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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 [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 + -e.a4 = 0
⊢ 1 * e.a1 * e.a3 - e.a4 = 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 [← add_zero (-1), ← 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
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 = -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 = -1 + 3 |
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 = 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 = -1 + 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] | rw [← add_zero 1, ← 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
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 = 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 + -1 * e.a4 = 0
⊢ 4 = 1 + 3 |
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 = 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 = 1 + 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] | 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] | rw [dweierstrass_dx] | 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_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
⊢ e.a1 * (pth_root (-e.a3 ^ 2 - e.a6), e.a1 * pth_root (-e.a3 ^ 2 - e.a6) + e.a3).snd -
(3 * (pth_root (-e.a3 ^ 2 - e.a6), e.a1 * pth_root (-e.a3 ^ 2 - e.a6) + e.a3).fst ^ 2 +
2 * e.a2 * (pth_root (-e.a3 ^ 2 - e.a6), e.a1 * pth_root (-e.a3 ^ 2 - e.a6) + e.a3).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] | rw [hchar'', 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
⊢ e.a1 * (pth_root (-e.a3 ^ 2 - e.a6), e.a1 * pth_root (-e.a3 ^ 2 - e.a6) + e.a3).snd -
(3 * (pth_root (-e.a3 ^ 2 - e.a6), e.a1 * pth_root (-e.a3 ^ 2 - e.a6) + e.a3).fst ^ 2 +
2 * e.a2 * (pth_root (-e.a3 ^ 2 - e.a6), e.a1 * pth_root (-e.a3 ^ 2 - e.a6) + e.a3).fst +
e.a4) =
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
⊢ e.a1 * (pth_root (-e.a3 ^ 2 - e.a6), e.a1 * pth_root (-e.a3 ^ 2 - e.a6) + e.a3).snd -
(2 * e.a2 * (pth_root (-e.a3 ^ 2 - e.a6), e.a1 * pth_root (-e.a3 ^ 2 - e.a6) + e.a3).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 | 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
⊢ e.a1 * (pth_root (-e.a3 ^ 2 - e.a6), e.a1 * pth_root (-e.a3 ^ 2 - e.a6) + e.a3).snd -
(2 * e.a2 * (pth_root (-e.a3 ^ 2 - e.a6), e.a1 * pth_root (-e.a3 ^ 2 - e.a6) + e.a3).fst + e.a4) =
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
⊢ 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) = 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 * (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] | 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
⊢ 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) = 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
⊢ (e.a1 * e.a1 - 2 * e.a2) * pth_root (-e.a3 ^ 2 - e.a6) + (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 (2 : K) = -1 by rw [← add_zero (-1), ← hchar'']; norm_num] at hb4 | 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
⊢ (e.a1 * e.a1 - 2 * e.a2) * pth_root (-e.a3 ^ 2 - e.a6) + (e.a1 * e.a3 - e.a4) = 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 + -1 * e.a4 = 0
⊢ (e.a1 * e.a1 - 2 * e.a2) * pth_root (-e.a3 ^ 2 - e.a6) + (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) = -2 by rw [← add_zero (-2), ← zero_mul (2 : K), ← hchar'']; norm_num] at hb2 | 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 + -1 * e.a4 = 0
⊢ (e.a1 * e.a1 - 2 * e.a2) * pth_root (-e.a3 ^ 2 - e.a6) + (e.a1 * e.a3 - e.a4) = 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 + -2 * e.a2 = 0
hb4 : e.a1 * e.a3 + -1 * e.a4 = 0
⊢ (e.a1 * e.a1 - 2 * e.a2) * pth_root (-e.a3 ^ 2 - e.a6) + (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, ← sub_eq_add_neg] at hb4 hb2 | 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 + -2 * e.a2 = 0
hb4 : e.a1 * e.a3 + -1 * e.a4 = 0
⊢ (e.a1 * e.a1 - 2 * e.a2) * pth_root (-e.a3 ^ 2 - e.a6) + (e.a1 * e.a3 - e.a4) = 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
hb4 : e.a1 * e.a3 - e.a4 = 0
hb2 : e.a1 * e.a1 - 2 * e.a2 = 0
⊢ (e.a1 * e.a1 - 2 * e.a2) * pth_root (-e.a3 ^ 2 - e.a6) + (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 [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
hb4 : e.a1 * e.a3 - e.a4 = 0
hb2 : e.a1 * e.a1 - 2 * e.a2 = 0
⊢ (e.a1 * e.a1 - 2 * e.a2) * pth_root (-e.a3 ^ 2 - e.a6) + (e.a1 * e.a3 - e.a4) = 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 * (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) | 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 [← add_zero (-2), ← zero_mul (2 : K), ← 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
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 = -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 + -1 * e.a4 = 0
⊢ 4 = -2 + 3 * 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] | 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 = -2 + 3 * 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 [dweierstrass_dy] | 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
⊢ 2 * (pth_root (-e.a3 ^ 2 - e.a6), e.a1 * pth_root (-e.a3 ^ 2 - e.a6) + e.a3).snd +
e.a1 * (pth_root (-e.a3 ^ 2 - e.a6), e.a1 * pth_root (-e.a3 ^ 2 - e.a6) + e.a3).fst +
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] | simp only | 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
⊢ 2 * (pth_root (-e.a3 ^ 2 - e.a6), e.a1 * pth_root (-e.a3 ^ 2 - e.a6) + e.a3).snd +
e.a1 * (pth_root (-e.a3 ^ 2 - e.a6), e.a1 * pth_root (-e.a3 ^ 2 - e.a6) + e.a3).fst +
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
⊢ 2 * (e.a1 * pth_root (-e.a3 ^ 2 - e.a6) + e.a3) + 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 [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] | 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
⊢ 2 * (e.a1 * pth_root (-e.a3 ^ 2 - e.a6) + e.a3) + 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
⊢ 3 * (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 [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
⊢ 3 * (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] | 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
⊢ 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) | 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] | rename_i hn2 hn3 | 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✝ : ℕ
hn2 : ringChar K = 2 → False
hn3 : 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] | 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✝ : ℕ
hn2 : ringChar K = 2 → False
hn3 : 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✝ : ℕ
hn2 : ringChar K = 2 → False
hn3 : ringChar K = 3 → False
⊢ weierstrass e (-b2 e / 12, -(-e.a1 * b2 e / 12 + e.a3) / 2) = 0 ∧
dweierstrass_dx e (-b2 e / 12, -(-e.a1 * b2 e / 12 + e.a3) / 2) = 0 ∧
dweierstrass_dy e (-b2 e / 12, -(-e.a1 * b2 e / 12 + 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] | have h2 : (2 : K) ≠ 0 := fun hh => hn2 (ringChar_eq_of_Prime hh (by norm_num)) | 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✝ : ℕ
hn2 : ringChar K = 2 → False
hn3 : ringChar K = 3 → False
⊢ weierstrass e (-b2 e / 12, -(-e.a1 * b2 e / 12 + e.a3) / 2) = 0 ∧
dweierstrass_dx e (-b2 e / 12, -(-e.a1 * b2 e / 12 + e.a3) / 2) = 0 ∧
dweierstrass_dy e (-b2 e / 12, -(-e.a1 * b2 e / 12 + e.a3) / 2) = 0 | 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✝ : ℕ
hn2 : ringChar K = 2 → False
hn3 : ringChar K = 3 → False
h2 : 2 ≠ 0
⊢ weierstrass e (-b2 e / 12, -(-e.a1 * b2 e / 12 + e.a3) / 2) = 0 ∧
dweierstrass_dx e (-b2 e / 12, -(-e.a1 * b2 e / 12 + e.a3) / 2) = 0 ∧
dweierstrass_dy e (-b2 e / 12, -(-e.a1 * b2 e / 12 + 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] | have h3 : (3 : K) ≠ 0 := fun hh => hn3 (ringChar_eq_of_Prime hh (by norm_num)) | 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✝ : ℕ
hn2 : ringChar K = 2 → False
hn3 : ringChar K = 3 → False
h2 : 2 ≠ 0
⊢ weierstrass e (-b2 e / 12, -(-e.a1 * b2 e / 12 + e.a3) / 2) = 0 ∧
dweierstrass_dx e (-b2 e / 12, -(-e.a1 * b2 e / 12 + e.a3) / 2) = 0 ∧
dweierstrass_dy e (-b2 e / 12, -(-e.a1 * b2 e / 12 + e.a3) / 2) = 0 | 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✝ : ℕ
hn2 : ringChar K = 2 → False
hn3 : ringChar K = 3 → False
h2 : 2 ≠ 0
h3 : 3 ≠ 0
⊢ weierstrass e (-b2 e / 12, -(-e.a1 * b2 e / 12 + e.a3) / 2) = 0 ∧
dweierstrass_dx e (-b2 e / 12, -(-e.a1 * b2 e / 12 + e.a3) / 2) = 0 ∧
dweierstrass_dy e (-b2 e / 12, -(-e.a1 * b2 e / 12 + 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] | have h12 : (12 : K) ≠ 0 := by
rw [show (12 : K) = 2 * 2 * 3 by norm_num]
repeat' apply mul_ne_zero
all_goals assumption | 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✝ : ℕ
hn2 : ringChar K = 2 → False
hn3 : ringChar K = 3 → False
h2 : 2 ≠ 0
h3 : 3 ≠ 0
⊢ weierstrass e (-b2 e / 12, -(-e.a1 * b2 e / 12 + e.a3) / 2) = 0 ∧
dweierstrass_dx e (-b2 e / 12, -(-e.a1 * b2 e / 12 + e.a3) / 2) = 0 ∧
dweierstrass_dy e (-b2 e / 12, -(-e.a1 * b2 e / 12 + e.a3) / 2) = 0 | 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✝ : ℕ
hn2 : ringChar K = 2 → False
hn3 : ringChar K = 3 → False
h2 : 2 ≠ 0
h3 : 3 ≠ 0
h12 : 12 ≠ 0
⊢ weierstrass e (-b2 e / 12, -(-e.a1 * b2 e / 12 + e.a3) / 2) = 0 ∧
dweierstrass_dx e (-b2 e / 12, -(-e.a1 * b2 e / 12 + e.a3) / 2) = 0 ∧
dweierstrass_dy e (-b2 e / 12, -(-e.a1 * b2 e / 12 + 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] | refine ⟨?_, ?_, ?_⟩ | 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✝ : ℕ
hn2 : ringChar K = 2 → False
hn3 : ringChar K = 3 → False
h2 : 2 ≠ 0
h3 : 3 ≠ 0
h12 : 12 ≠ 0
⊢ weierstrass e (-b2 e / 12, -(-e.a1 * b2 e / 12 + e.a3) / 2) = 0 ∧
dweierstrass_dx e (-b2 e / 12, -(-e.a1 * b2 e / 12 + e.a3) / 2) = 0 ∧
dweierstrass_dy e (-b2 e / 12, -(-e.a1 * b2 e / 12 + e.a3) / 2) = 0 | case pos.h_3.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
hn2 : ringChar K = 2 → False
hn3 : ringChar K = 3 → False
h2 : 2 ≠ 0
h3 : 3 ≠ 0
h12 : 12 ≠ 0
⊢ weierstrass e (-b2 e / 12, -(-e.a1 * b2 e / 12 + e.a3) / 2) = 0
case pos.h_3.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
hn2 : ringChar K = 2 → False
hn3 : ringChar K = 3 → False
h2 : 2 ≠ 0
h3 : 3 ≠ 0
h12 : 12 ≠ 0
⊢ dweierstrass_dx e (-b2 e / 12, -(-e.a1 * b2 e / 12 + e.a3) / 2) = 0
case pos.h_3.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
hn2 : ringChar K = 2 → False
hn3 : ringChar K = 3 → False
h2 : 2 ≠ 0
h3 : 3 ≠ 0
h12 : 12 ≠ 0
⊢ dweierstrass_dy e (-b2 e / 12, -(-e.a1 * b2 e / 12 + 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] | . 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 | case pos.h_3.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
hn2 : ringChar K = 2 → False
hn3 : ringChar K = 3 → False
h2 : 2 ≠ 0
h3 : 3 ≠ 0
h12 : 12 ≠ 0
⊢ weierstrass e (-b2 e / 12, -(-e.a1 * b2 e / 12 + e.a3) / 2) = 0
case pos.h_3.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
hn2 : ringChar K = 2 → False
hn3 : ringChar K = 3 → False
h2 : 2 ≠ 0
h3 : 3 ≠ 0
h12 : 12 ≠ 0
⊢ dweierstrass_dx e (-b2 e / 12, -(-e.a1 * b2 e / 12 + e.a3) / 2) = 0
case pos.h_3.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
hn2 : ringChar K = 2 → False
hn3 : ringChar K = 3 → False
h2 : 2 ≠ 0
h3 : 3 ≠ 0
h12 : 12 ≠ 0
⊢ dweierstrass_dy e (-b2 e / 12, -(-e.a1 * b2 e / 12 + e.a3) / 2) = 0 | case pos.h_3.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
hn2 : ringChar K = 2 → False
hn3 : ringChar K = 3 → False
h2 : 2 ≠ 0
h3 : 3 ≠ 0
h12 : 12 ≠ 0
⊢ dweierstrass_dx e (-b2 e / 12, -(-e.a1 * b2 e / 12 + e.a3) / 2) = 0
case pos.h_3.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
hn2 : ringChar K = 2 → False
hn3 : ringChar K = 3 → False
h2 : 2 ≠ 0
h3 : 3 ≠ 0
h12 : 12 ≠ 0
⊢ dweierstrass_dy e (-b2 e / 12, -(-e.a1 * b2 e / 12 + 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] | . 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 | case pos.h_3.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
hn2 : ringChar K = 2 → False
hn3 : ringChar K = 3 → False
h2 : 2 ≠ 0
h3 : 3 ≠ 0
h12 : 12 ≠ 0
⊢ dweierstrass_dx e (-b2 e / 12, -(-e.a1 * b2 e / 12 + e.a3) / 2) = 0
case pos.h_3.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
hn2 : ringChar K = 2 → False
hn3 : ringChar K = 3 → False
h2 : 2 ≠ 0
h3 : 3 ≠ 0
h12 : 12 ≠ 0
⊢ dweierstrass_dy e (-b2 e / 12, -(-e.a1 * b2 e / 12 + e.a3) / 2) = 0 | case pos.h_3.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
hn2 : ringChar K = 2 → False
hn3 : ringChar K = 3 → False
h2 : 2 ≠ 0
h3 : 3 ≠ 0
h12 : 12 ≠ 0
⊢ dweierstrass_dy e (-b2 e / 12, -(-e.a1 * b2 e / 12 + 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] | . 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.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
hn2 : ringChar K = 2 → False
hn3 : ringChar K = 3 → False
h2 : 2 ≠ 0
h3 : 3 ≠ 0
h12 : 12 ≠ 0
⊢ dweierstrass_dy e (-b2 e / 12, -(-e.a1 * b2 e / 12 + e.a3) / 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
hn2 : ringChar K = 2 → False
hn3 : ringChar K = 3 → False
hh : 2 = 0
⊢ Nat.Prime 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] | 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
hn2 : ringChar K = 2 → False
hn3 : ringChar K = 3 → False
h2 : 2 ≠ 0
hh : 3 = 0
⊢ Nat.Prime 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] | rw [show (12 : K) = 2 * 2 * 3 by 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
hn2 : ringChar K = 2 → False
hn3 : ringChar K = 3 → False
h2 : 2 ≠ 0
h3 : 3 ≠ 0
⊢ 12 ≠ 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
hn2 : ringChar K = 2 → False
hn3 : ringChar K = 3 → False
h2 : 2 ≠ 0
h3 : 3 ≠ 0
⊢ 2 * 2 * 3 ≠ 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] | repeat' apply mul_ne_zero | 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
hn2 : ringChar K = 2 → False
hn3 : ringChar K = 3 → False
h2 : 2 ≠ 0
h3 : 3 ≠ 0
⊢ 2 * 2 * 3 ≠ 0 | case ha.ha
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
hn2 : ringChar K = 2 → False
hn3 : ringChar K = 3 → False
h2 : 2 ≠ 0
h3 : 3 ≠ 0
⊢ 2 ≠ 0
case ha.hb
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
hn2 : ringChar K = 2 → False
hn3 : ringChar K = 3 → False
h2 : 2 ≠ 0
h3 : 3 ≠ 0
⊢ 2 ≠ 0
case hb
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
hn2 : ringChar K = 2 → False
hn3 : ringChar K = 3 → False
h2 : 2 ≠ 0
h3 : 3 ≠ 0
⊢ 3 ≠ 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] | all_goals assumption | case ha.ha
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
hn2 : ringChar K = 2 → False
hn3 : ringChar K = 3 → False
h2 : 2 ≠ 0
h3 : 3 ≠ 0
⊢ 2 ≠ 0
case ha.hb
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
hn2 : ringChar K = 2 → False
hn3 : ringChar K = 3 → False
h2 : 2 ≠ 0
h3 : 3 ≠ 0
⊢ 2 ≠ 0
case hb
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
hn2 : ringChar K = 2 → False
hn3 : ringChar K = 3 → False
h2 : 2 ≠ 0
h3 : 3 ≠ 0
⊢ 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] | 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
hn2 : ringChar K = 2 → False
hn3 : ringChar K = 3 → False
h2 : 2 ≠ 0
h3 : 3 ≠ 0
⊢ 12 = 2 * 2 * 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] | apply mul_ne_zero | case ha
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
hn2 : ringChar K = 2 → False
hn3 : ringChar K = 3 → False
h2 : 2 ≠ 0
h3 : 3 ≠ 0
⊢ 2 * 2 ≠ 0 | case ha.ha
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
hn2 : ringChar K = 2 → False
hn3 : ringChar K = 3 → False
h2 : 2 ≠ 0
h3 : 3 ≠ 0
⊢ 2 ≠ 0
case ha.hb
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
hn2 : ringChar K = 2 → False
hn3 : ringChar K = 3 → False
h2 : 2 ≠ 0
h3 : 3 ≠ 0
⊢ 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] | assumption | case hb
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
hn2 : ringChar K = 2 → False
hn3 : ringChar K = 3 → False
h2 : 2 ≠ 0
h3 : 3 ≠ 0
⊢ 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] | apply nzero_mul_left_cancel (12 ^ 3) _ _ (pow_ne_zero _ h12) | case pos.h_3.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
hn2 : ringChar K = 2 → False
hn3 : ringChar K = 3 → False
h2 : 2 ≠ 0
h3 : 3 ≠ 0
h12 : 12 ≠ 0
⊢ weierstrass e (-b2 e / 12, -(-e.a1 * b2 e / 12 + e.a3) / 2) = 0 | case pos.h_3.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
hn2 : ringChar K = 2 → False
hn3 : ringChar K = 3 → False
h2 : 2 ≠ 0
h3 : 3 ≠ 0
h12 : 12 ≠ 0
⊢ 12 ^ 3 * weierstrass e (-b2 e / 12, -(-e.a1 * b2 e / 12 + e.a3) / 2) = 12 ^ 3 * 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 [weierstrass, div_eq_mul_inv, mul_zero] | case pos.h_3.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
hn2 : ringChar K = 2 → False
hn3 : ringChar K = 3 → False
h2 : 2 ≠ 0
h3 : 3 ≠ 0
h12 : 12 ≠ 0
⊢ 12 ^ 3 * weierstrass e (-b2 e / 12, -(-e.a1 * b2 e / 12 + e.a3) / 2) = 12 ^ 3 * 0 | case pos.h_3.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
hn2 : ringChar K = 2 → False
hn3 : ringChar K = 3 → False
h2 : 2 ≠ 0
h3 : 3 ≠ 0
h12 : 12 ≠ 0
⊢ 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)) =
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
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] | case pos.h_3.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
hn2 : ringChar K = 2 → False
hn3 : ringChar K = 3 → False
h2 : 2 ≠ 0
h3 : 3 ≠ 0
h12 : 12 ≠ 0
⊢ 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)) =
0 | case pos.h_3.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
hn2 : ringChar K = 2 → False
hn3 : ringChar K = 3 → False
h2 : 2 ≠ 0
h3 : 3 ≠ 0
h12 : 12 ≠ 0
⊢ 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) =
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 [mul_inv_cancel h2, mul_inv_cancel h12, one_mul, mul_one] | case pos.h_3.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
hn2 : ringChar K = 2 → False
hn3 : ringChar K = 3 → False
h2 : 2 ≠ 0
h3 : 3 ≠ 0
h12 : 12 ≠ 0
⊢ 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) =
0 | case pos.h_3.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
hn2 : ringChar K = 2 → False
hn3 : ringChar K = 3 → False
h2 : 2 ≠ 0
h3 : 3 ≠ 0
h12 : 12 ≠ 0
⊢ 3 * (-(-e.a1 * b2 e + 12 * e.a3)) ^ 2 + e.a1 * -b2 e * (-(-e.a1 * b2 e + 12 * e.a3) * 6) +
12 * e.a3 * (-(-e.a1 * b2 e + 12 * e.a3) * 6) -
((-b2 e) ^ 3 + 12 * e.a2 * (-b2 e) ^ 2 + 12 ^ 2 * e.a4 * -b2 e + 12 ^ 3 * 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 [← mul_zero (2 : K), ← hc6] | case pos.h_3.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
hn2 : ringChar K = 2 → False
hn3 : ringChar K = 3 → False
h2 : 2 ≠ 0
h3 : 3 ≠ 0
h12 : 12 ≠ 0
⊢ 3 * (-(-e.a1 * b2 e + 12 * e.a3)) ^ 2 + e.a1 * -b2 e * (-(-e.a1 * b2 e + 12 * e.a3) * 6) +
12 * e.a3 * (-(-e.a1 * b2 e + 12 * e.a3) * 6) -
((-b2 e) ^ 3 + 12 * e.a2 * (-b2 e) ^ 2 + 12 ^ 2 * e.a4 * -b2 e + 12 ^ 3 * e.a6) =
0 | case pos.h_3.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
hn2 : ringChar K = 2 → False
hn3 : ringChar K = 3 → False
h2 : 2 ≠ 0
h3 : 3 ≠ 0
h12 : 12 ≠ 0
⊢ 3 * (-(-e.a1 * b2 e + 12 * e.a3)) ^ 2 + e.a1 * -b2 e * (-(-e.a1 * b2 e + 12 * e.a3) * 6) +
12 * e.a3 * (-(-e.a1 * b2 e + 12 * e.a3) * 6) -
((-b2 e) ^ 3 + 12 * e.a2 * (-b2 e) ^ 2 + 12 ^ 2 * e.a4 * -b2 e + 12 ^ 3 * e.a6) =
2 * c6 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] | simp only [c6, b2, b4, b6] | case pos.h_3.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
hn2 : ringChar K = 2 → False
hn3 : ringChar K = 3 → False
h2 : 2 ≠ 0
h3 : 3 ≠ 0
h12 : 12 ≠ 0
⊢ 3 * (-(-e.a1 * b2 e + 12 * e.a3)) ^ 2 + e.a1 * -b2 e * (-(-e.a1 * b2 e + 12 * e.a3) * 6) +
12 * e.a3 * (-(-e.a1 * b2 e + 12 * e.a3) * 6) -
((-b2 e) ^ 3 + 12 * e.a2 * (-b2 e) ^ 2 + 12 ^ 2 * e.a4 * -b2 e + 12 ^ 3 * e.a6) =
2 * c6 e | case pos.h_3.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
hn2 : ringChar K = 2 → False
hn3 : ringChar K = 3 → False
h2 : 2 ≠ 0
h3 : 3 ≠ 0
h12 : 12 ≠ 0
⊢ 3 * (-(-e.a1 * (e.a1 * e.a1 + 4 * e.a2) + 12 * e.a3)) ^ 2 +
e.a1 * -(e.a1 * e.a1 + 4 * e.a2) * (-(-e.a1 * (e.a1 * e.a1 + 4 * e.a2) + 12 * e.a3) * 6) +
12 * e.a3 * (-(-e.a1 * (e.a1 * e.a1 + 4 * e.a2) + 12 * e.a3) * 6) -
((-(e.a1 * e.a1 + 4 * e.a2)) ^ 3 + 12 * e.a2 * (-(e.a1 * e.a1 + 4 * e.a2)) ^ 2 +
12 ^ 2 * e.a4 * -(e.a1 * e.a1 + 4 * e.a2) +
12 ^ 3 * e.a6) =
2 *
(-(e.a1 * e.a1 + 4 * e.a2) ^ 3 + 36 * (e.a1 * e.a1 + 4 * e.a2) * (e.a1 * e.a3 + 2 * e.a4) -
216 * (e.a3 * e.a3 + 4 * e.a6)) |
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 | case pos.h_3.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
hn2 : ringChar K = 2 → False
hn3 : ringChar K = 3 → False
h2 : 2 ≠ 0
h3 : 3 ≠ 0
h12 : 12 ≠ 0
⊢ 3 * (-(-e.a1 * (e.a1 * e.a1 + 4 * e.a2) + 12 * e.a3)) ^ 2 +
e.a1 * -(e.a1 * e.a1 + 4 * e.a2) * (-(-e.a1 * (e.a1 * e.a1 + 4 * e.a2) + 12 * e.a3) * 6) +
12 * e.a3 * (-(-e.a1 * (e.a1 * e.a1 + 4 * e.a2) + 12 * e.a3) * 6) -
((-(e.a1 * e.a1 + 4 * e.a2)) ^ 3 + 12 * e.a2 * (-(e.a1 * e.a1 + 4 * e.a2)) ^ 2 +
12 ^ 2 * e.a4 * -(e.a1 * e.a1 + 4 * e.a2) +
12 ^ 3 * e.a6) =
2 *
(-(e.a1 * e.a1 + 4 * e.a2) ^ 3 + 36 * (e.a1 * e.a1 + 4 * e.a2) * (e.a1 * e.a3 + 2 * e.a4) -
216 * (e.a3 * e.a3 + 4 * e.a6)) | 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
hn2 : ringChar K = 2 → False
hn3 : ringChar K = 3 → False
h2 : 2 ≠ 0
h3 : 3 ≠ 0
h12 : 12 ≠ 0
⊢ 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) | 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] | apply nzero_mul_left_cancel (12 ^ 2) _ _ (pow_ne_zero _ h12) | case pos.h_3.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
hn2 : ringChar K = 2 → False
hn3 : ringChar K = 3 → False
h2 : 2 ≠ 0
h3 : 3 ≠ 0
h12 : 12 ≠ 0
⊢ dweierstrass_dx e (-b2 e / 12, -(-e.a1 * b2 e / 12 + e.a3) / 2) = 0 | case pos.h_3.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
hn2 : ringChar K = 2 → False
hn3 : ringChar K = 3 → False
h2 : 2 ≠ 0
h3 : 3 ≠ 0
h12 : 12 ≠ 0
⊢ 12 ^ 2 * dweierstrass_dx e (-b2 e / 12, -(-e.a1 * b2 e / 12 + e.a3) / 2) = 12 ^ 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 only [dweierstrass_dx, div_eq_mul_inv, mul_zero] | case pos.h_3.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
hn2 : ringChar K = 2 → False
hn3 : ringChar K = 3 → False
h2 : 2 ≠ 0
h3 : 3 ≠ 0
h12 : 12 ≠ 0
⊢ 12 ^ 2 * dweierstrass_dx e (-b2 e / 12, -(-e.a1 * b2 e / 12 + e.a3) / 2) = 12 ^ 2 * 0 | case pos.h_3.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
hn2 : ringChar K = 2 → False
hn3 : ringChar K = 3 → False
h2 : 2 ≠ 0
h3 : 3 ≠ 0
h12 : 12 ≠ 0
⊢ 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)) =
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
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] | case pos.h_3.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
hn2 : ringChar K = 2 → False
hn3 : ringChar K = 3 → False
h2 : 2 ≠ 0
h3 : 3 ≠ 0
h12 : 12 ≠ 0
⊢ 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)) =
0 | case pos.h_3.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
hn2 : ringChar K = 2 → False
hn3 : ringChar K = 3 → False
h2 : 2 ≠ 0
h3 : 3 ≠ 0
h12 : 12 ≠ 0
⊢ 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) =
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 [mul_inv_cancel h2, mul_inv_cancel h12, one_mul, mul_one] | case pos.h_3.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
hn2 : ringChar K = 2 → False
hn3 : ringChar K = 3 → False
h2 : 2 ≠ 0
h3 : 3 ≠ 0
h12 : 12 ≠ 0
⊢ 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) =
0 | case pos.h_3.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
hn2 : ringChar K = 2 → False
hn3 : ringChar K = 3 → False
h2 : 2 ≠ 0
h3 : 3 ≠ 0
h12 : 12 ≠ 0
⊢ e.a1 * (-(-e.a1 * b2 e + 12 * e.a3) * 6) - (3 * (-b2 e) ^ 2 + 24 * e.a2 * -b2 e + 144 * 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 [← mul_zero (3 : K), ← hc4] | case pos.h_3.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
hn2 : ringChar K = 2 → False
hn3 : ringChar K = 3 → False
h2 : 2 ≠ 0
h3 : 3 ≠ 0
h12 : 12 ≠ 0
⊢ e.a1 * (-(-e.a1 * b2 e + 12 * e.a3) * 6) - (3 * (-b2 e) ^ 2 + 24 * e.a2 * -b2 e + 144 * e.a4) = 0 | case pos.h_3.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
hn2 : ringChar K = 2 → False
hn3 : ringChar K = 3 → False
h2 : 2 ≠ 0
h3 : 3 ≠ 0
h12 : 12 ≠ 0
⊢ e.a1 * (-(-e.a1 * b2 e + 12 * e.a3) * 6) - (3 * (-b2 e) ^ 2 + 24 * e.a2 * -b2 e + 144 * e.a4) = 3 * 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] | simp only [c4, c6, b2, b4, b6] | case pos.h_3.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
hn2 : ringChar K = 2 → False
hn3 : ringChar K = 3 → False
h2 : 2 ≠ 0
h3 : 3 ≠ 0
h12 : 12 ≠ 0
⊢ e.a1 * (-(-e.a1 * b2 e + 12 * e.a3) * 6) - (3 * (-b2 e) ^ 2 + 24 * e.a2 * -b2 e + 144 * e.a4) = 3 * c4 e | case pos.h_3.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
hn2 : ringChar K = 2 → False
hn3 : ringChar K = 3 → False
h2 : 2 ≠ 0
h3 : 3 ≠ 0
h12 : 12 ≠ 0
⊢ e.a1 * (-(-e.a1 * (e.a1 * e.a1 + 4 * e.a2) + 12 * e.a3) * 6) -
(3 * (-(e.a1 * e.a1 + 4 * e.a2)) ^ 2 + 24 * e.a2 * -(e.a1 * e.a1 + 4 * e.a2) + 144 * e.a4) =
3 * ((e.a1 * e.a1 + 4 * e.a2) ^ 2 - 24 * (e.a1 * e.a3 + 2 * 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] | ring | case pos.h_3.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
hn2 : ringChar K = 2 → False
hn3 : ringChar K = 3 → False
h2 : 2 ≠ 0
h3 : 3 ≠ 0
h12 : 12 ≠ 0
⊢ e.a1 * (-(-e.a1 * (e.a1 * e.a1 + 4 * e.a2) + 12 * e.a3) * 6) -
(3 * (-(e.a1 * e.a1 + 4 * e.a2)) ^ 2 + 24 * e.a2 * -(e.a1 * e.a1 + 4 * e.a2) + 144 * e.a4) =
3 * ((e.a1 * e.a1 + 4 * e.a2) ^ 2 - 24 * (e.a1 * e.a3 + 2 * 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] | 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
hn2 : ringChar K = 2 → False
hn3 : ringChar K = 3 → False
h2 : 2 ≠ 0
h3 : 3 ≠ 0
h12 : 12 ≠ 0
⊢ 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) | 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] | apply nzero_mul_left_cancel 12 _ _ h12 | case pos.h_3.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
hn2 : ringChar K = 2 → False
hn3 : ringChar K = 3 → False
h2 : 2 ≠ 0
h3 : 3 ≠ 0
h12 : 12 ≠ 0
⊢ dweierstrass_dy e (-b2 e / 12, -(-e.a1 * b2 e / 12 + e.a3) / 2) = 0 | case pos.h_3.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
hn2 : ringChar K = 2 → False
hn3 : ringChar K = 3 → False
h2 : 2 ≠ 0
h3 : 3 ≠ 0
h12 : 12 ≠ 0
⊢ 12 * dweierstrass_dy e (-b2 e / 12, -(-e.a1 * b2 e / 12 + e.a3) / 2) = 12 * 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 [dweierstrass_dy, div_eq_mul_inv, mul_zero] | case pos.h_3.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
hn2 : ringChar K = 2 → False
hn3 : ringChar K = 3 → False
h2 : 2 ≠ 0
h3 : 3 ≠ 0
h12 : 12 ≠ 0
⊢ 12 * dweierstrass_dy e (-b2 e / 12, -(-e.a1 * b2 e / 12 + e.a3) / 2) = 12 * 0 | case pos.h_3.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
hn2 : ringChar K = 2 → False
hn3 : ringChar K = 3 → False
h2 : 2 ≠ 0
h3 : 3 ≠ 0
h12 : 12 ≠ 0
⊢ 12 * (2 * (-(-e.a1 * b2 e * 12⁻¹ + e.a3) * 2⁻¹) + e.a1 * (-b2 e * 12⁻¹) + 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 [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] | case pos.h_3.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
hn2 : ringChar K = 2 → False
hn3 : ringChar K = 3 → False
h2 : 2 ≠ 0
h3 : 3 ≠ 0
h12 : 12 ≠ 0
⊢ 12 * (2 * (-(-e.a1 * b2 e * 12⁻¹ + e.a3) * 2⁻¹) + e.a1 * (-b2 e * 12⁻¹) + e.a3) = 0 | case pos.h_3.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
hn2 : ringChar K = 2 → False
hn3 : ringChar K = 3 → False
h2 : 2 ≠ 0
h3 : 3 ≠ 0
h12 : 12 ≠ 0
⊢ -(-e.a1 * b2 e * (12 * 12⁻¹) + 12 * e.a3) * (2 * 2⁻¹) + e.a1 * (-b2 e * (12 * 12⁻¹)) + 12 * 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] | simp only [mul_inv_cancel h2, mul_inv_cancel h12, one_mul, mul_one] | case pos.h_3.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
hn2 : ringChar K = 2 → False
hn3 : ringChar K = 3 → False
h2 : 2 ≠ 0
h3 : 3 ≠ 0
h12 : 12 ≠ 0
⊢ -(-e.a1 * b2 e * (12 * 12⁻¹) + 12 * e.a3) * (2 * 2⁻¹) + e.a1 * (-b2 e * (12 * 12⁻¹)) + 12 * e.a3 = 0 | case pos.h_3.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
hn2 : ringChar K = 2 → False
hn3 : ringChar K = 3 → False
h2 : 2 ≠ 0
h3 : 3 ≠ 0
h12 : 12 ≠ 0
⊢ -(-e.a1 * b2 e + 12 * e.a3) + e.a1 * -b2 e + 12 * 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] | simp only [c6, b2, b4, b6] | case pos.h_3.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
hn2 : ringChar K = 2 → False
hn3 : ringChar K = 3 → False
h2 : 2 ≠ 0
h3 : 3 ≠ 0
h12 : 12 ≠ 0
⊢ -(-e.a1 * b2 e + 12 * e.a3) + e.a1 * -b2 e + 12 * e.a3 = 0 | case pos.h_3.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
hn2 : ringChar K = 2 → False
hn3 : ringChar K = 3 → False
h2 : 2 ≠ 0
h3 : 3 ≠ 0
h12 : 12 ≠ 0
⊢ -(-e.a1 * (e.a1 * e.a1 + 4 * e.a2) + 12 * e.a3) + e.a1 * -(e.a1 * e.a1 + 4 * e.a2) + 12 * 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] | ring | case pos.h_3.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
hn2 : ringChar K = 2 → False
hn3 : ringChar K = 3 → False
h2 : 2 ≠ 0
h3 : 3 ≠ 0
h12 : 12 ≠ 0
⊢ -(-e.a1 * (e.a1 * e.a1 + 4 * e.a2) + 12 * e.a3) + e.a1 * -(e.a1 * e.a1 + 4 * e.a2) + 12 * 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] | 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
hn2 : ringChar K = 2 → False
hn3 : ringChar K = 3 → False
h2 : 2 ≠ 0
h3 : 3 ≠ 0
h12 : 12 ≠ 0
⊢ 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 | 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 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
⊢ weierstrass e ((18 * b6 e - b2 e * b4 e) / c4 e, (b2 e * b5 e + 3 * b7 e) / c4 e) = 0 ∧
dweierstrass_dx e ((18 * b6 e - b2 e * b4 e) / c4 e, (b2 e * b5 e + 3 * b7 e) / c4 e) = 0 ∧
dweierstrass_dy e ((18 * b6 e - b2 e * b4 e) / c4 e, (b2 e * b5 e + 3 * b7 e) / c4 e) = 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 neg
K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : ¬c4 e = 0
⊢ weierstrass e ((18 * b6 e - b2 e * b4 e) / c4 e, (b2 e * b5 e + 3 * b7 e) / c4 e) = 0 ∧
dweierstrass_dx e ((18 * b6 e - b2 e * b4 e) / c4 e, (b2 e * b5 e + 3 * b7 e) / c4 e) = 0 ∧
dweierstrass_dy e ((18 * b6 e - b2 e * b4 e) / c4 e, (b2 e * b5 e + 3 * b7 e) / c4 e) = 0 | case neg.refine_1
K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : ¬c4 e = 0
⊢ weierstrass e ((18 * b6 e - b2 e * b4 e) / c4 e, (b2 e * b5 e + 3 * b7 e) / c4 e) = 0
case neg.refine_2
K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : ¬c4 e = 0
⊢ dweierstrass_dx e ((18 * b6 e - b2 e * b4 e) / c4 e, (b2 e * b5 e + 3 * b7 e) / c4 e) = 0
case neg.refine_3
K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : ¬c4 e = 0
⊢ dweierstrass_dy e ((18 * b6 e - b2 e * b4 e) / c4 e, (b2 e * b5 e + 3 * b7 e) / c4 e) = 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]
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 | case neg.refine_1
K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : ¬c4 e = 0
⊢ weierstrass e ((18 * b6 e - b2 e * b4 e) / c4 e, (b2 e * b5 e + 3 * b7 e) / c4 e) = 0
case neg.refine_2
K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : ¬c4 e = 0
⊢ dweierstrass_dx e ((18 * b6 e - b2 e * b4 e) / c4 e, (b2 e * b5 e + 3 * b7 e) / c4 e) = 0
case neg.refine_3
K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : ¬c4 e = 0
⊢ dweierstrass_dy e ((18 * b6 e - b2 e * b4 e) / c4 e, (b2 e * b5 e + 3 * b7 e) / c4 e) = 0 | case neg.refine_2
K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : ¬c4 e = 0
⊢ dweierstrass_dx e ((18 * b6 e - b2 e * b4 e) / c4 e, (b2 e * b5 e + 3 * b7 e) / c4 e) = 0
case neg.refine_3
K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : ¬c4 e = 0
⊢ dweierstrass_dy e ((18 * b6 e - b2 e * b4 e) / c4 e, (b2 e * b5 e + 3 * b7 e) / c4 e) = 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]
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 | case neg.refine_2
K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : ¬c4 e = 0
⊢ dweierstrass_dx e ((18 * b6 e - b2 e * b4 e) / c4 e, (b2 e * b5 e + 3 * b7 e) / c4 e) = 0
case neg.refine_3
K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : ¬c4 e = 0
⊢ dweierstrass_dy e ((18 * b6 e - b2 e * b4 e) / c4 e, (b2 e * b5 e + 3 * b7 e) / c4 e) = 0 | case neg.refine_3
K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : ¬c4 e = 0
⊢ dweierstrass_dy e ((18 * b6 e - b2 e * b4 e) / c4 e, (b2 e * b5 e + 3 * b7 e) / c4 e) = 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]
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.refine_3
K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : ¬c4 e = 0
⊢ dweierstrass_dy e ((18 * b6 e - b2 e * b4 e) / c4 e, (b2 e * b5 e + 3 * b7 e) / c4 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 neg.refine_1
K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : ¬c4 e = 0
⊢ weierstrass e ((18 * b6 e - b2 e * b4 e) / c4 e, (b2 e * b5 e + 3 * b7 e) / c4 e) = 0 | case neg.refine_1
K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : ¬c4 e = 0
⊢ ((18 * b6 e - b2 e * b4 e) / c4 e, (b2 e * b5 e + 3 * b7 e) / c4 e).snd ^ 2 +
e.a1 * ((18 * b6 e - b2 e * b4 e) / c4 e, (b2 e * b5 e + 3 * b7 e) / c4 e).fst *
((18 * b6 e - b2 e * b4 e) / c4 e, (b2 e * b5 e + 3 * b7 e) / c4 e).snd +
e.a3 * ((18 * b6 e - b2 e * b4 e) / c4 e, (b2 e * b5 e + 3 * b7 e) / c4 e).snd -
(((18 * b6 e - b2 e * b4 e) / c4 e, (b2 e * b5 e + 3 * b7 e) / c4 e).fst ^ 3 +
e.a2 * ((18 * b6 e - b2 e * b4 e) / c4 e, (b2 e * b5 e + 3 * b7 e) / c4 e).fst ^ 2 +
e.a4 * ((18 * b6 e - b2 e * b4 e) / c4 e, (b2 e * b5 e + 3 * b7 e) / c4 e).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] | apply nzero_mul_left_cancel (e.c4 ^ 3) _ _ (pow_ne_zero _ hc4) | case neg.refine_1
K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : ¬c4 e = 0
⊢ ((18 * b6 e - b2 e * b4 e) / c4 e, (b2 e * b5 e + 3 * b7 e) / c4 e).snd ^ 2 +
e.a1 * ((18 * b6 e - b2 e * b4 e) / c4 e, (b2 e * b5 e + 3 * b7 e) / c4 e).fst *
((18 * b6 e - b2 e * b4 e) / c4 e, (b2 e * b5 e + 3 * b7 e) / c4 e).snd +
e.a3 * ((18 * b6 e - b2 e * b4 e) / c4 e, (b2 e * b5 e + 3 * b7 e) / c4 e).snd -
(((18 * b6 e - b2 e * b4 e) / c4 e, (b2 e * b5 e + 3 * b7 e) / c4 e).fst ^ 3 +
e.a2 * ((18 * b6 e - b2 e * b4 e) / c4 e, (b2 e * b5 e + 3 * b7 e) / c4 e).fst ^ 2 +
e.a4 * ((18 * b6 e - b2 e * b4 e) / c4 e, (b2 e * b5 e + 3 * b7 e) / c4 e).fst +
e.a6) =
0 | case neg.refine_1
K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : ¬c4 e = 0
⊢ c4 e ^ 3 *
(((18 * b6 e - b2 e * b4 e) / c4 e, (b2 e * b5 e + 3 * b7 e) / c4 e).snd ^ 2 +
e.a1 * ((18 * b6 e - b2 e * b4 e) / c4 e, (b2 e * b5 e + 3 * b7 e) / c4 e).fst *
((18 * b6 e - b2 e * b4 e) / c4 e, (b2 e * b5 e + 3 * b7 e) / c4 e).snd +
e.a3 * ((18 * b6 e - b2 e * b4 e) / c4 e, (b2 e * b5 e + 3 * b7 e) / c4 e).snd -
(((18 * b6 e - b2 e * b4 e) / c4 e, (b2 e * b5 e + 3 * b7 e) / c4 e).fst ^ 3 +
e.a2 * ((18 * b6 e - b2 e * b4 e) / c4 e, (b2 e * b5 e + 3 * b7 e) / c4 e).fst ^ 2 +
e.a4 * ((18 * b6 e - b2 e * b4 e) / c4 e, (b2 e * b5 e + 3 * b7 e) / c4 e).fst +
e.a6)) =
c4 e ^ 3 * 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 [mul_zero] | case neg.refine_1
K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : ¬c4 e = 0
⊢ c4 e ^ 3 *
(((18 * b6 e - b2 e * b4 e) / c4 e, (b2 e * b5 e + 3 * b7 e) / c4 e).snd ^ 2 +
e.a1 * ((18 * b6 e - b2 e * b4 e) / c4 e, (b2 e * b5 e + 3 * b7 e) / c4 e).fst *
((18 * b6 e - b2 e * b4 e) / c4 e, (b2 e * b5 e + 3 * b7 e) / c4 e).snd +
e.a3 * ((18 * b6 e - b2 e * b4 e) / c4 e, (b2 e * b5 e + 3 * b7 e) / c4 e).snd -
(((18 * b6 e - b2 e * b4 e) / c4 e, (b2 e * b5 e + 3 * b7 e) / c4 e).fst ^ 3 +
e.a2 * ((18 * b6 e - b2 e * b4 e) / c4 e, (b2 e * b5 e + 3 * b7 e) / c4 e).fst ^ 2 +
e.a4 * ((18 * b6 e - b2 e * b4 e) / c4 e, (b2 e * b5 e + 3 * b7 e) / c4 e).fst +
e.a6)) =
c4 e ^ 3 * 0 | case neg.refine_1
K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : ¬c4 e = 0
⊢ c4 e ^ 3 *
(((18 * b6 e - b2 e * b4 e) / c4 e, (b2 e * b5 e + 3 * b7 e) / c4 e).snd ^ 2 +
e.a1 * ((18 * b6 e - b2 e * b4 e) / c4 e, (b2 e * b5 e + 3 * b7 e) / c4 e).fst *
((18 * b6 e - b2 e * b4 e) / c4 e, (b2 e * b5 e + 3 * b7 e) / c4 e).snd +
e.a3 * ((18 * b6 e - b2 e * b4 e) / c4 e, (b2 e * b5 e + 3 * b7 e) / c4 e).snd -
(((18 * b6 e - b2 e * b4 e) / c4 e, (b2 e * b5 e + 3 * b7 e) / c4 e).fst ^ 3 +
e.a2 * ((18 * b6 e - b2 e * b4 e) / c4 e, (b2 e * b5 e + 3 * b7 e) / c4 e).fst ^ 2 +
e.a4 * ((18 * b6 e - b2 e * b4 e) / c4 e, (b2 e * b5 e + 3 * b7 e) / c4 e).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 [div_eq_mul_inv] | case neg.refine_1
K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : ¬c4 e = 0
⊢ c4 e ^ 3 *
(((18 * b6 e - b2 e * b4 e) / c4 e, (b2 e * b5 e + 3 * b7 e) / c4 e).snd ^ 2 +
e.a1 * ((18 * b6 e - b2 e * b4 e) / c4 e, (b2 e * b5 e + 3 * b7 e) / c4 e).fst *
((18 * b6 e - b2 e * b4 e) / c4 e, (b2 e * b5 e + 3 * b7 e) / c4 e).snd +
e.a3 * ((18 * b6 e - b2 e * b4 e) / c4 e, (b2 e * b5 e + 3 * b7 e) / c4 e).snd -
(((18 * b6 e - b2 e * b4 e) / c4 e, (b2 e * b5 e + 3 * b7 e) / c4 e).fst ^ 3 +
e.a2 * ((18 * b6 e - b2 e * b4 e) / c4 e, (b2 e * b5 e + 3 * b7 e) / c4 e).fst ^ 2 +
e.a4 * ((18 * b6 e - b2 e * b4 e) / c4 e, (b2 e * b5 e + 3 * b7 e) / c4 e).fst +
e.a6)) =
0 | case neg.refine_1
K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : ¬c4 e = 0
⊢ 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)) =
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 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] | case neg.refine_1
K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : ¬c4 e = 0
⊢ 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)) =
0 | case neg.refine_1
K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : ¬c4 e = 0
⊢ 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) =
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 [mul_inv_cancel hc4, one_mul] | case neg.refine_1
K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : ¬c4 e = 0
⊢ 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) =
0 | case neg.refine_1
K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : ¬c4 e = 0
⊢ c4 e * (b2 e * b5 e + 3 * b7 e) ^ 2 + c4 e * e.a1 * (18 * b6 e - b2 e * b4 e) * (b2 e * b5 e + 3 * b7 e) +
c4 e * c4 e * e.a3 * (b2 e * b5 e + 3 * b7 e) -
((18 * b6 e - b2 e * b4 e) ^ 3 + c4 e * e.a2 * (18 * b6 e - b2 e * b4 e) ^ 2 +
c4 e * c4 e * e.a4 * (18 * b6 e - b2 e * b4 e) +
c4 e * c4 e * c4 e * 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 [b5, b7, c4, b2, b4, b6] | case neg.refine_1
K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : ¬c4 e = 0
⊢ c4 e * (b2 e * b5 e + 3 * b7 e) ^ 2 + c4 e * e.a1 * (18 * b6 e - b2 e * b4 e) * (b2 e * b5 e + 3 * b7 e) +
c4 e * c4 e * e.a3 * (b2 e * b5 e + 3 * b7 e) -
((18 * b6 e - b2 e * b4 e) ^ 3 + c4 e * e.a2 * (18 * b6 e - b2 e * b4 e) ^ 2 +
c4 e * c4 e * e.a4 * (18 * b6 e - b2 e * b4 e) +
c4 e * c4 e * c4 e * e.a6) =
0 | case neg.refine_1
K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : ¬c4 e = 0
⊢ ((e.a1 * e.a1 + 4 * e.a2) ^ 2 - 24 * (e.a1 * e.a3 + 2 * e.a4)) *
((e.a1 * e.a1 + 4 * e.a2) * (e.a1 * e.a4 - 2 * e.a2 * e.a3) +
3 * (e.a1 * (e.a3 ^ 2 - 12 * e.a6) + 8 * e.a3 * e.a4)) ^
2 +
((e.a1 * e.a1 + 4 * e.a2) ^ 2 - 24 * (e.a1 * e.a3 + 2 * e.a4)) * e.a1 *
(18 * (e.a3 * e.a3 + 4 * e.a6) - (e.a1 * e.a1 + 4 * e.a2) * (e.a1 * e.a3 + 2 * e.a4)) *
((e.a1 * e.a1 + 4 * e.a2) * (e.a1 * e.a4 - 2 * e.a2 * e.a3) +
3 * (e.a1 * (e.a3 ^ 2 - 12 * e.a6) + 8 * e.a3 * e.a4)) +
((e.a1 * e.a1 + 4 * e.a2) ^ 2 - 24 * (e.a1 * e.a3 + 2 * e.a4)) *
((e.a1 * e.a1 + 4 * e.a2) ^ 2 - 24 * (e.a1 * e.a3 + 2 * e.a4)) *
e.a3 *
((e.a1 * e.a1 + 4 * e.a2) * (e.a1 * e.a4 - 2 * e.a2 * e.a3) +
3 * (e.a1 * (e.a3 ^ 2 - 12 * e.a6) + 8 * e.a3 * e.a4)) -
((18 * (e.a3 * e.a3 + 4 * e.a6) - (e.a1 * e.a1 + 4 * e.a2) * (e.a1 * e.a3 + 2 * e.a4)) ^ 3 +
((e.a1 * e.a1 + 4 * e.a2) ^ 2 - 24 * (e.a1 * e.a3 + 2 * e.a4)) * e.a2 *
(18 * (e.a3 * e.a3 + 4 * e.a6) - (e.a1 * e.a1 + 4 * e.a2) * (e.a1 * e.a3 + 2 * e.a4)) ^ 2 +
((e.a1 * e.a1 + 4 * e.a2) ^ 2 - 24 * (e.a1 * e.a3 + 2 * e.a4)) *
((e.a1 * e.a1 + 4 * e.a2) ^ 2 - 24 * (e.a1 * e.a3 + 2 * e.a4)) *
e.a4 *
(18 * (e.a3 * e.a3 + 4 * e.a6) - (e.a1 * e.a1 + 4 * e.a2) * (e.a1 * e.a3 + 2 * e.a4)) +
((e.a1 * e.a1 + 4 * e.a2) ^ 2 - 24 * (e.a1 * e.a3 + 2 * e.a4)) *
((e.a1 * e.a1 + 4 * e.a2) ^ 2 - 24 * (e.a1 * e.a3 + 2 * e.a4)) *
((e.a1 * e.a1 + 4 * e.a2) ^ 2 - 24 * (e.a1 * e.a3 + 2 * 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 [← 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] | case neg.refine_1
K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : ¬c4 e = 0
⊢ ((e.a1 * e.a1 + 4 * e.a2) ^ 2 - 24 * (e.a1 * e.a3 + 2 * e.a4)) *
((e.a1 * e.a1 + 4 * e.a2) * (e.a1 * e.a4 - 2 * e.a2 * e.a3) +
3 * (e.a1 * (e.a3 ^ 2 - 12 * e.a6) + 8 * e.a3 * e.a4)) ^
2 +
((e.a1 * e.a1 + 4 * e.a2) ^ 2 - 24 * (e.a1 * e.a3 + 2 * e.a4)) * e.a1 *
(18 * (e.a3 * e.a3 + 4 * e.a6) - (e.a1 * e.a1 + 4 * e.a2) * (e.a1 * e.a3 + 2 * e.a4)) *
((e.a1 * e.a1 + 4 * e.a2) * (e.a1 * e.a4 - 2 * e.a2 * e.a3) +
3 * (e.a1 * (e.a3 ^ 2 - 12 * e.a6) + 8 * e.a3 * e.a4)) +
((e.a1 * e.a1 + 4 * e.a2) ^ 2 - 24 * (e.a1 * e.a3 + 2 * e.a4)) *
((e.a1 * e.a1 + 4 * e.a2) ^ 2 - 24 * (e.a1 * e.a3 + 2 * e.a4)) *
e.a3 *
((e.a1 * e.a1 + 4 * e.a2) * (e.a1 * e.a4 - 2 * e.a2 * e.a3) +
3 * (e.a1 * (e.a3 ^ 2 - 12 * e.a6) + 8 * e.a3 * e.a4)) -
((18 * (e.a3 * e.a3 + 4 * e.a6) - (e.a1 * e.a1 + 4 * e.a2) * (e.a1 * e.a3 + 2 * e.a4)) ^ 3 +
((e.a1 * e.a1 + 4 * e.a2) ^ 2 - 24 * (e.a1 * e.a3 + 2 * e.a4)) * e.a2 *
(18 * (e.a3 * e.a3 + 4 * e.a6) - (e.a1 * e.a1 + 4 * e.a2) * (e.a1 * e.a3 + 2 * e.a4)) ^ 2 +
((e.a1 * e.a1 + 4 * e.a2) ^ 2 - 24 * (e.a1 * e.a3 + 2 * e.a4)) *
((e.a1 * e.a1 + 4 * e.a2) ^ 2 - 24 * (e.a1 * e.a3 + 2 * e.a4)) *
e.a4 *
(18 * (e.a3 * e.a3 + 4 * e.a6) - (e.a1 * e.a1 + 4 * e.a2) * (e.a1 * e.a3 + 2 * e.a4)) +
((e.a1 * e.a1 + 4 * e.a2) ^ 2 - 24 * (e.a1 * e.a3 + 2 * e.a4)) *
((e.a1 * e.a1 + 4 * e.a2) ^ 2 - 24 * (e.a1 * e.a3 + 2 * e.a4)) *
((e.a1 * e.a1 + 4 * e.a2) ^ 2 - 24 * (e.a1 * e.a3 + 2 * e.a4)) *
e.a6) =
0 | case neg.refine_1
K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : ¬c4 e = 0
⊢ ((e.a1 * e.a1 + 4 * e.a2) ^ 2 - 24 * (e.a1 * e.a3 + 2 * e.a4)) *
((e.a1 * e.a1 + 4 * e.a2) * (e.a1 * e.a4 - 2 * e.a2 * e.a3) +
3 * (e.a1 * (e.a3 ^ 2 - 12 * e.a6) + 8 * e.a3 * e.a4)) ^
2 +
((e.a1 * e.a1 + 4 * e.a2) ^ 2 - 24 * (e.a1 * e.a3 + 2 * e.a4)) * e.a1 *
(18 * (e.a3 * e.a3 + 4 * e.a6) - (e.a1 * e.a1 + 4 * e.a2) * (e.a1 * e.a3 + 2 * e.a4)) *
((e.a1 * e.a1 + 4 * e.a2) * (e.a1 * e.a4 - 2 * e.a2 * e.a3) +
3 * (e.a1 * (e.a3 ^ 2 - 12 * e.a6) + 8 * e.a3 * e.a4)) +
((e.a1 * e.a1 + 4 * e.a2) ^ 2 - 24 * (e.a1 * e.a3 + 2 * e.a4)) *
((e.a1 * e.a1 + 4 * e.a2) ^ 2 - 24 * (e.a1 * e.a3 + 2 * e.a4)) *
e.a3 *
((e.a1 * e.a1 + 4 * e.a2) * (e.a1 * e.a4 - 2 * e.a2 * e.a3) +
3 * (e.a1 * (e.a3 ^ 2 - 12 * e.a6) + 8 * e.a3 * e.a4)) -
((18 * (e.a3 * e.a3 + 4 * e.a6) - (e.a1 * e.a1 + 4 * e.a2) * (e.a1 * e.a3 + 2 * e.a4)) ^ 3 +
((e.a1 * e.a1 + 4 * e.a2) ^ 2 - 24 * (e.a1 * e.a3 + 2 * e.a4)) * e.a2 *
(18 * (e.a3 * e.a3 + 4 * e.a6) - (e.a1 * e.a1 + 4 * e.a2) * (e.a1 * e.a3 + 2 * e.a4)) ^ 2 +
((e.a1 * e.a1 + 4 * e.a2) ^ 2 - 24 * (e.a1 * e.a3 + 2 * e.a4)) *
((e.a1 * e.a1 + 4 * e.a2) ^ 2 - 24 * (e.a1 * e.a3 + 2 * e.a4)) *
e.a4 *
(18 * (e.a3 * e.a3 + 4 * e.a6) - (e.a1 * e.a1 + 4 * e.a2) * (e.a1 * e.a3 + 2 * e.a4)) +
((e.a1 * e.a1 + 4 * e.a2) ^ 2 - 24 * (e.a1 * e.a3 + 2 * e.a4)) *
((e.a1 * e.a1 + 4 * e.a2) ^ 2 - 24 * (e.a1 * e.a3 + 2 * e.a4)) *
((e.a1 * e.a1 + 4 * e.a2) ^ 2 - 24 * (e.a1 * e.a3 + 2 * e.a4)) *
e.a6) =
(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) *
-(e.a1 ^ 4 * e.a2 * e.a3 ^ 2 - e.a1 ^ 5 * e.a3 * e.a4 + e.a1 ^ 6 * e.a6 + 8 * e.a1 ^ 2 * e.a2 ^ 2 * e.a3 ^ 2 -
e.a1 ^ 3 * e.a3 ^ 3 -
8 * e.a1 ^ 3 * e.a2 * e.a3 * e.a4 -
e.a1 ^ 4 * e.a4 ^ 2 +
12 * e.a1 ^ 4 * e.a2 * e.a6 +
16 * e.a2 ^ 3 * e.a3 ^ 2 -
36 * e.a1 * e.a2 * e.a3 ^ 3 -
16 * e.a1 * e.a2 ^ 2 * e.a3 * e.a4 +
30 * e.a1 ^ 2 * e.a3 ^ 2 * e.a4 -
8 * e.a1 ^ 2 * e.a2 * e.a4 ^ 2 +
48 * e.a1 ^ 2 * e.a2 ^ 2 * e.a6 -
36 * e.a1 ^ 3 * e.a3 * e.a6 +
27 * e.a3 ^ 4 -
72 * e.a2 * e.a3 ^ 2 * e.a4 -
16 * e.a2 ^ 2 * e.a4 ^ 2 +
96 * e.a1 * e.a3 * e.a4 ^ 2 +
64 * e.a2 ^ 3 * e.a6 -
144 * e.a1 * e.a2 * e.a3 * e.a6 -
72 * e.a1 ^ 2 * e.a4 * e.a6 +
64 * e.a4 ^ 3 +
216 * e.a3 ^ 2 * e.a6 -
288 * e.a2 * e.a4 * e.a6 +
432 * e.a6 ^ 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] | ring | case neg.refine_1
K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : ¬c4 e = 0
⊢ ((e.a1 * e.a1 + 4 * e.a2) ^ 2 - 24 * (e.a1 * e.a3 + 2 * e.a4)) *
((e.a1 * e.a1 + 4 * e.a2) * (e.a1 * e.a4 - 2 * e.a2 * e.a3) +
3 * (e.a1 * (e.a3 ^ 2 - 12 * e.a6) + 8 * e.a3 * e.a4)) ^
2 +
((e.a1 * e.a1 + 4 * e.a2) ^ 2 - 24 * (e.a1 * e.a3 + 2 * e.a4)) * e.a1 *
(18 * (e.a3 * e.a3 + 4 * e.a6) - (e.a1 * e.a1 + 4 * e.a2) * (e.a1 * e.a3 + 2 * e.a4)) *
((e.a1 * e.a1 + 4 * e.a2) * (e.a1 * e.a4 - 2 * e.a2 * e.a3) +
3 * (e.a1 * (e.a3 ^ 2 - 12 * e.a6) + 8 * e.a3 * e.a4)) +
((e.a1 * e.a1 + 4 * e.a2) ^ 2 - 24 * (e.a1 * e.a3 + 2 * e.a4)) *
((e.a1 * e.a1 + 4 * e.a2) ^ 2 - 24 * (e.a1 * e.a3 + 2 * e.a4)) *
e.a3 *
((e.a1 * e.a1 + 4 * e.a2) * (e.a1 * e.a4 - 2 * e.a2 * e.a3) +
3 * (e.a1 * (e.a3 ^ 2 - 12 * e.a6) + 8 * e.a3 * e.a4)) -
((18 * (e.a3 * e.a3 + 4 * e.a6) - (e.a1 * e.a1 + 4 * e.a2) * (e.a1 * e.a3 + 2 * e.a4)) ^ 3 +
((e.a1 * e.a1 + 4 * e.a2) ^ 2 - 24 * (e.a1 * e.a3 + 2 * e.a4)) * e.a2 *
(18 * (e.a3 * e.a3 + 4 * e.a6) - (e.a1 * e.a1 + 4 * e.a2) * (e.a1 * e.a3 + 2 * e.a4)) ^ 2 +
((e.a1 * e.a1 + 4 * e.a2) ^ 2 - 24 * (e.a1 * e.a3 + 2 * e.a4)) *
((e.a1 * e.a1 + 4 * e.a2) ^ 2 - 24 * (e.a1 * e.a3 + 2 * e.a4)) *
e.a4 *
(18 * (e.a3 * e.a3 + 4 * e.a6) - (e.a1 * e.a1 + 4 * e.a2) * (e.a1 * e.a3 + 2 * e.a4)) +
((e.a1 * e.a1 + 4 * e.a2) ^ 2 - 24 * (e.a1 * e.a3 + 2 * e.a4)) *
((e.a1 * e.a1 + 4 * e.a2) ^ 2 - 24 * (e.a1 * e.a3 + 2 * e.a4)) *
((e.a1 * e.a1 + 4 * e.a2) ^ 2 - 24 * (e.a1 * e.a3 + 2 * e.a4)) *
e.a6) =
(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) *
-(e.a1 ^ 4 * e.a2 * e.a3 ^ 2 - e.a1 ^ 5 * e.a3 * e.a4 + e.a1 ^ 6 * e.a6 + 8 * e.a1 ^ 2 * e.a2 ^ 2 * e.a3 ^ 2 -
e.a1 ^ 3 * e.a3 ^ 3 -
8 * e.a1 ^ 3 * e.a2 * e.a3 * e.a4 -
e.a1 ^ 4 * e.a4 ^ 2 +
12 * e.a1 ^ 4 * e.a2 * e.a6 +
16 * e.a2 ^ 3 * e.a3 ^ 2 -
36 * e.a1 * e.a2 * e.a3 ^ 3 -
16 * e.a1 * e.a2 ^ 2 * e.a3 * e.a4 +
30 * e.a1 ^ 2 * e.a3 ^ 2 * e.a4 -
8 * e.a1 ^ 2 * e.a2 * e.a4 ^ 2 +
48 * e.a1 ^ 2 * e.a2 ^ 2 * e.a6 -
36 * e.a1 ^ 3 * e.a3 * e.a6 +
27 * e.a3 ^ 4 -
72 * e.a2 * e.a3 ^ 2 * e.a4 -
16 * e.a2 ^ 2 * e.a4 ^ 2 +
96 * e.a1 * e.a3 * e.a4 ^ 2 +
64 * e.a2 ^ 3 * e.a6 -
144 * e.a1 * e.a2 * e.a3 * e.a6 -
72 * e.a1 ^ 2 * e.a4 * e.a6 +
64 * e.a4 ^ 3 +
216 * e.a3 ^ 2 * e.a6 -
288 * e.a2 * e.a4 * e.a6 +
432 * e.a6 ^ 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] | ring | K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : ¬c4 e = 0
⊢ 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) | 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 neg.refine_2
K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : ¬c4 e = 0
⊢ dweierstrass_dx e ((18 * b6 e - b2 e * b4 e) / c4 e, (b2 e * b5 e + 3 * b7 e) / c4 e) = 0 | case neg.refine_2
K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : ¬c4 e = 0
⊢ e.a1 * ((18 * b6 e - b2 e * b4 e) / c4 e, (b2 e * b5 e + 3 * b7 e) / c4 e).snd -
(3 * ((18 * b6 e - b2 e * b4 e) / c4 e, (b2 e * b5 e + 3 * b7 e) / c4 e).fst ^ 2 +
2 * e.a2 * ((18 * b6 e - b2 e * b4 e) / c4 e, (b2 e * b5 e + 3 * b7 e) / c4 e).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] | apply nzero_mul_left_cancel (e.c4 ^ 2) _ _ (pow_ne_zero _ hc4) | case neg.refine_2
K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : ¬c4 e = 0
⊢ e.a1 * ((18 * b6 e - b2 e * b4 e) / c4 e, (b2 e * b5 e + 3 * b7 e) / c4 e).snd -
(3 * ((18 * b6 e - b2 e * b4 e) / c4 e, (b2 e * b5 e + 3 * b7 e) / c4 e).fst ^ 2 +
2 * e.a2 * ((18 * b6 e - b2 e * b4 e) / c4 e, (b2 e * b5 e + 3 * b7 e) / c4 e).fst +
e.a4) =
0 | case neg.refine_2
K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : ¬c4 e = 0
⊢ c4 e ^ 2 *
(e.a1 * ((18 * b6 e - b2 e * b4 e) / c4 e, (b2 e * b5 e + 3 * b7 e) / c4 e).snd -
(3 * ((18 * b6 e - b2 e * b4 e) / c4 e, (b2 e * b5 e + 3 * b7 e) / c4 e).fst ^ 2 +
2 * e.a2 * ((18 * b6 e - b2 e * b4 e) / c4 e, (b2 e * b5 e + 3 * b7 e) / c4 e).fst +
e.a4)) =
c4 e ^ 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 [mul_zero, pow_two] | case neg.refine_2
K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : ¬c4 e = 0
⊢ c4 e ^ 2 *
(e.a1 * ((18 * b6 e - b2 e * b4 e) / c4 e, (b2 e * b5 e + 3 * b7 e) / c4 e).snd -
(3 * ((18 * b6 e - b2 e * b4 e) / c4 e, (b2 e * b5 e + 3 * b7 e) / c4 e).fst ^ 2 +
2 * e.a2 * ((18 * b6 e - b2 e * b4 e) / c4 e, (b2 e * b5 e + 3 * b7 e) / c4 e).fst +
e.a4)) =
c4 e ^ 2 * 0 | case neg.refine_2
K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : ¬c4 e = 0
⊢ c4 e * c4 e *
(e.a1 * ((18 * b6 e - b2 e * b4 e) / c4 e, (b2 e * b5 e + 3 * b7 e) / c4 e).snd -
(3 * ((18 * b6 e - b2 e * b4 e) / c4 e, (b2 e * b5 e + 3 * b7 e) / c4 e).fst ^ 2 +
2 * e.a2 * ((18 * b6 e - b2 e * b4 e) / c4 e, (b2 e * b5 e + 3 * b7 e) / c4 e).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 [div_eq_mul_inv] | case neg.refine_2
K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : ¬c4 e = 0
⊢ c4 e * c4 e *
(e.a1 * ((18 * b6 e - b2 e * b4 e) / c4 e, (b2 e * b5 e + 3 * b7 e) / c4 e).snd -
(3 * ((18 * b6 e - b2 e * b4 e) / c4 e, (b2 e * b5 e + 3 * b7 e) / c4 e).fst ^ 2 +
2 * e.a2 * ((18 * b6 e - b2 e * b4 e) / c4 e, (b2 e * b5 e + 3 * b7 e) / c4 e).fst +
e.a4)) =
0 | case neg.refine_2
K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : ¬c4 e = 0
⊢ 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)) =
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 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] | case neg.refine_2
K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : ¬c4 e = 0
⊢ 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)) =
0 | case neg.refine_2
K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : ¬c4 e = 0
⊢ 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 =
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 [mul_inv_cancel hc4, one_mul] | case neg.refine_2
K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : ¬c4 e = 0
⊢ 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 =
0 | case neg.refine_2
K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : ¬c4 e = 0
⊢ c4 e * (e.a1 * (b2 e * b5 e + 3 * b7 e) - 2 * e.a2 * (18 * b6 e - b2 e * b4 e)) - 3 * (18 * b6 e - b2 e * b4 e) ^ 2 -
e.a4 * c4 e * c4 e =
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 [b5, b7, c4, b2, b4, b6] | case neg.refine_2
K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : ¬c4 e = 0
⊢ c4 e * (e.a1 * (b2 e * b5 e + 3 * b7 e) - 2 * e.a2 * (18 * b6 e - b2 e * b4 e)) - 3 * (18 * b6 e - b2 e * b4 e) ^ 2 -
e.a4 * c4 e * c4 e =
0 | case neg.refine_2
K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : ¬c4 e = 0
⊢ ((e.a1 * e.a1 + 4 * e.a2) ^ 2 - 24 * (e.a1 * e.a3 + 2 * e.a4)) *
(e.a1 *
((e.a1 * e.a1 + 4 * e.a2) * (e.a1 * e.a4 - 2 * e.a2 * e.a3) +
3 * (e.a1 * (e.a3 ^ 2 - 12 * e.a6) + 8 * e.a3 * e.a4)) -
2 * e.a2 * (18 * (e.a3 * e.a3 + 4 * e.a6) - (e.a1 * e.a1 + 4 * e.a2) * (e.a1 * e.a3 + 2 * e.a4))) -
3 * (18 * (e.a3 * e.a3 + 4 * e.a6) - (e.a1 * e.a1 + 4 * e.a2) * (e.a1 * e.a3 + 2 * e.a4)) ^ 2 -
e.a4 * ((e.a1 * e.a1 + 4 * e.a2) ^ 2 - 24 * (e.a1 * e.a3 + 2 * e.a4)) *
((e.a1 * e.a1 + 4 * e.a2) ^ 2 - 24 * (e.a1 * e.a3 + 2 * 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 [← mul_zero (36 : K), ← h, discr_eq_neg_singular] | case neg.refine_2
K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : ¬c4 e = 0
⊢ ((e.a1 * e.a1 + 4 * e.a2) ^ 2 - 24 * (e.a1 * e.a3 + 2 * e.a4)) *
(e.a1 *
((e.a1 * e.a1 + 4 * e.a2) * (e.a1 * e.a4 - 2 * e.a2 * e.a3) +
3 * (e.a1 * (e.a3 ^ 2 - 12 * e.a6) + 8 * e.a3 * e.a4)) -
2 * e.a2 * (18 * (e.a3 * e.a3 + 4 * e.a6) - (e.a1 * e.a1 + 4 * e.a2) * (e.a1 * e.a3 + 2 * e.a4))) -
3 * (18 * (e.a3 * e.a3 + 4 * e.a6) - (e.a1 * e.a1 + 4 * e.a2) * (e.a1 * e.a3 + 2 * e.a4)) ^ 2 -
e.a4 * ((e.a1 * e.a1 + 4 * e.a2) ^ 2 - 24 * (e.a1 * e.a3 + 2 * e.a4)) *
((e.a1 * e.a1 + 4 * e.a2) ^ 2 - 24 * (e.a1 * e.a3 + 2 * e.a4)) =
0 | case neg.refine_2
K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : ¬c4 e = 0
⊢ ((e.a1 * e.a1 + 4 * e.a2) ^ 2 - 24 * (e.a1 * e.a3 + 2 * e.a4)) *
(e.a1 *
((e.a1 * e.a1 + 4 * e.a2) * (e.a1 * e.a4 - 2 * e.a2 * e.a3) +
3 * (e.a1 * (e.a3 ^ 2 - 12 * e.a6) + 8 * e.a3 * e.a4)) -
2 * e.a2 * (18 * (e.a3 * e.a3 + 4 * e.a6) - (e.a1 * e.a1 + 4 * e.a2) * (e.a1 * e.a3 + 2 * e.a4))) -
3 * (18 * (e.a3 * e.a3 + 4 * e.a6) - (e.a1 * e.a1 + 4 * e.a2) * (e.a1 * e.a3 + 2 * e.a4)) ^ 2 -
e.a4 * ((e.a1 * e.a1 + 4 * e.a2) ^ 2 - 24 * (e.a1 * e.a3 + 2 * e.a4)) *
((e.a1 * e.a1 + 4 * e.a2) ^ 2 - 24 * (e.a1 * e.a3 + 2 * e.a4)) =
36 *
-(e.a1 ^ 4 * e.a2 * e.a3 ^ 2 - e.a1 ^ 5 * e.a3 * e.a4 + e.a1 ^ 6 * e.a6 + 8 * e.a1 ^ 2 * e.a2 ^ 2 * e.a3 ^ 2 -
e.a1 ^ 3 * e.a3 ^ 3 -
8 * e.a1 ^ 3 * e.a2 * e.a3 * e.a4 -
e.a1 ^ 4 * e.a4 ^ 2 +
12 * e.a1 ^ 4 * e.a2 * e.a6 +
16 * e.a2 ^ 3 * e.a3 ^ 2 -
36 * e.a1 * e.a2 * e.a3 ^ 3 -
16 * e.a1 * e.a2 ^ 2 * e.a3 * e.a4 +
30 * e.a1 ^ 2 * e.a3 ^ 2 * e.a4 -
8 * e.a1 ^ 2 * e.a2 * e.a4 ^ 2 +
48 * e.a1 ^ 2 * e.a2 ^ 2 * e.a6 -
36 * e.a1 ^ 3 * e.a3 * e.a6 +
27 * e.a3 ^ 4 -
72 * e.a2 * e.a3 ^ 2 * e.a4 -
16 * e.a2 ^ 2 * e.a4 ^ 2 +
96 * e.a1 * e.a3 * e.a4 ^ 2 +
64 * e.a2 ^ 3 * e.a6 -
144 * e.a1 * e.a2 * e.a3 * e.a6 -
72 * e.a1 ^ 2 * e.a4 * e.a6 +
64 * e.a4 ^ 3 +
216 * e.a3 ^ 2 * e.a6 -
288 * e.a2 * e.a4 * e.a6 +
432 * e.a6 ^ 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] | ring | case neg.refine_2
K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : ¬c4 e = 0
⊢ ((e.a1 * e.a1 + 4 * e.a2) ^ 2 - 24 * (e.a1 * e.a3 + 2 * e.a4)) *
(e.a1 *
((e.a1 * e.a1 + 4 * e.a2) * (e.a1 * e.a4 - 2 * e.a2 * e.a3) +
3 * (e.a1 * (e.a3 ^ 2 - 12 * e.a6) + 8 * e.a3 * e.a4)) -
2 * e.a2 * (18 * (e.a3 * e.a3 + 4 * e.a6) - (e.a1 * e.a1 + 4 * e.a2) * (e.a1 * e.a3 + 2 * e.a4))) -
3 * (18 * (e.a3 * e.a3 + 4 * e.a6) - (e.a1 * e.a1 + 4 * e.a2) * (e.a1 * e.a3 + 2 * e.a4)) ^ 2 -
e.a4 * ((e.a1 * e.a1 + 4 * e.a2) ^ 2 - 24 * (e.a1 * e.a3 + 2 * e.a4)) *
((e.a1 * e.a1 + 4 * e.a2) ^ 2 - 24 * (e.a1 * e.a3 + 2 * e.a4)) =
36 *
-(e.a1 ^ 4 * e.a2 * e.a3 ^ 2 - e.a1 ^ 5 * e.a3 * e.a4 + e.a1 ^ 6 * e.a6 + 8 * e.a1 ^ 2 * e.a2 ^ 2 * e.a3 ^ 2 -
e.a1 ^ 3 * e.a3 ^ 3 -
8 * e.a1 ^ 3 * e.a2 * e.a3 * e.a4 -
e.a1 ^ 4 * e.a4 ^ 2 +
12 * e.a1 ^ 4 * e.a2 * e.a6 +
16 * e.a2 ^ 3 * e.a3 ^ 2 -
36 * e.a1 * e.a2 * e.a3 ^ 3 -
16 * e.a1 * e.a2 ^ 2 * e.a3 * e.a4 +
30 * e.a1 ^ 2 * e.a3 ^ 2 * e.a4 -
8 * e.a1 ^ 2 * e.a2 * e.a4 ^ 2 +
48 * e.a1 ^ 2 * e.a2 ^ 2 * e.a6 -
36 * e.a1 ^ 3 * e.a3 * e.a6 +
27 * e.a3 ^ 4 -
72 * e.a2 * e.a3 ^ 2 * e.a4 -
16 * e.a2 ^ 2 * e.a4 ^ 2 +
96 * e.a1 * e.a3 * e.a4 ^ 2 +
64 * e.a2 ^ 3 * e.a6 -
144 * e.a1 * e.a2 * e.a3 * e.a6 -
72 * e.a1 ^ 2 * e.a4 * e.a6 +
64 * e.a4 ^ 3 +
216 * e.a3 ^ 2 * e.a6 -
288 * e.a2 * e.a4 * e.a6 +
432 * e.a6 ^ 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] | ring | K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : ¬c4 e = 0
⊢ 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 | 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_dy] | case neg.refine_3
K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : ¬c4 e = 0
⊢ dweierstrass_dy e ((18 * b6 e - b2 e * b4 e) / c4 e, (b2 e * b5 e + 3 * b7 e) / c4 e) = 0 | case neg.refine_3
K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : ¬c4 e = 0
⊢ 2 * ((18 * b6 e - b2 e * b4 e) / c4 e, (b2 e * b5 e + 3 * b7 e) / c4 e).snd +
e.a1 * ((18 * b6 e - b2 e * b4 e) / c4 e, (b2 e * b5 e + 3 * b7 e) / c4 e).fst +
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] | apply nzero_mul_left_cancel e.c4 _ _ hc4 | case neg.refine_3
K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : ¬c4 e = 0
⊢ 2 * ((18 * b6 e - b2 e * b4 e) / c4 e, (b2 e * b5 e + 3 * b7 e) / c4 e).snd +
e.a1 * ((18 * b6 e - b2 e * b4 e) / c4 e, (b2 e * b5 e + 3 * b7 e) / c4 e).fst +
e.a3 =
0 | case neg.refine_3
K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : ¬c4 e = 0
⊢ c4 e *
(2 * ((18 * b6 e - b2 e * b4 e) / c4 e, (b2 e * b5 e + 3 * b7 e) / c4 e).snd +
e.a1 * ((18 * b6 e - b2 e * b4 e) / c4 e, (b2 e * b5 e + 3 * b7 e) / c4 e).fst +
e.a3) =
c4 e * 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 [div_eq_mul_inv, mul_zero] | case neg.refine_3
K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : ¬c4 e = 0
⊢ c4 e *
(2 * ((18 * b6 e - b2 e * b4 e) / c4 e, (b2 e * b5 e + 3 * b7 e) / c4 e).snd +
e.a1 * ((18 * b6 e - b2 e * b4 e) / c4 e, (b2 e * b5 e + 3 * b7 e) / c4 e).fst +
e.a3) =
c4 e * 0 | case neg.refine_3
K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : ¬c4 e = 0
⊢ 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) = 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 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] | case neg.refine_3
K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : ¬c4 e = 0
⊢ 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) = 0 | case neg.refine_3
K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : ¬c4 e = 0
⊢ 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 = 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 [mul_inv_cancel hc4, one_mul] | case neg.refine_3
K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : ¬c4 e = 0
⊢ 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 = 0 | case neg.refine_3
K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : ¬c4 e = 0
⊢ 2 * (b2 e * b5 e + 3 * b7 e) + e.a1 * (18 * b6 e - b2 e * b4 e) + c4 e * 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 [b5, b7, c4, b2, b4, b6] | case neg.refine_3
K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : ¬c4 e = 0
⊢ 2 * (b2 e * b5 e + 3 * b7 e) + e.a1 * (18 * b6 e - b2 e * b4 e) + c4 e * e.a3 = 0 | case neg.refine_3
K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : ¬c4 e = 0
⊢ 2 *
((e.a1 * e.a1 + 4 * e.a2) * (e.a1 * e.a4 - 2 * e.a2 * e.a3) +
3 * (e.a1 * (e.a3 ^ 2 - 12 * e.a6) + 8 * e.a3 * e.a4)) +
e.a1 * (18 * (e.a3 * e.a3 + 4 * e.a6) - (e.a1 * e.a1 + 4 * e.a2) * (e.a1 * e.a3 + 2 * e.a4)) +
((e.a1 * e.a1 + 4 * e.a2) ^ 2 - 24 * (e.a1 * e.a3 + 2 * e.a4)) * 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] | ring | case neg.refine_3
K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : ¬c4 e = 0
⊢ 2 *
((e.a1 * e.a1 + 4 * e.a2) * (e.a1 * e.a4 - 2 * e.a2 * e.a3) +
3 * (e.a1 * (e.a3 ^ 2 - 12 * e.a6) + 8 * e.a3 * e.a4)) +
e.a1 * (18 * (e.a3 * e.a3 + 4 * e.a6) - (e.a1 * e.a1 + 4 * e.a2) * (e.a1 * e.a3 + 2 * e.a4)) +
((e.a1 * e.a1 + 4 * e.a2) ^ 2 - 24 * (e.a1 * e.a3 + 2 * e.a4)) * 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] | ring | K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : ¬c4 e = 0
⊢ 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 | no goals |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/EllipticCurve/Model.lean | Model.Field.move_singular_point | [709, 1] | [715, 8] | rw [is_singular_point, weierstrass_iso_eq_var_change,
dweierstrass_dx_iso_eq_var_change, zero_mul, add_zero,
dweierstrass_dy_iso_eq_var_change, var_change_comp] | K : Type u
inst✝ : Field K
e : Model K
r t : K
P : K × K
h : is_singular_point e P
⊢ is_singular_point (rst_iso r 0 t e) (var_change (-r) 0 (-t) P) | K : Type u
inst✝ : Field K
e : Model K
r t : K
P : K × K
h : is_singular_point e P
⊢ weierstrass e (var_change (r + -r) (0 + 0) (t + -t + 0 * -r) P) = 0 ∧
dweierstrass_dx e (var_change (r + -r) (0 + 0) (t + -t + 0 * -r) P) = 0 ∧
dweierstrass_dy e (var_change (r + -r) (0 + 0) (t + -t + 0 * -r) P) = 0 |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/EllipticCurve/Model.lean | Model.Field.move_singular_point | [709, 1] | [715, 8] | simpa | K : Type u
inst✝ : Field K
e : Model K
r t : K
P : K × K
h : is_singular_point e P
⊢ weierstrass e (var_change (r + -r) (0 + 0) (t + -t + 0 * -r) P) = 0 ∧
dweierstrass_dx e (var_change (r + -r) (0 + 0) (t + -t + 0 * -r) P) = 0 ∧
dweierstrass_dy e (var_change (r + -r) (0 + 0) (t + -t + 0 * -r) P) = 0 | no goals |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/EllipticCurve/Model.lean | Model.Field.move_singular_point_to_origin | [717, 1] | [723, 20] | rw [move_singular_point_to_origin_iso, rst_triple, move_singular_point_to_origin_triple] | K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
⊢ is_singular_point (move_singular_point_to_origin_iso e) (0, 0) | K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
⊢ is_singular_point
(rst_iso ((singular_point e).fst, 0, (singular_point e).snd).fst
((singular_point e).fst, 0, (singular_point e).snd).snd.fst
((singular_point e).fst, 0, (singular_point e).snd).snd.snd e)
(0, 0) |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/EllipticCurve/Model.lean | Model.Field.move_singular_point_to_origin | [717, 1] | [723, 20] | convert move_singular_point e (singular_point e).fst (singular_point e).snd
(is_singular_point_singular_point e h) using 2 <;> simp [var_change] | K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
⊢ is_singular_point
(rst_iso ((singular_point e).fst, 0, (singular_point e).snd).fst
((singular_point e).fst, 0, (singular_point e).snd).snd.fst
((singular_point e).fst, 0, (singular_point e).snd).snd.snd e)
(0, 0) | no goals |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/EllipticCurve/Model.lean | Model.Field.move_singular_point_to_origin' | [725, 1] | [731, 43] | rintro ⟨P, hP⟩ | K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
⊢ (∃ P, is_singular_point e P) → is_singular_point (move_singular_point_to_origin_iso e) (0, 0) | case intro
K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
P : K × K
hP : is_singular_point e P
⊢ is_singular_point (move_singular_point_to_origin_iso e) (0, 0) |
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