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https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/ValuedRing.lean | Int.nat_val_aux_mul_eq_add | [790, 1] | [797, 75] | cases b with
| zero => simp [nat_valuation_aux]
| succ b =>
exact nat_val_aux'_mul_eq_add p prime prime.one_lt a.succ b.succ _ _ | case succ
R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p : ℕ
prime : Nat.Prime p
hp : optParam (1 < p) (_ : 1 < p)
b a : ℕ
⊢ nat_valuation_aux p hp (Nat.succ a * b) = nat_valuation_aux p hp (Nat.succ a) + nat_valuation_aux p hp b | no goals |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/ValuedRing.lean | Int.nat_val_aux_mul_eq_add | [790, 1] | [797, 75] | simp [nat_valuation_aux] | case succ.zero
R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p : ℕ
prime : Nat.Prime p
hp : optParam (1 < p) (_ : 1 < p)
a : ℕ
⊢ nat_valuation_aux p hp (Nat.succ a * Nat.zero) = nat_valuation_aux p hp (Nat.succ a) + nat_valuation_aux p hp Nat.zero | no goals |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/ValuedRing.lean | Int.nat_val_aux_mul_eq_add | [790, 1] | [797, 75] | exact nat_val_aux'_mul_eq_add p prime prime.one_lt a.succ b.succ _ _ | case succ.succ
R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p : ℕ
prime : Nat.Prime p
hp : optParam (1 < p) (_ : 1 < p)
a b : ℕ
⊢ nat_valuation_aux p hp (Nat.succ a * Nat.succ b) =
nat_valuation_aux p hp (Nat.succ a) + nat_valuation_aux p hp (Nat.succ b) | no goals |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/ValuedRing.lean | Int.nat_val_mul_eq_add | [799, 1] | [803, 48] | convert nat_val_aux_mul_eq_add p prime prime.one_lt a b <;>
ext <;>
rw [(nat_valuation_of_one_lt p _ prime.one_lt)] | R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p : ℕ
prime : Nat.Prime p
a b : ℕ
⊢ nat_valuation p (a * b) = nat_valuation p a + nat_valuation p b | no goals |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/ValuedRing.lean | Int.int_val_mul_eq_add | [805, 1] | [808, 57] | simp [int_val, natAbs_mul] | R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p : ℕ
prime : Nat.Prime p
a b : ℤ
⊢ int_val p (a * b) = int_val p a + int_val p b | R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p : ℕ
prime : Nat.Prime p
a b : ℤ
⊢ nat_valuation p (natAbs a * natAbs b) = nat_valuation p (natAbs a) + nat_valuation p (natAbs b) |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/ValuedRing.lean | Int.int_val_mul_eq_add | [805, 1] | [808, 57] | exact nat_val_mul_eq_add p prime (natAbs a) (natAbs b) | R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p : ℕ
prime : Nat.Prime p
a b : ℤ
⊢ nat_valuation p (natAbs a * natAbs b) = nat_valuation p (natAbs a) + nat_valuation p (natAbs b) | no goals |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/ValuedRing.lean | Int.nat_val_add_eq_min | [810, 1] | [811, 58] | sorry | R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p a b : ℕ
h : nat_valuation p a < nat_valuation p b
⊢ nat_valuation p (a + b) = nat_valuation p a | no goals |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/ValuedRing.lean | Int.nat_val_add_ge_min | [813, 1] | [832, 17] | cases lt_trichotomy (nat_valuation p a) (nat_valuation p b) with | inl h =>
simp only [min, if_pos (le_of_lt h)]
apply le_of_eq
rw [nat_val_add_eq_min p a b h]
rw [inf_eq_left]
exact h.le
| inr h => cases h with
| inl h =>
simp only [min, if_pos (le_of_eq h)]
sorry
| inr h =>
simp only [add_comm a b, min, if_neg (not_le_of_lt h)]
apply le_of_eq
rw [nat_val_add_eq_min p b a h]
simp
exact h.le | R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p a b : ℕ
⊢ min (nat_valuation p a) (nat_valuation p b) ≤ nat_valuation p (a + b) | no goals |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/ValuedRing.lean | Int.nat_val_add_ge_min | [813, 1] | [832, 17] | simp only [min, if_pos (le_of_lt h)] | case inl
R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p a b : ℕ
h : nat_valuation p a < nat_valuation p b
⊢ min (nat_valuation p a) (nat_valuation p b) ≤ nat_valuation p (a + b) | case inl
R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p a b : ℕ
h : nat_valuation p a < nat_valuation p b
⊢ nat_valuation p a ⊓ nat_valuation p b ≤ nat_valuation p (a + b) |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/ValuedRing.lean | Int.nat_val_add_ge_min | [813, 1] | [832, 17] | apply le_of_eq | case inl
R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p a b : ℕ
h : nat_valuation p a < nat_valuation p b
⊢ nat_valuation p a ⊓ nat_valuation p b ≤ nat_valuation p (a + b) | case inl.a
R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p a b : ℕ
h : nat_valuation p a < nat_valuation p b
⊢ nat_valuation p a ⊓ nat_valuation p b = nat_valuation p (a + b) |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/ValuedRing.lean | Int.nat_val_add_ge_min | [813, 1] | [832, 17] | rw [nat_val_add_eq_min p a b h] | case inl.a
R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p a b : ℕ
h : nat_valuation p a < nat_valuation p b
⊢ nat_valuation p a ⊓ nat_valuation p b = nat_valuation p (a + b) | case inl.a
R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p a b : ℕ
h : nat_valuation p a < nat_valuation p b
⊢ nat_valuation p a ⊓ nat_valuation p b = nat_valuation p a |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/ValuedRing.lean | Int.nat_val_add_ge_min | [813, 1] | [832, 17] | rw [inf_eq_left] | case inl.a
R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p a b : ℕ
h : nat_valuation p a < nat_valuation p b
⊢ nat_valuation p a ⊓ nat_valuation p b = nat_valuation p a | case inl.a
R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p a b : ℕ
h : nat_valuation p a < nat_valuation p b
⊢ nat_valuation p a ≤ nat_valuation p b |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/ValuedRing.lean | Int.nat_val_add_ge_min | [813, 1] | [832, 17] | exact h.le | case inl.a
R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p a b : ℕ
h : nat_valuation p a < nat_valuation p b
⊢ nat_valuation p a ≤ nat_valuation p b | no goals |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/ValuedRing.lean | Int.nat_val_add_ge_min | [813, 1] | [832, 17] | cases h with
| inl h =>
simp only [min, if_pos (le_of_eq h)]
sorry
| inr h =>
simp only [add_comm a b, min, if_neg (not_le_of_lt h)]
apply le_of_eq
rw [nat_val_add_eq_min p b a h]
simp
exact h.le | case inr
R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p a b : ℕ
h : nat_valuation p a = nat_valuation p b ∨ nat_valuation p b < nat_valuation p a
⊢ min (nat_valuation p a) (nat_valuation p b) ≤ nat_valuation p (a + b) | no goals |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/ValuedRing.lean | Int.nat_val_add_ge_min | [813, 1] | [832, 17] | simp only [min, if_pos (le_of_eq h)] | case inr.inl
R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p a b : ℕ
h : nat_valuation p a = nat_valuation p b
⊢ min (nat_valuation p a) (nat_valuation p b) ≤ nat_valuation p (a + b) | case inr.inl
R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p a b : ℕ
h : nat_valuation p a = nat_valuation p b
⊢ nat_valuation p a ⊓ nat_valuation p b ≤ nat_valuation p (a + b) |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/ValuedRing.lean | Int.nat_val_add_ge_min | [813, 1] | [832, 17] | sorry | case inr.inl
R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p a b : ℕ
h : nat_valuation p a = nat_valuation p b
⊢ nat_valuation p a ⊓ nat_valuation p b ≤ nat_valuation p (a + b) | no goals |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/ValuedRing.lean | Int.nat_val_add_ge_min | [813, 1] | [832, 17] | simp only [add_comm a b, min, if_neg (not_le_of_lt h)] | case inr.inr
R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p a b : ℕ
h : nat_valuation p b < nat_valuation p a
⊢ min (nat_valuation p a) (nat_valuation p b) ≤ nat_valuation p (a + b) | case inr.inr
R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p a b : ℕ
h : nat_valuation p b < nat_valuation p a
⊢ nat_valuation p a ⊓ nat_valuation p b ≤ nat_valuation p (b + a) |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/ValuedRing.lean | Int.nat_val_add_ge_min | [813, 1] | [832, 17] | apply le_of_eq | case inr.inr
R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p a b : ℕ
h : nat_valuation p b < nat_valuation p a
⊢ nat_valuation p a ⊓ nat_valuation p b ≤ nat_valuation p (b + a) | case inr.inr.a
R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p a b : ℕ
h : nat_valuation p b < nat_valuation p a
⊢ nat_valuation p a ⊓ nat_valuation p b = nat_valuation p (b + a) |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/ValuedRing.lean | Int.nat_val_add_ge_min | [813, 1] | [832, 17] | rw [nat_val_add_eq_min p b a h] | case inr.inr.a
R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p a b : ℕ
h : nat_valuation p b < nat_valuation p a
⊢ nat_valuation p a ⊓ nat_valuation p b = nat_valuation p (b + a) | case inr.inr.a
R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p a b : ℕ
h : nat_valuation p b < nat_valuation p a
⊢ nat_valuation p a ⊓ nat_valuation p b = nat_valuation p b |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/ValuedRing.lean | Int.nat_val_add_ge_min | [813, 1] | [832, 17] | simp | case inr.inr.a
R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p a b : ℕ
h : nat_valuation p b < nat_valuation p a
⊢ nat_valuation p a ⊓ nat_valuation p b = nat_valuation p b | case inr.inr.a
R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p a b : ℕ
h : nat_valuation p b < nat_valuation p a
⊢ nat_valuation p b ≤ nat_valuation p a |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/ValuedRing.lean | Int.nat_val_add_ge_min | [813, 1] | [832, 17] | exact h.le | case inr.inr.a
R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p a b : ℕ
h : nat_valuation p b < nat_valuation p a
⊢ nat_valuation p b ≤ nat_valuation p a | no goals |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/ValuedRing.lean | Int.natAbs_add | [836, 1] | [836, 112] | sorry | R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
a b : ℤ
⊢ natAbs (a + b) = max (natAbs a) (natAbs b) - min (natAbs a) (natAbs b) | no goals |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/ValuedRing.lean | Int.int_val_add_ge_min | [838, 1] | [841, 8] | simp [int_val, natAbs_add] | R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p : ℕ
a b : ℤ
⊢ min (int_val p a) (int_val p b) ≤ int_val p (a + b) | R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p : ℕ
a b : ℤ
⊢ nat_valuation p (natAbs a) ≤ nat_valuation p (max (natAbs a) (natAbs b) - min (natAbs a) (natAbs b)) ∨
nat_valuation p (natAbs b) ≤ nat_valuation p (max (natAbs a) (natAbs b) - min (natAbs a) (natAbs b)) |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/ValuedRing.lean | Int.int_val_add_ge_min | [838, 1] | [841, 8] | sorry | R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p : ℕ
a b : ℤ
⊢ nat_valuation p (natAbs a) ≤ nat_valuation p (max (natAbs a) (natAbs b) - min (natAbs a) (natAbs b)) ∨
nat_valuation p (natAbs b) ≤ nat_valuation p (max (natAbs a) (natAbs b) - min (natAbs a) (natAbs b)) | no goals |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/ValuedRing.lean | Int.int_val_add_eq_min | [843, 1] | [844, 46] | sorry | R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p : ℕ
a b : ℤ
h : int_val p a < int_val p b
⊢ int_val p (a + b) = int_val p a | no goals |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/ValuedRing.lean | Int.int_val_eq_top_iff_zero | [846, 1] | [866, 38] | apply Iff.intro | R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p : ℕ
gt1 : 1 < p
a : ℤ
⊢ int_val p a = ⊤ ↔ a = 0 | case mp
R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p : ℕ
gt1 : 1 < p
a : ℤ
⊢ int_val p a = ⊤ → a = 0
case mpr
R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p : ℕ
gt1 : 1 < p
a : ℤ
⊢ a = 0 → int_val p a = ⊤ |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/ValuedRing.lean | Int.int_val_eq_top_iff_zero | [846, 1] | [866, 38] | . intro hval
simp [int_val, nat_valuation] at hval
cases abs_a : (natAbs a) with
| zero => exact natAbs_eq_zero.1 abs_a
| succ n =>
cases hp : p with
| zero =>
rw [hp] at gt1
apply False.elim ((of_decide_eq_true rfl : ¬1 < 0) gt1)
| succ p' =>
cases hp' : p' with
| zero =>
rw [hp, hp'] at gt1
apply False.elim ((of_decide_eq_true rfl : ¬1 < 1) gt1)
| succ n =>
simp [hp, hp', abs_a, nat_valuation_aux, nat_valuation_aux'] at hval | case mp
R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p : ℕ
gt1 : 1 < p
a : ℤ
⊢ int_val p a = ⊤ → a = 0
case mpr
R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p : ℕ
gt1 : 1 < p
a : ℤ
⊢ a = 0 → int_val p a = ⊤ | case mpr
R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p : ℕ
gt1 : 1 < p
a : ℤ
⊢ a = 0 → int_val p a = ⊤ |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/ValuedRing.lean | Int.int_val_eq_top_iff_zero | [846, 1] | [866, 38] | . intro ha
simp [ha, int_val, nat_valuation] | case mpr
R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p : ℕ
gt1 : 1 < p
a : ℤ
⊢ a = 0 → int_val p a = ⊤ | no goals |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/ValuedRing.lean | Int.int_val_eq_top_iff_zero | [846, 1] | [866, 38] | intro hval | case mp
R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p : ℕ
gt1 : 1 < p
a : ℤ
⊢ int_val p a = ⊤ → a = 0 | case mp
R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p : ℕ
gt1 : 1 < p
a : ℤ
hval : int_val p a = ⊤
⊢ a = 0 |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/ValuedRing.lean | Int.int_val_eq_top_iff_zero | [846, 1] | [866, 38] | simp [int_val, nat_valuation] at hval | case mp
R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p : ℕ
gt1 : 1 < p
a : ℤ
hval : int_val p a = ⊤
⊢ a = 0 | case mp
R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p : ℕ
gt1 : 1 < p
a : ℤ
hval :
(match p, natAbs a with
| x, 0 => ⊤
| 0, Nat.succ n => 0
| 1, Nat.succ x => ⊤
| Nat.succ (Nat.succ q), Nat.succ m => nat_valuation_aux (q + 2) (_ : Nat.succ 0 < Nat.succ (q + 1)) (m + 1)) =
⊤
⊢ a = 0 |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/ValuedRing.lean | Int.int_val_eq_top_iff_zero | [846, 1] | [866, 38] | cases abs_a : (natAbs a) with
| zero => exact natAbs_eq_zero.1 abs_a
| succ n =>
cases hp : p with
| zero =>
rw [hp] at gt1
apply False.elim ((of_decide_eq_true rfl : ¬1 < 0) gt1)
| succ p' =>
cases hp' : p' with
| zero =>
rw [hp, hp'] at gt1
apply False.elim ((of_decide_eq_true rfl : ¬1 < 1) gt1)
| succ n =>
simp [hp, hp', abs_a, nat_valuation_aux, nat_valuation_aux'] at hval | case mp
R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p : ℕ
gt1 : 1 < p
a : ℤ
hval :
(match p, natAbs a with
| x, 0 => ⊤
| 0, Nat.succ n => 0
| 1, Nat.succ x => ⊤
| Nat.succ (Nat.succ q), Nat.succ m => nat_valuation_aux (q + 2) (_ : Nat.succ 0 < Nat.succ (q + 1)) (m + 1)) =
⊤
⊢ a = 0 | no goals |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/ValuedRing.lean | Int.int_val_eq_top_iff_zero | [846, 1] | [866, 38] | exact natAbs_eq_zero.1 abs_a | case mp.zero
R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p : ℕ
gt1 : 1 < p
a : ℤ
hval :
(match p, natAbs a with
| x, 0 => ⊤
| 0, Nat.succ n => 0
| 1, Nat.succ x => ⊤
| Nat.succ (Nat.succ q), Nat.succ m => nat_valuation_aux (q + 2) (_ : Nat.succ 0 < Nat.succ (q + 1)) (m + 1)) =
⊤
abs_a : natAbs a = Nat.zero
⊢ a = 0 | no goals |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/ValuedRing.lean | Int.int_val_eq_top_iff_zero | [846, 1] | [866, 38] | cases hp : p with
| zero =>
rw [hp] at gt1
apply False.elim ((of_decide_eq_true rfl : ¬1 < 0) gt1)
| succ p' =>
cases hp' : p' with
| zero =>
rw [hp, hp'] at gt1
apply False.elim ((of_decide_eq_true rfl : ¬1 < 1) gt1)
| succ n =>
simp [hp, hp', abs_a, nat_valuation_aux, nat_valuation_aux'] at hval | case mp.succ
R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p : ℕ
gt1 : 1 < p
a : ℤ
hval :
(match p, natAbs a with
| x, 0 => ⊤
| 0, Nat.succ n => 0
| 1, Nat.succ x => ⊤
| Nat.succ (Nat.succ q), Nat.succ m => nat_valuation_aux (q + 2) (_ : Nat.succ 0 < Nat.succ (q + 1)) (m + 1)) =
⊤
n : ℕ
abs_a : natAbs a = Nat.succ n
⊢ a = 0 | no goals |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/ValuedRing.lean | Int.int_val_eq_top_iff_zero | [846, 1] | [866, 38] | rw [hp] at gt1 | case mp.succ.zero
R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p : ℕ
gt1 : 1 < p
a : ℤ
hval :
(match p, natAbs a with
| x, 0 => ⊤
| 0, Nat.succ n => 0
| 1, Nat.succ x => ⊤
| Nat.succ (Nat.succ q), Nat.succ m => nat_valuation_aux (q + 2) (_ : Nat.succ 0 < Nat.succ (q + 1)) (m + 1)) =
⊤
n : ℕ
abs_a : natAbs a = Nat.succ n
hp : p = Nat.zero
⊢ a = 0 | case mp.succ.zero
R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p : ℕ
gt1 : 1 < Nat.zero
a : ℤ
hval :
(match p, natAbs a with
| x, 0 => ⊤
| 0, Nat.succ n => 0
| 1, Nat.succ x => ⊤
| Nat.succ (Nat.succ q), Nat.succ m => nat_valuation_aux (q + 2) (_ : Nat.succ 0 < Nat.succ (q + 1)) (m + 1)) =
⊤
n : ℕ
abs_a : natAbs a = Nat.succ n
hp : p = Nat.zero
⊢ a = 0 |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/ValuedRing.lean | Int.int_val_eq_top_iff_zero | [846, 1] | [866, 38] | apply False.elim ((of_decide_eq_true rfl : ¬1 < 0) gt1) | case mp.succ.zero
R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p : ℕ
gt1 : 1 < Nat.zero
a : ℤ
hval :
(match p, natAbs a with
| x, 0 => ⊤
| 0, Nat.succ n => 0
| 1, Nat.succ x => ⊤
| Nat.succ (Nat.succ q), Nat.succ m => nat_valuation_aux (q + 2) (_ : Nat.succ 0 < Nat.succ (q + 1)) (m + 1)) =
⊤
n : ℕ
abs_a : natAbs a = Nat.succ n
hp : p = Nat.zero
⊢ a = 0 | no goals |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/ValuedRing.lean | Int.int_val_eq_top_iff_zero | [846, 1] | [866, 38] | cases hp' : p' with
| zero =>
rw [hp, hp'] at gt1
apply False.elim ((of_decide_eq_true rfl : ¬1 < 1) gt1)
| succ n =>
simp [hp, hp', abs_a, nat_valuation_aux, nat_valuation_aux'] at hval | case mp.succ.succ
R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p : ℕ
gt1 : 1 < p
a : ℤ
hval :
(match p, natAbs a with
| x, 0 => ⊤
| 0, Nat.succ n => 0
| 1, Nat.succ x => ⊤
| Nat.succ (Nat.succ q), Nat.succ m => nat_valuation_aux (q + 2) (_ : Nat.succ 0 < Nat.succ (q + 1)) (m + 1)) =
⊤
n : ℕ
abs_a : natAbs a = Nat.succ n
p' : ℕ
hp : p = Nat.succ p'
⊢ a = 0 | no goals |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/ValuedRing.lean | Int.int_val_eq_top_iff_zero | [846, 1] | [866, 38] | rw [hp, hp'] at gt1 | case mp.succ.succ.zero
R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p : ℕ
gt1 : 1 < p
a : ℤ
hval :
(match p, natAbs a with
| x, 0 => ⊤
| 0, Nat.succ n => 0
| 1, Nat.succ x => ⊤
| Nat.succ (Nat.succ q), Nat.succ m => nat_valuation_aux (q + 2) (_ : Nat.succ 0 < Nat.succ (q + 1)) (m + 1)) =
⊤
n : ℕ
abs_a : natAbs a = Nat.succ n
p' : ℕ
hp : p = Nat.succ p'
hp' : p' = Nat.zero
⊢ a = 0 | case mp.succ.succ.zero
R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p : ℕ
a : ℤ
hval :
(match p, natAbs a with
| x, 0 => ⊤
| 0, Nat.succ n => 0
| 1, Nat.succ x => ⊤
| Nat.succ (Nat.succ q), Nat.succ m => nat_valuation_aux (q + 2) (_ : Nat.succ 0 < Nat.succ (q + 1)) (m + 1)) =
⊤
n : ℕ
abs_a : natAbs a = Nat.succ n
p' : ℕ
gt1 : 1 < Nat.succ Nat.zero
hp : p = Nat.succ p'
hp' : p' = Nat.zero
⊢ a = 0 |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/ValuedRing.lean | Int.int_val_eq_top_iff_zero | [846, 1] | [866, 38] | apply False.elim ((of_decide_eq_true rfl : ¬1 < 1) gt1) | case mp.succ.succ.zero
R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p : ℕ
a : ℤ
hval :
(match p, natAbs a with
| x, 0 => ⊤
| 0, Nat.succ n => 0
| 1, Nat.succ x => ⊤
| Nat.succ (Nat.succ q), Nat.succ m => nat_valuation_aux (q + 2) (_ : Nat.succ 0 < Nat.succ (q + 1)) (m + 1)) =
⊤
n : ℕ
abs_a : natAbs a = Nat.succ n
p' : ℕ
gt1 : 1 < Nat.succ Nat.zero
hp : p = Nat.succ p'
hp' : p' = Nat.zero
⊢ a = 0 | no goals |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/ValuedRing.lean | Int.int_val_eq_top_iff_zero | [846, 1] | [866, 38] | simp [hp, hp', abs_a, nat_valuation_aux, nat_valuation_aux'] at hval | case mp.succ.succ.succ
R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p : ℕ
gt1 : 1 < p
a : ℤ
hval :
(match p, natAbs a with
| x, 0 => ⊤
| 0, Nat.succ n => 0
| 1, Nat.succ x => ⊤
| Nat.succ (Nat.succ q), Nat.succ m => nat_valuation_aux (q + 2) (_ : Nat.succ 0 < Nat.succ (q + 1)) (m + 1)) =
⊤
n✝ : ℕ
abs_a : natAbs a = Nat.succ n✝
p' : ℕ
hp : p = Nat.succ p'
n : ℕ
hp' : p' = Nat.succ n
⊢ a = 0 | no goals |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/ValuedRing.lean | Int.int_val_eq_top_iff_zero | [846, 1] | [866, 38] | intro ha | case mpr
R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p : ℕ
gt1 : 1 < p
a : ℤ
⊢ a = 0 → int_val p a = ⊤ | case mpr
R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p : ℕ
gt1 : 1 < p
a : ℤ
ha : a = 0
⊢ int_val p a = ⊤ |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/ValuedRing.lean | Int.int_val_eq_top_iff_zero | [846, 1] | [866, 38] | simp [ha, int_val, nat_valuation] | case mpr
R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p : ℕ
gt1 : 1 < p
a : ℤ
ha : a = 0
⊢ int_val p a = ⊤ | no goals |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/ValuedRing.lean | Int.nat_valuation_eq_zero_iff | [882, 1] | [887, 8] | have := nat_val_aux_succ | R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p : ℕ
hp : 1 < p
k : ℕ
⊢ nat_valuation p k = 0 ↔ k % p ≠ 0 | R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p : ℕ
hp : 1 < p
k : ℕ
this :
∀ (q m : ℕ) (hq : 1 < q + 2),
nat_valuation_aux (q + 2) hq (m + 1) =
if (m + 1) % (q + 2) ≠ 0 then 0 else nat_valuation_aux (q + 2) hq ((m + 1) / (q + 2)) + 1
⊢ nat_valuation p k = 0 ↔ k % p ≠ 0 |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/ValuedRing.lean | Int.nat_valuation_eq_zero_iff | [882, 1] | [887, 8] | simp only [nat_valuation, ne_eq] | R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p : ℕ
hp : 1 < p
k : ℕ
this :
∀ (q m : ℕ) (hq : 1 < q + 2),
nat_valuation_aux (q + 2) hq (m + 1) =
if (m + 1) % (q + 2) ≠ 0 then 0 else nat_valuation_aux (q + 2) hq ((m + 1) / (q + 2)) + 1
⊢ nat_valuation p k = 0 ↔ k % p ≠ 0 | R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p : ℕ
hp : 1 < p
k : ℕ
this :
∀ (q m : ℕ) (hq : 1 < q + 2),
nat_valuation_aux (q + 2) hq (m + 1) =
if (m + 1) % (q + 2) ≠ 0 then 0 else nat_valuation_aux (q + 2) hq ((m + 1) / (q + 2)) + 1
⊢ (match p, k with
| x, 0 => ⊤
| 0, Nat.succ n => 0
| 1, Nat.succ x => ⊤
| Nat.succ (Nat.succ q), Nat.succ m => nat_valuation_aux (q + 2) (_ : Nat.succ 0 < Nat.succ (q + 1)) (m + 1)) =
0 ↔
¬k % p = 0 |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/ValuedRing.lean | Int.nat_valuation_eq_zero_iff | [882, 1] | [887, 8] | aesop | R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p : ℕ
hp : 1 < p
k : ℕ
this :
∀ (q m : ℕ) (hq : 1 < q + 2),
nat_valuation_aux (q + 2) hq (m + 1) =
if (m + 1) % (q + 2) ≠ 0 then 0 else nat_valuation_aux (q + 2) hq ((m + 1) / (q + 2)) + 1
⊢ (match p, k with
| x, 0 => ⊤
| 0, Nat.succ n => 0
| 1, Nat.succ x => ⊤
| Nat.succ (Nat.succ q), Nat.succ m => nat_valuation_aux (q + 2) (_ : Nat.succ 0 < Nat.succ (q + 1)) (m + 1)) =
0 ↔
¬k % p = 0 | no goals |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/ValuedRing.lean | Int.Int.neg_emod_eq_zero_iff | [890, 1] | [892, 36] | simp [← Int.dvd_iff_emod_eq_zero] | R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p k : ℤ
⊢ -p % k = 0 ↔ p % k = 0 | no goals |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/ValuedRing.lean | Int.int_valuation_eq_zero_iff | [896, 1] | [914, 8] | simp only [int_val] | R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p : ℕ
k : ℤ
hp : 1 < p
⊢ int_val p k = 0 ↔ k % ↑p ≠ 0 | R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p : ℕ
k : ℤ
hp : 1 < p
⊢ nat_valuation p (natAbs k) = 0 ↔ k % ↑p ≠ 0 |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/ValuedRing.lean | Int.int_valuation_eq_zero_iff | [896, 1] | [914, 8] | rw [nat_valuation_eq_zero_iff hp] | R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p : ℕ
k : ℤ
hp : 1 < p
⊢ nat_valuation p (natAbs k) = 0 ↔ k % ↑p ≠ 0 | R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p : ℕ
k : ℤ
hp : 1 < p
⊢ natAbs k % p ≠ 0 ↔ k % ↑p ≠ 0 |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/ValuedRing.lean | Int.int_valuation_eq_zero_iff | [896, 1] | [914, 8] | rw [not_iff_not] | R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p : ℕ
k : ℤ
hp : 1 < p
⊢ natAbs k % p ≠ 0 ↔ k % ↑p ≠ 0 | R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p : ℕ
k : ℤ
hp : 1 < p
⊢ natAbs k % p = 0 ↔ k % ↑p = 0 |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/ValuedRing.lean | Int.int_valuation_eq_zero_iff | [896, 1] | [914, 8] | aesop | R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p : ℕ
k : ℤ
hp : 1 < p
⊢ natAbs k % p = 0 ↔ k % ↑p = 0 | case mp
R : Type u
inst : CommRing R
p : ℕ
k : ℤ
inst_1 : IsDomain R
hp : 1 < p
a : natAbs k % p = 0
⊢ ↑p ∣ k
case mpr
R : Type u
inst : CommRing R
p : ℕ
k : ℤ
inst_1 : IsDomain R
hp : 1 < p
a : ↑p ∣ k
⊢ natAbs k % p = 0 |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/ValuedRing.lean | Int.int_valuation_eq_zero_iff | [896, 1] | [914, 8] | . cases k
. rwa [ofNat_eq_coe, dvd_iff_mod_eq_zero, ← Int.natAbs_eq_zero]
.
rw [dvd_iff_mod_eq_zero]
rw [mod]
simp at *
rw [dvd_iff_mod_eq_zero]
norm_cast
rw [Nat.succ_eq_add_one] at a
exact Iff.mp natAbs_eq_zero a | case mp
R : Type u
inst : CommRing R
p : ℕ
k : ℤ
inst_1 : IsDomain R
hp : 1 < p
a : natAbs k % p = 0
⊢ ↑p ∣ k
case mpr
R : Type u
inst : CommRing R
p : ℕ
k : ℤ
inst_1 : IsDomain R
hp : 1 < p
a : ↑p ∣ k
⊢ natAbs k % p = 0 | case mpr
R : Type u
inst : CommRing R
p : ℕ
k : ℤ
inst_1 : IsDomain R
hp : 1 < p
a : ↑p ∣ k
⊢ natAbs k % p = 0 |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/ValuedRing.lean | Int.int_valuation_eq_zero_iff | [896, 1] | [914, 8] | sorry | case mpr
R : Type u
inst : CommRing R
p : ℕ
k : ℤ
inst_1 : IsDomain R
hp : 1 < p
a : ↑p ∣ k
⊢ natAbs k % p = 0 | no goals |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/ValuedRing.lean | Int.int_valuation_eq_zero_iff | [896, 1] | [914, 8] | cases k | case mp
R : Type u
inst : CommRing R
p : ℕ
k : ℤ
inst_1 : IsDomain R
hp : 1 < p
a : natAbs k % p = 0
⊢ ↑p ∣ k | case mp.ofNat
R : Type u
inst : CommRing R
p : ℕ
inst_1 : IsDomain R
hp : 1 < p
a✝ : ℕ
a : natAbs (ofNat a✝) % p = 0
⊢ ↑p ∣ ofNat a✝
case mp.negSucc
R : Type u
inst : CommRing R
p : ℕ
inst_1 : IsDomain R
hp : 1 < p
a✝ : ℕ
a : natAbs -[a✝+1] % p = 0
⊢ ↑p ∣ -[a✝+1] |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/ValuedRing.lean | Int.int_valuation_eq_zero_iff | [896, 1] | [914, 8] | . rwa [ofNat_eq_coe, dvd_iff_mod_eq_zero, ← Int.natAbs_eq_zero] | case mp.ofNat
R : Type u
inst : CommRing R
p : ℕ
inst_1 : IsDomain R
hp : 1 < p
a✝ : ℕ
a : natAbs (ofNat a✝) % p = 0
⊢ ↑p ∣ ofNat a✝
case mp.negSucc
R : Type u
inst : CommRing R
p : ℕ
inst_1 : IsDomain R
hp : 1 < p
a✝ : ℕ
a : natAbs -[a✝+1] % p = 0
⊢ ↑p ∣ -[a✝+1] | case mp.negSucc
R : Type u
inst : CommRing R
p : ℕ
inst_1 : IsDomain R
hp : 1 < p
a✝ : ℕ
a : natAbs -[a✝+1] % p = 0
⊢ ↑p ∣ -[a✝+1] |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/ValuedRing.lean | Int.int_valuation_eq_zero_iff | [896, 1] | [914, 8] | .
rw [dvd_iff_mod_eq_zero]
rw [mod]
simp at *
rw [dvd_iff_mod_eq_zero]
norm_cast
rw [Nat.succ_eq_add_one] at a
exact Iff.mp natAbs_eq_zero a | case mp.negSucc
R : Type u
inst : CommRing R
p : ℕ
inst_1 : IsDomain R
hp : 1 < p
a✝ : ℕ
a : natAbs -[a✝+1] % p = 0
⊢ ↑p ∣ -[a✝+1] | no goals |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/ValuedRing.lean | Int.int_valuation_eq_zero_iff | [896, 1] | [914, 8] | rwa [ofNat_eq_coe, dvd_iff_mod_eq_zero, ← Int.natAbs_eq_zero] | case mp.ofNat
R : Type u
inst : CommRing R
p : ℕ
inst_1 : IsDomain R
hp : 1 < p
a✝ : ℕ
a : natAbs (ofNat a✝) % p = 0
⊢ ↑p ∣ ofNat a✝ | no goals |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/ValuedRing.lean | Int.int_valuation_eq_zero_iff | [896, 1] | [914, 8] | rw [dvd_iff_mod_eq_zero] | case mp.negSucc
R : Type u
inst : CommRing R
p : ℕ
inst_1 : IsDomain R
hp : 1 < p
a✝ : ℕ
a : natAbs -[a✝+1] % p = 0
⊢ ↑p ∣ -[a✝+1] | case mp.negSucc
R : Type u
inst : CommRing R
p : ℕ
inst_1 : IsDomain R
hp : 1 < p
a✝ : ℕ
a : natAbs -[a✝+1] % p = 0
⊢ mod -[a✝+1] ↑p = 0 |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/ValuedRing.lean | Int.int_valuation_eq_zero_iff | [896, 1] | [914, 8] | rw [mod] | case mp.negSucc
R : Type u
inst : CommRing R
p : ℕ
inst_1 : IsDomain R
hp : 1 < p
a✝ : ℕ
a : natAbs -[a✝+1] % p = 0
⊢ mod -[a✝+1] ↑p = 0 | case mp.negSucc
R : Type u
inst : CommRing R
p : ℕ
inst_1 : IsDomain R
hp : 1 < p
a✝ : ℕ
a : natAbs -[a✝+1] % p = 0
⊢ (match -[a✝+1], ↑p with
| ofNat m, ofNat n => ofNat (m % n)
| ofNat m, -[n+1] => ofNat (m % Nat.succ n)
| -[m+1], ofNat n => -ofNat (Nat.succ m % n)
| -[m+1], -[n+1] => -ofNat (Nat.succ m % Nat.succ n)) =
0 |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/ValuedRing.lean | Int.int_valuation_eq_zero_iff | [896, 1] | [914, 8] | simp at * | case mp.negSucc
R : Type u
inst : CommRing R
p : ℕ
inst_1 : IsDomain R
hp : 1 < p
a✝ : ℕ
a : natAbs -[a✝+1] % p = 0
⊢ (match -[a✝+1], ↑p with
| ofNat m, ofNat n => ofNat (m % n)
| ofNat m, -[n+1] => ofNat (m % Nat.succ n)
| -[m+1], ofNat n => -ofNat (Nat.succ m % n)
| -[m+1], -[n+1] => -ofNat (Nat.succ m % Nat.succ n)) =
0 | case mp.negSucc
R : Type u
inst : CommRing R
p : ℕ
inst_1 : IsDomain R
hp : 1 < p
a✝ : ℕ
a : Nat.succ a✝ % p = 0
⊢ ↑p ∣ ↑a✝ + 1 |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/ValuedRing.lean | Int.int_valuation_eq_zero_iff | [896, 1] | [914, 8] | rw [dvd_iff_mod_eq_zero] | case mp.negSucc
R : Type u
inst : CommRing R
p : ℕ
inst_1 : IsDomain R
hp : 1 < p
a✝ : ℕ
a : Nat.succ a✝ % p = 0
⊢ ↑p ∣ ↑a✝ + 1 | case mp.negSucc
R : Type u
inst : CommRing R
p : ℕ
inst_1 : IsDomain R
hp : 1 < p
a✝ : ℕ
a : Nat.succ a✝ % p = 0
⊢ mod (↑a✝ + 1) ↑p = 0 |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/ValuedRing.lean | Int.int_valuation_eq_zero_iff | [896, 1] | [914, 8] | norm_cast | case mp.negSucc
R : Type u
inst : CommRing R
p : ℕ
inst_1 : IsDomain R
hp : 1 < p
a✝ : ℕ
a : Nat.succ a✝ % p = 0
⊢ mod (↑a✝ + 1) ↑p = 0 | case mp.negSucc
R : Type u
inst : CommRing R
p : ℕ
inst_1 : IsDomain R
hp : 1 < p
a✝ : ℕ
a : Nat.succ a✝ % p = 0
⊢ mod ↑(a✝ + 1) ↑p = 0 |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/ValuedRing.lean | Int.int_valuation_eq_zero_iff | [896, 1] | [914, 8] | rw [Nat.succ_eq_add_one] at a | case mp.negSucc
R : Type u
inst : CommRing R
p : ℕ
inst_1 : IsDomain R
hp : 1 < p
a✝ : ℕ
a : Nat.succ a✝ % p = 0
⊢ mod ↑(a✝ + 1) ↑p = 0 | case mp.negSucc
R : Type u
inst : CommRing R
p : ℕ
inst_1 : IsDomain R
hp : 1 < p
a✝ : ℕ
a : (a✝ + 1) % p = 0
⊢ mod ↑(a✝ + 1) ↑p = 0 |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/ValuedRing.lean | Int.int_valuation_eq_zero_iff | [896, 1] | [914, 8] | exact Iff.mp natAbs_eq_zero a | case mp.negSucc
R : Type u
inst : CommRing R
p : ℕ
inst_1 : IsDomain R
hp : 1 < p
a✝ : ℕ
a : (a✝ + 1) % p = 0
⊢ mod ↑(a✝ + 1) ↑p = 0 | no goals |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/ValuedRing.lean | Int.primeVal_eq_zero_iff | [916, 1] | [918, 80] | rw [primeVal, int_valuation_eq_zero_iff hp.one_lt, Int.dvd_iff_emod_eq_zero] | R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p : ℕ
k : ℤ
hp : Nat.Prime p
⊢ SurjVal.v (primeVal hp) k = 0 ↔ ¬↑p ∣ k | no goals |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/ValuedRing.lean | Int.primeVal_pos_iff | [920, 1] | [922, 23] | simp [← not_iff_not] | R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p : ℕ
k : ℤ
hp : Nat.Prime p
⊢ 0 < SurjVal.v (primeVal hp) k ↔ ↑p ∣ k | no goals |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/ValuedRing.lean | Int.zero_valtn_decr_p | [924, 1] | [928, 8] | rw [decr_val_p] | R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p : ℕ
k : ℤ
hp : Nat.Prime p
h : SurjVal.v (primeVal hp) k = 0
⊢ decr_val_p p k = k | R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p : ℕ
k : ℤ
hp : Nat.Prime p
h : SurjVal.v (primeVal hp) k = 0
⊢ (if (k % ↑p == 0) = true then k / ↑p else k) = k |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/ValuedRing.lean | Int.zero_valtn_decr_p | [924, 1] | [928, 8] | aesop | R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p : ℕ
k : ℤ
hp : Nat.Prime p
h : SurjVal.v (primeVal hp) k = 0
⊢ (if (k % ↑p == 0) = true then k / ↑p else k) = k | no goals |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/ValuedRing.lean | Int.Int.ModEq.pow_card_eq_self | [945, 1] | [953, 76] | haveI : Fact p.Prime := ⟨hp⟩ | R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p : ℕ
hp : Nat.Prime p
n : ℤ
⊢ n ^ p ≡ n [ZMOD ↑p] | R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p : ℕ
hp : Nat.Prime p
n : ℤ
this : Fact (Nat.Prime p)
⊢ n ^ p ≡ n [ZMOD ↑p] |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/ValuedRing.lean | Int.Int.ModEq.pow_card_eq_self | [945, 1] | [953, 76] | by_cases this : (n : ZMod p) = 0 | R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p : ℕ
hp : Nat.Prime p
n : ℤ
this : Fact (Nat.Prime p)
⊢ n ^ p ≡ n [ZMOD ↑p] | case pos
R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p : ℕ
hp : Nat.Prime p
n : ℤ
this✝ : Fact (Nat.Prime p)
this : ↑n = 0
⊢ n ^ p ≡ n [ZMOD ↑p]
case neg
R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p : ℕ
hp : Nat.Prime p
n : ℤ
this✝ : Fact (Nat.Prime p)
this : ¬↑n = 0
⊢ n ^ p ≡ n [ZMOD ↑p] |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/ValuedRing.lean | Int.Int.ModEq.pow_card_eq_self | [945, 1] | [953, 76] | conv_lhs =>
rw [← show p - 1 + 1 = p from Nat.sub_add_cancel hp.one_lt.le] | case neg
R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p : ℕ
hp : Nat.Prime p
n : ℤ
this✝ : Fact (Nat.Prime p)
this : ¬↑n = 0
⊢ n ^ p ≡ n [ZMOD ↑p] | case neg
R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p : ℕ
hp : Nat.Prime p
n : ℤ
this✝ : Fact (Nat.Prime p)
this : ¬↑n = 0
⊢ n ^ (p - 1 + 1) ≡ n [ZMOD ↑p] |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/ValuedRing.lean | Int.Int.ModEq.pow_card_eq_self | [945, 1] | [953, 76] | rw [pow_add, pow_one] | case neg
R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p : ℕ
hp : Nat.Prime p
n : ℤ
this✝ : Fact (Nat.Prime p)
this : ¬↑n = 0
⊢ n ^ (p - 1 + 1) ≡ n [ZMOD ↑p] | case neg
R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p : ℕ
hp : Nat.Prime p
n : ℤ
this✝ : Fact (Nat.Prime p)
this : ¬↑n = 0
⊢ n ^ (p - 1) * n ≡ n [ZMOD ↑p] |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/ValuedRing.lean | Int.Int.ModEq.pow_card_eq_self | [945, 1] | [953, 76] | simp [← ZMod.int_cast_eq_int_cast_iff, ZMod.pow_card_sub_one_eq_one this] | case neg
R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p : ℕ
hp : Nat.Prime p
n : ℤ
this✝ : Fact (Nat.Prime p)
this : ¬↑n = 0
⊢ n ^ (p - 1) * n ≡ n [ZMOD ↑p] | no goals |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/ValuedRing.lean | Int.Int.ModEq.pow_card_eq_self | [945, 1] | [953, 76] | simp [← ZMod.int_cast_eq_int_cast_iff, this] | case pos
R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p : ℕ
hp : Nat.Prime p
n : ℤ
this✝ : Fact (Nat.Prime p)
this : ↑n = 0
⊢ n ^ p ≡ n [ZMOD ↑p] | no goals |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/ValuedRing.lean | Int.val_poly_of_double_root | [1028, 1] | [1031, 70] | sorry | R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p : ℕ
hp : Nat.Prime p
a b c : ℤ
H : has_double_root a b c hp
⊢ SurjVal.v (primeEVR hp).valtn (a * double_root a b c p ^ 2 + b * double_root a b c p + c) > 0 ∧
SurjVal.v (primeEVR hp).valtn (2 * a * double_root a b c p + b) > 0 | no goals |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/CharP/Basic.lean | add_pow_ringChar | [17, 1] | [23, 27] | have : NeZero (ringChar R) := ⟨h⟩ | R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDomain R
a b : R
h : ringChar R ≠ 0
⊢ (a + b) ^ ringChar R = a ^ ringChar R + b ^ ringChar R | R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDomain R
a b : R
h : ringChar R ≠ 0
this : NeZero (ringChar R)
⊢ (a + b) ^ ringChar R = a ^ ringChar R + b ^ ringChar R |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/CharP/Basic.lean | add_pow_ringChar | [17, 1] | [23, 27] | have := CharP.char_is_prime_of_pos R (ringChar R) | R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDomain R
a b : R
h : ringChar R ≠ 0
this : NeZero (ringChar R)
⊢ (a + b) ^ ringChar R = a ^ ringChar R + b ^ ringChar R | R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDomain R
a b : R
h : ringChar R ≠ 0
this✝ : NeZero (ringChar R)
this : Fact (Nat.Prime (ringChar R))
⊢ (a + b) ^ ringChar R = a ^ ringChar R + b ^ ringChar R |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/CharP/Basic.lean | add_pow_ringChar | [17, 1] | [23, 27] | exact add_pow_char _ _ _ | R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDomain R
a b : R
h : ringChar R ≠ 0
this✝ : NeZero (ringChar R)
this : Fact (Nat.Prime (ringChar R))
⊢ (a + b) ^ ringChar R = a ^ ringChar R + b ^ ringChar R | no goals |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/CharP/Basic.lean | sub_pow_ringChar | [26, 1] | [32, 27] | have : NeZero (ringChar R) := ⟨h⟩ | R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDomain R
a b : R
h : ringChar R ≠ 0
⊢ (a - b) ^ ringChar R = a ^ ringChar R - b ^ ringChar R | R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDomain R
a b : R
h : ringChar R ≠ 0
this : NeZero (ringChar R)
⊢ (a - b) ^ ringChar R = a ^ ringChar R - b ^ ringChar R |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/CharP/Basic.lean | sub_pow_ringChar | [26, 1] | [32, 27] | have := CharP.char_is_prime_of_pos R (ringChar R) | R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDomain R
a b : R
h : ringChar R ≠ 0
this : NeZero (ringChar R)
⊢ (a - b) ^ ringChar R = a ^ ringChar R - b ^ ringChar R | R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDomain R
a b : R
h : ringChar R ≠ 0
this✝ : NeZero (ringChar R)
this : Fact (Nat.Prime (ringChar R))
⊢ (a - b) ^ ringChar R = a ^ ringChar R - b ^ ringChar R |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/CharP/Basic.lean | sub_pow_ringChar | [26, 1] | [32, 27] | exact sub_pow_char _ _ _ | R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDomain R
a b : R
h : ringChar R ≠ 0
this✝ : NeZero (ringChar R)
this : Fact (Nat.Prime (ringChar R))
⊢ (a - b) ^ ringChar R = a ^ ringChar R - b ^ ringChar R | no goals |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/CharP/Basic.lean | pow_ringChar_injective | [35, 1] | [42, 22] | intros x y h | R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDomain R
hn : ringChar R ≠ 0
⊢ Function.Injective fun x => x ^ ringChar R | R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDomain R
hn : ringChar R ≠ 0
x y : R
h : (fun x => x ^ ringChar R) x = (fun x => x ^ ringChar R) y
⊢ x = y |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/CharP/Basic.lean | pow_ringChar_injective | [35, 1] | [42, 22] | rw [←sub_eq_zero] at * | R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDomain R
hn : ringChar R ≠ 0
x y : R
h : (fun x => x ^ ringChar R) x = (fun x => x ^ ringChar R) y
⊢ x = y | R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDomain R
hn : ringChar R ≠ 0
x y : R
h✝ : (fun x => x ^ ringChar R) x = (fun x => x ^ ringChar R) y
h : (fun x => x ^ ringChar R) x - (fun x => x ^ ringChar R) y = 0
⊢ x - y = 0 |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/CharP/Basic.lean | pow_ringChar_injective | [35, 1] | [42, 22] | rw [←sub_eq_zero] at * | R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDomain R
hn : ringChar R ≠ 0
x y : R
h✝ : (fun x => x ^ ringChar R) x = (fun x => x ^ ringChar R) y
h : (fun x => x ^ ringChar R) x - (fun x => x ^ ringChar R) y = 0
⊢ x - y = 0 | R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDomain R
hn : ringChar R ≠ 0
x y : R
h✝² : (fun x => x ^ ringChar R) x = (fun x => x ^ ringChar R) y
h✝¹ h✝ : (fun x => x ^ ringChar R) x - (fun x => x ^ ringChar R) y = 0
h : (fun x => x ^ ringChar R) x - (fun x => x ^ ringChar R) y - 0 = 0
⊢ x - y - 0 = 0 |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/CharP/Basic.lean | pow_ringChar_injective | [35, 1] | [42, 22] | simp only [sub_zero] at * | R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDomain R
hn : ringChar R ≠ 0
x y : R
h✝² : (fun x => x ^ ringChar R) x = (fun x => x ^ ringChar R) y
h✝¹ h✝ : (fun x => x ^ ringChar R) x - (fun x => x ^ ringChar R) y = 0
h : (fun x => x ^ ringChar R) x - (fun x => x ^ ringChar R) y - 0 = 0
⊢ x - y - 0 = 0 | R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDomain R
hn : ringChar R ≠ 0
x y : R
h✝² : x ^ ringChar R = y ^ ringChar R
h✝¹ h✝ h : x ^ ringChar R - y ^ ringChar R = 0
⊢ x - y = 0 |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/CharP/Basic.lean | pow_ringChar_injective | [35, 1] | [42, 22] | rw [← sub_pow_ringChar _ _ hn] at h | R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDomain R
hn : ringChar R ≠ 0
x y : R
h✝² : x ^ ringChar R = y ^ ringChar R
h✝¹ h✝ h : x ^ ringChar R - y ^ ringChar R = 0
⊢ x - y = 0 | R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDomain R
hn : ringChar R ≠ 0
x y : R
h✝² : x ^ ringChar R = y ^ ringChar R
h✝¹ h✝ : x ^ ringChar R - y ^ ringChar R = 0
h : (x - y) ^ ringChar R = 0
⊢ x - y = 0 |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/CharP/Basic.lean | pow_ringChar_injective | [35, 1] | [42, 22] | exact pow_eq_zero h | R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDomain R
hn : ringChar R ≠ 0
x y : R
h✝² : x ^ ringChar R = y ^ ringChar R
h✝¹ h✝ : x ^ ringChar R - y ^ ringChar R = 0
h : (x - y) ^ ringChar R = 0
⊢ x - y = 0 | no goals |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/EllipticCurve/AuxRingLemmas.lean | factorize1 | [9, 1] | [9, 150] | ring | R : Type u
inst✝ : CommRing R
root b p : R
q : ℕ
⊢ root * p ^ q * (p ^ q * b) + root * p ^ q * (root * p ^ q) = p ^ q * p ^ q * ((root + b) * root) | no goals |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/EllipticCurve/AuxRingLemmas.lean | factorize2 | [11, 1] | [11, 121] | ring | R : Type u
inst✝ : CommRing R
root a p : R
q : ℕ
⊢ 2 * (root * p ^ q) * (p ^ 1 * a) = p ^ q * p ^ 1 * (2 * a * root) | no goals |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/EllipticCurve/AuxRingLemmas.lean | factorize3 | [13, 1] | [13, 123] | ring | R : Type u
inst✝ : CommRing R
root p : R
q : ℕ
⊢ 3 * (root * p ^ q * (root * p ^ q)) = p ^ q * p ^ q * (3 * root * root) | no goals |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/EllipticCurve/AuxRingLemmas.lean | factorize4 | [15, 1] | [15, 245] | ring | R : Type u
inst✝ : CommRing R
root a b c p : R
q : ℕ
⊢ p ^ (2 * q + 1) * c + root * p ^ q * (p ^ (q + 1) * b) + (root * p ^ q) ^ 2 * (p ^ 1 * a) =
p ^ q * p ^ q * p ^ 1 * (a * root ^ 2) + p ^ q * p ^ (q + 1) * (b * root) + p ^ (2 * q + 1) * c | no goals |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/EllipticCurve/AuxRingLemmas.lean | factorize5 | [17, 1] | [17, 110] | ring | R : Type u
inst✝ : CommRing R
b c p : R
⊢ p ^ 1 * b * (p ^ 1 * b) + 4 * (p ^ 2 * c) = p ^ 2 * (b * b + 4 * c) | no goals |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/EllipticCurve/AuxRingLemmas.lean | factorize6 | [19, 1] | [19, 128] | ring | R : Type u
inst✝ : CommRing R
p x b c : R
⊢ p ^ 2 * x ^ 2 + p * x * (p ^ 1 * b) + p ^ 2 * -c = p ^ 2 * (1 * x ^ 2 + b * x + -c) | no goals |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/EllipticCurve/AuxRingLemmas.lean | factorize7 | [21, 1] | [21, 138] | ring | R : Type u
inst✝ : CommRing R
a b r p : R
⊢ p ^ 2 * a + 2 * (p * r) * (p ^ 1 * b) + 3 * (p * r) ^ 2 = p ^ 2 * (a + 2 * r * b + 3 * r ^ 2) | no goals |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/EllipticCurve/AuxRingLemmas.lean | factorize8 | [23, 1] | [23, 166] | ring | R : Type u
inst✝ : CommRing R
a b c r p : R
⊢ p ^ 3 * a + p * r * (p ^ 2 * b) + (p * r) ^ 2 * (p ^ 1 * c) + (p * r) ^ 3 = p ^ 3 * (a + r * b + r ^ 2 * c + r ^ 3) | no goals |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/EllipticCurve/AuxRingLemmas.lean | factorize9 | [25, 1] | [25, 308] | ring | R : Type u
inst✝ : CommRing R
a1 a2 a3 a4 a6 b8 p : R
⊢ p ^ 1 * a1 * (p ^ 1 * a1) * (p ^ 3 * a6) + p ^ 1 * a1 * (p ^ 2 * a3) * -a4 + 4 * a2 * (p ^ 3 * a6) +
a2 * (p ^ 2 * a3) * (p ^ 2 * a3) +
p ^ 3 * -b8 =
p ^ 3 * (p ^ 1 * a1 * (p ^ 1 * a1) * a6 + a1 * a3 * -a4 + 4 * a2 * a6 + a2 * a3 * (p ^ 1 * a3) + -b8) | no goals |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/Ring/Basic.lean | add_self_eq_mul_two | [10, 1] | [13, 11] | rw [←one_mul a, ←add_mul, one_mul] | R : Type u_1
inst✝ : Semiring R
a : R
⊢ a + a = 2 * a | R : Type u_1
inst✝ : Semiring R
a : R
⊢ (1 + 1) * a = 2 * a |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/Ring/Basic.lean | add_self_eq_mul_two | [10, 1] | [13, 11] | congr | R : Type u_1
inst✝ : Semiring R
a : R
⊢ (1 + 1) * a = 2 * a | case e_a
R : Type u_1
inst✝ : Semiring R
a : R
⊢ 1 + 1 = 2 |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/Ring/Basic.lean | add_self_eq_mul_two | [10, 1] | [13, 11] | norm_num | case e_a
R : Type u_1
inst✝ : Semiring R
a : R
⊢ 1 + 1 = 2 | no goals |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/Ring/Basic.lean | evenpow_neg | [48, 1] | [49, 35] | rw [h, pow_mul, pow_mul, neg_sq] | R : Type u_1
inst✝ : CommRing R
n m : ℕ
a : R
h : n = 2 * m
⊢ (-a) ^ n = a ^ n | no goals |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/Ring/Basic.lean | oddpow_neg | [51, 1] | [53, 17] | rw [h, pow_succ, evenpow_neg a (show 2 * m = 2 * m by rfl), ←neg_mul_eq_neg_mul, ←pow_succ,
Nat.add_one] | R : Type u_1
inst✝ : CommRing R
n m : ℕ
a : R
h : n = 2 * m + 1
⊢ (-a) ^ n = -a ^ n | no goals |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/Ring/Basic.lean | oddpow_neg | [51, 1] | [53, 17] | rfl | R : Type u_1
inst✝ : CommRing R
n m : ℕ
a : R
h : n = 2 * m + 1
⊢ 2 * m = 2 * m | no goals |
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