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stringclasses 147
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stringlengths 1
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https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/ValuedRing.lean | ENatValRing.sub_val_neg | [376, 1] | [387, 43] | simp [h'] | R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p : R
evr : ENatValRing p
x : R
n : ℕ
ih : sub_val evr n (-x) = -sub_val evr n x
h' : v evr.valtn x = 0
⊢ v evr.valtn (-x) = 0 | no goals |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/ValuedRing.lean | ENatValRing.sub_val_neg | [376, 1] | [387, 43] | rw [sub_val_val_pos_succ evr _ _, sub_val_val_pos_succ evr _ _, sub_val_decr_val_comm, ih,
decr_val_neg, sub_val_decr_val_comm] | case succ.inr
R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p : R
evr : ENatValRing p
x : R
n : ℕ
ih : sub_val evr n (-x) = -sub_val evr n x
h' : 0 < v evr.valtn x
⊢ sub_val evr (Nat.succ n) (-x) = -sub_val evr (Nat.succ n) x | no goals |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/ValuedRing.lean | ENatValRing.sub_val_add | [389, 1] | [395, 65] | apply nzero_mul_left_cancel (p ^ n) | R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p : R
evr : ENatValRing p
x y : R
n : ℕ
hx : ↑n ≤ v evr.valtn x
hy : ↑n ≤ v evr.valtn y
⊢ sub_val evr n (x + y) = sub_val evr n x + sub_val evr n y | case a
R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p : R
evr : ENatValRing p
x y : R
n : ℕ
hx : ↑n ≤ v evr.valtn x
hy : ↑n ≤ v evr.valtn y
⊢ p ^ n ≠ 0
case a
R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p : R
evr : ENatValRing p
x y : R
n : ℕ
hx : ↑n ≤ v evr.valtn x
hy : ↑n ≤ v evr.valtn y
⊢ p ^ n * sub_val evr n (x + y) = p ^ n * (sub_val evr n x + sub_val evr n y) |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/ValuedRing.lean | ENatValRing.sub_val_add | [389, 1] | [395, 65] | . exact pow_ne_zero n (p_non_zero evr.valtn) | case a
R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p : R
evr : ENatValRing p
x y : R
n : ℕ
hx : ↑n ≤ v evr.valtn x
hy : ↑n ≤ v evr.valtn y
⊢ p ^ n ≠ 0
case a
R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p : R
evr : ENatValRing p
x y : R
n : ℕ
hx : ↑n ≤ v evr.valtn x
hy : ↑n ≤ v evr.valtn y
⊢ p ^ n * sub_val evr n (x + y) = p ^ n * (sub_val evr n x + sub_val evr n y) | case a
R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p : R
evr : ENatValRing p
x y : R
n : ℕ
hx : ↑n ≤ v evr.valtn x
hy : ↑n ≤ v evr.valtn y
⊢ p ^ n * sub_val evr n (x + y) = p ^ n * (sub_val evr n x + sub_val evr n y) |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/ValuedRing.lean | ENatValRing.sub_val_add | [389, 1] | [395, 65] | . rw [← factor_p_of_le_val evr (_ : n ≤ evr.valtn (x + y)), mul_add, ← factor_p_of_le_val evr hx,
← factor_p_of_le_val evr hy]
exact le_trans (le_min hx hy) (evr.valtn.v_add_ge_min_v x y) | case a
R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p : R
evr : ENatValRing p
x y : R
n : ℕ
hx : ↑n ≤ v evr.valtn x
hy : ↑n ≤ v evr.valtn y
⊢ p ^ n * sub_val evr n (x + y) = p ^ n * (sub_val evr n x + sub_val evr n y) | no goals |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/ValuedRing.lean | ENatValRing.sub_val_add | [389, 1] | [395, 65] | exact pow_ne_zero n (p_non_zero evr.valtn) | case a
R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p : R
evr : ENatValRing p
x y : R
n : ℕ
hx : ↑n ≤ v evr.valtn x
hy : ↑n ≤ v evr.valtn y
⊢ p ^ n ≠ 0 | no goals |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/ValuedRing.lean | ENatValRing.sub_val_add | [389, 1] | [395, 65] | rw [← factor_p_of_le_val evr (_ : n ≤ evr.valtn (x + y)), mul_add, ← factor_p_of_le_val evr hx,
← factor_p_of_le_val evr hy] | case a
R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p : R
evr : ENatValRing p
x y : R
n : ℕ
hx : ↑n ≤ v evr.valtn x
hy : ↑n ≤ v evr.valtn y
⊢ p ^ n * sub_val evr n (x + y) = p ^ n * (sub_val evr n x + sub_val evr n y) | R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p : R
evr : ENatValRing p
x y : R
n : ℕ
hx : ↑n ≤ v evr.valtn x
hy : ↑n ≤ v evr.valtn y
⊢ ↑n ≤ v evr.valtn (x + y) |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/ValuedRing.lean | ENatValRing.sub_val_add | [389, 1] | [395, 65] | exact le_trans (le_min hx hy) (evr.valtn.v_add_ge_min_v x y) | R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p : R
evr : ENatValRing p
x y : R
n : ℕ
hx : ↑n ≤ v evr.valtn x
hy : ↑n ≤ v evr.valtn y
⊢ ↑n ≤ v evr.valtn (x + y) | no goals |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/ValuedRing.lean | ENatValRing.sub_val_sub | [397, 1] | [401, 8] | rw [sub_eq_add_neg, sub_eq_add_neg, sub_val_add evr hx, sub_val_neg] | R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p : R
evr : ENatValRing p
x y : R
n : ℕ
hx : ↑n ≤ v evr.valtn x
hy : ↑n ≤ v evr.valtn y
⊢ sub_val evr n (x - y) = sub_val evr n x - sub_val evr n y | R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p : R
evr : ENatValRing p
x y : R
n : ℕ
hx : ↑n ≤ v evr.valtn x
hy : ↑n ≤ v evr.valtn y
⊢ ↑n ≤ v evr.valtn (-y) |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/ValuedRing.lean | ENatValRing.sub_val_sub | [397, 1] | [401, 8] | simpa | R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p : R
evr : ENatValRing p
x y : R
n : ℕ
hx : ↑n ≤ v evr.valtn x
hy : ↑n ≤ v evr.valtn y
⊢ ↑n ≤ v evr.valtn (-y) | no goals |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/ValuedRing.lean | ENatValRing.sub_val_mul_left | [403, 1] | [409, 54] | apply nzero_mul_left_cancel (p ^ n) | R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p : R
evr : ENatValRing p
x y : R
n : ℕ
hx : ↑n ≤ v evr.valtn x
⊢ sub_val evr n (x * y) = sub_val evr n x * y | case a
R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p : R
evr : ENatValRing p
x y : R
n : ℕ
hx : ↑n ≤ v evr.valtn x
⊢ p ^ n ≠ 0
case a
R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p : R
evr : ENatValRing p
x y : R
n : ℕ
hx : ↑n ≤ v evr.valtn x
⊢ p ^ n * sub_val evr n (x * y) = p ^ n * (sub_val evr n x * y) |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/ValuedRing.lean | ENatValRing.sub_val_mul_left | [403, 1] | [409, 54] | . exact pow_ne_zero n (p_non_zero evr.valtn) | case a
R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p : R
evr : ENatValRing p
x y : R
n : ℕ
hx : ↑n ≤ v evr.valtn x
⊢ p ^ n ≠ 0
case a
R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p : R
evr : ENatValRing p
x y : R
n : ℕ
hx : ↑n ≤ v evr.valtn x
⊢ p ^ n * sub_val evr n (x * y) = p ^ n * (sub_val evr n x * y) | case a
R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p : R
evr : ENatValRing p
x y : R
n : ℕ
hx : ↑n ≤ v evr.valtn x
⊢ p ^ n * sub_val evr n (x * y) = p ^ n * (sub_val evr n x * y) |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/ValuedRing.lean | ENatValRing.sub_val_mul_left | [403, 1] | [409, 54] | . rw [← factor_p_of_le_val evr (_ : n ≤ evr.valtn (x * y)), ← mul_assoc,
← factor_p_of_le_val evr hx]
exact le_trans hx (val_mul_ge_left evr.valtn x y) | case a
R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p : R
evr : ENatValRing p
x y : R
n : ℕ
hx : ↑n ≤ v evr.valtn x
⊢ p ^ n * sub_val evr n (x * y) = p ^ n * (sub_val evr n x * y) | no goals |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/ValuedRing.lean | ENatValRing.sub_val_mul_left | [403, 1] | [409, 54] | exact pow_ne_zero n (p_non_zero evr.valtn) | case a
R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p : R
evr : ENatValRing p
x y : R
n : ℕ
hx : ↑n ≤ v evr.valtn x
⊢ p ^ n ≠ 0 | no goals |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/ValuedRing.lean | ENatValRing.sub_val_mul_left | [403, 1] | [409, 54] | rw [← factor_p_of_le_val evr (_ : n ≤ evr.valtn (x * y)), ← mul_assoc,
← factor_p_of_le_val evr hx] | case a
R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p : R
evr : ENatValRing p
x y : R
n : ℕ
hx : ↑n ≤ v evr.valtn x
⊢ p ^ n * sub_val evr n (x * y) = p ^ n * (sub_val evr n x * y) | R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p : R
evr : ENatValRing p
x y : R
n : ℕ
hx : ↑n ≤ v evr.valtn x
⊢ ↑n ≤ v evr.valtn (x * y) |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/ValuedRing.lean | ENatValRing.sub_val_mul_left | [403, 1] | [409, 54] | exact le_trans hx (val_mul_ge_left evr.valtn x y) | R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p : R
evr : ENatValRing p
x y : R
n : ℕ
hx : ↑n ≤ v evr.valtn x
⊢ ↑n ≤ v evr.valtn (x * y) | no goals |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/ValuedRing.lean | ENatValRing.sub_val_mul_right | [411, 1] | [413, 56] | rw [mul_comm x y, sub_val_mul_left evr hy, mul_comm] | R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p : R
evr : ENatValRing p
x y : R
n : ℕ
hy : ↑n ≤ v evr.valtn y
⊢ sub_val evr n (x * y) = x * sub_val evr n y | no goals |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/ValuedRing.lean | ENatValRing.sub_val_mul_sub_val | [415, 1] | [427, 27] | apply nzero_mul_left_cancel (p ^ (n + m)) _ _ (pow_ne_zero _ (p_non_zero evr.valtn)) | R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p : R
evr : ENatValRing p
x y : R
n m : ℕ
hx : ↑n ≤ v evr.valtn x
hy : ↑m ≤ v evr.valtn y
⊢ sub_val evr n x * sub_val evr m y = sub_val evr (n + m) (x * y) | R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p : R
evr : ENatValRing p
x y : R
n m : ℕ
hx : ↑n ≤ v evr.valtn x
hy : ↑m ≤ v evr.valtn y
⊢ p ^ (n + m) * (sub_val evr n x * sub_val evr m y) = p ^ (n + m) * sub_val evr (n + m) (x * y) |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/ValuedRing.lean | ENatValRing.sub_val_mul_sub_val | [415, 1] | [427, 27] | rw [←factor_p_of_le_val evr, pow_add, mul_assoc,
mul_comm (p ^ m), ← mul_assoc,
← mul_assoc,
←factor_p_of_le_val evr (_ : n ≤ evr.valtn x), mul_assoc, mul_comm _ (p ^ m),
←factor_p_of_le_val evr (_ : m ≤ evr.valtn y)] | R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p : R
evr : ENatValRing p
x y : R
n m : ℕ
hx : ↑n ≤ v evr.valtn x
hy : ↑m ≤ v evr.valtn y
⊢ p ^ (n + m) * (sub_val evr n x * sub_val evr m y) = p ^ (n + m) * sub_val evr (n + m) (x * y) | R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p : R
evr : ENatValRing p
x y : R
n m : ℕ
hx : ↑n ≤ v evr.valtn x
hy : ↑m ≤ v evr.valtn y
⊢ ↑m ≤ v evr.valtn y
R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p : R
evr : ENatValRing p
x y : R
n m : ℕ
hx : ↑n ≤ v evr.valtn x
hy : ↑m ≤ v evr.valtn y
⊢ ↑n ≤ v evr.valtn x
R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p : R
evr : ENatValRing p
x y : R
n m : ℕ
hx : ↑n ≤ v evr.valtn x
hy : ↑m ≤ v evr.valtn y
⊢ ↑(n + m) ≤ v evr.valtn (x * y) |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/ValuedRing.lean | ENatValRing.sub_val_mul_sub_val | [415, 1] | [427, 27] | . assumption | R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p : R
evr : ENatValRing p
x y : R
n m : ℕ
hx : ↑n ≤ v evr.valtn x
hy : ↑m ≤ v evr.valtn y
⊢ ↑m ≤ v evr.valtn y
R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p : R
evr : ENatValRing p
x y : R
n m : ℕ
hx : ↑n ≤ v evr.valtn x
hy : ↑m ≤ v evr.valtn y
⊢ ↑n ≤ v evr.valtn x
R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p : R
evr : ENatValRing p
x y : R
n m : ℕ
hx : ↑n ≤ v evr.valtn x
hy : ↑m ≤ v evr.valtn y
⊢ ↑(n + m) ≤ v evr.valtn (x * y) | R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p : R
evr : ENatValRing p
x y : R
n m : ℕ
hx : ↑n ≤ v evr.valtn x
hy : ↑m ≤ v evr.valtn y
⊢ ↑n ≤ v evr.valtn x
R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p : R
evr : ENatValRing p
x y : R
n m : ℕ
hx : ↑n ≤ v evr.valtn x
hy : ↑m ≤ v evr.valtn y
⊢ ↑(n + m) ≤ v evr.valtn (x * y) |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/ValuedRing.lean | ENatValRing.sub_val_mul_sub_val | [415, 1] | [427, 27] | . assumption | R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p : R
evr : ENatValRing p
x y : R
n m : ℕ
hx : ↑n ≤ v evr.valtn x
hy : ↑m ≤ v evr.valtn y
⊢ ↑n ≤ v evr.valtn x
R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p : R
evr : ENatValRing p
x y : R
n m : ℕ
hx : ↑n ≤ v evr.valtn x
hy : ↑m ≤ v evr.valtn y
⊢ ↑(n + m) ≤ v evr.valtn (x * y) | R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p : R
evr : ENatValRing p
x y : R
n m : ℕ
hx : ↑n ≤ v evr.valtn x
hy : ↑m ≤ v evr.valtn y
⊢ ↑(n + m) ≤ v evr.valtn (x * y) |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/ValuedRing.lean | ENatValRing.sub_val_mul_sub_val | [415, 1] | [427, 27] | . rw [SurjVal.v_mul_eq_add_v]
exact add_le_add hx hy | R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p : R
evr : ENatValRing p
x y : R
n m : ℕ
hx : ↑n ≤ v evr.valtn x
hy : ↑m ≤ v evr.valtn y
⊢ ↑(n + m) ≤ v evr.valtn (x * y) | no goals |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/ValuedRing.lean | ENatValRing.sub_val_mul_sub_val | [415, 1] | [427, 27] | assumption | R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p : R
evr : ENatValRing p
x y : R
n m : ℕ
hx : ↑n ≤ v evr.valtn x
hy : ↑m ≤ v evr.valtn y
⊢ ↑m ≤ v evr.valtn y | no goals |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/ValuedRing.lean | ENatValRing.sub_val_mul_sub_val | [415, 1] | [427, 27] | assumption | R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p : R
evr : ENatValRing p
x y : R
n m : ℕ
hx : ↑n ≤ v evr.valtn x
hy : ↑m ≤ v evr.valtn y
⊢ ↑n ≤ v evr.valtn x | no goals |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/ValuedRing.lean | ENatValRing.sub_val_mul_sub_val | [415, 1] | [427, 27] | rw [SurjVal.v_mul_eq_add_v] | R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p : R
evr : ENatValRing p
x y : R
n m : ℕ
hx : ↑n ≤ v evr.valtn x
hy : ↑m ≤ v evr.valtn y
⊢ ↑(n + m) ≤ v evr.valtn (x * y) | R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p : R
evr : ENatValRing p
x y : R
n m : ℕ
hx : ↑n ≤ v evr.valtn x
hy : ↑m ≤ v evr.valtn y
⊢ ↑(n + m) ≤ v evr.valtn x + v evr.valtn y |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/ValuedRing.lean | ENatValRing.sub_val_mul_sub_val | [415, 1] | [427, 27] | exact add_le_add hx hy | R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p : R
evr : ENatValRing p
x y : R
n m : ℕ
hx : ↑n ≤ v evr.valtn x
hy : ↑m ≤ v evr.valtn y
⊢ ↑(n + m) ≤ v evr.valtn x + v evr.valtn y | no goals |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/ValuedRing.lean | ENatValRing.sub_val_mul | [429, 1] | [432, 44] | rw [← h, sub_val_mul_sub_val _ _ _ hx hy] | R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p : R
evr : ENatValRing p
x y : R
n m nm : ℕ
h : n + m = nm
hx : ↑n ≤ v evr.valtn x
hy : ↑m ≤ v evr.valtn y
⊢ sub_val evr nm (x * y) = sub_val evr n x * sub_val evr m y | no goals |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/ValuedRing.lean | ENatValRing.sub_val_pow | [434, 1] | [445, 11] | simp [← h] | case zero
R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p : R
evr : ENatValRing p
x : R
n : ℕ
hx : ↑n ≤ v evr.valtn x
nm : ℕ
h : Nat.zero * n = nm
⊢ sub_val evr nm (x ^ Nat.zero) = sub_val evr n x ^ Nat.zero | no goals |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/ValuedRing.lean | ENatValRing.sub_val_pow | [434, 1] | [445, 11] | rw [pow_succ, sub_val_mul _ n (k * n), pow_succ, ← ih] | case succ
R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p : R
evr : ENatValRing p
x : R
n : ℕ
hx : ↑n ≤ v evr.valtn x
k : ℕ
ih : ∀ {nm : ℕ}, k * n = nm → sub_val evr nm (x ^ k) = sub_val evr n x ^ k
nm : ℕ
h : Nat.succ k * n = nm
⊢ sub_val evr nm (x ^ Nat.succ k) = sub_val evr n x ^ Nat.succ k | case succ
R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p : R
evr : ENatValRing p
x : R
n : ℕ
hx : ↑n ≤ v evr.valtn x
k : ℕ
ih : ∀ {nm : ℕ}, k * n = nm → sub_val evr nm (x ^ k) = sub_val evr n x ^ k
nm : ℕ
h : Nat.succ k * n = nm
⊢ k * n = k * n
case succ.h
R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p : R
evr : ENatValRing p
x : R
n : ℕ
hx : ↑n ≤ v evr.valtn x
k : ℕ
ih : ∀ {nm : ℕ}, k * n = nm → sub_val evr nm (x ^ k) = sub_val evr n x ^ k
nm : ℕ
h : Nat.succ k * n = nm
⊢ n + k * n = nm
case succ.hx
R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p : R
evr : ENatValRing p
x : R
n : ℕ
hx : ↑n ≤ v evr.valtn x
k : ℕ
ih : ∀ {nm : ℕ}, k * n = nm → sub_val evr nm (x ^ k) = sub_val evr n x ^ k
nm : ℕ
h : Nat.succ k * n = nm
⊢ ↑n ≤ v evr.valtn x
case succ.hy
R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p : R
evr : ENatValRing p
x : R
n : ℕ
hx : ↑n ≤ v evr.valtn x
k : ℕ
ih : ∀ {nm : ℕ}, k * n = nm → sub_val evr nm (x ^ k) = sub_val evr n x ^ k
nm : ℕ
h : Nat.succ k * n = nm
⊢ ↑(k * n) ≤ v evr.valtn (x ^ k) |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/ValuedRing.lean | ENatValRing.sub_val_pow | [434, 1] | [445, 11] | rfl | case succ
R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p : R
evr : ENatValRing p
x : R
n : ℕ
hx : ↑n ≤ v evr.valtn x
k : ℕ
ih : ∀ {nm : ℕ}, k * n = nm → sub_val evr nm (x ^ k) = sub_val evr n x ^ k
nm : ℕ
h : Nat.succ k * n = nm
⊢ k * n = k * n | no goals |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/ValuedRing.lean | ENatValRing.sub_val_pow | [434, 1] | [445, 11] | rw [← h, Nat.succ_mul, add_comm] | case succ.h
R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p : R
evr : ENatValRing p
x : R
n : ℕ
hx : ↑n ≤ v evr.valtn x
k : ℕ
ih : ∀ {nm : ℕ}, k * n = nm → sub_val evr nm (x ^ k) = sub_val evr n x ^ k
nm : ℕ
h : Nat.succ k * n = nm
⊢ n + k * n = nm | no goals |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/ValuedRing.lean | ENatValRing.sub_val_pow | [434, 1] | [445, 11] | exact hx | case succ.hx
R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p : R
evr : ENatValRing p
x : R
n : ℕ
hx : ↑n ≤ v evr.valtn x
k : ℕ
ih : ∀ {nm : ℕ}, k * n = nm → sub_val evr nm (x ^ k) = sub_val evr n x ^ k
nm : ℕ
h : Nat.succ k * n = nm
⊢ ↑n ≤ v evr.valtn x | no goals |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/ValuedRing.lean | ENatValRing.sub_val_pow | [434, 1] | [445, 11] | convert val_pow_ge_of_ge evr.valtn k hx | case succ.hy
R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p : R
evr : ENatValRing p
x : R
n : ℕ
hx : ↑n ≤ v evr.valtn x
k : ℕ
ih : ∀ {nm : ℕ}, k * n = nm → sub_val evr nm (x ^ k) = sub_val evr n x ^ k
nm : ℕ
h : Nat.succ k * n = nm
⊢ ↑(k * n) ≤ v evr.valtn (x ^ k) | case h.e'_3
R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p : R
evr : ENatValRing p
x : R
n : ℕ
hx : ↑n ≤ v evr.valtn x
k : ℕ
ih : ∀ {nm : ℕ}, k * n = nm → sub_val evr nm (x ^ k) = sub_val evr n x ^ k
nm : ℕ
h : Nat.succ k * n = nm
⊢ ↑(k * n) = k • ↑n |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/ValuedRing.lean | ENatValRing.sub_val_pow | [434, 1] | [445, 11] | simp | case h.e'_3
R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p : R
evr : ENatValRing p
x : R
n : ℕ
hx : ↑n ≤ v evr.valtn x
k : ℕ
ih : ∀ {nm : ℕ}, k * n = nm → sub_val evr nm (x ^ k) = sub_val evr n x ^ k
nm : ℕ
h : Nat.succ k * n = nm
⊢ ↑(k * n) = k • ↑n | no goals |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/ValuedRing.lean | ENatValRing.sub_val_sub_val | [447, 1] | [459, 18] | have general : ∀ y : R, sub_val evr n (sub_val evr m y) = sub_val evr (m + n) y := by
induction m with
| zero => simp [sub_val_x_zero]
| succ m ih =>
intro y
cases @eq_zero_or_pos _ _ (evr.valtn y) with
| inl h' => simp [sub_val_val_zero evr y _ h']
| inr h' =>
rw [sub_val_val_pos_succ evr y m, Nat.succ_add, sub_val_val_pos_succ evr y _]
exact ih (evr.decr_val y) | R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p : R
evr : ENatValRing p
x : R
m n : ℕ
⊢ sub_val evr n (sub_val evr m x) = sub_val evr (m + n) x | R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p : R
evr : ENatValRing p
x : R
m n : ℕ
general : ∀ (y : R), sub_val evr n (sub_val evr m y) = sub_val evr (m + n) y
⊢ sub_val evr n (sub_val evr m x) = sub_val evr (m + n) x |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/ValuedRing.lean | ENatValRing.sub_val_sub_val | [447, 1] | [459, 18] | exact general x | R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p : R
evr : ENatValRing p
x : R
m n : ℕ
general : ∀ (y : R), sub_val evr n (sub_val evr m y) = sub_val evr (m + n) y
⊢ sub_val evr n (sub_val evr m x) = sub_val evr (m + n) x | no goals |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/ValuedRing.lean | ENatValRing.sub_val_sub_val | [447, 1] | [459, 18] | induction m with
| zero => simp [sub_val_x_zero]
| succ m ih =>
intro y
cases @eq_zero_or_pos _ _ (evr.valtn y) with
| inl h' => simp [sub_val_val_zero evr y _ h']
| inr h' =>
rw [sub_val_val_pos_succ evr y m, Nat.succ_add, sub_val_val_pos_succ evr y _]
exact ih (evr.decr_val y) | R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p : R
evr : ENatValRing p
x : R
m n : ℕ
⊢ ∀ (y : R), sub_val evr n (sub_val evr m y) = sub_val evr (m + n) y | no goals |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/ValuedRing.lean | ENatValRing.sub_val_sub_val | [447, 1] | [459, 18] | simp [sub_val_x_zero] | case zero
R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p : R
evr : ENatValRing p
x : R
n : ℕ
⊢ ∀ (y : R), sub_val evr n (sub_val evr Nat.zero y) = sub_val evr (Nat.zero + n) y | no goals |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/ValuedRing.lean | ENatValRing.sub_val_sub_val | [447, 1] | [459, 18] | intro y | case succ
R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p : R
evr : ENatValRing p
x : R
n m : ℕ
ih : ∀ (y : R), sub_val evr n (sub_val evr m y) = sub_val evr (m + n) y
⊢ ∀ (y : R), sub_val evr n (sub_val evr (Nat.succ m) y) = sub_val evr (Nat.succ m + n) y | case succ
R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p : R
evr : ENatValRing p
x : R
n m : ℕ
ih : ∀ (y : R), sub_val evr n (sub_val evr m y) = sub_val evr (m + n) y
y : R
⊢ sub_val evr n (sub_val evr (Nat.succ m) y) = sub_val evr (Nat.succ m + n) y |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/ValuedRing.lean | ENatValRing.sub_val_sub_val | [447, 1] | [459, 18] | cases @eq_zero_or_pos _ _ (evr.valtn y) with
| inl h' => simp [sub_val_val_zero evr y _ h']
| inr h' =>
rw [sub_val_val_pos_succ evr y m, Nat.succ_add, sub_val_val_pos_succ evr y _]
exact ih (evr.decr_val y) | case succ
R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p : R
evr : ENatValRing p
x : R
n m : ℕ
ih : ∀ (y : R), sub_val evr n (sub_val evr m y) = sub_val evr (m + n) y
y : R
⊢ sub_val evr n (sub_val evr (Nat.succ m) y) = sub_val evr (Nat.succ m + n) y | no goals |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/ValuedRing.lean | ENatValRing.sub_val_sub_val | [447, 1] | [459, 18] | simp [sub_val_val_zero evr y _ h'] | case succ.inl
R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p : R
evr : ENatValRing p
x : R
n m : ℕ
ih : ∀ (y : R), sub_val evr n (sub_val evr m y) = sub_val evr (m + n) y
y : R
h' : v evr.valtn y = 0
⊢ sub_val evr n (sub_val evr (Nat.succ m) y) = sub_val evr (Nat.succ m + n) y | no goals |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/ValuedRing.lean | ENatValRing.sub_val_sub_val | [447, 1] | [459, 18] | rw [sub_val_val_pos_succ evr y m, Nat.succ_add, sub_val_val_pos_succ evr y _] | case succ.inr
R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p : R
evr : ENatValRing p
x : R
n m : ℕ
ih : ∀ (y : R), sub_val evr n (sub_val evr m y) = sub_val evr (m + n) y
y : R
h' : 0 < v evr.valtn y
⊢ sub_val evr n (sub_val evr (Nat.succ m) y) = sub_val evr (Nat.succ m + n) y | case succ.inr
R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p : R
evr : ENatValRing p
x : R
n m : ℕ
ih : ∀ (y : R), sub_val evr n (sub_val evr m y) = sub_val evr (m + n) y
y : R
h' : 0 < v evr.valtn y
⊢ sub_val evr n (sub_val evr m (decr_val evr y)) = sub_val evr (m + n) (decr_val evr y) |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/ValuedRing.lean | ENatValRing.sub_val_sub_val | [447, 1] | [459, 18] | exact ih (evr.decr_val y) | case succ.inr
R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p : R
evr : ENatValRing p
x : R
n m : ℕ
ih : ∀ (y : R), sub_val evr n (sub_val evr m y) = sub_val evr (m + n) y
y : R
h' : 0 < v evr.valtn y
⊢ sub_val evr n (sub_val evr m (decr_val evr y)) = sub_val evr (m + n) (decr_val evr y) | no goals |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/ValuedRing.lean | ENatValRing.val_poly_of_double_root | [470, 1] | [473, 62] | sorry | R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p : R
evr : ENatValRing p
a b c : R
H : has_double_root evr a b c
⊢ v evr.valtn (a * double_root evr a b c ^ 2 + b * double_root evr a b c + c) > 0 ∧
v evr.valtn (2 * a * double_root evr a b c + b) > 0 | no goals |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/ValuedRing.lean | ENatValRing.pth_root_pos_of_pos | [475, 1] | [486, 19] | suffices 0 < evr.valtn (evr.pth_root r ^ evr.residue_char) by
. simp at this
exact this.2 | R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p : R
evr : ENatValRing p
r : R
ha : 0 < v evr.valtn r
hchar : evr.residue_char ≠ 0
⊢ v evr.valtn (pth_root evr r) > 0 | R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p : R
evr : ENatValRing p
r : R
ha : 0 < v evr.valtn r
hchar : evr.residue_char ≠ 0
⊢ 0 < v evr.valtn (pth_root evr r ^ evr.residue_char) |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/ValuedRing.lean | ENatValRing.pth_root_pos_of_pos | [475, 1] | [486, 19] | have :
min (SurjVal.v evr.valtn (pth_root evr r ^ evr.residue_char - r)) (SurjVal.v evr.valtn r) > 0 :=
min_rec' (LT.lt 0) (evr.pth_root_spec.resolve_left hchar r) ha | R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p : R
evr : ENatValRing p
r : R
ha : 0 < v evr.valtn r
hchar : evr.residue_char ≠ 0
⊢ 0 < v evr.valtn (pth_root evr r ^ evr.residue_char) | R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p : R
evr : ENatValRing p
r : R
ha : 0 < v evr.valtn r
hchar : evr.residue_char ≠ 0
this : min (v evr.valtn (pth_root evr r ^ evr.residue_char - r)) (v evr.valtn r) > 0
⊢ 0 < v evr.valtn (pth_root evr r ^ evr.residue_char) |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/ValuedRing.lean | ENatValRing.pth_root_pos_of_pos | [475, 1] | [486, 19] | have := this.trans_le (evr.valtn.v_add_ge_min_v (evr.pth_root r ^ evr.residue_char - r) r) | R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p : R
evr : ENatValRing p
r : R
ha : 0 < v evr.valtn r
hchar : evr.residue_char ≠ 0
this : min (v evr.valtn (pth_root evr r ^ evr.residue_char - r)) (v evr.valtn r) > 0
⊢ 0 < v evr.valtn (pth_root evr r ^ evr.residue_char) | R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p : R
evr : ENatValRing p
r : R
ha : 0 < v evr.valtn r
hchar : evr.residue_char ≠ 0
this✝ : min (v evr.valtn (pth_root evr r ^ evr.residue_char - r)) (v evr.valtn r) > 0
this : 0 < v evr.valtn (pth_root evr r ^ evr.residue_char - r + r)
⊢ 0 < v evr.valtn (pth_root evr r ^ evr.residue_char) |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/ValuedRing.lean | ENatValRing.pth_root_pos_of_pos | [475, 1] | [486, 19] | simpa using this | R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p : R
evr : ENatValRing p
r : R
ha : 0 < v evr.valtn r
hchar : evr.residue_char ≠ 0
this✝ : min (v evr.valtn (pth_root evr r ^ evr.residue_char - r)) (v evr.valtn r) > 0
this : 0 < v evr.valtn (pth_root evr r ^ evr.residue_char - r + r)
⊢ 0 < v evr.valtn (pth_root evr r ^ evr.residue_char) | no goals |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/ValuedRing.lean | ENatValRing.pth_root_pos_of_pos | [475, 1] | [486, 19] | . simp at this
exact this.2 | R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p : R
evr : ENatValRing p
r : R
ha : 0 < v evr.valtn r
hchar : evr.residue_char ≠ 0
this : 0 < v evr.valtn (pth_root evr r ^ evr.residue_char)
⊢ v evr.valtn (pth_root evr r) > 0 | no goals |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/ValuedRing.lean | ENatValRing.pth_root_pos_of_pos | [475, 1] | [486, 19] | simp at this | R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p : R
evr : ENatValRing p
r : R
ha : 0 < v evr.valtn r
hchar : evr.residue_char ≠ 0
this : 0 < v evr.valtn (pth_root evr r ^ evr.residue_char)
⊢ v evr.valtn (pth_root evr r) > 0 | R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p : R
evr : ENatValRing p
r : R
ha : 0 < v evr.valtn r
hchar : evr.residue_char ≠ 0
this : 0 < evr.residue_char ∧ 0 < v evr.valtn (pth_root evr r)
⊢ v evr.valtn (pth_root evr r) > 0 |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/ValuedRing.lean | ENatValRing.pth_root_pos_of_pos | [475, 1] | [486, 19] | exact this.2 | R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p : R
evr : ENatValRing p
r : R
ha : 0 < v evr.valtn r
hchar : evr.residue_char ≠ 0
this : 0 < evr.residue_char ∧ 0 < v evr.valtn (pth_root evr r)
⊢ v evr.valtn (pth_root evr r) > 0 | no goals |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/ValuedRing.lean | ndiv_mul_left | [491, 1] | [494, 25] | intro hab ha | R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
a b p : ℕ
⊢ a * b % p ≠ 0 → a % p ≠ 0 | R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
a b p : ℕ
hab : a * b % p ≠ 0
ha : a % p = 0
⊢ False |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/ValuedRing.lean | ndiv_mul_left | [491, 1] | [494, 25] | apply hab | R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
a b p : ℕ
hab : a * b % p ≠ 0
ha : a % p = 0
⊢ False | R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
a b p : ℕ
hab : a * b % p ≠ 0
ha : a % p = 0
⊢ a * b % p = 0 |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/ValuedRing.lean | ndiv_mul_left | [491, 1] | [494, 25] | simp [Nat.mul_mod, ha] | R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
a b p : ℕ
hab : a * b % p ≠ 0
ha : a % p = 0
⊢ a * b % p = 0 | no goals |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/ValuedRing.lean | ndiv_mul_right | [496, 1] | [498, 28] | rw [Nat.mul_comm] | R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
a b p : ℕ
⊢ a * b % p ≠ 0 → b % p ≠ 0 | R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
a b p : ℕ
⊢ b * a % p ≠ 0 → b % p ≠ 0 |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/ValuedRing.lean | ndiv_mul_right | [496, 1] | [498, 28] | exact ndiv_mul_left b a p | R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
a b p : ℕ
⊢ b * a % p ≠ 0 → b % p ≠ 0 | no goals |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/ValuedRing.lean | nat_valuation_aux''_of_dvd_induction | [549, 1] | [576, 26] | intro M | R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
q : ℕ
hq : 1 < q
⊢ ∀ (M m : ℕ),
m ≤ M →
∀ (hm : 0 < m) (n : ℕ) (hmq : m % q = 0),
nat_valuation_aux'' q hq m hm n = nat_valuation_aux'' q hq (m / q) (_ : 0 < m / q) n + 1 | R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
q : ℕ
hq : 1 < q
M : ℕ
⊢ ∀ (m : ℕ),
m ≤ M →
∀ (hm : 0 < m) (n : ℕ) (hmq : m % q = 0),
nat_valuation_aux'' q hq m hm n = nat_valuation_aux'' q hq (m / q) (_ : 0 < m / q) n + 1 |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/ValuedRing.lean | nat_valuation_aux''_of_dvd_induction | [549, 1] | [576, 26] | induction M with
| zero =>
intro m mle0 hm n hmq
rw [Nat.le_zero] at mle0
exact ((ne_of_gt hm) mle0).elim
| succ M IH =>
intro m m_le_sM hm n hmq
cases LE.le.lt_or_eq m_le_sM with
| inl mltsM =>
exact IH m (Nat.le_of_lt_succ mltsM) hm n hmq
| inr meqsM =>
cases em ((m / q) % q == 0) with
| inl h =>
rw [nat_valuation_aux'', nat_valuation_aux'', dif_pos h]
simp only [beq_iff_eq, Nat.cast_succ]
rw [dif_pos hmq]
simp only [meqsM]
rw [meqsM] at hm h hmq
exact IH (M.succ/q) (Nat.le_of_lt_succ (Nat.div_lt_self hm hq))
(Nat.div_pos_of_mod hm hq hmq) (n+1) (by simpa using h)
| inr h =>
rw [nat_valuation_aux'', nat_valuation_aux'', dif_neg h, dif_pos, nat_valuation_aux'', dif_neg h]
simp
. simp only [hmq] | R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
q : ℕ
hq : 1 < q
M : ℕ
⊢ ∀ (m : ℕ),
m ≤ M →
∀ (hm : 0 < m) (n : ℕ) (hmq : m % q = 0),
nat_valuation_aux'' q hq m hm n = nat_valuation_aux'' q hq (m / q) (_ : 0 < m / q) n + 1 | no goals |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/ValuedRing.lean | nat_valuation_aux''_of_dvd_induction | [549, 1] | [576, 26] | intro m mle0 hm n hmq | case zero
R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
q : ℕ
hq : 1 < q
⊢ ∀ (m : ℕ),
m ≤ Nat.zero →
∀ (hm : 0 < m) (n : ℕ) (hmq : m % q = 0),
nat_valuation_aux'' q hq m hm n = nat_valuation_aux'' q hq (m / q) (_ : 0 < m / q) n + 1 | case zero
R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
q : ℕ
hq : 1 < q
m : ℕ
mle0 : m ≤ Nat.zero
hm : 0 < m
n : ℕ
hmq : m % q = 0
⊢ nat_valuation_aux'' q hq m hm n = nat_valuation_aux'' q hq (m / q) (_ : 0 < m / q) n + 1 |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/ValuedRing.lean | nat_valuation_aux''_of_dvd_induction | [549, 1] | [576, 26] | rw [Nat.le_zero] at mle0 | case zero
R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
q : ℕ
hq : 1 < q
m : ℕ
mle0 : m ≤ Nat.zero
hm : 0 < m
n : ℕ
hmq : m % q = 0
⊢ nat_valuation_aux'' q hq m hm n = nat_valuation_aux'' q hq (m / q) (_ : 0 < m / q) n + 1 | case zero
R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
q : ℕ
hq : 1 < q
m : ℕ
mle0 : m = 0
hm : 0 < m
n : ℕ
hmq : m % q = 0
⊢ nat_valuation_aux'' q hq m hm n = nat_valuation_aux'' q hq (m / q) (_ : 0 < m / q) n + 1 |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/ValuedRing.lean | nat_valuation_aux''_of_dvd_induction | [549, 1] | [576, 26] | exact ((ne_of_gt hm) mle0).elim | case zero
R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
q : ℕ
hq : 1 < q
m : ℕ
mle0 : m = 0
hm : 0 < m
n : ℕ
hmq : m % q = 0
⊢ nat_valuation_aux'' q hq m hm n = nat_valuation_aux'' q hq (m / q) (_ : 0 < m / q) n + 1 | no goals |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/ValuedRing.lean | nat_valuation_aux''_of_dvd_induction | [549, 1] | [576, 26] | intro m m_le_sM hm n hmq | case succ
R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
q : ℕ
hq : 1 < q
M : ℕ
IH :
∀ (m : ℕ),
m ≤ M →
∀ (hm : 0 < m) (n : ℕ) (hmq : m % q = 0),
nat_valuation_aux'' q hq m hm n = nat_valuation_aux'' q hq (m / q) (_ : 0 < m / q) n + 1
⊢ ∀ (m : ℕ),
m ≤ Nat.succ M →
∀ (hm : 0 < m) (n : ℕ) (hmq : m % q = 0),
nat_valuation_aux'' q hq m hm n = nat_valuation_aux'' q hq (m / q) (_ : 0 < m / q) n + 1 | case succ
R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
q : ℕ
hq : 1 < q
M : ℕ
IH :
∀ (m : ℕ),
m ≤ M →
∀ (hm : 0 < m) (n : ℕ) (hmq : m % q = 0),
nat_valuation_aux'' q hq m hm n = nat_valuation_aux'' q hq (m / q) (_ : 0 < m / q) n + 1
m : ℕ
m_le_sM : m ≤ Nat.succ M
hm : 0 < m
n : ℕ
hmq : m % q = 0
⊢ nat_valuation_aux'' q hq m hm n = nat_valuation_aux'' q hq (m / q) (_ : 0 < m / q) n + 1 |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/ValuedRing.lean | nat_valuation_aux''_of_dvd_induction | [549, 1] | [576, 26] | cases LE.le.lt_or_eq m_le_sM with
| inl mltsM =>
exact IH m (Nat.le_of_lt_succ mltsM) hm n hmq
| inr meqsM =>
cases em ((m / q) % q == 0) with
| inl h =>
rw [nat_valuation_aux'', nat_valuation_aux'', dif_pos h]
simp only [beq_iff_eq, Nat.cast_succ]
rw [dif_pos hmq]
simp only [meqsM]
rw [meqsM] at hm h hmq
exact IH (M.succ/q) (Nat.le_of_lt_succ (Nat.div_lt_self hm hq))
(Nat.div_pos_of_mod hm hq hmq) (n+1) (by simpa using h)
| inr h =>
rw [nat_valuation_aux'', nat_valuation_aux'', dif_neg h, dif_pos, nat_valuation_aux'', dif_neg h]
simp
. simp only [hmq] | case succ
R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
q : ℕ
hq : 1 < q
M : ℕ
IH :
∀ (m : ℕ),
m ≤ M →
∀ (hm : 0 < m) (n : ℕ) (hmq : m % q = 0),
nat_valuation_aux'' q hq m hm n = nat_valuation_aux'' q hq (m / q) (_ : 0 < m / q) n + 1
m : ℕ
m_le_sM : m ≤ Nat.succ M
hm : 0 < m
n : ℕ
hmq : m % q = 0
⊢ nat_valuation_aux'' q hq m hm n = nat_valuation_aux'' q hq (m / q) (_ : 0 < m / q) n + 1 | no goals |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/ValuedRing.lean | nat_valuation_aux''_of_dvd_induction | [549, 1] | [576, 26] | exact IH m (Nat.le_of_lt_succ mltsM) hm n hmq | case succ.inl
R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
q : ℕ
hq : 1 < q
M : ℕ
IH :
∀ (m : ℕ),
m ≤ M →
∀ (hm : 0 < m) (n : ℕ) (hmq : m % q = 0),
nat_valuation_aux'' q hq m hm n = nat_valuation_aux'' q hq (m / q) (_ : 0 < m / q) n + 1
m : ℕ
m_le_sM : m ≤ Nat.succ M
hm : 0 < m
n : ℕ
hmq : m % q = 0
mltsM : m < Nat.succ M
⊢ nat_valuation_aux'' q hq m hm n = nat_valuation_aux'' q hq (m / q) (_ : 0 < m / q) n + 1 | no goals |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/ValuedRing.lean | nat_valuation_aux''_of_dvd_induction | [549, 1] | [576, 26] | cases em ((m / q) % q == 0) with
| inl h =>
rw [nat_valuation_aux'', nat_valuation_aux'', dif_pos h]
simp only [beq_iff_eq, Nat.cast_succ]
rw [dif_pos hmq]
simp only [meqsM]
rw [meqsM] at hm h hmq
exact IH (M.succ/q) (Nat.le_of_lt_succ (Nat.div_lt_self hm hq))
(Nat.div_pos_of_mod hm hq hmq) (n+1) (by simpa using h)
| inr h =>
rw [nat_valuation_aux'', nat_valuation_aux'', dif_neg h, dif_pos, nat_valuation_aux'', dif_neg h]
simp
. simp only [hmq] | case succ.inr
R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
q : ℕ
hq : 1 < q
M : ℕ
IH :
∀ (m : ℕ),
m ≤ M →
∀ (hm : 0 < m) (n : ℕ) (hmq : m % q = 0),
nat_valuation_aux'' q hq m hm n = nat_valuation_aux'' q hq (m / q) (_ : 0 < m / q) n + 1
m : ℕ
m_le_sM : m ≤ Nat.succ M
hm : 0 < m
n : ℕ
hmq : m % q = 0
meqsM : m = Nat.succ M
⊢ nat_valuation_aux'' q hq m hm n = nat_valuation_aux'' q hq (m / q) (_ : 0 < m / q) n + 1 | no goals |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/ValuedRing.lean | nat_valuation_aux''_of_dvd_induction | [549, 1] | [576, 26] | rw [nat_valuation_aux'', nat_valuation_aux'', dif_pos h] | case succ.inr.inl
R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
q : ℕ
hq : 1 < q
M : ℕ
IH :
∀ (m : ℕ),
m ≤ M →
∀ (hm : 0 < m) (n : ℕ) (hmq : m % q = 0),
nat_valuation_aux'' q hq m hm n = nat_valuation_aux'' q hq (m / q) (_ : 0 < m / q) n + 1
m : ℕ
m_le_sM : m ≤ Nat.succ M
hm : 0 < m
n : ℕ
hmq : m % q = 0
meqsM : m = Nat.succ M
h : (m / q % q == 0) = true
⊢ nat_valuation_aux'' q hq m hm n = nat_valuation_aux'' q hq (m / q) (_ : 0 < m / q) n + 1 | case succ.inr.inl
R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
q : ℕ
hq : 1 < q
M : ℕ
IH :
∀ (m : ℕ),
m ≤ M →
∀ (hm : 0 < m) (n : ℕ) (hmq : m % q = 0),
nat_valuation_aux'' q hq m hm n = nat_valuation_aux'' q hq (m / q) (_ : 0 < m / q) n + 1
m : ℕ
m_le_sM : m ≤ Nat.succ M
hm : 0 < m
n : ℕ
hmq : m % q = 0
meqsM : m = Nat.succ M
h : (m / q % q == 0) = true
⊢ (if hmq : (m % q == 0) = true then nat_valuation_aux'' q hq (m / q) (_ : 0 < m / q) (n + 1) else n) =
nat_valuation_aux'' q hq (m / q / q) (_ : 0 < m / q / q) (n + 1) + 1 |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/ValuedRing.lean | nat_valuation_aux''_of_dvd_induction | [549, 1] | [576, 26] | simp only [beq_iff_eq, Nat.cast_succ] | case succ.inr.inl
R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
q : ℕ
hq : 1 < q
M : ℕ
IH :
∀ (m : ℕ),
m ≤ M →
∀ (hm : 0 < m) (n : ℕ) (hmq : m % q = 0),
nat_valuation_aux'' q hq m hm n = nat_valuation_aux'' q hq (m / q) (_ : 0 < m / q) n + 1
m : ℕ
m_le_sM : m ≤ Nat.succ M
hm : 0 < m
n : ℕ
hmq : m % q = 0
meqsM : m = Nat.succ M
h : (m / q % q == 0) = true
⊢ (if hmq : (m % q == 0) = true then nat_valuation_aux'' q hq (m / q) (_ : 0 < m / q) (n + 1) else n) =
nat_valuation_aux'' q hq (m / q / q) (_ : 0 < m / q / q) (n + 1) + 1 | case succ.inr.inl
R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
q : ℕ
hq : 1 < q
M : ℕ
IH :
∀ (m : ℕ),
m ≤ M →
∀ (hm : 0 < m) (n : ℕ) (hmq : m % q = 0),
nat_valuation_aux'' q hq m hm n = nat_valuation_aux'' q hq (m / q) (_ : 0 < m / q) n + 1
m : ℕ
m_le_sM : m ≤ Nat.succ M
hm : 0 < m
n : ℕ
hmq : m % q = 0
meqsM : m = Nat.succ M
h : (m / q % q == 0) = true
⊢ (if h : m % q = 0 then nat_valuation_aux'' q hq (m / q) (_ : 0 < m / q) (n + 1) else n) =
nat_valuation_aux'' q hq (m / q / q) (_ : 0 < m / q / q) (n + 1) + 1 |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/ValuedRing.lean | nat_valuation_aux''_of_dvd_induction | [549, 1] | [576, 26] | rw [dif_pos hmq] | case succ.inr.inl
R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
q : ℕ
hq : 1 < q
M : ℕ
IH :
∀ (m : ℕ),
m ≤ M →
∀ (hm : 0 < m) (n : ℕ) (hmq : m % q = 0),
nat_valuation_aux'' q hq m hm n = nat_valuation_aux'' q hq (m / q) (_ : 0 < m / q) n + 1
m : ℕ
m_le_sM : m ≤ Nat.succ M
hm : 0 < m
n : ℕ
hmq : m % q = 0
meqsM : m = Nat.succ M
h : (m / q % q == 0) = true
⊢ (if h : m % q = 0 then nat_valuation_aux'' q hq (m / q) (_ : 0 < m / q) (n + 1) else n) =
nat_valuation_aux'' q hq (m / q / q) (_ : 0 < m / q / q) (n + 1) + 1 | case succ.inr.inl
R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
q : ℕ
hq : 1 < q
M : ℕ
IH :
∀ (m : ℕ),
m ≤ M →
∀ (hm : 0 < m) (n : ℕ) (hmq : m % q = 0),
nat_valuation_aux'' q hq m hm n = nat_valuation_aux'' q hq (m / q) (_ : 0 < m / q) n + 1
m : ℕ
m_le_sM : m ≤ Nat.succ M
hm : 0 < m
n : ℕ
hmq : m % q = 0
meqsM : m = Nat.succ M
h : (m / q % q == 0) = true
⊢ nat_valuation_aux'' q hq (m / q) (_ : 0 < m / q) (n + 1) =
nat_valuation_aux'' q hq (m / q / q) (_ : 0 < m / q / q) (n + 1) + 1 |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/ValuedRing.lean | nat_valuation_aux''_of_dvd_induction | [549, 1] | [576, 26] | simp only [meqsM] | case succ.inr.inl
R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
q : ℕ
hq : 1 < q
M : ℕ
IH :
∀ (m : ℕ),
m ≤ M →
∀ (hm : 0 < m) (n : ℕ) (hmq : m % q = 0),
nat_valuation_aux'' q hq m hm n = nat_valuation_aux'' q hq (m / q) (_ : 0 < m / q) n + 1
m : ℕ
m_le_sM : m ≤ Nat.succ M
hm : 0 < m
n : ℕ
hmq : m % q = 0
meqsM : m = Nat.succ M
h : (m / q % q == 0) = true
⊢ nat_valuation_aux'' q hq (m / q) (_ : 0 < m / q) (n + 1) =
nat_valuation_aux'' q hq (m / q / q) (_ : 0 < m / q / q) (n + 1) + 1 | case succ.inr.inl
R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
q : ℕ
hq : 1 < q
M : ℕ
IH :
∀ (m : ℕ),
m ≤ M →
∀ (hm : 0 < m) (n : ℕ) (hmq : m % q = 0),
nat_valuation_aux'' q hq m hm n = nat_valuation_aux'' q hq (m / q) (_ : 0 < m / q) n + 1
m : ℕ
m_le_sM : m ≤ Nat.succ M
hm : 0 < m
n : ℕ
hmq : m % q = 0
meqsM : m = Nat.succ M
h : (m / q % q == 0) = true
⊢ nat_valuation_aux'' q hq (Nat.succ M / q) (_ : 0 < Nat.succ M / q) (n + 1) =
nat_valuation_aux'' q hq (Nat.succ M / q / q) (_ : 0 < Nat.succ M / q / q) (n + 1) + 1 |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/ValuedRing.lean | nat_valuation_aux''_of_dvd_induction | [549, 1] | [576, 26] | rw [meqsM] at hm h hmq | case succ.inr.inl
R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
q : ℕ
hq : 1 < q
M : ℕ
IH :
∀ (m : ℕ),
m ≤ M →
∀ (hm : 0 < m) (n : ℕ) (hmq : m % q = 0),
nat_valuation_aux'' q hq m hm n = nat_valuation_aux'' q hq (m / q) (_ : 0 < m / q) n + 1
m : ℕ
m_le_sM : m ≤ Nat.succ M
hm : 0 < m
n : ℕ
hmq : m % q = 0
meqsM : m = Nat.succ M
h : (m / q % q == 0) = true
⊢ nat_valuation_aux'' q hq (Nat.succ M / q) (_ : 0 < Nat.succ M / q) (n + 1) =
nat_valuation_aux'' q hq (Nat.succ M / q / q) (_ : 0 < Nat.succ M / q / q) (n + 1) + 1 | case succ.inr.inl
R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
q : ℕ
hq : 1 < q
M : ℕ
IH :
∀ (m : ℕ),
m ≤ M →
∀ (hm : 0 < m) (n : ℕ) (hmq : m % q = 0),
nat_valuation_aux'' q hq m hm n = nat_valuation_aux'' q hq (m / q) (_ : 0 < m / q) n + 1
m : ℕ
m_le_sM : m ≤ Nat.succ M
hm✝ : 0 < m
hm : 0 < Nat.succ M
n : ℕ
hmq✝ : m % q = 0
hmq : Nat.succ M % q = 0
meqsM : m = Nat.succ M
h✝ : (m / q % q == 0) = true
h : (Nat.succ M / q % q == 0) = true
⊢ nat_valuation_aux'' q hq (Nat.succ M / q) (_ : 0 < Nat.succ M / q) (n + 1) =
nat_valuation_aux'' q hq (Nat.succ M / q / q) (_ : 0 < Nat.succ M / q / q) (n + 1) + 1 |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/ValuedRing.lean | nat_valuation_aux''_of_dvd_induction | [549, 1] | [576, 26] | exact IH (M.succ/q) (Nat.le_of_lt_succ (Nat.div_lt_self hm hq))
(Nat.div_pos_of_mod hm hq hmq) (n+1) (by simpa using h) | case succ.inr.inl
R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
q : ℕ
hq : 1 < q
M : ℕ
IH :
∀ (m : ℕ),
m ≤ M →
∀ (hm : 0 < m) (n : ℕ) (hmq : m % q = 0),
nat_valuation_aux'' q hq m hm n = nat_valuation_aux'' q hq (m / q) (_ : 0 < m / q) n + 1
m : ℕ
m_le_sM : m ≤ Nat.succ M
hm✝ : 0 < m
hm : 0 < Nat.succ M
n : ℕ
hmq✝ : m % q = 0
hmq : Nat.succ M % q = 0
meqsM : m = Nat.succ M
h✝ : (m / q % q == 0) = true
h : (Nat.succ M / q % q == 0) = true
⊢ nat_valuation_aux'' q hq (Nat.succ M / q) (_ : 0 < Nat.succ M / q) (n + 1) =
nat_valuation_aux'' q hq (Nat.succ M / q / q) (_ : 0 < Nat.succ M / q / q) (n + 1) + 1 | no goals |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/ValuedRing.lean | nat_valuation_aux''_of_dvd_induction | [549, 1] | [576, 26] | simpa using h | R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
q : ℕ
hq : 1 < q
M : ℕ
IH :
∀ (m : ℕ),
m ≤ M →
∀ (hm : 0 < m) (n : ℕ) (hmq : m % q = 0),
nat_valuation_aux'' q hq m hm n = nat_valuation_aux'' q hq (m / q) (_ : 0 < m / q) n + 1
m : ℕ
m_le_sM : m ≤ Nat.succ M
hm✝ : 0 < m
hm : 0 < Nat.succ M
n : ℕ
hmq✝ : m % q = 0
hmq : Nat.succ M % q = 0
meqsM : m = Nat.succ M
h✝ : (m / q % q == 0) = true
h : (Nat.succ M / q % q == 0) = true
⊢ Nat.succ M / q % q = 0 | no goals |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/ValuedRing.lean | nat_valuation_aux''_of_dvd_induction | [549, 1] | [576, 26] | rw [nat_valuation_aux'', nat_valuation_aux'', dif_neg h, dif_pos, nat_valuation_aux'', dif_neg h] | case succ.inr.inr
R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
q : ℕ
hq : 1 < q
M : ℕ
IH :
∀ (m : ℕ),
m ≤ M →
∀ (hm : 0 < m) (n : ℕ) (hmq : m % q = 0),
nat_valuation_aux'' q hq m hm n = nat_valuation_aux'' q hq (m / q) (_ : 0 < m / q) n + 1
m : ℕ
m_le_sM : m ≤ Nat.succ M
hm : 0 < m
n : ℕ
hmq : m % q = 0
meqsM : m = Nat.succ M
h : ¬(m / q % q == 0) = true
⊢ nat_valuation_aux'' q hq m hm n = nat_valuation_aux'' q hq (m / q) (_ : 0 < m / q) n + 1 | case succ.inr.inr.hc
R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
q : ℕ
hq : 1 < q
M : ℕ
IH :
∀ (m : ℕ),
m ≤ M →
∀ (hm : 0 < m) (n : ℕ) (hmq : m % q = 0),
nat_valuation_aux'' q hq m hm n = nat_valuation_aux'' q hq (m / q) (_ : 0 < m / q) n + 1
m : ℕ
m_le_sM : m ≤ Nat.succ M
hm : 0 < m
n : ℕ
hmq : m % q = 0
meqsM : m = Nat.succ M
h : ¬(m / q % q == 0) = true
⊢ (m % q == 0) = true
case succ.inr.inr.hc
R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
q : ℕ
hq : 1 < q
M : ℕ
IH :
∀ (m : ℕ),
m ≤ M →
∀ (hm : 0 < m) (n : ℕ) (hmq : m % q = 0),
nat_valuation_aux'' q hq m hm n = nat_valuation_aux'' q hq (m / q) (_ : 0 < m / q) n + 1
m : ℕ
m_le_sM : m ≤ Nat.succ M
hm : 0 < m
n : ℕ
hmq : m % q = 0
meqsM : m = Nat.succ M
h : ¬(m / q % q == 0) = true
⊢ (m % q == 0) = true
case succ.inr.inr.hc
R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
q : ℕ
hq : 1 < q
M : ℕ
IH :
∀ (m : ℕ),
m ≤ M →
∀ (hm : 0 < m) (n : ℕ) (hmq : m % q = 0),
nat_valuation_aux'' q hq m hm n = nat_valuation_aux'' q hq (m / q) (_ : 0 < m / q) n + 1
m : ℕ
m_le_sM : m ≤ Nat.succ M
hm : 0 < m
n : ℕ
hmq : m % q = 0
meqsM : m = Nat.succ M
h : ¬(m / q % q == 0) = true
⊢ (m % q == 0) = true |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/ValuedRing.lean | nat_valuation_aux''_of_dvd_induction | [549, 1] | [576, 26] | simp | case succ.inr.inr.hc
R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
q : ℕ
hq : 1 < q
M : ℕ
IH :
∀ (m : ℕ),
m ≤ M →
∀ (hm : 0 < m) (n : ℕ) (hmq : m % q = 0),
nat_valuation_aux'' q hq m hm n = nat_valuation_aux'' q hq (m / q) (_ : 0 < m / q) n + 1
m : ℕ
m_le_sM : m ≤ Nat.succ M
hm : 0 < m
n : ℕ
hmq : m % q = 0
meqsM : m = Nat.succ M
h : ¬(m / q % q == 0) = true
⊢ (m % q == 0) = true
case succ.inr.inr.hc
R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
q : ℕ
hq : 1 < q
M : ℕ
IH :
∀ (m : ℕ),
m ≤ M →
∀ (hm : 0 < m) (n : ℕ) (hmq : m % q = 0),
nat_valuation_aux'' q hq m hm n = nat_valuation_aux'' q hq (m / q) (_ : 0 < m / q) n + 1
m : ℕ
m_le_sM : m ≤ Nat.succ M
hm : 0 < m
n : ℕ
hmq : m % q = 0
meqsM : m = Nat.succ M
h : ¬(m / q % q == 0) = true
⊢ (m % q == 0) = true
case succ.inr.inr.hc
R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
q : ℕ
hq : 1 < q
M : ℕ
IH :
∀ (m : ℕ),
m ≤ M →
∀ (hm : 0 < m) (n : ℕ) (hmq : m % q = 0),
nat_valuation_aux'' q hq m hm n = nat_valuation_aux'' q hq (m / q) (_ : 0 < m / q) n + 1
m : ℕ
m_le_sM : m ≤ Nat.succ M
hm : 0 < m
n : ℕ
hmq : m % q = 0
meqsM : m = Nat.succ M
h : ¬(m / q % q == 0) = true
⊢ (m % q == 0) = true | case succ.inr.inr.hc
R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
q : ℕ
hq : 1 < q
M : ℕ
IH :
∀ (m : ℕ),
m ≤ M →
∀ (hm : 0 < m) (n : ℕ) (hmq : m % q = 0),
nat_valuation_aux'' q hq m hm n = nat_valuation_aux'' q hq (m / q) (_ : 0 < m / q) n + 1
m : ℕ
m_le_sM : m ≤ Nat.succ M
hm : 0 < m
n : ℕ
hmq : m % q = 0
meqsM : m = Nat.succ M
h : ¬(m / q % q == 0) = true
⊢ m % q = 0 |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/ValuedRing.lean | nat_valuation_aux''_of_dvd_induction | [549, 1] | [576, 26] | . simp only [hmq] | case succ.inr.inr.hc
R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
q : ℕ
hq : 1 < q
M : ℕ
IH :
∀ (m : ℕ),
m ≤ M →
∀ (hm : 0 < m) (n : ℕ) (hmq : m % q = 0),
nat_valuation_aux'' q hq m hm n = nat_valuation_aux'' q hq (m / q) (_ : 0 < m / q) n + 1
m : ℕ
m_le_sM : m ≤ Nat.succ M
hm : 0 < m
n : ℕ
hmq : m % q = 0
meqsM : m = Nat.succ M
h : ¬(m / q % q == 0) = true
⊢ m % q = 0 | no goals |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/ValuedRing.lean | nat_valuation_aux''_of_dvd_induction | [549, 1] | [576, 26] | simp only [hmq] | case succ.inr.inr.hc
R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
q : ℕ
hq : 1 < q
M : ℕ
IH :
∀ (m : ℕ),
m ≤ M →
∀ (hm : 0 < m) (n : ℕ) (hmq : m % q = 0),
nat_valuation_aux'' q hq m hm n = nat_valuation_aux'' q hq (m / q) (_ : 0 < m / q) n + 1
m : ℕ
m_le_sM : m ≤ Nat.succ M
hm : 0 < m
n : ℕ
hmq : m % q = 0
meqsM : m = Nat.succ M
h : ¬(m / q % q == 0) = true
⊢ m % q = 0 | no goals |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/ValuedRing.lean | nat_valuation_aux''_of_not_dvd | [582, 1] | [588, 45] | have hmq_bool : ¬m % q == 0 := by
intro H
apply hmq (eq_of_beq H) | R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
q : ℕ
hq : 1 < q
m : ℕ
hm : 0 < m
hmq : m % q ≠ 0
⊢ nat_valuation_aux'' q hq m hm 0 = 0 | R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
q : ℕ
hq : 1 < q
m : ℕ
hm : 0 < m
hmq : m % q ≠ 0
hmq_bool : ¬(m % q == 0) = true
⊢ nat_valuation_aux'' q hq m hm 0 = 0 |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/ValuedRing.lean | nat_valuation_aux''_of_not_dvd | [582, 1] | [588, 45] | rw [nat_valuation_aux'', dif_neg hmq_bool] | R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
q : ℕ
hq : 1 < q
m : ℕ
hm : 0 < m
hmq : m % q ≠ 0
hmq_bool : ¬(m % q == 0) = true
⊢ nat_valuation_aux'' q hq m hm 0 = 0 | no goals |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/ValuedRing.lean | nat_valuation_aux''_of_not_dvd | [582, 1] | [588, 45] | intro H | R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
q : ℕ
hq : 1 < q
m : ℕ
hm : 0 < m
hmq : m % q ≠ 0
⊢ ¬(m % q == 0) = true | R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
q : ℕ
hq : 1 < q
m : ℕ
hm : 0 < m
hmq : m % q ≠ 0
H : (m % q == 0) = true
⊢ False |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/ValuedRing.lean | nat_valuation_aux''_of_not_dvd | [582, 1] | [588, 45] | apply hmq (eq_of_beq H) | R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
q : ℕ
hq : 1 < q
m : ℕ
hm : 0 < m
hmq : m % q ≠ 0
H : (m % q == 0) = true
⊢ False | no goals |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/ValuedRing.lean | nat_valuation_aux'_of_not_dvd | [594, 1] | [598, 54] | rw [nat_valuation_aux'] | R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
q : ℕ
hq : 1 < q
m : ℕ
hm : 0 < m
hmq : m % q ≠ 0
⊢ nat_valuation_aux' q hq m hm = 0 | R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
q : ℕ
hq : 1 < q
m : ℕ
hm : 0 < m
hmq : m % q ≠ 0
⊢ (match m, hm with
| m, hm => ↑(nat_valuation_aux'' q hq m hm 0)) =
0 |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/ValuedRing.lean | nat_valuation_aux'_of_not_dvd | [594, 1] | [598, 54] | simp [nat_valuation_aux''_of_not_dvd q hq m hm hmq] | R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
q : ℕ
hq : 1 < q
m : ℕ
hm : 0 < m
hmq : m % q ≠ 0
⊢ (match m, hm with
| m, hm => ↑(nat_valuation_aux'' q hq m hm 0)) =
0 | no goals |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/ValuedRing.lean | nat_valuation_aux'_of_dvd | [600, 1] | [604, 72] | simp [nat_valuation_aux', nat_valuation_aux''_of_dvd q hq m hm 0 hmq] | R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
q : ℕ
hq : 1 < q
m : ℕ
hm : 0 < m
hmq : m % q = 0
⊢ nat_valuation_aux' q hq m hm = nat_valuation_aux' q hq (m / q) (_ : 0 < m / q) + 1 | no goals |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/ValuedRing.lean | nat_val_aux'_succ | [606, 1] | [617, 50] | simp only [Nat.succ_ne_zero, dite_false, ne_eq, ite_not] | R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
q m : ℕ
hq : 1 < q + 2
⊢ nat_valuation_aux' (q + 2) hq (m + 1) (_ : 0 < Nat.succ m) =
if hmq : (m + 1) % (q + 2) ≠ 0 then 0
else nat_valuation_aux' (q + 2) hq ((m + 1) / (q + 2)) (_ : 0 < (m + 1) / (q + 2)) + 1 | R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
q m : ℕ
hq : 1 < q + 2
⊢ nat_valuation_aux' (q + 2) hq (m + 1) (_ : 0 < Nat.succ m) =
if hmq : ¬(m + 1) % (q + 2) = 0 then 0
else nat_valuation_aux' (q + 2) hq ((m + 1) / (q + 2)) (_ : 0 < (m + 1) / (q + 2)) + 1 |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/ValuedRing.lean | nat_val_aux'_succ | [606, 1] | [617, 50] | cases em ((m + 1) % (q + 2) = 0) with
| inl h =>
rw [dif_neg (not_not_intro h)]
exact nat_valuation_aux'_of_dvd _ _ _ _ h
| inr h =>
rw [dif_pos h]
exact nat_valuation_aux'_of_not_dvd _ _ _ _ h | R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
q m : ℕ
hq : 1 < q + 2
⊢ nat_valuation_aux' (q + 2) hq (m + 1) (_ : 0 < Nat.succ m) =
if hmq : ¬(m + 1) % (q + 2) = 0 then 0
else nat_valuation_aux' (q + 2) hq ((m + 1) / (q + 2)) (_ : 0 < (m + 1) / (q + 2)) + 1 | no goals |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/ValuedRing.lean | nat_val_aux'_succ | [606, 1] | [617, 50] | rw [dif_neg (not_not_intro h)] | case inl
R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
q m : ℕ
hq : 1 < q + 2
h : (m + 1) % (q + 2) = 0
⊢ nat_valuation_aux' (q + 2) hq (m + 1) (_ : 0 < Nat.succ m) =
if hmq : ¬(m + 1) % (q + 2) = 0 then 0
else nat_valuation_aux' (q + 2) hq ((m + 1) / (q + 2)) (_ : 0 < (m + 1) / (q + 2)) + 1 | case inl
R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
q m : ℕ
hq : 1 < q + 2
h : (m + 1) % (q + 2) = 0
⊢ nat_valuation_aux' (q + 2) hq (m + 1) (_ : 0 < Nat.succ m) =
nat_valuation_aux' (q + 2) hq ((m + 1) / (q + 2)) (_ : 0 < (m + 1) / (q + 2)) + 1 |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/ValuedRing.lean | nat_val_aux'_succ | [606, 1] | [617, 50] | exact nat_valuation_aux'_of_dvd _ _ _ _ h | case inl
R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
q m : ℕ
hq : 1 < q + 2
h : (m + 1) % (q + 2) = 0
⊢ nat_valuation_aux' (q + 2) hq (m + 1) (_ : 0 < Nat.succ m) =
nat_valuation_aux' (q + 2) hq ((m + 1) / (q + 2)) (_ : 0 < (m + 1) / (q + 2)) + 1 | no goals |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/ValuedRing.lean | nat_val_aux'_succ | [606, 1] | [617, 50] | rw [dif_pos h] | case inr
R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
q m : ℕ
hq : 1 < q + 2
h : ¬(m + 1) % (q + 2) = 0
⊢ nat_valuation_aux' (q + 2) hq (m + 1) (_ : 0 < Nat.succ m) =
if hmq : ¬(m + 1) % (q + 2) = 0 then 0
else nat_valuation_aux' (q + 2) hq ((m + 1) / (q + 2)) (_ : 0 < (m + 1) / (q + 2)) + 1 | case inr
R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
q m : ℕ
hq : 1 < q + 2
h : ¬(m + 1) % (q + 2) = 0
⊢ nat_valuation_aux' (q + 2) hq (m + 1) (_ : 0 < Nat.succ m) = 0 |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/ValuedRing.lean | nat_val_aux'_succ | [606, 1] | [617, 50] | exact nat_valuation_aux'_of_not_dvd _ _ _ _ h | case inr
R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
q m : ℕ
hq : 1 < q + 2
h : ¬(m + 1) % (q + 2) = 0
⊢ nat_valuation_aux' (q + 2) hq (m + 1) (_ : 0 < Nat.succ m) = 0 | no goals |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/ValuedRing.lean | nat_val_aux_zero | [622, 1] | [624, 27] | simp [nat_valuation_aux] | R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p : ℕ
hp : 1 < p
⊢ nat_valuation_aux p hp 0 = ⊤ | no goals |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/ValuedRing.lean | nat_val_aux_succ | [628, 1] | [638, 61] | simp only [nat_valuation_aux, Nat.succ_ne_zero, dite_false, ne_eq, ite_not] | R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
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 | R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
q m : ℕ
hq : 1 < q + 2
⊢ nat_valuation_aux' (q + 2) hq (m + 1) (_ : 0 < m + 1) =
if (m + 1) % (q + 2) = 0 then
(if hm : (m + 1) / (q + 2) = 0 then ⊤
else nat_valuation_aux' (q + 2) hq ((m + 1) / (q + 2)) (_ : 0 < (m + 1) / (q + 2))) +
1
else 0 |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/ValuedRing.lean | nat_val_aux_succ | [628, 1] | [638, 61] | by_cases hmq : (m + 1) % (q + 2) = 0 | R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
q m : ℕ
hq : 1 < q + 2
⊢ nat_valuation_aux' (q + 2) hq (m + 1) (_ : 0 < m + 1) =
if (m + 1) % (q + 2) = 0 then
(if hm : (m + 1) / (q + 2) = 0 then ⊤
else nat_valuation_aux' (q + 2) hq ((m + 1) / (q + 2)) (_ : 0 < (m + 1) / (q + 2))) +
1
else 0 | case pos
R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
q m : ℕ
hq : 1 < q + 2
hmq : (m + 1) % (q + 2) = 0
⊢ nat_valuation_aux' (q + 2) hq (m + 1) (_ : 0 < m + 1) =
if (m + 1) % (q + 2) = 0 then
(if hm : (m + 1) / (q + 2) = 0 then ⊤
else nat_valuation_aux' (q + 2) hq ((m + 1) / (q + 2)) (_ : 0 < (m + 1) / (q + 2))) +
1
else 0
case neg
R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
q m : ℕ
hq : 1 < q + 2
hmq : ¬(m + 1) % (q + 2) = 0
⊢ nat_valuation_aux' (q + 2) hq (m + 1) (_ : 0 < m + 1) =
if (m + 1) % (q + 2) = 0 then
(if hm : (m + 1) / (q + 2) = 0 then ⊤
else nat_valuation_aux' (q + 2) hq ((m + 1) / (q + 2)) (_ : 0 < (m + 1) / (q + 2))) +
1
else 0 |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/ValuedRing.lean | nat_val_aux_succ | [628, 1] | [638, 61] | . have h : (m + 1) / (q + 2) ≠ 0 := by
apply Nat.ne_of_gt
exact Nat.div_pos (x' hmq) (lt_trans (Nat.lt_succ_self 0) hq)
rw [if_pos hmq, dif_neg h]
exact nat_valuation_aux'_of_dvd (q+2) hq (m+1) _ hmq | case pos
R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
q m : ℕ
hq : 1 < q + 2
hmq : (m + 1) % (q + 2) = 0
⊢ nat_valuation_aux' (q + 2) hq (m + 1) (_ : 0 < m + 1) =
if (m + 1) % (q + 2) = 0 then
(if hm : (m + 1) / (q + 2) = 0 then ⊤
else nat_valuation_aux' (q + 2) hq ((m + 1) / (q + 2)) (_ : 0 < (m + 1) / (q + 2))) +
1
else 0
case neg
R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
q m : ℕ
hq : 1 < q + 2
hmq : ¬(m + 1) % (q + 2) = 0
⊢ nat_valuation_aux' (q + 2) hq (m + 1) (_ : 0 < m + 1) =
if (m + 1) % (q + 2) = 0 then
(if hm : (m + 1) / (q + 2) = 0 then ⊤
else nat_valuation_aux' (q + 2) hq ((m + 1) / (q + 2)) (_ : 0 < (m + 1) / (q + 2))) +
1
else 0 | case neg
R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
q m : ℕ
hq : 1 < q + 2
hmq : ¬(m + 1) % (q + 2) = 0
⊢ nat_valuation_aux' (q + 2) hq (m + 1) (_ : 0 < m + 1) =
if (m + 1) % (q + 2) = 0 then
(if hm : (m + 1) / (q + 2) = 0 then ⊤
else nat_valuation_aux' (q + 2) hq ((m + 1) / (q + 2)) (_ : 0 < (m + 1) / (q + 2))) +
1
else 0 |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/ValuedRing.lean | nat_val_aux_succ | [628, 1] | [638, 61] | . rw [if_neg hmq]
exact nat_valuation_aux'_of_not_dvd (q+2) hq (m+1) _ hmq | case neg
R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
q m : ℕ
hq : 1 < q + 2
hmq : ¬(m + 1) % (q + 2) = 0
⊢ nat_valuation_aux' (q + 2) hq (m + 1) (_ : 0 < m + 1) =
if (m + 1) % (q + 2) = 0 then
(if hm : (m + 1) / (q + 2) = 0 then ⊤
else nat_valuation_aux' (q + 2) hq ((m + 1) / (q + 2)) (_ : 0 < (m + 1) / (q + 2))) +
1
else 0 | no goals |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/ValuedRing.lean | nat_val_aux_succ | [628, 1] | [638, 61] | have h : (m + 1) / (q + 2) ≠ 0 := by
apply Nat.ne_of_gt
exact Nat.div_pos (x' hmq) (lt_trans (Nat.lt_succ_self 0) hq) | case pos
R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
q m : ℕ
hq : 1 < q + 2
hmq : (m + 1) % (q + 2) = 0
⊢ nat_valuation_aux' (q + 2) hq (m + 1) (_ : 0 < m + 1) =
if (m + 1) % (q + 2) = 0 then
(if hm : (m + 1) / (q + 2) = 0 then ⊤
else nat_valuation_aux' (q + 2) hq ((m + 1) / (q + 2)) (_ : 0 < (m + 1) / (q + 2))) +
1
else 0 | case pos
R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
q m : ℕ
hq : 1 < q + 2
hmq : (m + 1) % (q + 2) = 0
h : (m + 1) / (q + 2) ≠ 0
⊢ nat_valuation_aux' (q + 2) hq (m + 1) (_ : 0 < m + 1) =
if (m + 1) % (q + 2) = 0 then
(if hm : (m + 1) / (q + 2) = 0 then ⊤
else nat_valuation_aux' (q + 2) hq ((m + 1) / (q + 2)) (_ : 0 < (m + 1) / (q + 2))) +
1
else 0 |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/ValuedRing.lean | nat_val_aux_succ | [628, 1] | [638, 61] | rw [if_pos hmq, dif_neg h] | case pos
R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
q m : ℕ
hq : 1 < q + 2
hmq : (m + 1) % (q + 2) = 0
h : (m + 1) / (q + 2) ≠ 0
⊢ nat_valuation_aux' (q + 2) hq (m + 1) (_ : 0 < m + 1) =
if (m + 1) % (q + 2) = 0 then
(if hm : (m + 1) / (q + 2) = 0 then ⊤
else nat_valuation_aux' (q + 2) hq ((m + 1) / (q + 2)) (_ : 0 < (m + 1) / (q + 2))) +
1
else 0 | case pos
R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
q m : ℕ
hq : 1 < q + 2
hmq : (m + 1) % (q + 2) = 0
h : (m + 1) / (q + 2) ≠ 0
⊢ nat_valuation_aux' (q + 2) hq (m + 1) (_ : 0 < m + 1) =
nat_valuation_aux' (q + 2) hq ((m + 1) / (q + 2)) (_ : 0 < (m + 1) / (q + 2)) + 1 |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/ValuedRing.lean | nat_val_aux_succ | [628, 1] | [638, 61] | exact nat_valuation_aux'_of_dvd (q+2) hq (m+1) _ hmq | case pos
R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
q m : ℕ
hq : 1 < q + 2
hmq : (m + 1) % (q + 2) = 0
h : (m + 1) / (q + 2) ≠ 0
⊢ nat_valuation_aux' (q + 2) hq (m + 1) (_ : 0 < m + 1) =
nat_valuation_aux' (q + 2) hq ((m + 1) / (q + 2)) (_ : 0 < (m + 1) / (q + 2)) + 1 | no goals |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/ValuedRing.lean | nat_val_aux_succ | [628, 1] | [638, 61] | apply Nat.ne_of_gt | R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
q m : ℕ
hq : 1 < q + 2
hmq : (m + 1) % (q + 2) = 0
⊢ (m + 1) / (q + 2) ≠ 0 | case h
R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
q m : ℕ
hq : 1 < q + 2
hmq : (m + 1) % (q + 2) = 0
⊢ 0 < (m + 1) / (q + 2) |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/ValuedRing.lean | nat_val_aux_succ | [628, 1] | [638, 61] | exact Nat.div_pos (x' hmq) (lt_trans (Nat.lt_succ_self 0) hq) | case h
R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
q m : ℕ
hq : 1 < q + 2
hmq : (m + 1) % (q + 2) = 0
⊢ 0 < (m + 1) / (q + 2) | no goals |
https://github.com/KisaraBlue/ec-tate-lean.git | b9d36a5b70bb0958bf9741ae6216a43b35c87ed4 | ECTate/Algebra/ValuedRing.lean | nat_val_aux_succ | [628, 1] | [638, 61] | rw [if_neg hmq] | case neg
R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
q m : ℕ
hq : 1 < q + 2
hmq : ¬(m + 1) % (q + 2) = 0
⊢ nat_valuation_aux' (q + 2) hq (m + 1) (_ : 0 < m + 1) =
if (m + 1) % (q + 2) = 0 then
(if hm : (m + 1) / (q + 2) = 0 then ⊤
else nat_valuation_aux' (q + 2) hq ((m + 1) / (q + 2)) (_ : 0 < (m + 1) / (q + 2))) +
1
else 0 | case neg
R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
q m : ℕ
hq : 1 < q + 2
hmq : ¬(m + 1) % (q + 2) = 0
⊢ nat_valuation_aux' (q + 2) hq (m + 1) (_ : 0 < m + 1) = 0 |
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