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/RatAlgebra.lean | DividedPowers.OfInvertibleFactorial.dpow_comp_dif_pos | [156, 1] | [172, 39] | rw [mul_pow, ← pow_mul, mul_comm k, ← mul_assoc, ← mul_assoc] | case neg
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n : ℕ
hn_fac : IsUnit ↑(n - 1)!
m k : ℕ
hk : k ≠ 0
hkm : m * k < n
x : A
hx : x ∈ I
hmn : m < n
hm0 : ¬m = 0
hkn : k < n
⊢ Ring.inverse ↑m ! * (Ring.inverse ↑k ! * x ^ k) ^ m = ↑(mchoose m k) * (Ring.inverse ↑(m * k)! * x ^ (m * k)) | case neg
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n : ℕ
hn_fac : IsUnit ↑(n - 1)!
m k : ℕ
hk : k ≠ 0
hkm : m * k < n
x : A
hx : x ∈ I
hmn : m < n
hm0 : ¬m = 0
hkn : k < n
⊢ Ring.inverse ↑m ! * Ring.inverse ↑k ! ^ m * x ^ (m * k) = ↑(mchoose m k) * Ring.inverse ↑(m * k)! * x ^ (m * k) | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n : ℕ
hn_fac : IsUnit ↑(n - 1)!
m k : ℕ
hk : k ≠ 0
hkm : m * k < n
x : A
hx : x ∈ I
hmn : m < n
hm0 : ¬m = 0
hkn : k < n
⊢ Ring.inverse ↑m ! * (Ring.inverse ↑k ! * x ^ k) ^ m = ↑(mchoose m k) * (Ring.in... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/RatAlgebra.lean | DividedPowers.OfInvertibleFactorial.dpow_comp_dif_pos | [156, 1] | [172, 39] | apply congr_arg₂ _ _ rfl | case neg
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n : ℕ
hn_fac : IsUnit ↑(n - 1)!
m k : ℕ
hk : k ≠ 0
hkm : m * k < n
x : A
hx : x ∈ I
hmn : m < n
hm0 : ¬m = 0
hkn : k < n
⊢ Ring.inverse ↑m ! * Ring.inverse ↑k ! ^ m * x ^ (m * k) = ↑(mchoose m k) * Ring.inverse ↑(m * k)! * x ^ (m * k) | A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n : ℕ
hn_fac : IsUnit ↑(n - 1)!
m k : ℕ
hk : k ≠ 0
hkm : m * k < n
x : A
hx : x ∈ I
hmn : m < n
hm0 : ¬m = 0
hkn : k < n
⊢ Ring.inverse ↑m ! * Ring.inverse ↑k ! ^ m = ↑(mchoose m k) * Ring.inverse ↑(m * k)! | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n : ℕ
hn_fac : IsUnit ↑(n - 1)!
m k : ℕ
hk : k ≠ 0
hkm : m * k < n
x : A
hx : x ∈ I
hmn : m < n
hm0 : ¬m = 0
hkn : k < n
⊢ Ring.inverse ↑m ! * Ring.inverse ↑k ! ^ m * x ^ (m * k) = ↑(mchoose m k) * Ring... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/RatAlgebra.lean | DividedPowers.OfInvertibleFactorial.dpow_comp_dif_pos | [156, 1] | [172, 39] | rw [Ring.eq_mul_inverse_iff_mul_eq _ _ _ (factorial_isUnit hn_fac hkm),
mul_assoc, Ring.inverse_mul_eq_iff_eq_mul _ _ _ (factorial_isUnit hn_fac hmn)] | A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n : ℕ
hn_fac : IsUnit ↑(n - 1)!
m k : ℕ
hk : k ≠ 0
hkm : m * k < n
x : A
hx : x ∈ I
hmn : m < n
hm0 : ¬m = 0
hkn : k < n
⊢ Ring.inverse ↑m ! * Ring.inverse ↑k ! ^ m = ↑(mchoose m k) * Ring.inverse ↑(m * k)! | A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n : ℕ
hn_fac : IsUnit ↑(n - 1)!
m k : ℕ
hk : k ≠ 0
hkm : m * k < n
x : A
hx : x ∈ I
hmn : m < n
hm0 : ¬m = 0
hkn : k < n
⊢ Ring.inverse ↑k ! ^ m * ↑(m * k)! = ↑m ! * ↑(mchoose m k) | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n : ℕ
hn_fac : IsUnit ↑(n - 1)!
m k : ℕ
hk : k ≠ 0
hkm : m * k < n
x : A
hx : x ∈ I
hmn : m < n
hm0 : ¬m = 0
hkn : k < n
⊢ Ring.inverse ↑m ! * Ring.inverse ↑k ! ^ m = ↑(mchoose m k) * Ring.inverse ↑(m * k)!
TACT... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/RatAlgebra.lean | DividedPowers.OfInvertibleFactorial.dpow_comp_dif_pos | [156, 1] | [172, 39] | rw [Ring.inverse_pow_mul_eq_iff_eq_mul _ _ (factorial_isUnit hn_fac hkn)] | A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n : ℕ
hn_fac : IsUnit ↑(n - 1)!
m k : ℕ
hk : k ≠ 0
hkm : m * k < n
x : A
hx : x ∈ I
hmn : m < n
hm0 : ¬m = 0
hkn : k < n
⊢ Ring.inverse ↑k ! ^ m * ↑(m * k)! = ↑m ! * ↑(mchoose m k) | A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n : ℕ
hn_fac : IsUnit ↑(n - 1)!
m k : ℕ
hk : k ≠ 0
hkm : m * k < n
x : A
hx : x ∈ I
hmn : m < n
hm0 : ¬m = 0
hkn : k < n
⊢ ↑(m * k)! = ↑k ! ^ m * (↑m ! * ↑(mchoose m k)) | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n : ℕ
hn_fac : IsUnit ↑(n - 1)!
m k : ℕ
hk : k ≠ 0
hkm : m * k < n
x : A
hx : x ∈ I
hmn : m < n
hm0 : ¬m = 0
hkn : k < n
⊢ Ring.inverse ↑k ! ^ m * ↑(m * k)! = ↑m ! * ↑(mchoose m k)
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/RatAlgebra.lean | DividedPowers.OfInvertibleFactorial.dpow_comp_dif_pos | [156, 1] | [172, 39] | rw [← mchoose_lemma _ hk] | A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n : ℕ
hn_fac : IsUnit ↑(n - 1)!
m k : ℕ
hk : k ≠ 0
hkm : m * k < n
x : A
hx : x ∈ I
hmn : m < n
hm0 : ¬m = 0
hkn : k < n
⊢ ↑(m * k)! = ↑k ! ^ m * (↑m ! * ↑(mchoose m k)) | A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n : ℕ
hn_fac : IsUnit ↑(n - 1)!
m k : ℕ
hk : k ≠ 0
hkm : m * k < n
x : A
hx : x ∈ I
hmn : m < n
hm0 : ¬m = 0
hkn : k < n
⊢ ↑(m ! * k ! ^ m * mchoose m k) = ↑k ! ^ m * (↑m ! * ↑(mchoose m k)) | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n : ℕ
hn_fac : IsUnit ↑(n - 1)!
m k : ℕ
hk : k ≠ 0
hkm : m * k < n
x : A
hx : x ∈ I
hmn : m < n
hm0 : ¬m = 0
hkn : k < n
⊢ ↑(m * k)! = ↑k ! ^ m * (↑m ! * ↑(mchoose m k))
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/RatAlgebra.lean | DividedPowers.OfInvertibleFactorial.dpow_comp_dif_pos | [156, 1] | [172, 39] | simp only [Nat.cast_mul, Nat.cast_pow] | A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n : ℕ
hn_fac : IsUnit ↑(n - 1)!
m k : ℕ
hk : k ≠ 0
hkm : m * k < n
x : A
hx : x ∈ I
hmn : m < n
hm0 : ¬m = 0
hkn : k < n
⊢ ↑(m ! * k ! ^ m * mchoose m k) = ↑k ! ^ m * (↑m ! * ↑(mchoose m k)) | A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n : ℕ
hn_fac : IsUnit ↑(n - 1)!
m k : ℕ
hk : k ≠ 0
hkm : m * k < n
x : A
hx : x ∈ I
hmn : m < n
hm0 : ¬m = 0
hkn : k < n
⊢ ↑m ! * ↑k ! ^ m * ↑(mchoose m k) = ↑k ! ^ m * (↑m ! * ↑(mchoose m k)) | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n : ℕ
hn_fac : IsUnit ↑(n - 1)!
m k : ℕ
hk : k ≠ 0
hkm : m * k < n
x : A
hx : x ∈ I
hmn : m < n
hm0 : ¬m = 0
hkn : k < n
⊢ ↑(m ! * k ! ^ m * mchoose m k) = ↑k ! ^ m * (↑m ! * ↑(mchoose m k))
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/RatAlgebra.lean | DividedPowers.OfInvertibleFactorial.dpow_comp_dif_pos | [156, 1] | [172, 39] | rw [mul_comm (m ! : A), mul_assoc] | A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n : ℕ
hn_fac : IsUnit ↑(n - 1)!
m k : ℕ
hk : k ≠ 0
hkm : m * k < n
x : A
hx : x ∈ I
hmn : m < n
hm0 : ¬m = 0
hkn : k < n
⊢ ↑m ! * ↑k ! ^ m * ↑(mchoose m k) = ↑k ! ^ m * (↑m ! * ↑(mchoose m k)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n : ℕ
hn_fac : IsUnit ↑(n - 1)!
m k : ℕ
hk : k ≠ 0
hkm : m * k < n
x : A
hx : x ∈ I
hmn : m < n
hm0 : ¬m = 0
hkn : k < n
⊢ ↑m ! * ↑k ! ^ m * ↑(mchoose m k) = ↑k ! ^ m * (↑m ! * ↑(mchoose m k))
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/RatAlgebra.lean | DividedPowers.OfInvertibleFactorial.dpow_comp | [175, 1] | [183, 75] | by_cases hmk : m * k < n | A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n : ℕ
hn_fac : IsUnit ↑(n - 1)!
hnI : I ^ n = 0
m k : ℕ
hk : k ≠ 0
x : A
hx : x ∈ I
⊢ dpow I m (dpow I k x) = ↑(mchoose m k) * dpow I (m * k) x | case pos
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n : ℕ
hn_fac : IsUnit ↑(n - 1)!
hnI : I ^ n = 0
m k : ℕ
hk : k ≠ 0
x : A
hx : x ∈ I
hmk : m * k < n
⊢ dpow I m (dpow I k x) = ↑(mchoose m k) * dpow I (m * k) x
case neg
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n : ℕ
hn_fac : IsUnit ↑(n - 1)!
hnI : I ^ n =... | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n : ℕ
hn_fac : IsUnit ↑(n - 1)!
hnI : I ^ n = 0
m k : ℕ
hk : k ≠ 0
x : A
hx : x ∈ I
⊢ dpow I m (dpow I k x) = ↑(mchoose m k) * dpow I (m * k) x
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/RatAlgebra.lean | DividedPowers.OfInvertibleFactorial.dpow_comp | [175, 1] | [183, 75] | exact dpow_comp_dif_pos hn_fac hk hmk hx | case pos
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n : ℕ
hn_fac : IsUnit ↑(n - 1)!
hnI : I ^ n = 0
m k : ℕ
hk : k ≠ 0
x : A
hx : x ∈ I
hmk : m * k < n
⊢ dpow I m (dpow I k x) = ↑(mchoose m k) * dpow I (m * k) x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n : ℕ
hn_fac : IsUnit ↑(n - 1)!
hnI : I ^ n = 0
m k : ℕ
hk : k ≠ 0
x : A
hx : x ∈ I
hmk : m * k < n
⊢ dpow I m (dpow I k x) = ↑(mchoose m k) * dpow I (m * k) x
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/RatAlgebra.lean | DividedPowers.OfInvertibleFactorial.dpow_comp | [175, 1] | [183, 75] | have hxmk : x ^ (m * k) = 0 := Ideal.mem_pow_eq_zero n (m * k) hnI (not_lt.mp hmk) hx | case neg
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n : ℕ
hn_fac : IsUnit ↑(n - 1)!
hnI : I ^ n = 0
m k : ℕ
hk : k ≠ 0
x : A
hx : x ∈ I
hmk : ¬m * k < n
⊢ dpow I m (dpow I k x) = ↑(mchoose m k) * dpow I (m * k) x | case neg
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n : ℕ
hn_fac : IsUnit ↑(n - 1)!
hnI : I ^ n = 0
m k : ℕ
hk : k ≠ 0
x : A
hx : x ∈ I
hmk : ¬m * k < n
hxmk : x ^ (m * k) = 0
⊢ dpow I m (dpow I k x) = ↑(mchoose m k) * dpow I (m * k) x | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n : ℕ
hn_fac : IsUnit ↑(n - 1)!
hnI : I ^ n = 0
m k : ℕ
hk : k ≠ 0
x : A
hx : x ∈ I
hmk : ¬m * k < n
⊢ dpow I m (dpow I k x) = ↑(mchoose m k) * dpow I (m * k) x
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/RatAlgebra.lean | DividedPowers.OfInvertibleFactorial.dpow_comp | [175, 1] | [183, 75] | rw [dpow_eq_of_mem _ (dpow_mem hk hx), dpow_eq_of_mem _ hx, dpow_eq_of_mem _ hx,
mul_pow, ← pow_mul, ← mul_assoc, mul_comm k, hxmk,
MulZeroClass.mul_zero, MulZeroClass.mul_zero, MulZeroClass.mul_zero] | case neg
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n : ℕ
hn_fac : IsUnit ↑(n - 1)!
hnI : I ^ n = 0
m k : ℕ
hk : k ≠ 0
x : A
hx : x ∈ I
hmk : ¬m * k < n
hxmk : x ^ (m * k) = 0
⊢ dpow I m (dpow I k x) = ↑(mchoose m k) * dpow I (m * k) x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
n : ℕ
hn_fac : IsUnit ↑(n - 1)!
hnI : I ^ n = 0
m k : ℕ
hk : k ≠ 0
x : A
hx : x ∈ I
hmk : ¬m * k < n
hxmk : x ^ (m * k) = 0
⊢ dpow I m (dpow I k x) = ↑(mchoose m k) * dpow I (m * k) x
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/RatAlgebra.lean | DividedPowers.OfSquareZero.dpow_of_two_le | [215, 1] | [220, 44] | simp only [dividedPowers, OfInvertibleFactorial.dpow_def, ite_eq_right_iff] | A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI2 : I ^ 2 = 0
n : ℕ
hn : 2 ≤ n
a : A
⊢ (dividedPowers hI2).dpow n a = 0 | A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI2 : I ^ 2 = 0
n : ℕ
hn : 2 ≤ n
a : A
⊢ a ∈ I → Ring.inverse ↑n ! * a ^ n = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI2 : I ^ 2 = 0
n : ℕ
hn : 2 ≤ n
a : A
⊢ (dividedPowers hI2).dpow n a = 0
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/RatAlgebra.lean | DividedPowers.OfSquareZero.dpow_of_two_le | [215, 1] | [220, 44] | intro ha | A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI2 : I ^ 2 = 0
n : ℕ
hn : 2 ≤ n
a : A
⊢ a ∈ I → Ring.inverse ↑n ! * a ^ n = 0 | A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI2 : I ^ 2 = 0
n : ℕ
hn : 2 ≤ n
a : A
ha : a ∈ I
⊢ Ring.inverse ↑n ! * a ^ n = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI2 : I ^ 2 = 0
n : ℕ
hn : 2 ≤ n
a : A
⊢ a ∈ I → Ring.inverse ↑n ! * a ^ n = 0
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/RatAlgebra.lean | DividedPowers.OfSquareZero.dpow_of_two_le | [215, 1] | [220, 44] | suffices h : a ^ n = 0 by
rw [h, mul_zero] | A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI2 : I ^ 2 = 0
n : ℕ
hn : 2 ≤ n
a : A
ha : a ∈ I
⊢ Ring.inverse ↑n ! * a ^ n = 0 | A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI2 : I ^ 2 = 0
n : ℕ
hn : 2 ≤ n
a : A
ha : a ∈ I
⊢ a ^ n = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI2 : I ^ 2 = 0
n : ℕ
hn : 2 ≤ n
a : A
ha : a ∈ I
⊢ Ring.inverse ↑n ! * a ^ n = 0
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/RatAlgebra.lean | DividedPowers.OfSquareZero.dpow_of_two_le | [215, 1] | [220, 44] | exact Ideal.mem_pow_eq_zero 2 n hI2 hn ha | A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI2 : I ^ 2 = 0
n : ℕ
hn : 2 ≤ n
a : A
ha : a ∈ I
⊢ a ^ n = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI2 : I ^ 2 = 0
n : ℕ
hn : 2 ≤ n
a : A
ha : a ∈ I
⊢ a ^ n = 0
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/RatAlgebra.lean | DividedPowers.OfSquareZero.dpow_of_two_le | [215, 1] | [220, 44] | rw [h, mul_zero] | A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI2 : I ^ 2 = 0
n : ℕ
hn : 2 ≤ n
a : A
ha : a ∈ I
h : a ^ n = 0
⊢ Ring.inverse ↑n ! * a ^ n = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI2 : I ^ 2 = 0
n : ℕ
hn : 2 ≤ n
a : A
ha : a ∈ I
h : a ^ n = 0
⊢ Ring.inverse ↑n ! * a ^ n = 0
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/RatAlgebra.lean | DividedPowers.RatAlgebra.dpow_eq_of_mem | [236, 1] | [238, 55] | rw [dpow, OfInvertibleFactorial.dpow_eq_of_mem _ hx] | R : Type u_1
inst✝ : CommSemiring R
I : Ideal R
n : ℕ
x : R
hx : x ∈ I
⊢ dpow I n x = Ring.inverse ↑n ! * x ^ n | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝ : CommSemiring R
I : Ideal R
n : ℕ
x : R
hx : x ∈ I
⊢ dpow I n x = Ring.inverse ↑n ! * x ^ n
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/RatAlgebra.lean | DividedPowers.RatAlgebra.dpow_eq_inv_fact_smul | [266, 1] | [282, 30] | simp only [Ring.inverse_eq_inv'] | R : Type u_1
inst✝¹ : CommSemiring R
I : Ideal R
inst✝ : Algebra ℚ R
hI : DividedPowers I
n : ℕ
x : R
hx : x ∈ I
⊢ hI.dpow n x = Ring.inverse ↑n ! • x ^ n | R : Type u_1
inst✝¹ : CommSemiring R
I : Ideal R
inst✝ : Algebra ℚ R
hI : DividedPowers I
n : ℕ
x : R
hx : x ∈ I
⊢ hI.dpow n x = (↑n !)⁻¹ • x ^ n | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝¹ : CommSemiring R
I : Ideal R
inst✝ : Algebra ℚ R
hI : DividedPowers I
n : ℕ
x : R
hx : x ∈ I
⊢ hI.dpow n x = Ring.inverse ↑n ! • x ^ n
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/RatAlgebra.lean | DividedPowers.RatAlgebra.dpow_eq_inv_fact_smul | [266, 1] | [282, 30] | rw [← factorial_mul_dpow_eq_pow hI n x hx] | R : Type u_1
inst✝¹ : CommSemiring R
I : Ideal R
inst✝ : Algebra ℚ R
hI : DividedPowers I
n : ℕ
x : R
hx : x ∈ I
⊢ hI.dpow n x = (↑n !)⁻¹ • x ^ n | R : Type u_1
inst✝¹ : CommSemiring R
I : Ideal R
inst✝ : Algebra ℚ R
hI : DividedPowers I
n : ℕ
x : R
hx : x ∈ I
⊢ hI.dpow n x = (↑n !)⁻¹ • (↑n ! * hI.dpow n x) | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝¹ : CommSemiring R
I : Ideal R
inst✝ : Algebra ℚ R
hI : DividedPowers I
n : ℕ
x : R
hx : x ∈ I
⊢ hI.dpow n x = (↑n !)⁻¹ • x ^ n
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/RatAlgebra.lean | DividedPowers.RatAlgebra.dpow_eq_inv_fact_smul | [266, 1] | [282, 30] | rw [← smul_eq_mul] | R : Type u_1
inst✝¹ : CommSemiring R
I : Ideal R
inst✝ : Algebra ℚ R
hI : DividedPowers I
n : ℕ
x : R
hx : x ∈ I
⊢ hI.dpow n x = (↑n !)⁻¹ • (↑n ! * hI.dpow n x) | R : Type u_1
inst✝¹ : CommSemiring R
I : Ideal R
inst✝ : Algebra ℚ R
hI : DividedPowers I
n : ℕ
x : R
hx : x ∈ I
⊢ hI.dpow n x = (↑n !)⁻¹ • ↑n ! • hI.dpow n x | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝¹ : CommSemiring R
I : Ideal R
inst✝ : Algebra ℚ R
hI : DividedPowers I
n : ℕ
x : R
hx : x ∈ I
⊢ hI.dpow n x = (↑n !)⁻¹ • (↑n ! * hI.dpow n x)
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/RatAlgebra.lean | DividedPowers.RatAlgebra.dpow_eq_inv_fact_smul | [266, 1] | [282, 30] | rw [← smul_assoc] | R : Type u_1
inst✝¹ : CommSemiring R
I : Ideal R
inst✝ : Algebra ℚ R
hI : DividedPowers I
n : ℕ
x : R
hx : x ∈ I
⊢ hI.dpow n x = (↑n !)⁻¹ • ↑n ! • hI.dpow n x | R : Type u_1
inst✝¹ : CommSemiring R
I : Ideal R
inst✝ : Algebra ℚ R
hI : DividedPowers I
n : ℕ
x : R
hx : x ∈ I
⊢ hI.dpow n x = ((↑n !)⁻¹ • ↑n !) • hI.dpow n x | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝¹ : CommSemiring R
I : Ideal R
inst✝ : Algebra ℚ R
hI : DividedPowers I
n : ℕ
x : R
hx : x ∈ I
⊢ hI.dpow n x = (↑n !)⁻¹ • ↑n ! • hI.dpow n x
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/RatAlgebra.lean | DividedPowers.RatAlgebra.dpow_eq_inv_fact_smul | [266, 1] | [282, 30] | nth_rewrite 1 [← one_smul R (hI.dpow n x)] | R : Type u_1
inst✝¹ : CommSemiring R
I : Ideal R
inst✝ : Algebra ℚ R
hI : DividedPowers I
n : ℕ
x : R
hx : x ∈ I
⊢ hI.dpow n x = ((↑n !)⁻¹ • ↑n !) • hI.dpow n x | R : Type u_1
inst✝¹ : CommSemiring R
I : Ideal R
inst✝ : Algebra ℚ R
hI : DividedPowers I
n : ℕ
x : R
hx : x ∈ I
⊢ 1 • hI.dpow n x = ((↑n !)⁻¹ • ↑n !) • hI.dpow n x | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝¹ : CommSemiring R
I : Ideal R
inst✝ : Algebra ℚ R
hI : DividedPowers I
n : ℕ
x : R
hx : x ∈ I
⊢ hI.dpow n x = ((↑n !)⁻¹ • ↑n !) • hI.dpow n x
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/RatAlgebra.lean | DividedPowers.RatAlgebra.dpow_eq_inv_fact_smul | [266, 1] | [282, 30] | congr | R : Type u_1
inst✝¹ : CommSemiring R
I : Ideal R
inst✝ : Algebra ℚ R
hI : DividedPowers I
n : ℕ
x : R
hx : x ∈ I
⊢ 1 • hI.dpow n x = ((↑n !)⁻¹ • ↑n !) • hI.dpow n x | case e_a
R : Type u_1
inst✝¹ : CommSemiring R
I : Ideal R
inst✝ : Algebra ℚ R
hI : DividedPowers I
n : ℕ
x : R
hx : x ∈ I
⊢ 1 = (↑n !)⁻¹ • ↑n ! | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝¹ : CommSemiring R
I : Ideal R
inst✝ : Algebra ℚ R
hI : DividedPowers I
n : ℕ
x : R
hx : x ∈ I
⊢ 1 • hI.dpow n x = ((↑n !)⁻¹ • ↑n !) • hI.dpow n x
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/RatAlgebra.lean | DividedPowers.RatAlgebra.dpow_eq_inv_fact_smul | [266, 1] | [282, 30] | have this_rat : ((n !) : R) = (n ! : ℚ) • (1 : R) := by
rw [← nsmul_eq_smul_cast, nsmul_eq_mul, mul_one] | case e_a
R : Type u_1
inst✝¹ : CommSemiring R
I : Ideal R
inst✝ : Algebra ℚ R
hI : DividedPowers I
n : ℕ
x : R
hx : x ∈ I
⊢ 1 = (↑n !)⁻¹ • ↑n ! | case e_a
R : Type u_1
inst✝¹ : CommSemiring R
I : Ideal R
inst✝ : Algebra ℚ R
hI : DividedPowers I
n : ℕ
x : R
hx : x ∈ I
this_rat : ↑n ! = ↑n ! • 1
⊢ 1 = (↑n !)⁻¹ • ↑n ! | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a
R : Type u_1
inst✝¹ : CommSemiring R
I : Ideal R
inst✝ : Algebra ℚ R
hI : DividedPowers I
n : ℕ
x : R
hx : x ∈ I
⊢ 1 = (↑n !)⁻¹ • ↑n !
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/RatAlgebra.lean | DividedPowers.RatAlgebra.dpow_eq_inv_fact_smul | [266, 1] | [282, 30] | rw [this_rat, ← mul_smul] | case e_a
R : Type u_1
inst✝¹ : CommSemiring R
I : Ideal R
inst✝ : Algebra ℚ R
hI : DividedPowers I
n : ℕ
x : R
hx : x ∈ I
this_rat : ↑n ! = ↑n ! • 1
⊢ 1 = (↑n !)⁻¹ • ↑n ! | case e_a
R : Type u_1
inst✝¹ : CommSemiring R
I : Ideal R
inst✝ : Algebra ℚ R
hI : DividedPowers I
n : ℕ
x : R
hx : x ∈ I
this_rat : ↑n ! = ↑n ! • 1
⊢ 1 = ((↑n !)⁻¹ * ↑n !) • 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a
R : Type u_1
inst✝¹ : CommSemiring R
I : Ideal R
inst✝ : Algebra ℚ R
hI : DividedPowers I
n : ℕ
x : R
hx : x ∈ I
this_rat : ↑n ! = ↑n ! • 1
⊢ 1 = (↑n !)⁻¹ • ↑n !
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/RatAlgebra.lean | DividedPowers.RatAlgebra.dpow_eq_inv_fact_smul | [266, 1] | [282, 30] | suffices (n ! : ℚ)⁻¹ * (n !) = 1 by
rw [this, one_smul] | case e_a
R : Type u_1
inst✝¹ : CommSemiring R
I : Ideal R
inst✝ : Algebra ℚ R
hI : DividedPowers I
n : ℕ
x : R
hx : x ∈ I
this_rat : ↑n ! = ↑n ! • 1
⊢ 1 = ((↑n !)⁻¹ * ↑n !) • 1 | case e_a
R : Type u_1
inst✝¹ : CommSemiring R
I : Ideal R
inst✝ : Algebra ℚ R
hI : DividedPowers I
n : ℕ
x : R
hx : x ∈ I
this_rat : ↑n ! = ↑n ! • 1
⊢ (↑n !)⁻¹ * ↑n ! = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a
R : Type u_1
inst✝¹ : CommSemiring R
I : Ideal R
inst✝ : Algebra ℚ R
hI : DividedPowers I
n : ℕ
x : R
hx : x ∈ I
this_rat : ↑n ! = ↑n ! • 1
⊢ 1 = ((↑n !)⁻¹ * ↑n !) • 1
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/RatAlgebra.lean | DividedPowers.RatAlgebra.dpow_eq_inv_fact_smul | [266, 1] | [282, 30] | apply Rat.inv_mul_cancel | case e_a
R : Type u_1
inst✝¹ : CommSemiring R
I : Ideal R
inst✝ : Algebra ℚ R
hI : DividedPowers I
n : ℕ
x : R
hx : x ∈ I
this_rat : ↑n ! = ↑n ! • 1
⊢ (↑n !)⁻¹ * ↑n ! = 1 | case e_a.h
R : Type u_1
inst✝¹ : CommSemiring R
I : Ideal R
inst✝ : Algebra ℚ R
hI : DividedPowers I
n : ℕ
x : R
hx : x ∈ I
this_rat : ↑n ! = ↑n ! • 1
⊢ ↑n ! ≠ 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a
R : Type u_1
inst✝¹ : CommSemiring R
I : Ideal R
inst✝ : Algebra ℚ R
hI : DividedPowers I
n : ℕ
x : R
hx : x ∈ I
this_rat : ↑n ! = ↑n ! • 1
⊢ (↑n !)⁻¹ * ↑n ! = 1
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/RatAlgebra.lean | DividedPowers.RatAlgebra.dpow_eq_inv_fact_smul | [266, 1] | [282, 30] | rw [← cast_zero, ne_eq] | case e_a.h
R : Type u_1
inst✝¹ : CommSemiring R
I : Ideal R
inst✝ : Algebra ℚ R
hI : DividedPowers I
n : ℕ
x : R
hx : x ∈ I
this_rat : ↑n ! = ↑n ! • 1
⊢ ↑n ! ≠ 0 | case e_a.h
R : Type u_1
inst✝¹ : CommSemiring R
I : Ideal R
inst✝ : Algebra ℚ R
hI : DividedPowers I
n : ℕ
x : R
hx : x ∈ I
this_rat : ↑n ! = ↑n ! • 1
⊢ ¬↑n ! = ↑0 | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a.h
R : Type u_1
inst✝¹ : CommSemiring R
I : Ideal R
inst✝ : Algebra ℚ R
hI : DividedPowers I
n : ℕ
x : R
hx : x ∈ I
this_rat : ↑n ! = ↑n ! • 1
⊢ ↑n ! ≠ 0
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/RatAlgebra.lean | DividedPowers.RatAlgebra.dpow_eq_inv_fact_smul | [266, 1] | [282, 30] | simp only [cast_zero, cast_eq_zero] | case e_a.h
R : Type u_1
inst✝¹ : CommSemiring R
I : Ideal R
inst✝ : Algebra ℚ R
hI : DividedPowers I
n : ℕ
x : R
hx : x ∈ I
this_rat : ↑n ! = ↑n ! • 1
⊢ ¬↑n ! = ↑0 | case e_a.h
R : Type u_1
inst✝¹ : CommSemiring R
I : Ideal R
inst✝ : Algebra ℚ R
hI : DividedPowers I
n : ℕ
x : R
hx : x ∈ I
this_rat : ↑n ! = ↑n ! • 1
⊢ ¬n ! = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a.h
R : Type u_1
inst✝¹ : CommSemiring R
I : Ideal R
inst✝ : Algebra ℚ R
hI : DividedPowers I
n : ℕ
x : R
hx : x ∈ I
this_rat : ↑n ! = ↑n ! • 1
⊢ ¬↑n ! = ↑0
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/RatAlgebra.lean | DividedPowers.RatAlgebra.dpow_eq_inv_fact_smul | [266, 1] | [282, 30] | apply Nat.factorial_ne_zero | case e_a.h
R : Type u_1
inst✝¹ : CommSemiring R
I : Ideal R
inst✝ : Algebra ℚ R
hI : DividedPowers I
n : ℕ
x : R
hx : x ∈ I
this_rat : ↑n ! = ↑n ! • 1
⊢ ¬n ! = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a.h
R : Type u_1
inst✝¹ : CommSemiring R
I : Ideal R
inst✝ : Algebra ℚ R
hI : DividedPowers I
n : ℕ
x : R
hx : x ∈ I
this_rat : ↑n ! = ↑n ! • 1
⊢ ¬n ! = 0
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/RatAlgebra.lean | DividedPowers.RatAlgebra.dpow_eq_inv_fact_smul | [266, 1] | [282, 30] | rw [← nsmul_eq_smul_cast, nsmul_eq_mul, mul_one] | R : Type u_1
inst✝¹ : CommSemiring R
I : Ideal R
inst✝ : Algebra ℚ R
hI : DividedPowers I
n : ℕ
x : R
hx : x ∈ I
⊢ ↑n ! = ↑n ! • 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝¹ : CommSemiring R
I : Ideal R
inst✝ : Algebra ℚ R
hI : DividedPowers I
n : ℕ
x : R
hx : x ∈ I
⊢ ↑n ! = ↑n ! • 1
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/RatAlgebra.lean | DividedPowers.RatAlgebra.dpow_eq_inv_fact_smul | [266, 1] | [282, 30] | rw [this, one_smul] | R : Type u_1
inst✝¹ : CommSemiring R
I : Ideal R
inst✝ : Algebra ℚ R
hI : DividedPowers I
n : ℕ
x : R
hx : x ∈ I
this_rat : ↑n ! = ↑n ! • 1
this : (↑n !)⁻¹ * ↑n ! = 1
⊢ 1 = ((↑n !)⁻¹ * ↑n !) • 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝¹ : CommSemiring R
I : Ideal R
inst✝ : Algebra ℚ R
hI : DividedPowers I
n : ℕ
x : R
hx : x ∈ I
this_rat : ↑n ! = ↑n ! • 1
this : (↑n !)⁻¹ * ↑n ! = 1
⊢ 1 = ((↑n !)⁻¹ * ↑n !) • 1
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/RatAlgebra.lean | DividedPowers.RatAlgebra.dividedPowers_unique | [287, 1] | [293, 40] | apply eq_of_eq_on_ideal | R : Type u_1
inst✝¹ : CommSemiring R
I : Ideal R
inst✝ : Algebra ℚ R
hI : DividedPowers I
⊢ hI = dividedPowers I | case h_eq
R : Type u_1
inst✝¹ : CommSemiring R
I : Ideal R
inst✝ : Algebra ℚ R
hI : DividedPowers I
⊢ ∀ (n : ℕ) {x : R}, x ∈ I → hI.dpow n x = (dividedPowers I).dpow n x | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝¹ : CommSemiring R
I : Ideal R
inst✝ : Algebra ℚ R
hI : DividedPowers I
⊢ hI = dividedPowers I
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/RatAlgebra.lean | DividedPowers.RatAlgebra.dividedPowers_unique | [287, 1] | [293, 40] | intro n x hx | case h_eq
R : Type u_1
inst✝¹ : CommSemiring R
I : Ideal R
inst✝ : Algebra ℚ R
hI : DividedPowers I
⊢ ∀ (n : ℕ) {x : R}, x ∈ I → hI.dpow n x = (dividedPowers I).dpow n x | case h_eq
R : Type u_1
inst✝¹ : CommSemiring R
I : Ideal R
inst✝ : Algebra ℚ R
hI : DividedPowers I
n : ℕ
x : R
hx : x ∈ I
⊢ hI.dpow n x = (dividedPowers I).dpow n x | Please generate a tactic in lean4 to solve the state.
STATE:
case h_eq
R : Type u_1
inst✝¹ : CommSemiring R
I : Ideal R
inst✝ : Algebra ℚ R
hI : DividedPowers I
⊢ ∀ (n : ℕ) {x : R}, x ∈ I → hI.dpow n x = (dividedPowers I).dpow n x
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/RatAlgebra.lean | DividedPowers.RatAlgebra.dividedPowers_unique | [287, 1] | [293, 40] | have hn : IsUnit (n.factorial : R) := Factorial.isUnit n | case h_eq
R : Type u_1
inst✝¹ : CommSemiring R
I : Ideal R
inst✝ : Algebra ℚ R
hI : DividedPowers I
n : ℕ
x : R
hx : x ∈ I
⊢ hI.dpow n x = (dividedPowers I).dpow n x | case h_eq
R : Type u_1
inst✝¹ : CommSemiring R
I : Ideal R
inst✝ : Algebra ℚ R
hI : DividedPowers I
n : ℕ
x : R
hx : x ∈ I
hn : IsUnit ↑n !
⊢ hI.dpow n x = (dividedPowers I).dpow n x | Please generate a tactic in lean4 to solve the state.
STATE:
case h_eq
R : Type u_1
inst✝¹ : CommSemiring R
I : Ideal R
inst✝ : Algebra ℚ R
hI : DividedPowers I
n : ℕ
x : R
hx : x ∈ I
⊢ hI.dpow n x = (dividedPowers I).dpow n x
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/RatAlgebra.lean | DividedPowers.RatAlgebra.dividedPowers_unique | [287, 1] | [293, 40] | rw [dividedPowers_dpow_apply, dpow_eq_of_mem n hx, eq_comm, Ring.inverse_mul_eq_iff_eq_mul _ _ _ hn,
factorial_mul_dpow_eq_pow _ _ _ hx] | case h_eq
R : Type u_1
inst✝¹ : CommSemiring R
I : Ideal R
inst✝ : Algebra ℚ R
hI : DividedPowers I
n : ℕ
x : R
hx : x ∈ I
hn : IsUnit ↑n !
⊢ hI.dpow n x = (dividedPowers I).dpow n x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h_eq
R : Type u_1
inst✝¹ : CommSemiring R
I : Ideal R
inst✝ : Algebra ℚ R
hI : DividedPowers I
n : ℕ
x : R
hx : x ∈ I
hn : IsUnit ↑n !
⊢ hI.dpow n x = (dividedPowers I).dpow n x
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/RobyLemma9.lean | RingHom.ker_eq_ideal_iff | [11, 1] | [21, 27] | constructor | A : Type u_1
B : Type u_2
inst✝¹ : CommRing A
inst✝ : CommRing B
f : A →+* B
I : Ideal A
⊢ ker f = I ↔ ∃ (h : I ≤ ker f), Function.Injective ⇑(Ideal.Quotient.lift I f h) | case mp
A : Type u_1
B : Type u_2
inst✝¹ : CommRing A
inst✝ : CommRing B
f : A →+* B
I : Ideal A
⊢ ker f = I → ∃ (h : I ≤ ker f), Function.Injective ⇑(Ideal.Quotient.lift I f h)
case mpr
A : Type u_1
B : Type u_2
inst✝¹ : CommRing A
inst✝ : CommRing B
f : A →+* B
I : Ideal A
⊢ (∃ (h : I ≤ ker f), Function.Injective ⇑(... | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
B : Type u_2
inst✝¹ : CommRing A
inst✝ : CommRing B
f : A →+* B
I : Ideal A
⊢ ker f = I ↔ ∃ (h : I ≤ ker f), Function.Injective ⇑(Ideal.Quotient.lift I f h)
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/RobyLemma9.lean | RingHom.ker_eq_ideal_iff | [11, 1] | [21, 27] | intro hI | case mp
A : Type u_1
B : Type u_2
inst✝¹ : CommRing A
inst✝ : CommRing B
f : A →+* B
I : Ideal A
⊢ ker f = I → ∃ (h : I ≤ ker f), Function.Injective ⇑(Ideal.Quotient.lift I f h) | case mp
A : Type u_1
B : Type u_2
inst✝¹ : CommRing A
inst✝ : CommRing B
f : A →+* B
I : Ideal A
hI : ker f = I
⊢ ∃ (h : I ≤ ker f), Function.Injective ⇑(Ideal.Quotient.lift I f h) | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
A : Type u_1
B : Type u_2
inst✝¹ : CommRing A
inst✝ : CommRing B
f : A →+* B
I : Ideal A
⊢ ker f = I → ∃ (h : I ≤ ker f), Function.Injective ⇑(Ideal.Quotient.lift I f h)
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/RobyLemma9.lean | RingHom.ker_eq_ideal_iff | [11, 1] | [21, 27] | use le_of_eq hI.symm | case mp
A : Type u_1
B : Type u_2
inst✝¹ : CommRing A
inst✝ : CommRing B
f : A →+* B
I : Ideal A
hI : ker f = I
⊢ ∃ (h : I ≤ ker f), Function.Injective ⇑(Ideal.Quotient.lift I f h) | case h
A : Type u_1
B : Type u_2
inst✝¹ : CommRing A
inst✝ : CommRing B
f : A →+* B
I : Ideal A
hI : ker f = I
⊢ Function.Injective ⇑(Ideal.Quotient.lift I f ⋯) | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
A : Type u_1
B : Type u_2
inst✝¹ : CommRing A
inst✝ : CommRing B
f : A →+* B
I : Ideal A
hI : ker f = I
⊢ ∃ (h : I ≤ ker f), Function.Injective ⇑(Ideal.Quotient.lift I f h)
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/RobyLemma9.lean | RingHom.ker_eq_ideal_iff | [11, 1] | [21, 27] | apply RingHom.lift_injective_of_ker_le_ideal | case h
A : Type u_1
B : Type u_2
inst✝¹ : CommRing A
inst✝ : CommRing B
f : A →+* B
I : Ideal A
hI : ker f = I
⊢ Function.Injective ⇑(Ideal.Quotient.lift I f ⋯) | case h.hI
A : Type u_1
B : Type u_2
inst✝¹ : CommRing A
inst✝ : CommRing B
f : A →+* B
I : Ideal A
hI : ker f = I
⊢ ker f ≤ I | Please generate a tactic in lean4 to solve the state.
STATE:
case h
A : Type u_1
B : Type u_2
inst✝¹ : CommRing A
inst✝ : CommRing B
f : A →+* B
I : Ideal A
hI : ker f = I
⊢ Function.Injective ⇑(Ideal.Quotient.lift I f ⋯)
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/RobyLemma9.lean | RingHom.ker_eq_ideal_iff | [11, 1] | [21, 27] | simp only [hI, le_refl] | case h.hI
A : Type u_1
B : Type u_2
inst✝¹ : CommRing A
inst✝ : CommRing B
f : A →+* B
I : Ideal A
hI : ker f = I
⊢ ker f ≤ I | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.hI
A : Type u_1
B : Type u_2
inst✝¹ : CommRing A
inst✝ : CommRing B
f : A →+* B
I : Ideal A
hI : ker f = I
⊢ ker f ≤ I
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/RobyLemma9.lean | RingHom.ker_eq_ideal_iff | [11, 1] | [21, 27] | rintro ⟨hI, h⟩ | case mpr
A : Type u_1
B : Type u_2
inst✝¹ : CommRing A
inst✝ : CommRing B
f : A →+* B
I : Ideal A
⊢ (∃ (h : I ≤ ker f), Function.Injective ⇑(Ideal.Quotient.lift I f h)) → ker f = I | case mpr.intro
A : Type u_1
B : Type u_2
inst✝¹ : CommRing A
inst✝ : CommRing B
f : A →+* B
I : Ideal A
hI : I ≤ ker f
h : Function.Injective ⇑(Ideal.Quotient.lift I f hI)
⊢ ker f = I | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
A : Type u_1
B : Type u_2
inst✝¹ : CommRing A
inst✝ : CommRing B
f : A →+* B
I : Ideal A
⊢ (∃ (h : I ≤ ker f), Function.Injective ⇑(Ideal.Quotient.lift I f h)) → ker f = I
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/RobyLemma9.lean | RingHom.ker_eq_ideal_iff | [11, 1] | [21, 27] | simp only [RingHom.injective_iff_ker_eq_bot, Ideal.ker_quotient_lift f hI,
Ideal.map_eq_bot_iff_le_ker, Ideal.mk_ker] at h | case mpr.intro
A : Type u_1
B : Type u_2
inst✝¹ : CommRing A
inst✝ : CommRing B
f : A →+* B
I : Ideal A
hI : I ≤ ker f
h : Function.Injective ⇑(Ideal.Quotient.lift I f hI)
⊢ ker f = I | case mpr.intro
A : Type u_1
B : Type u_2
inst✝¹ : CommRing A
inst✝ : CommRing B
f : A →+* B
I : Ideal A
hI : I ≤ ker f
h : ker f ≤ I
⊢ ker f = I | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.intro
A : Type u_1
B : Type u_2
inst✝¹ : CommRing A
inst✝ : CommRing B
f : A →+* B
I : Ideal A
hI : I ≤ ker f
h : Function.Injective ⇑(Ideal.Quotient.lift I f hI)
⊢ ker f = I
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/RobyLemma9.lean | RingHom.ker_eq_ideal_iff | [11, 1] | [21, 27] | exact le_antisymm h hI | case mpr.intro
A : Type u_1
B : Type u_2
inst✝¹ : CommRing A
inst✝ : CommRing B
f : A →+* B
I : Ideal A
hI : I ≤ ker f
h : ker f ≤ I
⊢ ker f = I | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.intro
A : Type u_1
B : Type u_2
inst✝¹ : CommRing A
inst✝ : CommRing B
f : A →+* B
I : Ideal A
hI : I ≤ ker f
h : ker f ≤ I
⊢ ker f = I
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/RobyLemma9.lean | AlgHom.ker_eq_ideal_iff | [28, 1] | [45, 54] | have : RingHom.ker f = RingHom.ker f.toRingHom := rfl | R : Type u_1
A : Type u_2
B : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing A
inst✝² : Algebra R A
inst✝¹ : CommRing B
inst✝ : Algebra R B
f : A →ₐ[R] B
I : Ideal A
⊢ RingHom.ker f = I ↔ ∃ (h : I ≤ RingHom.ker f), Function.Injective ⇑(Ideal.Quotient.liftₐ I f h) | R : Type u_1
A : Type u_2
B : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing A
inst✝² : Algebra R A
inst✝¹ : CommRing B
inst✝ : Algebra R B
f : A →ₐ[R] B
I : Ideal A
this : RingHom.ker f = RingHom.ker f.toRingHom
⊢ RingHom.ker f = I ↔ ∃ (h : I ≤ RingHom.ker f), Function.Injective ⇑(Ideal.Quotient.liftₐ I f h) | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
A : Type u_2
B : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing A
inst✝² : Algebra R A
inst✝¹ : CommRing B
inst✝ : Algebra R B
f : A →ₐ[R] B
I : Ideal A
⊢ RingHom.ker f = I ↔ ∃ (h : I ≤ RingHom.ker f), Function.Injective ⇑(Ideal.Quotient.liftₐ I ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/RobyLemma9.lean | AlgHom.ker_eq_ideal_iff | [28, 1] | [45, 54] | constructor | R : Type u_1
A : Type u_2
B : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing A
inst✝² : Algebra R A
inst✝¹ : CommRing B
inst✝ : Algebra R B
f : A →ₐ[R] B
I : Ideal A
this : RingHom.ker f = RingHom.ker f.toRingHom
⊢ RingHom.ker f = I ↔ ∃ (h : I ≤ RingHom.ker f), Function.Injective ⇑(Ideal.Quotient.liftₐ I f h) | case mp
R : Type u_1
A : Type u_2
B : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing A
inst✝² : Algebra R A
inst✝¹ : CommRing B
inst✝ : Algebra R B
f : A →ₐ[R] B
I : Ideal A
this : RingHom.ker f = RingHom.ker f.toRingHom
⊢ RingHom.ker f = I → ∃ (h : I ≤ RingHom.ker f), Function.Injective ⇑(Ideal.Quotient.liftₐ I f h)
... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
A : Type u_2
B : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing A
inst✝² : Algebra R A
inst✝¹ : CommRing B
inst✝ : Algebra R B
f : A →ₐ[R] B
I : Ideal A
this : RingHom.ker f = RingHom.ker f.toRingHom
⊢ RingHom.ker f = I ↔ ∃ (h : I ≤ RingHom.ker f... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/RobyLemma9.lean | AlgHom.ker_eq_ideal_iff | [28, 1] | [45, 54] | intro hI | case mp
R : Type u_1
A : Type u_2
B : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing A
inst✝² : Algebra R A
inst✝¹ : CommRing B
inst✝ : Algebra R B
f : A →ₐ[R] B
I : Ideal A
this : RingHom.ker f = RingHom.ker f.toRingHom
⊢ RingHom.ker f = I → ∃ (h : I ≤ RingHom.ker f), Function.Injective ⇑(Ideal.Quotient.liftₐ I f h) | case mp
R : Type u_1
A : Type u_2
B : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing A
inst✝² : Algebra R A
inst✝¹ : CommRing B
inst✝ : Algebra R B
f : A →ₐ[R] B
I : Ideal A
this : RingHom.ker f = RingHom.ker f.toRingHom
hI : RingHom.ker f = I
⊢ ∃ (h : I ≤ RingHom.ker f), Function.Injective ⇑(Ideal.Quotient.liftₐ I f h... | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
R : Type u_1
A : Type u_2
B : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing A
inst✝² : Algebra R A
inst✝¹ : CommRing B
inst✝ : Algebra R B
f : A →ₐ[R] B
I : Ideal A
this : RingHom.ker f = RingHom.ker f.toRingHom
⊢ RingHom.ker f = I → ∃ (h : I ≤ RingH... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/RobyLemma9.lean | AlgHom.ker_eq_ideal_iff | [28, 1] | [45, 54] | use le_of_eq hI.symm | case mp
R : Type u_1
A : Type u_2
B : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing A
inst✝² : Algebra R A
inst✝¹ : CommRing B
inst✝ : Algebra R B
f : A →ₐ[R] B
I : Ideal A
this : RingHom.ker f = RingHom.ker f.toRingHom
hI : RingHom.ker f = I
⊢ ∃ (h : I ≤ RingHom.ker f), Function.Injective ⇑(Ideal.Quotient.liftₐ I f h... | case h
R : Type u_1
A : Type u_2
B : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing A
inst✝² : Algebra R A
inst✝¹ : CommRing B
inst✝ : Algebra R B
f : A →ₐ[R] B
I : Ideal A
this : RingHom.ker f = RingHom.ker f.toRingHom
hI : RingHom.ker f = I
⊢ Function.Injective ⇑(Ideal.Quotient.liftₐ I f ⋯) | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
R : Type u_1
A : Type u_2
B : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing A
inst✝² : Algebra R A
inst✝¹ : CommRing B
inst✝ : Algebra R B
f : A →ₐ[R] B
I : Ideal A
this : RingHom.ker f = RingHom.ker f.toRingHom
hI : RingHom.ker f = I
⊢ ∃ (h : I ≤ Ri... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/RobyLemma9.lean | AlgHom.ker_eq_ideal_iff | [28, 1] | [45, 54] | suffices h : Function.Injective (Ideal.Quotient.lift I f.toRingHom (le_of_eq hI.symm)) by
intro x y hxy; apply h
simpa only [Ideal.Quotient.liftₐ_apply] using hxy | case h
R : Type u_1
A : Type u_2
B : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing A
inst✝² : Algebra R A
inst✝¹ : CommRing B
inst✝ : Algebra R B
f : A →ₐ[R] B
I : Ideal A
this : RingHom.ker f = RingHom.ker f.toRingHom
hI : RingHom.ker f = I
⊢ Function.Injective ⇑(Ideal.Quotient.liftₐ I f ⋯) | case h
R : Type u_1
A : Type u_2
B : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing A
inst✝² : Algebra R A
inst✝¹ : CommRing B
inst✝ : Algebra R B
f : A →ₐ[R] B
I : Ideal A
this : RingHom.ker f = RingHom.ker f.toRingHom
hI : RingHom.ker f = I
⊢ Function.Injective ⇑(Ideal.Quotient.lift I f.toRingHom ⋯) | Please generate a tactic in lean4 to solve the state.
STATE:
case h
R : Type u_1
A : Type u_2
B : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing A
inst✝² : Algebra R A
inst✝¹ : CommRing B
inst✝ : Algebra R B
f : A →ₐ[R] B
I : Ideal A
this : RingHom.ker f = RingHom.ker f.toRingHom
hI : RingHom.ker f = I
⊢ Function.Injec... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/RobyLemma9.lean | AlgHom.ker_eq_ideal_iff | [28, 1] | [45, 54] | apply RingHom.lift_injective_of_ker_le_ideal | case h
R : Type u_1
A : Type u_2
B : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing A
inst✝² : Algebra R A
inst✝¹ : CommRing B
inst✝ : Algebra R B
f : A →ₐ[R] B
I : Ideal A
this : RingHom.ker f = RingHom.ker f.toRingHom
hI : RingHom.ker f = I
⊢ Function.Injective ⇑(Ideal.Quotient.lift I f.toRingHom ⋯) | case h.hI
R : Type u_1
A : Type u_2
B : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing A
inst✝² : Algebra R A
inst✝¹ : CommRing B
inst✝ : Algebra R B
f : A →ₐ[R] B
I : Ideal A
this : RingHom.ker f = RingHom.ker f.toRingHom
hI : RingHom.ker f = I
⊢ RingHom.ker f.toRingHom ≤ I | Please generate a tactic in lean4 to solve the state.
STATE:
case h
R : Type u_1
A : Type u_2
B : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing A
inst✝² : Algebra R A
inst✝¹ : CommRing B
inst✝ : Algebra R B
f : A →ₐ[R] B
I : Ideal A
this : RingHom.ker f = RingHom.ker f.toRingHom
hI : RingHom.ker f = I
⊢ Function.Injec... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/RobyLemma9.lean | AlgHom.ker_eq_ideal_iff | [28, 1] | [45, 54] | rw [← hI, ← this] | case h.hI
R : Type u_1
A : Type u_2
B : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing A
inst✝² : Algebra R A
inst✝¹ : CommRing B
inst✝ : Algebra R B
f : A →ₐ[R] B
I : Ideal A
this : RingHom.ker f = RingHom.ker f.toRingHom
hI : RingHom.ker f = I
⊢ RingHom.ker f.toRingHom ≤ I | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.hI
R : Type u_1
A : Type u_2
B : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing A
inst✝² : Algebra R A
inst✝¹ : CommRing B
inst✝ : Algebra R B
f : A →ₐ[R] B
I : Ideal A
this : RingHom.ker f = RingHom.ker f.toRingHom
hI : RingHom.ker f = I
⊢ RingHom.ker... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/RobyLemma9.lean | AlgHom.ker_eq_ideal_iff | [28, 1] | [45, 54] | intro x y hxy | R : Type u_1
A : Type u_2
B : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing A
inst✝² : Algebra R A
inst✝¹ : CommRing B
inst✝ : Algebra R B
f : A →ₐ[R] B
I : Ideal A
this : RingHom.ker f = RingHom.ker f.toRingHom
hI : RingHom.ker f = I
h : Function.Injective ⇑(Ideal.Quotient.lift I f.toRingHom ⋯)
⊢ Function.Injective ⇑... | R : Type u_1
A : Type u_2
B : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing A
inst✝² : Algebra R A
inst✝¹ : CommRing B
inst✝ : Algebra R B
f : A →ₐ[R] B
I : Ideal A
this : RingHom.ker f = RingHom.ker f.toRingHom
hI : RingHom.ker f = I
h : Function.Injective ⇑(Ideal.Quotient.lift I f.toRingHom ⋯)
x y : A ⧸ I
hxy : (Ide... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
A : Type u_2
B : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing A
inst✝² : Algebra R A
inst✝¹ : CommRing B
inst✝ : Algebra R B
f : A →ₐ[R] B
I : Ideal A
this : RingHom.ker f = RingHom.ker f.toRingHom
hI : RingHom.ker f = I
h : Function.Injective ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/RobyLemma9.lean | AlgHom.ker_eq_ideal_iff | [28, 1] | [45, 54] | apply h | R : Type u_1
A : Type u_2
B : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing A
inst✝² : Algebra R A
inst✝¹ : CommRing B
inst✝ : Algebra R B
f : A →ₐ[R] B
I : Ideal A
this : RingHom.ker f = RingHom.ker f.toRingHom
hI : RingHom.ker f = I
h : Function.Injective ⇑(Ideal.Quotient.lift I f.toRingHom ⋯)
x y : A ⧸ I
hxy : (Ide... | case a
R : Type u_1
A : Type u_2
B : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing A
inst✝² : Algebra R A
inst✝¹ : CommRing B
inst✝ : Algebra R B
f : A →ₐ[R] B
I : Ideal A
this : RingHom.ker f = RingHom.ker f.toRingHom
hI : RingHom.ker f = I
h : Function.Injective ⇑(Ideal.Quotient.lift I f.toRingHom ⋯)
x y : A ⧸ I
hxy... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
A : Type u_2
B : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing A
inst✝² : Algebra R A
inst✝¹ : CommRing B
inst✝ : Algebra R B
f : A →ₐ[R] B
I : Ideal A
this : RingHom.ker f = RingHom.ker f.toRingHom
hI : RingHom.ker f = I
h : Function.Injective ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/RobyLemma9.lean | AlgHom.ker_eq_ideal_iff | [28, 1] | [45, 54] | simpa only [Ideal.Quotient.liftₐ_apply] using hxy | case a
R : Type u_1
A : Type u_2
B : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing A
inst✝² : Algebra R A
inst✝¹ : CommRing B
inst✝ : Algebra R B
f : A →ₐ[R] B
I : Ideal A
this : RingHom.ker f = RingHom.ker f.toRingHom
hI : RingHom.ker f = I
h : Function.Injective ⇑(Ideal.Quotient.lift I f.toRingHom ⋯)
x y : A ⧸ I
hxy... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a
R : Type u_1
A : Type u_2
B : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing A
inst✝² : Algebra R A
inst✝¹ : CommRing B
inst✝ : Algebra R B
f : A →ₐ[R] B
I : Ideal A
this : RingHom.ker f = RingHom.ker f.toRingHom
hI : RingHom.ker f = I
h : Function.Inj... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/RobyLemma9.lean | AlgHom.ker_eq_ideal_iff | [28, 1] | [45, 54] | rintro ⟨hI, h⟩ | case mpr
R : Type u_1
A : Type u_2
B : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing A
inst✝² : Algebra R A
inst✝¹ : CommRing B
inst✝ : Algebra R B
f : A →ₐ[R] B
I : Ideal A
this : RingHom.ker f = RingHom.ker f.toRingHom
⊢ (∃ (h : I ≤ RingHom.ker f), Function.Injective ⇑(Ideal.Quotient.liftₐ I f h)) → RingHom.ker f = ... | case mpr.intro
R : Type u_1
A : Type u_2
B : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing A
inst✝² : Algebra R A
inst✝¹ : CommRing B
inst✝ : Algebra R B
f : A →ₐ[R] B
I : Ideal A
this : RingHom.ker f = RingHom.ker f.toRingHom
hI : I ≤ RingHom.ker f
h : Function.Injective ⇑(Ideal.Quotient.liftₐ I f hI)
⊢ RingHom.ker f... | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
R : Type u_1
A : Type u_2
B : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing A
inst✝² : Algebra R A
inst✝¹ : CommRing B
inst✝ : Algebra R B
f : A →ₐ[R] B
I : Ideal A
this : RingHom.ker f = RingHom.ker f.toRingHom
⊢ (∃ (h : I ≤ RingHom.ker f), Functio... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/RobyLemma9.lean | AlgHom.ker_eq_ideal_iff | [28, 1] | [45, 54] | rw [this] | case mpr.intro
R : Type u_1
A : Type u_2
B : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing A
inst✝² : Algebra R A
inst✝¹ : CommRing B
inst✝ : Algebra R B
f : A →ₐ[R] B
I : Ideal A
this : RingHom.ker f = RingHom.ker f.toRingHom
hI : I ≤ RingHom.ker f
h : Function.Injective ⇑(Ideal.Quotient.liftₐ I f hI)
⊢ RingHom.ker f... | case mpr.intro
R : Type u_1
A : Type u_2
B : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing A
inst✝² : Algebra R A
inst✝¹ : CommRing B
inst✝ : Algebra R B
f : A →ₐ[R] B
I : Ideal A
this : RingHom.ker f = RingHom.ker f.toRingHom
hI : I ≤ RingHom.ker f
h : Function.Injective ⇑(Ideal.Quotient.liftₐ I f hI)
⊢ RingHom.ker f... | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.intro
R : Type u_1
A : Type u_2
B : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing A
inst✝² : Algebra R A
inst✝¹ : CommRing B
inst✝ : Algebra R B
f : A →ₐ[R] B
I : Ideal A
this : RingHom.ker f = RingHom.ker f.toRingHom
hI : I ≤ RingHom.ker f
h : Func... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/RobyLemma9.lean | AlgHom.ker_eq_ideal_iff | [28, 1] | [45, 54] | rw [RingHom.ker_eq_ideal_iff] | case mpr.intro
R : Type u_1
A : Type u_2
B : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing A
inst✝² : Algebra R A
inst✝¹ : CommRing B
inst✝ : Algebra R B
f : A →ₐ[R] B
I : Ideal A
this : RingHom.ker f = RingHom.ker f.toRingHom
hI : I ≤ RingHom.ker f
h : Function.Injective ⇑(Ideal.Quotient.liftₐ I f hI)
⊢ RingHom.ker f... | case mpr.intro
R : Type u_1
A : Type u_2
B : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing A
inst✝² : Algebra R A
inst✝¹ : CommRing B
inst✝ : Algebra R B
f : A →ₐ[R] B
I : Ideal A
this : RingHom.ker f = RingHom.ker f.toRingHom
hI : I ≤ RingHom.ker f
h : Function.Injective ⇑(Ideal.Quotient.liftₐ I f hI)
⊢ ∃ (h : I ≤ Ri... | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.intro
R : Type u_1
A : Type u_2
B : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing A
inst✝² : Algebra R A
inst✝¹ : CommRing B
inst✝ : Algebra R B
f : A →ₐ[R] B
I : Ideal A
this : RingHom.ker f = RingHom.ker f.toRingHom
hI : I ≤ RingHom.ker f
h : Func... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/RobyLemma9.lean | AlgHom.ker_eq_ideal_iff | [28, 1] | [45, 54] | rw [this] at hI | case mpr.intro
R : Type u_1
A : Type u_2
B : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing A
inst✝² : Algebra R A
inst✝¹ : CommRing B
inst✝ : Algebra R B
f : A →ₐ[R] B
I : Ideal A
this : RingHom.ker f = RingHom.ker f.toRingHom
hI : I ≤ RingHom.ker f
h : Function.Injective ⇑(Ideal.Quotient.liftₐ I f hI)
⊢ ∃ (h : I ≤ Ri... | case mpr.intro
R : Type u_1
A : Type u_2
B : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing A
inst✝² : Algebra R A
inst✝¹ : CommRing B
inst✝ : Algebra R B
f : A →ₐ[R] B
I : Ideal A
this : RingHom.ker f = RingHom.ker f.toRingHom
hI✝ : I ≤ RingHom.ker f
hI : I ≤ RingHom.ker f.toRingHom
h : Function.Injective ⇑(Ideal.Quot... | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.intro
R : Type u_1
A : Type u_2
B : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing A
inst✝² : Algebra R A
inst✝¹ : CommRing B
inst✝ : Algebra R B
f : A →ₐ[R] B
I : Ideal A
this : RingHom.ker f = RingHom.ker f.toRingHom
hI : I ≤ RingHom.ker f
h : Func... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/RobyLemma9.lean | AlgHom.ker_eq_ideal_iff | [28, 1] | [45, 54] | use hI | case mpr.intro
R : Type u_1
A : Type u_2
B : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing A
inst✝² : Algebra R A
inst✝¹ : CommRing B
inst✝ : Algebra R B
f : A →ₐ[R] B
I : Ideal A
this : RingHom.ker f = RingHom.ker f.toRingHom
hI✝ : I ≤ RingHom.ker f
hI : I ≤ RingHom.ker f.toRingHom
h : Function.Injective ⇑(Ideal.Quot... | case h
R : Type u_1
A : Type u_2
B : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing A
inst✝² : Algebra R A
inst✝¹ : CommRing B
inst✝ : Algebra R B
f : A →ₐ[R] B
I : Ideal A
this : RingHom.ker f = RingHom.ker f.toRingHom
hI✝ : I ≤ RingHom.ker f
hI : I ≤ RingHom.ker f.toRingHom
h : Function.Injective ⇑(Ideal.Quotient.lif... | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.intro
R : Type u_1
A : Type u_2
B : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing A
inst✝² : Algebra R A
inst✝¹ : CommRing B
inst✝ : Algebra R B
f : A →ₐ[R] B
I : Ideal A
this : RingHom.ker f = RingHom.ker f.toRingHom
hI✝ : I ≤ RingHom.ker f
hI : I ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/RobyLemma9.lean | AlgHom.ker_eq_ideal_iff | [28, 1] | [45, 54] | intro x y hxy | case h
R : Type u_1
A : Type u_2
B : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing A
inst✝² : Algebra R A
inst✝¹ : CommRing B
inst✝ : Algebra R B
f : A →ₐ[R] B
I : Ideal A
this : RingHom.ker f = RingHom.ker f.toRingHom
hI✝ : I ≤ RingHom.ker f
hI : I ≤ RingHom.ker f.toRingHom
h : Function.Injective ⇑(Ideal.Quotient.lif... | case h
R : Type u_1
A : Type u_2
B : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing A
inst✝² : Algebra R A
inst✝¹ : CommRing B
inst✝ : Algebra R B
f : A →ₐ[R] B
I : Ideal A
this : RingHom.ker f = RingHom.ker f.toRingHom
hI✝ : I ≤ RingHom.ker f
hI : I ≤ RingHom.ker f.toRingHom
h : Function.Injective ⇑(Ideal.Quotient.lif... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
R : Type u_1
A : Type u_2
B : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing A
inst✝² : Algebra R A
inst✝¹ : CommRing B
inst✝ : Algebra R B
f : A →ₐ[R] B
I : Ideal A
this : RingHom.ker f = RingHom.ker f.toRingHom
hI✝ : I ≤ RingHom.ker f
hI : I ≤ RingHo... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/RobyLemma9.lean | AlgHom.ker_eq_ideal_iff | [28, 1] | [45, 54] | apply h | case h
R : Type u_1
A : Type u_2
B : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing A
inst✝² : Algebra R A
inst✝¹ : CommRing B
inst✝ : Algebra R B
f : A →ₐ[R] B
I : Ideal A
this : RingHom.ker f = RingHom.ker f.toRingHom
hI✝ : I ≤ RingHom.ker f
hI : I ≤ RingHom.ker f.toRingHom
h : Function.Injective ⇑(Ideal.Quotient.lif... | case h.a
R : Type u_1
A : Type u_2
B : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing A
inst✝² : Algebra R A
inst✝¹ : CommRing B
inst✝ : Algebra R B
f : A →ₐ[R] B
I : Ideal A
this : RingHom.ker f = RingHom.ker f.toRingHom
hI✝ : I ≤ RingHom.ker f
hI : I ≤ RingHom.ker f.toRingHom
h : Function.Injective ⇑(Ideal.Quotient.l... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
R : Type u_1
A : Type u_2
B : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing A
inst✝² : Algebra R A
inst✝¹ : CommRing B
inst✝ : Algebra R B
f : A →ₐ[R] B
I : Ideal A
this : RingHom.ker f = RingHom.ker f.toRingHom
hI✝ : I ≤ RingHom.ker f
hI : I ≤ RingHo... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/RobyLemma9.lean | AlgHom.ker_eq_ideal_iff | [28, 1] | [45, 54] | simpa only [Ideal.Quotient.liftₐ_apply] using hxy | case h.a
R : Type u_1
A : Type u_2
B : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing A
inst✝² : Algebra R A
inst✝¹ : CommRing B
inst✝ : Algebra R B
f : A →ₐ[R] B
I : Ideal A
this : RingHom.ker f = RingHom.ker f.toRingHom
hI✝ : I ≤ RingHom.ker f
hI : I ≤ RingHom.ker f.toRingHom
h : Function.Injective ⇑(Ideal.Quotient.l... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.a
R : Type u_1
A : Type u_2
B : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing A
inst✝² : Algebra R A
inst✝¹ : CommRing B
inst✝ : Algebra R B
f : A →ₐ[R] B
I : Ideal A
this : RingHom.ker f = RingHom.ker f.toRingHom
hI✝ : I ≤ RingHom.ker f
hI : I ≤ Ring... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/RobyLemma9.lean | φ_apply | [66, 1] | [68, 71] | simp only [φ, productMap_apply_tmul, AlgHom.coe_restrictScalars', includeLeft_apply,
includeRight_apply, tmul_mul_tmul, _root_.mul_one, _root_.one_mul] | R : Type u_4
inst✝¹⁰ : CommRing R
S : Type u_3
inst✝⁹ : CommRing S
M : Type u_1
inst✝⁸ : CommRing M
inst✝⁷ : Algebra R M
inst✝⁶ : Algebra S M
N : Type u_2
inst✝⁵ : CommRing N
inst✝⁴ : Algebra R N
inst✝³ : Algebra S N
inst✝² : Algebra R S
inst✝¹ : IsScalarTower R S M
inst✝ : IsScalarTower R S N
m : M
n : N
⊢ (φ R S M N)... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_4
inst✝¹⁰ : CommRing R
S : Type u_3
inst✝⁹ : CommRing S
M : Type u_1
inst✝⁸ : CommRing M
inst✝⁷ : Algebra R M
inst✝⁶ : Algebra S M
N : Type u_2
inst✝⁵ : CommRing N
inst✝⁴ : Algebra R N
inst✝³ : Algebra S N
inst✝² : Algebra R S
inst✝¹ : IsScalarTowe... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/RobyLemma9.lean | φ_surjective | [71, 1] | [79, 35] | intro z | R : Type u_3
inst✝¹⁰ : CommRing R
S : Type u_4
inst✝⁹ : CommRing S
M : Type u_1
inst✝⁸ : CommRing M
inst✝⁷ : Algebra R M
inst✝⁶ : Algebra S M
N : Type u_2
inst✝⁵ : CommRing N
inst✝⁴ : Algebra R N
inst✝³ : Algebra S N
inst✝² : Algebra R S
inst✝¹ : IsScalarTower R S M
inst✝ : IsScalarTower R S N
⊢ Function.Surjective ⇑(φ... | R : Type u_3
inst✝¹⁰ : CommRing R
S : Type u_4
inst✝⁹ : CommRing S
M : Type u_1
inst✝⁸ : CommRing M
inst✝⁷ : Algebra R M
inst✝⁶ : Algebra S M
N : Type u_2
inst✝⁵ : CommRing N
inst✝⁴ : Algebra R N
inst✝³ : Algebra S N
inst✝² : Algebra R S
inst✝¹ : IsScalarTower R S M
inst✝ : IsScalarTower R S N
z : M ⊗[S] N
⊢ ∃ a, (φ R ... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_3
inst✝¹⁰ : CommRing R
S : Type u_4
inst✝⁹ : CommRing S
M : Type u_1
inst✝⁸ : CommRing M
inst✝⁷ : Algebra R M
inst✝⁶ : Algebra S M
N : Type u_2
inst✝⁵ : CommRing N
inst✝⁴ : Algebra R N
inst✝³ : Algebra S N
inst✝² : Algebra R S
inst✝¹ : IsScalarTowe... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/RobyLemma9.lean | φ_surjective | [71, 1] | [79, 35] | induction z using TensorProduct.induction_on with
| zero => use 0; simp only [map_zero]
| tmul m n => use m ⊗ₜ n; simp only [φ_apply]
| add x y hx hy =>
obtain ⟨a, rfl⟩ := hx
obtain ⟨b, rfl⟩ := hy
exact ⟨a + b, map_add _ _ _⟩ | R : Type u_3
inst✝¹⁰ : CommRing R
S : Type u_4
inst✝⁹ : CommRing S
M : Type u_1
inst✝⁸ : CommRing M
inst✝⁷ : Algebra R M
inst✝⁶ : Algebra S M
N : Type u_2
inst✝⁵ : CommRing N
inst✝⁴ : Algebra R N
inst✝³ : Algebra S N
inst✝² : Algebra R S
inst✝¹ : IsScalarTower R S M
inst✝ : IsScalarTower R S N
z : M ⊗[S] N
⊢ ∃ a, (φ R ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_3
inst✝¹⁰ : CommRing R
S : Type u_4
inst✝⁹ : CommRing S
M : Type u_1
inst✝⁸ : CommRing M
inst✝⁷ : Algebra R M
inst✝⁶ : Algebra S M
N : Type u_2
inst✝⁵ : CommRing N
inst✝⁴ : Algebra R N
inst✝³ : Algebra S N
inst✝² : Algebra R S
inst✝¹ : IsScalarTowe... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/RobyLemma9.lean | φ_surjective | [71, 1] | [79, 35] | use 0 | case zero
R : Type u_3
inst✝¹⁰ : CommRing R
S : Type u_4
inst✝⁹ : CommRing S
M : Type u_1
inst✝⁸ : CommRing M
inst✝⁷ : Algebra R M
inst✝⁶ : Algebra S M
N : Type u_2
inst✝⁵ : CommRing N
inst✝⁴ : Algebra R N
inst✝³ : Algebra S N
inst✝² : Algebra R S
inst✝¹ : IsScalarTower R S M
inst✝ : IsScalarTower R S N
⊢ ∃ a, (φ R S M... | case h
R : Type u_3
inst✝¹⁰ : CommRing R
S : Type u_4
inst✝⁹ : CommRing S
M : Type u_1
inst✝⁸ : CommRing M
inst✝⁷ : Algebra R M
inst✝⁶ : Algebra S M
N : Type u_2
inst✝⁵ : CommRing N
inst✝⁴ : Algebra R N
inst✝³ : Algebra S N
inst✝² : Algebra R S
inst✝¹ : IsScalarTower R S M
inst✝ : IsScalarTower R S N
⊢ (φ R S M N) 0 = ... | Please generate a tactic in lean4 to solve the state.
STATE:
case zero
R : Type u_3
inst✝¹⁰ : CommRing R
S : Type u_4
inst✝⁹ : CommRing S
M : Type u_1
inst✝⁸ : CommRing M
inst✝⁷ : Algebra R M
inst✝⁶ : Algebra S M
N : Type u_2
inst✝⁵ : CommRing N
inst✝⁴ : Algebra R N
inst✝³ : Algebra S N
inst✝² : Algebra R S
inst✝¹ : Is... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/RobyLemma9.lean | φ_surjective | [71, 1] | [79, 35] | simp only [map_zero] | case h
R : Type u_3
inst✝¹⁰ : CommRing R
S : Type u_4
inst✝⁹ : CommRing S
M : Type u_1
inst✝⁸ : CommRing M
inst✝⁷ : Algebra R M
inst✝⁶ : Algebra S M
N : Type u_2
inst✝⁵ : CommRing N
inst✝⁴ : Algebra R N
inst✝³ : Algebra S N
inst✝² : Algebra R S
inst✝¹ : IsScalarTower R S M
inst✝ : IsScalarTower R S N
⊢ (φ R S M N) 0 = ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
R : Type u_3
inst✝¹⁰ : CommRing R
S : Type u_4
inst✝⁹ : CommRing S
M : Type u_1
inst✝⁸ : CommRing M
inst✝⁷ : Algebra R M
inst✝⁶ : Algebra S M
N : Type u_2
inst✝⁵ : CommRing N
inst✝⁴ : Algebra R N
inst✝³ : Algebra S N
inst✝² : Algebra R S
inst✝¹ : IsSca... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/RobyLemma9.lean | φ_surjective | [71, 1] | [79, 35] | use m ⊗ₜ n | case tmul
R : Type u_3
inst✝¹⁰ : CommRing R
S : Type u_4
inst✝⁹ : CommRing S
M : Type u_1
inst✝⁸ : CommRing M
inst✝⁷ : Algebra R M
inst✝⁶ : Algebra S M
N : Type u_2
inst✝⁵ : CommRing N
inst✝⁴ : Algebra R N
inst✝³ : Algebra S N
inst✝² : Algebra R S
inst✝¹ : IsScalarTower R S M
inst✝ : IsScalarTower R S N
m : M
n : N
⊢ ∃... | case h
R : Type u_3
inst✝¹⁰ : CommRing R
S : Type u_4
inst✝⁹ : CommRing S
M : Type u_1
inst✝⁸ : CommRing M
inst✝⁷ : Algebra R M
inst✝⁶ : Algebra S M
N : Type u_2
inst✝⁵ : CommRing N
inst✝⁴ : Algebra R N
inst✝³ : Algebra S N
inst✝² : Algebra R S
inst✝¹ : IsScalarTower R S M
inst✝ : IsScalarTower R S N
m : M
n : N
⊢ (φ R... | Please generate a tactic in lean4 to solve the state.
STATE:
case tmul
R : Type u_3
inst✝¹⁰ : CommRing R
S : Type u_4
inst✝⁹ : CommRing S
M : Type u_1
inst✝⁸ : CommRing M
inst✝⁷ : Algebra R M
inst✝⁶ : Algebra S M
N : Type u_2
inst✝⁵ : CommRing N
inst✝⁴ : Algebra R N
inst✝³ : Algebra S N
inst✝² : Algebra R S
inst✝¹ : Is... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/RobyLemma9.lean | φ_surjective | [71, 1] | [79, 35] | simp only [φ_apply] | case h
R : Type u_3
inst✝¹⁰ : CommRing R
S : Type u_4
inst✝⁹ : CommRing S
M : Type u_1
inst✝⁸ : CommRing M
inst✝⁷ : Algebra R M
inst✝⁶ : Algebra S M
N : Type u_2
inst✝⁵ : CommRing N
inst✝⁴ : Algebra R N
inst✝³ : Algebra S N
inst✝² : Algebra R S
inst✝¹ : IsScalarTower R S M
inst✝ : IsScalarTower R S N
m : M
n : N
⊢ (φ R... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
R : Type u_3
inst✝¹⁰ : CommRing R
S : Type u_4
inst✝⁹ : CommRing S
M : Type u_1
inst✝⁸ : CommRing M
inst✝⁷ : Algebra R M
inst✝⁶ : Algebra S M
N : Type u_2
inst✝⁵ : CommRing N
inst✝⁴ : Algebra R N
inst✝³ : Algebra S N
inst✝² : Algebra R S
inst✝¹ : IsSca... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/RobyLemma9.lean | φ_surjective | [71, 1] | [79, 35] | obtain ⟨a, rfl⟩ := hx | case add
R : Type u_3
inst✝¹⁰ : CommRing R
S : Type u_4
inst✝⁹ : CommRing S
M : Type u_1
inst✝⁸ : CommRing M
inst✝⁷ : Algebra R M
inst✝⁶ : Algebra S M
N : Type u_2
inst✝⁵ : CommRing N
inst✝⁴ : Algebra R N
inst✝³ : Algebra S N
inst✝² : Algebra R S
inst✝¹ : IsScalarTower R S M
inst✝ : IsScalarTower R S N
x y : M ⊗[S] N
h... | case add.intro
R : Type u_3
inst✝¹⁰ : CommRing R
S : Type u_4
inst✝⁹ : CommRing S
M : Type u_1
inst✝⁸ : CommRing M
inst✝⁷ : Algebra R M
inst✝⁶ : Algebra S M
N : Type u_2
inst✝⁵ : CommRing N
inst✝⁴ : Algebra R N
inst✝³ : Algebra S N
inst✝² : Algebra R S
inst✝¹ : IsScalarTower R S M
inst✝ : IsScalarTower R S N
y : M ⊗[S]... | Please generate a tactic in lean4 to solve the state.
STATE:
case add
R : Type u_3
inst✝¹⁰ : CommRing R
S : Type u_4
inst✝⁹ : CommRing S
M : Type u_1
inst✝⁸ : CommRing M
inst✝⁷ : Algebra R M
inst✝⁶ : Algebra S M
N : Type u_2
inst✝⁵ : CommRing N
inst✝⁴ : Algebra R N
inst✝³ : Algebra S N
inst✝² : Algebra R S
inst✝¹ : IsS... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/RobyLemma9.lean | φ_surjective | [71, 1] | [79, 35] | obtain ⟨b, rfl⟩ := hy | case add.intro
R : Type u_3
inst✝¹⁰ : CommRing R
S : Type u_4
inst✝⁹ : CommRing S
M : Type u_1
inst✝⁸ : CommRing M
inst✝⁷ : Algebra R M
inst✝⁶ : Algebra S M
N : Type u_2
inst✝⁵ : CommRing N
inst✝⁴ : Algebra R N
inst✝³ : Algebra S N
inst✝² : Algebra R S
inst✝¹ : IsScalarTower R S M
inst✝ : IsScalarTower R S N
y : M ⊗[S]... | case add.intro.intro
R : Type u_3
inst✝¹⁰ : CommRing R
S : Type u_4
inst✝⁹ : CommRing S
M : Type u_1
inst✝⁸ : CommRing M
inst✝⁷ : Algebra R M
inst✝⁶ : Algebra S M
N : Type u_2
inst✝⁵ : CommRing N
inst✝⁴ : Algebra R N
inst✝³ : Algebra S N
inst✝² : Algebra R S
inst✝¹ : IsScalarTower R S M
inst✝ : IsScalarTower R S N
a b ... | Please generate a tactic in lean4 to solve the state.
STATE:
case add.intro
R : Type u_3
inst✝¹⁰ : CommRing R
S : Type u_4
inst✝⁹ : CommRing S
M : Type u_1
inst✝⁸ : CommRing M
inst✝⁷ : Algebra R M
inst✝⁶ : Algebra S M
N : Type u_2
inst✝⁵ : CommRing N
inst✝⁴ : Algebra R N
inst✝³ : Algebra S N
inst✝² : Algebra R S
inst✝¹... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/RobyLemma9.lean | φ_surjective | [71, 1] | [79, 35] | exact ⟨a + b, map_add _ _ _⟩ | case add.intro.intro
R : Type u_3
inst✝¹⁰ : CommRing R
S : Type u_4
inst✝⁹ : CommRing S
M : Type u_1
inst✝⁸ : CommRing M
inst✝⁷ : Algebra R M
inst✝⁶ : Algebra S M
N : Type u_2
inst✝⁵ : CommRing N
inst✝⁴ : Algebra R N
inst✝³ : Algebra S N
inst✝² : Algebra R S
inst✝¹ : IsScalarTower R S M
inst✝ : IsScalarTower R S N
a b ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case add.intro.intro
R : Type u_3
inst✝¹⁰ : CommRing R
S : Type u_4
inst✝⁹ : CommRing S
M : Type u_1
inst✝⁸ : CommRing M
inst✝⁷ : Algebra R M
inst✝⁶ : Algebra S M
N : Type u_2
inst✝⁵ : CommRing N
inst✝⁴ : Algebra R N
inst✝³ : Algebra S N
inst✝² : Algebra R S
... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/RobyLemma9.lean | ψ_apply | [126, 1] | [133, 60] | simp only [ψ, ψLeft, ψRight, AlgHom.toRingHom_eq_coe, productMap_apply_tmul, AlgHom.coe_mk,
RingHom.coe_coe, AlgHom.coe_comp, AlgHom.coe_restrictScalars', Ideal.Quotient.mkₐ_eq_mk,
Function.comp_apply, includeLeft_apply, includeRight_apply] | R : Type u_3
inst✝¹⁰ : CommRing R
S : Type u_4
inst✝⁹ : CommRing S
M : Type u_1
inst✝⁸ : CommRing M
inst✝⁷ : Algebra R M
inst✝⁶ : Algebra S M
N : Type u_2
inst✝⁵ : CommRing N
inst✝⁴ : Algebra R N
inst✝³ : Algebra S N
inst✝² : Algebra R S
inst✝¹ : IsScalarTower R S M
inst✝ : IsScalarTower R S N
m : M
n : N
⊢ (ψ R S M N)... | R : Type u_3
inst✝¹⁰ : CommRing R
S : Type u_4
inst✝⁹ : CommRing S
M : Type u_1
inst✝⁸ : CommRing M
inst✝⁷ : Algebra R M
inst✝⁶ : Algebra S M
N : Type u_2
inst✝⁵ : CommRing N
inst✝⁴ : Algebra R N
inst✝³ : Algebra S N
inst✝² : Algebra R S
inst✝¹ : IsScalarTower R S M
inst✝ : IsScalarTower R S N
m : M
n : N
⊢ (Ideal.Quot... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_3
inst✝¹⁰ : CommRing R
S : Type u_4
inst✝⁹ : CommRing S
M : Type u_1
inst✝⁸ : CommRing M
inst✝⁷ : Algebra R M
inst✝⁶ : Algebra S M
N : Type u_2
inst✝⁵ : CommRing N
inst✝⁴ : Algebra R N
inst✝³ : Algebra S N
inst✝² : Algebra R S
inst✝¹ : IsScalarTowe... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/RobyLemma9.lean | ψ_apply | [126, 1] | [133, 60] | rw [← RingHom.map_mul] | R : Type u_3
inst✝¹⁰ : CommRing R
S : Type u_4
inst✝⁹ : CommRing S
M : Type u_1
inst✝⁸ : CommRing M
inst✝⁷ : Algebra R M
inst✝⁶ : Algebra S M
N : Type u_2
inst✝⁵ : CommRing N
inst✝⁴ : Algebra R N
inst✝³ : Algebra S N
inst✝² : Algebra R S
inst✝¹ : IsScalarTower R S M
inst✝ : IsScalarTower R S N
m : M
n : N
⊢ (Ideal.Quot... | R : Type u_3
inst✝¹⁰ : CommRing R
S : Type u_4
inst✝⁹ : CommRing S
M : Type u_1
inst✝⁸ : CommRing M
inst✝⁷ : Algebra R M
inst✝⁶ : Algebra S M
N : Type u_2
inst✝⁵ : CommRing N
inst✝⁴ : Algebra R N
inst✝³ : Algebra S N
inst✝² : Algebra R S
inst✝¹ : IsScalarTower R S M
inst✝ : IsScalarTower R S N
m : M
n : N
⊢ (Ideal.Quot... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_3
inst✝¹⁰ : CommRing R
S : Type u_4
inst✝⁹ : CommRing S
M : Type u_1
inst✝⁸ : CommRing M
inst✝⁷ : Algebra R M
inst✝⁶ : Algebra S M
N : Type u_2
inst✝⁵ : CommRing N
inst✝⁴ : Algebra R N
inst✝³ : Algebra S N
inst✝² : Algebra R S
inst✝¹ : IsScalarTowe... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/RobyLemma9.lean | ψ_apply | [126, 1] | [133, 60] | simp only [tmul_mul_tmul, _root_.mul_one, _root_.one_mul] | R : Type u_3
inst✝¹⁰ : CommRing R
S : Type u_4
inst✝⁹ : CommRing S
M : Type u_1
inst✝⁸ : CommRing M
inst✝⁷ : Algebra R M
inst✝⁶ : Algebra S M
N : Type u_2
inst✝⁵ : CommRing N
inst✝⁴ : Algebra R N
inst✝³ : Algebra S N
inst✝² : Algebra R S
inst✝¹ : IsScalarTower R S M
inst✝ : IsScalarTower R S N
m : M
n : N
⊢ (Ideal.Quot... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_3
inst✝¹⁰ : CommRing R
S : Type u_4
inst✝⁹ : CommRing S
M : Type u_1
inst✝⁸ : CommRing M
inst✝⁷ : Algebra R M
inst✝⁶ : Algebra S M
N : Type u_2
inst✝⁵ : CommRing N
inst✝⁴ : Algebra R N
inst✝³ : Algebra S N
inst✝² : Algebra R S
inst✝¹ : IsScalarTowe... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/RobyLemma9.lean | kerφ_eq | [136, 1] | [157, 6] | suffices h : kerφ R S M N ≤ RingHom.ker (φ R S M N).toRingHom by
rw [RingHom.ker_eq_ideal_iff]
use h
apply Function.HasLeftInverse.injective
use ψ R S M N
intro z
obtain ⟨y, rfl⟩ := Ideal.Quotient.mk_surjective z
simp only [AlgHom.toRingHom_eq_coe, Ideal.Quotient.lift_mk, AlgHom.coe_toRingHom]
induction... | R : Type u_3
inst✝¹⁰ : CommRing R
S : Type u_4
inst✝⁹ : CommRing S
M : Type u_1
inst✝⁸ : CommRing M
inst✝⁷ : Algebra R M
inst✝⁶ : Algebra S M
N : Type u_2
inst✝⁵ : CommRing N
inst✝⁴ : Algebra R N
inst✝³ : Algebra S N
inst✝² : Algebra R S
inst✝¹ : IsScalarTower R S M
inst✝ : IsScalarTower R S N
⊢ RingHom.ker (φ R S M N)... | R : Type u_3
inst✝¹⁰ : CommRing R
S : Type u_4
inst✝⁹ : CommRing S
M : Type u_1
inst✝⁸ : CommRing M
inst✝⁷ : Algebra R M
inst✝⁶ : Algebra S M
N : Type u_2
inst✝⁵ : CommRing N
inst✝⁴ : Algebra R N
inst✝³ : Algebra S N
inst✝² : Algebra R S
inst✝¹ : IsScalarTower R S M
inst✝ : IsScalarTower R S N
⊢ kerφ R S M N ≤ RingHom.... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_3
inst✝¹⁰ : CommRing R
S : Type u_4
inst✝⁹ : CommRing S
M : Type u_1
inst✝⁸ : CommRing M
inst✝⁷ : Algebra R M
inst✝⁶ : Algebra S M
N : Type u_2
inst✝⁵ : CommRing N
inst✝⁴ : Algebra R N
inst✝³ : Algebra S N
inst✝² : Algebra R S
inst✝¹ : IsScalarTowe... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/RobyLemma9.lean | kerφ_eq | [136, 1] | [157, 6] | simp only [kerφ] | R : Type u_3
inst✝¹⁰ : CommRing R
S : Type u_4
inst✝⁹ : CommRing S
M : Type u_1
inst✝⁸ : CommRing M
inst✝⁷ : Algebra R M
inst✝⁶ : Algebra S M
N : Type u_2
inst✝⁵ : CommRing N
inst✝⁴ : Algebra R N
inst✝³ : Algebra S N
inst✝² : Algebra R S
inst✝¹ : IsScalarTower R S M
inst✝ : IsScalarTower R S N
⊢ kerφ R S M N ≤ RingHom.... | R : Type u_3
inst✝¹⁰ : CommRing R
S : Type u_4
inst✝⁹ : CommRing S
M : Type u_1
inst✝⁸ : CommRing M
inst✝⁷ : Algebra R M
inst✝⁶ : Algebra S M
N : Type u_2
inst✝⁵ : CommRing N
inst✝⁴ : Algebra R N
inst✝³ : Algebra S N
inst✝² : Algebra R S
inst✝¹ : IsScalarTower R S M
inst✝ : IsScalarTower R S N
⊢ Ideal.span ((fun r => (... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_3
inst✝¹⁰ : CommRing R
S : Type u_4
inst✝⁹ : CommRing S
M : Type u_1
inst✝⁸ : CommRing M
inst✝⁷ : Algebra R M
inst✝⁶ : Algebra S M
N : Type u_2
inst✝⁵ : CommRing N
inst✝⁴ : Algebra R N
inst✝³ : Algebra S N
inst✝² : Algebra R S
inst✝¹ : IsScalarTowe... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/RobyLemma9.lean | kerφ_eq | [136, 1] | [157, 6] | rw [Ideal.span_le] | R : Type u_3
inst✝¹⁰ : CommRing R
S : Type u_4
inst✝⁹ : CommRing S
M : Type u_1
inst✝⁸ : CommRing M
inst✝⁷ : Algebra R M
inst✝⁶ : Algebra S M
N : Type u_2
inst✝⁵ : CommRing N
inst✝⁴ : Algebra R N
inst✝³ : Algebra S N
inst✝² : Algebra R S
inst✝¹ : IsScalarTower R S M
inst✝ : IsScalarTower R S N
⊢ Ideal.span ((fun r => (... | R : Type u_3
inst✝¹⁰ : CommRing R
S : Type u_4
inst✝⁹ : CommRing S
M : Type u_1
inst✝⁸ : CommRing M
inst✝⁷ : Algebra R M
inst✝⁶ : Algebra S M
N : Type u_2
inst✝⁵ : CommRing N
inst✝⁴ : Algebra R N
inst✝³ : Algebra S N
inst✝² : Algebra R S
inst✝¹ : IsScalarTower R S M
inst✝ : IsScalarTower R S N
⊢ (fun r => (r • 1) ⊗ₜ[R]... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_3
inst✝¹⁰ : CommRing R
S : Type u_4
inst✝⁹ : CommRing S
M : Type u_1
inst✝⁸ : CommRing M
inst✝⁷ : Algebra R M
inst✝⁶ : Algebra S M
N : Type u_2
inst✝⁵ : CommRing N
inst✝⁴ : Algebra R N
inst✝³ : Algebra S N
inst✝² : Algebra R S
inst✝¹ : IsScalarTowe... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/RobyLemma9.lean | kerφ_eq | [136, 1] | [157, 6] | intro z hz | R : Type u_3
inst✝¹⁰ : CommRing R
S : Type u_4
inst✝⁹ : CommRing S
M : Type u_1
inst✝⁸ : CommRing M
inst✝⁷ : Algebra R M
inst✝⁶ : Algebra S M
N : Type u_2
inst✝⁵ : CommRing N
inst✝⁴ : Algebra R N
inst✝³ : Algebra S N
inst✝² : Algebra R S
inst✝¹ : IsScalarTower R S M
inst✝ : IsScalarTower R S N
⊢ (fun r => (r • 1) ⊗ₜ[R]... | R : Type u_3
inst✝¹⁰ : CommRing R
S : Type u_4
inst✝⁹ : CommRing S
M : Type u_1
inst✝⁸ : CommRing M
inst✝⁷ : Algebra R M
inst✝⁶ : Algebra S M
N : Type u_2
inst✝⁵ : CommRing N
inst✝⁴ : Algebra R N
inst✝³ : Algebra S N
inst✝² : Algebra R S
inst✝¹ : IsScalarTower R S M
inst✝ : IsScalarTower R S N
z : M ⊗[R] N
hz : z ∈ (fu... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_3
inst✝¹⁰ : CommRing R
S : Type u_4
inst✝⁹ : CommRing S
M : Type u_1
inst✝⁸ : CommRing M
inst✝⁷ : Algebra R M
inst✝⁶ : Algebra S M
N : Type u_2
inst✝⁵ : CommRing N
inst✝⁴ : Algebra R N
inst✝³ : Algebra S N
inst✝² : Algebra R S
inst✝¹ : IsScalarTowe... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/RobyLemma9.lean | kerφ_eq | [136, 1] | [157, 6] | simp only [Set.top_eq_univ, Set.image_univ, Set.mem_range] at hz | R : Type u_3
inst✝¹⁰ : CommRing R
S : Type u_4
inst✝⁹ : CommRing S
M : Type u_1
inst✝⁸ : CommRing M
inst✝⁷ : Algebra R M
inst✝⁶ : Algebra S M
N : Type u_2
inst✝⁵ : CommRing N
inst✝⁴ : Algebra R N
inst✝³ : Algebra S N
inst✝² : Algebra R S
inst✝¹ : IsScalarTower R S M
inst✝ : IsScalarTower R S N
z : M ⊗[R] N
hz : z ∈ (fu... | R : Type u_3
inst✝¹⁰ : CommRing R
S : Type u_4
inst✝⁹ : CommRing S
M : Type u_1
inst✝⁸ : CommRing M
inst✝⁷ : Algebra R M
inst✝⁶ : Algebra S M
N : Type u_2
inst✝⁵ : CommRing N
inst✝⁴ : Algebra R N
inst✝³ : Algebra S N
inst✝² : Algebra R S
inst✝¹ : IsScalarTower R S M
inst✝ : IsScalarTower R S N
z : M ⊗[R] N
hz : ∃ y, (y... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_3
inst✝¹⁰ : CommRing R
S : Type u_4
inst✝⁹ : CommRing S
M : Type u_1
inst✝⁸ : CommRing M
inst✝⁷ : Algebra R M
inst✝⁶ : Algebra S M
N : Type u_2
inst✝⁵ : CommRing N
inst✝⁴ : Algebra R N
inst✝³ : Algebra S N
inst✝² : Algebra R S
inst✝¹ : IsScalarTowe... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/RobyLemma9.lean | kerφ_eq | [136, 1] | [157, 6] | obtain ⟨r, rfl⟩ := hz | R : Type u_3
inst✝¹⁰ : CommRing R
S : Type u_4
inst✝⁹ : CommRing S
M : Type u_1
inst✝⁸ : CommRing M
inst✝⁷ : Algebra R M
inst✝⁶ : Algebra S M
N : Type u_2
inst✝⁵ : CommRing N
inst✝⁴ : Algebra R N
inst✝³ : Algebra S N
inst✝² : Algebra R S
inst✝¹ : IsScalarTower R S M
inst✝ : IsScalarTower R S N
z : M ⊗[R] N
hz : ∃ y, (y... | case intro
R : Type u_3
inst✝¹⁰ : CommRing R
S : Type u_4
inst✝⁹ : CommRing S
M : Type u_1
inst✝⁸ : CommRing M
inst✝⁷ : Algebra R M
inst✝⁶ : Algebra S M
N : Type u_2
inst✝⁵ : CommRing N
inst✝⁴ : Algebra R N
inst✝³ : Algebra S N
inst✝² : Algebra R S
inst✝¹ : IsScalarTower R S M
inst✝ : IsScalarTower R S N
r : S
⊢ (r • 1... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_3
inst✝¹⁰ : CommRing R
S : Type u_4
inst✝⁹ : CommRing S
M : Type u_1
inst✝⁸ : CommRing M
inst✝⁷ : Algebra R M
inst✝⁶ : Algebra S M
N : Type u_2
inst✝⁵ : CommRing N
inst✝⁴ : Algebra R N
inst✝³ : Algebra S N
inst✝² : Algebra R S
inst✝¹ : IsScalarTowe... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/RobyLemma9.lean | kerφ_eq | [136, 1] | [157, 6] | simp only [SetLike.mem_coe, RingHom.sub_mem_ker_iff,AlgHom.toRingHom_eq_coe, AlgHom.coe_toRingHom,
φ_apply, TensorProduct.tmul_smul] | case intro
R : Type u_3
inst✝¹⁰ : CommRing R
S : Type u_4
inst✝⁹ : CommRing S
M : Type u_1
inst✝⁸ : CommRing M
inst✝⁷ : Algebra R M
inst✝⁶ : Algebra S M
N : Type u_2
inst✝⁵ : CommRing N
inst✝⁴ : Algebra R N
inst✝³ : Algebra S N
inst✝² : Algebra R S
inst✝¹ : IsScalarTower R S M
inst✝ : IsScalarTower R S N
r : S
⊢ (r • 1... | case intro
R : Type u_3
inst✝¹⁰ : CommRing R
S : Type u_4
inst✝⁹ : CommRing S
M : Type u_1
inst✝⁸ : CommRing M
inst✝⁷ : Algebra R M
inst✝⁶ : Algebra S M
N : Type u_2
inst✝⁵ : CommRing N
inst✝⁴ : Algebra R N
inst✝³ : Algebra S N
inst✝² : Algebra R S
inst✝¹ : IsScalarTower R S M
inst✝ : IsScalarTower R S N
r : S
⊢ (r • 1... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
R : Type u_3
inst✝¹⁰ : CommRing R
S : Type u_4
inst✝⁹ : CommRing S
M : Type u_1
inst✝⁸ : CommRing M
inst✝⁷ : Algebra R M
inst✝⁶ : Algebra S M
N : Type u_2
inst✝⁵ : CommRing N
inst✝⁴ : Algebra R N
inst✝³ : Algebra S N
inst✝² : Algebra R S
inst✝¹ : I... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/RobyLemma9.lean | kerφ_eq | [136, 1] | [157, 6] | rfl | case intro
R : Type u_3
inst✝¹⁰ : CommRing R
S : Type u_4
inst✝⁹ : CommRing S
M : Type u_1
inst✝⁸ : CommRing M
inst✝⁷ : Algebra R M
inst✝⁶ : Algebra S M
N : Type u_2
inst✝⁵ : CommRing N
inst✝⁴ : Algebra R N
inst✝³ : Algebra S N
inst✝² : Algebra R S
inst✝¹ : IsScalarTower R S M
inst✝ : IsScalarTower R S N
r : S
⊢ (r • 1... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
R : Type u_3
inst✝¹⁰ : CommRing R
S : Type u_4
inst✝⁹ : CommRing S
M : Type u_1
inst✝⁸ : CommRing M
inst✝⁷ : Algebra R M
inst✝⁶ : Algebra S M
N : Type u_2
inst✝⁵ : CommRing N
inst✝⁴ : Algebra R N
inst✝³ : Algebra S N
inst✝² : Algebra R S
inst✝¹ : I... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/RobyLemma9.lean | kerφ_eq | [136, 1] | [157, 6] | rw [RingHom.ker_eq_ideal_iff] | R : Type u_3
inst✝¹⁰ : CommRing R
S : Type u_4
inst✝⁹ : CommRing S
M : Type u_1
inst✝⁸ : CommRing M
inst✝⁷ : Algebra R M
inst✝⁶ : Algebra S M
N : Type u_2
inst✝⁵ : CommRing N
inst✝⁴ : Algebra R N
inst✝³ : Algebra S N
inst✝² : Algebra R S
inst✝¹ : IsScalarTower R S M
inst✝ : IsScalarTower R S N
h : kerφ R S M N ≤ RingHo... | R : Type u_3
inst✝¹⁰ : CommRing R
S : Type u_4
inst✝⁹ : CommRing S
M : Type u_1
inst✝⁸ : CommRing M
inst✝⁷ : Algebra R M
inst✝⁶ : Algebra S M
N : Type u_2
inst✝⁵ : CommRing N
inst✝⁴ : Algebra R N
inst✝³ : Algebra S N
inst✝² : Algebra R S
inst✝¹ : IsScalarTower R S M
inst✝ : IsScalarTower R S N
h : kerφ R S M N ≤ RingHo... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_3
inst✝¹⁰ : CommRing R
S : Type u_4
inst✝⁹ : CommRing S
M : Type u_1
inst✝⁸ : CommRing M
inst✝⁷ : Algebra R M
inst✝⁶ : Algebra S M
N : Type u_2
inst✝⁵ : CommRing N
inst✝⁴ : Algebra R N
inst✝³ : Algebra S N
inst✝² : Algebra R S
inst✝¹ : IsScalarTowe... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/RobyLemma9.lean | kerφ_eq | [136, 1] | [157, 6] | use h | R : Type u_3
inst✝¹⁰ : CommRing R
S : Type u_4
inst✝⁹ : CommRing S
M : Type u_1
inst✝⁸ : CommRing M
inst✝⁷ : Algebra R M
inst✝⁶ : Algebra S M
N : Type u_2
inst✝⁵ : CommRing N
inst✝⁴ : Algebra R N
inst✝³ : Algebra S N
inst✝² : Algebra R S
inst✝¹ : IsScalarTower R S M
inst✝ : IsScalarTower R S N
h : kerφ R S M N ≤ RingHo... | case h
R : Type u_3
inst✝¹⁰ : CommRing R
S : Type u_4
inst✝⁹ : CommRing S
M : Type u_1
inst✝⁸ : CommRing M
inst✝⁷ : Algebra R M
inst✝⁶ : Algebra S M
N : Type u_2
inst✝⁵ : CommRing N
inst✝⁴ : Algebra R N
inst✝³ : Algebra S N
inst✝² : Algebra R S
inst✝¹ : IsScalarTower R S M
inst✝ : IsScalarTower R S N
h : kerφ R S M N ≤... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_3
inst✝¹⁰ : CommRing R
S : Type u_4
inst✝⁹ : CommRing S
M : Type u_1
inst✝⁸ : CommRing M
inst✝⁷ : Algebra R M
inst✝⁶ : Algebra S M
N : Type u_2
inst✝⁵ : CommRing N
inst✝⁴ : Algebra R N
inst✝³ : Algebra S N
inst✝² : Algebra R S
inst✝¹ : IsScalarTowe... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/RobyLemma9.lean | kerφ_eq | [136, 1] | [157, 6] | apply Function.HasLeftInverse.injective | case h
R : Type u_3
inst✝¹⁰ : CommRing R
S : Type u_4
inst✝⁹ : CommRing S
M : Type u_1
inst✝⁸ : CommRing M
inst✝⁷ : Algebra R M
inst✝⁶ : Algebra S M
N : Type u_2
inst✝⁵ : CommRing N
inst✝⁴ : Algebra R N
inst✝³ : Algebra S N
inst✝² : Algebra R S
inst✝¹ : IsScalarTower R S M
inst✝ : IsScalarTower R S N
h : kerφ R S M N ≤... | case h.a
R : Type u_3
inst✝¹⁰ : CommRing R
S : Type u_4
inst✝⁹ : CommRing S
M : Type u_1
inst✝⁸ : CommRing M
inst✝⁷ : Algebra R M
inst✝⁶ : Algebra S M
N : Type u_2
inst✝⁵ : CommRing N
inst✝⁴ : Algebra R N
inst✝³ : Algebra S N
inst✝² : Algebra R S
inst✝¹ : IsScalarTower R S M
inst✝ : IsScalarTower R S N
h : kerφ R S M N... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
R : Type u_3
inst✝¹⁰ : CommRing R
S : Type u_4
inst✝⁹ : CommRing S
M : Type u_1
inst✝⁸ : CommRing M
inst✝⁷ : Algebra R M
inst✝⁶ : Algebra S M
N : Type u_2
inst✝⁵ : CommRing N
inst✝⁴ : Algebra R N
inst✝³ : Algebra S N
inst✝² : Algebra R S
inst✝¹ : IsSca... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/RobyLemma9.lean | kerφ_eq | [136, 1] | [157, 6] | use ψ R S M N | case h.a
R : Type u_3
inst✝¹⁰ : CommRing R
S : Type u_4
inst✝⁹ : CommRing S
M : Type u_1
inst✝⁸ : CommRing M
inst✝⁷ : Algebra R M
inst✝⁶ : Algebra S M
N : Type u_2
inst✝⁵ : CommRing N
inst✝⁴ : Algebra R N
inst✝³ : Algebra S N
inst✝² : Algebra R S
inst✝¹ : IsScalarTower R S M
inst✝ : IsScalarTower R S N
h : kerφ R S M N... | case h
R : Type u_3
inst✝¹⁰ : CommRing R
S : Type u_4
inst✝⁹ : CommRing S
M : Type u_1
inst✝⁸ : CommRing M
inst✝⁷ : Algebra R M
inst✝⁶ : Algebra S M
N : Type u_2
inst✝⁵ : CommRing N
inst✝⁴ : Algebra R N
inst✝³ : Algebra S N
inst✝² : Algebra R S
inst✝¹ : IsScalarTower R S M
inst✝ : IsScalarTower R S N
h : kerφ R S M N ≤... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.a
R : Type u_3
inst✝¹⁰ : CommRing R
S : Type u_4
inst✝⁹ : CommRing S
M : Type u_1
inst✝⁸ : CommRing M
inst✝⁷ : Algebra R M
inst✝⁶ : Algebra S M
N : Type u_2
inst✝⁵ : CommRing N
inst✝⁴ : Algebra R N
inst✝³ : Algebra S N
inst✝² : Algebra R S
inst✝¹ : IsS... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/RobyLemma9.lean | kerφ_eq | [136, 1] | [157, 6] | intro z | case h
R : Type u_3
inst✝¹⁰ : CommRing R
S : Type u_4
inst✝⁹ : CommRing S
M : Type u_1
inst✝⁸ : CommRing M
inst✝⁷ : Algebra R M
inst✝⁶ : Algebra S M
N : Type u_2
inst✝⁵ : CommRing N
inst✝⁴ : Algebra R N
inst✝³ : Algebra S N
inst✝² : Algebra R S
inst✝¹ : IsScalarTower R S M
inst✝ : IsScalarTower R S N
h : kerφ R S M N ≤... | case h
R : Type u_3
inst✝¹⁰ : CommRing R
S : Type u_4
inst✝⁹ : CommRing S
M : Type u_1
inst✝⁸ : CommRing M
inst✝⁷ : Algebra R M
inst✝⁶ : Algebra S M
N : Type u_2
inst✝⁵ : CommRing N
inst✝⁴ : Algebra R N
inst✝³ : Algebra S N
inst✝² : Algebra R S
inst✝¹ : IsScalarTower R S M
inst✝ : IsScalarTower R S N
h : kerφ R S M N ≤... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
R : Type u_3
inst✝¹⁰ : CommRing R
S : Type u_4
inst✝⁹ : CommRing S
M : Type u_1
inst✝⁸ : CommRing M
inst✝⁷ : Algebra R M
inst✝⁶ : Algebra S M
N : Type u_2
inst✝⁵ : CommRing N
inst✝⁴ : Algebra R N
inst✝³ : Algebra S N
inst✝² : Algebra R S
inst✝¹ : IsSca... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/RobyLemma9.lean | kerφ_eq | [136, 1] | [157, 6] | obtain ⟨y, rfl⟩ := Ideal.Quotient.mk_surjective z | case h
R : Type u_3
inst✝¹⁰ : CommRing R
S : Type u_4
inst✝⁹ : CommRing S
M : Type u_1
inst✝⁸ : CommRing M
inst✝⁷ : Algebra R M
inst✝⁶ : Algebra S M
N : Type u_2
inst✝⁵ : CommRing N
inst✝⁴ : Algebra R N
inst✝³ : Algebra S N
inst✝² : Algebra R S
inst✝¹ : IsScalarTower R S M
inst✝ : IsScalarTower R S N
h : kerφ R S M N ≤... | case h.intro
R : Type u_3
inst✝¹⁰ : CommRing R
S : Type u_4
inst✝⁹ : CommRing S
M : Type u_1
inst✝⁸ : CommRing M
inst✝⁷ : Algebra R M
inst✝⁶ : Algebra S M
N : Type u_2
inst✝⁵ : CommRing N
inst✝⁴ : Algebra R N
inst✝³ : Algebra S N
inst✝² : Algebra R S
inst✝¹ : IsScalarTower R S M
inst✝ : IsScalarTower R S N
h : kerφ R S... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
R : Type u_3
inst✝¹⁰ : CommRing R
S : Type u_4
inst✝⁹ : CommRing S
M : Type u_1
inst✝⁸ : CommRing M
inst✝⁷ : Algebra R M
inst✝⁶ : Algebra S M
N : Type u_2
inst✝⁵ : CommRing N
inst✝⁴ : Algebra R N
inst✝³ : Algebra S N
inst✝² : Algebra R S
inst✝¹ : IsSca... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/RobyLemma9.lean | kerφ_eq | [136, 1] | [157, 6] | simp only [AlgHom.toRingHom_eq_coe, Ideal.Quotient.lift_mk, AlgHom.coe_toRingHom] | case h.intro
R : Type u_3
inst✝¹⁰ : CommRing R
S : Type u_4
inst✝⁹ : CommRing S
M : Type u_1
inst✝⁸ : CommRing M
inst✝⁷ : Algebra R M
inst✝⁶ : Algebra S M
N : Type u_2
inst✝⁵ : CommRing N
inst✝⁴ : Algebra R N
inst✝³ : Algebra S N
inst✝² : Algebra R S
inst✝¹ : IsScalarTower R S M
inst✝ : IsScalarTower R S N
h : kerφ R S... | case h.intro
R : Type u_3
inst✝¹⁰ : CommRing R
S : Type u_4
inst✝⁹ : CommRing S
M : Type u_1
inst✝⁸ : CommRing M
inst✝⁷ : Algebra R M
inst✝⁶ : Algebra S M
N : Type u_2
inst✝⁵ : CommRing N
inst✝⁴ : Algebra R N
inst✝³ : Algebra S N
inst✝² : Algebra R S
inst✝¹ : IsScalarTower R S M
inst✝ : IsScalarTower R S N
h : kerφ R S... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.intro
R : Type u_3
inst✝¹⁰ : CommRing R
S : Type u_4
inst✝⁹ : CommRing S
M : Type u_1
inst✝⁸ : CommRing M
inst✝⁷ : Algebra R M
inst✝⁶ : Algebra S M
N : Type u_2
inst✝⁵ : CommRing N
inst✝⁴ : Algebra R N
inst✝³ : Algebra S N
inst✝² : Algebra R S
inst✝¹ :... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/RobyLemma9.lean | kerφ_eq | [136, 1] | [157, 6] | induction y using TensorProduct.induction_on with
| zero => simp only [RingHom.map_zero, AlgHom.map_zero]
| tmul m n => simp only [ψ_apply, φ_apply]
| add x y hx hy =>
simp only [RingHom.map_add, AlgHom.map_add, ← Ideal.Quotient.mkₐ_eq_mk, hx, hy] | case h.intro
R : Type u_3
inst✝¹⁰ : CommRing R
S : Type u_4
inst✝⁹ : CommRing S
M : Type u_1
inst✝⁸ : CommRing M
inst✝⁷ : Algebra R M
inst✝⁶ : Algebra S M
N : Type u_2
inst✝⁵ : CommRing N
inst✝⁴ : Algebra R N
inst✝³ : Algebra S N
inst✝² : Algebra R S
inst✝¹ : IsScalarTower R S M
inst✝ : IsScalarTower R S N
h : kerφ R S... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.intro
R : Type u_3
inst✝¹⁰ : CommRing R
S : Type u_4
inst✝⁹ : CommRing S
M : Type u_1
inst✝⁸ : CommRing M
inst✝⁷ : Algebra R M
inst✝⁶ : Algebra S M
N : Type u_2
inst✝⁵ : CommRing N
inst✝⁴ : Algebra R N
inst✝³ : Algebra S N
inst✝² : Algebra R S
inst✝¹ :... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/RobyLemma9.lean | kerφ_eq | [136, 1] | [157, 6] | simp only [RingHom.map_zero, AlgHom.map_zero] | case h.intro.zero
R : Type u_3
inst✝¹⁰ : CommRing R
S : Type u_4
inst✝⁹ : CommRing S
M : Type u_1
inst✝⁸ : CommRing M
inst✝⁷ : Algebra R M
inst✝⁶ : Algebra S M
N : Type u_2
inst✝⁵ : CommRing N
inst✝⁴ : Algebra R N
inst✝³ : Algebra S N
inst✝² : Algebra R S
inst✝¹ : IsScalarTower R S M
inst✝ : IsScalarTower R S N
h : ker... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.intro.zero
R : Type u_3
inst✝¹⁰ : CommRing R
S : Type u_4
inst✝⁹ : CommRing S
M : Type u_1
inst✝⁸ : CommRing M
inst✝⁷ : Algebra R M
inst✝⁶ : Algebra S M
N : Type u_2
inst✝⁵ : CommRing N
inst✝⁴ : Algebra R N
inst✝³ : Algebra S N
inst✝² : Algebra R S
ins... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/RobyLemma9.lean | kerφ_eq | [136, 1] | [157, 6] | simp only [ψ_apply, φ_apply] | case h.intro.tmul
R : Type u_3
inst✝¹⁰ : CommRing R
S : Type u_4
inst✝⁹ : CommRing S
M : Type u_1
inst✝⁸ : CommRing M
inst✝⁷ : Algebra R M
inst✝⁶ : Algebra S M
N : Type u_2
inst✝⁵ : CommRing N
inst✝⁴ : Algebra R N
inst✝³ : Algebra S N
inst✝² : Algebra R S
inst✝¹ : IsScalarTower R S M
inst✝ : IsScalarTower R S N
h : ker... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.intro.tmul
R : Type u_3
inst✝¹⁰ : CommRing R
S : Type u_4
inst✝⁹ : CommRing S
M : Type u_1
inst✝⁸ : CommRing M
inst✝⁷ : Algebra R M
inst✝⁶ : Algebra S M
N : Type u_2
inst✝⁵ : CommRing N
inst✝⁴ : Algebra R N
inst✝³ : Algebra S N
inst✝² : Algebra R S
ins... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/RobyLemma9.lean | kerφ_eq | [136, 1] | [157, 6] | simp only [RingHom.map_add, AlgHom.map_add, ← Ideal.Quotient.mkₐ_eq_mk, hx, hy] | case h.intro.add
R : Type u_3
inst✝¹⁰ : CommRing R
S : Type u_4
inst✝⁹ : CommRing S
M : Type u_1
inst✝⁸ : CommRing M
inst✝⁷ : Algebra R M
inst✝⁶ : Algebra S M
N : Type u_2
inst✝⁵ : CommRing N
inst✝⁴ : Algebra R N
inst✝³ : Algebra S N
inst✝² : Algebra R S
inst✝¹ : IsScalarTower R S M
inst✝ : IsScalarTower R S N
h : kerφ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.intro.add
R : Type u_3
inst✝¹⁰ : CommRing R
S : Type u_4
inst✝⁹ : CommRing S
M : Type u_1
inst✝⁸ : CommRing M
inst✝⁷ : Algebra R M
inst✝⁶ : Algebra S M
N : Type u_2
inst✝⁵ : CommRing N
inst✝⁴ : Algebra R N
inst✝³ : Algebra S N
inst✝² : Algebra R S
inst... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/Homogeneous.lean | MvPolynomial.degree_eq_weightedDegree | [18, 1] | [20, 60] | simp [degree, weightedDegree, Finsupp.total, Finsupp.sum] | σ : Type u_1
τ : Type ?u.203
R : Type ?u.206
inst✝ : CommSemiring R
S : Type ?u.212
d : σ →₀ ℕ
⊢ degree d = ∑ i ∈ d.support, d i | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
σ : Type u_1
τ : Type ?u.203
R : Type ?u.206
inst✝ : CommSemiring R
S : Type ?u.212
d : σ →₀ ℕ
⊢ degree d = ∑ i ∈ d.support, d i
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/Homogeneous.lean | MvPolynomial.totalDegree_eq_weightedTotalDegree | [33, 1] | [37, 44] | simp only [totalDegree, weightedTotalDegree, weightedDegree, LinearMap.toAddMonoidHom_coe,
Finsupp.total, Pi.one_apply, Finsupp.coe_lsum, LinearMap.coe_smulRight, LinearMap.id_coe,
id_eq, Algebra.id.smul_eq_mul, mul_one] | σ : Type u_2
τ : Type ?u.7209
R : Type u_1
inst✝ : CommSemiring R
S : Type ?u.7218
φ : MvPolynomial σ R
⊢ φ.totalDegree = weightedTotalDegree 1 φ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
σ : Type u_2
τ : Type ?u.7209
R : Type u_1
inst✝ : CommSemiring R
S : Type ?u.7218
φ : MvPolynomial σ R
⊢ φ.totalDegree = weightedTotalDegree 1 φ
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/Homogeneous.lean | MvPolynomial.homogeneousSubmodule_eq_finsupp_supported | [59, 1] | [61, 66] | convert weightedHomogeneousSubmodule_eq_finsupp_supported R 1 n | σ : Type u_1
τ : Type ?u.9961
R : Type u_2
inst✝ : CommSemiring R
S : Type ?u.9970
n : ℕ
⊢ homogeneousSubmodule σ R n = Finsupp.supported R R {d | degree d = n} | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
σ : Type u_1
τ : Type ?u.9961
R : Type u_2
inst✝ : CommSemiring R
S : Type ?u.9970
n : ℕ
⊢ homogeneousSubmodule σ R n = Finsupp.supported R R {d | degree d = n}
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/Homogeneous.lean | MvPolynomial.totalDegree_eq_zero_iff | [83, 1] | [86, 40] | rw [totalDegree_eq_weightedTotalDegree, weightedTotalDegree_eq_zero_iff _ p] | σ : Type u_2
τ : Type ?u.26920
R : Type u_1
inst✝ : CommSemiring R
S : Type ?u.26929
p : MvPolynomial σ R
⊢ p.totalDegree = 0 ↔ ∀ m ∈ p.support, ∀ (x : σ), m x = 0 | σ : Type u_2
τ : Type ?u.26920
R : Type u_1
inst✝ : CommSemiring R
S : Type ?u.26929
p : MvPolynomial σ R
⊢ NonTorsionWeight 1 | Please generate a tactic in lean4 to solve the state.
STATE:
σ : Type u_2
τ : Type ?u.26920
R : Type u_1
inst✝ : CommSemiring R
S : Type ?u.26929
p : MvPolynomial σ R
⊢ p.totalDegree = 0 ↔ ∀ m ∈ p.support, ∀ (x : σ), m x = 0
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/Homogeneous.lean | MvPolynomial.totalDegree_eq_zero_iff | [83, 1] | [86, 40] | exact unit_weight_is_nonTrivialWeight | σ : Type u_2
τ : Type ?u.26920
R : Type u_1
inst✝ : CommSemiring R
S : Type ?u.26929
p : MvPolynomial σ R
⊢ NonTorsionWeight 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
σ : Type u_2
τ : Type ?u.26920
R : Type u_1
inst✝ : CommSemiring R
S : Type ?u.26929
p : MvPolynomial σ R
⊢ NonTorsionWeight 1
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/Homogeneous.lean | MvPolynomial.isHomogeneous_of_totalDegree_zero_iff | [100, 1] | [103, 81] | rw [totalDegree_eq_weightedTotalDegree,
← isWeightedHomogeneous_zero_iff_weightedTotalDegree_eq_zero, IsHomogeneous] | σ : Type u_2
τ : Type ?u.27316
R : Type u_1
inst✝ : CommSemiring R
S : Type ?u.27325
p : MvPolynomial σ R
⊢ p.totalDegree = 0 ↔ p.IsHomogeneous 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
σ : Type u_2
τ : Type ?u.27316
R : Type u_1
inst✝ : CommSemiring R
S : Type ?u.27325
p : MvPolynomial σ R
⊢ p.totalDegree = 0 ↔ p.IsHomogeneous 0
TACTIC:
|
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