K ≠ 0
ha1 : e.a1 = 0
ha3 : e.a3 = 0
⊢ dweierstrass_dx e (pth_root e.a4, pth_root (e.a2 * e.a4 + e.a6)) = 0
case pos.h_1.refine_3
K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : c4 e = 0
hc6 : c6 e = 0
x✝ : ℕ
heq✝ : ringChar K = 2
hchar : ringChar K = 2
hchar' : ↑(ringChar K) = 2
hchar'' : 2 = 0
hcharne : ringChar K ≠ 0
ha1 : e.a1 = 0
ha3 : e.a3 = 0
⊢ dweierstrass_dy e (pth_root e.a4, pth_root (e.a2 * e.a4 + e.a6)) = 0 |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/EllipticCurve/Model.lean | Model.Field.is_singular_point_singular_point | [509, 1] | [694, 11] | . rw [dweierstrass_dx]
simp only [ha1, zero_mul, hchar'', add_zero, zero_sub, neg_add_rev]
rw [← hchar, pth_root_pow_char hcharne, ← sub_eq_add_neg]
simp only [add_sub_add_right_eq_sub, sub_eq_iff_eq_add, zero_add]
rw [show (3 : K) = 2 * 2 - 1 by norm_num]
rw [hchar'']
simp [← neg_mul_eq_neg_mul] | case pos.h_1.refine_2
K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : c4 e = 0
hc6 : c6 e = 0
x✝ : ℕ
heq✝ : ringChar K = 2
hchar : ringChar K = 2
hchar' : ↑(ringChar K) = 2
hchar'' : 2 = 0
hcharne : ringChar K ≠ 0
ha1 : e.a1 = 0
ha3 : e.a3 = 0
⊢ dweierstrass_dx e (pth_root e.a4, pth_root (e.a2 * e.a4 + e.a6)) = 0
case pos.h_1.refine_3
K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : c4 e = 0
hc6 : c6 e = 0
x✝ : ℕ
heq✝ : ringChar K = 2
hchar : ringChar K = 2
hchar' : ↑(ringChar K) = 2
hchar'' : 2 = 0
hcharne : ringChar K ≠ 0
ha1 : e.a1 = 0
ha3 : e.a3 = 0
⊢ dweierstrass_dy e (pth_root e.a4, pth_root (e.a2 * e.a4 + e.a6)) = 0 | case pos.h_1.refine_3
K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : c4 e = 0
hc6 : c6 e = 0
x✝ : ℕ
heq✝ : ringChar K = 2
hchar : ringChar K = 2
hchar' : ↑(ringChar K) = 2
hchar'' : 2 = 0
hcharne : ringChar K ≠ 0
ha1 : e.a1 = 0
ha3 : e.a3 = 0
⊢ dweierstrass_dy e (pth_root e.a4, pth_root (e.a2 * e.a4 + e.a6)) = 0 |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/EllipticCurve/Model.lean | Model.Field.is_singular_point_singular_point | [509, 1] | [694, 11] | . simp [dweierstrass_dy, ha1, ha3, hchar''] | case pos.h_1.refine_3
K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : c4 e = 0
hc6 : c6 e = 0
x✝ : ℕ
heq✝ : ringChar K = 2
hchar : ringChar K = 2
hchar' : ↑(ringChar K) = 2
hchar'' : 2 = 0
hcharne : ringChar K ≠ 0
ha1 : e.a1 = 0
ha3 : e.a3 = 0
⊢ dweierstrass_dy e (pth_root e.a4, pth_root (e.a2 * e.a4 + e.a6)) = 0 | no goals |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/EllipticCurve/Model.lean | Model.Field.is_singular_point_singular_point | [509, 1] | [694, 11] | assumption | K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : c4 e = 0
hc6 : c6 e = 0
x✝ : ℕ
heq✝ : ringChar K = 2
⊢ ringChar K = 2 | no goals |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/EllipticCurve/Model.lean | Model.Field.is_singular_point_singular_point | [509, 1] | [694, 11] | simp [hchar] | K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : c4 e = 0
hc6 : c6 e = 0
x✝ : ℕ
heq✝ : ringChar K = 2
hchar : ringChar K = 2
⊢ ↑(ringChar K) = 2 | no goals |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/EllipticCurve/Model.lean | Model.Field.is_singular_point_singular_point | [509, 1] | [694, 11] | simp [← hchar', ringChar.Nat.cast_ringChar] | K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : c4 e = 0
hc6 : c6 e = 0
x✝ : ℕ
heq✝ : ringChar K = 2
hchar : ringChar K = 2
hchar' : ↑(ringChar K) = 2
⊢ 2 = 0 | no goals |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/EllipticCurve/Model.lean | Model.Field.is_singular_point_singular_point | [509, 1] | [694, 11] | simp [hchar] | K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : c4 e = 0
hc6 : c6 e = 0
x✝ : ℕ
heq✝ : ringChar K = 2
hchar : ringChar K = 2
hchar' : ↑(ringChar K) = 2
hchar'' : 2 = 0
⊢ ringChar K ≠ 0 | no goals |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/EllipticCurve/Model.lean | Model.Field.is_singular_point_singular_point | [509, 1] | [694, 11] | simpa [c4_zero_iff_a1_zero_of_char_two e hchar] using hc4 | K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : c4 e = 0
hc6 : c6 e = 0
x✝ : ℕ
heq✝ : ringChar K = 2
hchar : ringChar K = 2
hchar' : ↑(ringChar K) = 2
hchar'' : 2 = 0
hcharne : ringChar K ≠ 0
⊢ e.a1 = 0 | no goals |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/EllipticCurve/Model.lean | Model.Field.is_singular_point_singular_point | [509, 1] | [694, 11] | rw [weierstrass] | case pos.h_1.refine_1
K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : c4 e = 0
hc6 : c6 e = 0
x✝ : ℕ
heq✝ : ringChar K = 2
hchar : ringChar K = 2
hchar' : ↑(ringChar K) = 2
hchar'' : 2 = 0
hcharne : ringChar K ≠ 0
ha1 : e.a1 = 0
ha3 : e.a3 = 0
⊢ weierstrass e (pth_root e.a4, pth_root (e.a2 * e.a4 + e.a6)) = 0 | case pos.h_1.refine_1
K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : c4 e = 0
hc6 : c6 e = 0
x✝ : ℕ
heq✝ : ringChar K = 2
hchar : ringChar K = 2
hchar' : ↑(ringChar K) = 2
hchar'' : 2 = 0
hcharne : ringChar K ≠ 0
ha1 : e.a1 = 0
ha3 : e.a3 = 0
⊢ (pth_root e.a4, pth_root (e.a2 * e.a4 + e.a6)).snd ^ 2 +
e.a1 * (pth_root e.a4, pth_root (e.a2 * e.a4 + e.a6)).fst *
(pth_root e.a4, pth_root (e.a2 * e.a4 + e.a6)).snd +
e.a3 * (pth_root e.a4, pth_root (e.a2 * e.a4 + e.a6)).snd -
((pth_root e.a4, pth_root (e.a2 * e.a4 + e.a6)).fst ^ 3 +
e.a2 * (pth_root e.a4, pth_root (e.a2 * e.a4 + e.a6)).fst ^ 2 +
e.a4 * (pth_root e.a4, pth_root (e.a2 * e.a4 + e.a6)).fst +
e.a6) =
0 |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/EllipticCurve/Model.lean | Model.Field.is_singular_point_singular_point | [509, 1] | [694, 11] | simp only [ha1, ha3, mul_zero, zero_add, zero_mul, add_zero] | case pos.h_1.refine_1
K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : c4 e = 0
hc6 : c6 e = 0
x✝ : ℕ
heq✝ : ringChar K = 2
hchar : ringChar K = 2
hchar' : ↑(ringChar K) = 2
hchar'' : 2 = 0
hcharne : ringChar K ≠ 0
ha1 : e.a1 = 0
ha3 : e.a3 = 0
⊢ (pth_root e.a4, pth_root (e.a2 * e.a4 + e.a6)).snd ^ 2 +
e.a1 * (pth_root e.a4, pth_root (e.a2 * e.a4 + e.a6)).fst *
(pth_root e.a4, pth_root (e.a2 * e.a4 + e.a6)).snd +
e.a3 * (pth_root e.a4, pth_root (e.a2 * e.a4 + e.a6)).snd -
((pth_root e.a4, pth_root (e.a2 * e.a4 + e.a6)).fst ^ 3 +
e.a2 * (pth_root e.a4, pth_root (e.a2 * e.a4 + e.a6)).fst ^ 2 +
e.a4 * (pth_root e.a4, pth_root (e.a2 * e.a4 + e.a6)).fst +
e.a6) =
0 | case pos.h_1.refine_1
K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : c4 e = 0
hc6 : c6 e = 0
x✝ : ℕ
heq✝ : ringChar K = 2
hchar : ringChar K = 2
hchar' : ↑(ringChar K) = 2
hchar'' : 2 = 0
hcharne : ringChar K ≠ 0
ha1 : e.a1 = 0
ha3 : e.a3 = 0
⊢ pth_root (e.a2 * e.a4 + e.a6) ^ 2 - (pth_root e.a4 ^ 3 + e.a2 * pth_root e.a4 ^ 2 + e.a4 * pth_root e.a4 + e.a6) = 0 |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/EllipticCurve/Model.lean | Model.Field.is_singular_point_singular_point | [509, 1] | [694, 11] | rw [show 3 = 2 + 1 by norm_num] | case pos.h_1.refine_1
K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : c4 e = 0
hc6 : c6 e = 0
x✝ : ℕ
heq✝ : ringChar K = 2
hchar : ringChar K = 2
hchar' : ↑(ringChar K) = 2
hchar'' : 2 = 0
hcharne : ringChar K ≠ 0
ha1 : e.a1 = 0
ha3 : e.a3 = 0
⊢ pth_root (e.a2 * e.a4 + e.a6) ^ 2 - (pth_root e.a4 ^ 3 + e.a2 * pth_root e.a4 ^ 2 + e.a4 * pth_root e.a4 + e.a6) = 0 | case pos.h_1.refine_1
K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : c4 e = 0
hc6 : c6 e = 0
x✝ : ℕ
heq✝ : ringChar K = 2
hchar : ringChar K = 2
hchar' : ↑(ringChar K) = 2
hchar'' : 2 = 0
hcharne : ringChar K ≠ 0
ha1 : e.a1 = 0
ha3 : e.a3 = 0
⊢ pth_root (e.a2 * e.a4 + e.a6) ^ 2 -
(pth_root e.a4 ^ (2 + 1) + e.a2 * pth_root e.a4 ^ 2 + e.a4 * pth_root e.a4 + e.a6) =
0 |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/EllipticCurve/Model.lean | Model.Field.is_singular_point_singular_point | [509, 1] | [694, 11] | rw [pow_succ _ 2] | case pos.h_1.refine_1
K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : c4 e = 0
hc6 : c6 e = 0
x✝ : ℕ
heq✝ : ringChar K = 2
hchar : ringChar K = 2
hchar' : ↑(ringChar K) = 2
hchar'' : 2 = 0
hcharne : ringChar K ≠ 0
ha1 : e.a1 = 0
ha3 : e.a3 = 0
⊢ pth_root (e.a2 * e.a4 + e.a6) ^ 2 -
(pth_root e.a4 ^ (2 + 1) + e.a2 * pth_root e.a4 ^ 2 + e.a4 * pth_root e.a4 + e.a6) =
0 | case pos.h_1.refine_1
K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : c4 e = 0
hc6 : c6 e = 0
x✝ : ℕ
heq✝ : ringChar K = 2
hchar : ringChar K = 2
hchar' : ↑(ringChar K) = 2
hchar'' : 2 = 0
hcharne : ringChar K ≠ 0
ha1 : e.a1 = 0
ha3 : e.a3 = 0
⊢ pth_root (e.a2 * e.a4 + e.a6) ^ 2 -
(pth_root e.a4 * pth_root e.a4 ^ 2 + e.a2 * pth_root e.a4 ^ 2 + e.a4 * pth_root e.a4 + e.a6) =
0 |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/EllipticCurve/Model.lean | Model.Field.is_singular_point_singular_point | [509, 1] | [694, 11] | rw [← hchar, pth_root_pow_char hcharne] | case pos.h_1.refine_1
K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : c4 e = 0
hc6 : c6 e = 0
x✝ : ℕ
heq✝ : ringChar K = 2
hchar : ringChar K = 2
hchar' : ↑(ringChar K) = 2
hchar'' : 2 = 0
hcharne : ringChar K ≠ 0
ha1 : e.a1 = 0
ha3 : e.a3 = 0
⊢ pth_root (e.a2 * e.a4 + e.a6) ^ 2 -
(pth_root e.a4 * pth_root e.a4 ^ 2 + e.a2 * pth_root e.a4 ^ 2 + e.a4 * pth_root e.a4 + e.a6) =
0 | case pos.h_1.refine_1
K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : c4 e = 0
hc6 : c6 e = 0
x✝ : ℕ
heq✝ : ringChar K = 2
hchar : ringChar K = 2
hchar' : ↑(ringChar K) = 2
hchar'' : 2 = 0
hcharne : ringChar K ≠ 0
ha1 : e.a1 = 0
ha3 : e.a3 = 0
⊢ e.a2 * e.a4 + e.a6 -
(pth_root e.a4 * pth_root e.a4 ^ ringChar K + e.a2 * pth_root e.a4 ^ ringChar K + e.a4 * pth_root e.a4 + e.a6) =
0 |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/EllipticCurve/Model.lean | Model.Field.is_singular_point_singular_point | [509, 1] | [694, 11] | rw [pth_root_pow_char hcharne] | case pos.h_1.refine_1
K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : c4 e = 0
hc6 : c6 e = 0
x✝ : ℕ
heq✝ : ringChar K = 2
hchar : ringChar K = 2
hchar' : ↑(ringChar K) = 2
hchar'' : 2 = 0
hcharne : ringChar K ≠ 0
ha1 : e.a1 = 0
ha3 : e.a3 = 0
⊢ e.a2 * e.a4 + e.a6 -
(pth_root e.a4 * pth_root e.a4 ^ ringChar K + e.a2 * pth_root e.a4 ^ ringChar K + e.a4 * pth_root e.a4 + e.a6) =
0 | case pos.h_1.refine_1
K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : c4 e = 0
hc6 : c6 e = 0
x✝ : ℕ
heq✝ : ringChar K = 2
hchar : ringChar K = 2
hchar' : ↑(ringChar K) = 2
hchar'' : 2 = 0
hcharne : ringChar K ≠ 0
ha1 : e.a1 = 0
ha3 : e.a3 = 0
⊢ e.a2 * e.a4 + e.a6 - (pth_root e.a4 * e.a4 + e.a2 * e.a4 + e.a4 * pth_root e.a4 + e.a6) = 0 |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/EllipticCurve/Model.lean | Model.Field.is_singular_point_singular_point | [509, 1] | [694, 11] | ring_nf | case pos.h_1.refine_1
K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : c4 e = 0
hc6 : c6 e = 0
x✝ : ℕ
heq✝ : ringChar K = 2
hchar : ringChar K = 2
hchar' : ↑(ringChar K) = 2
hchar'' : 2 = 0
hcharne : ringChar K ≠ 0
ha1 : e.a1 = 0
ha3 : e.a3 = 0
⊢ e.a2 * e.a4 + e.a6 - (pth_root e.a4 * e.a4 + e.a2 * e.a4 + e.a4 * pth_root e.a4 + e.a6) = 0 | case pos.h_1.refine_1
K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : c4 e = 0
hc6 : c6 e = 0
x✝ : ℕ
heq✝ : ringChar K = 2
hchar : ringChar K = 2
hchar' : ↑(ringChar K) = 2
hchar'' : 2 = 0
hcharne : ringChar K ≠ 0
ha1 : e.a1 = 0
ha3 : e.a3 = 0
⊢ -(e.a4 * pth_root e.a4 * 2) = 0 |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/EllipticCurve/Model.lean | Model.Field.is_singular_point_singular_point | [509, 1] | [694, 11] | simp [hchar''] | case pos.h_1.refine_1
K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : c4 e = 0
hc6 : c6 e = 0
x✝ : ℕ
heq✝ : ringChar K = 2
hchar : ringChar K = 2
hchar' : ↑(ringChar K) = 2
hchar'' : 2 = 0
hcharne : ringChar K ≠ 0
ha1 : e.a1 = 0
ha3 : e.a3 = 0
⊢ -(e.a4 * pth_root e.a4 * 2) = 0 | no goals |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/EllipticCurve/Model.lean | Model.Field.is_singular_point_singular_point | [509, 1] | [694, 11] | norm_num | K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : c4 e = 0
hc6 : c6 e = 0
x✝ : ℕ
heq✝ : ringChar K = 2
hchar : ringChar K = 2
hchar' : ↑(ringChar K) = 2
hchar'' : 2 = 0
hcharne : ringChar K ≠ 0
ha1 : e.a1 = 0
ha3 : e.a3 = 0
⊢ 3 = 2 + 1 | no goals |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/EllipticCurve/Model.lean | Model.Field.is_singular_point_singular_point | [509, 1] | [694, 11] | rw [dweierstrass_dx] | case pos.h_1.refine_2
K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : c4 e = 0
hc6 : c6 e = 0
x✝ : ℕ
heq✝ : ringChar K = 2
hchar : ringChar K = 2
hchar' : ↑(ringChar K) = 2
hchar'' : 2 = 0
hcharne : ringChar K ≠ 0
ha1 : e.a1 = 0
ha3 : e.a3 = 0
⊢ dweierstrass_dx e (pth_root e.a4, pth_root (e.a2 * e.a4 + e.a6)) = 0 | case pos.h_1.refine_2
K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : c4 e = 0
hc6 : c6 e = 0
x✝ : ℕ
heq✝ : ringChar K = 2
hchar : ringChar K = 2
hchar' : ↑(ringChar K) = 2
hchar'' : 2 = 0
hcharne : ringChar K ≠ 0
ha1 : e.a1 = 0
ha3 : e.a3 = 0
⊢ e.a1 * (pth_root e.a4, pth_root (e.a2 * e.a4 + e.a6)).snd -
(3 * (pth_root e.a4, pth_root (e.a2 * e.a4 + e.a6)).fst ^ 2 +
2 * e.a2 * (pth_root e.a4, pth_root (e.a2 * e.a4 + e.a6)).fst +
e.a4) =
0 |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/EllipticCurve/Model.lean | Model.Field.is_singular_point_singular_point | [509, 1] | [694, 11] | simp only [ha1, zero_mul, hchar'', add_zero, zero_sub, neg_add_rev] | case pos.h_1.refine_2
K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : c4 e = 0
hc6 : c6 e = 0
x✝ : ℕ
heq✝ : ringChar K = 2
hchar : ringChar K = 2
hchar' : ↑(ringChar K) = 2
hchar'' : 2 = 0
hcharne : ringChar K ≠ 0
ha1 : e.a1 = 0
ha3 : e.a3 = 0
⊢ e.a1 * (pth_root e.a4, pth_root (e.a2 * e.a4 + e.a6)).snd -
(3 * (pth_root e.a4, pth_root (e.a2 * e.a4 + e.a6)).fst ^ 2 +
2 * e.a2 * (pth_root e.a4, pth_root (e.a2 * e.a4 + e.a6)).fst +
e.a4) =
0 | case pos.h_1.refine_2
K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : c4 e = 0
hc6 : c6 e = 0
x✝ : ℕ
heq✝ : ringChar K = 2
hchar : ringChar K = 2
hchar' : ↑(ringChar K) = 2
hchar'' : 2 = 0
hcharne : ringChar K ≠ 0
ha1 : e.a1 = 0
ha3 : e.a3 = 0
⊢ -e.a4 + -(3 * pth_root e.a4 ^ 2) = 0 |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/EllipticCurve/Model.lean | Model.Field.is_singular_point_singular_point | [509, 1] | [694, 11] | rw [← hchar, pth_root_pow_char hcharne, ← sub_eq_add_neg] | case pos.h_1.refine_2
K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : c4 e = 0
hc6 : c6 e = 0
x✝ : ℕ
heq✝ : ringChar K = 2
hchar : ringChar K = 2
hchar' : ↑(ringChar K) = 2
hchar'' : 2 = 0
hcharne : ringChar K ≠ 0
ha1 : e.a1 = 0
ha3 : e.a3 = 0
⊢ -e.a4 + -(3 * pth_root e.a4 ^ 2) = 0 | case pos.h_1.refine_2
K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : c4 e = 0
hc6 : c6 e = 0
x✝ : ℕ
heq✝ : ringChar K = 2
hchar : ringChar K = 2
hchar' : ↑(ringChar K) = 2
hchar'' : 2 = 0
hcharne : ringChar K ≠ 0
ha1 : e.a1 = 0
ha3 : e.a3 = 0
⊢ -e.a4 - 3 * e.a4 = 0 |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/EllipticCurve/Model.lean | Model.Field.is_singular_point_singular_point | [509, 1] | [694, 11] | simp only [add_sub_add_right_eq_sub, sub_eq_iff_eq_add, zero_add] | case pos.h_1.refine_2
K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : c4 e = 0
hc6 : c6 e = 0
x✝ : ℕ
heq✝ : ringChar K = 2
hchar : ringChar K = 2
hchar' : ↑(ringChar K) = 2
hchar'' : 2 = 0
hcharne : ringChar K ≠ 0
ha1 : e.a1 = 0
ha3 : e.a3 = 0
⊢ -e.a4 - 3 * e.a4 = 0 | case pos.h_1.refine_2
K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : c4 e = 0
hc6 : c6 e = 0
x✝ : ℕ
heq✝ : ringChar K = 2
hchar : ringChar K = 2
hchar' : ↑(ringChar K) = 2
hchar'' : 2 = 0
hcharne : ringChar K ≠ 0
ha1 : e.a1 = 0
ha3 : e.a3 = 0
⊢ -e.a4 = 3 * e.a4 |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/EllipticCurve/Model.lean | Model.Field.is_singular_point_singular_point | [509, 1] | [694, 11] | rw [show (3 : K) = 2 * 2 - 1 by norm_num] | case pos.h_1.refine_2
K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : c4 e = 0
hc6 : c6 e = 0
x✝ : ℕ
heq✝ : ringChar K = 2
hchar : ringChar K = 2
hchar' : ↑(ringChar K) = 2
hchar'' : 2 = 0
hcharne : ringChar K ≠ 0
ha1 : e.a1 = 0
ha3 : e.a3 = 0
⊢ -e.a4 = 3 * e.a4 | case pos.h_1.refine_2
K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : c4 e = 0
hc6 : c6 e = 0
x✝ : ℕ
heq✝ : ringChar K = 2
hchar : ringChar K = 2
hchar' : ↑(ringChar K) = 2
hchar'' : 2 = 0
hcharne : ringChar K ≠ 0
ha1 : e.a1 = 0
ha3 : e.a3 = 0
⊢ -e.a4 = (2 * 2 - 1) * e.a4 |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/EllipticCurve/Model.lean | Model.Field.is_singular_point_singular_point | [509, 1] | [694, 11] | rw [hchar''] | case pos.h_1.refine_2
K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : c4 e = 0
hc6 : c6 e = 0
x✝ : ℕ
heq✝ : ringChar K = 2
hchar : ringChar K = 2
hchar' : ↑(ringChar K) = 2
hchar'' : 2 = 0
hcharne : ringChar K ≠ 0
ha1 : e.a1 = 0
ha3 : e.a3 = 0
⊢ -e.a4 = (2 * 2 - 1) * e.a4 | case pos.h_1.refine_2
K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : c4 e = 0
hc6 : c6 e = 0
x✝ : ℕ
heq✝ : ringChar K = 2
hchar : ringChar K = 2
hchar' : ↑(ringChar K) = 2
hchar'' : 2 = 0
hcharne : ringChar K ≠ 0
ha1 : e.a1 = 0
ha3 : e.a3 = 0
⊢ -e.a4 = (0 * 0 - 1) * e.a4 |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/EllipticCurve/Model.lean | Model.Field.is_singular_point_singular_point | [509, 1] | [694, 11] | simp [← neg_mul_eq_neg_mul] | case pos.h_1.refine_2
K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : c4 e = 0
hc6 : c6 e = 0
x✝ : ℕ
heq✝ : ringChar K = 2
hchar : ringChar K = 2
hchar' : ↑(ringChar K) = 2
hchar'' : 2 = 0
hcharne : ringChar K ≠ 0
ha1 : e.a1 = 0
ha3 : e.a3 = 0
⊢ -e.a4 = (0 * 0 - 1) * e.a4 | no goals |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/EllipticCurve/Model.lean | Model.Field.is_singular_point_singular_point | [509, 1] | [694, 11] | norm_num | K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : c4 e = 0
hc6 : c6 e = 0
x✝ : ℕ
heq✝ : ringChar K = 2
hchar : ringChar K = 2
hchar' : ↑(ringChar K) = 2
hchar'' : 2 = 0
hcharne : ringChar K ≠ 0
ha1 : e.a1 = 0
ha3 : e.a3 = 0
⊢ 3 = 2 * 2 - 1 | no goals |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/EllipticCurve/Model.lean | Model.Field.is_singular_point_singular_point | [509, 1] | [694, 11] | simp [dweierstrass_dy, ha1, ha3, hchar''] | case pos.h_1.refine_3
K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : c4 e = 0
hc6 : c6 e = 0
x✝ : ℕ
heq✝ : ringChar K = 2
hchar : ringChar K = 2
hchar' : ↑(ringChar K) = 2
hchar'' : 2 = 0
hcharne : ringChar K ≠ 0
ha1 : e.a1 = 0
ha3 : e.a3 = 0
⊢ dweierstrass_dy e (pth_root e.a4, pth_root (e.a2 * e.a4 + e.a6)) = 0 | no goals |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/EllipticCurve/Model.lean | Model.Field.is_singular_point_singular_point | [509, 1] | [694, 11] | rw [is_singular_point] | case pos.h_2
K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : c4 e = 0
hc6 : c6 e = 0
x✝ : ℕ
heq✝ : ringChar K = 3
⊢ is_singular_point e (pth_root (-e.a3 ^ 2 - e.a6), e.a1 * pth_root (-e.a3 ^ 2 - e.a6) + e.a3) | case pos.h_2
K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : c4 e = 0
hc6 : c6 e = 0
x✝ : ℕ
heq✝ : ringChar K = 3
⊢ weierstrass e (pth_root (-e.a3 ^ 2 - e.a6), e.a1 * pth_root (-e.a3 ^ 2 - e.a6) + e.a3) = 0 ∧
dweierstrass_dx e (pth_root (-e.a3 ^ 2 - e.a6), e.a1 * pth_root (-e.a3 ^ 2 - e.a6) + e.a3) = 0 ∧
dweierstrass_dy e (pth_root (-e.a3 ^ 2 - e.a6), e.a1 * pth_root (-e.a3 ^ 2 - e.a6) + e.a3) = 0 |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/EllipticCurve/Model.lean | Model.Field.is_singular_point_singular_point | [509, 1] | [694, 11] | have hchar : ringChar K = 3 := by assumption | case pos.h_2
K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : c4 e = 0
hc6 : c6 e = 0
x✝ : ℕ
heq✝ : ringChar K = 3
⊢ weierstrass e (pth_root (-e.a3 ^ 2 - e.a6), e.a1 * pth_root (-e.a3 ^ 2 - e.a6) + e.a3) = 0 ∧
dweierstrass_dx e (pth_root (-e.a3 ^ 2 - e.a6), e.a1 * pth_root (-e.a3 ^ 2 - e.a6) + e.a3) = 0 ∧
dweierstrass_dy e (pth_root (-e.a3 ^ 2 - e.a6), e.a1 * pth_root (-e.a3 ^ 2 - e.a6) + e.a3) = 0 | case pos.h_2
K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : c4 e = 0
hc6 : c6 e = 0
x✝ : ℕ
heq✝ : ringChar K = 3
hchar : ringChar K = 3
⊢ weierstrass e (pth_root (-e.a3 ^ 2 - e.a6), e.a1 * pth_root (-e.a3 ^ 2 - e.a6) + e.a3) = 0 ∧
dweierstrass_dx e (pth_root (-e.a3 ^ 2 - e.a6), e.a1 * pth_root (-e.a3 ^ 2 - e.a6) + e.a3) = 0 ∧
dweierstrass_dy e (pth_root (-e.a3 ^ 2 - e.a6), e.a1 * pth_root (-e.a3 ^ 2 - e.a6) + e.a3) = 0 |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/EllipticCurve/Model.lean | Model.Field.is_singular_point_singular_point | [509, 1] | [694, 11] | have hcharne : ringChar K ≠ 0 := by simp [hchar] | case pos.h_2
K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : c4 e = 0
hc6 : c6 e = 0
x✝ : ℕ
heq✝ : ringChar K = 3
hchar : ringChar K = 3
⊢ weierstrass e (pth_root (-e.a3 ^ 2 - e.a6), e.a1 * pth_root (-e.a3 ^ 2 - e.a6) + e.a3) = 0 ∧
dweierstrass_dx e (pth_root (-e.a3 ^ 2 - e.a6), e.a1 * pth_root (-e.a3 ^ 2 - e.a6) + e.a3) = 0 ∧
dweierstrass_dy e (pth_root (-e.a3 ^ 2 - e.a6), e.a1 * pth_root (-e.a3 ^ 2 - e.a6) + e.a3) = 0 | case pos.h_2
K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : c4 e = 0
hc6 : c6 e = 0
x✝ : ℕ
heq✝ : ringChar K = 3
hchar : ringChar K = 3
hcharne : ringChar K ≠ 0
⊢ weierstrass e (pth_root (-e.a3 ^ 2 - e.a6), e.a1 * pth_root (-e.a3 ^ 2 - e.a6) + e.a3) = 0 ∧
dweierstrass_dx e (pth_root (-e.a3 ^ 2 - e.a6), e.a1 * pth_root (-e.a3 ^ 2 - e.a6) + e.a3) = 0 ∧
dweierstrass_dy e (pth_root (-e.a3 ^ 2 - e.a6), e.a1 * pth_root (-e.a3 ^ 2 - e.a6) + e.a3) = 0 |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/EllipticCurve/Model.lean | Model.Field.is_singular_point_singular_point | [509, 1] | [694, 11] | have hchar' : (ringChar K : K) = 3 := by simp [hchar] | case pos.h_2
K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : c4 e = 0
hc6 : c6 e = 0
x✝ : ℕ
heq✝ : ringChar K = 3
hchar : ringChar K = 3
hcharne : ringChar K ≠ 0
⊢ weierstrass e (pth_root (-e.a3 ^ 2 - e.a6), e.a1 * pth_root (-e.a3 ^ 2 - e.a6) + e.a3) = 0 ∧
dweierstrass_dx e (pth_root (-e.a3 ^ 2 - e.a6), e.a1 * pth_root (-e.a3 ^ 2 - e.a6) + e.a3) = 0 ∧
dweierstrass_dy e (pth_root (-e.a3 ^ 2 - e.a6), e.a1 * pth_root (-e.a3 ^ 2 - e.a6) + e.a3) = 0 | case pos.h_2
K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : c4 e = 0
hc6 : c6 e = 0
x✝ : ℕ
heq✝ : ringChar K = 3
hchar : ringChar K = 3
hcharne : ringChar K ≠ 0
hchar' : ↑(ringChar K) = 3
⊢ weierstrass e (pth_root (-e.a3 ^ 2 - e.a6), e.a1 * pth_root (-e.a3 ^ 2 - e.a6) + e.a3) = 0 ∧
dweierstrass_dx e (pth_root (-e.a3 ^ 2 - e.a6), e.a1 * pth_root (-e.a3 ^ 2 - e.a6) + e.a3) = 0 ∧
dweierstrass_dy e (pth_root (-e.a3 ^ 2 - e.a6), e.a1 * pth_root (-e.a3 ^ 2 - e.a6) + e.a3) = 0 |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/EllipticCurve/Model.lean | Model.Field.is_singular_point_singular_point | [509, 1] | [694, 11] | have hchar'' : (3 : K) = 0 := by simp [← hchar', ringChar.Nat.cast_ringChar] | case pos.h_2
K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : c4 e = 0
hc6 : c6 e = 0
x✝ : ℕ
heq✝ : ringChar K = 3
hchar : ringChar K = 3
hcharne : ringChar K ≠ 0
hchar' : ↑(ringChar K) = 3
⊢ weierstrass e (pth_root (-e.a3 ^ 2 - e.a6), e.a1 * pth_root (-e.a3 ^ 2 - e.a6) + e.a3) = 0 ∧
dweierstrass_dx e (pth_root (-e.a3 ^ 2 - e.a6), e.a1 * pth_root (-e.a3 ^ 2 - e.a6) + e.a3) = 0 ∧
dweierstrass_dy e (pth_root (-e.a3 ^ 2 - e.a6), e.a1 * pth_root (-e.a3 ^ 2 - e.a6) + e.a3) = 0 | case pos.h_2
K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : c4 e = 0
hc6 : c6 e = 0
x✝ : ℕ
heq✝ : ringChar K = 3
hchar : ringChar K = 3
hcharne : ringChar K ≠ 0
hchar' : ↑(ringChar K) = 3
hchar'' : 3 = 0
⊢ weierstrass e (pth_root (-e.a3 ^ 2 - e.a6), e.a1 * pth_root (-e.a3 ^ 2 - e.a6) + e.a3) = 0 ∧
dweierstrass_dx e (pth_root (-e.a3 ^ 2 - e.a6), e.a1 * pth_root (-e.a3 ^ 2 - e.a6) + e.a3) = 0 ∧
dweierstrass_dy e (pth_root (-e.a3 ^ 2 - e.a6), e.a1 * pth_root (-e.a3 ^ 2 - e.a6) + e.a3) = 0 |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/EllipticCurve/Model.lean | Model.Field.is_singular_point_singular_point | [509, 1] | [694, 11] | have hb2 : e.b2 = 0 := by simpa [c4_zero_iff_b2_zero_of_char_three e hchar] using hc4 | case pos.h_2
K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : c4 e = 0
hc6 : c6 e = 0
x✝ : ℕ
heq✝ : ringChar K = 3
hchar : ringChar K = 3
hcharne : ringChar K ≠ 0
hchar' : ↑(ringChar K) = 3
hchar'' : 3 = 0
⊢ weierstrass e (pth_root (-e.a3 ^ 2 - e.a6), e.a1 * pth_root (-e.a3 ^ 2 - e.a6) + e.a3) = 0 ∧
dweierstrass_dx e (pth_root (-e.a3 ^ 2 - e.a6), e.a1 * pth_root (-e.a3 ^ 2 - e.a6) + e.a3) = 0 ∧
dweierstrass_dy e (pth_root (-e.a3 ^ 2 - e.a6), e.a1 * pth_root (-e.a3 ^ 2 - e.a6) + e.a3) = 0 | case pos.h_2
K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : c4 e = 0
hc6 : c6 e = 0
x✝ : ℕ
heq✝ : ringChar K = 3
hchar : ringChar K = 3
hcharne : ringChar K ≠ 0
hchar' : ↑(ringChar K) = 3
hchar'' : 3 = 0
hb2 : b2 e = 0
⊢ weierstrass e (pth_root (-e.a3 ^ 2 - e.a6), e.a1 * pth_root (-e.a3 ^ 2 - e.a6) + e.a3) = 0 ∧
dweierstrass_dx e (pth_root (-e.a3 ^ 2 - e.a6), e.a1 * pth_root (-e.a3 ^ 2 - e.a6) + e.a3) = 0 ∧
dweierstrass_dy e (pth_root (-e.a3 ^ 2 - e.a6), e.a1 * pth_root (-e.a3 ^ 2 - e.a6) + e.a3) = 0 |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/EllipticCurve/Model.lean | Model.Field.is_singular_point_singular_point | [509, 1] | [694, 11] | have hb4 : e.b4 = 0 := b4_zero_of_b2_zero_of_disc_zero_of_char_three e hchar h hb2 | case pos.h_2
K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : c4 e = 0
hc6 : c6 e = 0
x✝ : ℕ
heq✝ : ringChar K = 3
hchar : ringChar K = 3
hcharne : ringChar K ≠ 0
hchar' : ↑(ringChar K) = 3
hchar'' : 3 = 0
hb2 : b2 e = 0
⊢ weierstrass e (pth_root (-e.a3 ^ 2 - e.a6), e.a1 * pth_root (-e.a3 ^ 2 - e.a6) + e.a3) = 0 ∧
dweierstrass_dx e (pth_root (-e.a3 ^ 2 - e.a6), e.a1 * pth_root (-e.a3 ^ 2 - e.a6) + e.a3) = 0 ∧
dweierstrass_dy e (pth_root (-e.a3 ^ 2 - e.a6), e.a1 * pth_root (-e.a3 ^ 2 - e.a6) + e.a3) = 0 | case pos.h_2
K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : c4 e = 0
hc6 : c6 e = 0
x✝ : ℕ
heq✝ : ringChar K = 3
hchar : ringChar K = 3
hcharne : ringChar K ≠ 0
hchar' : ↑(ringChar K) = 3
hchar'' : 3 = 0
hb2 : b2 e = 0
hb4 : b4 e = 0
⊢ weierstrass e (pth_root (-e.a3 ^ 2 - e.a6), e.a1 * pth_root (-e.a3 ^ 2 - e.a6) + e.a3) = 0 ∧
dweierstrass_dx e (pth_root (-e.a3 ^ 2 - e.a6), e.a1 * pth_root (-e.a3 ^ 2 - e.a6) + e.a3) = 0 ∧
dweierstrass_dy e (pth_root (-e.a3 ^ 2 - e.a6), e.a1 * pth_root (-e.a3 ^ 2 - e.a6) + e.a3) = 0 |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/EllipticCurve/Model.lean | Model.Field.is_singular_point_singular_point | [509, 1] | [694, 11] | rw [b2] at hb2 | case pos.h_2
K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : c4 e = 0
hc6 : c6 e = 0
x✝ : ℕ
heq✝ : ringChar K = 3
hchar : ringChar K = 3
hcharne : ringChar K ≠ 0
hchar' : ↑(ringChar K) = 3
hchar'' : 3 = 0
hb2 : b2 e = 0
hb4 : b4 e = 0
⊢ weierstrass e (pth_root (-e.a3 ^ 2 - e.a6), e.a1 * pth_root (-e.a3 ^ 2 - e.a6) + e.a3) = 0 ∧
dweierstrass_dx e (pth_root (-e.a3 ^ 2 - e.a6), e.a1 * pth_root (-e.a3 ^ 2 - e.a6) + e.a3) = 0 ∧
dweierstrass_dy e (pth_root (-e.a3 ^ 2 - e.a6), e.a1 * pth_root (-e.a3 ^ 2 - e.a6) + e.a3) = 0 | case pos.h_2
K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : c4 e = 0
hc6 : c6 e = 0
x✝ : ℕ
heq✝ : ringChar K = 3
hchar : ringChar K = 3
hcharne : ringChar K ≠ 0
hchar' : ↑(ringChar K) = 3
hchar'' : 3 = 0
hb2 : e.a1 * e.a1 + 4 * e.a2 = 0
hb4 : b4 e = 0
⊢ weierstrass e (pth_root (-e.a3 ^ 2 - e.a6), e.a1 * pth_root (-e.a3 ^ 2 - e.a6) + e.a3) = 0 ∧
dweierstrass_dx e (pth_root (-e.a3 ^ 2 - e.a6), e.a1 * pth_root (-e.a3 ^ 2 - e.a6) + e.a3) = 0 ∧
dweierstrass_dy e (pth_root (-e.a3 ^ 2 - e.a6), e.a1 * pth_root (-e.a3 ^ 2 - e.a6) + e.a3) = 0 |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/EllipticCurve/Model.lean | Model.Field.is_singular_point_singular_point | [509, 1] | [694, 11] | rw [b4] at hb4 | case pos.h_2
K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : c4 e = 0
hc6 : c6 e = 0
x✝ : ℕ
heq✝ : ringChar K = 3
hchar : ringChar K = 3
hcharne : ringChar K ≠ 0
hchar' : ↑(ringChar K) = 3
hchar'' : 3 = 0
hb2 : e.a1 * e.a1 + 4 * e.a2 = 0
hb4 : b4 e = 0
⊢ weierstrass e (pth_root (-e.a3 ^ 2 - e.a6), e.a1 * pth_root (-e.a3 ^ 2 - e.a6) + e.a3) = 0 ∧
dweierstrass_dx e (pth_root (-e.a3 ^ 2 - e.a6), e.a1 * pth_root (-e.a3 ^ 2 - e.a6) + e.a3) = 0 ∧
dweierstrass_dy e (pth_root (-e.a3 ^ 2 - e.a6), e.a1 * pth_root (-e.a3 ^ 2 - e.a6) + e.a3) = 0 | case pos.h_2
K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : c4 e = 0
hc6 : c6 e = 0
x✝ : ℕ
heq✝ : ringChar K = 3
hchar : ringChar K = 3
hcharne : ringChar K ≠ 0
hchar' : ↑(ringChar K) = 3
hchar'' : 3 = 0
hb2 : e.a1 * e.a1 + 4 * e.a2 = 0
hb4 : e.a1 * e.a3 + 2 * e.a4 = 0
⊢ weierstrass e (pth_root (-e.a3 ^ 2 - e.a6), e.a1 * pth_root (-e.a3 ^ 2 - e.a6) + e.a3) = 0 ∧
dweierstrass_dx e (pth_root (-e.a3 ^ 2 - e.a6), e.a1 * pth_root (-e.a3 ^ 2 - e.a6) + e.a3) = 0 ∧
dweierstrass_dy e (pth_root (-e.a3 ^ 2 - e.a6), e.a1 * pth_root (-e.a3 ^ 2 - e.a6) + e.a3) = 0 |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/EllipticCurve/Model.lean | Model.Field.is_singular_point_singular_point | [509, 1] | [694, 11] | refine ⟨?_, ?_, ?_⟩ | case pos.h_2
K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : c4 e = 0
hc6 : c6 e = 0
x✝ : ℕ
heq✝ : ringChar K = 3
hchar : ringChar K = 3
hcharne : ringChar K ≠ 0
hchar' : ↑(ringChar K) = 3
hchar'' : 3 = 0
hb2 : e.a1 * e.a1 + 4 * e.a2 = 0
hb4 : e.a1 * e.a3 + 2 * e.a4 = 0
⊢ weierstrass e (pth_root (-e.a3 ^ 2 - e.a6), e.a1 * pth_root (-e.a3 ^ 2 - e.a6) + e.a3) = 0 ∧
dweierstrass_dx e (pth_root (-e.a3 ^ 2 - e.a6), e.a1 * pth_root (-e.a3 ^ 2 - e.a6) + e.a3) = 0 ∧
dweierstrass_dy e (pth_root (-e.a3 ^ 2 - e.a6), e.a1 * pth_root (-e.a3 ^ 2 - e.a6) + e.a3) = 0 | case pos.h_2.refine_1
K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : c4 e = 0
hc6 : c6 e = 0
x✝ : ℕ
heq✝ : ringChar K = 3
hchar : ringChar K = 3
hcharne : ringChar K ≠ 0
hchar' : ↑(ringChar K) = 3
hchar'' : 3 = 0
hb2 : e.a1 * e.a1 + 4 * e.a2 = 0
hb4 : e.a1 * e.a3 + 2 * e.a4 = 0
⊢ weierstrass e (pth_root (-e.a3 ^ 2 - e.a6), e.a1 * pth_root (-e.a3 ^ 2 - e.a6) + e.a3) = 0
case pos.h_2.refine_2
K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : c4 e = 0
hc6 : c6 e = 0
x✝ : ℕ
heq✝ : ringChar K = 3
hchar : ringChar K = 3
hcharne : ringChar K ≠ 0
hchar' : ↑(ringChar K) = 3
hchar'' : 3 = 0
hb2 : e.a1 * e.a1 + 4 * e.a2 = 0
hb4 : e.a1 * e.a3 + 2 * e.a4 = 0
⊢ dweierstrass_dx e (pth_root (-e.a3 ^ 2 - e.a6), e.a1 * pth_root (-e.a3 ^ 2 - e.a6) + e.a3) = 0
case pos.h_2.refine_3
K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : c4 e = 0
hc6 : c6 e = 0
x✝ : ℕ
heq✝ : ringChar K = 3
hchar : ringChar K = 3
hcharne : ringChar K ≠ 0
hchar' : ↑(ringChar K) = 3
hchar'' : 3 = 0
hb2 : e.a1 * e.a1 + 4 * e.a2 = 0
hb4 : e.a1 * e.a3 + 2 * e.a4 = 0
⊢ dweierstrass_dy e (pth_root (-e.a3 ^ 2 - e.a6), e.a1 * pth_root (-e.a3 ^ 2 - e.a6) + e.a3) = 0 |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/EllipticCurve/Model.lean | Model.Field.is_singular_point_singular_point | [509, 1] | [694, 11] | . rw [weierstrass]
rw [← hchar, pth_root_pow_char hcharne]
simp only
rw [show
(e.a1 * pth_root (-(e.a3 ^ 2) - e.a6) + e.a3) ^ 2 +
e.a1 * pth_root (-(e.a3 ^ 2) - e.a6) * (e.a1 * pth_root (-(e.a3 ^ 2) - e.a6) + e.a3) +
e.a3 * (e.a1 * pth_root (-(e.a3 ^ 2) - e.a6) + e.a3) -
(-(e.a3 ^ 2) - e.a6 + e.a2 * pth_root (-(e.a3 ^ 2) - e.a6) ^ 2 + e.a4 * pth_root (-(e.a3 ^ 2) - e.a6) + e.a6) =
(2 * e.a1 ^ 2 - e.a2) * pth_root (-(e.a3 ^ 2) - e.a6) ^ 2 +
(4 * e.a1 * e.a3 - e.a4) * pth_root (-(e.a3 ^ 2) - e.a6) +
3 * e.a3 ^ 2
by ring]
rw [show 2 * e.a1 ^ 2 - e.a2 = 0 from ?_]
rw [show 4 * e.a1 * e.a3 - e.a4 = 0 from ?_]
simp only [zero_mul, add_zero, hchar'']
. rw [show (2 : K) = -1 by rw [← add_zero (-1), ← hchar'']; norm_num] at hb4
rw [show (4 : K) = 1 by rw [← add_zero 1, ← hchar'']; norm_num]
simp only [neg_mul, one_mul] at hb4
simp [sub_eq_add_neg, hb4]
. linear_combination (norm := ring_nf <;> simp [hchar'']) -hb2 | case pos.h_2.refine_1
K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : c4 e = 0
hc6 : c6 e = 0
x✝ : ℕ
heq✝ : ringChar K = 3
hchar : ringChar K = 3
hcharne : ringChar K ≠ 0
hchar' : ↑(ringChar K) = 3
hchar'' : 3 = 0
hb2 : e.a1 * e.a1 + 4 * e.a2 = 0
hb4 : e.a1 * e.a3 + 2 * e.a4 = 0
⊢ weierstrass e (pth_root (-e.a3 ^ 2 - e.a6), e.a1 * pth_root (-e.a3 ^ 2 - e.a6) + e.a3) = 0
case pos.h_2.refine_2
K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : c4 e = 0
hc6 : c6 e = 0
x✝ : ℕ
heq✝ : ringChar K = 3
hchar : ringChar K = 3
hcharne : ringChar K ≠ 0
hchar' : ↑(ringChar K) = 3
hchar'' : 3 = 0
hb2 : e.a1 * e.a1 + 4 * e.a2 = 0
hb4 : e.a1 * e.a3 + 2 * e.a4 = 0
⊢ dweierstrass_dx e (pth_root (-e.a3 ^ 2 - e.a6), e.a1 * pth_root (-e.a3 ^ 2 - e.a6) + e.a3) = 0
case pos.h_2.refine_3
K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : c4 e = 0
hc6 : c6 e = 0
x✝ : ℕ
heq✝ : ringChar K = 3
hchar : ringChar K = 3
hcharne : ringChar K ≠ 0
hchar' : ↑(ringChar K) = 3
hchar'' : 3 = 0
hb2 : e.a1 * e.a1 + 4 * e.a2 = 0
hb4 : e.a1 * e.a3 + 2 * e.a4 = 0
⊢ dweierstrass_dy e (pth_root (-e.a3 ^ 2 - e.a6), e.a1 * pth_root (-e.a3 ^ 2 - e.a6) + e.a3) = 0 | case pos.h_2.refine_2
K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : c4 e = 0
hc6 : c6 e = 0
x✝ : ℕ
heq✝ : ringChar K = 3
hchar : ringChar K = 3
hcharne : ringChar K ≠ 0
hchar' : ↑(ringChar K) = 3
hchar'' : 3 = 0
hb2 : e.a1 * e.a1 + 4 * e.a2 = 0
hb4 : e.a1 * e.a3 + 2 * e.a4 = 0
⊢ dweierstrass_dx e (pth_root (-e.a3 ^ 2 - e.a6), e.a1 * pth_root (-e.a3 ^ 2 - e.a6) + e.a3) = 0
case pos.h_2.refine_3
K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : c4 e = 0
hc6 : c6 e = 0
x✝ : ℕ
heq✝ : ringChar K = 3
hchar : ringChar K = 3
hcharne : ringChar K ≠ 0
hchar' : ↑(ringChar K) = 3
hchar'' : 3 = 0
hb2 : e.a1 * e.a1 + 4 * e.a2 = 0
hb4 : e.a1 * e.a3 + 2 * e.a4 = 0
⊢ dweierstrass_dy e (pth_root (-e.a3 ^ 2 - e.a6), e.a1 * pth_root (-e.a3 ^ 2 - e.a6) + e.a3) = 0 |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/EllipticCurve/Model.lean | Model.Field.is_singular_point_singular_point | [509, 1] | [694, 11] | . rw [dweierstrass_dx]
rw [hchar'', zero_mul, zero_add]
simp only
rw [show e.a1 * (e.a1 * pth_root (-(e.a3 ^ 2) - e.a6) + e.a3) - (2 * e.a2 * pth_root (-(e.a3 ^ 2) - e.a6) + e.a4) =
(e.a1 * e.a1 - 2 * e.a2) * pth_root (-(e.a3 ^ 2) - e.a6) + (e.a1 * e.a3 - e.a4)
by ring]
rw [show (2 : K) = -1 by rw [← add_zero (-1), ← hchar'']; norm_num] at hb4
rw [show (4 : K) = -2 by rw [← add_zero (-2), ← zero_mul (2 : K), ← hchar'']; norm_num] at hb2
simp only [neg_mul, one_mul, ← sub_eq_add_neg] at hb4 hb2
rw [hb4, hb2, zero_mul, zero_add] | case pos.h_2.refine_2
K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : c4 e = 0
hc6 : c6 e = 0
x✝ : ℕ
heq✝ : ringChar K = 3
hchar : ringChar K = 3
hcharne : ringChar K ≠ 0
hchar' : ↑(ringChar K) = 3
hchar'' : 3 = 0
hb2 : e.a1 * e.a1 + 4 * e.a2 = 0
hb4 : e.a1 * e.a3 + 2 * e.a4 = 0
⊢ dweierstrass_dx e (pth_root (-e.a3 ^ 2 - e.a6), e.a1 * pth_root (-e.a3 ^ 2 - e.a6) + e.a3) = 0
case pos.h_2.refine_3
K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : c4 e = 0
hc6 : c6 e = 0
x✝ : ℕ
heq✝ : ringChar K = 3
hchar : ringChar K = 3
hcharne : ringChar K ≠ 0
hchar' : ↑(ringChar K) = 3
hchar'' : 3 = 0
hb2 : e.a1 * e.a1 + 4 * e.a2 = 0
hb4 : e.a1 * e.a3 + 2 * e.a4 = 0
⊢ dweierstrass_dy e (pth_root (-e.a3 ^ 2 - e.a6), e.a1 * pth_root (-e.a3 ^ 2 - e.a6) + e.a3) = 0 | case pos.h_2.refine_3
K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : c4 e = 0
hc6 : c6 e = 0
x✝ : ℕ
heq✝ : ringChar K = 3
hchar : ringChar K = 3
hcharne : ringChar K ≠ 0
hchar' : ↑(ringChar K) = 3
hchar'' : 3 = 0
hb2 : e.a1 * e.a1 + 4 * e.a2 = 0
hb4 : e.a1 * e.a3 + 2 * e.a4 = 0
⊢ dweierstrass_dy e (pth_root (-e.a3 ^ 2 - e.a6), e.a1 * pth_root (-e.a3 ^ 2 - e.a6) + e.a3) = 0 |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/EllipticCurve/Model.lean | Model.Field.is_singular_point_singular_point | [509, 1] | [694, 11] | . rw [dweierstrass_dy]
simp only
rw [show 2 * (e.a1 * pth_root (-(e.a3 ^ 2) - e.a6) + e.a3)
+ e.a1 * pth_root (- (e.a3 ^ 2) - e.a6) + e.a3 = 3 * ((e.a1 * pth_root (-(e.a3 ^ 2) - e.a6)) + e.a3) by ring]
rw [hchar'', zero_mul] | case pos.h_2.refine_3
K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : c4 e = 0
hc6 : c6 e = 0
x✝ : ℕ
heq✝ : ringChar K = 3
hchar : ringChar K = 3
hcharne : ringChar K ≠ 0
hchar' : ↑(ringChar K) = 3
hchar'' : 3 = 0
hb2 : e.a1 * e.a1 + 4 * e.a2 = 0
hb4 : e.a1 * e.a3 + 2 * e.a4 = 0
⊢ dweierstrass_dy e (pth_root (-e.a3 ^ 2 - e.a6), e.a1 * pth_root (-e.a3 ^ 2 - e.a6) + e.a3) = 0 | no goals |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/EllipticCurve/Model.lean | Model.Field.is_singular_point_singular_point | [509, 1] | [694, 11] | assumption | K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : c4 e = 0
hc6 : c6 e = 0
x✝ : ℕ
heq✝ : ringChar K = 3
⊢ ringChar K = 3 | no goals |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/EllipticCurve/Model.lean | Model.Field.is_singular_point_singular_point | [509, 1] | [694, 11] | simp [hchar] | K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : c4 e = 0
hc6 : c6 e = 0
x✝ : ℕ
heq✝ : ringChar K = 3
hchar : ringChar K = 3
⊢ ringChar K ≠ 0 | no goals |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/EllipticCurve/Model.lean | Model.Field.is_singular_point_singular_point | [509, 1] | [694, 11] | simp [hchar] | K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : c4 e = 0
hc6 : c6 e = 0
x✝ : ℕ
heq✝ : ringChar K = 3
hchar : ringChar K = 3
hcharne : ringChar K ≠ 0
⊢ ↑(ringChar K) = 3 | no goals |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/EllipticCurve/Model.lean | Model.Field.is_singular_point_singular_point | [509, 1] | [694, 11] | simp [← hchar', ringChar.Nat.cast_ringChar] | K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : c4 e = 0
hc6 : c6 e = 0
x✝ : ℕ
heq✝ : ringChar K = 3
hchar : ringChar K = 3
hcharne : ringChar K ≠ 0
hchar' : ↑(ringChar K) = 3
⊢ 3 = 0 | no goals |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/EllipticCurve/Model.lean | Model.Field.is_singular_point_singular_point | [509, 1] | [694, 11] | simpa [c4_zero_iff_b2_zero_of_char_three e hchar] using hc4 | K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : c4 e = 0
hc6 : c6 e = 0
x✝ : ℕ
heq✝ : ringChar K = 3
hchar : ringChar K = 3
hcharne : ringChar K ≠ 0
hchar' : ↑(ringChar K) = 3
hchar'' : 3 = 0
⊢ b2 e = 0 | no goals |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/EllipticCurve/Model.lean | Model.Field.is_singular_point_singular_point | [509, 1] | [694, 11] | rw [weierstrass] | case pos.h_2.refine_1
K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : c4 e = 0
hc6 : c6 e = 0
x✝ : ℕ
heq✝ : ringChar K = 3
hchar : ringChar K = 3
hcharne : ringChar K ≠ 0
hchar' : ↑(ringChar K) = 3
hchar'' : 3 = 0
hb2 : e.a1 * e.a1 + 4 * e.a2 = 0
hb4 : e.a1 * e.a3 + 2 * e.a4 = 0
⊢ weierstrass e (pth_root (-e.a3 ^ 2 - e.a6), e.a1 * pth_root (-e.a3 ^ 2 - e.a6) + e.a3) = 0 | case pos.h_2.refine_1
K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : c4 e = 0
hc6 : c6 e = 0
x✝ : ℕ
heq✝ : ringChar K = 3
hchar : ringChar K = 3
hcharne : ringChar K ≠ 0
hchar' : ↑(ringChar K) = 3
hchar'' : 3 = 0
hb2 : e.a1 * e.a1 + 4 * e.a2 = 0
hb4 : e.a1 * e.a3 + 2 * e.a4 = 0
⊢ (pth_root (-e.a3 ^ 2 - e.a6), e.a1 * pth_root (-e.a3 ^ 2 - e.a6) + e.a3).snd ^ 2 +
e.a1 * (pth_root (-e.a3 ^ 2 - e.a6), e.a1 * pth_root (-e.a3 ^ 2 - e.a6) + e.a3).fst *
(pth_root (-e.a3 ^ 2 - e.a6), e.a1 * pth_root (-e.a3 ^ 2 - e.a6) + e.a3).snd +
e.a3 * (pth_root (-e.a3 ^ 2 - e.a6), e.a1 * pth_root (-e.a3 ^ 2 - e.a6) + e.a3).snd -
((pth_root (-e.a3 ^ 2 - e.a6), e.a1 * pth_root (-e.a3 ^ 2 - e.a6) + e.a3).fst ^ 3 +
e.a2 * (pth_root (-e.a3 ^ 2 - e.a6), e.a1 * pth_root (-e.a3 ^ 2 - e.a6) + e.a3).fst ^ 2 +
e.a4 * (pth_root (-e.a3 ^ 2 - e.a6), e.a1 * pth_root (-e.a3 ^ 2 - e.a6) + e.a3).fst +
e.a6) =
0 |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/EllipticCurve/Model.lean | Model.Field.is_singular_point_singular_point | [509, 1] | [694, 11] | rw [← hchar, pth_root_pow_char hcharne] | case pos.h_2.refine_1
K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : c4 e = 0
hc6 : c6 e = 0
x✝ : ℕ
heq✝ : ringChar K = 3
hchar : ringChar K = 3
hcharne : ringChar K ≠ 0
hchar' : ↑(ringChar K) = 3
hchar'' : 3 = 0
hb2 : e.a1 * e.a1 + 4 * e.a2 = 0
hb4 : e.a1 * e.a3 + 2 * e.a4 = 0
⊢ (pth_root (-e.a3 ^ 2 - e.a6), e.a1 * pth_root (-e.a3 ^ 2 - e.a6) + e.a3).snd ^ 2 +
e.a1 * (pth_root (-e.a3 ^ 2 - e.a6), e.a1 * pth_root (-e.a3 ^ 2 - e.a6) + e.a3).fst *
(pth_root (-e.a3 ^ 2 - e.a6), e.a1 * pth_root (-e.a3 ^ 2 - e.a6) + e.a3).snd +
e.a3 * (pth_root (-e.a3 ^ 2 - e.a6), e.a1 * pth_root (-e.a3 ^ 2 - e.a6) + e.a3).snd -
((pth_root (-e.a3 ^ 2 - e.a6), e.a1 * pth_root (-e.a3 ^ 2 - e.a6) + e.a3).fst ^ 3 +
e.a2 * (pth_root (-e.a3 ^ 2 - e.a6), e.a1 * pth_root (-e.a3 ^ 2 - e.a6) + e.a3).fst ^ 2 +
e.a4 * (pth_root (-e.a3 ^ 2 - e.a6), e.a1 * pth_root (-e.a3 ^ 2 - e.a6) + e.a3).fst +
e.a6) =
0 | case pos.h_2.refine_1
K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : c4 e = 0
hc6 : c6 e = 0
x✝ : ℕ
heq✝ : ringChar K = 3
hchar : ringChar K = 3
hcharne : ringChar K ≠ 0
hchar' : ↑(ringChar K) = 3
hchar'' : 3 = 0
hb2 : e.a1 * e.a1 + 4 * e.a2 = 0
hb4 : e.a1 * e.a3 + 2 * e.a4 = 0
⊢ (pth_root (-e.a3 ^ 2 - e.a6), e.a1 * pth_root (-e.a3 ^ 2 - e.a6) + e.a3).snd ^ 2 +
e.a1 * (pth_root (-e.a3 ^ 2 - e.a6), e.a1 * pth_root (-e.a3 ^ 2 - e.a6) + e.a3).fst *
(pth_root (-e.a3 ^ 2 - e.a6), e.a1 * pth_root (-e.a3 ^ 2 - e.a6) + e.a3).snd +
e.a3 * (pth_root (-e.a3 ^ 2 - e.a6), e.a1 * pth_root (-e.a3 ^ 2 - e.a6) + e.a3).snd -
(-e.a3 ^ 2 - e.a6 + e.a2 * (pth_root (-e.a3 ^ 2 - e.a6), e.a1 * pth_root (-e.a3 ^ 2 - e.a6) + e.a3).fst ^ 2 +
e.a4 * (pth_root (-e.a3 ^ 2 - e.a6), e.a1 * pth_root (-e.a3 ^ 2 - e.a6) + e.a3).fst +
e.a6) =
0 |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/EllipticCurve/Model.lean | Model.Field.is_singular_point_singular_point | [509, 1] | [694, 11] | simp only | case pos.h_2.refine_1
K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : c4 e = 0
hc6 : c6 e = 0
x✝ : ℕ
heq✝ : ringChar K = 3
hchar : ringChar K = 3
hcharne : ringChar K ≠ 0
hchar' : ↑(ringChar K) = 3
hchar'' : 3 = 0
hb2 : e.a1 * e.a1 + 4 * e.a2 = 0
hb4 : e.a1 * e.a3 + 2 * e.a4 = 0
⊢ (pth_root (-e.a3 ^ 2 - e.a6), e.a1 * pth_root (-e.a3 ^ 2 - e.a6) + e.a3).snd ^ 2 +
e.a1 * (pth_root (-e.a3 ^ 2 - e.a6), e.a1 * pth_root (-e.a3 ^ 2 - e.a6) + e.a3).fst *
(pth_root (-e.a3 ^ 2 - e.a6), e.a1 * pth_root (-e.a3 ^ 2 - e.a6) + e.a3).snd +
e.a3 * (pth_root (-e.a3 ^ 2 - e.a6), e.a1 * pth_root (-e.a3 ^ 2 - e.a6) + e.a3).snd -
(-e.a3 ^ 2 - e.a6 + e.a2 * (pth_root (-e.a3 ^ 2 - e.a6), e.a1 * pth_root (-e.a3 ^ 2 - e.a6) + e.a3).fst ^ 2 +
e.a4 * (pth_root (-e.a3 ^ 2 - e.a6), e.a1 * pth_root (-e.a3 ^ 2 - e.a6) + e.a3).fst +
e.a6) =
0 | case pos.h_2.refine_1
K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : c4 e = 0
hc6 : c6 e = 0
x✝ : ℕ
heq✝ : ringChar K = 3
hchar : ringChar K = 3
hcharne : ringChar K ≠ 0
hchar' : ↑(ringChar K) = 3
hchar'' : 3 = 0
hb2 : e.a1 * e.a1 + 4 * e.a2 = 0
hb4 : e.a1 * e.a3 + 2 * e.a4 = 0
⊢ (e.a1 * pth_root (-e.a3 ^ 2 - e.a6) + e.a3) ^ 2 +
e.a1 * pth_root (-e.a3 ^ 2 - e.a6) * (e.a1 * pth_root (-e.a3 ^ 2 - e.a6) + e.a3) +
e.a3 * (e.a1 * pth_root (-e.a3 ^ 2 - e.a6) + e.a3) -
(-e.a3 ^ 2 - e.a6 + e.a2 * pth_root (-e.a3 ^ 2 - e.a6) ^ 2 + e.a4 * pth_root (-e.a3 ^ 2 - e.a6) + e.a6) =
0 |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/EllipticCurve/Model.lean | Model.Field.is_singular_point_singular_point | [509, 1] | [694, 11] | rw [show
(e.a1 * pth_root (-(e.a3 ^ 2) - e.a6) + e.a3) ^ 2 +
e.a1 * pth_root (-(e.a3 ^ 2) - e.a6) * (e.a1 * pth_root (-(e.a3 ^ 2) - e.a6) + e.a3) +
e.a3 * (e.a1 * pth_root (-(e.a3 ^ 2) - e.a6) + e.a3) -
(-(e.a3 ^ 2) - e.a6 + e.a2 * pth_root (-(e.a3 ^ 2) - e.a6) ^ 2 + e.a4 * pth_root (-(e.a3 ^ 2) - e.a6) + e.a6) =
(2 * e.a1 ^ 2 - e.a2) * pth_root (-(e.a3 ^ 2) - e.a6) ^ 2 +
(4 * e.a1 * e.a3 - e.a4) * pth_root (-(e.a3 ^ 2) - e.a6) +
3 * e.a3 ^ 2
by ring] | case pos.h_2.refine_1
K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : c4 e = 0
hc6 : c6 e = 0
x✝ : ℕ
heq✝ : ringChar K = 3
hchar : ringChar K = 3
hcharne : ringChar K ≠ 0
hchar' : ↑(ringChar K) = 3
hchar'' : 3 = 0
hb2 : e.a1 * e.a1 + 4 * e.a2 = 0
hb4 : e.a1 * e.a3 + 2 * e.a4 = 0
⊢ (e.a1 * pth_root (-e.a3 ^ 2 - e.a6) + e.a3) ^ 2 +
e.a1 * pth_root (-e.a3 ^ 2 - e.a6) * (e.a1 * pth_root (-e.a3 ^ 2 - e.a6) + e.a3) +
e.a3 * (e.a1 * pth_root (-e.a3 ^ 2 - e.a6) + e.a3) -
(-e.a3 ^ 2 - e.a6 + e.a2 * pth_root (-e.a3 ^ 2 - e.a6) ^ 2 + e.a4 * pth_root (-e.a3 ^ 2 - e.a6) + e.a6) =
0 | case pos.h_2.refine_1
K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : c4 e = 0
hc6 : c6 e = 0
x✝ : ℕ
heq✝ : ringChar K = 3
hchar : ringChar K = 3
hcharne : ringChar K ≠ 0
hchar' : ↑(ringChar K) = 3
hchar'' : 3 = 0
hb2 : e.a1 * e.a1 + 4 * e.a2 = 0
hb4 : e.a1 * e.a3 + 2 * e.a4 = 0
⊢ (2 * e.a1 ^ 2 - e.a2) * pth_root (-e.a3 ^ 2 - e.a6) ^ 2 + (4 * e.a1 * e.a3 - e.a4) * pth_root (-e.a3 ^ 2 - e.a6) +
3 * e.a3 ^ 2 =
0 |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/EllipticCurve/Model.lean | Model.Field.is_singular_point_singular_point | [509, 1] | [694, 11] | rw [show 2 * e.a1 ^ 2 - e.a2 = 0 from ?_] | case pos.h_2.refine_1
K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : c4 e = 0
hc6 : c6 e = 0
x✝ : ℕ
heq✝ : ringChar K = 3
hchar : ringChar K = 3
hcharne : ringChar K ≠ 0
hchar' : ↑(ringChar K) = 3
hchar'' : 3 = 0
hb2 : e.a1 * e.a1 + 4 * e.a2 = 0
hb4 : e.a1 * e.a3 + 2 * e.a4 = 0
⊢ (2 * e.a1 ^ 2 - e.a2) * pth_root (-e.a3 ^ 2 - e.a6) ^ 2 + (4 * e.a1 * e.a3 - e.a4) * pth_root (-e.a3 ^ 2 - e.a6) +
3 * e.a3 ^ 2 =
0 | case pos.h_2.refine_1
K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : c4 e = 0
hc6 : c6 e = 0
x✝ : ℕ
heq✝ : ringChar K = 3
hchar : ringChar K = 3
hcharne : ringChar K ≠ 0
hchar' : ↑(ringChar K) = 3
hchar'' : 3 = 0
hb2 : e.a1 * e.a1 + 4 * e.a2 = 0
hb4 : e.a1 * e.a3 + 2 * e.a4 = 0
⊢ 0 * pth_root (-e.a3 ^ 2 - e.a6) ^ 2 + (4 * e.a1 * e.a3 - e.a4) * pth_root (-e.a3 ^ 2 - e.a6) + 3 * e.a3 ^ 2 = 0
K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : c4 e = 0
hc6 : c6 e = 0
x✝ : ℕ
heq✝ : ringChar K = 3
hchar : ringChar K = 3
hcharne : ringChar K ≠ 0
hchar' : ↑(ringChar K) = 3
hchar'' : 3 = 0
hb2 : e.a1 * e.a1 + 4 * e.a2 = 0
hb4 : e.a1 * e.a3 + 2 * e.a4 = 0
⊢ 2 * e.a1 ^ 2 - e.a2 = 0 |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/EllipticCurve/Model.lean | Model.Field.is_singular_point_singular_point | [509, 1] | [694, 11] | rw [show 4 * e.a1 * e.a3 - e.a4 = 0 from ?_] | case pos.h_2.refine_1
K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : c4 e = 0
hc6 : c6 e = 0
x✝ : ℕ
heq✝ : ringChar K = 3
hchar : ringChar K = 3
hcharne : ringChar K ≠ 0
hchar' : ↑(ringChar K) = 3
hchar'' : 3 = 0
hb2 : e.a1 * e.a1 + 4 * e.a2 = 0
hb4 : e.a1 * e.a3 + 2 * e.a4 = 0
⊢ 0 * pth_root (-e.a3 ^ 2 - e.a6) ^ 2 + (4 * e.a1 * e.a3 - e.a4) * pth_root (-e.a3 ^ 2 - e.a6) + 3 * e.a3 ^ 2 = 0
K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : c4 e = 0
hc6 : c6 e = 0
x✝ : ℕ
heq✝ : ringChar K = 3
hchar : ringChar K = 3
hcharne : ringChar K ≠ 0
hchar' : ↑(ringChar K) = 3
hchar'' : 3 = 0
hb2 : e.a1 * e.a1 + 4 * e.a2 = 0
hb4 : e.a1 * e.a3 + 2 * e.a4 = 0
⊢ 2 * e.a1 ^ 2 - e.a2 = 0 | case pos.h_2.refine_1
K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : c4 e = 0
hc6 : c6 e = 0
x✝ : ℕ
heq✝ : ringChar K = 3
hchar : ringChar K = 3
hcharne : ringChar K ≠ 0
hchar' : ↑(ringChar K) = 3
hchar'' : 3 = 0
hb2 : e.a1 * e.a1 + 4 * e.a2 = 0
hb4 : e.a1 * e.a3 + 2 * e.a4 = 0
⊢ 0 * pth_root (-e.a3 ^ 2 - e.a6) ^ 2 + 0 * pth_root (-e.a3 ^ 2 - e.a6) + 3 * e.a3 ^ 2 = 0
K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : c4 e = 0
hc6 : c6 e = 0
x✝ : ℕ
heq✝ : ringChar K = 3
hchar : ringChar K = 3
hcharne : ringChar K ≠ 0
hchar' : ↑(ringChar K) = 3
hchar'' : 3 = 0
hb2 : e.a1 * e.a1 + 4 * e.a2 = 0
hb4 : e.a1 * e.a3 + 2 * e.a4 = 0
⊢ 4 * e.a1 * e.a3 - e.a4 = 0
K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : c4 e = 0
hc6 : c6 e = 0
x✝ : ℕ
heq✝ : ringChar K = 3
hchar : ringChar K = 3
hcharne : ringChar K ≠ 0
hchar' : ↑(ringChar K) = 3
hchar'' : 3 = 0
hb2 : e.a1 * e.a1 + 4 * e.a2 = 0
hb4 : e.a1 * e.a3 + 2 * e.a4 = 0
⊢ 2 * e.a1 ^ 2 - e.a2 = 0 |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/EllipticCurve/Model.lean | Model.Field.is_singular_point_singular_point | [509, 1] | [694, 11] | simp only [zero_mul, add_zero, hchar''] | case pos.h_2.refine_1
K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : c4 e = 0
hc6 : c6 e = 0
x✝ : ℕ
heq✝ : ringChar K = 3
hchar : ringChar K = 3
hcharne : ringChar K ≠ 0
hchar' : ↑(ringChar K) = 3
hchar'' : 3 = 0
hb2 : e.a1 * e.a1 + 4 * e.a2 = 0
hb4 : e.a1 * e.a3 + 2 * e.a4 = 0
⊢ 0 * pth_root (-e.a3 ^ 2 - e.a6) ^ 2 + 0 * pth_root (-e.a3 ^ 2 - e.a6) + 3 * e.a3 ^ 2 = 0
K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : c4 e = 0
hc6 : c6 e = 0
x✝ : ℕ
heq✝ : ringChar K = 3
hchar : ringChar K = 3
hcharne : ringChar K ≠ 0
hchar' : ↑(ringChar K) = 3
hchar'' : 3 = 0
hb2 : e.a1 * e.a1 + 4 * e.a2 = 0
hb4 : e.a1 * e.a3 + 2 * e.a4 = 0
⊢ 4 * e.a1 * e.a3 - e.a4 = 0
K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : c4 e = 0
hc6 : c6 e = 0
x✝ : ℕ
heq✝ : ringChar K = 3
hchar : ringChar K = 3
hcharne : ringChar K ≠ 0
hchar' : ↑(ringChar K) = 3
hchar'' : 3 = 0
hb2 : e.a1 * e.a1 + 4 * e.a2 = 0
hb4 : e.a1 * e.a3 + 2 * e.a4 = 0
⊢ 2 * e.a1 ^ 2 - e.a2 = 0 | K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : c4 e = 0
hc6 : c6 e = 0
x✝ : ℕ
heq✝ : ringChar K = 3
hchar : ringChar K = 3
hcharne : ringChar K ≠ 0
hchar' : ↑(ringChar K) = 3
hchar'' : 3 = 0
hb2 : e.a1 * e.a1 + 4 * e.a2 = 0
hb4 : e.a1 * e.a3 + 2 * e.a4 = 0
⊢ 4 * e.a1 * e.a3 - e.a4 = 0
K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : c4 e = 0
hc6 : c6 e = 0
x✝ : ℕ
heq✝ : ringChar K = 3
hchar : ringChar K = 3
hcharne : ringChar K ≠ 0
hchar' : ↑(ringChar K) = 3
hchar'' : 3 = 0
hb2 : e.a1 * e.a1 + 4 * e.a2 = 0
hb4 : e.a1 * e.a3 + 2 * e.a4 = 0
⊢ 2 * e.a1 ^ 2 - e.a2 = 0 |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/EllipticCurve/Model.lean | Model.Field.is_singular_point_singular_point | [509, 1] | [694, 11] | . rw [show (2 : K) = -1 by rw [← add_zero (-1), ← hchar'']; norm_num] at hb4
rw [show (4 : K) = 1 by rw [← add_zero 1, ← hchar'']; norm_num]
simp only [neg_mul, one_mul] at hb4
simp [sub_eq_add_neg, hb4] | K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : c4 e = 0
hc6 : c6 e = 0
x✝ : ℕ
heq✝ : ringChar K = 3
hchar : ringChar K = 3
hcharne : ringChar K ≠ 0
hchar' : ↑(ringChar K) = 3
hchar'' : 3 = 0
hb2 : e.a1 * e.a1 + 4 * e.a2 = 0
hb4 : e.a1 * e.a3 + 2 * e.a4 = 0
⊢ 4 * e.a1 * e.a3 - e.a4 = 0
K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : c4 e = 0
hc6 : c6 e = 0
x✝ : ℕ
heq✝ : ringChar K = 3
hchar : ringChar K = 3
hcharne : ringChar K ≠ 0
hchar' : ↑(ringChar K) = 3
hchar'' : 3 = 0
hb2 : e.a1 * e.a1 + 4 * e.a2 = 0
hb4 : e.a1 * e.a3 + 2 * e.a4 = 0
⊢ 2 * e.a1 ^ 2 - e.a2 = 0 | K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : c4 e = 0
hc6 : c6 e = 0
x✝ : ℕ
heq✝ : ringChar K = 3
hchar : ringChar K = 3
hcharne : ringChar K ≠ 0
hchar' : ↑(ringChar K) = 3
hchar'' : 3 = 0
hb2 : e.a1 * e.a1 + 4 * e.a2 = 0
hb4 : e.a1 * e.a3 + 2 * e.a4 = 0
⊢ 2 * e.a1 ^ 2 - e.a2 = 0 |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/EllipticCurve/Model.lean | Model.Field.is_singular_point_singular_point | [509, 1] | [694, 11] | . linear_combination (norm := ring_nf <;> simp [hchar'']) -hb2 | K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : c4 e = 0
hc6 : c6 e = 0
x✝ : ℕ
heq✝ : ringChar K = 3
hchar : ringChar K = 3
hcharne : ringChar K ≠ 0
hchar' : ↑(ringChar K) = 3
hchar'' : 3 = 0
hb2 : e.a1 * e.a1 + 4 * e.a2 = 0
hb4 : e.a1 * e.a3 + 2 * e.a4 = 0
⊢ 2 * e.a1 ^ 2 - e.a2 = 0 | no goals |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/EllipticCurve/Model.lean | Model.Field.is_singular_point_singular_point | [509, 1] | [694, 11] | ring | K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : c4 e = 0
hc6 : c6 e = 0
x✝ : ℕ
heq✝ : ringChar K = 3
hchar : ringChar K = 3
hcharne : ringChar K ≠ 0
hchar' : ↑(ringChar K) = 3
hchar'' : 3 = 0
hb2 : e.a1 * e.a1 + 4 * e.a2 = 0
hb4 : e.a1 * e.a3 + 2 * e.a4 = 0
⊢ (e.a1 * pth_root (-e.a3 ^ 2 - e.a6) + e.a3) ^ 2 +
e.a1 * pth_root (-e.a3 ^ 2 - e.a6) * (e.a1 * pth_root (-e.a3 ^ 2 - e.a6) + e.a3) +
e.a3 * (e.a1 * pth_root (-e.a3 ^ 2 - e.a6) + e.a3) -
(-e.a3 ^ 2 - e.a6 + e.a2 * pth_root (-e.a3 ^ 2 - e.a6) ^ 2 + e.a4 * pth_root (-e.a3 ^ 2 - e.a6) + e.a6) =
(2 * e.a1 ^ 2 - e.a2) * pth_root (-e.a3 ^ 2 - e.a6) ^ 2 + (4 * e.a1 * e.a3 - e.a4) * pth_root (-e.a3 ^ 2 - e.a6) +
3 * e.a3 ^ 2 | no goals |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/EllipticCurve/Model.lean | Model.Field.is_singular_point_singular_point | [509, 1] | [694, 11] | rw [show (2 : K) = -1 by rw [← add_zero (-1), ← hchar'']; norm_num] at hb4 | K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : c4 e = 0
hc6 : c6 e = 0
x✝ : ℕ
heq✝ : ringChar K = 3
hchar : ringChar K = 3
hcharne : ringChar K ≠ 0
hchar' : ↑(ringChar K) = 3
hchar'' : 3 = 0
hb2 : e.a1 * e.a1 + 4 * e.a2 = 0
hb4 : e.a1 * e.a3 + 2 * e.a4 = 0
⊢ 4 * e.a1 * e.a3 - e.a4 = 0 | K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : c4 e = 0
hc6 : c6 e = 0
x✝ : ℕ
heq✝ : ringChar K = 3
hchar : ringChar K = 3
hcharne : ringChar K ≠ 0
hchar' : ↑(ringChar K) = 3
hchar'' : 3 = 0
hb2 : e.a1 * e.a1 + 4 * e.a2 = 0
hb4 : e.a1 * e.a3 + -1 * e.a4 = 0
⊢ 4 * e.a1 * e.a3 - e.a4 = 0 |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/EllipticCurve/Model.lean | Model.Field.is_singular_point_singular_point | [509, 1] | [694, 11] | rw [show (4 : K) = 1 by rw [← add_zero 1, ← hchar'']; norm_num] | K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : c4 e = 0
hc6 : c6 e = 0
x✝ : ℕ
heq✝ : ringChar K = 3
hchar : ringChar K = 3
hcharne : ringChar K ≠ 0
hchar' : ↑(ringChar K) = 3
hchar'' : 3 = 0
hb2 : e.a1 * e.a1 + 4 * e.a2 = 0
hb4 : e.a1 * e.a3 + -1 * e.a4 = 0
⊢ 4 * e.a1 * e.a3 - e.a4 = 0 | K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : c4 e = 0
hc6 : c6 e = 0
x✝ : ℕ
heq✝ : ringChar K = 3
hchar : ringChar K = 3
hcharne : ringChar K ≠ 0
hchar' : ↑(ringChar K) = 3
hchar'' : 3 = 0
hb2 : e.a1 * e.a1 + 4 * e.a2 = 0
hb4 : e.a1 * e.a3 + -1 * e.a4 = 0
⊢ 1 * e.a1 * e.a3 - e.a4 = 0 |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/EllipticCurve/Model.lean | Model.Field.is_singular_point_singular_point | [509, 1] | [694, 11] | simp only [neg_mul, one_mul] at hb4 | K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : c4 e = 0
hc6 : c6 e = 0
x✝ : ℕ
heq✝ : ringChar K = 3
hchar : ringChar K = 3
hcharne : ringChar K ≠ 0
hchar' : ↑(ringChar K) = 3
hchar'' : 3 = 0
hb2 : e.a1 * e.a1 + 4 * e.a2 = 0
hb4 : e.a1 * e.a3 + -1 * e.a4 = 0
⊢ 1 * e.a1 * e.a3 - e.a4 = 0 | K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : c4 e = 0
hc6 : c6 e = 0
x✝ : ℕ
heq✝ : ringChar K = 3
hchar : ringChar K = 3
hcharne : ringChar K ≠ 0
hchar' : ↑(ringChar K) = 3
hchar'' : 3 = 0
hb2 : e.a1 * e.a1 + 4 * e.a2 = 0
hb4 : e.a1 * e.a3 + -e.a4 = 0
⊢ 1 * e.a1 * e.a3 - e.a4 = 0 |
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