url stringclasses 147
values | commit stringclasses 147
values | file_path stringlengths 7 101 | full_name stringlengths 1 94 | start stringlengths 6 10 | end stringlengths 6 11 | tactic stringlengths 1 11.2k | state_before stringlengths 3 2.09M | state_after stringlengths 6 2.09M | input stringlengths 73 2.09M |
|---|---|---|---|---|---|---|---|---|---|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.Polynomial.inv_C_eq_C_inv | [490, 1] | [511, 31] | simp only [Ring.inverse, dif_neg ha, map_zero] | case neg
A : Type ?u.83499
inst✝¹ : CommRing A
I : Ideal A
hI : DividedPowers I
R : Type u_1
inst✝ : CommSemiring R
a : R
ha : ¬IsUnit a
⊢ (if h : IsUnit (C a) then ↑h.unit⁻¹ else 0) = C (if h : IsUnit a then ↑h.unit⁻¹ else 0) | case neg
A : Type ?u.83499
inst✝¹ : CommRing A
I : Ideal A
hI : DividedPowers I
R : Type u_1
inst✝ : CommSemiring R
a : R
ha : ¬IsUnit a
⊢ (if h : IsUnit (C a) then ↑h.unit⁻¹ else 0) = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
A : Type ?u.83499
inst✝¹ : CommRing A
I : Ideal A
hI : DividedPowers I
R : Type u_1
inst✝ : CommSemiring R
a : R
ha : ¬IsUnit a
⊢ (if h : IsUnit (C a) then ↑h.unit⁻¹ else 0) = C (if h : IsUnit a then ↑h.unit⁻¹ else 0)
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.Polynomial.inv_C_eq_C_inv | [490, 1] | [511, 31] | rw [dif_neg] | case neg
A : Type ?u.83499
inst✝¹ : CommRing A
I : Ideal A
hI : DividedPowers I
R : Type u_1
inst✝ : CommSemiring R
a : R
ha : ¬IsUnit a
⊢ (if h : IsUnit (C a) then ↑h.unit⁻¹ else 0) = 0 | case neg.hnc
A : Type ?u.83499
inst✝¹ : CommRing A
I : Ideal A
hI : DividedPowers I
R : Type u_1
inst✝ : CommSemiring R
a : R
ha : ¬IsUnit a
⊢ ¬IsUnit (C a) | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
A : Type ?u.83499
inst✝¹ : CommRing A
I : Ideal A
hI : DividedPowers I
R : Type u_1
inst✝ : CommSemiring R
a : R
ha : ¬IsUnit a
⊢ (if h : IsUnit (C a) then ↑h.unit⁻¹ else 0) = 0
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.Polynomial.inv_C_eq_C_inv | [490, 1] | [511, 31] | intro hCa | case neg.hnc
A : Type ?u.83499
inst✝¹ : CommRing A
I : Ideal A
hI : DividedPowers I
R : Type u_1
inst✝ : CommSemiring R
a : R
ha : ¬IsUnit a
⊢ ¬IsUnit (C a) | case neg.hnc
A : Type ?u.83499
inst✝¹ : CommRing A
I : Ideal A
hI : DividedPowers I
R : Type u_1
inst✝ : CommSemiring R
a : R
ha : ¬IsUnit a
hCa : IsUnit (C a)
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.hnc
A : Type ?u.83499
inst✝¹ : CommRing A
I : Ideal A
hI : DividedPowers I
R : Type u_1
inst✝ : CommSemiring R
a : R
ha : ¬IsUnit a
⊢ ¬IsUnit (C a)
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.Polynomial.inv_C_eq_C_inv | [490, 1] | [511, 31] | apply ha | case neg.hnc
A : Type ?u.83499
inst✝¹ : CommRing A
I : Ideal A
hI : DividedPowers I
R : Type u_1
inst✝ : CommSemiring R
a : R
ha : ¬IsUnit a
hCa : IsUnit (C a)
⊢ False | case neg.hnc
A : Type ?u.83499
inst✝¹ : CommRing A
I : Ideal A
hI : DividedPowers I
R : Type u_1
inst✝ : CommSemiring R
a : R
ha : ¬IsUnit a
hCa : IsUnit (C a)
⊢ IsUnit a | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.hnc
A : Type ?u.83499
inst✝¹ : CommRing A
I : Ideal A
hI : DividedPowers I
R : Type u_1
inst✝ : CommSemiring R
a : R
ha : ¬IsUnit a
hCa : IsUnit (C a)
⊢ False
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.Polynomial.inv_C_eq_C_inv | [490, 1] | [511, 31] | rw [isUnit_iff_exists_inv] at hCa ⊢ | case neg.hnc
A : Type ?u.83499
inst✝¹ : CommRing A
I : Ideal A
hI : DividedPowers I
R : Type u_1
inst✝ : CommSemiring R
a : R
ha : ¬IsUnit a
hCa : IsUnit (C a)
⊢ IsUnit a | case neg.hnc
A : Type ?u.83499
inst✝¹ : CommRing A
I : Ideal A
hI : DividedPowers I
R : Type u_1
inst✝ : CommSemiring R
a : R
ha : ¬IsUnit a
hCa : ∃ b, C a * b = 1
⊢ ∃ b, a * b = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.hnc
A : Type ?u.83499
inst✝¹ : CommRing A
I : Ideal A
hI : DividedPowers I
R : Type u_1
inst✝ : CommSemiring R
a : R
ha : ¬IsUnit a
hCa : IsUnit (C a)
⊢ IsUnit a
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.Polynomial.inv_C_eq_C_inv | [490, 1] | [511, 31] | obtain ⟨b, hb⟩ := hCa | case neg.hnc
A : Type ?u.83499
inst✝¹ : CommRing A
I : Ideal A
hI : DividedPowers I
R : Type u_1
inst✝ : CommSemiring R
a : R
ha : ¬IsUnit a
hCa : ∃ b, C a * b = 1
⊢ ∃ b, a * b = 1 | case neg.hnc.intro
A : Type ?u.83499
inst✝¹ : CommRing A
I : Ideal A
hI : DividedPowers I
R : Type u_1
inst✝ : CommSemiring R
a : R
ha : ¬IsUnit a
b : R[X]
hb : C a * b = 1
⊢ ∃ b, a * b = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.hnc
A : Type ?u.83499
inst✝¹ : CommRing A
I : Ideal A
hI : DividedPowers I
R : Type u_1
inst✝ : CommSemiring R
a : R
ha : ¬IsUnit a
hCa : ∃ b, C a * b = 1
⊢ ∃ b, a * b = 1
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.Polynomial.inv_C_eq_C_inv | [490, 1] | [511, 31] | use b.coeff 0 | case neg.hnc.intro
A : Type ?u.83499
inst✝¹ : CommRing A
I : Ideal A
hI : DividedPowers I
R : Type u_1
inst✝ : CommSemiring R
a : R
ha : ¬IsUnit a
b : R[X]
hb : C a * b = 1
⊢ ∃ b, a * b = 1 | case h
A : Type ?u.83499
inst✝¹ : CommRing A
I : Ideal A
hI : DividedPowers I
R : Type u_1
inst✝ : CommSemiring R
a : R
ha : ¬IsUnit a
b : R[X]
hb : C a * b = 1
⊢ a * b.coeff 0 = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.hnc.intro
A : Type ?u.83499
inst✝¹ : CommRing A
I : Ideal A
hI : DividedPowers I
R : Type u_1
inst✝ : CommSemiring R
a : R
ha : ¬IsUnit a
b : R[X]
hb : C a * b = 1
⊢ ∃ b, a * b = 1
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.Polynomial.inv_C_eq_C_inv | [490, 1] | [511, 31] | convert congr_arg₂ coeff hb rfl | case h
A : Type ?u.83499
inst✝¹ : CommRing A
I : Ideal A
hI : DividedPowers I
R : Type u_1
inst✝ : CommSemiring R
a : R
ha : ¬IsUnit a
b : R[X]
hb : C a * b = 1
⊢ a * b.coeff 0 = 1 | case h.e'_2
A : Type ?u.83499
inst✝¹ : CommRing A
I : Ideal A
hI : DividedPowers I
R : Type u_1
inst✝ : CommSemiring R
a : R
ha : ¬IsUnit a
b : R[X]
hb : C a * b = 1
⊢ a * b.coeff 0 = (C a * b).coeff ?h
case h.e'_3
A : Type ?u.83499
inst✝¹ : CommRing A
I : Ideal A
hI : DividedPowers I
R : Type u_1
inst✝ : CommSemiring... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
A : Type ?u.83499
inst✝¹ : CommRing A
I : Ideal A
hI : DividedPowers I
R : Type u_1
inst✝ : CommSemiring R
a : R
ha : ¬IsUnit a
b : R[X]
hb : C a * b = 1
⊢ a * b.coeff 0 = 1
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.Polynomial.inv_C_eq_C_inv | [490, 1] | [511, 31] | rw [Polynomial.coeff_C_mul] | case h.e'_2
A : Type ?u.83499
inst✝¹ : CommRing A
I : Ideal A
hI : DividedPowers I
R : Type u_1
inst✝ : CommSemiring R
a : R
ha : ¬IsUnit a
b : R[X]
hb : C a * b = 1
⊢ a * b.coeff 0 = (C a * b).coeff ?h
case h.e'_3
A : Type ?u.83499
inst✝¹ : CommRing A
I : Ideal A
hI : DividedPowers I
R : Type u_1
inst✝ : CommSemiring... | case h.e'_3
A : Type ?u.83499
inst✝¹ : CommRing A
I : Ideal A
hI : DividedPowers I
R : Type u_1
inst✝ : CommSemiring R
a : R
ha : ¬IsUnit a
b : R[X]
hb : C a * b = 1
⊢ 1 = coeff 1 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_2
A : Type ?u.83499
inst✝¹ : CommRing A
I : Ideal A
hI : DividedPowers I
R : Type u_1
inst✝ : CommSemiring R
a : R
ha : ¬IsUnit a
b : R[X]
hb : C a * b = 1
⊢ a * b.coeff 0 = (C a * b).coeff ?h
case h.e'_3
A : Type ?u.83499
inst✝¹ : CommRing A
I : I... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.Polynomial.inv_C_eq_C_inv | [490, 1] | [511, 31] | simp only [coeff_one_zero] | case h.e'_3
A : Type ?u.83499
inst✝¹ : CommRing A
I : Ideal A
hI : DividedPowers I
R : Type u_1
inst✝ : CommSemiring R
a : R
ha : ¬IsUnit a
b : R[X]
hb : C a * b = 1
⊢ 1 = coeff 1 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_3
A : Type ?u.83499
inst✝¹ : CommRing A
I : Ideal A
hI : DividedPowers I
R : Type u_1
inst✝ : CommSemiring R
a : R
ha : ¬IsUnit a
b : R[X]
hb : C a * b = 1
⊢ 1 = coeff 1 0
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_comp_coeffs | [517, 1] | [585, 18] | rw [← mul_left_inj' (pos_iff_ne_zero.mp (Nat.choose_pos hp))] | A : Type ?u.93141
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
m n p : ℕ
hn : n ≠ 0
hp : p ≤ m * n
⊢ mchoose m n =
∑ x ∈ filter (fun l => ∑ i ∈ range (n + 1), Multiset.count i ↑l * i = p) ((range (n + 1)).sym m),
(∏ i ∈ range (n + 1), DividedPowers.IdealAdd.cnik n i ↑x) *
((Nat.multinomial (ran... | A : Type ?u.93141
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
m n p : ℕ
hn : n ≠ 0
hp : p ≤ m * n
⊢ mchoose m n * (m * n).choose p =
(∑ x ∈ filter (fun l => ∑ i ∈ range (n + 1), Multiset.count i ↑l * i = p) ((range (n + 1)).sym m),
(∏ i ∈ range (n + 1), DividedPowers.IdealAdd.cnik n i ↑x) *
... | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type ?u.93141
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
m n p : ℕ
hn : n ≠ 0
hp : p ≤ m * n
⊢ mchoose m n =
∑ x ∈ filter (fun l => ∑ i ∈ range (n + 1), Multiset.count i ↑l * i = p) ((range (n + 1)).sym m),
(∏ i ∈ range (n + 1), Divided... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_comp_coeffs | [517, 1] | [585, 18] | apply @Nat.cast_injective ℚ | A : Type ?u.93141
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
m n p : ℕ
hn : n ≠ 0
hp : p ≤ m * n
⊢ mchoose m n * (m * n).choose p =
(∑ x ∈ filter (fun l => ∑ i ∈ range (n + 1), Multiset.count i ↑l * i = p) ((range (n + 1)).sym m),
(∏ i ∈ range (n + 1), DividedPowers.IdealAdd.cnik n i ↑x) *
... | case a
A : Type ?u.93141
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
m n p : ℕ
hn : n ≠ 0
hp : p ≤ m * n
⊢ ↑(mchoose m n * (m * n).choose p) =
↑((∑ x ∈ filter (fun l => ∑ i ∈ range (n + 1), Multiset.count i ↑l * i = p) ((range (n + 1)).sym m),
(∏ i ∈ range (n + 1), DividedPowers.IdealAdd.cnik n i ... | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type ?u.93141
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
m n p : ℕ
hn : n ≠ 0
hp : p ≤ m * n
⊢ mchoose m n * (m * n).choose p =
(∑ x ∈ filter (fun l => ∑ i ∈ range (n + 1), Multiset.count i ↑l * i = p) ((range (n + 1)).sym m),
(∏ i ∈ ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_comp_coeffs | [517, 1] | [585, 18] | simp only [Sym.mem_coe, mem_sym_iff, mem_range, ge_iff_le,
Nat.cast_sum, Nat.cast_mul, Nat.cast_prod, Nat.cast_eq_zero] | case a
A : Type ?u.93141
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
m n p : ℕ
hn : n ≠ 0
hp : p ≤ m * n
⊢ ↑(mchoose m n * (m * n).choose p) =
↑((∑ x ∈ filter (fun l => ∑ i ∈ range (n + 1), Multiset.count i ↑l * i = p) ((range (n + 1)).sym m),
(∏ i ∈ range (n + 1), DividedPowers.IdealAdd.cnik n i ... | case a
A : Type ?u.93141
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
m n p : ℕ
hn : n ≠ 0
hp : p ≤ m * n
⊢ ↑(mchoose m n) * ↑((m * n).choose p) =
(∑ x ∈ filter (fun l => ∑ i ∈ range (n + 1), Multiset.count i ↑l * i = p) ((range (n + 1)).sym m),
(∏ i ∈ range (n + 1), ↑(DividedPowers.IdealAdd.cnik n i... | Please generate a tactic in lean4 to solve the state.
STATE:
case a
A : Type ?u.93141
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
m n p : ℕ
hn : n ≠ 0
hp : p ≤ m * n
⊢ ↑(mchoose m n * (m * n).choose p) =
↑((∑ x ∈ filter (fun l => ∑ i ∈ range (n + 1), Multiset.count i ↑l * i = p) ((range (n + 1)).sym m),
... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_comp_coeffs | [517, 1] | [585, 18] | conv_lhs => rw [← Polynomial.coeff_X_add_one_pow ℚ (m * n) p] | case a
A : Type ?u.93141
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
m n p : ℕ
hn : n ≠ 0
hp : p ≤ m * n
⊢ ↑(mchoose m n) * ↑((m * n).choose p) =
(∑ x ∈ filter (fun l => ∑ i ∈ range (n + 1), Multiset.count i ↑l * i = p) ((range (n + 1)).sym m),
(∏ i ∈ range (n + 1), ↑(DividedPowers.IdealAdd.cnik n i... | case a
A : Type ?u.93141
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
m n p : ℕ
hn : n ≠ 0
hp : p ≤ m * n
⊢ ↑(mchoose m n) * ((X + 1) ^ (m * n)).coeff p =
(∑ x ∈ filter (fun l => ∑ i ∈ range (n + 1), Multiset.count i ↑l * i = p) ((range (n + 1)).sym m),
(∏ i ∈ range (n + 1), ↑(DividedPowers.IdealAdd.... | Please generate a tactic in lean4 to solve the state.
STATE:
case a
A : Type ?u.93141
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
m n p : ℕ
hn : n ≠ 0
hp : p ≤ m * n
⊢ ↑(mchoose m n) * ↑((m * n).choose p) =
(∑ x ∈ filter (fun l => ∑ i ∈ range (n + 1), Multiset.count i ↑l * i = p) ((range (n + 1)).sym m),
... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_comp_coeffs | [517, 1] | [585, 18] | let A := ℚ[X] | case a
A : Type ?u.93141
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
m n p : ℕ
hn : n ≠ 0
hp : p ≤ m * n
⊢ ↑(mchoose m n) * ((X + 1) ^ (m * n)).coeff p =
(∑ x ∈ filter (fun l => ∑ i ∈ range (n + 1), Multiset.count i ↑l * i = p) ((range (n + 1)).sym m),
(∏ i ∈ range (n + 1), ↑(DividedPowers.IdealAdd.... | case a
A✝ : Type ?u.93141
inst✝ : CommRing A✝
I : Ideal A✝
hI : DividedPowers I
m n p : ℕ
hn : n ≠ 0
hp : p ≤ m * n
A : Type := ℚ[X]
⊢ ↑(mchoose m n) * ((X + 1) ^ (m * n)).coeff p =
(∑ x ∈ filter (fun l => ∑ i ∈ range (n + 1), Multiset.count i ↑l * i = p) ((range (n + 1)).sym m),
(∏ i ∈ range (n + 1), ↑(Div... | Please generate a tactic in lean4 to solve the state.
STATE:
case a
A : Type ?u.93141
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
m n p : ℕ
hn : n ≠ 0
hp : p ≤ m * n
⊢ ↑(mchoose m n) * ((X + 1) ^ (m * n)).coeff p =
(∑ x ∈ filter (fun l => ∑ i ∈ range (n + 1), Multiset.count i ↑l * i = p) ((range (n + 1)).sy... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_comp_coeffs | [517, 1] | [585, 18] | let I : Ideal A := ⊤ | case a
A✝ : Type ?u.93141
inst✝ : CommRing A✝
I : Ideal A✝
hI : DividedPowers I
m n p : ℕ
hn : n ≠ 0
hp : p ≤ m * n
A : Type := ℚ[X]
⊢ ↑(mchoose m n) * ((X + 1) ^ (m * n)).coeff p =
(∑ x ∈ filter (fun l => ∑ i ∈ range (n + 1), Multiset.count i ↑l * i = p) ((range (n + 1)).sym m),
(∏ i ∈ range (n + 1), ↑(Div... | case a
A✝ : Type ?u.93141
inst✝ : CommRing A✝
I✝ : Ideal A✝
hI : DividedPowers I✝
m n p : ℕ
hn : n ≠ 0
hp : p ≤ m * n
A : Type := ℚ[X]
I : Ideal A := ⊤
⊢ ↑(mchoose m n) * ((X + 1) ^ (m * n)).coeff p =
(∑ x ∈ filter (fun l => ∑ i ∈ range (n + 1), Multiset.count i ↑l * i = p) ((range (n + 1)).sym m),
(∏ i ∈ r... | Please generate a tactic in lean4 to solve the state.
STATE:
case a
A✝ : Type ?u.93141
inst✝ : CommRing A✝
I : Ideal A✝
hI : DividedPowers I
m n p : ℕ
hn : n ≠ 0
hp : p ≤ m * n
A : Type := ℚ[X]
⊢ ↑(mchoose m n) * ((X + 1) ^ (m * n)).coeff p =
(∑ x ∈ filter (fun l => ∑ i ∈ range (n + 1), Multiset.count i ↑l * i = p)... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_comp_coeffs | [517, 1] | [585, 18] | let hI : DividedPowers I := RatAlgebra.dividedPowers ⊤ | case a
A✝ : Type ?u.93141
inst✝ : CommRing A✝
I✝ : Ideal A✝
hI : DividedPowers I✝
m n p : ℕ
hn : n ≠ 0
hp : p ≤ m * n
A : Type := ℚ[X]
I : Ideal A := ⊤
⊢ ↑(mchoose m n) * ((X + 1) ^ (m * n)).coeff p =
(∑ x ∈ filter (fun l => ∑ i ∈ range (n + 1), Multiset.count i ↑l * i = p) ((range (n + 1)).sym m),
(∏ i ∈ r... | case a
A✝ : Type ?u.93141
inst✝ : CommRing A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
m n p : ℕ
hn : n ≠ 0
hp : p ≤ m * n
A : Type := ℚ[X]
I : Ideal A := ⊤
hI : DividedPowers I := RatAlgebra.dividedPowers ⊤
⊢ ↑(mchoose m n) * ((X + 1) ^ (m * n)).coeff p =
(∑ x ∈ filter (fun l => ∑ i ∈ range (n + 1), Multiset.count i ↑... | Please generate a tactic in lean4 to solve the state.
STATE:
case a
A✝ : Type ?u.93141
inst✝ : CommRing A✝
I✝ : Ideal A✝
hI : DividedPowers I✝
m n p : ℕ
hn : n ≠ 0
hp : p ≤ m * n
A : Type := ℚ[X]
I : Ideal A := ⊤
⊢ ↑(mchoose m n) * ((X + 1) ^ (m * n)).coeff p =
(∑ x ∈ filter (fun l => ∑ i ∈ range (n + 1), Multiset.... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_comp_coeffs | [517, 1] | [585, 18] | let hII : ∀ (n : ℕ) (a : A), a ∈ I ⊓ I → hI.dpow n a = hI.dpow n a := fun n a _ => rfl | case a
A✝ : Type ?u.93141
inst✝ : CommRing A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
m n p : ℕ
hn : n ≠ 0
hp : p ≤ m * n
A : Type := ℚ[X]
I : Ideal A := ⊤
hI : DividedPowers I := RatAlgebra.dividedPowers ⊤
⊢ ↑(mchoose m n) * ((X + 1) ^ (m * n)).coeff p =
(∑ x ∈ filter (fun l => ∑ i ∈ range (n + 1), Multiset.count i ↑... | case a
A✝ : Type ?u.93141
inst✝ : CommRing A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
m n p : ℕ
hn : n ≠ 0
hp : p ≤ m * n
A : Type := ℚ[X]
I : Ideal A := ⊤
hI : DividedPowers I := RatAlgebra.dividedPowers ⊤
hII : ∀ (n : ℕ), ∀ a ∈ I ⊓ I, hI.dpow n a = hI.dpow n a := fun n a x => rfl
⊢ ↑(mchoose m n) * ((X + 1) ^ (m * n)).c... | Please generate a tactic in lean4 to solve the state.
STATE:
case a
A✝ : Type ?u.93141
inst✝ : CommRing A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
m n p : ℕ
hn : n ≠ 0
hp : p ≤ m * n
A : Type := ℚ[X]
I : Ideal A := ⊤
hI : DividedPowers I := RatAlgebra.dividedPowers ⊤
⊢ ↑(mchoose m n) * ((X + 1) ^ (m * n)).coeff p =
(∑... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_comp_coeffs | [517, 1] | [585, 18] | let h1 : (1 : A) ∈ I := Submodule.mem_top | case a
A✝ : Type ?u.93141
inst✝ : CommRing A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
m n p : ℕ
hn : n ≠ 0
hp : p ≤ m * n
A : Type := ℚ[X]
I : Ideal A := ⊤
hI : DividedPowers I := RatAlgebra.dividedPowers ⊤
hII : ∀ (n : ℕ), ∀ a ∈ I ⊓ I, hI.dpow n a = hI.dpow n a := fun n a x => rfl
⊢ ↑(mchoose m n) * ((X + 1) ^ (m * n)).c... | case a
A✝ : Type ?u.93141
inst✝ : CommRing A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
m n p : ℕ
hn : n ≠ 0
hp : p ≤ m * n
A : Type := ℚ[X]
I : Ideal A := ⊤
hI : DividedPowers I := RatAlgebra.dividedPowers ⊤
hII : ∀ (n : ℕ), ∀ a ∈ I ⊓ I, hI.dpow n a = hI.dpow n a := fun n a x => rfl
h1 : 1 ∈ I := Submodule.mem_top
⊢ ↑(mcho... | Please generate a tactic in lean4 to solve the state.
STATE:
case a
A✝ : Type ?u.93141
inst✝ : CommRing A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
m n p : ℕ
hn : n ≠ 0
hp : p ≤ m * n
A : Type := ℚ[X]
I : Ideal A := ⊤
hI : DividedPowers I := RatAlgebra.dividedPowers ⊤
hII : ∀ (n : ℕ), ∀ a ∈ I ⊓ I, hI.dpow n a = hI.dpow n a... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_comp_coeffs | [517, 1] | [585, 18] | let hX : X ∈ I := Submodule.mem_top | case a
A✝ : Type ?u.93141
inst✝ : CommRing A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
m n p : ℕ
hn : n ≠ 0
hp : p ≤ m * n
A : Type := ℚ[X]
I : Ideal A := ⊤
hI : DividedPowers I := RatAlgebra.dividedPowers ⊤
hII : ∀ (n : ℕ), ∀ a ∈ I ⊓ I, hI.dpow n a = hI.dpow n a := fun n a x => rfl
h1 : 1 ∈ I := Submodule.mem_top
⊢ ↑(mcho... | case a
A✝ : Type ?u.93141
inst✝ : CommRing A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
m n p : ℕ
hn : n ≠ 0
hp : p ≤ m * n
A : Type := ℚ[X]
I : Ideal A := ⊤
hI : DividedPowers I := RatAlgebra.dividedPowers ⊤
hII : ∀ (n : ℕ), ∀ a ∈ I ⊓ I, hI.dpow n a = hI.dpow n a := fun n a x => rfl
h1 : 1 ∈ I := Submodule.mem_top
hX : X ∈... | Please generate a tactic in lean4 to solve the state.
STATE:
case a
A✝ : Type ?u.93141
inst✝ : CommRing A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
m n p : ℕ
hn : n ≠ 0
hp : p ≤ m * n
A : Type := ℚ[X]
I : Ideal A := ⊤
hI : DividedPowers I := RatAlgebra.dividedPowers ⊤
hII : ∀ (n : ℕ), ∀ a ∈ I ⊓ I, hI.dpow n a = hI.dpow n a... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_comp_coeffs | [517, 1] | [585, 18] | rw [← hI.factorial_mul_dpow_eq_pow (m * n) (X + 1) Submodule.mem_top] | case a
A✝ : Type ?u.93141
inst✝ : CommRing A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
m n p : ℕ
hn : n ≠ 0
hp : p ≤ m * n
A : Type := ℚ[X]
I : Ideal A := ⊤
hI : DividedPowers I := RatAlgebra.dividedPowers ⊤
hII : ∀ (n : ℕ), ∀ a ∈ I ⊓ I, hI.dpow n a = hI.dpow n a := fun n a x => rfl
h1 : 1 ∈ I := Submodule.mem_top
hX : X ∈... | case a
A✝ : Type ?u.93141
inst✝ : CommRing A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
m n p : ℕ
hn : n ≠ 0
hp : p ≤ m * n
A : Type := ℚ[X]
I : Ideal A := ⊤
hI : DividedPowers I := RatAlgebra.dividedPowers ⊤
hII : ∀ (n : ℕ), ∀ a ∈ I ⊓ I, hI.dpow n a = hI.dpow n a := fun n a x => rfl
h1 : 1 ∈ I := Submodule.mem_top
hX : X ∈... | Please generate a tactic in lean4 to solve the state.
STATE:
case a
A✝ : Type ?u.93141
inst✝ : CommRing A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
m n p : ℕ
hn : n ≠ 0
hp : p ≤ m * n
A : Type := ℚ[X]
I : Ideal A := ⊤
hI : DividedPowers I := RatAlgebra.dividedPowers ⊤
hII : ∀ (n : ℕ), ∀ a ∈ I ⊓ I, hI.dpow n a = hI.dpow n a... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_comp_coeffs | [517, 1] | [585, 18] | rw [← Polynomial.coeff_C_mul] | case a
A✝ : Type ?u.93141
inst✝ : CommRing A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
m n p : ℕ
hn : n ≠ 0
hp : p ≤ m * n
A : Type := ℚ[X]
I : Ideal A := ⊤
hI : DividedPowers I := RatAlgebra.dividedPowers ⊤
hII : ∀ (n : ℕ), ∀ a ∈ I ⊓ I, hI.dpow n a = hI.dpow n a := fun n a x => rfl
h1 : 1 ∈ I := Submodule.mem_top
hX : X ∈... | case a
A✝ : Type ?u.93141
inst✝ : CommRing A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
m n p : ℕ
hn : n ≠ 0
hp : p ≤ m * n
A : Type := ℚ[X]
I : Ideal A := ⊤
hI : DividedPowers I := RatAlgebra.dividedPowers ⊤
hII : ∀ (n : ℕ), ∀ a ∈ I ⊓ I, hI.dpow n a = hI.dpow n a := fun n a x => rfl
h1 : 1 ∈ I := Submodule.mem_top
hX : X ∈... | Please generate a tactic in lean4 to solve the state.
STATE:
case a
A✝ : Type ?u.93141
inst✝ : CommRing A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
m n p : ℕ
hn : n ≠ 0
hp : p ≤ m * n
A : Type := ℚ[X]
I : Ideal A := ⊤
hI : DividedPowers I := RatAlgebra.dividedPowers ⊤
hII : ∀ (n : ℕ), ∀ a ∈ I ⊓ I, hI.dpow n a = hI.dpow n a... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_comp_coeffs | [517, 1] | [585, 18] | rw [← mul_assoc, mul_comm (C ((mchoose m n) : ℚ)), mul_assoc] | case a
A✝ : Type ?u.93141
inst✝ : CommRing A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
m n p : ℕ
hn : n ≠ 0
hp : p ≤ m * n
A : Type := ℚ[X]
I : Ideal A := ⊤
hI : DividedPowers I := RatAlgebra.dividedPowers ⊤
hII : ∀ (n : ℕ), ∀ a ∈ I ⊓ I, hI.dpow n a = hI.dpow n a := fun n a x => rfl
h1 : 1 ∈ I := Submodule.mem_top
hX : X ∈... | case a
A✝ : Type ?u.93141
inst✝ : CommRing A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
m n p : ℕ
hn : n ≠ 0
hp : p ≤ m * n
A : Type := ℚ[X]
I : Ideal A := ⊤
hI : DividedPowers I := RatAlgebra.dividedPowers ⊤
hII : ∀ (n : ℕ), ∀ a ∈ I ⊓ I, hI.dpow n a = hI.dpow n a := fun n a x => rfl
h1 : 1 ∈ I := Submodule.mem_top
hX : X ∈... | Please generate a tactic in lean4 to solve the state.
STATE:
case a
A✝ : Type ?u.93141
inst✝ : CommRing A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
m n p : ℕ
hn : n ≠ 0
hp : p ≤ m * n
A : Type := ℚ[X]
I : Ideal A := ⊤
hI : DividedPowers I := RatAlgebra.dividedPowers ⊤
hII : ∀ (n : ℕ), ∀ a ∈ I ⊓ I, hI.dpow n a = hI.dpow n a... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_comp_coeffs | [517, 1] | [585, 18] | simp only [C_eq_natCast] | case a
A✝ : Type ?u.93141
inst✝ : CommRing A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
m n p : ℕ
hn : n ≠ 0
hp : p ≤ m * n
A : Type := ℚ[X]
I : Ideal A := ⊤
hI : DividedPowers I := RatAlgebra.dividedPowers ⊤
hII : ∀ (n : ℕ), ∀ a ∈ I ⊓ I, hI.dpow n a = hI.dpow n a := fun n a x => rfl
h1 : 1 ∈ I := Submodule.mem_top
hX : X ∈... | case a
A✝ : Type ?u.93141
inst✝ : CommRing A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
m n p : ℕ
hn : n ≠ 0
hp : p ≤ m * n
A : Type := ℚ[X]
I : Ideal A := ⊤
hI : DividedPowers I := RatAlgebra.dividedPowers ⊤
hII : ∀ (n : ℕ), ∀ a ∈ I ⊓ I, hI.dpow n a = hI.dpow n a := fun n a x => rfl
h1 : 1 ∈ I := Submodule.mem_top
hX : X ∈... | Please generate a tactic in lean4 to solve the state.
STATE:
case a
A✝ : Type ?u.93141
inst✝ : CommRing A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
m n p : ℕ
hn : n ≠ 0
hp : p ≤ m * n
A : Type := ℚ[X]
I : Ideal A := ⊤
hI : DividedPowers I := RatAlgebra.dividedPowers ⊤
hII : ∀ (n : ℕ), ∀ a ∈ I ⊓ I, hI.dpow n a = hI.dpow n a... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_comp_coeffs | [517, 1] | [585, 18] | rw [← hI.dpow_comp m hn Submodule.mem_top] | case a
A✝ : Type ?u.93141
inst✝ : CommRing A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
m n p : ℕ
hn : n ≠ 0
hp : p ≤ m * n
A : Type := ℚ[X]
I : Ideal A := ⊤
hI : DividedPowers I := RatAlgebra.dividedPowers ⊤
hII : ∀ (n : ℕ), ∀ a ∈ I ⊓ I, hI.dpow n a = hI.dpow n a := fun n a x => rfl
h1 : 1 ∈ I := Submodule.mem_top
hX : X ∈... | case a
A✝ : Type ?u.93141
inst✝ : CommRing A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
m n p : ℕ
hn : n ≠ 0
hp : p ≤ m * n
A : Type := ℚ[X]
I : Ideal A := ⊤
hI : DividedPowers I := RatAlgebra.dividedPowers ⊤
hII : ∀ (n : ℕ), ∀ a ∈ I ⊓ I, hI.dpow n a = hI.dpow n a := fun n a x => rfl
h1 : 1 ∈ I := Submodule.mem_top
hX : X ∈... | Please generate a tactic in lean4 to solve the state.
STATE:
case a
A✝ : Type ?u.93141
inst✝ : CommRing A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
m n p : ℕ
hn : n ≠ 0
hp : p ≤ m * n
A : Type := ℚ[X]
I : Ideal A := ⊤
hI : DividedPowers I := RatAlgebra.dividedPowers ⊤
hII : ∀ (n : ℕ), ∀ a ∈ I ⊓ I, hI.dpow n a = hI.dpow n a... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_comp_coeffs | [517, 1] | [585, 18] | rw [← dpow_eq_of_mem_left hI hI hII n Submodule.mem_top,
← dpow_eq_of_mem_left hI hI hII m Submodule.mem_top] | case a
A✝ : Type ?u.93141
inst✝ : CommRing A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
m n p : ℕ
hn : n ≠ 0
hp : p ≤ m * n
A : Type := ℚ[X]
I : Ideal A := ⊤
hI : DividedPowers I := RatAlgebra.dividedPowers ⊤
hII : ∀ (n : ℕ), ∀ a ∈ I ⊓ I, hI.dpow n a = hI.dpow n a := fun n a x => rfl
h1 : 1 ∈ I := Submodule.mem_top
hX : X ∈... | case a
A✝ : Type ?u.93141
inst✝ : CommRing A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
m n p : ℕ
hn : n ≠ 0
hp : p ≤ m * n
A : Type := ℚ[X]
I : Ideal A := ⊤
hI : DividedPowers I := RatAlgebra.dividedPowers ⊤
hII : ∀ (n : ℕ), ∀ a ∈ I ⊓ I, hI.dpow n a = hI.dpow n a := fun n a x => rfl
h1 : 1 ∈ I := Submodule.mem_top
hX : X ∈... | Please generate a tactic in lean4 to solve the state.
STATE:
case a
A✝ : Type ?u.93141
inst✝ : CommRing A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
m n p : ℕ
hn : n ≠ 0
hp : p ≤ m * n
A : Type := ℚ[X]
I : Ideal A := ⊤
hI : DividedPowers I := RatAlgebra.dividedPowers ⊤
hII : ∀ (n : ℕ), ∀ a ∈ I ⊓ I, hI.dpow n a = hI.dpow n a... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_comp_coeffs | [517, 1] | [585, 18] | rw [dpow_comp_aux hI hI hII m hn hX h1] | case a
A✝ : Type ?u.93141
inst✝ : CommRing A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
m n p : ℕ
hn : n ≠ 0
hp : p ≤ m * n
A : Type := ℚ[X]
I : Ideal A := ⊤
hI : DividedPowers I := RatAlgebra.dividedPowers ⊤
hII : ∀ (n : ℕ), ∀ a ∈ I ⊓ I, hI.dpow n a = hI.dpow n a := fun n a x => rfl
h1 : 1 ∈ I := Submodule.mem_top
hX : X ∈... | case a
A✝ : Type ?u.93141
inst✝ : CommRing A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
m n p : ℕ
hn : n ≠ 0
hp : p ≤ m * n
A : Type := ℚ[X]
I : Ideal A := ⊤
hI : DividedPowers I := RatAlgebra.dividedPowers ⊤
hII : ∀ (n : ℕ), ∀ a ∈ I ⊓ I, hI.dpow n a = hI.dpow n a := fun n a x => rfl
h1 : 1 ∈ I := Submodule.mem_top
hX : X ∈... | Please generate a tactic in lean4 to solve the state.
STATE:
case a
A✝ : Type ?u.93141
inst✝ : CommRing A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
m n p : ℕ
hn : n ≠ 0
hp : p ≤ m * n
A : Type := ℚ[X]
I : Ideal A := ⊤
hI : DividedPowers I := RatAlgebra.dividedPowers ⊤
hII : ∀ (n : ℕ), ∀ a ∈ I ⊓ I, hI.dpow n a = hI.dpow n a... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_comp_coeffs | [517, 1] | [585, 18] | rw [← C_eq_natCast] | case a
A✝ : Type ?u.93141
inst✝ : CommRing A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
m n p : ℕ
hn : n ≠ 0
hp : p ≤ m * n
A : Type := ℚ[X]
I : Ideal A := ⊤
hI : DividedPowers I := RatAlgebra.dividedPowers ⊤
hII : ∀ (n : ℕ), ∀ a ∈ I ⊓ I, hI.dpow n a = hI.dpow n a := fun n a x => rfl
h1 : 1 ∈ I := Submodule.mem_top
hX : X ∈... | case a
A✝ : Type ?u.93141
inst✝ : CommRing A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
m n p : ℕ
hn : n ≠ 0
hp : p ≤ m * n
A : Type := ℚ[X]
I : Ideal A := ⊤
hI : DividedPowers I := RatAlgebra.dividedPowers ⊤
hII : ∀ (n : ℕ), ∀ a ∈ I ⊓ I, hI.dpow n a = hI.dpow n a := fun n a x => rfl
h1 : 1 ∈ I := Submodule.mem_top
hX : X ∈... | Please generate a tactic in lean4 to solve the state.
STATE:
case a
A✝ : Type ?u.93141
inst✝ : CommRing A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
m n p : ℕ
hn : n ≠ 0
hp : p ≤ m * n
A : Type := ℚ[X]
I : Ideal A := ⊤
hI : DividedPowers I := RatAlgebra.dividedPowers ⊤
hII : ∀ (n : ℕ), ∀ a ∈ I ⊓ I, hI.dpow n a = hI.dpow n a... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_comp_coeffs | [517, 1] | [585, 18] | simp only [Finset.mul_sum] | case a
A✝ : Type ?u.93141
inst✝ : CommRing A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
m n p : ℕ
hn : n ≠ 0
hp : p ≤ m * n
A : Type := ℚ[X]
I : Ideal A := ⊤
hI : DividedPowers I := RatAlgebra.dividedPowers ⊤
hII : ∀ (n : ℕ), ∀ a ∈ I ⊓ I, hI.dpow n a = hI.dpow n a := fun n a x => rfl
h1 : 1 ∈ I := Submodule.mem_top
hX : X ∈... | case a
A✝ : Type ?u.93141
inst✝ : CommRing A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
m n p : ℕ
hn : n ≠ 0
hp : p ≤ m * n
A : Type := ℚ[X]
I : Ideal A := ⊤
hI : DividedPowers I := RatAlgebra.dividedPowers ⊤
hII : ∀ (n : ℕ), ∀ a ∈ I ⊓ I, hI.dpow n a = hI.dpow n a := fun n a x => rfl
h1 : 1 ∈ I := Submodule.mem_top
hX : X ∈... | Please generate a tactic in lean4 to solve the state.
STATE:
case a
A✝ : Type ?u.93141
inst✝ : CommRing A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
m n p : ℕ
hn : n ≠ 0
hp : p ≤ m * n
A : Type := ℚ[X]
I : Ideal A := ⊤
hI : DividedPowers I := RatAlgebra.dividedPowers ⊤
hII : ∀ (n : ℕ), ∀ a ∈ I ⊓ I, hI.dpow n a = hI.dpow n a... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_comp_coeffs | [517, 1] | [585, 18] | simp only [finset_sum_coeff] | case a
A✝ : Type ?u.93141
inst✝ : CommRing A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
m n p : ℕ
hn : n ≠ 0
hp : p ≤ m * n
A : Type := ℚ[X]
I : Ideal A := ⊤
hI : DividedPowers I := RatAlgebra.dividedPowers ⊤
hII : ∀ (n : ℕ), ∀ a ∈ I ⊓ I, hI.dpow n a = hI.dpow n a := fun n a x => rfl
h1 : 1 ∈ I := Submodule.mem_top
hX : X ∈... | case a
A✝ : Type ?u.93141
inst✝ : CommRing A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
m n p : ℕ
hn : n ≠ 0
hp : p ≤ m * n
A : Type := ℚ[X]
I : Ideal A := ⊤
hI : DividedPowers I := RatAlgebra.dividedPowers ⊤
hII : ∀ (n : ℕ), ∀ a ∈ I ⊓ I, hI.dpow n a = hI.dpow n a := fun n a x => rfl
h1 : 1 ∈ I := Submodule.mem_top
hX : X ∈... | Please generate a tactic in lean4 to solve the state.
STATE:
case a
A✝ : Type ?u.93141
inst✝ : CommRing A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
m n p : ℕ
hn : n ≠ 0
hp : p ≤ m * n
A : Type := ℚ[X]
I : Ideal A := ⊤
hI : DividedPowers I := RatAlgebra.dividedPowers ⊤
hII : ∀ (n : ℕ), ∀ a ∈ I ⊓ I, hI.dpow n a = hI.dpow n a... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_comp_coeffs | [517, 1] | [585, 18] | simp only [hI] | case a
A✝ : Type ?u.93141
inst✝ : CommRing A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
m n p : ℕ
hn : n ≠ 0
hp : p ≤ m * n
A : Type := ℚ[X]
I : Ideal A := ⊤
hI : DividedPowers I := RatAlgebra.dividedPowers ⊤
hII : ∀ (n : ℕ), ∀ a ∈ I ⊓ I, hI.dpow n a = hI.dpow n a := fun n a x => rfl
h1 : 1 ∈ I := Submodule.mem_top
hX : X ∈... | case a
A✝ : Type ?u.93141
inst✝ : CommRing A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
m n p : ℕ
hn : n ≠ 0
hp : p ≤ m * n
A : Type := ℚ[X]
I : Ideal A := ⊤
hI : DividedPowers I := RatAlgebra.dividedPowers ⊤
hII : ∀ (n : ℕ), ∀ a ∈ I ⊓ I, hI.dpow n a = hI.dpow n a := fun n a x => rfl
h1 : 1 ∈ I := Submodule.mem_top
hX : X ∈... | Please generate a tactic in lean4 to solve the state.
STATE:
case a
A✝ : Type ?u.93141
inst✝ : CommRing A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
m n p : ℕ
hn : n ≠ 0
hp : p ≤ m * n
A : Type := ℚ[X]
I : Ideal A := ⊤
hI : DividedPowers I := RatAlgebra.dividedPowers ⊤
hII : ∀ (n : ℕ), ∀ a ∈ I ⊓ I, hI.dpow n a = hI.dpow n a... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_comp_coeffs | [517, 1] | [585, 18] | simp only [RatAlgebra.dpow_eq_inv_fact_smul _ _ Submodule.mem_top] | case a
A✝ : Type ?u.93141
inst✝ : CommRing A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
m n p : ℕ
hn : n ≠ 0
hp : p ≤ m * n
A : Type := ℚ[X]
I : Ideal A := ⊤
hI : DividedPowers I := RatAlgebra.dividedPowers ⊤
hII : ∀ (n : ℕ), ∀ a ∈ I ⊓ I, hI.dpow n a = hI.dpow n a := fun n a x => rfl
h1 : 1 ∈ I := Submodule.mem_top
hX : X ∈... | case a
A✝ : Type ?u.93141
inst✝ : CommRing A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
m n p : ℕ
hn : n ≠ 0
hp : p ≤ m * n
A : Type := ℚ[X]
I : Ideal A := ⊤
hI : DividedPowers I := RatAlgebra.dividedPowers ⊤
hII : ∀ (n : ℕ), ∀ a ∈ I ⊓ I, hI.dpow n a = hI.dpow n a := fun n a x => rfl
h1 : 1 ∈ I := Submodule.mem_top
hX : X ∈... | Please generate a tactic in lean4 to solve the state.
STATE:
case a
A✝ : Type ?u.93141
inst✝ : CommRing A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
m n p : ℕ
hn : n ≠ 0
hp : p ≤ m * n
A : Type := ℚ[X]
I : Ideal A := ⊤
hI : DividedPowers I := RatAlgebra.dividedPowers ⊤
hII : ∀ (n : ℕ), ∀ a ∈ I ⊓ I, hI.dpow n a = hI.dpow n a... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_comp_coeffs | [517, 1] | [585, 18] | simp only [map_natCast, Nat.cast_sum, Nat.cast_mul, Nat.cast_prod,
Ring.inverse_eq_inv', Algebra.mul_smul_comm, one_pow, mul_one, coeff_smul,
coeff_natCast_mul, smul_eq_mul] | case a
A✝ : Type ?u.93141
inst✝ : CommRing A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
m n p : ℕ
hn : n ≠ 0
hp : p ≤ m * n
A : Type := ℚ[X]
I : Ideal A := ⊤
hI : DividedPowers I := RatAlgebra.dividedPowers ⊤
hII : ∀ (n : ℕ), ∀ a ∈ I ⊓ I, hI.dpow n a = hI.dpow n a := fun n a x => rfl
h1 : 1 ∈ I := Submodule.mem_top
hX : X ∈... | case a
A✝ : Type ?u.93141
inst✝ : CommRing A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
m n p : ℕ
hn : n ≠ 0
hp : p ≤ m * n
A : Type := ℚ[X]
I : Ideal A := ⊤
hI : DividedPowers I := RatAlgebra.dividedPowers ⊤
hII : ∀ (n : ℕ), ∀ a ∈ I ⊓ I, hI.dpow n a = hI.dpow n a := fun n a x => rfl
h1 : 1 ∈ I := Submodule.mem_top
hX : X ∈... | Please generate a tactic in lean4 to solve the state.
STATE:
case a
A✝ : Type ?u.93141
inst✝ : CommRing A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
m n p : ℕ
hn : n ≠ 0
hp : p ≤ m * n
A : Type := ℚ[X]
I : Ideal A := ⊤
hI : DividedPowers I := RatAlgebra.dividedPowers ⊤
hII : ∀ (n : ℕ), ∀ a ∈ I ⊓ I, hI.dpow n a = hI.dpow n a... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_comp_coeffs | [517, 1] | [585, 18] | simp only [← Nat.cast_prod, ← Nat.cast_mul, ← Nat.cast_sum] | case a
A✝ : Type ?u.93141
inst✝ : CommRing A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
m n p : ℕ
hn : n ≠ 0
hp : p ≤ m * n
A : Type := ℚ[X]
I : Ideal A := ⊤
hI : DividedPowers I := RatAlgebra.dividedPowers ⊤
hII : ∀ (n : ℕ), ∀ a ∈ I ⊓ I, hI.dpow n a = hI.dpow n a := fun n a x => rfl
h1 : 1 ∈ I := Submodule.mem_top
hX : X ∈... | case a
A✝ : Type ?u.93141
inst✝ : CommRing A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
m n p : ℕ
hn : n ≠ 0
hp : p ≤ m * n
A : Type := ℚ[X]
I : Ideal A := ⊤
hI : DividedPowers I := RatAlgebra.dividedPowers ⊤
hII : ∀ (n : ℕ), ∀ a ∈ I ⊓ I, hI.dpow n a = hI.dpow n a := fun n a x => rfl
h1 : 1 ∈ I := Submodule.mem_top
hX : X ∈... | Please generate a tactic in lean4 to solve the state.
STATE:
case a
A✝ : Type ?u.93141
inst✝ : CommRing A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
m n p : ℕ
hn : n ≠ 0
hp : p ≤ m * n
A : Type := ℚ[X]
I : Ideal A := ⊤
hI : DividedPowers I := RatAlgebra.dividedPowers ⊤
hII : ∀ (n : ℕ), ∀ a ∈ I ⊓ I, hI.dpow n a = hI.dpow n a... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_comp_coeffs | [517, 1] | [585, 18] | rw [Finset.sum_eq_single p] | case a
A✝ : Type ?u.93141
inst✝ : CommRing A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
m n p : ℕ
hn : n ≠ 0
hp : p ≤ m * n
A : Type := ℚ[X]
I : Ideal A := ⊤
hI : DividedPowers I := RatAlgebra.dividedPowers ⊤
hII : ∀ (n : ℕ), ∀ a ∈ I ⊓ I, hI.dpow n a = hI.dpow n a := fun n a x => rfl
h1 : 1 ∈ I := Submodule.mem_top
hX : X ∈... | case a
A✝ : Type ?u.93141
inst✝ : CommRing A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
m n p : ℕ
hn : n ≠ 0
hp : p ≤ m * n
A : Type := ℚ[X]
I : Ideal A := ⊤
hI : DividedPowers I := RatAlgebra.dividedPowers ⊤
hII : ∀ (n : ℕ), ∀ a ∈ I ⊓ I, hI.dpow n a = hI.dpow n a := fun n a x => rfl
h1 : 1 ∈ I := Submodule.mem_top
hX : X ∈... | Please generate a tactic in lean4 to solve the state.
STATE:
case a
A✝ : Type ?u.93141
inst✝ : CommRing A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
m n p : ℕ
hn : n ≠ 0
hp : p ≤ m * n
A : Type := ℚ[X]
I : Ideal A := ⊤
hI : DividedPowers I := RatAlgebra.dividedPowers ⊤
hII : ∀ (n : ℕ), ∀ a ∈ I ⊓ I, hI.dpow n a = hI.dpow n a... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_comp_coeffs | [517, 1] | [585, 18] | conv_lhs =>
rw [coeff_natCast_mul, coeff_X_pow, if_pos, mul_one]
simp only [← Nat.cast_sum, ← Nat.cast_mul, ← Nat.cast_prod]
rw [← mul_assoc, mul_comm]
rw [Nat.cast_mul]
simp only [mul_assoc]
rw [mul_comm] | case a
A✝ : Type ?u.93141
inst✝ : CommRing A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
m n p : ℕ
hn : n ≠ 0
hp : p ≤ m * n
A : Type := ℚ[X]
I : Ideal A := ⊤
hI : DividedPowers I := RatAlgebra.dividedPowers ⊤
hII : ∀ (n : ℕ), ∀ a ∈ I ⊓ I, hI.dpow n a = hI.dpow n a := fun n a x => rfl
h1 : 1 ∈ I := Submodule.mem_top
hX : X ∈... | case a
A✝ : Type ?u.93141
inst✝ : CommRing A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
m n p : ℕ
hn : n ≠ 0
hp : p ≤ m * n
A : Type := ℚ[X]
I : Ideal A := ⊤
hI : DividedPowers I := RatAlgebra.dividedPowers ⊤
hII : ∀ (n : ℕ), ∀ a ∈ I ⊓ I, hI.dpow n a = hI.dpow n a := fun n a x => rfl
h1 : 1 ∈ I := Submodule.mem_top
hX : X ∈... | Please generate a tactic in lean4 to solve the state.
STATE:
case a
A✝ : Type ?u.93141
inst✝ : CommRing A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
m n p : ℕ
hn : n ≠ 0
hp : p ≤ m * n
A : Type := ℚ[X]
I : Ideal A := ⊤
hI : DividedPowers I := RatAlgebra.dividedPowers ⊤
hII : ∀ (n : ℕ), ∀ a ∈ I ⊓ I, hI.dpow n a = hI.dpow n a... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_comp_coeffs | [517, 1] | [585, 18] | simp only [Nat.cast_sum, Nat.cast_mul, Nat.cast_prod] | case a
A✝ : Type ?u.93141
inst✝ : CommRing A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
m n p : ℕ
hn : n ≠ 0
hp : p ≤ m * n
A : Type := ℚ[X]
I : Ideal A := ⊤
hI : DividedPowers I := RatAlgebra.dividedPowers ⊤
hII : ∀ (n : ℕ), ∀ a ∈ I ⊓ I, hI.dpow n a = hI.dpow n a := fun n a x => rfl
h1 : 1 ∈ I := Submodule.mem_top
hX : X ∈... | case a
A✝ : Type ?u.93141
inst✝ : CommRing A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
m n p : ℕ
hn : n ≠ 0
hp : p ≤ m * n
A : Type := ℚ[X]
I : Ideal A := ⊤
hI : DividedPowers I := RatAlgebra.dividedPowers ⊤
hII : ∀ (n : ℕ), ∀ a ∈ I ⊓ I, hI.dpow n a = hI.dpow n a := fun n a x => rfl
h1 : 1 ∈ I := Submodule.mem_top
hX : X ∈... | Please generate a tactic in lean4 to solve the state.
STATE:
case a
A✝ : Type ?u.93141
inst✝ : CommRing A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
m n p : ℕ
hn : n ≠ 0
hp : p ≤ m * n
A : Type := ℚ[X]
I : Ideal A := ⊤
hI : DividedPowers I := RatAlgebra.dividedPowers ⊤
hII : ∀ (n : ℕ), ∀ a ∈ I ⊓ I, hI.dpow n a = hI.dpow n a... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_comp_coeffs | [517, 1] | [585, 18] | simp only [Finset.sum_mul] | case a
A✝ : Type ?u.93141
inst✝ : CommRing A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
m n p : ℕ
hn : n ≠ 0
hp : p ≤ m * n
A : Type := ℚ[X]
I : Ideal A := ⊤
hI : DividedPowers I := RatAlgebra.dividedPowers ⊤
hII : ∀ (n : ℕ), ∀ a ∈ I ⊓ I, hI.dpow n a = hI.dpow n a := fun n a x => rfl
h1 : 1 ∈ I := Submodule.mem_top
hX : X ∈... | case a
A✝ : Type ?u.93141
inst✝ : CommRing A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
m n p : ℕ
hn : n ≠ 0
hp : p ≤ m * n
A : Type := ℚ[X]
I : Ideal A := ⊤
hI : DividedPowers I := RatAlgebra.dividedPowers ⊤
hII : ∀ (n : ℕ), ∀ a ∈ I ⊓ I, hI.dpow n a = hI.dpow n a := fun n a x => rfl
h1 : 1 ∈ I := Submodule.mem_top
hX : X ∈... | Please generate a tactic in lean4 to solve the state.
STATE:
case a
A✝ : Type ?u.93141
inst✝ : CommRing A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
m n p : ℕ
hn : n ≠ 0
hp : p ≤ m * n
A : Type := ℚ[X]
I : Ideal A := ⊤
hI : DividedPowers I := RatAlgebra.dividedPowers ⊤
hII : ∀ (n : ℕ), ∀ a ∈ I ⊓ I, hI.dpow n a = hI.dpow n a... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_comp_coeffs | [517, 1] | [585, 18] | apply Finset.sum_congr rfl | case a
A✝ : Type ?u.93141
inst✝ : CommRing A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
m n p : ℕ
hn : n ≠ 0
hp : p ≤ m * n
A : Type := ℚ[X]
I : Ideal A := ⊤
hI : DividedPowers I := RatAlgebra.dividedPowers ⊤
hII : ∀ (n : ℕ), ∀ a ∈ I ⊓ I, hI.dpow n a = hI.dpow n a := fun n a x => rfl
h1 : 1 ∈ I := Submodule.mem_top
hX : X ∈... | case a
A✝ : Type ?u.93141
inst✝ : CommRing A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
m n p : ℕ
hn : n ≠ 0
hp : p ≤ m * n
A : Type := ℚ[X]
I : Ideal A := ⊤
hI : DividedPowers I := RatAlgebra.dividedPowers ⊤
hII : ∀ (n : ℕ), ∀ a ∈ I ⊓ I, hI.dpow n a = hI.dpow n a := fun n a x => rfl
h1 : 1 ∈ I := Submodule.mem_top
hX : X ∈... | Please generate a tactic in lean4 to solve the state.
STATE:
case a
A✝ : Type ?u.93141
inst✝ : CommRing A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
m n p : ℕ
hn : n ≠ 0
hp : p ≤ m * n
A : Type := ℚ[X]
I : Ideal A := ⊤
hI : DividedPowers I := RatAlgebra.dividedPowers ⊤
hII : ∀ (n : ℕ), ∀ a ∈ I ⊓ I, hI.dpow n a = hI.dpow n a... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_comp_coeffs | [517, 1] | [585, 18] | intro x _ | case a
A✝ : Type ?u.93141
inst✝ : CommRing A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
m n p : ℕ
hn : n ≠ 0
hp : p ≤ m * n
A : Type := ℚ[X]
I : Ideal A := ⊤
hI : DividedPowers I := RatAlgebra.dividedPowers ⊤
hII : ∀ (n : ℕ), ∀ a ∈ I ⊓ I, hI.dpow n a = hI.dpow n a := fun n a x => rfl
h1 : 1 ∈ I := Submodule.mem_top
hX : X ∈... | case a
A✝ : Type ?u.93141
inst✝ : CommRing A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
m n p : ℕ
hn : n ≠ 0
hp : p ≤ m * n
A : Type := ℚ[X]
I : Ideal A := ⊤
hI : DividedPowers I := RatAlgebra.dividedPowers ⊤
hII : ∀ (n : ℕ), ∀ a ∈ I ⊓ I, hI.dpow n a = hI.dpow n a := fun n a x => rfl
h1 : 1 ∈ I := Submodule.mem_top
hX : X ∈... | Please generate a tactic in lean4 to solve the state.
STATE:
case a
A✝ : Type ?u.93141
inst✝ : CommRing A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
m n p : ℕ
hn : n ≠ 0
hp : p ≤ m * n
A : Type := ℚ[X]
I : Ideal A := ⊤
hI : DividedPowers I := RatAlgebra.dividedPowers ⊤
hII : ∀ (n : ℕ), ∀ a ∈ I ⊓ I, hI.dpow n a = hI.dpow n a... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_comp_coeffs | [517, 1] | [585, 18] | simp only [mul_assoc] | case a
A✝ : Type ?u.93141
inst✝ : CommRing A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
m n p : ℕ
hn : n ≠ 0
hp : p ≤ m * n
A : Type := ℚ[X]
I : Ideal A := ⊤
hI : DividedPowers I := RatAlgebra.dividedPowers ⊤
hII : ∀ (n : ℕ), ∀ a ∈ I ⊓ I, hI.dpow n a = hI.dpow n a := fun n a x => rfl
h1 : 1 ∈ I := Submodule.mem_top
hX : X ∈... | case a
A✝ : Type ?u.93141
inst✝ : CommRing A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
m n p : ℕ
hn : n ≠ 0
hp : p ≤ m * n
A : Type := ℚ[X]
I : Ideal A := ⊤
hI : DividedPowers I := RatAlgebra.dividedPowers ⊤
hII : ∀ (n : ℕ), ∀ a ∈ I ⊓ I, hI.dpow n a = hI.dpow n a := fun n a x => rfl
h1 : 1 ∈ I := Submodule.mem_top
hX : X ∈... | Please generate a tactic in lean4 to solve the state.
STATE:
case a
A✝ : Type ?u.93141
inst✝ : CommRing A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
m n p : ℕ
hn : n ≠ 0
hp : p ≤ m * n
A : Type := ℚ[X]
I : Ideal A := ⊤
hI : DividedPowers I := RatAlgebra.dividedPowers ⊤
hII : ∀ (n : ℕ), ∀ a ∈ I ⊓ I, hI.dpow n a = hI.dpow n a... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_comp_coeffs | [517, 1] | [585, 18] | congr | case a
A✝ : Type ?u.93141
inst✝ : CommRing A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
m n p : ℕ
hn : n ≠ 0
hp : p ≤ m * n
A : Type := ℚ[X]
I : Ideal A := ⊤
hI : DividedPowers I := RatAlgebra.dividedPowers ⊤
hII : ∀ (n : ℕ), ∀ a ∈ I ⊓ I, hI.dpow n a = hI.dpow n a := fun n a x => rfl
h1 : 1 ∈ I := Submodule.mem_top
hX : X ∈... | case a.e_a.e_a.e_a
A✝ : Type ?u.93141
inst✝ : CommRing A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
m n p : ℕ
hn : n ≠ 0
hp : p ≤ m * n
A : Type := ℚ[X]
I : Ideal A := ⊤
hI : DividedPowers I := RatAlgebra.dividedPowers ⊤
hII : ∀ (n : ℕ), ∀ a ∈ I ⊓ I, hI.dpow n a = hI.dpow n a := fun n a x => rfl
h1 : 1 ∈ I := Submodule.mem_... | Please generate a tactic in lean4 to solve the state.
STATE:
case a
A✝ : Type ?u.93141
inst✝ : CommRing A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
m n p : ℕ
hn : n ≠ 0
hp : p ≤ m * n
A : Type := ℚ[X]
I : Ideal A := ⊤
hI : DividedPowers I := RatAlgebra.dividedPowers ⊤
hII : ∀ (n : ℕ), ∀ a ∈ I ⊓ I, hI.dpow n a = hI.dpow n a... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_comp_coeffs | [517, 1] | [585, 18] | ring_nf | case a.e_a.e_a.e_a
A✝ : Type ?u.93141
inst✝ : CommRing A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
m n p : ℕ
hn : n ≠ 0
hp : p ≤ m * n
A : Type := ℚ[X]
I : Ideal A := ⊤
hI : DividedPowers I := RatAlgebra.dividedPowers ⊤
hII : ∀ (n : ℕ), ∀ a ∈ I ⊓ I, hI.dpow n a = hI.dpow n a := fun n a x => rfl
h1 : 1 ∈ I := Submodule.mem_... | case a.e_a.e_a.e_a
A✝ : Type ?u.93141
inst✝ : CommRing A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
m n p : ℕ
hn : n ≠ 0
hp : p ≤ m * n
A : Type := ℚ[X]
I : Ideal A := ⊤
hI : DividedPowers I := RatAlgebra.dividedPowers ⊤
hII : ∀ (n : ℕ), ∀ a ∈ I ⊓ I, hI.dpow n a = hI.dpow n a := fun n a x => rfl
h1 : 1 ∈ I := Submodule.mem_... | Please generate a tactic in lean4 to solve the state.
STATE:
case a.e_a.e_a.e_a
A✝ : Type ?u.93141
inst✝ : CommRing A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
m n p : ℕ
hn : n ≠ 0
hp : p ≤ m * n
A : Type := ℚ[X]
I : Ideal A := ⊤
hI : DividedPowers I := RatAlgebra.dividedPowers ⊤
hII : ∀ (n : ℕ), ∀ a ∈ I ⊓ I, hI.dpow n a =... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_comp_coeffs | [517, 1] | [585, 18] | simp only [mul_assoc] | case a.e_a.e_a.e_a
A✝ : Type ?u.93141
inst✝ : CommRing A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
m n p : ℕ
hn : n ≠ 0
hp : p ≤ m * n
A : Type := ℚ[X]
I : Ideal A := ⊤
hI : DividedPowers I := RatAlgebra.dividedPowers ⊤
hII : ∀ (n : ℕ), ∀ a ∈ I ⊓ I, hI.dpow n a = hI.dpow n a := fun n a x => rfl
h1 : 1 ∈ I := Submodule.mem_... | case a.e_a.e_a.e_a
A✝ : Type ?u.93141
inst✝ : CommRing A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
m n p : ℕ
hn : n ≠ 0
hp : p ≤ m * n
A : Type := ℚ[X]
I : Ideal A := ⊤
hI : DividedPowers I := RatAlgebra.dividedPowers ⊤
hII : ∀ (n : ℕ), ∀ a ∈ I ⊓ I, hI.dpow n a = hI.dpow n a := fun n a x => rfl
h1 : 1 ∈ I := Submodule.mem_... | Please generate a tactic in lean4 to solve the state.
STATE:
case a.e_a.e_a.e_a
A✝ : Type ?u.93141
inst✝ : CommRing A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
m n p : ℕ
hn : n ≠ 0
hp : p ≤ m * n
A : Type := ℚ[X]
I : Ideal A := ⊤
hI : DividedPowers I := RatAlgebra.dividedPowers ⊤
hII : ∀ (n : ℕ), ∀ a ∈ I ⊓ I, hI.dpow n a =... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_comp_coeffs | [517, 1] | [585, 18] | rw [inv_mul_eq_iff_eq_mul₀] | case a.e_a.e_a.e_a
A✝ : Type ?u.93141
inst✝ : CommRing A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
m n p : ℕ
hn : n ≠ 0
hp : p ≤ m * n
A : Type := ℚ[X]
I : Ideal A := ⊤
hI : DividedPowers I := RatAlgebra.dividedPowers ⊤
hII : ∀ (n : ℕ), ∀ a ∈ I ⊓ I, hI.dpow n a = hI.dpow n a := fun n a x => rfl
h1 : 1 ∈ I := Submodule.mem_... | case a.e_a.e_a.e_a
A✝ : Type ?u.93141
inst✝ : CommRing A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
m n p : ℕ
hn : n ≠ 0
hp : p ≤ m * n
A : Type := ℚ[X]
I : Ideal A := ⊤
hI : DividedPowers I := RatAlgebra.dividedPowers ⊤
hII : ∀ (n : ℕ), ∀ a ∈ I ⊓ I, hI.dpow n a = hI.dpow n a := fun n a x => rfl
h1 : 1 ∈ I := Submodule.mem_... | Please generate a tactic in lean4 to solve the state.
STATE:
case a.e_a.e_a.e_a
A✝ : Type ?u.93141
inst✝ : CommRing A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
m n p : ℕ
hn : n ≠ 0
hp : p ≤ m * n
A : Type := ℚ[X]
I : Ideal A := ⊤
hI : DividedPowers I := RatAlgebra.dividedPowers ⊤
hII : ∀ (n : ℕ), ∀ a ∈ I ⊓ I, hI.dpow n a =... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_comp_coeffs | [517, 1] | [585, 18] | rw [inv_mul_eq_iff_eq_mul₀] | case a.e_a.e_a.e_a
A✝ : Type ?u.93141
inst✝ : CommRing A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
m n p : ℕ
hn : n ≠ 0
hp : p ≤ m * n
A : Type := ℚ[X]
I : Ideal A := ⊤
hI : DividedPowers I := RatAlgebra.dividedPowers ⊤
hII : ∀ (n : ℕ), ∀ a ∈ I ⊓ I, hI.dpow n a = hI.dpow n a := fun n a x => rfl
h1 : 1 ∈ I := Submodule.mem_... | case a.e_a.e_a.e_a
A✝ : Type ?u.93141
inst✝ : CommRing A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
m n p : ℕ
hn : n ≠ 0
hp : p ≤ m * n
A : Type := ℚ[X]
I : Ideal A := ⊤
hI : DividedPowers I := RatAlgebra.dividedPowers ⊤
hII : ∀ (n : ℕ), ∀ a ∈ I ⊓ I, hI.dpow n a = hI.dpow n a := fun n a x => rfl
h1 : 1 ∈ I := Submodule.mem_... | Please generate a tactic in lean4 to solve the state.
STATE:
case a.e_a.e_a.e_a
A✝ : Type ?u.93141
inst✝ : CommRing A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
m n p : ℕ
hn : n ≠ 0
hp : p ≤ m * n
A : Type := ℚ[X]
I : Ideal A := ⊤
hI : DividedPowers I := RatAlgebra.dividedPowers ⊤
hII : ∀ (n : ℕ), ∀ a ∈ I ⊓ I, hI.dpow n a =... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_comp_coeffs | [517, 1] | [585, 18] | rw [← Nat.choose_mul_factorial_mul_factorial hp] | case a.e_a.e_a.e_a
A✝ : Type ?u.93141
inst✝ : CommRing A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
m n p : ℕ
hn : n ≠ 0
hp : p ≤ m * n
A : Type := ℚ[X]
I : Ideal A := ⊤
hI : DividedPowers I := RatAlgebra.dividedPowers ⊤
hII : ∀ (n : ℕ), ∀ a ∈ I ⊓ I, hI.dpow n a = hI.dpow n a := fun n a x => rfl
h1 : 1 ∈ I := Submodule.mem_... | case a.e_a.e_a.e_a
A✝ : Type ?u.93141
inst✝ : CommRing A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
m n p : ℕ
hn : n ≠ 0
hp : p ≤ m * n
A : Type := ℚ[X]
I : Ideal A := ⊤
hI : DividedPowers I := RatAlgebra.dividedPowers ⊤
hII : ∀ (n : ℕ), ∀ a ∈ I ⊓ I, hI.dpow n a = hI.dpow n a := fun n a x => rfl
h1 : 1 ∈ I := Submodule.mem_... | Please generate a tactic in lean4 to solve the state.
STATE:
case a.e_a.e_a.e_a
A✝ : Type ?u.93141
inst✝ : CommRing A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
m n p : ℕ
hn : n ≠ 0
hp : p ≤ m * n
A : Type := ℚ[X]
I : Ideal A := ⊤
hI : DividedPowers I := RatAlgebra.dividedPowers ⊤
hII : ∀ (n : ℕ), ∀ a ∈ I ⊓ I, hI.dpow n a =... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_comp_coeffs | [517, 1] | [585, 18] | simp only [Nat.cast_mul] | case a.e_a.e_a.e_a
A✝ : Type ?u.93141
inst✝ : CommRing A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
m n p : ℕ
hn : n ≠ 0
hp : p ≤ m * n
A : Type := ℚ[X]
I : Ideal A := ⊤
hI : DividedPowers I := RatAlgebra.dividedPowers ⊤
hII : ∀ (n : ℕ), ∀ a ∈ I ⊓ I, hI.dpow n a = hI.dpow n a := fun n a x => rfl
h1 : 1 ∈ I := Submodule.mem_... | case a.e_a.e_a.e_a
A✝ : Type ?u.93141
inst✝ : CommRing A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
m n p : ℕ
hn : n ≠ 0
hp : p ≤ m * n
A : Type := ℚ[X]
I : Ideal A := ⊤
hI : DividedPowers I := RatAlgebra.dividedPowers ⊤
hII : ∀ (n : ℕ), ∀ a ∈ I ⊓ I, hI.dpow n a = hI.dpow n a := fun n a x => rfl
h1 : 1 ∈ I := Submodule.mem_... | Please generate a tactic in lean4 to solve the state.
STATE:
case a.e_a.e_a.e_a
A✝ : Type ?u.93141
inst✝ : CommRing A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
m n p : ℕ
hn : n ≠ 0
hp : p ≤ m * n
A : Type := ℚ[X]
I : Ideal A := ⊤
hI : DividedPowers I := RatAlgebra.dividedPowers ⊤
hII : ∀ (n : ℕ), ∀ a ∈ I ⊓ I, hI.dpow n a =... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_comp_coeffs | [517, 1] | [585, 18] | ring | case a.e_a.e_a.e_a
A✝ : Type ?u.93141
inst✝ : CommRing A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
m n p : ℕ
hn : n ≠ 0
hp : p ≤ m * n
A : Type := ℚ[X]
I : Ideal A := ⊤
hI : DividedPowers I := RatAlgebra.dividedPowers ⊤
hII : ∀ (n : ℕ), ∀ a ∈ I ⊓ I, hI.dpow n a = hI.dpow n a := fun n a x => rfl
h1 : 1 ∈ I := Submodule.mem_... | case a.e_a.e_a.e_a
A✝ : Type ?u.93141
inst✝ : CommRing A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
m n p : ℕ
hn : n ≠ 0
hp : p ≤ m * n
A : Type := ℚ[X]
I : Ideal A := ⊤
hI : DividedPowers I := RatAlgebra.dividedPowers ⊤
hII : ∀ (n : ℕ), ∀ a ∈ I ⊓ I, hI.dpow n a = hI.dpow n a := fun n a x => rfl
h1 : 1 ∈ I := Submodule.mem_... | Please generate a tactic in lean4 to solve the state.
STATE:
case a.e_a.e_a.e_a
A✝ : Type ?u.93141
inst✝ : CommRing A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
m n p : ℕ
hn : n ≠ 0
hp : p ≤ m * n
A : Type := ℚ[X]
I : Ideal A := ⊤
hI : DividedPowers I := RatAlgebra.dividedPowers ⊤
hII : ∀ (n : ℕ), ∀ a ∈ I ⊓ I, hI.dpow n a =... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_comp_coeffs | [517, 1] | [585, 18] | all_goals
simp only [ne_eq, Nat.cast_eq_zero]
apply Nat.factorial_ne_zero | case a.e_a.e_a.e_a
A✝ : Type ?u.93141
inst✝ : CommRing A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
m n p : ℕ
hn : n ≠ 0
hp : p ≤ m * n
A : Type := ℚ[X]
I : Ideal A := ⊤
hI : DividedPowers I := RatAlgebra.dividedPowers ⊤
hII : ∀ (n : ℕ), ∀ a ∈ I ⊓ I, hI.dpow n a = hI.dpow n a := fun n a x => rfl
h1 : 1 ∈ I := Submodule.mem_... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a.e_a.e_a.e_a
A✝ : Type ?u.93141
inst✝ : CommRing A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
m n p : ℕ
hn : n ≠ 0
hp : p ≤ m * n
A : Type := ℚ[X]
I : Ideal A := ⊤
hI : DividedPowers I := RatAlgebra.dividedPowers ⊤
hII : ∀ (n : ℕ), ∀ a ∈ I ⊓ I, hI.dpow n a =... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_comp_coeffs | [517, 1] | [585, 18] | simp only [ne_eq, Nat.cast_eq_zero] | case a.e_a.e_a.e_a
A✝ : Type ?u.93141
inst✝ : CommRing A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
m n p : ℕ
hn : n ≠ 0
hp : p ≤ m * n
A : Type := ℚ[X]
I : Ideal A := ⊤
hI : DividedPowers I := RatAlgebra.dividedPowers ⊤
hII : ∀ (n : ℕ), ∀ a ∈ I ⊓ I, hI.dpow n a = hI.dpow n a := fun n a x => rfl
h1 : 1 ∈ I := Submodule.mem_... | case a.e_a.e_a.e_a
A✝ : Type ?u.93141
inst✝ : CommRing A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
m n p : ℕ
hn : n ≠ 0
hp : p ≤ m * n
A : Type := ℚ[X]
I : Ideal A := ⊤
hI : DividedPowers I := RatAlgebra.dividedPowers ⊤
hII : ∀ (n : ℕ), ∀ a ∈ I ⊓ I, hI.dpow n a = hI.dpow n a := fun n a x => rfl
h1 : 1 ∈ I := Submodule.mem_... | Please generate a tactic in lean4 to solve the state.
STATE:
case a.e_a.e_a.e_a
A✝ : Type ?u.93141
inst✝ : CommRing A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
m n p : ℕ
hn : n ≠ 0
hp : p ≤ m * n
A : Type := ℚ[X]
I : Ideal A := ⊤
hI : DividedPowers I := RatAlgebra.dividedPowers ⊤
hII : ∀ (n : ℕ), ∀ a ∈ I ⊓ I, hI.dpow n a =... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_comp_coeffs | [517, 1] | [585, 18] | apply Nat.factorial_ne_zero | case a.e_a.e_a.e_a
A✝ : Type ?u.93141
inst✝ : CommRing A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
m n p : ℕ
hn : n ≠ 0
hp : p ≤ m * n
A : Type := ℚ[X]
I : Ideal A := ⊤
hI : DividedPowers I := RatAlgebra.dividedPowers ⊤
hII : ∀ (n : ℕ), ∀ a ∈ I ⊓ I, hI.dpow n a = hI.dpow n a := fun n a x => rfl
h1 : 1 ∈ I := Submodule.mem_... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a.e_a.e_a.e_a
A✝ : Type ?u.93141
inst✝ : CommRing A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
m n p : ℕ
hn : n ≠ 0
hp : p ≤ m * n
A : Type := ℚ[X]
I : Ideal A := ⊤
hI : DividedPowers I := RatAlgebra.dividedPowers ⊤
hII : ∀ (n : ℕ), ∀ a ∈ I ⊓ I, hI.dpow n a =... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_comp_coeffs | [517, 1] | [585, 18] | intro b _ hb | case a.h₀
A✝ : Type ?u.93141
inst✝ : CommRing A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
m n p : ℕ
hn : n ≠ 0
hp : p ≤ m * n
A : Type := ℚ[X]
I : Ideal A := ⊤
hI : DividedPowers I := RatAlgebra.dividedPowers ⊤
hII : ∀ (n : ℕ), ∀ a ∈ I ⊓ I, hI.dpow n a = hI.dpow n a := fun n a x => rfl
h1 : 1 ∈ I := Submodule.mem_top
hX : ... | case a.h₀
A✝ : Type ?u.93141
inst✝ : CommRing A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
m n p : ℕ
hn : n ≠ 0
hp : p ≤ m * n
A : Type := ℚ[X]
I : Ideal A := ⊤
hI : DividedPowers I := RatAlgebra.dividedPowers ⊤
hII : ∀ (n : ℕ), ∀ a ∈ I ⊓ I, hI.dpow n a = hI.dpow n a := fun n a x => rfl
h1 : 1 ∈ I := Submodule.mem_top
hX : ... | Please generate a tactic in lean4 to solve the state.
STATE:
case a.h₀
A✝ : Type ?u.93141
inst✝ : CommRing A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
m n p : ℕ
hn : n ≠ 0
hp : p ≤ m * n
A : Type := ℚ[X]
I : Ideal A := ⊤
hI : DividedPowers I := RatAlgebra.dividedPowers ⊤
hII : ∀ (n : ℕ), ∀ a ∈ I ⊓ I, hI.dpow n a = hI.dpow ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_comp_coeffs | [517, 1] | [585, 18] | rw [coeff_natCast_mul, coeff_X_pow, if_neg hb.symm] | case a.h₀
A✝ : Type ?u.93141
inst✝ : CommRing A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
m n p : ℕ
hn : n ≠ 0
hp : p ≤ m * n
A : Type := ℚ[X]
I : Ideal A := ⊤
hI : DividedPowers I := RatAlgebra.dividedPowers ⊤
hII : ∀ (n : ℕ), ∀ a ∈ I ⊓ I, hI.dpow n a = hI.dpow n a := fun n a x => rfl
h1 : 1 ∈ I := Submodule.mem_top
hX : ... | case a.h₀
A✝ : Type ?u.93141
inst✝ : CommRing A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
m n p : ℕ
hn : n ≠ 0
hp : p ≤ m * n
A : Type := ℚ[X]
I : Ideal A := ⊤
hI : DividedPowers I := RatAlgebra.dividedPowers ⊤
hII : ∀ (n : ℕ), ∀ a ∈ I ⊓ I, hI.dpow n a = hI.dpow n a := fun n a x => rfl
h1 : 1 ∈ I := Submodule.mem_top
hX : ... | Please generate a tactic in lean4 to solve the state.
STATE:
case a.h₀
A✝ : Type ?u.93141
inst✝ : CommRing A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
m n p : ℕ
hn : n ≠ 0
hp : p ≤ m * n
A : Type := ℚ[X]
I : Ideal A := ⊤
hI : DividedPowers I := RatAlgebra.dividedPowers ⊤
hII : ∀ (n : ℕ), ∀ a ∈ I ⊓ I, hI.dpow n a = hI.dpow ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_comp_coeffs | [517, 1] | [585, 18] | simp only [mul_zero] | case a.h₀
A✝ : Type ?u.93141
inst✝ : CommRing A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
m n p : ℕ
hn : n ≠ 0
hp : p ≤ m * n
A : Type := ℚ[X]
I : Ideal A := ⊤
hI : DividedPowers I := RatAlgebra.dividedPowers ⊤
hII : ∀ (n : ℕ), ∀ a ∈ I ⊓ I, hI.dpow n a = hI.dpow n a := fun n a x => rfl
h1 : 1 ∈ I := Submodule.mem_top
hX : ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a.h₀
A✝ : Type ?u.93141
inst✝ : CommRing A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
m n p : ℕ
hn : n ≠ 0
hp : p ≤ m * n
A : Type := ℚ[X]
I : Ideal A := ⊤
hI : DividedPowers I := RatAlgebra.dividedPowers ⊤
hII : ∀ (n : ℕ), ∀ a ∈ I ⊓ I, hI.dpow n a = hI.dpow ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_comp_coeffs | [517, 1] | [585, 18] | intro hp' | case a.h₁
A✝ : Type ?u.93141
inst✝ : CommRing A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
m n p : ℕ
hn : n ≠ 0
hp : p ≤ m * n
A : Type := ℚ[X]
I : Ideal A := ⊤
hI : DividedPowers I := RatAlgebra.dividedPowers ⊤
hII : ∀ (n : ℕ), ∀ a ∈ I ⊓ I, hI.dpow n a = hI.dpow n a := fun n a x => rfl
h1 : 1 ∈ I := Submodule.mem_top
hX : ... | case a.h₁
A✝ : Type ?u.93141
inst✝ : CommRing A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
m n p : ℕ
hn : n ≠ 0
hp : p ≤ m * n
A : Type := ℚ[X]
I : Ideal A := ⊤
hI : DividedPowers I := RatAlgebra.dividedPowers ⊤
hII : ∀ (n : ℕ), ∀ a ∈ I ⊓ I, hI.dpow n a = hI.dpow n a := fun n a x => rfl
h1 : 1 ∈ I := Submodule.mem_top
hX : ... | Please generate a tactic in lean4 to solve the state.
STATE:
case a.h₁
A✝ : Type ?u.93141
inst✝ : CommRing A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
m n p : ℕ
hn : n ≠ 0
hp : p ≤ m * n
A : Type := ℚ[X]
I : Ideal A := ⊤
hI : DividedPowers I := RatAlgebra.dividedPowers ⊤
hII : ∀ (n : ℕ), ∀ a ∈ I ⊓ I, hI.dpow n a = hI.dpow ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_comp_coeffs | [517, 1] | [585, 18] | simp only [mem_range, Nat.lt_succ_iff] at hp' | case a.h₁
A✝ : Type ?u.93141
inst✝ : CommRing A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
m n p : ℕ
hn : n ≠ 0
hp : p ≤ m * n
A : Type := ℚ[X]
I : Ideal A := ⊤
hI : DividedPowers I := RatAlgebra.dividedPowers ⊤
hII : ∀ (n : ℕ), ∀ a ∈ I ⊓ I, hI.dpow n a = hI.dpow n a := fun n a x => rfl
h1 : 1 ∈ I := Submodule.mem_top
hX : ... | case a.h₁
A✝ : Type ?u.93141
inst✝ : CommRing A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
m n p : ℕ
hn : n ≠ 0
hp : p ≤ m * n
A : Type := ℚ[X]
I : Ideal A := ⊤
hI : DividedPowers I := RatAlgebra.dividedPowers ⊤
hII : ∀ (n : ℕ), ∀ a ∈ I ⊓ I, hI.dpow n a = hI.dpow n a := fun n a x => rfl
h1 : 1 ∈ I := Submodule.mem_top
hX : ... | Please generate a tactic in lean4 to solve the state.
STATE:
case a.h₁
A✝ : Type ?u.93141
inst✝ : CommRing A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
m n p : ℕ
hn : n ≠ 0
hp : p ≤ m * n
A : Type := ℚ[X]
I : Ideal A := ⊤
hI : DividedPowers I := RatAlgebra.dividedPowers ⊤
hII : ∀ (n : ℕ), ∀ a ∈ I ⊓ I, hI.dpow n a = hI.dpow ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_comp_coeffs | [517, 1] | [585, 18] | contradiction | case a.h₁
A✝ : Type ?u.93141
inst✝ : CommRing A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
m n p : ℕ
hn : n ≠ 0
hp : p ≤ m * n
A : Type := ℚ[X]
I : Ideal A := ⊤
hI : DividedPowers I := RatAlgebra.dividedPowers ⊤
hII : ∀ (n : ℕ), ∀ a ∈ I ⊓ I, hI.dpow n a = hI.dpow n a := fun n a x => rfl
h1 : 1 ∈ I := Submodule.mem_top
hX : ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a.h₁
A✝ : Type ?u.93141
inst✝ : CommRing A✝
I✝ : Ideal A✝
hI✝ : DividedPowers I✝
m n p : ℕ
hn : n ≠ 0
hp : p ≤ m * n
A : Type := ℚ[X]
I : Ideal A := ⊤
hI : DividedPowers I := RatAlgebra.dividedPowers ⊤
hII : ∀ (n : ℕ), ∀ a ∈ I ⊓ I, hI.dpow n a = hI.dpow ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_comp | [588, 1] | [603, 73] | rw [Ideal.add_eq_sup, Submodule.mem_sup] at hx | A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
x : A
hn : n ≠ 0
hx : x ∈ I + J
⊢ dpow hI hJ m (dpow hI hJ n x) = ↑(mchoose m n) * dpow hI hJ (m * n) x | A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
x : A
hn : n ≠ 0
hx : ∃ y ∈ I, ∃ z ∈ J, y + z = x
⊢ dpow hI hJ m (dpow hI hJ n x) = ↑(mchoose m n) * dpow hI hJ (m * n) x | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
x : A
hn : n ≠ 0
hx : x ∈ I + J
⊢ dpow hI hJ m (dpow hI hJ n x) = ↑(mchoose m n) * dpow hI hJ (m ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_comp | [588, 1] | [603, 73] | obtain ⟨a, ha, b, hb, rfl⟩ := hx | A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
x : A
hn : n ≠ 0
hx : ∃ y ∈ I, ∃ z ∈ J, y + z = x
⊢ dpow hI hJ m (dpow hI hJ n x) = ↑(mchoose m n) * dpow hI hJ (m * n) x | case intro.intro.intro.intro
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
⊢ dpow hI hJ m (dpow hI hJ n (a + b)) = ↑(mchoose m n) * dpow hI hJ (m * n) (a + b) | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
x : A
hn : n ≠ 0
hx : ∃ y ∈ I, ∃ z ∈ J, y + z = x
⊢ dpow hI hJ m (dpow hI hJ n x) = ↑(mchoose m n... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_comp | [588, 1] | [603, 73] | rw [dpow_comp_aux hI hJ hIJ m hn ha hb,
dpow_add' hI hJ hIJ _ (Submodule.mem_sup_left ha) (Submodule.mem_sup_right hb), Finset.mul_sum] | case intro.intro.intro.intro
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
⊢ dpow hI hJ m (dpow hI hJ n (a + b)) = ↑(mchoose m n) * dpow hI hJ (m * n) (a + b) | case intro.intro.intro.intro
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
⊢ ∑ p ∈ range (m * n + 1),
↑(∑ x ∈ filter (fun l => ∑ i ∈ range (n + 1), Multi... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
⊢ dpow hI hJ m (dpow hI... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_comp | [588, 1] | [603, 73] | apply Finset.sum_congr rfl | case intro.intro.intro.intro
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
⊢ ∑ p ∈ range (m * n + 1),
↑(∑ x ∈ filter (fun l => ∑ i ∈ range (n + 1), Multi... | case intro.intro.intro.intro
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
⊢ ∀ x ∈ range (m * n + 1),
↑(∑ x ∈ filter (fun l => ∑ i ∈ range (n + 1), Multise... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
⊢ ∑ p ∈ range (m * n + ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_comp | [588, 1] | [603, 73] | intro p hp | case intro.intro.intro.intro
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
⊢ ∀ x ∈ range (m * n + 1),
↑(∑ x ∈ filter (fun l => ∑ i ∈ range (n + 1), Multise... | case intro.intro.intro.intro
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
p : ℕ
hp : p ∈ range (m * n + 1)
⊢ ↑(∑ x ∈ filter (fun l => ∑ i ∈ range (n + 1), Mul... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
⊢ ∀ x ∈ range (m * n + ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_comp | [588, 1] | [603, 73] | rw [dpow_eq_of_mem_left hI hJ hIJ _ ha] | case intro.intro.intro.intro
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
p : ℕ
hp : p ∈ range (m * n + 1)
⊢ ↑(∑ x ∈ filter (fun l => ∑ i ∈ range (n + 1), Mul... | case intro.intro.intro.intro
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
p : ℕ
hp : p ∈ range (m * n + 1)
⊢ ↑(∑ x ∈ filter (fun l => ∑ i ∈ range (n + 1), Mul... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
p : ℕ
hp : p ∈ range (m... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_comp | [588, 1] | [603, 73] | rw [dpow_eq_of_mem_right hI hJ hIJ _ hb] | case intro.intro.intro.intro
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
p : ℕ
hp : p ∈ range (m * n + 1)
⊢ ↑(∑ x ∈ filter (fun l => ∑ i ∈ range (n + 1), Mul... | case intro.intro.intro.intro
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
p : ℕ
hp : p ∈ range (m * n + 1)
⊢ ↑(∑ x ∈ filter (fun l => ∑ i ∈ range (n + 1), Mul... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
p : ℕ
hp : p ∈ range (m... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_comp | [588, 1] | [603, 73] | simp only [mul_assoc] | case intro.intro.intro.intro
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
p : ℕ
hp : p ∈ range (m * n + 1)
⊢ ↑(∑ x ∈ filter (fun l => ∑ i ∈ range (n + 1), Mul... | case intro.intro.intro.intro
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
p : ℕ
hp : p ∈ range (m * n + 1)
⊢ ↑(∑ x ∈ filter (fun l => ∑ i ∈ range (n + 1), Mul... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
p : ℕ
hp : p ∈ range (m... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_comp | [588, 1] | [603, 73] | rw [dpow_comp_coeffs hn (Nat.lt_succ_iff.mp (Finset.mem_range.mp hp))] | A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
p : ℕ
hp : p ∈ range (m * n + 1)
⊢ ↑(∑ x ∈ filter (fun l => ∑ i ∈ range (n + 1), Multiset.count i ↑l * i = p) ((r... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
p : ℕ
hp : p ∈ range (m * n + 1)
⊢ ↑(∑ x ∈ filter (f... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_null | [606, 1] | [612, 36] | simp only [dpow] | A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
n : ℕ
x : A
hx : x ∉ I + J
⊢ dpow hI hJ n x = 0 | A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
n : ℕ
x : A
hx : x ∉ I + J
⊢ Function.extend (fun x => ↑x.1 + ↑x.2) (fun x => ∑ k ∈ range (n + 1), hI.dpow k ↑x.1 * hJ.dpow (n - k) ↑x.2)
(Function.const A 0) x =
0 | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
n : ℕ
x : A
hx : x ∉ I + J
⊢ dpow hI hJ n x = 0
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_null | [606, 1] | [612, 36] | rw [Function.extend_apply'] | A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
n : ℕ
x : A
hx : x ∉ I + J
⊢ Function.extend (fun x => ↑x.1 + ↑x.2) (fun x => ∑ k ∈ range (n + 1), hI.dpow k ↑x.1 * hJ.dpow (n - k) ↑x.2)
(Function.const A 0) x =
0 | A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
n : ℕ
x : A
hx : x ∉ I + J
⊢ Function.const A 0 x = 0
case hb
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
n : ℕ
x : A
hx : x ∉ I + J
⊢ ¬∃ a, ↑a.1 + ↑a.2 = x | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
n : ℕ
x : A
hx : x ∉ I + J
⊢ Function.extend (fun x => ↑x.1 + ↑x.2) (fun x => ∑ k ∈ range (n + 1), hI.dpow k ↑x.1 * hJ.dpow (n - k) ↑x.2)
(Function.const A... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_null | [606, 1] | [612, 36] | rfl | A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
n : ℕ
x : A
hx : x ∉ I + J
⊢ Function.const A 0 x = 0
case hb
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
n : ℕ
x : A
hx : x ∉ I + J
⊢ ¬∃ a, ↑a.1 + ↑a.2 = x | case hb
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
n : ℕ
x : A
hx : x ∉ I + J
⊢ ¬∃ a, ↑a.1 + ↑a.2 = x | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
n : ℕ
x : A
hx : x ∉ I + J
⊢ Function.const A 0 x = 0
case hb
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_null | [606, 1] | [612, 36] | rintro ⟨⟨⟨a, ha⟩, ⟨b, hb⟩⟩, h⟩ | case hb
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
n : ℕ
x : A
hx : x ∉ I + J
⊢ ¬∃ a, ↑a.1 + ↑a.2 = x | case hb.intro.mk.mk.mk
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
n : ℕ
x : A
hx : x ∉ I + J
a : A
ha : a ∈ I
b : A
hb : b ∈ J
h : ↑(⟨a, ha⟩, ⟨b, hb⟩).1 + ↑(⟨a, ha⟩, ⟨b, hb⟩).2 = x
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case hb
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
n : ℕ
x : A
hx : x ∉ I + J
⊢ ¬∃ a, ↑a.1 + ↑a.2 = x
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_null | [606, 1] | [612, 36] | apply hx | case hb.intro.mk.mk.mk
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
n : ℕ
x : A
hx : x ∉ I + J
a : A
ha : a ∈ I
b : A
hb : b ∈ J
h : ↑(⟨a, ha⟩, ⟨b, hb⟩).1 + ↑(⟨a, ha⟩, ⟨b, hb⟩).2 = x
⊢ False | case hb.intro.mk.mk.mk
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
n : ℕ
x : A
hx : x ∉ I + J
a : A
ha : a ∈ I
b : A
hb : b ∈ J
h : ↑(⟨a, ha⟩, ⟨b, hb⟩).1 + ↑(⟨a, ha⟩, ⟨b, hb⟩).2 = x
⊢ x ∈ I + J | Please generate a tactic in lean4 to solve the state.
STATE:
case hb.intro.mk.mk.mk
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
n : ℕ
x : A
hx : x ∉ I + J
a : A
ha : a ∈ I
b : A
hb : b ∈ J
h : ↑(⟨a, ha⟩, ⟨b, hb⟩).1 + ↑(⟨a, ha⟩, ⟨b, hb⟩).2 = x
⊢ False
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_null | [606, 1] | [612, 36] | rw [← h] | case hb.intro.mk.mk.mk
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
n : ℕ
x : A
hx : x ∉ I + J
a : A
ha : a ∈ I
b : A
hb : b ∈ J
h : ↑(⟨a, ha⟩, ⟨b, hb⟩).1 + ↑(⟨a, ha⟩, ⟨b, hb⟩).2 = x
⊢ x ∈ I + J | case hb.intro.mk.mk.mk
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
n : ℕ
x : A
hx : x ∉ I + J
a : A
ha : a ∈ I
b : A
hb : b ∈ J
h : ↑(⟨a, ha⟩, ⟨b, hb⟩).1 + ↑(⟨a, ha⟩, ⟨b, hb⟩).2 = x
⊢ ↑(⟨a, ha⟩, ⟨b, hb⟩).1 + ↑(⟨a, ha⟩, ⟨b, hb⟩).2 ∈ I + J | Please generate a tactic in lean4 to solve the state.
STATE:
case hb.intro.mk.mk.mk
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
n : ℕ
x : A
hx : x ∉ I + J
a : A
ha : a ∈ I
b : A
hb : b ∈ J
h : ↑(⟨a, ha⟩, ⟨b, hb⟩).1 + ↑(⟨a, ha⟩, ⟨b, hb⟩).2 = x
⊢ x ∈ I + J
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_null | [606, 1] | [612, 36] | exact Submodule.add_mem_sup ha hb | case hb.intro.mk.mk.mk
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
n : ℕ
x : A
hx : x ∉ I + J
a : A
ha : a ∈ I
b : A
hb : b ∈ J
h : ↑(⟨a, ha⟩, ⟨b, hb⟩).1 + ↑(⟨a, ha⟩, ⟨b, hb⟩).2 = x
⊢ ↑(⟨a, ha⟩, ⟨b, hb⟩).1 + ↑(⟨a, ha⟩, ⟨b, hb⟩).2 ∈ I + J | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hb.intro.mk.mk.mk
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
n : ℕ
x : A
hx : x ∉ I + J
a : A
ha : a ∈ I
b : A
hb : b ∈ J
h : ↑(⟨a, ha⟩, ⟨b, hb⟩).1 + ↑(⟨a, ha⟩, ⟨b, hb⟩).2 = x
⊢ ↑(⟨a, ha⟩, ⟨b, hb⟩).1... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_one | [614, 1] | [627, 73] | rw [Ideal.add_eq_sup, Submodule.mem_sup] at hx | A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
x : A
hx : x ∈ I + J
⊢ dpow hI hJ 1 x = x | A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
x : A
hx : ∃ y ∈ I, ∃ z ∈ J, y + z = x
⊢ dpow hI hJ 1 x = x | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
x : A
hx : x ∈ I + J
⊢ dpow hI hJ 1 x = x
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_one | [614, 1] | [627, 73] | obtain ⟨a, ha, b, hb, rfl⟩ := hx | A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
x : A
hx : ∃ y ∈ I, ∃ z ∈ J, y + z = x
⊢ dpow hI hJ 1 x = x | case intro.intro.intro.intro
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
a : A
ha : a ∈ I
b : A
hb : b ∈ J
⊢ dpow hI hJ 1 (a + b) = a + b | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
x : A
hx : ∃ y ∈ I, ∃ z ∈ J, y + z = x
⊢ dpow hI hJ 1 x = x
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_one | [614, 1] | [627, 73] | rw [dpow_eq hI hJ hIJ _ ha hb] | case intro.intro.intro.intro
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
a : A
ha : a ∈ I
b : A
hb : b ∈ J
⊢ dpow hI hJ 1 (a + b) = a + b | case intro.intro.intro.intro
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
a : A
ha : a ∈ I
b : A
hb : b ∈ J
⊢ ∑ k ∈ range (1 + 1), hI.dpow k a * hJ.dpow (1 - k) b = a + b | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
a : A
ha : a ∈ I
b : A
hb : b ∈ J
⊢ dpow hI hJ 1 (a + b) = a + b
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_one | [614, 1] | [627, 73] | suffices h : Finset.range (1 + 1) = {0, 1} by
rw [h]
simp only [Finset.sum_insert, Finset.mem_singleton, Nat.zero_ne_one, not_false_iff, tsub_zero,
Finset.sum_singleton, tsub_self]
rw [hI.dpow_zero ha, hI.dpow_one ha, hJ.dpow_zero hb, hJ.dpow_one hb]
ring | case intro.intro.intro.intro
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
a : A
ha : a ∈ I
b : A
hb : b ∈ J
⊢ ∑ k ∈ range (1 + 1), hI.dpow k a * hJ.dpow (1 - k) b = a + b | case intro.intro.intro.intro
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
a : A
ha : a ∈ I
b : A
hb : b ∈ J
⊢ range (1 + 1) = {0, 1} | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
a : A
ha : a ∈ I
b : A
hb : b ∈ J
⊢ ∑ k ∈ range (1 + 1), hI.dpow k a * hJ.dp... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_one | [614, 1] | [627, 73] | rw [h] | A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
a : A
ha : a ∈ I
b : A
hb : b ∈ J
h : range (1 + 1) = {0, 1}
⊢ ∑ k ∈ range (1 + 1), hI.dpow k a * hJ.dpow (1 - k) b = a + b | A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
a : A
ha : a ∈ I
b : A
hb : b ∈ J
h : range (1 + 1) = {0, 1}
⊢ ∑ k ∈ {0, 1}, hI.dpow k a * hJ.dpow (1 - k) b = a + b | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
a : A
ha : a ∈ I
b : A
hb : b ∈ J
h : range (1 + 1) = {0, 1}
⊢ ∑ k ∈ range (1 + 1), hI.dpow k a * hJ.dpow... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_one | [614, 1] | [627, 73] | simp only [Finset.sum_insert, Finset.mem_singleton, Nat.zero_ne_one, not_false_iff, tsub_zero,
Finset.sum_singleton, tsub_self] | A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
a : A
ha : a ∈ I
b : A
hb : b ∈ J
h : range (1 + 1) = {0, 1}
⊢ ∑ k ∈ {0, 1}, hI.dpow k a * hJ.dpow (1 - k) b = a + b | A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
a : A
ha : a ∈ I
b : A
hb : b ∈ J
h : range (1 + 1) = {0, 1}
⊢ hI.dpow 0 a * hJ.dpow 1 b + hI.dpow 1 a * hJ.dpow 0 b = a + b | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
a : A
ha : a ∈ I
b : A
hb : b ∈ J
h : range (1 + 1) = {0, 1}
⊢ ∑ k ∈ {0, 1}, hI.dpow k a * hJ.dpow (1 - k... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_one | [614, 1] | [627, 73] | rw [hI.dpow_zero ha, hI.dpow_one ha, hJ.dpow_zero hb, hJ.dpow_one hb] | A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
a : A
ha : a ∈ I
b : A
hb : b ∈ J
h : range (1 + 1) = {0, 1}
⊢ hI.dpow 0 a * hJ.dpow 1 b + hI.dpow 1 a * hJ.dpow 0 b = a + b | A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
a : A
ha : a ∈ I
b : A
hb : b ∈ J
h : range (1 + 1) = {0, 1}
⊢ 1 * b + a * 1 = a + b | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
a : A
ha : a ∈ I
b : A
hb : b ∈ J
h : range (1 + 1) = {0, 1}
⊢ hI.dpow 0 a * hJ.dpow 1 b + hI.dpow 1 a * ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_one | [614, 1] | [627, 73] | ring | A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
a : A
ha : a ∈ I
b : A
hb : b ∈ J
h : range (1 + 1) = {0, 1}
⊢ 1 * b + a * 1 = a + b | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
a : A
ha : a ∈ I
b : A
hb : b ∈ J
h : range (1 + 1) = {0, 1}
⊢ 1 * b + a * 1 = a + b
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_one | [614, 1] | [627, 73] | rw [Finset.range_succ, Finset.range_one] | case intro.intro.intro.intro
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
a : A
ha : a ∈ I
b : A
hb : b ∈ J
⊢ range (1 + 1) = {0, 1} | case intro.intro.intro.intro
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
a : A
ha : a ∈ I
b : A
hb : b ∈ J
⊢ {1, 0} = {0, 1} | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
a : A
ha : a ∈ I
b : A
hb : b ∈ J
⊢ range (1 + 1) = {0, 1}
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_one | [614, 1] | [627, 73] | ext k | case intro.intro.intro.intro
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
a : A
ha : a ∈ I
b : A
hb : b ∈ J
⊢ {1, 0} = {0, 1} | case intro.intro.intro.intro.a
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
a : A
ha : a ∈ I
b : A
hb : b ∈ J
k : ℕ
⊢ k ∈ {1, 0} ↔ k ∈ {0, 1} | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
a : A
ha : a ∈ I
b : A
hb : b ∈ J
⊢ {1, 0} = {0, 1}
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_one | [614, 1] | [627, 73] | simp | case intro.intro.intro.intro.a
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
a : A
ha : a ∈ I
b : A
hb : b ∈ J
k : ℕ
⊢ k ∈ {1, 0} ↔ k ∈ {0, 1} | case intro.intro.intro.intro.a
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
a : A
ha : a ∈ I
b : A
hb : b ∈ J
k : ℕ
⊢ k = 1 ∨ k = 0 ↔ k = 0 ∨ k = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.a
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
a : A
ha : a ∈ I
b : A
hb : b ∈ J
k : ℕ
⊢ k ∈ {1, 0} ↔ k ∈ {0, 1}
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_one | [614, 1] | [627, 73] | exact or_comm | case intro.intro.intro.intro.a
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
a : A
ha : a ∈ I
b : A
hb : b ∈ J
k : ℕ
⊢ k = 1 ∨ k = 0 ↔ k = 0 ∨ k = 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.a
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
a : A
ha : a ∈ I
b : A
hb : b ∈ J
k : ℕ
⊢ k = 1 ∨ k = 0 ↔ k = 0 ∨ k = 1
TA... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_unique | [643, 1] | [660, 63] | rw [Submodule.mem_inf] at ha | A : Type ?u.129773
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hsup : DividedPowers (I + J)
hI' : ∀ (n : ℕ), ∀ a ∈ I, hI.dpow n a = hsup.dpow n a
hJ' : ∀ (n : ℕ), ∀ b ∈ J, hJ.dpow n b = hsup.dpow n b
n : ℕ
a : A
ha : a ∈ I ⊓ J
⊢ hI.dpow n a = hJ.dpow n a | A : Type ?u.129773
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hsup : DividedPowers (I + J)
hI' : ∀ (n : ℕ), ∀ a ∈ I, hI.dpow n a = hsup.dpow n a
hJ' : ∀ (n : ℕ), ∀ b ∈ J, hJ.dpow n b = hsup.dpow n b
n : ℕ
a : A
ha : a ∈ I ∧ a ∈ J
⊢ hI.dpow n a = hJ.dpow n a | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type ?u.129773
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hsup : DividedPowers (I + J)
hI' : ∀ (n : ℕ), ∀ a ∈ I, hI.dpow n a = hsup.dpow n a
hJ' : ∀ (n : ℕ), ∀ b ∈ J, hJ.dpow n b = hsup.dpow n b
n : ℕ
a : A
ha : a... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_unique | [643, 1] | [660, 63] | rw [hI' _ _ ha.1, hJ' _ _ ha.2] | A : Type ?u.129773
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hsup : DividedPowers (I + J)
hI' : ∀ (n : ℕ), ∀ a ∈ I, hI.dpow n a = hsup.dpow n a
hJ' : ∀ (n : ℕ), ∀ b ∈ J, hJ.dpow n b = hsup.dpow n b
n : ℕ
a : A
ha : a ∈ I ∧ a ∈ J
⊢ hI.dpow n a = hJ.dpow n a | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type ?u.129773
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hsup : DividedPowers (I + J)
hI' : ∀ (n : ℕ), ∀ a ∈ I, hI.dpow n a = hsup.dpow n a
hJ' : ∀ (n : ℕ), ∀ b ∈ J, hJ.dpow n b = hsup.dpow n b
n : ℕ
a : A
ha : a... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_unique | [643, 1] | [660, 63] | intro hIJ | A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hsup : DividedPowers (I + J)
hI' : ∀ (n : ℕ), ∀ a ∈ I, hI.dpow n a = hsup.dpow n a
hJ' : ∀ (n : ℕ), ∀ b ∈ J, hJ.dpow n b = hsup.dpow n b
⊢ let hIJ := ⋯;
hsup = dividedPowers hI hJ hIJ | A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hsup : DividedPowers (I + J)
hI' : ∀ (n : ℕ), ∀ a ∈ I, hI.dpow n a = hsup.dpow n a
hJ' : ∀ (n : ℕ), ∀ b ∈ J, hJ.dpow n b = hsup.dpow n b
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a :=
fun n a ha =>
Eq.mpr... | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hsup : DividedPowers (I + J)
hI' : ∀ (n : ℕ), ∀ a ∈ I, hI.dpow n a = hsup.dpow n a
hJ' : ∀ (n : ℕ), ∀ b ∈ J, hJ.dpow n b = hsup.dpow n b
⊢ let hIJ := ⋯;
hsup =... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_unique | [643, 1] | [660, 63] | apply eq_of_eq_on_ideal | A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hsup : DividedPowers (I + J)
hI' : ∀ (n : ℕ), ∀ a ∈ I, hI.dpow n a = hsup.dpow n a
hJ' : ∀ (n : ℕ), ∀ b ∈ J, hJ.dpow n b = hsup.dpow n b
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a :=
fun n a ha =>
Eq.mpr... | case h_eq
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hsup : DividedPowers (I + J)
hI' : ∀ (n : ℕ), ∀ a ∈ I, hI.dpow n a = hsup.dpow n a
hJ' : ∀ (n : ℕ), ∀ b ∈ J, hJ.dpow n b = hsup.dpow n b
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a :=
fun n a ha =>
... | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hsup : DividedPowers (I + J)
hI' : ∀ (n : ℕ), ∀ a ∈ I, hI.dpow n a = hsup.dpow n a
hJ' : ∀ (n : ℕ), ∀ b ∈ J, hJ.dpow n b = hsup.dpow n b
hIJ : ∀ (n : ℕ), ∀ a ∈ I... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_unique | [643, 1] | [660, 63] | intro n x hx | case h_eq
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hsup : DividedPowers (I + J)
hI' : ∀ (n : ℕ), ∀ a ∈ I, hI.dpow n a = hsup.dpow n a
hJ' : ∀ (n : ℕ), ∀ b ∈ J, hJ.dpow n b = hsup.dpow n b
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a :=
fun n a ha =>
... | case h_eq
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hsup : DividedPowers (I + J)
hI' : ∀ (n : ℕ), ∀ a ∈ I, hI.dpow n a = hsup.dpow n a
hJ' : ∀ (n : ℕ), ∀ b ∈ J, hJ.dpow n b = hsup.dpow n b
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a :=
fun n a ha =>
... | Please generate a tactic in lean4 to solve the state.
STATE:
case h_eq
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hsup : DividedPowers (I + J)
hI' : ∀ (n : ℕ), ∀ a ∈ I, hI.dpow n a = hsup.dpow n a
hJ' : ∀ (n : ℕ), ∀ b ∈ J, hJ.dpow n b = hsup.dpow n b
hIJ : ∀ (n : ℕ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_unique | [643, 1] | [660, 63] | rw [Ideal.add_eq_sup, Submodule.mem_sup] at hx | case h_eq
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hsup : DividedPowers (I + J)
hI' : ∀ (n : ℕ), ∀ a ∈ I, hI.dpow n a = hsup.dpow n a
hJ' : ∀ (n : ℕ), ∀ b ∈ J, hJ.dpow n b = hsup.dpow n b
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a :=
fun n a ha =>
... | case h_eq
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hsup : DividedPowers (I + J)
hI' : ∀ (n : ℕ), ∀ a ∈ I, hI.dpow n a = hsup.dpow n a
hJ' : ∀ (n : ℕ), ∀ b ∈ J, hJ.dpow n b = hsup.dpow n b
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a :=
fun n a ha =>
... | Please generate a tactic in lean4 to solve the state.
STATE:
case h_eq
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hsup : DividedPowers (I + J)
hI' : ∀ (n : ℕ), ∀ a ∈ I, hI.dpow n a = hsup.dpow n a
hJ' : ∀ (n : ℕ), ∀ b ∈ J, hJ.dpow n b = hsup.dpow n b
hIJ : ∀ (n : ℕ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_unique | [643, 1] | [660, 63] | obtain ⟨a, ha, b, hb, rfl⟩ := hx | case h_eq
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hsup : DividedPowers (I + J)
hI' : ∀ (n : ℕ), ∀ a ∈ I, hI.dpow n a = hsup.dpow n a
hJ' : ∀ (n : ℕ), ∀ b ∈ J, hJ.dpow n b = hsup.dpow n b
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a :=
fun n a ha =>
... | case h_eq.intro.intro.intro.intro
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hsup : DividedPowers (I + J)
hI' : ∀ (n : ℕ), ∀ a ∈ I, hI.dpow n a = hsup.dpow n a
hJ' : ∀ (n : ℕ), ∀ b ∈ J, hJ.dpow n b = hsup.dpow n b
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow... | Please generate a tactic in lean4 to solve the state.
STATE:
case h_eq
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hsup : DividedPowers (I + J)
hI' : ∀ (n : ℕ), ∀ a ∈ I, hI.dpow n a = hsup.dpow n a
hJ' : ∀ (n : ℕ), ∀ b ∈ J, hJ.dpow n b = hsup.dpow n b
hIJ : ∀ (n : ℕ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_unique | [643, 1] | [660, 63] | rw [hsup.dpow_add' n (Submodule.mem_sup_left ha) (Submodule.mem_sup_right hb)] | case h_eq.intro.intro.intro.intro
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hsup : DividedPowers (I + J)
hI' : ∀ (n : ℕ), ∀ a ∈ I, hI.dpow n a = hsup.dpow n a
hJ' : ∀ (n : ℕ), ∀ b ∈ J, hJ.dpow n b = hsup.dpow n b
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow... | case h_eq.intro.intro.intro.intro
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hsup : DividedPowers (I + J)
hI' : ∀ (n : ℕ), ∀ a ∈ I, hI.dpow n a = hsup.dpow n a
hJ' : ∀ (n : ℕ), ∀ b ∈ J, hJ.dpow n b = hsup.dpow n b
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow... | Please generate a tactic in lean4 to solve the state.
STATE:
case h_eq.intro.intro.intro.intro
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hsup : DividedPowers (I + J)
hI' : ∀ (n : ℕ), ∀ a ∈ I, hI.dpow n a = hsup.dpow n a
hJ' : ∀ (n : ℕ), ∀ b ∈ J, hJ.dpow n b = hsup... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_unique | [643, 1] | [660, 63] | simp only [IdealAdd.dividedPowers, dpow_eq hI hJ hIJ n ha hb] | case h_eq.intro.intro.intro.intro
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hsup : DividedPowers (I + J)
hI' : ∀ (n : ℕ), ∀ a ∈ I, hI.dpow n a = hsup.dpow n a
hJ' : ∀ (n : ℕ), ∀ b ∈ J, hJ.dpow n b = hsup.dpow n b
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow... | case h_eq.intro.intro.intro.intro
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hsup : DividedPowers (I + J)
hI' : ∀ (n : ℕ), ∀ a ∈ I, hI.dpow n a = hsup.dpow n a
hJ' : ∀ (n : ℕ), ∀ b ∈ J, hJ.dpow n b = hsup.dpow n b
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow... | Please generate a tactic in lean4 to solve the state.
STATE:
case h_eq.intro.intro.intro.intro
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hsup : DividedPowers (I + J)
hI' : ∀ (n : ℕ), ∀ a ∈ I, hI.dpow n a = hsup.dpow n a
hJ' : ∀ (n : ℕ), ∀ b ∈ J, hJ.dpow n b = hsup... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_unique | [643, 1] | [660, 63] | apply Finset.sum_congr rfl | case h_eq.intro.intro.intro.intro
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hsup : DividedPowers (I + J)
hI' : ∀ (n : ℕ), ∀ a ∈ I, hI.dpow n a = hsup.dpow n a
hJ' : ∀ (n : ℕ), ∀ b ∈ J, hJ.dpow n b = hsup.dpow n b
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow... | case h_eq.intro.intro.intro.intro
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hsup : DividedPowers (I + J)
hI' : ∀ (n : ℕ), ∀ a ∈ I, hI.dpow n a = hsup.dpow n a
hJ' : ∀ (n : ℕ), ∀ b ∈ J, hJ.dpow n b = hsup.dpow n b
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow... | Please generate a tactic in lean4 to solve the state.
STATE:
case h_eq.intro.intro.intro.intro
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hsup : DividedPowers (I + J)
hI' : ∀ (n : ℕ), ∀ a ∈ I, hI.dpow n a = hsup.dpow n a
hJ' : ∀ (n : ℕ), ∀ b ∈ J, hJ.dpow n b = hsup... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_unique | [643, 1] | [660, 63] | intro k _ | case h_eq.intro.intro.intro.intro
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hsup : DividedPowers (I + J)
hI' : ∀ (n : ℕ), ∀ a ∈ I, hI.dpow n a = hsup.dpow n a
hJ' : ∀ (n : ℕ), ∀ b ∈ J, hJ.dpow n b = hsup.dpow n b
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow... | case h_eq.intro.intro.intro.intro
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hsup : DividedPowers (I + J)
hI' : ∀ (n : ℕ), ∀ a ∈ I, hI.dpow n a = hsup.dpow n a
hJ' : ∀ (n : ℕ), ∀ b ∈ J, hJ.dpow n b = hsup.dpow n b
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow... | Please generate a tactic in lean4 to solve the state.
STATE:
case h_eq.intro.intro.intro.intro
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hsup : DividedPowers (I + J)
hI' : ∀ (n : ℕ), ∀ a ∈ I, hI.dpow n a = hsup.dpow n a
hJ' : ∀ (n : ℕ), ∀ b ∈ J, hJ.dpow n b = hsup... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/Topology/LinearTopology.lean | TopologicalSpace.le_iff_nhds_le | [34, 1] | [48, 58] | rw [le_iff_nhds] | α : Type u_1
τ τ' : TopologicalSpace α
⊢ τ ≤ τ' ↔ ∀ (s : Set α), ∀ a ∈ s, s ∈ nhds a → s ∈ nhds a | α : Type u_1
τ τ' : TopologicalSpace α
⊢ (∀ (x : α), nhds x ≤ nhds x) ↔ ∀ (s : Set α), ∀ a ∈ s, s ∈ nhds a → s ∈ nhds a | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
τ τ' : TopologicalSpace α
⊢ τ ≤ τ' ↔ ∀ (s : Set α), ∀ a ∈ s, s ∈ nhds a → s ∈ nhds a
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/Topology/LinearTopology.lean | TopologicalSpace.le_iff_nhds_le | [34, 1] | [48, 58] | rw [forall_comm] | α : Type u_1
τ τ' : TopologicalSpace α
⊢ (∀ (x : α), nhds x ≤ nhds x) ↔ ∀ (s : Set α), ∀ a ∈ s, s ∈ nhds a → s ∈ nhds a | α : Type u_1
τ τ' : TopologicalSpace α
⊢ (∀ (x : α), nhds x ≤ nhds x) ↔ ∀ (b : α) (a : Set α), b ∈ a → a ∈ nhds b → a ∈ nhds b | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
τ τ' : TopologicalSpace α
⊢ (∀ (x : α), nhds x ≤ nhds x) ↔ ∀ (s : Set α), ∀ a ∈ s, s ∈ nhds a → s ∈ nhds a
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/Topology/LinearTopology.lean | TopologicalSpace.le_iff_nhds_le | [34, 1] | [48, 58] | apply forall_congr' | α : Type u_1
τ τ' : TopologicalSpace α
⊢ (∀ (x : α), nhds x ≤ nhds x) ↔ ∀ (b : α) (a : Set α), b ∈ a → a ∈ nhds b → a ∈ nhds b | case h
α : Type u_1
τ τ' : TopologicalSpace α
⊢ ∀ (a : α), nhds a ≤ nhds a ↔ ∀ (a_1 : Set α), a ∈ a_1 → a_1 ∈ nhds a → a_1 ∈ nhds a | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
τ τ' : TopologicalSpace α
⊢ (∀ (x : α), nhds x ≤ nhds x) ↔ ∀ (b : α) (a : Set α), b ∈ a → a ∈ nhds b → a ∈ nhds b
TACTIC:
|
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