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https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/Ring/Basic.lean | nzero_mul_left_cancel | [62, 1] | [69, 12] | intro a_ne_z ab_eq_ac | R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDomain R
a b c : R
⊢ a ≠ 0 → a * b = a * c → b = c | R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDomain R
a b c : R
a_ne_z : a ≠ 0
ab_eq_ac : a * b = a * c
⊢ b = c |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/Ring/Basic.lean | nzero_mul_left_cancel | [62, 1] | [69, 12] | rw [←add_left_inj (-(a * c)), add_neg_self (a * c), neg_mul_eq_mul_neg, ←mul_add] at ab_eq_ac | R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDomain R
a b c : R
a_ne_z : a ≠ 0
ab_eq_ac : a * b = a * c
⊢ b = c | R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDomain R
a b c : R
a_ne_z : a ≠ 0
ab_eq_ac : a * (b + -c) = 0
⊢ b = c |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/Ring/Basic.lean | nzero_mul_left_cancel | [62, 1] | [69, 12] | cases (@mul_eq_zero _ _ _ a (b + -c)).1 ab_eq_ac with
| inl h => exact False.elim (a_ne_z h)
| inr h =>
rw [←add_left_inj (-c), add_neg_self c]
exact h | R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDomain R
a b c : R
a_ne_z : a ≠ 0
ab_eq_ac : a * (b + -c) = 0
⊢ b = c | no goals |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/Ring/Basic.lean | nzero_mul_left_cancel | [62, 1] | [69, 12] | exact False.elim (a_ne_z h) | case inl
R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDomain R
a b c : R
a_ne_z : a ≠ 0
ab_eq_ac : a * (b + -c) = 0
h : a = 0
⊢ b = c | no goals |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/Ring/Basic.lean | nzero_mul_left_cancel | [62, 1] | [69, 12] | rw [←add_left_inj (-c), add_neg_self c] | case inr
R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDomain R
a b c : R
a_ne_z : a ≠ 0
ab_eq_ac : a * (b + -c) = 0
h : b + -c = 0
⊢ b = c | case inr
R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDomain R
a b c : R
a_ne_z : a ≠ 0
ab_eq_ac : a * (b + -c) = 0
h : b + -c = 0
⊢ b + -c = 0 |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/Ring/Basic.lean | nzero_mul_left_cancel | [62, 1] | [69, 12] | exact h | case inr
R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDomain R
a b c : R
a_ne_z : a ≠ 0
ab_eq_ac : a * (b + -c) = 0
h : b + -c = 0
⊢ b + -c = 0 | no goals |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/QuadRing/Valuations.lean | QuadRing.gaussian_val_uniformizer | [11, 1] | [14, 13] | rw [gaussian_val] | p : ℕ
gt1 : 1 < p
⊢ gaussian_val p ↑p = 1 | p : ℕ
gt1 : 1 < p
⊢ min (Int.int_val p (↑p).b1) (Int.int_val p (↑p).b2) = 1 |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/QuadRing/Valuations.lean | QuadRing.gaussian_val_uniformizer | [11, 1] | [14, 13] | simp [gt1] | p : ℕ
gt1 : 1 < p
⊢ min (Int.int_val p (↑p).b1) (Int.int_val p (↑p).b2) = 1 | no goals |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/QuadRing/Valuations.lean | QuadRing.gaussian_val_mul_eq_add | [16, 1] | [19, 8] | simp [gaussian_val] | p : ℕ
prime : Nat.Prime p
a b : QuadRing ℤ 0 (-1)
⊢ gaussian_val p (a * b) = gaussian_val p a + gaussian_val p b | p : ℕ
prime : Nat.Prime p
a b : QuadRing ℤ 0 (-1)
⊢ min (Int.int_val p (a.b1 * b.b1 + -(a.b2 * b.b2))) (Int.int_val p (a.b2 * b.b1 + a.b1 * b.b2)) =
min (Int.int_val p a.b1) (Int.int_val p a.b2) + min (Int.int_val p b.b1) (Int.int_val p b.b2) |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/QuadRing/Valuations.lean | QuadRing.gaussian_val_mul_eq_add | [16, 1] | [19, 8] | sorry | p : ℕ
prime : Nat.Prime p
a b : QuadRing ℤ 0 (-1)
⊢ min (Int.int_val p (a.b1 * b.b1 + -(a.b2 * b.b2))) (Int.int_val p (a.b2 * b.b1 + a.b1 * b.b2)) =
min (Int.int_val p a.b1) (Int.int_val p a.b2) + min (Int.int_val p b.b1) (Int.int_val p b.b2) | no goals |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/QuadRing/Valuations.lean | QuadRing.gaussian_val_add_ge_min | [21, 1] | [24, 8] | simp [gaussian_val] | p : ℕ
a b : QuadRing ℤ 0 (-1)
⊢ gaussian_val p (a + b) ≥ min (gaussian_val p a) (gaussian_val p b) | p : ℕ
a b : QuadRing ℤ 0 (-1)
⊢ ((Int.int_val p a.b1 ≤ Int.int_val p (a.b1 + b.b1) ∨ Int.int_val p a.b2 ≤ Int.int_val p (a.b1 + b.b1)) ∨
Int.int_val p b.b1 ≤ Int.int_val p (a.b1 + b.b1) ∨ Int.int_val p b.b2 ≤ Int.int_val p (a.b1 + b.b1)) ∧
((Int.int_val p a.b1 ≤ Int.int_val p (a.b2 + b.b2) ∨ Int.int_val p a.b2 ≤ Int.int_val p (a.b2 + b.b2)) ∨
Int.int_val p b.b1 ≤ Int.int_val p (a.b2 + b.b2) ∨ Int.int_val p b.b2 ≤ Int.int_val p (a.b2 + b.b2)) |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/QuadRing/Valuations.lean | QuadRing.gaussian_val_add_ge_min | [21, 1] | [24, 8] | sorry | p : ℕ
a b : QuadRing ℤ 0 (-1)
⊢ ((Int.int_val p a.b1 ≤ Int.int_val p (a.b1 + b.b1) ∨ Int.int_val p a.b2 ≤ Int.int_val p (a.b1 + b.b1)) ∨
Int.int_val p b.b1 ≤ Int.int_val p (a.b1 + b.b1) ∨ Int.int_val p b.b2 ≤ Int.int_val p (a.b1 + b.b1)) ∧
((Int.int_val p a.b1 ≤ Int.int_val p (a.b2 + b.b2) ∨ Int.int_val p a.b2 ≤ Int.int_val p (a.b2 + b.b2)) ∨
Int.int_val p b.b1 ≤ Int.int_val p (a.b2 + b.b2) ∨ Int.int_val p b.b2 ≤ Int.int_val p (a.b2 + b.b2)) | no goals |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/QuadRing/Valuations.lean | QuadRing.eq_top_of_min_eq_top | [26, 1] | [28, 8] | sorry | a b : ℕ∞
⊢ min a b = ⊤ → a = ⊤ | no goals |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/QuadRing/Valuations.lean | QuadRing.min_eq_top_iff | [30, 1] | [34, 8] | sorry | a b : ℕ∞
⊢ min a b = ⊤ ↔ a = ⊤ ∧ b = ⊤ | no goals |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/QuadRing/Valuations.lean | QuadRing.gaussian_val_eq_top_iff_zero | [36, 1] | [38, 46] | simp [gaussian_val, gt1, QuadRing.ext_iff] | p : ℕ
gt1 : 1 < p
a : QuadRing ℤ 0 (-1)
⊢ gaussian_val p a = ⊤ ↔ a = 0 | no goals |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/ResidueRing.lean | ENatValRing.key | [350, 1] | [351, 8] | sorry | R✝ : Type u
inst✝³ : CommRing R✝
inst✝² : IsDomain R✝
R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p : R
evr : ENatValRing p
⊢ ringChar (R ⧸ SurjVal.ideal evr.valtn) = evr.residue_char | no goals |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | test/testelin.lean | v_b6_of_v_a3_a | [6, 1] | [7, 12] | elinarith | q : ℕ
h : ↑q ≥ 2
⊢ ↑q ≥ 1 | no goals |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/EllipticCurve/Kronecker.lean | div_succ_le_succ_div | [15, 1] | [20, 31] | rw [succ_div] | m n : ℕ
⊢ succ m / succ n ≤ succ (m / succ n) | m n : ℕ
⊢ (m / succ n + if succ n ∣ m + 1 then 1 else 0) ≤ succ (m / succ n) |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/EllipticCurve/Kronecker.lean | div_succ_le_succ_div | [15, 1] | [20, 31] | split | m n : ℕ
⊢ (m / succ n + if succ n ∣ m + 1 then 1 else 0) ≤ succ (m / succ n) | case inl
m n : ℕ
h✝ : succ n ∣ m + 1
⊢ m / succ n + 1 ≤ succ (m / succ n)
case inr
m n : ℕ
h✝ : ¬succ n ∣ m + 1
⊢ m / succ n + 0 ≤ succ (m / succ n) |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/EllipticCurve/Kronecker.lean | div_succ_le_succ_div | [15, 1] | [20, 31] | . rw [succ_eq_add_one] | case inl
m n : ℕ
h✝ : succ n ∣ m + 1
⊢ m / succ n + 1 ≤ succ (m / succ n)
case inr
m n : ℕ
h✝ : ¬succ n ∣ m + 1
⊢ m / succ n + 0 ≤ succ (m / succ n) | case inr
m n : ℕ
h✝ : ¬succ n ∣ m + 1
⊢ m / succ n + 0 ≤ succ (m / succ n) |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/EllipticCurve/Kronecker.lean | div_succ_le_succ_div | [15, 1] | [20, 31] | . rw [add_zero]
exact le_succ (m / succ n) | case inr
m n : ℕ
h✝ : ¬succ n ∣ m + 1
⊢ m / succ n + 0 ≤ succ (m / succ n) | no goals |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/EllipticCurve/Kronecker.lean | div_succ_le_succ_div | [15, 1] | [20, 31] | rw [succ_eq_add_one] | case inl
m n : ℕ
h✝ : succ n ∣ m + 1
⊢ m / succ n + 1 ≤ succ (m / succ n) | no goals |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/EllipticCurve/Kronecker.lean | div_succ_le_succ_div | [15, 1] | [20, 31] | rw [add_zero] | case inr
m n : ℕ
h✝ : ¬succ n ∣ m + 1
⊢ m / succ n + 0 ≤ succ (m / succ n) | case inr
m n : ℕ
h✝ : ¬succ n ∣ m + 1
⊢ m / succ n ≤ succ (m / succ n) |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/EllipticCurve/Kronecker.lean | div_succ_le_succ_div | [15, 1] | [20, 31] | exact le_succ (m / succ n) | case inr
m n : ℕ
h✝ : ¬succ n ∣ m + 1
⊢ m / succ n ≤ succ (m / succ n) | no goals |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/EllipticCurve/Kronecker.lean | div2_succ_succ_eq_succ_div2 | [22, 1] | [25, 34] | rw [succ_eq_add_one, succ_eq_add_one, add_assoc, add_div_eq_of_add_mod_lt, Nat.div_self (zero_lt_succ 1)] | n : ℕ
⊢ succ (succ n) / 2 = succ (n / 2) | n : ℕ
⊢ n % 2 + (1 + 1) % 2 < 2 |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/EllipticCurve/Kronecker.lean | div2_succ_succ_eq_succ_div2 | [22, 1] | [25, 34] | rw [mod_self, add_zero] | n : ℕ
⊢ n % 2 + (1 + 1) % 2 < 2 | n : ℕ
⊢ n % 2 < 2 |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/EllipticCurve/Kronecker.lean | div2_succ_succ_eq_succ_div2 | [22, 1] | [25, 34] | exact mod_lt n (zero_lt_succ 1) | n : ℕ
⊢ n % 2 < 2 | no goals |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/EllipticCurve/Kronecker.lean | Int.div2_succ_le_self | [33, 1] | [37, 98] | cases x with
| zero => simp
| succ x =>
exact (div_succ_le_succ_div (Nat.succ x) 1).trans (succ_le_of_lt (div2_lt_self (succ_pos x))) | x : ℕ
⊢ Nat.succ x / 2 ≤ x | no goals |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/EllipticCurve/Kronecker.lean | Int.div2_succ_le_self | [33, 1] | [37, 98] | simp | case zero
⊢ Nat.succ zero / 2 ≤ zero | no goals |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/EllipticCurve/Kronecker.lean | Int.div2_succ_le_self | [33, 1] | [37, 98] | exact (div_succ_le_succ_div (Nat.succ x) 1).trans (succ_le_of_lt (div2_lt_self (succ_pos x))) | case succ
x : ℕ
⊢ Nat.succ (Nat.succ x) / 2 ≤ Nat.succ x | no goals |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/EllipticCurve/Kronecker.lean | Int.odd_part_le_self_nat | [63, 1] | [104, 26] | have acc : ∀ y, y ≤ x → (val_bin_nat y).2 ≤ x := by
induction x with
| zero =>
intro y h
rw [le_zero_eq] at h
rw [h]
exact Nat.zero_le 0
| succ x ih =>
intro y y_le_sx
cases y_le_sx with
| refl =>
cases mod_two_eq_zero_or_one (Nat.succ x) with
| inl even =>
have h : 0 < Nat.succ x ∧ Nat.succ x % 2 = 0 :=
And.intro (zero_lt_succ x) even;
show (if 0 < Nat.succ x ∧ Nat.succ x % 2 = 0 then
let vbn_half := val_bin_nat (Nat.succ x / 2);
(vbn_half.fst + 1, vbn_half.snd)
else if 0 < Nat.succ x then
(0, Nat.succ x)
else
(0, 0)).snd ≤ Nat.succ x
rw [if_pos h]
have h' := ih (Nat.succ x / 2) (div2_succ_le_self x);
exact le_trans h' (le_succ x)
| inr odd =>
show (if 0 < Nat.succ x ∧ Nat.succ x % 2 = 0 then
let vbn_half := val_bin_nat (Nat.succ x / 2);
(vbn_half.fst + 1, vbn_half.snd)
else if 0 < Nat.succ x then
(0, Nat.succ x)
else
(0, 0)).snd ≤ Nat.succ x
have not_even : Nat.succ x % 2 ≠ 0 := by
rw [odd]
simp;
rw [if_neg (not_and_of_not_right (0 < Nat.succ x) not_even)]
rw [if_pos (zero_lt_succ x)]
| step y_le_x => exact le_trans (ih y y_le_x) (le_succ x); | x : ℕ
⊢ (val_bin_nat x).snd ≤ x | x : ℕ
acc : ∀ (y : ℕ), y ≤ x → (val_bin_nat y).snd ≤ x
⊢ (val_bin_nat x).snd ≤ x |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/EllipticCurve/Kronecker.lean | Int.odd_part_le_self_nat | [63, 1] | [104, 26] | exact acc x (le_refl x) | x : ℕ
acc : ∀ (y : ℕ), y ≤ x → (val_bin_nat y).snd ≤ x
⊢ (val_bin_nat x).snd ≤ x | no goals |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/EllipticCurve/Kronecker.lean | Int.odd_part_le_self_nat | [63, 1] | [104, 26] | induction x with
| zero =>
intro y h
rw [le_zero_eq] at h
rw [h]
exact Nat.zero_le 0
| succ x ih =>
intro y y_le_sx
cases y_le_sx with
| refl =>
cases mod_two_eq_zero_or_one (Nat.succ x) with
| inl even =>
have h : 0 < Nat.succ x ∧ Nat.succ x % 2 = 0 :=
And.intro (zero_lt_succ x) even;
show (if 0 < Nat.succ x ∧ Nat.succ x % 2 = 0 then
let vbn_half := val_bin_nat (Nat.succ x / 2);
(vbn_half.fst + 1, vbn_half.snd)
else if 0 < Nat.succ x then
(0, Nat.succ x)
else
(0, 0)).snd ≤ Nat.succ x
rw [if_pos h]
have h' := ih (Nat.succ x / 2) (div2_succ_le_self x);
exact le_trans h' (le_succ x)
| inr odd =>
show (if 0 < Nat.succ x ∧ Nat.succ x % 2 = 0 then
let vbn_half := val_bin_nat (Nat.succ x / 2);
(vbn_half.fst + 1, vbn_half.snd)
else if 0 < Nat.succ x then
(0, Nat.succ x)
else
(0, 0)).snd ≤ Nat.succ x
have not_even : Nat.succ x % 2 ≠ 0 := by
rw [odd]
simp;
rw [if_neg (not_and_of_not_right (0 < Nat.succ x) not_even)]
rw [if_pos (zero_lt_succ x)]
| step y_le_x => exact le_trans (ih y y_le_x) (le_succ x); | x : ℕ
⊢ ∀ (y : ℕ), y ≤ x → (val_bin_nat y).snd ≤ x | no goals |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/EllipticCurve/Kronecker.lean | Int.odd_part_le_self_nat | [63, 1] | [104, 26] | intro y h | case zero
⊢ ∀ (y : ℕ), y ≤ zero → (val_bin_nat y).snd ≤ zero | case zero
y : ℕ
h : y ≤ zero
⊢ (val_bin_nat y).snd ≤ zero |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/EllipticCurve/Kronecker.lean | Int.odd_part_le_self_nat | [63, 1] | [104, 26] | rw [le_zero_eq] at h | case zero
y : ℕ
h : y ≤ zero
⊢ (val_bin_nat y).snd ≤ zero | case zero
y : ℕ
h : y = 0
⊢ (val_bin_nat y).snd ≤ zero |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/EllipticCurve/Kronecker.lean | Int.odd_part_le_self_nat | [63, 1] | [104, 26] | rw [h] | case zero
y : ℕ
h : y = 0
⊢ (val_bin_nat y).snd ≤ zero | case zero
y : ℕ
h : y = 0
⊢ (val_bin_nat 0).snd ≤ zero |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/EllipticCurve/Kronecker.lean | Int.odd_part_le_self_nat | [63, 1] | [104, 26] | exact Nat.zero_le 0 | case zero
y : ℕ
h : y = 0
⊢ (val_bin_nat 0).snd ≤ zero | no goals |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/EllipticCurve/Kronecker.lean | Int.odd_part_le_self_nat | [63, 1] | [104, 26] | intro y y_le_sx | case succ
x : ℕ
ih : ∀ (y : ℕ), y ≤ x → (val_bin_nat y).snd ≤ x
⊢ ∀ (y : ℕ), y ≤ Nat.succ x → (val_bin_nat y).snd ≤ Nat.succ x | case succ
x : ℕ
ih : ∀ (y : ℕ), y ≤ x → (val_bin_nat y).snd ≤ x
y : ℕ
y_le_sx : y ≤ Nat.succ x
⊢ (val_bin_nat y).snd ≤ Nat.succ x |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/EllipticCurve/Kronecker.lean | Int.odd_part_le_self_nat | [63, 1] | [104, 26] | cases y_le_sx with
| refl =>
cases mod_two_eq_zero_or_one (Nat.succ x) with
| inl even =>
have h : 0 < Nat.succ x ∧ Nat.succ x % 2 = 0 :=
And.intro (zero_lt_succ x) even;
show (if 0 < Nat.succ x ∧ Nat.succ x % 2 = 0 then
let vbn_half := val_bin_nat (Nat.succ x / 2);
(vbn_half.fst + 1, vbn_half.snd)
else if 0 < Nat.succ x then
(0, Nat.succ x)
else
(0, 0)).snd ≤ Nat.succ x
rw [if_pos h]
have h' := ih (Nat.succ x / 2) (div2_succ_le_self x);
exact le_trans h' (le_succ x)
| inr odd =>
show (if 0 < Nat.succ x ∧ Nat.succ x % 2 = 0 then
let vbn_half := val_bin_nat (Nat.succ x / 2);
(vbn_half.fst + 1, vbn_half.snd)
else if 0 < Nat.succ x then
(0, Nat.succ x)
else
(0, 0)).snd ≤ Nat.succ x
have not_even : Nat.succ x % 2 ≠ 0 := by
rw [odd]
simp;
rw [if_neg (not_and_of_not_right (0 < Nat.succ x) not_even)]
rw [if_pos (zero_lt_succ x)]
| step y_le_x => exact le_trans (ih y y_le_x) (le_succ x); | case succ
x : ℕ
ih : ∀ (y : ℕ), y ≤ x → (val_bin_nat y).snd ≤ x
y : ℕ
y_le_sx : y ≤ Nat.succ x
⊢ (val_bin_nat y).snd ≤ Nat.succ x | no goals |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/EllipticCurve/Kronecker.lean | Int.odd_part_le_self_nat | [63, 1] | [104, 26] | cases mod_two_eq_zero_or_one (Nat.succ x) with
| inl even =>
have h : 0 < Nat.succ x ∧ Nat.succ x % 2 = 0 :=
And.intro (zero_lt_succ x) even;
show (if 0 < Nat.succ x ∧ Nat.succ x % 2 = 0 then
let vbn_half := val_bin_nat (Nat.succ x / 2);
(vbn_half.fst + 1, vbn_half.snd)
else if 0 < Nat.succ x then
(0, Nat.succ x)
else
(0, 0)).snd ≤ Nat.succ x
rw [if_pos h]
have h' := ih (Nat.succ x / 2) (div2_succ_le_self x);
exact le_trans h' (le_succ x)
| inr odd =>
show (if 0 < Nat.succ x ∧ Nat.succ x % 2 = 0 then
let vbn_half := val_bin_nat (Nat.succ x / 2);
(vbn_half.fst + 1, vbn_half.snd)
else if 0 < Nat.succ x then
(0, Nat.succ x)
else
(0, 0)).snd ≤ Nat.succ x
have not_even : Nat.succ x % 2 ≠ 0 := by
rw [odd]
simp;
rw [if_neg (not_and_of_not_right (0 < Nat.succ x) not_even)]
rw [if_pos (zero_lt_succ x)] | case succ.refl
x : ℕ
ih : ∀ (y : ℕ), y ≤ x → (val_bin_nat y).snd ≤ x
⊢ (val_bin_nat (Nat.succ x)).snd ≤ Nat.succ x | no goals |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/EllipticCurve/Kronecker.lean | Int.odd_part_le_self_nat | [63, 1] | [104, 26] | have h : 0 < Nat.succ x ∧ Nat.succ x % 2 = 0 :=
And.intro (zero_lt_succ x) even | case succ.refl.inl
x : ℕ
ih : ∀ (y : ℕ), y ≤ x → (val_bin_nat y).snd ≤ x
even : Nat.succ x % 2 = 0
⊢ (val_bin_nat (Nat.succ x)).snd ≤ Nat.succ x | case succ.refl.inl
x : ℕ
ih : ∀ (y : ℕ), y ≤ x → (val_bin_nat y).snd ≤ x
even : Nat.succ x % 2 = 0
h : 0 < Nat.succ x ∧ Nat.succ x % 2 = 0
⊢ (val_bin_nat (Nat.succ x)).snd ≤ Nat.succ x |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/EllipticCurve/Kronecker.lean | Int.odd_part_le_self_nat | [63, 1] | [104, 26] | show (if 0 < Nat.succ x ∧ Nat.succ x % 2 = 0 then
let vbn_half := val_bin_nat (Nat.succ x / 2);
(vbn_half.fst + 1, vbn_half.snd)
else if 0 < Nat.succ x then
(0, Nat.succ x)
else
(0, 0)).snd ≤ Nat.succ x | case succ.refl.inl
x : ℕ
ih : ∀ (y : ℕ), y ≤ x → (val_bin_nat y).snd ≤ x
even : Nat.succ x % 2 = 0
h : 0 < Nat.succ x ∧ Nat.succ x % 2 = 0
⊢ (val_bin_nat (Nat.succ x)).snd ≤ Nat.succ x | case succ.refl.inl
x : ℕ
ih : ∀ (y : ℕ), y ≤ x → (val_bin_nat y).snd ≤ x
even : Nat.succ x % 2 = 0
h : 0 < Nat.succ x ∧ Nat.succ x % 2 = 0
⊢ (if 0 < Nat.succ x ∧ Nat.succ x % 2 = 0 then
let vbn_half := val_bin_nat (Nat.succ x / 2);
(vbn_half.fst + 1, vbn_half.snd)
else if 0 < Nat.succ x then (0, Nat.succ x) else (0, 0)).snd ≤
Nat.succ x |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/EllipticCurve/Kronecker.lean | Int.odd_part_le_self_nat | [63, 1] | [104, 26] | rw [if_pos h] | case succ.refl.inl
x : ℕ
ih : ∀ (y : ℕ), y ≤ x → (val_bin_nat y).snd ≤ x
even : Nat.succ x % 2 = 0
h : 0 < Nat.succ x ∧ Nat.succ x % 2 = 0
⊢ (if 0 < Nat.succ x ∧ Nat.succ x % 2 = 0 then
let vbn_half := val_bin_nat (Nat.succ x / 2);
(vbn_half.fst + 1, vbn_half.snd)
else if 0 < Nat.succ x then (0, Nat.succ x) else (0, 0)).snd ≤
Nat.succ x | case succ.refl.inl
x : ℕ
ih : ∀ (y : ℕ), y ≤ x → (val_bin_nat y).snd ≤ x
even : Nat.succ x % 2 = 0
h : 0 < Nat.succ x ∧ Nat.succ x % 2 = 0
⊢ (let vbn_half := val_bin_nat (Nat.succ x / 2);
(vbn_half.fst + 1, vbn_half.snd)).snd ≤
Nat.succ x |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/EllipticCurve/Kronecker.lean | Int.odd_part_le_self_nat | [63, 1] | [104, 26] | have h' := ih (Nat.succ x / 2) (div2_succ_le_self x) | case succ.refl.inl
x : ℕ
ih : ∀ (y : ℕ), y ≤ x → (val_bin_nat y).snd ≤ x
even : Nat.succ x % 2 = 0
h : 0 < Nat.succ x ∧ Nat.succ x % 2 = 0
⊢ (let vbn_half := val_bin_nat (Nat.succ x / 2);
(vbn_half.fst + 1, vbn_half.snd)).snd ≤
Nat.succ x | case succ.refl.inl
x : ℕ
ih : ∀ (y : ℕ), y ≤ x → (val_bin_nat y).snd ≤ x
even : Nat.succ x % 2 = 0
h : 0 < Nat.succ x ∧ Nat.succ x % 2 = 0
h' : (val_bin_nat (Nat.succ x / 2)).snd ≤ x
⊢ (let vbn_half := val_bin_nat (Nat.succ x / 2);
(vbn_half.fst + 1, vbn_half.snd)).snd ≤
Nat.succ x |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/EllipticCurve/Kronecker.lean | Int.odd_part_le_self_nat | [63, 1] | [104, 26] | exact le_trans h' (le_succ x) | case succ.refl.inl
x : ℕ
ih : ∀ (y : ℕ), y ≤ x → (val_bin_nat y).snd ≤ x
even : Nat.succ x % 2 = 0
h : 0 < Nat.succ x ∧ Nat.succ x % 2 = 0
h' : (val_bin_nat (Nat.succ x / 2)).snd ≤ x
⊢ (let vbn_half := val_bin_nat (Nat.succ x / 2);
(vbn_half.fst + 1, vbn_half.snd)).snd ≤
Nat.succ x | no goals |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/EllipticCurve/Kronecker.lean | Int.odd_part_le_self_nat | [63, 1] | [104, 26] | show (if 0 < Nat.succ x ∧ Nat.succ x % 2 = 0 then
let vbn_half := val_bin_nat (Nat.succ x / 2);
(vbn_half.fst + 1, vbn_half.snd)
else if 0 < Nat.succ x then
(0, Nat.succ x)
else
(0, 0)).snd ≤ Nat.succ x | case succ.refl.inr
x : ℕ
ih : ∀ (y : ℕ), y ≤ x → (val_bin_nat y).snd ≤ x
odd : Nat.succ x % 2 = 1
⊢ (val_bin_nat (Nat.succ x)).snd ≤ Nat.succ x | case succ.refl.inr
x : ℕ
ih : ∀ (y : ℕ), y ≤ x → (val_bin_nat y).snd ≤ x
odd : Nat.succ x % 2 = 1
⊢ (if 0 < Nat.succ x ∧ Nat.succ x % 2 = 0 then
let vbn_half := val_bin_nat (Nat.succ x / 2);
(vbn_half.fst + 1, vbn_half.snd)
else if 0 < Nat.succ x then (0, Nat.succ x) else (0, 0)).snd ≤
Nat.succ x |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/EllipticCurve/Kronecker.lean | Int.odd_part_le_self_nat | [63, 1] | [104, 26] | have not_even : Nat.succ x % 2 ≠ 0 := by
rw [odd]
simp; | case succ.refl.inr
x : ℕ
ih : ∀ (y : ℕ), y ≤ x → (val_bin_nat y).snd ≤ x
odd : Nat.succ x % 2 = 1
⊢ (if 0 < Nat.succ x ∧ Nat.succ x % 2 = 0 then
let vbn_half := val_bin_nat (Nat.succ x / 2);
(vbn_half.fst + 1, vbn_half.snd)
else if 0 < Nat.succ x then (0, Nat.succ x) else (0, 0)).snd ≤
Nat.succ x | case succ.refl.inr
x : ℕ
ih : ∀ (y : ℕ), y ≤ x → (val_bin_nat y).snd ≤ x
odd : Nat.succ x % 2 = 1
not_even : Nat.succ x % 2 ≠ 0
⊢ (if 0 < Nat.succ x ∧ Nat.succ x % 2 = 0 then
let vbn_half := val_bin_nat (Nat.succ x / 2);
(vbn_half.fst + 1, vbn_half.snd)
else if 0 < Nat.succ x then (0, Nat.succ x) else (0, 0)).snd ≤
Nat.succ x |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/EllipticCurve/Kronecker.lean | Int.odd_part_le_self_nat | [63, 1] | [104, 26] | rw [if_neg (not_and_of_not_right (0 < Nat.succ x) not_even)] | case succ.refl.inr
x : ℕ
ih : ∀ (y : ℕ), y ≤ x → (val_bin_nat y).snd ≤ x
odd : Nat.succ x % 2 = 1
not_even : Nat.succ x % 2 ≠ 0
⊢ (if 0 < Nat.succ x ∧ Nat.succ x % 2 = 0 then
let vbn_half := val_bin_nat (Nat.succ x / 2);
(vbn_half.fst + 1, vbn_half.snd)
else if 0 < Nat.succ x then (0, Nat.succ x) else (0, 0)).snd ≤
Nat.succ x | case succ.refl.inr
x : ℕ
ih : ∀ (y : ℕ), y ≤ x → (val_bin_nat y).snd ≤ x
odd : Nat.succ x % 2 = 1
not_even : Nat.succ x % 2 ≠ 0
⊢ (if 0 < Nat.succ x then (0, Nat.succ x) else (0, 0)).snd ≤ Nat.succ x |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/EllipticCurve/Kronecker.lean | Int.odd_part_le_self_nat | [63, 1] | [104, 26] | rw [if_pos (zero_lt_succ x)] | case succ.refl.inr
x : ℕ
ih : ∀ (y : ℕ), y ≤ x → (val_bin_nat y).snd ≤ x
odd : Nat.succ x % 2 = 1
not_even : Nat.succ x % 2 ≠ 0
⊢ (if 0 < Nat.succ x then (0, Nat.succ x) else (0, 0)).snd ≤ Nat.succ x | no goals |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/EllipticCurve/Kronecker.lean | Int.odd_part_le_self_nat | [63, 1] | [104, 26] | rw [odd] | x : ℕ
ih : ∀ (y : ℕ), y ≤ x → (val_bin_nat y).snd ≤ x
odd : Nat.succ x % 2 = 1
⊢ Nat.succ x % 2 ≠ 0 | x : ℕ
ih : ∀ (y : ℕ), y ≤ x → (val_bin_nat y).snd ≤ x
odd : Nat.succ x % 2 = 1
⊢ 1 ≠ 0 |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/EllipticCurve/Kronecker.lean | Int.odd_part_le_self_nat | [63, 1] | [104, 26] | simp | x : ℕ
ih : ∀ (y : ℕ), y ≤ x → (val_bin_nat y).snd ≤ x
odd : Nat.succ x % 2 = 1
⊢ 1 ≠ 0 | no goals |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/EllipticCurve/Kronecker.lean | Int.odd_part_le_self_nat | [63, 1] | [104, 26] | exact le_trans (ih y y_le_x) (le_succ x) | case succ.step
x : ℕ
ih : ∀ (y : ℕ), y ≤ x → (val_bin_nat y).snd ≤ x
y : ℕ
y_le_x : Nat.le y x
⊢ (val_bin_nat y).snd ≤ Nat.succ x | no goals |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/EllipticCurve/Kronecker.lean | Int.odd_part_succ_pos | [106, 1] | [146, 26] | have acc : ∀ y, (y ≤ x → 0 < (val_bin_nat (Nat.succ y)).2) := by
induction x with
| zero =>
intro y y_le_x
rw [le_zero_eq] at y_le_x
rw [y_le_x]
exact Nat.lt_succ_self 0 | succ x ih =>
intro y y_le_sx
cases y_le_sx with
| refl =>
cases mod_two_eq_zero_or_one (Nat.succ (Nat.succ x)) with
| inl even =>
have h : 0 < Nat.succ (Nat.succ x) ∧ Nat.succ (Nat.succ x) % 2 = 0 :=
And.intro (zero_lt_succ (Nat.succ x)) even;
show 0 < (if 0 < Nat.succ (Nat.succ x) ∧ Nat.succ (Nat.succ x) % 2 = 0 then
let vbn_half := val_bin_nat (Nat.succ (Nat.succ x) / 2);
(vbn_half.fst + 1, vbn_half.snd)
else if 0 < Nat.succ (Nat.succ x) then
(0, Nat.succ (Nat.succ x))
else
(0, 0)).snd
rw [if_pos h, div2_succ_succ_eq_succ_div2]
exact ih (x / 2) (Nat.div_le_self x 2)
| inr odd =>
show 0 < (if 0 < Nat.succ (Nat.succ x) ∧ Nat.succ (Nat.succ x) % 2 = 0 then
let vbn_half := val_bin_nat (Nat.succ (Nat.succ x) / 2);
(vbn_half.fst + 1, vbn_half.snd)
else if 0 < Nat.succ (Nat.succ x) then
(0, Nat.succ (Nat.succ x))
else
(0, 0)).snd
have not_even : Nat.succ (Nat.succ x) % 2 ≠ 0 := by
rw [odd]
simp;
rw [if_neg (not_and_of_not_right (0 < Nat.succ (Nat.succ x)) not_even)]
rw [if_pos (zero_lt_succ (Nat.succ x))]
exact zero_lt_succ (Nat.succ x)
| step y_le_x => exact ih y y_le_x; | x : ℕ
⊢ 0 < (val_bin_nat (Nat.succ x)).snd | x : ℕ
acc : ∀ (y : ℕ), y ≤ x → 0 < (val_bin_nat (Nat.succ y)).snd
⊢ 0 < (val_bin_nat (Nat.succ x)).snd |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/EllipticCurve/Kronecker.lean | Int.odd_part_succ_pos | [106, 1] | [146, 26] | exact acc x (le_refl x) | x : ℕ
acc : ∀ (y : ℕ), y ≤ x → 0 < (val_bin_nat (Nat.succ y)).snd
⊢ 0 < (val_bin_nat (Nat.succ x)).snd | no goals |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/EllipticCurve/Kronecker.lean | Int.odd_part_succ_pos | [106, 1] | [146, 26] | induction x with
| zero =>
intro y y_le_x
rw [le_zero_eq] at y_le_x
rw [y_le_x]
exact Nat.lt_succ_self 0 | succ x ih =>
intro y y_le_sx
cases y_le_sx with
| refl =>
cases mod_two_eq_zero_or_one (Nat.succ (Nat.succ x)) with
| inl even =>
have h : 0 < Nat.succ (Nat.succ x) ∧ Nat.succ (Nat.succ x) % 2 = 0 :=
And.intro (zero_lt_succ (Nat.succ x)) even;
show 0 < (if 0 < Nat.succ (Nat.succ x) ∧ Nat.succ (Nat.succ x) % 2 = 0 then
let vbn_half := val_bin_nat (Nat.succ (Nat.succ x) / 2);
(vbn_half.fst + 1, vbn_half.snd)
else if 0 < Nat.succ (Nat.succ x) then
(0, Nat.succ (Nat.succ x))
else
(0, 0)).snd
rw [if_pos h, div2_succ_succ_eq_succ_div2]
exact ih (x / 2) (Nat.div_le_self x 2)
| inr odd =>
show 0 < (if 0 < Nat.succ (Nat.succ x) ∧ Nat.succ (Nat.succ x) % 2 = 0 then
let vbn_half := val_bin_nat (Nat.succ (Nat.succ x) / 2);
(vbn_half.fst + 1, vbn_half.snd)
else if 0 < Nat.succ (Nat.succ x) then
(0, Nat.succ (Nat.succ x))
else
(0, 0)).snd
have not_even : Nat.succ (Nat.succ x) % 2 ≠ 0 := by
rw [odd]
simp;
rw [if_neg (not_and_of_not_right (0 < Nat.succ (Nat.succ x)) not_even)]
rw [if_pos (zero_lt_succ (Nat.succ x))]
exact zero_lt_succ (Nat.succ x)
| step y_le_x => exact ih y y_le_x; | x : ℕ
⊢ ∀ (y : ℕ), y ≤ x → 0 < (val_bin_nat (Nat.succ y)).snd | no goals |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/EllipticCurve/Kronecker.lean | Int.odd_part_succ_pos | [106, 1] | [146, 26] | intro y y_le_x | case zero
⊢ ∀ (y : ℕ), y ≤ zero → 0 < (val_bin_nat (Nat.succ y)).snd | case zero
y : ℕ
y_le_x : y ≤ zero
⊢ 0 < (val_bin_nat (Nat.succ y)).snd |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/EllipticCurve/Kronecker.lean | Int.odd_part_succ_pos | [106, 1] | [146, 26] | rw [le_zero_eq] at y_le_x | case zero
y : ℕ
y_le_x : y ≤ zero
⊢ 0 < (val_bin_nat (Nat.succ y)).snd | case zero
y : ℕ
y_le_x : y = 0
⊢ 0 < (val_bin_nat (Nat.succ y)).snd |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/EllipticCurve/Kronecker.lean | Int.odd_part_succ_pos | [106, 1] | [146, 26] | rw [y_le_x] | case zero
y : ℕ
y_le_x : y = 0
⊢ 0 < (val_bin_nat (Nat.succ y)).snd | case zero
y : ℕ
y_le_x : y = 0
⊢ 0 < (val_bin_nat (Nat.succ 0)).snd |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/EllipticCurve/Kronecker.lean | Int.odd_part_succ_pos | [106, 1] | [146, 26] | exact Nat.lt_succ_self 0 | case zero
y : ℕ
y_le_x : y = 0
⊢ 0 < (val_bin_nat (Nat.succ 0)).snd | no goals |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/EllipticCurve/Kronecker.lean | Int.odd_part_succ_pos | [106, 1] | [146, 26] | intro y y_le_sx | case succ
x : ℕ
ih : ∀ (y : ℕ), y ≤ x → 0 < (val_bin_nat (Nat.succ y)).snd
⊢ ∀ (y : ℕ), y ≤ Nat.succ x → 0 < (val_bin_nat (Nat.succ y)).snd | case succ
x : ℕ
ih : ∀ (y : ℕ), y ≤ x → 0 < (val_bin_nat (Nat.succ y)).snd
y : ℕ
y_le_sx : y ≤ Nat.succ x
⊢ 0 < (val_bin_nat (Nat.succ y)).snd |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/EllipticCurve/Kronecker.lean | Int.odd_part_succ_pos | [106, 1] | [146, 26] | cases y_le_sx with
| refl =>
cases mod_two_eq_zero_or_one (Nat.succ (Nat.succ x)) with
| inl even =>
have h : 0 < Nat.succ (Nat.succ x) ∧ Nat.succ (Nat.succ x) % 2 = 0 :=
And.intro (zero_lt_succ (Nat.succ x)) even;
show 0 < (if 0 < Nat.succ (Nat.succ x) ∧ Nat.succ (Nat.succ x) % 2 = 0 then
let vbn_half := val_bin_nat (Nat.succ (Nat.succ x) / 2);
(vbn_half.fst + 1, vbn_half.snd)
else if 0 < Nat.succ (Nat.succ x) then
(0, Nat.succ (Nat.succ x))
else
(0, 0)).snd
rw [if_pos h, div2_succ_succ_eq_succ_div2]
exact ih (x / 2) (Nat.div_le_self x 2)
| inr odd =>
show 0 < (if 0 < Nat.succ (Nat.succ x) ∧ Nat.succ (Nat.succ x) % 2 = 0 then
let vbn_half := val_bin_nat (Nat.succ (Nat.succ x) / 2);
(vbn_half.fst + 1, vbn_half.snd)
else if 0 < Nat.succ (Nat.succ x) then
(0, Nat.succ (Nat.succ x))
else
(0, 0)).snd
have not_even : Nat.succ (Nat.succ x) % 2 ≠ 0 := by
rw [odd]
simp;
rw [if_neg (not_and_of_not_right (0 < Nat.succ (Nat.succ x)) not_even)]
rw [if_pos (zero_lt_succ (Nat.succ x))]
exact zero_lt_succ (Nat.succ x)
| step y_le_x => exact ih y y_le_x; | case succ
x : ℕ
ih : ∀ (y : ℕ), y ≤ x → 0 < (val_bin_nat (Nat.succ y)).snd
y : ℕ
y_le_sx : y ≤ Nat.succ x
⊢ 0 < (val_bin_nat (Nat.succ y)).snd | no goals |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/EllipticCurve/Kronecker.lean | Int.odd_part_succ_pos | [106, 1] | [146, 26] | cases mod_two_eq_zero_or_one (Nat.succ (Nat.succ x)) with
| inl even =>
have h : 0 < Nat.succ (Nat.succ x) ∧ Nat.succ (Nat.succ x) % 2 = 0 :=
And.intro (zero_lt_succ (Nat.succ x)) even;
show 0 < (if 0 < Nat.succ (Nat.succ x) ∧ Nat.succ (Nat.succ x) % 2 = 0 then
let vbn_half := val_bin_nat (Nat.succ (Nat.succ x) / 2);
(vbn_half.fst + 1, vbn_half.snd)
else if 0 < Nat.succ (Nat.succ x) then
(0, Nat.succ (Nat.succ x))
else
(0, 0)).snd
rw [if_pos h, div2_succ_succ_eq_succ_div2]
exact ih (x / 2) (Nat.div_le_self x 2)
| inr odd =>
show 0 < (if 0 < Nat.succ (Nat.succ x) ∧ Nat.succ (Nat.succ x) % 2 = 0 then
let vbn_half := val_bin_nat (Nat.succ (Nat.succ x) / 2);
(vbn_half.fst + 1, vbn_half.snd)
else if 0 < Nat.succ (Nat.succ x) then
(0, Nat.succ (Nat.succ x))
else
(0, 0)).snd
have not_even : Nat.succ (Nat.succ x) % 2 ≠ 0 := by
rw [odd]
simp;
rw [if_neg (not_and_of_not_right (0 < Nat.succ (Nat.succ x)) not_even)]
rw [if_pos (zero_lt_succ (Nat.succ x))]
exact zero_lt_succ (Nat.succ x) | case succ.refl
x : ℕ
ih : ∀ (y : ℕ), y ≤ x → 0 < (val_bin_nat (Nat.succ y)).snd
⊢ 0 < (val_bin_nat (Nat.succ (Nat.succ x))).snd | no goals |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/EllipticCurve/Kronecker.lean | Int.odd_part_succ_pos | [106, 1] | [146, 26] | have h : 0 < Nat.succ (Nat.succ x) ∧ Nat.succ (Nat.succ x) % 2 = 0 :=
And.intro (zero_lt_succ (Nat.succ x)) even | case succ.refl.inl
x : ℕ
ih : ∀ (y : ℕ), y ≤ x → 0 < (val_bin_nat (Nat.succ y)).snd
even : Nat.succ (Nat.succ x) % 2 = 0
⊢ 0 < (val_bin_nat (Nat.succ (Nat.succ x))).snd | case succ.refl.inl
x : ℕ
ih : ∀ (y : ℕ), y ≤ x → 0 < (val_bin_nat (Nat.succ y)).snd
even : Nat.succ (Nat.succ x) % 2 = 0
h : 0 < Nat.succ (Nat.succ x) ∧ Nat.succ (Nat.succ x) % 2 = 0
⊢ 0 < (val_bin_nat (Nat.succ (Nat.succ x))).snd |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/EllipticCurve/Kronecker.lean | Int.odd_part_succ_pos | [106, 1] | [146, 26] | show 0 < (if 0 < Nat.succ (Nat.succ x) ∧ Nat.succ (Nat.succ x) % 2 = 0 then
let vbn_half := val_bin_nat (Nat.succ (Nat.succ x) / 2);
(vbn_half.fst + 1, vbn_half.snd)
else if 0 < Nat.succ (Nat.succ x) then
(0, Nat.succ (Nat.succ x))
else
(0, 0)).snd | case succ.refl.inl
x : ℕ
ih : ∀ (y : ℕ), y ≤ x → 0 < (val_bin_nat (Nat.succ y)).snd
even : Nat.succ (Nat.succ x) % 2 = 0
h : 0 < Nat.succ (Nat.succ x) ∧ Nat.succ (Nat.succ x) % 2 = 0
⊢ 0 < (val_bin_nat (Nat.succ (Nat.succ x))).snd | case succ.refl.inl
x : ℕ
ih : ∀ (y : ℕ), y ≤ x → 0 < (val_bin_nat (Nat.succ y)).snd
even : Nat.succ (Nat.succ x) % 2 = 0
h : 0 < Nat.succ (Nat.succ x) ∧ Nat.succ (Nat.succ x) % 2 = 0
⊢ 0 <
(if 0 < Nat.succ (Nat.succ x) ∧ Nat.succ (Nat.succ x) % 2 = 0 then
let vbn_half := val_bin_nat (Nat.succ (Nat.succ x) / 2);
(vbn_half.fst + 1, vbn_half.snd)
else if 0 < Nat.succ (Nat.succ x) then (0, Nat.succ (Nat.succ x)) else (0, 0)).snd |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/EllipticCurve/Kronecker.lean | Int.odd_part_succ_pos | [106, 1] | [146, 26] | rw [if_pos h, div2_succ_succ_eq_succ_div2] | case succ.refl.inl
x : ℕ
ih : ∀ (y : ℕ), y ≤ x → 0 < (val_bin_nat (Nat.succ y)).snd
even : Nat.succ (Nat.succ x) % 2 = 0
h : 0 < Nat.succ (Nat.succ x) ∧ Nat.succ (Nat.succ x) % 2 = 0
⊢ 0 <
(if 0 < Nat.succ (Nat.succ x) ∧ Nat.succ (Nat.succ x) % 2 = 0 then
let vbn_half := val_bin_nat (Nat.succ (Nat.succ x) / 2);
(vbn_half.fst + 1, vbn_half.snd)
else if 0 < Nat.succ (Nat.succ x) then (0, Nat.succ (Nat.succ x)) else (0, 0)).snd | case succ.refl.inl
x : ℕ
ih : ∀ (y : ℕ), y ≤ x → 0 < (val_bin_nat (Nat.succ y)).snd
even : Nat.succ (Nat.succ x) % 2 = 0
h : 0 < Nat.succ (Nat.succ x) ∧ Nat.succ (Nat.succ x) % 2 = 0
⊢ 0 <
(let vbn_half := val_bin_nat (Nat.succ (x / 2));
(vbn_half.fst + 1, vbn_half.snd)).snd |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/EllipticCurve/Kronecker.lean | Int.odd_part_succ_pos | [106, 1] | [146, 26] | exact ih (x / 2) (Nat.div_le_self x 2) | case succ.refl.inl
x : ℕ
ih : ∀ (y : ℕ), y ≤ x → 0 < (val_bin_nat (Nat.succ y)).snd
even : Nat.succ (Nat.succ x) % 2 = 0
h : 0 < Nat.succ (Nat.succ x) ∧ Nat.succ (Nat.succ x) % 2 = 0
⊢ 0 <
(let vbn_half := val_bin_nat (Nat.succ (x / 2));
(vbn_half.fst + 1, vbn_half.snd)).snd | no goals |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/EllipticCurve/Kronecker.lean | Int.odd_part_succ_pos | [106, 1] | [146, 26] | show 0 < (if 0 < Nat.succ (Nat.succ x) ∧ Nat.succ (Nat.succ x) % 2 = 0 then
let vbn_half := val_bin_nat (Nat.succ (Nat.succ x) / 2);
(vbn_half.fst + 1, vbn_half.snd)
else if 0 < Nat.succ (Nat.succ x) then
(0, Nat.succ (Nat.succ x))
else
(0, 0)).snd | case succ.refl.inr
x : ℕ
ih : ∀ (y : ℕ), y ≤ x → 0 < (val_bin_nat (Nat.succ y)).snd
odd : Nat.succ (Nat.succ x) % 2 = 1
⊢ 0 < (val_bin_nat (Nat.succ (Nat.succ x))).snd | case succ.refl.inr
x : ℕ
ih : ∀ (y : ℕ), y ≤ x → 0 < (val_bin_nat (Nat.succ y)).snd
odd : Nat.succ (Nat.succ x) % 2 = 1
⊢ 0 <
(if 0 < Nat.succ (Nat.succ x) ∧ Nat.succ (Nat.succ x) % 2 = 0 then
let vbn_half := val_bin_nat (Nat.succ (Nat.succ x) / 2);
(vbn_half.fst + 1, vbn_half.snd)
else if 0 < Nat.succ (Nat.succ x) then (0, Nat.succ (Nat.succ x)) else (0, 0)).snd |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/EllipticCurve/Kronecker.lean | Int.odd_part_succ_pos | [106, 1] | [146, 26] | have not_even : Nat.succ (Nat.succ x) % 2 ≠ 0 := by
rw [odd]
simp; | case succ.refl.inr
x : ℕ
ih : ∀ (y : ℕ), y ≤ x → 0 < (val_bin_nat (Nat.succ y)).snd
odd : Nat.succ (Nat.succ x) % 2 = 1
⊢ 0 <
(if 0 < Nat.succ (Nat.succ x) ∧ Nat.succ (Nat.succ x) % 2 = 0 then
let vbn_half := val_bin_nat (Nat.succ (Nat.succ x) / 2);
(vbn_half.fst + 1, vbn_half.snd)
else if 0 < Nat.succ (Nat.succ x) then (0, Nat.succ (Nat.succ x)) else (0, 0)).snd | case succ.refl.inr
x : ℕ
ih : ∀ (y : ℕ), y ≤ x → 0 < (val_bin_nat (Nat.succ y)).snd
odd : Nat.succ (Nat.succ x) % 2 = 1
not_even : Nat.succ (Nat.succ x) % 2 ≠ 0
⊢ 0 <
(if 0 < Nat.succ (Nat.succ x) ∧ Nat.succ (Nat.succ x) % 2 = 0 then
let vbn_half := val_bin_nat (Nat.succ (Nat.succ x) / 2);
(vbn_half.fst + 1, vbn_half.snd)
else if 0 < Nat.succ (Nat.succ x) then (0, Nat.succ (Nat.succ x)) else (0, 0)).snd |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/EllipticCurve/Kronecker.lean | Int.odd_part_succ_pos | [106, 1] | [146, 26] | rw [if_neg (not_and_of_not_right (0 < Nat.succ (Nat.succ x)) not_even)] | case succ.refl.inr
x : ℕ
ih : ∀ (y : ℕ), y ≤ x → 0 < (val_bin_nat (Nat.succ y)).snd
odd : Nat.succ (Nat.succ x) % 2 = 1
not_even : Nat.succ (Nat.succ x) % 2 ≠ 0
⊢ 0 <
(if 0 < Nat.succ (Nat.succ x) ∧ Nat.succ (Nat.succ x) % 2 = 0 then
let vbn_half := val_bin_nat (Nat.succ (Nat.succ x) / 2);
(vbn_half.fst + 1, vbn_half.snd)
else if 0 < Nat.succ (Nat.succ x) then (0, Nat.succ (Nat.succ x)) else (0, 0)).snd | case succ.refl.inr
x : ℕ
ih : ∀ (y : ℕ), y ≤ x → 0 < (val_bin_nat (Nat.succ y)).snd
odd : Nat.succ (Nat.succ x) % 2 = 1
not_even : Nat.succ (Nat.succ x) % 2 ≠ 0
⊢ 0 < (if 0 < Nat.succ (Nat.succ x) then (0, Nat.succ (Nat.succ x)) else (0, 0)).snd |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/EllipticCurve/Kronecker.lean | Int.odd_part_succ_pos | [106, 1] | [146, 26] | rw [if_pos (zero_lt_succ (Nat.succ x))] | case succ.refl.inr
x : ℕ
ih : ∀ (y : ℕ), y ≤ x → 0 < (val_bin_nat (Nat.succ y)).snd
odd : Nat.succ (Nat.succ x) % 2 = 1
not_even : Nat.succ (Nat.succ x) % 2 ≠ 0
⊢ 0 < (if 0 < Nat.succ (Nat.succ x) then (0, Nat.succ (Nat.succ x)) else (0, 0)).snd | case succ.refl.inr
x : ℕ
ih : ∀ (y : ℕ), y ≤ x → 0 < (val_bin_nat (Nat.succ y)).snd
odd : Nat.succ (Nat.succ x) % 2 = 1
not_even : Nat.succ (Nat.succ x) % 2 ≠ 0
⊢ 0 < (0, Nat.succ (Nat.succ x)).snd |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/EllipticCurve/Kronecker.lean | Int.odd_part_succ_pos | [106, 1] | [146, 26] | exact zero_lt_succ (Nat.succ x) | case succ.refl.inr
x : ℕ
ih : ∀ (y : ℕ), y ≤ x → 0 < (val_bin_nat (Nat.succ y)).snd
odd : Nat.succ (Nat.succ x) % 2 = 1
not_even : Nat.succ (Nat.succ x) % 2 ≠ 0
⊢ 0 < (0, Nat.succ (Nat.succ x)).snd | no goals |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/EllipticCurve/Kronecker.lean | Int.odd_part_succ_pos | [106, 1] | [146, 26] | rw [odd] | x : ℕ
ih : ∀ (y : ℕ), y ≤ x → 0 < (val_bin_nat (Nat.succ y)).snd
odd : Nat.succ (Nat.succ x) % 2 = 1
⊢ Nat.succ (Nat.succ x) % 2 ≠ 0 | x : ℕ
ih : ∀ (y : ℕ), y ≤ x → 0 < (val_bin_nat (Nat.succ y)).snd
odd : Nat.succ (Nat.succ x) % 2 = 1
⊢ 1 ≠ 0 |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/EllipticCurve/Kronecker.lean | Int.odd_part_succ_pos | [106, 1] | [146, 26] | simp | x : ℕ
ih : ∀ (y : ℕ), y ≤ x → 0 < (val_bin_nat (Nat.succ y)).snd
odd : Nat.succ (Nat.succ x) % 2 = 1
⊢ 1 ≠ 0 | no goals |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/EllipticCurve/Kronecker.lean | Int.odd_part_succ_pos | [106, 1] | [146, 26] | exact ih y y_le_x | case succ.step
x : ℕ
ih : ∀ (y : ℕ), y ≤ x → 0 < (val_bin_nat (Nat.succ y)).snd
y : ℕ
y_le_x : Nat.le y x
⊢ 0 < (val_bin_nat (Nat.succ y)).snd | no goals |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/EllipticCurve/Kronecker.lean | Int.odd_part_nat_pos | [148, 1] | [157, 30] | cases x with
| zero =>
intro absurd
apply False.elim
apply absurd
rfl
| succ x =>
intro _
exact odd_part_succ_pos x | x : ℕ
⊢ x ≠ 0 → 0 < (val_bin_nat x).snd | no goals |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/EllipticCurve/Kronecker.lean | Int.odd_part_nat_pos | [148, 1] | [157, 30] | intro absurd | case zero
⊢ zero ≠ 0 → 0 < (val_bin_nat zero).snd | case zero
absurd : zero ≠ 0
⊢ 0 < (val_bin_nat zero).snd |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/EllipticCurve/Kronecker.lean | Int.odd_part_nat_pos | [148, 1] | [157, 30] | apply False.elim | case zero
absurd : zero ≠ 0
⊢ 0 < (val_bin_nat zero).snd | case zero.h
absurd : zero ≠ 0
⊢ False |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/EllipticCurve/Kronecker.lean | Int.odd_part_nat_pos | [148, 1] | [157, 30] | apply absurd | case zero.h
absurd : zero ≠ 0
⊢ False | case zero.h
absurd : zero ≠ 0
⊢ zero = 0 |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/EllipticCurve/Kronecker.lean | Int.odd_part_nat_pos | [148, 1] | [157, 30] | rfl | case zero.h
absurd : zero ≠ 0
⊢ zero = 0 | no goals |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/EllipticCurve/Kronecker.lean | Int.odd_part_nat_pos | [148, 1] | [157, 30] | intro _ | case succ
x : ℕ
⊢ Nat.succ x ≠ 0 → 0 < (val_bin_nat (Nat.succ x)).snd | case succ
x : ℕ
a✝ : Nat.succ x ≠ 0
⊢ 0 < (val_bin_nat (Nat.succ x)).snd |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/EllipticCurve/Kronecker.lean | Int.odd_part_nat_pos | [148, 1] | [157, 30] | exact odd_part_succ_pos x | case succ
x : ℕ
a✝ : Nat.succ x ≠ 0
⊢ 0 < (val_bin_nat (Nat.succ x)).snd | no goals |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/QuadRing/Basic.lean | QuadRing.coe_inj | [37, 9] | [38, 25] | aesop | F : Type u_1
a b : F
inst✝ : Zero F
z w : F
⊢ z = w → { b1 := z, b2 := 0 } = { b1 := w, b2 := 0 } | no goals |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/QuadRing/Basic.lean | QuadRing.coe_add | [77, 1] | [78, 35] | simp | F : Type u_1
a b : F
inst✝ : CommRing F
r s : F
⊢ { b1 := r + s, b2 := 0 }.b1 = ({ b1 := r, b2 := 0 } + { b1 := s, b2 := 0 }).b1 ∧
{ b1 := r + s, b2 := 0 }.b2 = ({ b1 := r, b2 := 0 } + { b1 := s, b2 := 0 }).b2 | no goals |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/QuadRing/Basic.lean | QuadRing.coe_neg | [85, 1] | [85, 87] | ext | F : Type u_1
a b : F
inst✝ : CommRing F
n : F
⊢ { b1 := -n, b2 := 0 } = -{ b1 := n, b2 := 0 } | case b1
F : Type u_1
a b : F
inst✝ : CommRing F
n : F
⊢ { b1 := -n, b2 := 0 }.b1 = (-{ b1 := n, b2 := 0 }).b1
case b2
F : Type u_1
a b : F
inst✝ : CommRing F
n : F
⊢ { b1 := -n, b2 := 0 }.b2 = (-{ b1 := n, b2 := 0 }).b2 |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/QuadRing/Basic.lean | QuadRing.coe_neg | [85, 1] | [85, 87] | simp | case b1
F : Type u_1
a b : F
inst✝ : CommRing F
n : F
⊢ { b1 := -n, b2 := 0 }.b1 = (-{ b1 := n, b2 := 0 }).b1
case b2
F : Type u_1
a b : F
inst✝ : CommRing F
n : F
⊢ { b1 := -n, b2 := 0 }.b2 = (-{ b1 := n, b2 := 0 }).b2 | case b2
F : Type u_1
a b : F
inst✝ : CommRing F
n : F
⊢ { b1 := -n, b2 := 0 }.b2 = (-{ b1 := n, b2 := 0 }).b2 |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/QuadRing/Basic.lean | QuadRing.coe_neg | [85, 1] | [85, 87] | sorry | case b2
F : Type u_1
a b : F
inst✝ : CommRing F
n : F
⊢ { b1 := -n, b2 := 0 }.b2 = (-{ b1 := n, b2 := 0 }).b2 | no goals |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/QuadRing/Basic.lean | QuadRing.coe_sub | [92, 1] | [92, 95] | ext | F : Type u_1
a b : F
inst✝ : CommRing F
z w : F
⊢ { b1 := z - w, b2 := 0 } = { b1 := z, b2 := 0 } - { b1 := w, b2 := 0 } | case b1
F : Type u_1
a b : F
inst✝ : CommRing F
z w : F
⊢ { b1 := z - w, b2 := 0 }.b1 = ({ b1 := z, b2 := 0 } - { b1 := w, b2 := 0 }).b1
case b2
F : Type u_1
a b : F
inst✝ : CommRing F
z w : F
⊢ { b1 := z - w, b2 := 0 }.b2 = ({ b1 := z, b2 := 0 } - { b1 := w, b2 := 0 }).b2 |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/QuadRing/Basic.lean | QuadRing.coe_sub | [92, 1] | [92, 95] | simp | case b1
F : Type u_1
a b : F
inst✝ : CommRing F
z w : F
⊢ { b1 := z - w, b2 := 0 }.b1 = ({ b1 := z, b2 := 0 } - { b1 := w, b2 := 0 }).b1
case b2
F : Type u_1
a b : F
inst✝ : CommRing F
z w : F
⊢ { b1 := z - w, b2 := 0 }.b2 = ({ b1 := z, b2 := 0 } - { b1 := w, b2 := 0 }).b2 | case b2
F : Type u_1
a b : F
inst✝ : CommRing F
z w : F
⊢ { b1 := z - w, b2 := 0 }.b2 = ({ b1 := z, b2 := 0 } - { b1 := w, b2 := 0 }).b2 |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/QuadRing/Basic.lean | QuadRing.coe_sub | [92, 1] | [92, 95] | sorry | case b2
F : Type u_1
a b : F
inst✝ : CommRing F
z w : F
⊢ { b1 := z - w, b2 := 0 }.b2 = ({ b1 := z, b2 := 0 } - { b1 := w, b2 := 0 }).b2 | no goals |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/QuadRing/Basic.lean | QuadRing.coe_nat_add | [150, 1] | [151, 86] | ext <;> simp only [coe_nat_b1, coe_nat_b2, add_b1, add_b2, Nat.cast_add, add_zero] | F : Type u_1
a b : F
inst✝ : CommRing F
r s : ℕ
⊢ ↑(r + s) = ↑r + ↑s | no goals |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/QuadRing/Basic.lean | QuadRing.coe_int_add | [162, 1] | [163, 86] | ext <;> simp only [coe_int_b1, coe_int_b2, add_b1, add_b2, Int.cast_add, add_zero] | F : Type u_1
a b : F
inst✝ : CommRing F
r s : ℤ
⊢ ↑(r + s) = ↑r + ↑s | no goals |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/QuadRing/Basic.lean | QuadRing.coe_mul | [173, 1] | [174, 51] | apply (QuadRing.ext_iff _ _).2 | F : Type u_1
a b : F
inst✝ : CommRing F
r s : F
⊢ { b1 := r * s, b2 := 0 } = { b1 := r, b2 := 0 } * { b1 := s, b2 := 0 } | F : Type u_1
a b : F
inst✝ : CommRing F
r s : F
⊢ { b1 := r * s, b2 := 0 }.b1 = ({ b1 := r, b2 := 0 } * { b1 := s, b2 := 0 }).b1 ∧
{ b1 := r * s, b2 := 0 }.b2 = ({ b1 := r, b2 := 0 } * { b1 := s, b2 := 0 }).b2 |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/QuadRing/Basic.lean | QuadRing.coe_mul | [173, 1] | [174, 51] | simp [mul_comm] | F : Type u_1
a b : F
inst✝ : CommRing F
r s : F
⊢ { b1 := r * s, b2 := 0 }.b1 = ({ b1 := r, b2 := 0 } * { b1 := s, b2 := 0 }).b1 ∧
{ b1 := r * s, b2 := 0 }.b2 = ({ b1 := r, b2 := 0 } * { b1 := s, b2 := 0 }).b2 | no goals |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/FieldTheory/PerfectClosure.lean | ECTate.PerfectRing.pth_root_pow_char | [29, 1] | [33, 82] | simp only [pth_root, h, dite_false] | R : Type u_1
inst✝¹ : CommSemiring R
inst✝ : PerfectRing R
h : ringChar R ≠ 0
x : R
⊢ pth_root x ^ ringChar R = x | R : Type u_1
inst✝¹ : CommSemiring R
inst✝ : PerfectRing R
h : ringChar R ≠ 0
x : R
⊢ Function.surjInv (_ : Function.Surjective fun x => x ^ ringChar R) x ^ ringChar R = x |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/FieldTheory/PerfectClosure.lean | ECTate.PerfectRing.pth_root_pow_char | [29, 1] | [33, 82] | exact Function.rightInverse_surjInv (pth_power_bijective_of_char_nonzero h).2 x | R : Type u_1
inst✝¹ : CommSemiring R
inst✝ : PerfectRing R
h : ringChar R ≠ 0
x : R
⊢ Function.surjInv (_ : Function.Surjective fun x => x ^ ringChar R) x ^ ringChar R = x | no goals |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/FieldTheory/PerfectClosure.lean | ECTate.PerfectRing.pth_root_pow_eq | [35, 1] | [43, 55] | by_cases h : ringChar R = 0 | R : Type u_1
inst✝¹ : CommSemiring R
n : ℕ
inst✝ : PerfectRing R
x : R
⊢ pth_root x ^ n = x ^ (n / ringChar R) * pth_root x ^ (n % ringChar R) | case pos
R : Type u_1
inst✝¹ : CommSemiring R
n : ℕ
inst✝ : PerfectRing R
x : R
h : ringChar R = 0
⊢ pth_root x ^ n = x ^ (n / ringChar R) * pth_root x ^ (n % ringChar R)
case neg
R : Type u_1
inst✝¹ : CommSemiring R
n : ℕ
inst✝ : PerfectRing R
x : R
h : ¬ringChar R = 0
⊢ pth_root x ^ n = x ^ (n / ringChar R) * pth_root x ^ (n % ringChar R) |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/FieldTheory/PerfectClosure.lean | ECTate.PerfectRing.pth_root_pow_eq | [35, 1] | [43, 55] | . simp [h] | case pos
R : Type u_1
inst✝¹ : CommSemiring R
n : ℕ
inst✝ : PerfectRing R
x : R
h : ringChar R = 0
⊢ pth_root x ^ n = x ^ (n / ringChar R) * pth_root x ^ (n % ringChar R)
case neg
R : Type u_1
inst✝¹ : CommSemiring R
n : ℕ
inst✝ : PerfectRing R
x : R
h : ¬ringChar R = 0
⊢ pth_root x ^ n = x ^ (n / ringChar R) * pth_root x ^ (n % ringChar R) | case neg
R : Type u_1
inst✝¹ : CommSemiring R
n : ℕ
inst✝ : PerfectRing R
x : R
h : ¬ringChar R = 0
⊢ pth_root x ^ n = x ^ (n / ringChar R) * pth_root x ^ (n % ringChar R) |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/FieldTheory/PerfectClosure.lean | ECTate.PerfectRing.pth_root_pow_eq | [35, 1] | [43, 55] | conv =>
lhs
rw [← Nat.mod_add_div n (ringChar R)] | case neg
R : Type u_1
inst✝¹ : CommSemiring R
n : ℕ
inst✝ : PerfectRing R
x : R
h : ¬ringChar R = 0
⊢ pth_root x ^ n = x ^ (n / ringChar R) * pth_root x ^ (n % ringChar R) | case neg
R : Type u_1
inst✝¹ : CommSemiring R
n : ℕ
inst✝ : PerfectRing R
x : R
h : ¬ringChar R = 0
⊢ pth_root x ^ (n % ringChar R + ringChar R * (n / ringChar R)) = x ^ (n / ringChar R) * pth_root x ^ (n % ringChar R) |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/FieldTheory/PerfectClosure.lean | ECTate.PerfectRing.pth_root_pow_eq | [35, 1] | [43, 55] | rw [pow_add, pow_mul, pth_root_pow_char h, mul_comm] | case neg
R : Type u_1
inst✝¹ : CommSemiring R
n : ℕ
inst✝ : PerfectRing R
x : R
h : ¬ringChar R = 0
⊢ pth_root x ^ (n % ringChar R + ringChar R * (n / ringChar R)) = x ^ (n / ringChar R) * pth_root x ^ (n % ringChar R) | no goals |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/FieldTheory/PerfectClosure.lean | ECTate.PerfectRing.pth_root_pow_eq | [35, 1] | [43, 55] | simp [h] | case pos
R : Type u_1
inst✝¹ : CommSemiring R
n : ℕ
inst✝ : PerfectRing R
x : R
h : ringChar R = 0
⊢ pth_root x ^ n = x ^ (n / ringChar R) * pth_root x ^ (n % ringChar R) | no goals |
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