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 | unused_files/Init_copy.lean | DividedPowerAlgebra.dp_mul | [213, 1] | [216, 31] | simp only [dp_def, ← _root_.map_mul, ← map_nsmul] | R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
n p : ℕ
m : M
⊢ dp R n m * dp R p m = (n + p).choose n • dp R (n + p) m | R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
n p : ℕ
m : M
⊢ (mkAlgHom R (Rel R M)) (X (n, m)) * (mkAlgHom R (Rel R M)) (X (p, m)) =
(n + p).choose n • (mkAlgHom R (Rel R M)) (X (n + p, m)) | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
n p : ℕ
m : M
⊢ dp R n m * dp R p m = (n + p).choose n • dp R (n + p) m
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | unused_files/Init_copy.lean | DividedPowerAlgebra.dp_add | [219, 1] | [224, 31] | simp only [dp_def] | R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
n : ℕ
x y : M
⊢ dp R n (x + y) = ∑ k ∈ range (n + 1), dp R k x * dp R (n - k) y | R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
n : ℕ
x y : M
⊢ (mkAlgHom R (Rel R M)) (X (n, x + y)) =
∑ x_1 ∈ range (n + 1), (mkAlgHom R (Rel R M)) (X (x_1, x)) * (mkAlgHom R (Rel R M)) (X (n - x_1, y)) | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
n : ℕ
x y : M
⊢ dp R n (x + y) = ∑ k ∈ range (n + 1), dp R k x * dp R (n - k) y
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | unused_files/Init_copy.lean | DividedPowerAlgebra.ext_iff | [247, 1] | [257, 54] | constructor | R : Type u_2
M : Type u_3
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
A : Type u_1
inst✝¹ : CommRing A
inst✝ : Algebra R A
f g : DividedPowerAlgebra R M →ₐ[R] A
⊢ f = g ↔ ∀ (n : ℕ) (m : M), f (dp R n m) = g (dp R n m) | case mp
R : Type u_2
M : Type u_3
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
A : Type u_1
inst✝¹ : CommRing A
inst✝ : Algebra R A
f g : DividedPowerAlgebra R M →ₐ[R] A
⊢ f = g → ∀ (n : ℕ) (m : M), f (dp R n m) = g (dp R n m)
case mpr
R : Type u_2
M : Type u_3
inst✝⁴ : CommRing R
inst✝³ : AddCommM... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
M : Type u_3
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
A : Type u_1
inst✝¹ : CommRing A
inst✝ : Algebra R A
f g : DividedPowerAlgebra R M →ₐ[R] A
⊢ f = g ↔ ∀ (n : ℕ) (m : M), f (dp R n m) = g (dp R n m)
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | unused_files/Init_copy.lean | DividedPowerAlgebra.ext_iff | [247, 1] | [257, 54] | . intro h n m
rw [h] | case mp
R : Type u_2
M : Type u_3
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
A : Type u_1
inst✝¹ : CommRing A
inst✝ : Algebra R A
f g : DividedPowerAlgebra R M →ₐ[R] A
⊢ f = g → ∀ (n : ℕ) (m : M), f (dp R n m) = g (dp R n m)
case mpr
R : Type u_2
M : Type u_3
inst✝⁴ : CommRing R
inst✝³ : AddCommM... | case mpr
R : Type u_2
M : Type u_3
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
A : Type u_1
inst✝¹ : CommRing A
inst✝ : Algebra R A
f g : DividedPowerAlgebra R M →ₐ[R] A
⊢ (∀ (n : ℕ) (m : M), f (dp R n m) = g (dp R n m)) → f = g | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
R : Type u_2
M : Type u_3
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
A : Type u_1
inst✝¹ : CommRing A
inst✝ : Algebra R A
f g : DividedPowerAlgebra R M →ₐ[R] A
⊢ f = g → ∀ (n : ℕ) (m : M), f (dp R n m) = g (dp R n m)
case mpr
R ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | unused_files/Init_copy.lean | DividedPowerAlgebra.ext_iff | [247, 1] | [257, 54] | . intro h
rw [FunLike.ext'_iff]
apply Function.Surjective.injective_comp_right (mkAlgHom_surjective R (Rel R M))
simp only [← AlgHom.coe_comp, ← FunLike.ext'_iff]
exact MvPolynomial.algHom_ext fun ⟨n, m⟩ => h n m | case mpr
R : Type u_2
M : Type u_3
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
A : Type u_1
inst✝¹ : CommRing A
inst✝ : Algebra R A
f g : DividedPowerAlgebra R M →ₐ[R] A
⊢ (∀ (n : ℕ) (m : M), f (dp R n m) = g (dp R n m)) → f = g | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
R : Type u_2
M : Type u_3
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
A : Type u_1
inst✝¹ : CommRing A
inst✝ : Algebra R A
f g : DividedPowerAlgebra R M →ₐ[R] A
⊢ (∀ (n : ℕ) (m : M), f (dp R n m) = g (dp R n m)) → f = g
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | unused_files/Init_copy.lean | DividedPowerAlgebra.ext_iff | [247, 1] | [257, 54] | intro h n m | case mp
R : Type u_2
M : Type u_3
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
A : Type u_1
inst✝¹ : CommRing A
inst✝ : Algebra R A
f g : DividedPowerAlgebra R M →ₐ[R] A
⊢ f = g → ∀ (n : ℕ) (m : M), f (dp R n m) = g (dp R n m) | case mp
R : Type u_2
M : Type u_3
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
A : Type u_1
inst✝¹ : CommRing A
inst✝ : Algebra R A
f g : DividedPowerAlgebra R M →ₐ[R] A
h : f = g
n : ℕ
m : M
⊢ f (dp R n m) = g (dp R n m) | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
R : Type u_2
M : Type u_3
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
A : Type u_1
inst✝¹ : CommRing A
inst✝ : Algebra R A
f g : DividedPowerAlgebra R M →ₐ[R] A
⊢ f = g → ∀ (n : ℕ) (m : M), f (dp R n m) = g (dp R n m)
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | unused_files/Init_copy.lean | DividedPowerAlgebra.ext_iff | [247, 1] | [257, 54] | rw [h] | case mp
R : Type u_2
M : Type u_3
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
A : Type u_1
inst✝¹ : CommRing A
inst✝ : Algebra R A
f g : DividedPowerAlgebra R M →ₐ[R] A
h : f = g
n : ℕ
m : M
⊢ f (dp R n m) = g (dp R n m) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
R : Type u_2
M : Type u_3
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
A : Type u_1
inst✝¹ : CommRing A
inst✝ : Algebra R A
f g : DividedPowerAlgebra R M →ₐ[R] A
h : f = g
n : ℕ
m : M
⊢ f (dp R n m) = g (dp R n m)
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | unused_files/Init_copy.lean | DividedPowerAlgebra.ext_iff | [247, 1] | [257, 54] | intro h | case mpr
R : Type u_2
M : Type u_3
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
A : Type u_1
inst✝¹ : CommRing A
inst✝ : Algebra R A
f g : DividedPowerAlgebra R M →ₐ[R] A
⊢ (∀ (n : ℕ) (m : M), f (dp R n m) = g (dp R n m)) → f = g | case mpr
R : Type u_2
M : Type u_3
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
A : Type u_1
inst✝¹ : CommRing A
inst✝ : Algebra R A
f g : DividedPowerAlgebra R M →ₐ[R] A
h : ∀ (n : ℕ) (m : M), f (dp R n m) = g (dp R n m)
⊢ f = g | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
R : Type u_2
M : Type u_3
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
A : Type u_1
inst✝¹ : CommRing A
inst✝ : Algebra R A
f g : DividedPowerAlgebra R M →ₐ[R] A
⊢ (∀ (n : ℕ) (m : M), f (dp R n m) = g (dp R n m)) → f = g
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | unused_files/Init_copy.lean | DividedPowerAlgebra.lift'_imp | [292, 1] | [304, 17] | cases' h with a r n a m n a n a b <;>
simp only [eval₂AlgHom_X', map_one, map_zero, map_smul, AlgHom.map_mul, map_nsmul, AlgHom.map_sum] | R : Type u_3
M : Type u_1
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
A : Type u_2
inst✝¹ : CommRing A
inst✝ : Algebra R A
f : ℕ × M → A
hf_zero : ∀ (m : M), f (0, m) = 1
hf_smul : ∀ (n : ℕ) (r : R) (m : M), f (n, r • m) = r ^ n • f (n, m)
hf_mul : ∀ (n p : ℕ) (m : M), f (n, m) * f (p, m) = (n + p)... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_3
M : Type u_1
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
A : Type u_2
inst✝¹ : CommRing A
inst✝ : Algebra R A
f : ℕ × M → A
hf_zero : ∀ (m : M), f (0, m) = 1
hf_smul : ∀ (n : ℕ) (r : R) (m : M), f (n, r • m) = r ^ n • f (n, m... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | unused_files/Init_copy.lean | DividedPowerAlgebra.lift'AlgHom_apply | [317, 1] | [325, 6] | rw [mk, lift', RingQuot.liftAlgHom_mkAlgHom_apply, coe_eval₂AlgHom] | R : Type u_3
M : Type u_1
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
A : Type u_2
inst✝¹ : CommRing A
inst✝ : Algebra R A
f : ℕ × M → A
hf_zero : ∀ (m : M), f (0, m) = 1
hf_smul : ∀ (n : ℕ) (r : R) (m : M), f (n, r • m) = r ^ n • f (n, m)
hf_mul : ∀ (n p : ℕ) (m : M), f (n, m) * f (p, m) = (n + p)... | R : Type u_3
M : Type u_1
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
A : Type u_2
inst✝¹ : CommRing A
inst✝ : Algebra R A
f : ℕ × M → A
hf_zero : ∀ (m : M), f (0, m) = 1
hf_smul : ∀ (n : ℕ) (r : R) (m : M), f (n, r • m) = r ^ n • f (n, m)
hf_mul : ∀ (n p : ℕ) (m : M), f (n, m) * f (p, m) = (n + p)... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_3
M : Type u_1
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
A : Type u_2
inst✝¹ : CommRing A
inst✝ : Algebra R A
f : ℕ × M → A
hf_zero : ∀ (m : M), f (0, m) = 1
hf_smul : ∀ (n : ℕ) (r : R) (m : M), f (n, r • m) = r ^ n • f (n, m... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | unused_files/Init_copy.lean | DividedPowerAlgebra.lift'AlgHom_apply_dp | [329, 1] | [335, 80] | rw [dp_def, ← mk, lift'AlgHom_apply f hf_zero hf_smul hf_mul hf_add, aeval_X] | R : Type u_3
M : Type u_1
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
A : Type u_2
inst✝¹ : CommRing A
inst✝ : Algebra R A
f : ℕ × M → A
hf_zero : ∀ (m : M), f (0, m) = 1
hf_smul : ∀ (n : ℕ) (r : R) (m : M), f (n, r • m) = r ^ n • f (n, m)
hf_mul : ∀ (n p : ℕ) (m : M), f (n, m) * f (p, m) = (n + p)... | R : Type u_3
M : Type u_1
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
A : Type u_2
inst✝¹ : CommRing A
inst✝ : Algebra R A
f : ℕ × M → A
hf_zero : ∀ (m : M), f (0, m) = 1
hf_smul : ∀ (n : ℕ) (r : R) (m : M), f (n, r • m) = r ^ n • f (n, m)
hf_mul : ∀ (n p : ℕ) (m : M), f (n, m) * f (p, m) = (n + p)... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_3
M : Type u_1
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
A : Type u_2
inst✝¹ : CommRing A
inst✝ : Algebra R A
f : ℕ × M → A
hf_zero : ∀ (m : M), f (0, m) = 1
hf_smul : ∀ (n : ℕ) (r : R) (m : M), f (n, r • m) = r ^ n • f (n, m... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | unused_files/Init_copy.lean | DividedPowerAlgebra.liftAlgHom_apply | [360, 1] | [363, 34] | rw [lift, lift'AlgHom_apply] | R : Type u_1
M : Type u_2
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
A : Type u_3
inst✝¹ : CommRing A
inst✝ : Algebra R A
I : Ideal A
hI : DividedPowers I
φ : M →ₗ[R] A
hφ : ∀ (m : M), φ m ∈ I
p : MvPolynomial (ℕ × M) R
⊢ (lift hI φ hφ) (mk p) = (aeval fun nm => hI.dpow nm.1 (φ nm.2)) p | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
M : Type u_2
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
A : Type u_3
inst✝¹ : CommRing A
inst✝ : Algebra R A
I : Ideal A
hI : DividedPowers I
φ : M →ₗ[R] A
hφ : ∀ (m : M), φ m ∈ I
p : MvPolynomial (ℕ × M) R
⊢ (lift hI φ hφ) ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | unused_files/Init_copy.lean | DividedPowerAlgebra.liftAlgHom_apply_dp | [373, 1] | [375, 34] | rw [lift, lift'AlgHom_apply_dp] | R : Type u_2
M : Type u_3
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
A : Type u_1
inst✝¹ : CommRing A
inst✝ : Algebra R A
I : Ideal A
hI : DividedPowers I
φ : M →ₗ[R] A
hφ : ∀ (m : M), φ m ∈ I
n : ℕ
m : M
⊢ (lift hI φ hφ) (dp R n m) = hI.dpow n (φ m) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
M : Type u_3
inst✝⁴ : CommRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
A : Type u_1
inst✝¹ : CommRing A
inst✝ : Algebra R A
I : Ideal A
hI : DividedPowers I
φ : M →ₗ[R] A
hφ : ∀ (m : M), φ m ∈ I
n : ℕ
m : M
⊢ (lift hI φ hφ) (dp R n m) = hI... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | unused_files/Init_copy.lean | DividedPowerAlgebra.LinearMap.dp_smul | [425, 1] | [429, 44] | rw [f.map_smul, algebra_compatible_smul S r (f a)] | R : Type u_4
M : Type u_3
inst✝¹⁰ : CommRing R
inst✝⁹ : AddCommMonoid M
inst✝⁸ : Module R M
S : Type u_1
inst✝⁷ : CommRing S
inst✝⁶ : Algebra R S
N : Type u_2
inst✝⁵ : AddCommMonoid N
inst✝⁴ : Module R N
inst✝³ : Module S N
inst✝² : IsScalarTower R S N
inst✝¹ : Algebra R (DividedPowerAlgebra S N)
inst✝ : IsScalarTower ... | R : Type u_4
M : Type u_3
inst✝¹⁰ : CommRing R
inst✝⁹ : AddCommMonoid M
inst✝⁸ : Module R M
S : Type u_1
inst✝⁷ : CommRing S
inst✝⁶ : Algebra R S
N : Type u_2
inst✝⁵ : AddCommMonoid N
inst✝⁴ : Module R N
inst✝³ : Module S N
inst✝² : IsScalarTower R S N
inst✝¹ : Algebra R (DividedPowerAlgebra S N)
inst✝ : IsScalarTower ... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_4
M : Type u_3
inst✝¹⁰ : CommRing R
inst✝⁹ : AddCommMonoid M
inst✝⁸ : Module R M
S : Type u_1
inst✝⁷ : CommRing S
inst✝⁶ : Algebra R S
N : Type u_2
inst✝⁵ : AddCommMonoid N
inst✝⁴ : Module R N
inst✝³ : Module S N
inst✝² : IsScalarTower R S N
inst✝¹... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | unused_files/Init_copy.lean | DividedPowerAlgebra.LinearMap.dp_smul | [425, 1] | [429, 44] | rw [DividedPowerAlgebra.dp_smul S ((algebraMap R S) r) n (f a)] | R : Type u_4
M : Type u_3
inst✝¹⁰ : CommRing R
inst✝⁹ : AddCommMonoid M
inst✝⁸ : Module R M
S : Type u_1
inst✝⁷ : CommRing S
inst✝⁶ : Algebra R S
N : Type u_2
inst✝⁵ : AddCommMonoid N
inst✝⁴ : Module R N
inst✝³ : Module S N
inst✝² : IsScalarTower R S N
inst✝¹ : Algebra R (DividedPowerAlgebra S N)
inst✝ : IsScalarTower ... | R : Type u_4
M : Type u_3
inst✝¹⁰ : CommRing R
inst✝⁹ : AddCommMonoid M
inst✝⁸ : Module R M
S : Type u_1
inst✝⁷ : CommRing S
inst✝⁶ : Algebra R S
N : Type u_2
inst✝⁵ : AddCommMonoid N
inst✝⁴ : Module R N
inst✝³ : Module S N
inst✝² : IsScalarTower R S N
inst✝¹ : Algebra R (DividedPowerAlgebra S N)
inst✝ : IsScalarTower ... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_4
M : Type u_3
inst✝¹⁰ : CommRing R
inst✝⁹ : AddCommMonoid M
inst✝⁸ : Module R M
S : Type u_1
inst✝⁷ : CommRing S
inst✝⁶ : Algebra R S
N : Type u_2
inst✝⁵ : AddCommMonoid N
inst✝⁴ : Module R N
inst✝³ : Module S N
inst✝² : IsScalarTower R S N
inst✝¹... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | unused_files/Init_copy.lean | DividedPowerAlgebra.LinearMap.dp_smul | [425, 1] | [429, 44] | rw [← map_pow, ← algebra_compatible_smul] | R : Type u_4
M : Type u_3
inst✝¹⁰ : CommRing R
inst✝⁹ : AddCommMonoid M
inst✝⁸ : Module R M
S : Type u_1
inst✝⁷ : CommRing S
inst✝⁶ : Algebra R S
N : Type u_2
inst✝⁵ : AddCommMonoid N
inst✝⁴ : Module R N
inst✝³ : Module S N
inst✝² : IsScalarTower R S N
inst✝¹ : Algebra R (DividedPowerAlgebra S N)
inst✝ : IsScalarTower ... | R : Type u_4
M : Type u_3
inst✝¹⁰ : CommRing R
inst✝⁹ : AddCommMonoid M
inst✝⁸ : Module R M
S : Type u_1
inst✝⁷ : CommRing S
inst✝⁶ : Algebra R S
N : Type u_2
inst✝⁵ : AddCommMonoid N
inst✝⁴ : Module R N
inst✝³ : Module S N
inst✝² : IsScalarTower R S N
inst✝¹ : Algebra R (DividedPowerAlgebra S N)
inst✝ : IsScalarTower ... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_4
M : Type u_3
inst✝¹⁰ : CommRing R
inst✝⁹ : AddCommMonoid M
inst✝⁸ : Module R M
S : Type u_1
inst✝⁷ : CommRing S
inst✝⁶ : Algebra R S
N : Type u_2
inst✝⁵ : AddCommMonoid N
inst✝⁴ : Module R N
inst✝³ : Module S N
inst✝² : IsScalarTower R S N
inst✝¹... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | unused_files/Init_copy.lean | DividedPowerAlgebra.LinearMap.dp_add | [435, 1] | [438, 43] | rw [map_add, DividedPowerAlgebra.dp_add] | R : Type u_4
M : Type u_3
inst✝¹⁰ : CommRing R
inst✝⁹ : AddCommMonoid M
inst✝⁸ : Module R M
S : Type u_1
inst✝⁷ : CommRing S
inst✝⁶ : Algebra R S
N : Type u_2
inst✝⁵ : AddCommMonoid N
inst✝⁴ : Module R N
inst✝³ : Module S N
inst✝² : IsScalarTower R S N
inst✝¹ : Algebra R (DividedPowerAlgebra S N)
inst✝ : IsScalarTower ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_4
M : Type u_3
inst✝¹⁰ : CommRing R
inst✝⁹ : AddCommMonoid M
inst✝⁸ : Module R M
S : Type u_1
inst✝⁷ : CommRing S
inst✝⁶ : Algebra R S
N : Type u_2
inst✝⁵ : AddCommMonoid N
inst✝⁴ : Module R N
inst✝³ : Module S N
inst✝² : IsScalarTower R S N
inst✝¹... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/RingTheory/AugmentationIdeal.lean | Submodule.sup_restrictScalars | [41, 1] | [44, 85] | apply Submodule.toAddSubmonoid_injective | R✝ : Type u_1
inst✝⁹ : CommRing R✝
A✝ : Type u_2
inst✝⁸ : CommRing A✝
inst✝⁷ : Algebra R✝ A✝
J : Ideal A✝
A : Type u_3
inst✝⁶ : CommSemiring A
R : Type u_4
inst✝⁵ : Ring R
inst✝⁴ : Algebra A R
M : Type u_5
inst✝³ : AddCommGroup M
inst✝² : Module A M
inst✝¹ : Module R M
inst✝ : IsScalarTower A R M
M₁ M₂ : Submodule R M
... | case a
R✝ : Type u_1
inst✝⁹ : CommRing R✝
A✝ : Type u_2
inst✝⁸ : CommRing A✝
inst✝⁷ : Algebra R✝ A✝
J : Ideal A✝
A : Type u_3
inst✝⁶ : CommSemiring A
R : Type u_4
inst✝⁵ : Ring R
inst✝⁴ : Algebra A R
M : Type u_5
inst✝³ : AddCommGroup M
inst✝² : Module A M
inst✝¹ : Module R M
inst✝ : IsScalarTower A R M
M₁ M₂ : Submodu... | Please generate a tactic in lean4 to solve the state.
STATE:
R✝ : Type u_1
inst✝⁹ : CommRing R✝
A✝ : Type u_2
inst✝⁸ : CommRing A✝
inst✝⁷ : Algebra R✝ A✝
J : Ideal A✝
A : Type u_3
inst✝⁶ : CommSemiring A
R : Type u_4
inst✝⁵ : Ring R
inst✝⁴ : Algebra A R
M : Type u_5
inst✝³ : AddCommGroup M
inst✝² : Module A M
inst✝¹ : ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/RingTheory/AugmentationIdeal.lean | Submodule.sup_restrictScalars | [41, 1] | [44, 85] | simp only [Submodule.toAddSubmonoid_restrictScalars, Submodule.sup_toAddSubmonoid] | case a
R✝ : Type u_1
inst✝⁹ : CommRing R✝
A✝ : Type u_2
inst✝⁸ : CommRing A✝
inst✝⁷ : Algebra R✝ A✝
J : Ideal A✝
A : Type u_3
inst✝⁶ : CommSemiring A
R : Type u_4
inst✝⁵ : Ring R
inst✝⁴ : Algebra A R
M : Type u_5
inst✝³ : AddCommGroup M
inst✝² : Module A M
inst✝¹ : Module R M
inst✝ : IsScalarTower A R M
M₁ M₂ : Submodu... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a
R✝ : Type u_1
inst✝⁹ : CommRing R✝
A✝ : Type u_2
inst✝⁸ : CommRing A✝
inst✝⁷ : Algebra R✝ A✝
J : Ideal A✝
A : Type u_3
inst✝⁶ : CommSemiring A
R : Type u_4
inst✝⁵ : Ring R
inst✝⁴ : Algebra A R
M : Type u_5
inst✝³ : AddCommGroup M
inst✝² : Module A M
in... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/RingTheory/AugmentationIdeal.lean | Submodule.codisjoint_restrictScalars_iff | [46, 1] | [49, 100] | simp only [codisjoint_iff, ← Submodule.sup_restrictScalars, Submodule.restrictScalars_eq_top_iff] | R✝ : Type u_1
inst✝⁹ : CommRing R✝
A✝ : Type u_2
inst✝⁸ : CommRing A✝
inst✝⁷ : Algebra R✝ A✝
J : Ideal A✝
A : Type u_3
inst✝⁶ : CommSemiring A
R : Type u_4
inst✝⁵ : Ring R
inst✝⁴ : Algebra A R
M : Type u_5
inst✝³ : AddCommGroup M
inst✝² : Module A M
inst✝¹ : Module R M
inst✝ : IsScalarTower A R M
M₁ M₂ : Submodule R M
... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R✝ : Type u_1
inst✝⁹ : CommRing R✝
A✝ : Type u_2
inst✝⁸ : CommRing A✝
inst✝⁷ : Algebra R✝ A✝
J : Ideal A✝
A : Type u_3
inst✝⁶ : CommSemiring A
R : Type u_4
inst✝⁵ : Ring R
inst✝⁴ : Algebra A R
M : Type u_5
inst✝³ : AddCommGroup M
inst✝² : Module A M
inst✝¹ : ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/RingTheory/AugmentationIdeal.lean | Submodule.disjoint_restrictScalars_iff | [51, 1] | [54, 68] | simp only [Submodule.disjoint_def, Submodule.restrictScalars_mem] | R✝ : Type u_1
inst✝⁹ : CommRing R✝
A✝ : Type u_2
inst✝⁸ : CommRing A✝
inst✝⁷ : Algebra R✝ A✝
J : Ideal A✝
A : Type u_3
inst✝⁶ : CommSemiring A
R : Type u_4
inst✝⁵ : Ring R
inst✝⁴ : Algebra A R
M : Type u_5
inst✝³ : AddCommGroup M
inst✝² : Module A M
inst✝¹ : Module R M
inst✝ : IsScalarTower A R M
M₁ M₂ : Submodule R M
... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R✝ : Type u_1
inst✝⁹ : CommRing R✝
A✝ : Type u_2
inst✝⁸ : CommRing A✝
inst✝⁷ : Algebra R✝ A✝
J : Ideal A✝
A : Type u_3
inst✝⁶ : CommSemiring A
R : Type u_4
inst✝⁵ : Ring R
inst✝⁴ : Algebra A R
M : Type u_5
inst✝³ : AddCommGroup M
inst✝² : Module A M
inst✝¹ : ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/RingTheory/AugmentationIdeal.lean | Submodule.isCompl_restrictScalars_iff | [56, 1] | [58, 108] | simp only [isCompl_iff, Submodule.disjoint_restrictScalars_iff, Submodule.codisjoint_restrictScalars_iff] | R✝ : Type u_1
inst✝⁹ : CommRing R✝
A✝ : Type u_2
inst✝⁸ : CommRing A✝
inst✝⁷ : Algebra R✝ A✝
J : Ideal A✝
A : Type u_3
inst✝⁶ : CommSemiring A
R : Type u_4
inst✝⁵ : Ring R
inst✝⁴ : Algebra A R
M : Type u_5
inst✝³ : AddCommGroup M
inst✝² : Module A M
inst✝¹ : Module R M
inst✝ : IsScalarTower A R M
M₁ M₂ : Submodule R M
... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R✝ : Type u_1
inst✝⁹ : CommRing R✝
A✝ : Type u_2
inst✝⁸ : CommRing A✝
inst✝⁷ : Algebra R✝ A✝
J : Ideal A✝
A : Type u_3
inst✝⁶ : CommSemiring A
R : Type u_4
inst✝⁵ : Ring R
inst✝⁴ : Algebra A R
M : Type u_5
inst✝³ : AddCommGroup M
inst✝² : Module A M
inst✝¹ : ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/RingTheory/AugmentationIdeal.lean | LinearMap.ker_lTensor_of_linearProjOfIsCompl | [73, 1] | [81, 56] | rw [← exact_iff] | R : Type u_1
inst✝⁶ : CommRing R
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R A
J : Ideal A
M : Type u_3
inst✝³ : AddCommGroup M
inst✝² : Module A M
M₁ M₂ : Submodule A M
hM : IsCompl M₁ M₂
Q : Type u_4
inst✝¹ : AddCommGroup Q
inst✝ : Module A Q
⊢ ker (lTensor Q (M₁.linearProjOfIsCompl M₂ hM)) = range (lTensor Q... | R : Type u_1
inst✝⁶ : CommRing R
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R A
J : Ideal A
M : Type u_3
inst✝³ : AddCommGroup M
inst✝² : Module A M
M₁ M₂ : Submodule A M
hM : IsCompl M₁ M₂
Q : Type u_4
inst✝¹ : AddCommGroup Q
inst✝ : Module A Q
⊢ Function.Exact ⇑(lTensor Q M₂.subtype) ⇑(lTensor Q (M₁.linearProj... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝⁶ : CommRing R
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R A
J : Ideal A
M : Type u_3
inst✝³ : AddCommGroup M
inst✝² : Module A M
M₁ M₂ : Submodule A M
hM : IsCompl M₁ M₂
Q : Type u_4
inst✝¹ : AddCommGroup Q
inst✝ : Module A Q
⊢ ker ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/RingTheory/AugmentationIdeal.lean | LinearMap.ker_lTensor_of_linearProjOfIsCompl | [73, 1] | [81, 56] | apply lTensor_exact | R : Type u_1
inst✝⁶ : CommRing R
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R A
J : Ideal A
M : Type u_3
inst✝³ : AddCommGroup M
inst✝² : Module A M
M₁ M₂ : Submodule A M
hM : IsCompl M₁ M₂
Q : Type u_4
inst✝¹ : AddCommGroup Q
inst✝ : Module A Q
⊢ Function.Exact ⇑(lTensor Q M₂.subtype) ⇑(lTensor Q (M₁.linearProj... | case hfg
R : Type u_1
inst✝⁶ : CommRing R
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R A
J : Ideal A
M : Type u_3
inst✝³ : AddCommGroup M
inst✝² : Module A M
M₁ M₂ : Submodule A M
hM : IsCompl M₁ M₂
Q : Type u_4
inst✝¹ : AddCommGroup Q
inst✝ : Module A Q
⊢ Function.Exact ⇑M₂.subtype ⇑(M₁.linearProjOfIsCompl M₂ h... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝⁶ : CommRing R
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R A
J : Ideal A
M : Type u_3
inst✝³ : AddCommGroup M
inst✝² : Module A M
M₁ M₂ : Submodule A M
hM : IsCompl M₁ M₂
Q : Type u_4
inst✝¹ : AddCommGroup Q
inst✝ : Module A Q
⊢ Func... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/RingTheory/AugmentationIdeal.lean | LinearMap.ker_lTensor_of_linearProjOfIsCompl | [73, 1] | [81, 56] | simp only [exact_iff, range_subtype, linearProjOfIsCompl_ker] | case hfg
R : Type u_1
inst✝⁶ : CommRing R
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R A
J : Ideal A
M : Type u_3
inst✝³ : AddCommGroup M
inst✝² : Module A M
M₁ M₂ : Submodule A M
hM : IsCompl M₁ M₂
Q : Type u_4
inst✝¹ : AddCommGroup Q
inst✝ : Module A Q
⊢ Function.Exact ⇑M₂.subtype ⇑(M₁.linearProjOfIsCompl M₂ h... | case hg
R : Type u_1
inst✝⁶ : CommRing R
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R A
J : Ideal A
M : Type u_3
inst✝³ : AddCommGroup M
inst✝² : Module A M
M₁ M₂ : Submodule A M
hM : IsCompl M₁ M₂
Q : Type u_4
inst✝¹ : AddCommGroup Q
inst✝ : Module A Q
⊢ Function.Surjective ⇑(M₁.linearProjOfIsCompl M₂ hM) | Please generate a tactic in lean4 to solve the state.
STATE:
case hfg
R : Type u_1
inst✝⁶ : CommRing R
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R A
J : Ideal A
M : Type u_3
inst✝³ : AddCommGroup M
inst✝² : Module A M
M₁ M₂ : Submodule A M
hM : IsCompl M₁ M₂
Q : Type u_4
inst✝¹ : AddCommGroup Q
inst✝ : Module A... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/RingTheory/AugmentationIdeal.lean | LinearMap.ker_lTensor_of_linearProjOfIsCompl | [73, 1] | [81, 56] | simp only [← range_eq_top, linearProjOfIsCompl_range] | case hg
R : Type u_1
inst✝⁶ : CommRing R
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R A
J : Ideal A
M : Type u_3
inst✝³ : AddCommGroup M
inst✝² : Module A M
M₁ M₂ : Submodule A M
hM : IsCompl M₁ M₂
Q : Type u_4
inst✝¹ : AddCommGroup Q
inst✝ : Module A Q
⊢ Function.Surjective ⇑(M₁.linearProjOfIsCompl M₂ hM) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hg
R : Type u_1
inst✝⁶ : CommRing R
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R A
J : Ideal A
M : Type u_3
inst✝³ : AddCommGroup M
inst✝² : Module A M
M₁ M₂ : Submodule A M
hM : IsCompl M₁ M₂
Q : Type u_4
inst✝¹ : AddCommGroup Q
inst✝ : Module A ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/RingTheory/AugmentationIdeal.lean | LinearMap.ker_baseChange_of_linearProjOfIsCompl | [83, 1] | [88, 73] | simpa only [← exact_iff] using ker_lTensor_of_linearProjOfIsCompl hM R | R✝ : Type u_1
inst✝⁶ : CommRing R✝
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R✝ A
J : Ideal A
M : Type u_3
inst✝³ : AddCommGroup M
inst✝² : Module A M
M₁ M₂ : Submodule A M
hM : IsCompl M₁ M₂
R : Type u_4
inst✝¹ : CommRing R
inst✝ : Algebra A R
⊢ ker (baseChange R (M₁.linearProjOfIsCompl M₂ hM)) = range (baseCh... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R✝ : Type u_1
inst✝⁶ : CommRing R✝
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R✝ A
J : Ideal A
M : Type u_3
inst✝³ : AddCommGroup M
inst✝² : Module A M
M₁ M₂ : Submodule A M
hM : IsCompl M₁ M₂
R : Type u_4
inst✝¹ : CommRing R
inst✝ : Algebra A R
⊢ ker ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/RingTheory/AugmentationIdeal.lean | LinearMap.isCompl_lTensor | [90, 1] | [119, 53] | have hq :
M₁.subtype.comp (linearProjOfIsCompl _ _ hM)
+ M₂.subtype.comp (linearProjOfIsCompl _ _ hM.symm) = id := by
ext x
simp only [add_apply, coe_comp, coeSubtype, Function.comp_apply,
id_coe, id_eq]
rw [linear_proj_add_linearProjOfIsCompl_eq_self] | R : Type u_1
inst✝⁶ : CommRing R
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R A
J : Ideal A
M : Type u_3
inst✝³ : AddCommGroup M
inst✝² : Module A M
M₁ M₂ : Submodule A M
hM : IsCompl M₁ M₂
Q : Type u_4
inst✝¹ : AddCommGroup Q
inst✝ : Module A Q
⊢ IsCompl (range (lTensor Q M₁.subtype)) (range (lTensor Q M₂.subty... | R : Type u_1
inst✝⁶ : CommRing R
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R A
J : Ideal A
M : Type u_3
inst✝³ : AddCommGroup M
inst✝² : Module A M
M₁ M₂ : Submodule A M
hM : IsCompl M₁ M₂
Q : Type u_4
inst✝¹ : AddCommGroup Q
inst✝ : Module A Q
hq : M₁.subtype ∘ₗ M₁.linearProjOfIsCompl M₂ hM + M₂.subtype ∘ₗ M₂.... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝⁶ : CommRing R
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R A
J : Ideal A
M : Type u_3
inst✝³ : AddCommGroup M
inst✝² : Module A M
M₁ M₂ : Submodule A M
hM : IsCompl M₁ M₂
Q : Type u_4
inst✝¹ : AddCommGroup Q
inst✝ : Module A Q
⊢ IsCo... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/RingTheory/AugmentationIdeal.lean | LinearMap.isCompl_lTensor | [90, 1] | [119, 53] | apply IsCompl.mk | R : Type u_1
inst✝⁶ : CommRing R
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R A
J : Ideal A
M : Type u_3
inst✝³ : AddCommGroup M
inst✝² : Module A M
M₁ M₂ : Submodule A M
hM : IsCompl M₁ M₂
Q : Type u_4
inst✝¹ : AddCommGroup Q
inst✝ : Module A Q
hq : M₁.subtype ∘ₗ M₁.linearProjOfIsCompl M₂ hM + M₂.subtype ∘ₗ M₂.... | case disjoint
R : Type u_1
inst✝⁶ : CommRing R
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R A
J : Ideal A
M : Type u_3
inst✝³ : AddCommGroup M
inst✝² : Module A M
M₁ M₂ : Submodule A M
hM : IsCompl M₁ M₂
Q : Type u_4
inst✝¹ : AddCommGroup Q
inst✝ : Module A Q
hq : M₁.subtype ∘ₗ M₁.linearProjOfIsCompl M₂ hM + M₂.... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝⁶ : CommRing R
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R A
J : Ideal A
M : Type u_3
inst✝³ : AddCommGroup M
inst✝² : Module A M
M₁ M₂ : Submodule A M
hM : IsCompl M₁ M₂
Q : Type u_4
inst✝¹ : AddCommGroup Q
inst✝ : Module A Q
hq : M... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/RingTheory/AugmentationIdeal.lean | LinearMap.isCompl_lTensor | [90, 1] | [119, 53] | ext x | R : Type u_1
inst✝⁶ : CommRing R
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R A
J : Ideal A
M : Type u_3
inst✝³ : AddCommGroup M
inst✝² : Module A M
M₁ M₂ : Submodule A M
hM : IsCompl M₁ M₂
Q : Type u_4
inst✝¹ : AddCommGroup Q
inst✝ : Module A Q
⊢ M₁.subtype ∘ₗ M₁.linearProjOfIsCompl M₂ hM + M₂.subtype ∘ₗ M₂.lin... | case h
R : Type u_1
inst✝⁶ : CommRing R
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R A
J : Ideal A
M : Type u_3
inst✝³ : AddCommGroup M
inst✝² : Module A M
M₁ M₂ : Submodule A M
hM : IsCompl M₁ M₂
Q : Type u_4
inst✝¹ : AddCommGroup Q
inst✝ : Module A Q
x : M
⊢ (M₁.subtype ∘ₗ M₁.linearProjOfIsCompl M₂ hM + M₂.sub... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝⁶ : CommRing R
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R A
J : Ideal A
M : Type u_3
inst✝³ : AddCommGroup M
inst✝² : Module A M
M₁ M₂ : Submodule A M
hM : IsCompl M₁ M₂
Q : Type u_4
inst✝¹ : AddCommGroup Q
inst✝ : Module A Q
⊢ M₁.s... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/RingTheory/AugmentationIdeal.lean | LinearMap.isCompl_lTensor | [90, 1] | [119, 53] | simp only [add_apply, coe_comp, coeSubtype, Function.comp_apply,
id_coe, id_eq] | case h
R : Type u_1
inst✝⁶ : CommRing R
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R A
J : Ideal A
M : Type u_3
inst✝³ : AddCommGroup M
inst✝² : Module A M
M₁ M₂ : Submodule A M
hM : IsCompl M₁ M₂
Q : Type u_4
inst✝¹ : AddCommGroup Q
inst✝ : Module A Q
x : M
⊢ (M₁.subtype ∘ₗ M₁.linearProjOfIsCompl M₂ hM + M₂.sub... | case h
R : Type u_1
inst✝⁶ : CommRing R
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R A
J : Ideal A
M : Type u_3
inst✝³ : AddCommGroup M
inst✝² : Module A M
M₁ M₂ : Submodule A M
hM : IsCompl M₁ M₂
Q : Type u_4
inst✝¹ : AddCommGroup Q
inst✝ : Module A Q
x : M
⊢ ↑((M₁.linearProjOfIsCompl M₂ hM) x) + ↑((M₂.linearPr... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
R : Type u_1
inst✝⁶ : CommRing R
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R A
J : Ideal A
M : Type u_3
inst✝³ : AddCommGroup M
inst✝² : Module A M
M₁ M₂ : Submodule A M
hM : IsCompl M₁ M₂
Q : Type u_4
inst✝¹ : AddCommGroup Q
inst✝ : Module A Q... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/RingTheory/AugmentationIdeal.lean | LinearMap.isCompl_lTensor | [90, 1] | [119, 53] | rw [linear_proj_add_linearProjOfIsCompl_eq_self] | case h
R : Type u_1
inst✝⁶ : CommRing R
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R A
J : Ideal A
M : Type u_3
inst✝³ : AddCommGroup M
inst✝² : Module A M
M₁ M₂ : Submodule A M
hM : IsCompl M₁ M₂
Q : Type u_4
inst✝¹ : AddCommGroup Q
inst✝ : Module A Q
x : M
⊢ ↑((M₁.linearProjOfIsCompl M₂ hM) x) + ↑((M₂.linearPr... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
R : Type u_1
inst✝⁶ : CommRing R
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R A
J : Ideal A
M : Type u_3
inst✝³ : AddCommGroup M
inst✝² : Module A M
M₁ M₂ : Submodule A M
hM : IsCompl M₁ M₂
Q : Type u_4
inst✝¹ : AddCommGroup Q
inst✝ : Module A Q... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/RingTheory/AugmentationIdeal.lean | LinearMap.isCompl_lTensor | [90, 1] | [119, 53] | rw [disjoint_def] | case disjoint
R : Type u_1
inst✝⁶ : CommRing R
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R A
J : Ideal A
M : Type u_3
inst✝³ : AddCommGroup M
inst✝² : Module A M
M₁ M₂ : Submodule A M
hM : IsCompl M₁ M₂
Q : Type u_4
inst✝¹ : AddCommGroup Q
inst✝ : Module A Q
hq : M₁.subtype ∘ₗ M₁.linearProjOfIsCompl M₂ hM + M₂.... | case disjoint
R : Type u_1
inst✝⁶ : CommRing R
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R A
J : Ideal A
M : Type u_3
inst✝³ : AddCommGroup M
inst✝² : Module A M
M₁ M₂ : Submodule A M
hM : IsCompl M₁ M₂
Q : Type u_4
inst✝¹ : AddCommGroup Q
inst✝ : Module A Q
hq : M₁.subtype ∘ₗ M₁.linearProjOfIsCompl M₂ hM + M₂.... | Please generate a tactic in lean4 to solve the state.
STATE:
case disjoint
R : Type u_1
inst✝⁶ : CommRing R
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R A
J : Ideal A
M : Type u_3
inst✝³ : AddCommGroup M
inst✝² : Module A M
M₁ M₂ : Submodule A M
hM : IsCompl M₁ M₂
Q : Type u_4
inst✝¹ : AddCommGroup Q
inst✝ : Mod... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/RingTheory/AugmentationIdeal.lean | LinearMap.isCompl_lTensor | [90, 1] | [119, 53] | intro x h h' | case disjoint
R : Type u_1
inst✝⁶ : CommRing R
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R A
J : Ideal A
M : Type u_3
inst✝³ : AddCommGroup M
inst✝² : Module A M
M₁ M₂ : Submodule A M
hM : IsCompl M₁ M₂
Q : Type u_4
inst✝¹ : AddCommGroup Q
inst✝ : Module A Q
hq : M₁.subtype ∘ₗ M₁.linearProjOfIsCompl M₂ hM + M₂.... | case disjoint
R : Type u_1
inst✝⁶ : CommRing R
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R A
J : Ideal A
M : Type u_3
inst✝³ : AddCommGroup M
inst✝² : Module A M
M₁ M₂ : Submodule A M
hM : IsCompl M₁ M₂
Q : Type u_4
inst✝¹ : AddCommGroup Q
inst✝ : Module A Q
hq : M₁.subtype ∘ₗ M₁.linearProjOfIsCompl M₂ hM + M₂.... | Please generate a tactic in lean4 to solve the state.
STATE:
case disjoint
R : Type u_1
inst✝⁶ : CommRing R
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R A
J : Ideal A
M : Type u_3
inst✝³ : AddCommGroup M
inst✝² : Module A M
M₁ M₂ : Submodule A M
hM : IsCompl M₁ M₂
Q : Type u_4
inst✝¹ : AddCommGroup Q
inst✝ : Mod... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/RingTheory/AugmentationIdeal.lean | LinearMap.isCompl_lTensor | [90, 1] | [119, 53] | rw [← id_apply x (R := A), ← lTensor_id, ← hq] | case disjoint
R : Type u_1
inst✝⁶ : CommRing R
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R A
J : Ideal A
M : Type u_3
inst✝³ : AddCommGroup M
inst✝² : Module A M
M₁ M₂ : Submodule A M
hM : IsCompl M₁ M₂
Q : Type u_4
inst✝¹ : AddCommGroup Q
inst✝ : Module A Q
hq : M₁.subtype ∘ₗ M₁.linearProjOfIsCompl M₂ hM + M₂.... | case disjoint
R : Type u_1
inst✝⁶ : CommRing R
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R A
J : Ideal A
M : Type u_3
inst✝³ : AddCommGroup M
inst✝² : Module A M
M₁ M₂ : Submodule A M
hM : IsCompl M₁ M₂
Q : Type u_4
inst✝¹ : AddCommGroup Q
inst✝ : Module A Q
hq : M₁.subtype ∘ₗ M₁.linearProjOfIsCompl M₂ hM + M₂.... | Please generate a tactic in lean4 to solve the state.
STATE:
case disjoint
R : Type u_1
inst✝⁶ : CommRing R
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R A
J : Ideal A
M : Type u_3
inst✝³ : AddCommGroup M
inst✝² : Module A M
M₁ M₂ : Submodule A M
hM : IsCompl M₁ M₂
Q : Type u_4
inst✝¹ : AddCommGroup Q
inst✝ : Mod... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/RingTheory/AugmentationIdeal.lean | LinearMap.isCompl_lTensor | [90, 1] | [119, 53] | simp only [lTensor_add, lTensor_comp,
LinearMap.add_apply, LinearMap.coe_comp, Function.comp_apply] | case disjoint
R : Type u_1
inst✝⁶ : CommRing R
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R A
J : Ideal A
M : Type u_3
inst✝³ : AddCommGroup M
inst✝² : Module A M
M₁ M₂ : Submodule A M
hM : IsCompl M₁ M₂
Q : Type u_4
inst✝¹ : AddCommGroup Q
inst✝ : Module A Q
hq : M₁.subtype ∘ₗ M₁.linearProjOfIsCompl M₂ hM + M₂.... | case disjoint
R : Type u_1
inst✝⁶ : CommRing R
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R A
J : Ideal A
M : Type u_3
inst✝³ : AddCommGroup M
inst✝² : Module A M
M₁ M₂ : Submodule A M
hM : IsCompl M₁ M₂
Q : Type u_4
inst✝¹ : AddCommGroup Q
inst✝ : Module A Q
hq : M₁.subtype ∘ₗ M₁.linearProjOfIsCompl M₂ hM + M₂.... | Please generate a tactic in lean4 to solve the state.
STATE:
case disjoint
R : Type u_1
inst✝⁶ : CommRing R
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R A
J : Ideal A
M : Type u_3
inst✝³ : AddCommGroup M
inst✝² : Module A M
M₁ M₂ : Submodule A M
hM : IsCompl M₁ M₂
Q : Type u_4
inst✝¹ : AddCommGroup Q
inst✝ : Mod... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/RingTheory/AugmentationIdeal.lean | LinearMap.isCompl_lTensor | [90, 1] | [119, 53] | rw [← ker_lTensor_of_linearProjOfIsCompl hM Q] at h' | case disjoint
R : Type u_1
inst✝⁶ : CommRing R
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R A
J : Ideal A
M : Type u_3
inst✝³ : AddCommGroup M
inst✝² : Module A M
M₁ M₂ : Submodule A M
hM : IsCompl M₁ M₂
Q : Type u_4
inst✝¹ : AddCommGroup Q
inst✝ : Module A Q
hq : M₁.subtype ∘ₗ M₁.linearProjOfIsCompl M₂ hM + M₂.... | case disjoint
R : Type u_1
inst✝⁶ : CommRing R
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R A
J : Ideal A
M : Type u_3
inst✝³ : AddCommGroup M
inst✝² : Module A M
M₁ M₂ : Submodule A M
hM : IsCompl M₁ M₂
Q : Type u_4
inst✝¹ : AddCommGroup Q
inst✝ : Module A Q
hq : M₁.subtype ∘ₗ M₁.linearProjOfIsCompl M₂ hM + M₂.... | Please generate a tactic in lean4 to solve the state.
STATE:
case disjoint
R : Type u_1
inst✝⁶ : CommRing R
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R A
J : Ideal A
M : Type u_3
inst✝³ : AddCommGroup M
inst✝² : Module A M
M₁ M₂ : Submodule A M
hM : IsCompl M₁ M₂
Q : Type u_4
inst✝¹ : AddCommGroup Q
inst✝ : Mod... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/RingTheory/AugmentationIdeal.lean | LinearMap.isCompl_lTensor | [90, 1] | [119, 53] | rw [← ker_lTensor_of_linearProjOfIsCompl hM.symm Q] at h | case disjoint
R : Type u_1
inst✝⁶ : CommRing R
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R A
J : Ideal A
M : Type u_3
inst✝³ : AddCommGroup M
inst✝² : Module A M
M₁ M₂ : Submodule A M
hM : IsCompl M₁ M₂
Q : Type u_4
inst✝¹ : AddCommGroup Q
inst✝ : Module A Q
hq : M₁.subtype ∘ₗ M₁.linearProjOfIsCompl M₂ hM + M₂.... | case disjoint
R : Type u_1
inst✝⁶ : CommRing R
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R A
J : Ideal A
M : Type u_3
inst✝³ : AddCommGroup M
inst✝² : Module A M
M₁ M₂ : Submodule A M
hM : IsCompl M₁ M₂
Q : Type u_4
inst✝¹ : AddCommGroup Q
inst✝ : Module A Q
hq : M₁.subtype ∘ₗ M₁.linearProjOfIsCompl M₂ hM + M₂.... | Please generate a tactic in lean4 to solve the state.
STATE:
case disjoint
R : Type u_1
inst✝⁶ : CommRing R
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R A
J : Ideal A
M : Type u_3
inst✝³ : AddCommGroup M
inst✝² : Module A M
M₁ M₂ : Submodule A M
hM : IsCompl M₁ M₂
Q : Type u_4
inst✝¹ : AddCommGroup Q
inst✝ : Mod... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/RingTheory/AugmentationIdeal.lean | LinearMap.isCompl_lTensor | [90, 1] | [119, 53] | rw [mem_ker] at h h' | case disjoint
R : Type u_1
inst✝⁶ : CommRing R
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R A
J : Ideal A
M : Type u_3
inst✝³ : AddCommGroup M
inst✝² : Module A M
M₁ M₂ : Submodule A M
hM : IsCompl M₁ M₂
Q : Type u_4
inst✝¹ : AddCommGroup Q
inst✝ : Module A Q
hq : M₁.subtype ∘ₗ M₁.linearProjOfIsCompl M₂ hM + M₂.... | case disjoint
R : Type u_1
inst✝⁶ : CommRing R
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R A
J : Ideal A
M : Type u_3
inst✝³ : AddCommGroup M
inst✝² : Module A M
M₁ M₂ : Submodule A M
hM : IsCompl M₁ M₂
Q : Type u_4
inst✝¹ : AddCommGroup Q
inst✝ : Module A Q
hq : M₁.subtype ∘ₗ M₁.linearProjOfIsCompl M₂ hM + M₂.... | Please generate a tactic in lean4 to solve the state.
STATE:
case disjoint
R : Type u_1
inst✝⁶ : CommRing R
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R A
J : Ideal A
M : Type u_3
inst✝³ : AddCommGroup M
inst✝² : Module A M
M₁ M₂ : Submodule A M
hM : IsCompl M₁ M₂
Q : Type u_4
inst✝¹ : AddCommGroup Q
inst✝ : Mod... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/RingTheory/AugmentationIdeal.lean | LinearMap.isCompl_lTensor | [90, 1] | [119, 53] | simp only [h', _root_.map_zero, h, add_zero] | case disjoint
R : Type u_1
inst✝⁶ : CommRing R
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R A
J : Ideal A
M : Type u_3
inst✝³ : AddCommGroup M
inst✝² : Module A M
M₁ M₂ : Submodule A M
hM : IsCompl M₁ M₂
Q : Type u_4
inst✝¹ : AddCommGroup Q
inst✝ : Module A Q
hq : M₁.subtype ∘ₗ M₁.linearProjOfIsCompl M₂ hM + M₂.... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case disjoint
R : Type u_1
inst✝⁶ : CommRing R
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R A
J : Ideal A
M : Type u_3
inst✝³ : AddCommGroup M
inst✝² : Module A M
M₁ M₂ : Submodule A M
hM : IsCompl M₁ M₂
Q : Type u_4
inst✝¹ : AddCommGroup Q
inst✝ : Mod... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/RingTheory/AugmentationIdeal.lean | LinearMap.isCompl_lTensor | [90, 1] | [119, 53] | rw [codisjoint_iff] | case codisjoint
R : Type u_1
inst✝⁶ : CommRing R
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R A
J : Ideal A
M : Type u_3
inst✝³ : AddCommGroup M
inst✝² : Module A M
M₁ M₂ : Submodule A M
hM : IsCompl M₁ M₂
Q : Type u_4
inst✝¹ : AddCommGroup Q
inst✝ : Module A Q
hq : M₁.subtype ∘ₗ M₁.linearProjOfIsCompl M₂ hM + M... | case codisjoint
R : Type u_1
inst✝⁶ : CommRing R
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R A
J : Ideal A
M : Type u_3
inst✝³ : AddCommGroup M
inst✝² : Module A M
M₁ M₂ : Submodule A M
hM : IsCompl M₁ M₂
Q : Type u_4
inst✝¹ : AddCommGroup Q
inst✝ : Module A Q
hq : M₁.subtype ∘ₗ M₁.linearProjOfIsCompl M₂ hM + M... | Please generate a tactic in lean4 to solve the state.
STATE:
case codisjoint
R : Type u_1
inst✝⁶ : CommRing R
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R A
J : Ideal A
M : Type u_3
inst✝³ : AddCommGroup M
inst✝² : Module A M
M₁ M₂ : Submodule A M
hM : IsCompl M₁ M₂
Q : Type u_4
inst✝¹ : AddCommGroup Q
inst✝ : M... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/RingTheory/AugmentationIdeal.lean | LinearMap.isCompl_lTensor | [90, 1] | [119, 53] | rw [eq_top_iff] | case codisjoint
R : Type u_1
inst✝⁶ : CommRing R
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R A
J : Ideal A
M : Type u_3
inst✝³ : AddCommGroup M
inst✝² : Module A M
M₁ M₂ : Submodule A M
hM : IsCompl M₁ M₂
Q : Type u_4
inst✝¹ : AddCommGroup Q
inst✝ : Module A Q
hq : M₁.subtype ∘ₗ M₁.linearProjOfIsCompl M₂ hM + M... | case codisjoint
R : Type u_1
inst✝⁶ : CommRing R
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R A
J : Ideal A
M : Type u_3
inst✝³ : AddCommGroup M
inst✝² : Module A M
M₁ M₂ : Submodule A M
hM : IsCompl M₁ M₂
Q : Type u_4
inst✝¹ : AddCommGroup Q
inst✝ : Module A Q
hq : M₁.subtype ∘ₗ M₁.linearProjOfIsCompl M₂ hM + M... | Please generate a tactic in lean4 to solve the state.
STATE:
case codisjoint
R : Type u_1
inst✝⁶ : CommRing R
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R A
J : Ideal A
M : Type u_3
inst✝³ : AddCommGroup M
inst✝² : Module A M
M₁ M₂ : Submodule A M
hM : IsCompl M₁ M₂
Q : Type u_4
inst✝¹ : AddCommGroup Q
inst✝ : M... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/RingTheory/AugmentationIdeal.lean | LinearMap.isCompl_lTensor | [90, 1] | [119, 53] | intro x _ | case codisjoint
R : Type u_1
inst✝⁶ : CommRing R
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R A
J : Ideal A
M : Type u_3
inst✝³ : AddCommGroup M
inst✝² : Module A M
M₁ M₂ : Submodule A M
hM : IsCompl M₁ M₂
Q : Type u_4
inst✝¹ : AddCommGroup Q
inst✝ : Module A Q
hq : M₁.subtype ∘ₗ M₁.linearProjOfIsCompl M₂ hM + M... | case codisjoint
R : Type u_1
inst✝⁶ : CommRing R
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R A
J : Ideal A
M : Type u_3
inst✝³ : AddCommGroup M
inst✝² : Module A M
M₁ M₂ : Submodule A M
hM : IsCompl M₁ M₂
Q : Type u_4
inst✝¹ : AddCommGroup Q
inst✝ : Module A Q
hq : M₁.subtype ∘ₗ M₁.linearProjOfIsCompl M₂ hM + M... | Please generate a tactic in lean4 to solve the state.
STATE:
case codisjoint
R : Type u_1
inst✝⁶ : CommRing R
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R A
J : Ideal A
M : Type u_3
inst✝³ : AddCommGroup M
inst✝² : Module A M
M₁ M₂ : Submodule A M
hM : IsCompl M₁ M₂
Q : Type u_4
inst✝¹ : AddCommGroup Q
inst✝ : M... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/RingTheory/AugmentationIdeal.lean | LinearMap.isCompl_lTensor | [90, 1] | [119, 53] | rw [← lTensor_id_apply Q _ x, ← hq] | case codisjoint
R : Type u_1
inst✝⁶ : CommRing R
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R A
J : Ideal A
M : Type u_3
inst✝³ : AddCommGroup M
inst✝² : Module A M
M₁ M₂ : Submodule A M
hM : IsCompl M₁ M₂
Q : Type u_4
inst✝¹ : AddCommGroup Q
inst✝ : Module A Q
hq : M₁.subtype ∘ₗ M₁.linearProjOfIsCompl M₂ hM + M... | case codisjoint
R : Type u_1
inst✝⁶ : CommRing R
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R A
J : Ideal A
M : Type u_3
inst✝³ : AddCommGroup M
inst✝² : Module A M
M₁ M₂ : Submodule A M
hM : IsCompl M₁ M₂
Q : Type u_4
inst✝¹ : AddCommGroup Q
inst✝ : Module A Q
hq : M₁.subtype ∘ₗ M₁.linearProjOfIsCompl M₂ hM + M... | Please generate a tactic in lean4 to solve the state.
STATE:
case codisjoint
R : Type u_1
inst✝⁶ : CommRing R
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R A
J : Ideal A
M : Type u_3
inst✝³ : AddCommGroup M
inst✝² : Module A M
M₁ M₂ : Submodule A M
hM : IsCompl M₁ M₂
Q : Type u_4
inst✝¹ : AddCommGroup Q
inst✝ : M... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/RingTheory/AugmentationIdeal.lean | LinearMap.isCompl_lTensor | [90, 1] | [119, 53] | simp only [lTensor_add, lTensor_comp, add_apply, coe_comp, Function.comp_apply] | case codisjoint
R : Type u_1
inst✝⁶ : CommRing R
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R A
J : Ideal A
M : Type u_3
inst✝³ : AddCommGroup M
inst✝² : Module A M
M₁ M₂ : Submodule A M
hM : IsCompl M₁ M₂
Q : Type u_4
inst✝¹ : AddCommGroup Q
inst✝ : Module A Q
hq : M₁.subtype ∘ₗ M₁.linearProjOfIsCompl M₂ hM + M... | case codisjoint
R : Type u_1
inst✝⁶ : CommRing R
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R A
J : Ideal A
M : Type u_3
inst✝³ : AddCommGroup M
inst✝² : Module A M
M₁ M₂ : Submodule A M
hM : IsCompl M₁ M₂
Q : Type u_4
inst✝¹ : AddCommGroup Q
inst✝ : Module A Q
hq : M₁.subtype ∘ₗ M₁.linearProjOfIsCompl M₂ hM + M... | Please generate a tactic in lean4 to solve the state.
STATE:
case codisjoint
R : Type u_1
inst✝⁶ : CommRing R
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R A
J : Ideal A
M : Type u_3
inst✝³ : AddCommGroup M
inst✝² : Module A M
M₁ M₂ : Submodule A M
hM : IsCompl M₁ M₂
Q : Type u_4
inst✝¹ : AddCommGroup Q
inst✝ : M... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/RingTheory/AugmentationIdeal.lean | LinearMap.isCompl_lTensor | [90, 1] | [119, 53] | exact Submodule.add_mem _
(mem_sup_left (LinearMap.mem_range_self _ _))
(mem_sup_right (LinearMap.mem_range_self _ _)) | case codisjoint
R : Type u_1
inst✝⁶ : CommRing R
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R A
J : Ideal A
M : Type u_3
inst✝³ : AddCommGroup M
inst✝² : Module A M
M₁ M₂ : Submodule A M
hM : IsCompl M₁ M₂
Q : Type u_4
inst✝¹ : AddCommGroup Q
inst✝ : Module A Q
hq : M₁.subtype ∘ₗ M₁.linearProjOfIsCompl M₂ hM + M... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case codisjoint
R : Type u_1
inst✝⁶ : CommRing R
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R A
J : Ideal A
M : Type u_3
inst✝³ : AddCommGroup M
inst✝² : Module A M
M₁ M₂ : Submodule A M
hM : IsCompl M₁ M₂
Q : Type u_4
inst✝¹ : AddCommGroup Q
inst✝ : M... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/RingTheory/AugmentationIdeal.lean | LinearMap.isCompl_baseChange | [121, 1] | [127, 29] | rw [← isCompl_restrictScalars_iff A] | R✝ : Type u_1
inst✝⁶ : CommRing R✝
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R✝ A
J : Ideal A
M : Type u_3
inst✝³ : AddCommGroup M
inst✝² : Module A M
M₁ M₂ : Submodule A M
hM : IsCompl M₁ M₂
R : Type u_4
inst✝¹ : CommRing R
inst✝ : Algebra A R
⊢ IsCompl (range (baseChange R M₁.subtype)) (range (baseChange R M₂... | R✝ : Type u_1
inst✝⁶ : CommRing R✝
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R✝ A
J : Ideal A
M : Type u_3
inst✝³ : AddCommGroup M
inst✝² : Module A M
M₁ M₂ : Submodule A M
hM : IsCompl M₁ M₂
R : Type u_4
inst✝¹ : CommRing R
inst✝ : Algebra A R
⊢ IsCompl (Submodule.restrictScalars A (range (baseChange R M₁.subt... | Please generate a tactic in lean4 to solve the state.
STATE:
R✝ : Type u_1
inst✝⁶ : CommRing R✝
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R✝ A
J : Ideal A
M : Type u_3
inst✝³ : AddCommGroup M
inst✝² : Module A M
M₁ M₂ : Submodule A M
hM : IsCompl M₁ M₂
R : Type u_4
inst✝¹ : CommRing R
inst✝ : Algebra A R
⊢ IsCo... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/RingTheory/AugmentationIdeal.lean | LinearMap.isCompl_baseChange | [121, 1] | [127, 29] | exact isCompl_lTensor hM R | R✝ : Type u_1
inst✝⁶ : CommRing R✝
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R✝ A
J : Ideal A
M : Type u_3
inst✝³ : AddCommGroup M
inst✝² : Module A M
M₁ M₂ : Submodule A M
hM : IsCompl M₁ M₂
R : Type u_4
inst✝¹ : CommRing R
inst✝ : Algebra A R
⊢ IsCompl (Submodule.restrictScalars A (range (baseChange R M₁.subt... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R✝ : Type u_1
inst✝⁶ : CommRing R✝
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R✝ A
J : Ideal A
M : Type u_3
inst✝³ : AddCommGroup M
inst✝² : Module A M
M₁ M₂ : Submodule A M
hM : IsCompl M₁ M₂
R : Type u_4
inst✝¹ : CommRing R
inst✝ : Algebra A R
⊢ IsCo... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/RingTheory/AugmentationIdeal.lean | Algebra.baseChange_bot | [129, 1] | [157, 44] | ext x | R✝ : Type u_1
inst✝⁶ : CommRing R✝
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R✝ A
J : Ideal A
R : Type u_3
S : Type u_4
inst✝³ : CommRing R
inst✝² : Algebra A R
inst✝¹ : CommRing S
inst✝ : Algebra A S
⊢ Subalgebra.toSubmodule ⊥ = LinearMap.range (LinearMap.baseChange R (Subalgebra.toSubmodule ⊥).subtype) | case h
R✝ : Type u_1
inst✝⁶ : CommRing R✝
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R✝ A
J : Ideal A
R : Type u_3
S : Type u_4
inst✝³ : CommRing R
inst✝² : Algebra A R
inst✝¹ : CommRing S
inst✝ : Algebra A S
x : R ⊗[A] S
⊢ x ∈ Subalgebra.toSubmodule ⊥ ↔ x ∈ LinearMap.range (LinearMap.baseChange R (Subalgebra.to... | Please generate a tactic in lean4 to solve the state.
STATE:
R✝ : Type u_1
inst✝⁶ : CommRing R✝
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R✝ A
J : Ideal A
R : Type u_3
S : Type u_4
inst✝³ : CommRing R
inst✝² : Algebra A R
inst✝¹ : CommRing S
inst✝ : Algebra A S
⊢ Subalgebra.toSubmodule ⊥ = LinearMap.range (Line... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/RingTheory/AugmentationIdeal.lean | Algebra.baseChange_bot | [129, 1] | [157, 44] | simp only [Subalgebra.mem_toSubmodule, Algebra.mem_bot] | case h
R✝ : Type u_1
inst✝⁶ : CommRing R✝
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R✝ A
J : Ideal A
R : Type u_3
S : Type u_4
inst✝³ : CommRing R
inst✝² : Algebra A R
inst✝¹ : CommRing S
inst✝ : Algebra A S
x : R ⊗[A] S
⊢ x ∈ Subalgebra.toSubmodule ⊥ ↔ x ∈ LinearMap.range (LinearMap.baseChange R (Subalgebra.to... | case h
R✝ : Type u_1
inst✝⁶ : CommRing R✝
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R✝ A
J : Ideal A
R : Type u_3
S : Type u_4
inst✝³ : CommRing R
inst✝² : Algebra A R
inst✝¹ : CommRing S
inst✝ : Algebra A S
x : R ⊗[A] S
⊢ x ∈ Set.range ⇑(algebraMap R (R ⊗[A] S)) ↔
x ∈ LinearMap.range (LinearMap.baseChange ... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
R✝ : Type u_1
inst✝⁶ : CommRing R✝
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R✝ A
J : Ideal A
R : Type u_3
S : Type u_4
inst✝³ : CommRing R
inst✝² : Algebra A R
inst✝¹ : CommRing S
inst✝ : Algebra A S
x : R ⊗[A] S
⊢ x ∈ Subalgebra.toSubmodule ⊥... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/RingTheory/AugmentationIdeal.lean | Algebra.baseChange_bot | [129, 1] | [157, 44] | simp only [Set.mem_range, LinearMap.mem_range] | case h
R✝ : Type u_1
inst✝⁶ : CommRing R✝
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R✝ A
J : Ideal A
R : Type u_3
S : Type u_4
inst✝³ : CommRing R
inst✝² : Algebra A R
inst✝¹ : CommRing S
inst✝ : Algebra A S
x : R ⊗[A] S
⊢ x ∈ Set.range ⇑(algebraMap R (R ⊗[A] S)) ↔
x ∈ LinearMap.range (LinearMap.baseChange ... | case h
R✝ : Type u_1
inst✝⁶ : CommRing R✝
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R✝ A
J : Ideal A
R : Type u_3
S : Type u_4
inst✝³ : CommRing R
inst✝² : Algebra A R
inst✝¹ : CommRing S
inst✝ : Algebra A S
x : R ⊗[A] S
⊢ (∃ y, (algebraMap R (R ⊗[A] S)) y = x) ↔ ∃ y, (LinearMap.baseChange R (Subalgebra.toSubmo... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
R✝ : Type u_1
inst✝⁶ : CommRing R✝
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R✝ A
J : Ideal A
R : Type u_3
S : Type u_4
inst✝³ : CommRing R
inst✝² : Algebra A R
inst✝¹ : CommRing S
inst✝ : Algebra A S
x : R ⊗[A] S
⊢ x ∈ Set.range ⇑(algebraMap R... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/RingTheory/AugmentationIdeal.lean | Algebra.baseChange_bot | [129, 1] | [157, 44] | constructor | case h
R✝ : Type u_1
inst✝⁶ : CommRing R✝
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R✝ A
J : Ideal A
R : Type u_3
S : Type u_4
inst✝³ : CommRing R
inst✝² : Algebra A R
inst✝¹ : CommRing S
inst✝ : Algebra A S
x : R ⊗[A] S
⊢ (∃ y, (algebraMap R (R ⊗[A] S)) y = x) ↔ ∃ y, (LinearMap.baseChange R (Subalgebra.toSubmo... | case h.mp
R✝ : Type u_1
inst✝⁶ : CommRing R✝
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R✝ A
J : Ideal A
R : Type u_3
S : Type u_4
inst✝³ : CommRing R
inst✝² : Algebra A R
inst✝¹ : CommRing S
inst✝ : Algebra A S
x : R ⊗[A] S
⊢ (∃ y, (algebraMap R (R ⊗[A] S)) y = x) → ∃ y, (LinearMap.baseChange R (Subalgebra.toSu... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
R✝ : Type u_1
inst✝⁶ : CommRing R✝
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R✝ A
J : Ideal A
R : Type u_3
S : Type u_4
inst✝³ : CommRing R
inst✝² : Algebra A R
inst✝¹ : CommRing S
inst✝ : Algebra A S
x : R ⊗[A] S
⊢ (∃ y, (algebraMap R (R ⊗[A] ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/RingTheory/AugmentationIdeal.lean | Algebra.baseChange_bot | [129, 1] | [157, 44] | rintro ⟨y, rfl⟩ | case h.mp
R✝ : Type u_1
inst✝⁶ : CommRing R✝
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R✝ A
J : Ideal A
R : Type u_3
S : Type u_4
inst✝³ : CommRing R
inst✝² : Algebra A R
inst✝¹ : CommRing S
inst✝ : Algebra A S
x : R ⊗[A] S
⊢ (∃ y, (algebraMap R (R ⊗[A] S)) y = x) → ∃ y, (LinearMap.baseChange R (Subalgebra.toSu... | case h.mp.intro
R✝ : Type u_1
inst✝⁶ : CommRing R✝
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R✝ A
J : Ideal A
R : Type u_3
S : Type u_4
inst✝³ : CommRing R
inst✝² : Algebra A R
inst✝¹ : CommRing S
inst✝ : Algebra A S
y : R
⊢ ∃ y_1, (LinearMap.baseChange R (Subalgebra.toSubmodule ⊥).subtype) y_1 = (algebraMap R ... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp
R✝ : Type u_1
inst✝⁶ : CommRing R✝
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R✝ A
J : Ideal A
R : Type u_3
S : Type u_4
inst✝³ : CommRing R
inst✝² : Algebra A R
inst✝¹ : CommRing S
inst✝ : Algebra A S
x : R ⊗[A] S
⊢ (∃ y, (algebraMap R (R ⊗[... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/RingTheory/AugmentationIdeal.lean | Algebra.baseChange_bot | [129, 1] | [157, 44] | exact ⟨y ⊗ₜ[A] ⟨1, (Subalgebra.mem_toSubmodule ⊥).mpr (one_mem ⊥)⟩, rfl⟩ | case h.mp.intro
R✝ : Type u_1
inst✝⁶ : CommRing R✝
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R✝ A
J : Ideal A
R : Type u_3
S : Type u_4
inst✝³ : CommRing R
inst✝² : Algebra A R
inst✝¹ : CommRing S
inst✝ : Algebra A S
y : R
⊢ ∃ y_1, (LinearMap.baseChange R (Subalgebra.toSubmodule ⊥).subtype) y_1 = (algebraMap R ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp.intro
R✝ : Type u_1
inst✝⁶ : CommRing R✝
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R✝ A
J : Ideal A
R : Type u_3
S : Type u_4
inst✝³ : CommRing R
inst✝² : Algebra A R
inst✝¹ : CommRing S
inst✝ : Algebra A S
y : R
⊢ ∃ y_1, (LinearMap.baseChan... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/RingTheory/AugmentationIdeal.lean | Algebra.baseChange_bot | [129, 1] | [157, 44] | rintro ⟨y, rfl⟩ | case h.mpr
R✝ : Type u_1
inst✝⁶ : CommRing R✝
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R✝ A
J : Ideal A
R : Type u_3
S : Type u_4
inst✝³ : CommRing R
inst✝² : Algebra A R
inst✝¹ : CommRing S
inst✝ : Algebra A S
x : R ⊗[A] S
⊢ (∃ y, (LinearMap.baseChange R (Subalgebra.toSubmodule ⊥).subtype) y = x) → ∃ y, (alge... | case h.mpr.intro
R✝ : Type u_1
inst✝⁶ : CommRing R✝
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R✝ A
J : Ideal A
R : Type u_3
S : Type u_4
inst✝³ : CommRing R
inst✝² : Algebra A R
inst✝¹ : CommRing S
inst✝ : Algebra A S
y : R ⊗[A] ↥(Subalgebra.toSubmodule ⊥)
⊢ ∃ y_1, (algebraMap R (R ⊗[A] S)) y_1 = (LinearMap.bas... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr
R✝ : Type u_1
inst✝⁶ : CommRing R✝
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R✝ A
J : Ideal A
R : Type u_3
S : Type u_4
inst✝³ : CommRing R
inst✝² : Algebra A R
inst✝¹ : CommRing S
inst✝ : Algebra A S
x : R ⊗[A] S
⊢ (∃ y, (LinearMap.baseCha... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/RingTheory/AugmentationIdeal.lean | Algebra.baseChange_bot | [129, 1] | [157, 44] | induction y using TensorProduct.induction_on with
| zero =>
use 0
simp only [zero_tmul, LinearMap.map_zero]
simp
| tmul r s =>
rcases s with ⟨s, hs⟩
simp only [Subalgebra.mem_toSubmodule] at hs
obtain ⟨a, rfl⟩ := hs
use a • r
simp only [Algebra.TensorProduct.algebraMap_apply, Algebra.id.map_eq_id, RingH... | case h.mpr.intro
R✝ : Type u_1
inst✝⁶ : CommRing R✝
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R✝ A
J : Ideal A
R : Type u_3
S : Type u_4
inst✝³ : CommRing R
inst✝² : Algebra A R
inst✝¹ : CommRing S
inst✝ : Algebra A S
y : R ⊗[A] ↥(Subalgebra.toSubmodule ⊥)
⊢ ∃ y_1, (algebraMap R (R ⊗[A] S)) y_1 = (LinearMap.bas... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr.intro
R✝ : Type u_1
inst✝⁶ : CommRing R✝
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R✝ A
J : Ideal A
R : Type u_3
S : Type u_4
inst✝³ : CommRing R
inst✝² : Algebra A R
inst✝¹ : CommRing S
inst✝ : Algebra A S
y : R ⊗[A] ↥(Subalgebra.toSubmodu... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/RingTheory/AugmentationIdeal.lean | Algebra.baseChange_bot | [129, 1] | [157, 44] | use 0 | case h.mpr.intro.zero
R✝ : Type u_1
inst✝⁶ : CommRing R✝
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R✝ A
J : Ideal A
R : Type u_3
S : Type u_4
inst✝³ : CommRing R
inst✝² : Algebra A R
inst✝¹ : CommRing S
inst✝ : Algebra A S
⊢ ∃ y, (algebraMap R (R ⊗[A] S)) y = (LinearMap.baseChange R (Subalgebra.toSubmodule ⊥).s... | case h
R✝ : Type u_1
inst✝⁶ : CommRing R✝
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R✝ A
J : Ideal A
R : Type u_3
S : Type u_4
inst✝³ : CommRing R
inst✝² : Algebra A R
inst✝¹ : CommRing S
inst✝ : Algebra A S
⊢ (algebraMap R (R ⊗[A] S)) 0 = (LinearMap.baseChange R (Subalgebra.toSubmodule ⊥).subtype) 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr.intro.zero
R✝ : Type u_1
inst✝⁶ : CommRing R✝
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R✝ A
J : Ideal A
R : Type u_3
S : Type u_4
inst✝³ : CommRing R
inst✝² : Algebra A R
inst✝¹ : CommRing S
inst✝ : Algebra A S
⊢ ∃ y, (algebraMap R (R ⊗[A]... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/RingTheory/AugmentationIdeal.lean | Algebra.baseChange_bot | [129, 1] | [157, 44] | simp only [zero_tmul, LinearMap.map_zero] | case h
R✝ : Type u_1
inst✝⁶ : CommRing R✝
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R✝ A
J : Ideal A
R : Type u_3
S : Type u_4
inst✝³ : CommRing R
inst✝² : Algebra A R
inst✝¹ : CommRing S
inst✝ : Algebra A S
⊢ (algebraMap R (R ⊗[A] S)) 0 = (LinearMap.baseChange R (Subalgebra.toSubmodule ⊥).subtype) 0 | case h
R✝ : Type u_1
inst✝⁶ : CommRing R✝
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R✝ A
J : Ideal A
R : Type u_3
S : Type u_4
inst✝³ : CommRing R
inst✝² : Algebra A R
inst✝¹ : CommRing S
inst✝ : Algebra A S
⊢ (algebraMap R (R ⊗[A] S)) 0 = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case h
R✝ : Type u_1
inst✝⁶ : CommRing R✝
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R✝ A
J : Ideal A
R : Type u_3
S : Type u_4
inst✝³ : CommRing R
inst✝² : Algebra A R
inst✝¹ : CommRing S
inst✝ : Algebra A S
⊢ (algebraMap R (R ⊗[A] S)) 0 = (LinearMap.... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/RingTheory/AugmentationIdeal.lean | Algebra.baseChange_bot | [129, 1] | [157, 44] | simp | case h
R✝ : Type u_1
inst✝⁶ : CommRing R✝
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R✝ A
J : Ideal A
R : Type u_3
S : Type u_4
inst✝³ : CommRing R
inst✝² : Algebra A R
inst✝¹ : CommRing S
inst✝ : Algebra A S
⊢ (algebraMap R (R ⊗[A] S)) 0 = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
R✝ : Type u_1
inst✝⁶ : CommRing R✝
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R✝ A
J : Ideal A
R : Type u_3
S : Type u_4
inst✝³ : CommRing R
inst✝² : Algebra A R
inst✝¹ : CommRing S
inst✝ : Algebra A S
⊢ (algebraMap R (R ⊗[A] S)) 0 = 0
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/RingTheory/AugmentationIdeal.lean | Algebra.baseChange_bot | [129, 1] | [157, 44] | rcases s with ⟨s, hs⟩ | case h.mpr.intro.tmul
R✝ : Type u_1
inst✝⁶ : CommRing R✝
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R✝ A
J : Ideal A
R : Type u_3
S : Type u_4
inst✝³ : CommRing R
inst✝² : Algebra A R
inst✝¹ : CommRing S
inst✝ : Algebra A S
r : R
s : ↥(Subalgebra.toSubmodule ⊥)
⊢ ∃ y, (algebraMap R (R ⊗[A] S)) y = (LinearMap.bas... | case h.mpr.intro.tmul.mk
R✝ : Type u_1
inst✝⁶ : CommRing R✝
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R✝ A
J : Ideal A
R : Type u_3
S : Type u_4
inst✝³ : CommRing R
inst✝² : Algebra A R
inst✝¹ : CommRing S
inst✝ : Algebra A S
r : R
s : S
hs : s ∈ Subalgebra.toSubmodule ⊥
⊢ ∃ y, (algebraMap R (R ⊗[A] S)) y = (Li... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr.intro.tmul
R✝ : Type u_1
inst✝⁶ : CommRing R✝
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R✝ A
J : Ideal A
R : Type u_3
S : Type u_4
inst✝³ : CommRing R
inst✝² : Algebra A R
inst✝¹ : CommRing S
inst✝ : Algebra A S
r : R
s : ↥(Subalgebra.toSub... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/RingTheory/AugmentationIdeal.lean | Algebra.baseChange_bot | [129, 1] | [157, 44] | simp only [Subalgebra.mem_toSubmodule] at hs | case h.mpr.intro.tmul.mk
R✝ : Type u_1
inst✝⁶ : CommRing R✝
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R✝ A
J : Ideal A
R : Type u_3
S : Type u_4
inst✝³ : CommRing R
inst✝² : Algebra A R
inst✝¹ : CommRing S
inst✝ : Algebra A S
r : R
s : S
hs : s ∈ Subalgebra.toSubmodule ⊥
⊢ ∃ y, (algebraMap R (R ⊗[A] S)) y = (Li... | case h.mpr.intro.tmul.mk
R✝ : Type u_1
inst✝⁶ : CommRing R✝
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R✝ A
J : Ideal A
R : Type u_3
S : Type u_4
inst✝³ : CommRing R
inst✝² : Algebra A R
inst✝¹ : CommRing S
inst✝ : Algebra A S
r : R
s : S
hs✝ : s ∈ Subalgebra.toSubmodule ⊥
hs : s ∈ ⊥
⊢ ∃ y, (algebraMap R (R ⊗[A]... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr.intro.tmul.mk
R✝ : Type u_1
inst✝⁶ : CommRing R✝
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R✝ A
J : Ideal A
R : Type u_3
S : Type u_4
inst✝³ : CommRing R
inst✝² : Algebra A R
inst✝¹ : CommRing S
inst✝ : Algebra A S
r : R
s : S
hs : s ∈ Suba... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/RingTheory/AugmentationIdeal.lean | Algebra.baseChange_bot | [129, 1] | [157, 44] | obtain ⟨a, rfl⟩ := hs | case h.mpr.intro.tmul.mk
R✝ : Type u_1
inst✝⁶ : CommRing R✝
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R✝ A
J : Ideal A
R : Type u_3
S : Type u_4
inst✝³ : CommRing R
inst✝² : Algebra A R
inst✝¹ : CommRing S
inst✝ : Algebra A S
r : R
s : S
hs✝ : s ∈ Subalgebra.toSubmodule ⊥
hs : s ∈ ⊥
⊢ ∃ y, (algebraMap R (R ⊗[A]... | case h.mpr.intro.tmul.mk.intro
R✝ : Type u_1
inst✝⁶ : CommRing R✝
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R✝ A
J : Ideal A
R : Type u_3
S : Type u_4
inst✝³ : CommRing R
inst✝² : Algebra A R
inst✝¹ : CommRing S
inst✝ : Algebra A S
r : R
a : A
hs : (ofId A S).toRingHom a ∈ Subalgebra.toSubmodule ⊥
⊢ ∃ y,
(a... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr.intro.tmul.mk
R✝ : Type u_1
inst✝⁶ : CommRing R✝
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R✝ A
J : Ideal A
R : Type u_3
S : Type u_4
inst✝³ : CommRing R
inst✝² : Algebra A R
inst✝¹ : CommRing S
inst✝ : Algebra A S
r : R
s : S
hs✝ : s ∈ Sub... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/RingTheory/AugmentationIdeal.lean | Algebra.baseChange_bot | [129, 1] | [157, 44] | use a • r | case h.mpr.intro.tmul.mk.intro
R✝ : Type u_1
inst✝⁶ : CommRing R✝
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R✝ A
J : Ideal A
R : Type u_3
S : Type u_4
inst✝³ : CommRing R
inst✝² : Algebra A R
inst✝¹ : CommRing S
inst✝ : Algebra A S
r : R
a : A
hs : (ofId A S).toRingHom a ∈ Subalgebra.toSubmodule ⊥
⊢ ∃ y,
(a... | case h
R✝ : Type u_1
inst✝⁶ : CommRing R✝
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R✝ A
J : Ideal A
R : Type u_3
S : Type u_4
inst✝³ : CommRing R
inst✝² : Algebra A R
inst✝¹ : CommRing S
inst✝ : Algebra A S
r : R
a : A
hs : (ofId A S).toRingHom a ∈ Subalgebra.toSubmodule ⊥
⊢ (algebraMap R (R ⊗[A] S)) (a • r) =... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr.intro.tmul.mk.intro
R✝ : Type u_1
inst✝⁶ : CommRing R✝
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R✝ A
J : Ideal A
R : Type u_3
S : Type u_4
inst✝³ : CommRing R
inst✝² : Algebra A R
inst✝¹ : CommRing S
inst✝ : Algebra A S
r : R
a : A
hs : (o... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/RingTheory/AugmentationIdeal.lean | Algebra.baseChange_bot | [129, 1] | [157, 44] | simp only [Algebra.TensorProduct.algebraMap_apply, Algebra.id.map_eq_id, RingHom.id_apply,
toRingHom_eq_coe, coe_coe, baseChange_tmul, coeSubtype] | case h
R✝ : Type u_1
inst✝⁶ : CommRing R✝
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R✝ A
J : Ideal A
R : Type u_3
S : Type u_4
inst✝³ : CommRing R
inst✝² : Algebra A R
inst✝¹ : CommRing S
inst✝ : Algebra A S
r : R
a : A
hs : (ofId A S).toRingHom a ∈ Subalgebra.toSubmodule ⊥
⊢ (algebraMap R (R ⊗[A] S)) (a • r) =... | case h
R✝ : Type u_1
inst✝⁶ : CommRing R✝
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R✝ A
J : Ideal A
R : Type u_3
S : Type u_4
inst✝³ : CommRing R
inst✝² : Algebra A R
inst✝¹ : CommRing S
inst✝ : Algebra A S
r : R
a : A
hs : (ofId A S).toRingHom a ∈ Subalgebra.toSubmodule ⊥
⊢ (a • r) ⊗ₜ[A] 1 = r ⊗ₜ[A] (ofId A S... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
R✝ : Type u_1
inst✝⁶ : CommRing R✝
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R✝ A
J : Ideal A
R : Type u_3
S : Type u_4
inst✝³ : CommRing R
inst✝² : Algebra A R
inst✝¹ : CommRing S
inst✝ : Algebra A S
r : R
a : A
hs : (ofId A S).toRingHom a ∈ S... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/RingTheory/AugmentationIdeal.lean | Algebra.baseChange_bot | [129, 1] | [157, 44] | simp only [smul_tmul] | case h
R✝ : Type u_1
inst✝⁶ : CommRing R✝
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R✝ A
J : Ideal A
R : Type u_3
S : Type u_4
inst✝³ : CommRing R
inst✝² : Algebra A R
inst✝¹ : CommRing S
inst✝ : Algebra A S
r : R
a : A
hs : (ofId A S).toRingHom a ∈ Subalgebra.toSubmodule ⊥
⊢ (a • r) ⊗ₜ[A] 1 = r ⊗ₜ[A] (ofId A S... | case h
R✝ : Type u_1
inst✝⁶ : CommRing R✝
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R✝ A
J : Ideal A
R : Type u_3
S : Type u_4
inst✝³ : CommRing R
inst✝² : Algebra A R
inst✝¹ : CommRing S
inst✝ : Algebra A S
r : R
a : A
hs : (ofId A S).toRingHom a ∈ Subalgebra.toSubmodule ⊥
⊢ r ⊗ₜ[A] (a • 1) = r ⊗ₜ[A] (ofId A S... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
R✝ : Type u_1
inst✝⁶ : CommRing R✝
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R✝ A
J : Ideal A
R : Type u_3
S : Type u_4
inst✝³ : CommRing R
inst✝² : Algebra A R
inst✝¹ : CommRing S
inst✝ : Algebra A S
r : R
a : A
hs : (ofId A S).toRingHom a ∈ S... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/RingTheory/AugmentationIdeal.lean | Algebra.baseChange_bot | [129, 1] | [157, 44] | rw [Algebra.ofId_apply, Algebra.algebraMap_eq_smul_one] | case h
R✝ : Type u_1
inst✝⁶ : CommRing R✝
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R✝ A
J : Ideal A
R : Type u_3
S : Type u_4
inst✝³ : CommRing R
inst✝² : Algebra A R
inst✝¹ : CommRing S
inst✝ : Algebra A S
r : R
a : A
hs : (ofId A S).toRingHom a ∈ Subalgebra.toSubmodule ⊥
⊢ r ⊗ₜ[A] (a • 1) = r ⊗ₜ[A] (ofId A S... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
R✝ : Type u_1
inst✝⁶ : CommRing R✝
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R✝ A
J : Ideal A
R : Type u_3
S : Type u_4
inst✝³ : CommRing R
inst✝² : Algebra A R
inst✝¹ : CommRing S
inst✝ : Algebra A S
r : R
a : A
hs : (ofId A S).toRingHom a ∈ S... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/RingTheory/AugmentationIdeal.lean | Algebra.baseChange_bot | [129, 1] | [157, 44] | obtain ⟨x', hx⟩ := hx | case h.mpr.intro.add
R✝ : Type u_1
inst✝⁶ : CommRing R✝
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R✝ A
J : Ideal A
R : Type u_3
S : Type u_4
inst✝³ : CommRing R
inst✝² : Algebra A R
inst✝¹ : CommRing S
inst✝ : Algebra A S
x y : R ⊗[A] ↥(Subalgebra.toSubmodule ⊥)
hx : ∃ y, (algebraMap R (R ⊗[A] S)) y = (LinearMa... | case h.mpr.intro.add.intro
R✝ : Type u_1
inst✝⁶ : CommRing R✝
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R✝ A
J : Ideal A
R : Type u_3
S : Type u_4
inst✝³ : CommRing R
inst✝² : Algebra A R
inst✝¹ : CommRing S
inst✝ : Algebra A S
x y : R ⊗[A] ↥(Subalgebra.toSubmodule ⊥)
hy : ∃ y_1, (algebraMap R (R ⊗[A] S)) y_1 =... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr.intro.add
R✝ : Type u_1
inst✝⁶ : CommRing R✝
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R✝ A
J : Ideal A
R : Type u_3
S : Type u_4
inst✝³ : CommRing R
inst✝² : Algebra A R
inst✝¹ : CommRing S
inst✝ : Algebra A S
x y : R ⊗[A] ↥(Subalgebra.toS... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/RingTheory/AugmentationIdeal.lean | Algebra.baseChange_bot | [129, 1] | [157, 44] | obtain ⟨y', hy⟩ := hy | case h.mpr.intro.add.intro
R✝ : Type u_1
inst✝⁶ : CommRing R✝
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R✝ A
J : Ideal A
R : Type u_3
S : Type u_4
inst✝³ : CommRing R
inst✝² : Algebra A R
inst✝¹ : CommRing S
inst✝ : Algebra A S
x y : R ⊗[A] ↥(Subalgebra.toSubmodule ⊥)
hy : ∃ y_1, (algebraMap R (R ⊗[A] S)) y_1 =... | case h.mpr.intro.add.intro.intro
R✝ : Type u_1
inst✝⁶ : CommRing R✝
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R✝ A
J : Ideal A
R : Type u_3
S : Type u_4
inst✝³ : CommRing R
inst✝² : Algebra A R
inst✝¹ : CommRing S
inst✝ : Algebra A S
x y : R ⊗[A] ↥(Subalgebra.toSubmodule ⊥)
x' : R
hx : (algebraMap R (R ⊗[A] S))... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr.intro.add.intro
R✝ : Type u_1
inst✝⁶ : CommRing R✝
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R✝ A
J : Ideal A
R : Type u_3
S : Type u_4
inst✝³ : CommRing R
inst✝² : Algebra A R
inst✝¹ : CommRing S
inst✝ : Algebra A S
x y : R ⊗[A] ↥(Subalgeb... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/RingTheory/AugmentationIdeal.lean | Algebra.baseChange_bot | [129, 1] | [157, 44] | use x' + y' | case h.mpr.intro.add.intro.intro
R✝ : Type u_1
inst✝⁶ : CommRing R✝
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R✝ A
J : Ideal A
R : Type u_3
S : Type u_4
inst✝³ : CommRing R
inst✝² : Algebra A R
inst✝¹ : CommRing S
inst✝ : Algebra A S
x y : R ⊗[A] ↥(Subalgebra.toSubmodule ⊥)
x' : R
hx : (algebraMap R (R ⊗[A] S))... | case h
R✝ : Type u_1
inst✝⁶ : CommRing R✝
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R✝ A
J : Ideal A
R : Type u_3
S : Type u_4
inst✝³ : CommRing R
inst✝² : Algebra A R
inst✝¹ : CommRing S
inst✝ : Algebra A S
x y : R ⊗[A] ↥(Subalgebra.toSubmodule ⊥)
x' : R
hx : (algebraMap R (R ⊗[A] S)) x' = (LinearMap.baseChang... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr.intro.add.intro.intro
R✝ : Type u_1
inst✝⁶ : CommRing R✝
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R✝ A
J : Ideal A
R : Type u_3
S : Type u_4
inst✝³ : CommRing R
inst✝² : Algebra A R
inst✝¹ : CommRing S
inst✝ : Algebra A S
x y : R ⊗[A] ↥(Su... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/RingTheory/AugmentationIdeal.lean | Algebra.baseChange_bot | [129, 1] | [157, 44] | simp only [add_tmul, hx, hy, map_add] | case h
R✝ : Type u_1
inst✝⁶ : CommRing R✝
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R✝ A
J : Ideal A
R : Type u_3
S : Type u_4
inst✝³ : CommRing R
inst✝² : Algebra A R
inst✝¹ : CommRing S
inst✝ : Algebra A S
x y : R ⊗[A] ↥(Subalgebra.toSubmodule ⊥)
x' : R
hx : (algebraMap R (R ⊗[A] S)) x' = (LinearMap.baseChang... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
R✝ : Type u_1
inst✝⁶ : CommRing R✝
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R✝ A
J : Ideal A
R : Type u_3
S : Type u_4
inst✝³ : CommRing R
inst✝² : Algebra A R
inst✝¹ : CommRing S
inst✝ : Algebra A S
x y : R ⊗[A] ↥(Subalgebra.toSubmodule ⊥)
x'... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/RingTheory/AugmentationIdeal.lean | Algebra.TensorProduct.map_includeRight_eq_range_baseChange | [159, 1] | [209, 34] | ext x | R✝ : Type u_1
inst✝⁶ : CommRing R✝
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R✝ A
J : Ideal A
S : Type u_3
inst✝³ : CommRing S
inst✝² : Algebra A S
I : Ideal S
R : Type u_4
inst✝¹ : CommRing R
inst✝ : Algebra A R
⊢ Submodule.restrictScalars R (Ideal.map includeRight I) =
LinearMap.range (LinearMap.baseChang... | case h
R✝ : Type u_1
inst✝⁶ : CommRing R✝
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R✝ A
J : Ideal A
S : Type u_3
inst✝³ : CommRing S
inst✝² : Algebra A S
I : Ideal S
R : Type u_4
inst✝¹ : CommRing R
inst✝ : Algebra A R
x : R ⊗[A] S
⊢ x ∈ Submodule.restrictScalars R (Ideal.map includeRight I) ↔
x ∈ LinearMa... | Please generate a tactic in lean4 to solve the state.
STATE:
R✝ : Type u_1
inst✝⁶ : CommRing R✝
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R✝ A
J : Ideal A
S : Type u_3
inst✝³ : CommRing S
inst✝² : Algebra A S
I : Ideal S
R : Type u_4
inst✝¹ : CommRing R
inst✝ : Algebra A R
⊢ Submodule.restrictScalars R (Ideal.m... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/RingTheory/AugmentationIdeal.lean | Algebra.TensorProduct.map_includeRight_eq_range_baseChange | [159, 1] | [209, 34] | simp only [restrictScalars_mem, LinearMap.mem_range] | case h
R✝ : Type u_1
inst✝⁶ : CommRing R✝
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R✝ A
J : Ideal A
S : Type u_3
inst✝³ : CommRing S
inst✝² : Algebra A S
I : Ideal S
R : Type u_4
inst✝¹ : CommRing R
inst✝ : Algebra A R
x : R ⊗[A] S
⊢ x ∈ Submodule.restrictScalars R (Ideal.map includeRight I) ↔
x ∈ LinearMa... | case h
R✝ : Type u_1
inst✝⁶ : CommRing R✝
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R✝ A
J : Ideal A
S : Type u_3
inst✝³ : CommRing S
inst✝² : Algebra A S
I : Ideal S
R : Type u_4
inst✝¹ : CommRing R
inst✝ : Algebra A R
x : R ⊗[A] S
⊢ x ∈ Ideal.map includeRight I ↔ ∃ y, (LinearMap.baseChange R (Submodule.restri... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
R✝ : Type u_1
inst✝⁶ : CommRing R✝
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R✝ A
J : Ideal A
S : Type u_3
inst✝³ : CommRing S
inst✝² : Algebra A S
I : Ideal S
R : Type u_4
inst✝¹ : CommRing R
inst✝ : Algebra A R
x : R ⊗[A] S
⊢ x ∈ Submodule.re... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/RingTheory/AugmentationIdeal.lean | Algebra.TensorProduct.map_includeRight_eq_range_baseChange | [159, 1] | [209, 34] | constructor | case h
R✝ : Type u_1
inst✝⁶ : CommRing R✝
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R✝ A
J : Ideal A
S : Type u_3
inst✝³ : CommRing S
inst✝² : Algebra A S
I : Ideal S
R : Type u_4
inst✝¹ : CommRing R
inst✝ : Algebra A R
x : R ⊗[A] S
⊢ x ∈ Ideal.map includeRight I ↔ ∃ y, (LinearMap.baseChange R (Submodule.restri... | case h.mp
R✝ : Type u_1
inst✝⁶ : CommRing R✝
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R✝ A
J : Ideal A
S : Type u_3
inst✝³ : CommRing S
inst✝² : Algebra A S
I : Ideal S
R : Type u_4
inst✝¹ : CommRing R
inst✝ : Algebra A R
x : R ⊗[A] S
⊢ x ∈ Ideal.map includeRight I → ∃ y, (LinearMap.baseChange R (Submodule.res... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
R✝ : Type u_1
inst✝⁶ : CommRing R✝
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R✝ A
J : Ideal A
S : Type u_3
inst✝³ : CommRing S
inst✝² : Algebra A S
I : Ideal S
R : Type u_4
inst✝¹ : CommRing R
inst✝ : Algebra A R
x : R ⊗[A] S
⊢ x ∈ Ideal.map in... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/RingTheory/AugmentationIdeal.lean | Algebra.TensorProduct.map_includeRight_eq_range_baseChange | [159, 1] | [209, 34] | intro hx | case h.mp
R✝ : Type u_1
inst✝⁶ : CommRing R✝
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R✝ A
J : Ideal A
S : Type u_3
inst✝³ : CommRing S
inst✝² : Algebra A S
I : Ideal S
R : Type u_4
inst✝¹ : CommRing R
inst✝ : Algebra A R
x : R ⊗[A] S
⊢ x ∈ Ideal.map includeRight I → ∃ y, (LinearMap.baseChange R (Submodule.res... | case h.mp
R✝ : Type u_1
inst✝⁶ : CommRing R✝
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R✝ A
J : Ideal A
S : Type u_3
inst✝³ : CommRing S
inst✝² : Algebra A S
I : Ideal S
R : Type u_4
inst✝¹ : CommRing R
inst✝ : Algebra A R
x : R ⊗[A] S
hx : x ∈ Ideal.map includeRight I
⊢ ∃ y, (LinearMap.baseChange R (Submodule.... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp
R✝ : Type u_1
inst✝⁶ : CommRing R✝
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R✝ A
J : Ideal A
S : Type u_3
inst✝³ : CommRing S
inst✝² : Algebra A S
I : Ideal S
R : Type u_4
inst✝¹ : CommRing R
inst✝ : Algebra A R
x : R ⊗[A] S
⊢ x ∈ Ideal.map... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/RingTheory/AugmentationIdeal.lean | Algebra.TensorProduct.map_includeRight_eq_range_baseChange | [159, 1] | [209, 34] | apply Submodule.span_induction hx
(p := fun x ↦ ∃ y, (LinearMap.baseChange R (Submodule.restrictScalars A I).subtype) y = x ) | case h.mp
R✝ : Type u_1
inst✝⁶ : CommRing R✝
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R✝ A
J : Ideal A
S : Type u_3
inst✝³ : CommRing S
inst✝² : Algebra A S
I : Ideal S
R : Type u_4
inst✝¹ : CommRing R
inst✝ : Algebra A R
x : R ⊗[A] S
hx : x ∈ Ideal.map includeRight I
⊢ ∃ y, (LinearMap.baseChange R (Submodule.... | case h.mp.mem
R✝ : Type u_1
inst✝⁶ : CommRing R✝
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R✝ A
J : Ideal A
S : Type u_3
inst✝³ : CommRing S
inst✝² : Algebra A S
I : Ideal S
R : Type u_4
inst✝¹ : CommRing R
inst✝ : Algebra A R
x : R ⊗[A] S
hx : x ∈ Ideal.map includeRight I
⊢ ∀ x ∈ ⇑includeRight '' ↑I, ∃ y, (Lin... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp
R✝ : Type u_1
inst✝⁶ : CommRing R✝
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R✝ A
J : Ideal A
S : Type u_3
inst✝³ : CommRing S
inst✝² : Algebra A S
I : Ideal S
R : Type u_4
inst✝¹ : CommRing R
inst✝ : Algebra A R
x : R ⊗[A] S
hx : x ∈ Ideal.... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/RingTheory/AugmentationIdeal.lean | Algebra.TensorProduct.map_includeRight_eq_range_baseChange | [159, 1] | [209, 34] | rintro x ⟨s, hs, rfl⟩ | case h.mp.mem
R✝ : Type u_1
inst✝⁶ : CommRing R✝
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R✝ A
J : Ideal A
S : Type u_3
inst✝³ : CommRing S
inst✝² : Algebra A S
I : Ideal S
R : Type u_4
inst✝¹ : CommRing R
inst✝ : Algebra A R
x : R ⊗[A] S
hx : x ∈ Ideal.map includeRight I
⊢ ∀ x ∈ ⇑includeRight '' ↑I, ∃ y, (Lin... | case h.mp.mem.intro.intro
R✝ : Type u_1
inst✝⁶ : CommRing R✝
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R✝ A
J : Ideal A
S : Type u_3
inst✝³ : CommRing S
inst✝² : Algebra A S
I : Ideal S
R : Type u_4
inst✝¹ : CommRing R
inst✝ : Algebra A R
x : R ⊗[A] S
hx : x ∈ Ideal.map includeRight I
s : S
hs : s ∈ ↑I
⊢ ∃ y, (... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp.mem
R✝ : Type u_1
inst✝⁶ : CommRing R✝
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R✝ A
J : Ideal A
S : Type u_3
inst✝³ : CommRing S
inst✝² : Algebra A S
I : Ideal S
R : Type u_4
inst✝¹ : CommRing R
inst✝ : Algebra A R
x : R ⊗[A] S
hx : x ∈ Id... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/RingTheory/AugmentationIdeal.lean | Algebra.TensorProduct.map_includeRight_eq_range_baseChange | [159, 1] | [209, 34] | use 1 ⊗ₜ[A] ⟨s, hs⟩ | case h.mp.mem.intro.intro
R✝ : Type u_1
inst✝⁶ : CommRing R✝
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R✝ A
J : Ideal A
S : Type u_3
inst✝³ : CommRing S
inst✝² : Algebra A S
I : Ideal S
R : Type u_4
inst✝¹ : CommRing R
inst✝ : Algebra A R
x : R ⊗[A] S
hx : x ∈ Ideal.map includeRight I
s : S
hs : s ∈ ↑I
⊢ ∃ y, (... | case h
R✝ : Type u_1
inst✝⁶ : CommRing R✝
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R✝ A
J : Ideal A
S : Type u_3
inst✝³ : CommRing S
inst✝² : Algebra A S
I : Ideal S
R : Type u_4
inst✝¹ : CommRing R
inst✝ : Algebra A R
x : R ⊗[A] S
hx : x ∈ Ideal.map includeRight I
s : S
hs : s ∈ ↑I
⊢ (LinearMap.baseChange R (... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp.mem.intro.intro
R✝ : Type u_1
inst✝⁶ : CommRing R✝
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R✝ A
J : Ideal A
S : Type u_3
inst✝³ : CommRing S
inst✝² : Algebra A S
I : Ideal S
R : Type u_4
inst✝¹ : CommRing R
inst✝ : Algebra A R
x : R ⊗[A] S... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/RingTheory/AugmentationIdeal.lean | Algebra.TensorProduct.map_includeRight_eq_range_baseChange | [159, 1] | [209, 34] | rfl | case h
R✝ : Type u_1
inst✝⁶ : CommRing R✝
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R✝ A
J : Ideal A
S : Type u_3
inst✝³ : CommRing S
inst✝² : Algebra A S
I : Ideal S
R : Type u_4
inst✝¹ : CommRing R
inst✝ : Algebra A R
x : R ⊗[A] S
hx : x ∈ Ideal.map includeRight I
s : S
hs : s ∈ ↑I
⊢ (LinearMap.baseChange R (... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
R✝ : Type u_1
inst✝⁶ : CommRing R✝
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R✝ A
J : Ideal A
S : Type u_3
inst✝³ : CommRing S
inst✝² : Algebra A S
I : Ideal S
R : Type u_4
inst✝¹ : CommRing R
inst✝ : Algebra A R
x : R ⊗[A] S
hx : x ∈ Ideal.map... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/RingTheory/AugmentationIdeal.lean | Algebra.TensorProduct.map_includeRight_eq_range_baseChange | [159, 1] | [209, 34] | use 0 | case h.mp.zero
R✝ : Type u_1
inst✝⁶ : CommRing R✝
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R✝ A
J : Ideal A
S : Type u_3
inst✝³ : CommRing S
inst✝² : Algebra A S
I : Ideal S
R : Type u_4
inst✝¹ : CommRing R
inst✝ : Algebra A R
x : R ⊗[A] S
hx : x ∈ Ideal.map includeRight I
⊢ ∃ y, (LinearMap.baseChange R (Submo... | case h
R✝ : Type u_1
inst✝⁶ : CommRing R✝
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R✝ A
J : Ideal A
S : Type u_3
inst✝³ : CommRing S
inst✝² : Algebra A S
I : Ideal S
R : Type u_4
inst✝¹ : CommRing R
inst✝ : Algebra A R
x : R ⊗[A] S
hx : x ∈ Ideal.map includeRight I
⊢ (LinearMap.baseChange R (Submodule.restrict... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp.zero
R✝ : Type u_1
inst✝⁶ : CommRing R✝
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R✝ A
J : Ideal A
S : Type u_3
inst✝³ : CommRing S
inst✝² : Algebra A S
I : Ideal S
R : Type u_4
inst✝¹ : CommRing R
inst✝ : Algebra A R
x : R ⊗[A] S
hx : x ∈ I... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/RingTheory/AugmentationIdeal.lean | Algebra.TensorProduct.map_includeRight_eq_range_baseChange | [159, 1] | [209, 34] | simp only [_root_.map_zero] | case h
R✝ : Type u_1
inst✝⁶ : CommRing R✝
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R✝ A
J : Ideal A
S : Type u_3
inst✝³ : CommRing S
inst✝² : Algebra A S
I : Ideal S
R : Type u_4
inst✝¹ : CommRing R
inst✝ : Algebra A R
x : R ⊗[A] S
hx : x ∈ Ideal.map includeRight I
⊢ (LinearMap.baseChange R (Submodule.restrict... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
R✝ : Type u_1
inst✝⁶ : CommRing R✝
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R✝ A
J : Ideal A
S : Type u_3
inst✝³ : CommRing S
inst✝² : Algebra A S
I : Ideal S
R : Type u_4
inst✝¹ : CommRing R
inst✝ : Algebra A R
x : R ⊗[A] S
hx : x ∈ Ideal.map... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/RingTheory/AugmentationIdeal.lean | Algebra.TensorProduct.map_includeRight_eq_range_baseChange | [159, 1] | [209, 34] | rintro _ _ ⟨x, rfl⟩ ⟨y, rfl⟩ | case h.mp.add
R✝ : Type u_1
inst✝⁶ : CommRing R✝
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R✝ A
J : Ideal A
S : Type u_3
inst✝³ : CommRing S
inst✝² : Algebra A S
I : Ideal S
R : Type u_4
inst✝¹ : CommRing R
inst✝ : Algebra A R
x : R ⊗[A] S
hx : x ∈ Ideal.map includeRight I
⊢ ∀ (x y : R ⊗[A] S),
(∃ y, (Linea... | case h.mp.add.intro.intro
R✝ : Type u_1
inst✝⁶ : CommRing R✝
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R✝ A
J : Ideal A
S : Type u_3
inst✝³ : CommRing S
inst✝² : Algebra A S
I : Ideal S
R : Type u_4
inst✝¹ : CommRing R
inst✝ : Algebra A R
x✝ : R ⊗[A] S
hx : x✝ ∈ Ideal.map includeRight I
x y : R ⊗[A] ↥(Submodule... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp.add
R✝ : Type u_1
inst✝⁶ : CommRing R✝
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R✝ A
J : Ideal A
S : Type u_3
inst✝³ : CommRing S
inst✝² : Algebra A S
I : Ideal S
R : Type u_4
inst✝¹ : CommRing R
inst✝ : Algebra A R
x : R ⊗[A] S
hx : x ∈ Id... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/RingTheory/AugmentationIdeal.lean | Algebra.TensorProduct.map_includeRight_eq_range_baseChange | [159, 1] | [209, 34] | use x + y | case h.mp.add.intro.intro
R✝ : Type u_1
inst✝⁶ : CommRing R✝
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R✝ A
J : Ideal A
S : Type u_3
inst✝³ : CommRing S
inst✝² : Algebra A S
I : Ideal S
R : Type u_4
inst✝¹ : CommRing R
inst✝ : Algebra A R
x✝ : R ⊗[A] S
hx : x✝ ∈ Ideal.map includeRight I
x y : R ⊗[A] ↥(Submodule... | case h
R✝ : Type u_1
inst✝⁶ : CommRing R✝
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R✝ A
J : Ideal A
S : Type u_3
inst✝³ : CommRing S
inst✝² : Algebra A S
I : Ideal S
R : Type u_4
inst✝¹ : CommRing R
inst✝ : Algebra A R
x✝ : R ⊗[A] S
hx : x✝ ∈ Ideal.map includeRight I
x y : R ⊗[A] ↥(Submodule.restrictScalars A ... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp.add.intro.intro
R✝ : Type u_1
inst✝⁶ : CommRing R✝
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R✝ A
J : Ideal A
S : Type u_3
inst✝³ : CommRing S
inst✝² : Algebra A S
I : Ideal S
R : Type u_4
inst✝¹ : CommRing R
inst✝ : Algebra A R
x✝ : R ⊗[A] ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/RingTheory/AugmentationIdeal.lean | Algebra.TensorProduct.map_includeRight_eq_range_baseChange | [159, 1] | [209, 34] | simp only [map_add] | case h
R✝ : Type u_1
inst✝⁶ : CommRing R✝
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R✝ A
J : Ideal A
S : Type u_3
inst✝³ : CommRing S
inst✝² : Algebra A S
I : Ideal S
R : Type u_4
inst✝¹ : CommRing R
inst✝ : Algebra A R
x✝ : R ⊗[A] S
hx : x✝ ∈ Ideal.map includeRight I
x y : R ⊗[A] ↥(Submodule.restrictScalars A ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
R✝ : Type u_1
inst✝⁶ : CommRing R✝
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R✝ A
J : Ideal A
S : Type u_3
inst✝³ : CommRing S
inst✝² : Algebra A S
I : Ideal S
R : Type u_4
inst✝¹ : CommRing R
inst✝ : Algebra A R
x✝ : R ⊗[A] S
hx : x✝ ∈ Ideal.m... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/RingTheory/AugmentationIdeal.lean | Algebra.TensorProduct.map_includeRight_eq_range_baseChange | [159, 1] | [209, 34] | rintro a _ ⟨x, rfl⟩ | case h.mp.smul
R✝ : Type u_1
inst✝⁶ : CommRing R✝
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R✝ A
J : Ideal A
S : Type u_3
inst✝³ : CommRing S
inst✝² : Algebra A S
I : Ideal S
R : Type u_4
inst✝¹ : CommRing R
inst✝ : Algebra A R
x : R ⊗[A] S
hx : x ∈ Ideal.map includeRight I
⊢ ∀ (a x : R ⊗[A] S),
(∃ y, (Line... | case h.mp.smul.intro
R✝ : Type u_1
inst✝⁶ : CommRing R✝
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R✝ A
J : Ideal A
S : Type u_3
inst✝³ : CommRing S
inst✝² : Algebra A S
I : Ideal S
R : Type u_4
inst✝¹ : CommRing R
inst✝ : Algebra A R
x✝ : R ⊗[A] S
hx : x✝ ∈ Ideal.map includeRight I
a : R ⊗[A] S
x : R ⊗[A] ↥(Sub... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp.smul
R✝ : Type u_1
inst✝⁶ : CommRing R✝
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R✝ A
J : Ideal A
S : Type u_3
inst✝³ : CommRing S
inst✝² : Algebra A S
I : Ideal S
R : Type u_4
inst✝¹ : CommRing R
inst✝ : Algebra A R
x : R ⊗[A] S
hx : x ∈ I... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/RingTheory/AugmentationIdeal.lean | Algebra.TensorProduct.map_includeRight_eq_range_baseChange | [159, 1] | [209, 34] | induction x using TensorProduct.induction_on with
| zero => use 0; simp only [_root_.map_zero, smul_eq_mul, mul_zero]
| tmul r s =>
induction a using TensorProduct.induction_on with
| zero =>
use 0
simp only [_root_.map_zero, baseChange_tmul, coeSubtype, smul_eq_mul, zero_mul]
| tmul u v =>
use (u * r... | case h.mp.smul.intro
R✝ : Type u_1
inst✝⁶ : CommRing R✝
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R✝ A
J : Ideal A
S : Type u_3
inst✝³ : CommRing S
inst✝² : Algebra A S
I : Ideal S
R : Type u_4
inst✝¹ : CommRing R
inst✝ : Algebra A R
x✝ : R ⊗[A] S
hx : x✝ ∈ Ideal.map includeRight I
a : R ⊗[A] S
x : R ⊗[A] ↥(Sub... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp.smul.intro
R✝ : Type u_1
inst✝⁶ : CommRing R✝
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R✝ A
J : Ideal A
S : Type u_3
inst✝³ : CommRing S
inst✝² : Algebra A S
I : Ideal S
R : Type u_4
inst✝¹ : CommRing R
inst✝ : Algebra A R
x✝ : R ⊗[A] S
hx ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/RingTheory/AugmentationIdeal.lean | Algebra.TensorProduct.map_includeRight_eq_range_baseChange | [159, 1] | [209, 34] | use 0 | case h.mp.smul.intro.zero
R✝ : Type u_1
inst✝⁶ : CommRing R✝
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R✝ A
J : Ideal A
S : Type u_3
inst✝³ : CommRing S
inst✝² : Algebra A S
I : Ideal S
R : Type u_4
inst✝¹ : CommRing R
inst✝ : Algebra A R
x : R ⊗[A] S
hx : x ∈ Ideal.map includeRight I
a : R ⊗[A] S
⊢ ∃ y,
(L... | case h
R✝ : Type u_1
inst✝⁶ : CommRing R✝
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R✝ A
J : Ideal A
S : Type u_3
inst✝³ : CommRing S
inst✝² : Algebra A S
I : Ideal S
R : Type u_4
inst✝¹ : CommRing R
inst✝ : Algebra A R
x : R ⊗[A] S
hx : x ∈ Ideal.map includeRight I
a : R ⊗[A] S
⊢ (LinearMap.baseChange R (Submo... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp.smul.intro.zero
R✝ : Type u_1
inst✝⁶ : CommRing R✝
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R✝ A
J : Ideal A
S : Type u_3
inst✝³ : CommRing S
inst✝² : Algebra A S
I : Ideal S
R : Type u_4
inst✝¹ : CommRing R
inst✝ : Algebra A R
x : R ⊗[A] S... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/RingTheory/AugmentationIdeal.lean | Algebra.TensorProduct.map_includeRight_eq_range_baseChange | [159, 1] | [209, 34] | simp only [_root_.map_zero, smul_eq_mul, mul_zero] | case h
R✝ : Type u_1
inst✝⁶ : CommRing R✝
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R✝ A
J : Ideal A
S : Type u_3
inst✝³ : CommRing S
inst✝² : Algebra A S
I : Ideal S
R : Type u_4
inst✝¹ : CommRing R
inst✝ : Algebra A R
x : R ⊗[A] S
hx : x ∈ Ideal.map includeRight I
a : R ⊗[A] S
⊢ (LinearMap.baseChange R (Submo... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
R✝ : Type u_1
inst✝⁶ : CommRing R✝
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R✝ A
J : Ideal A
S : Type u_3
inst✝³ : CommRing S
inst✝² : Algebra A S
I : Ideal S
R : Type u_4
inst✝¹ : CommRing R
inst✝ : Algebra A R
x : R ⊗[A] S
hx : x ∈ Ideal.map... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/RingTheory/AugmentationIdeal.lean | Algebra.TensorProduct.map_includeRight_eq_range_baseChange | [159, 1] | [209, 34] | induction a using TensorProduct.induction_on with
| zero =>
use 0
simp only [_root_.map_zero, baseChange_tmul, coeSubtype, smul_eq_mul, zero_mul]
| tmul u v =>
use (u * r) ⊗ₜ[A] (v • s)
simp only [baseChange_tmul, coeSubtype, smul_eq_mul,
Algebra.TensorProduct.tmul_mul_tmul]
rw [Submodule.coe_smul, smul_e... | case h.mp.smul.intro.tmul
R✝ : Type u_1
inst✝⁶ : CommRing R✝
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R✝ A
J : Ideal A
S : Type u_3
inst✝³ : CommRing S
inst✝² : Algebra A S
I : Ideal S
R : Type u_4
inst✝¹ : CommRing R
inst✝ : Algebra A R
x : R ⊗[A] S
hx : x ∈ Ideal.map includeRight I
a : R ⊗[A] S
r : R
s : ↥(S... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp.smul.intro.tmul
R✝ : Type u_1
inst✝⁶ : CommRing R✝
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R✝ A
J : Ideal A
S : Type u_3
inst✝³ : CommRing S
inst✝² : Algebra A S
I : Ideal S
R : Type u_4
inst✝¹ : CommRing R
inst✝ : Algebra A R
x : R ⊗[A] S... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/RingTheory/AugmentationIdeal.lean | Algebra.TensorProduct.map_includeRight_eq_range_baseChange | [159, 1] | [209, 34] | use 0 | case h.mp.smul.intro.tmul.zero
R✝ : Type u_1
inst✝⁶ : CommRing R✝
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R✝ A
J : Ideal A
S : Type u_3
inst✝³ : CommRing S
inst✝² : Algebra A S
I : Ideal S
R : Type u_4
inst✝¹ : CommRing R
inst✝ : Algebra A R
x : R ⊗[A] S
hx : x ∈ Ideal.map includeRight I
r : R
s : ↥(Submodule... | case h
R✝ : Type u_1
inst✝⁶ : CommRing R✝
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R✝ A
J : Ideal A
S : Type u_3
inst✝³ : CommRing S
inst✝² : Algebra A S
I : Ideal S
R : Type u_4
inst✝¹ : CommRing R
inst✝ : Algebra A R
x : R ⊗[A] S
hx : x ∈ Ideal.map includeRight I
r : R
s : ↥(Submodule.restrictScalars A I)
⊢ ... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp.smul.intro.tmul.zero
R✝ : Type u_1
inst✝⁶ : CommRing R✝
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R✝ A
J : Ideal A
S : Type u_3
inst✝³ : CommRing S
inst✝² : Algebra A S
I : Ideal S
R : Type u_4
inst✝¹ : CommRing R
inst✝ : Algebra A R
x : R ⊗... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/RingTheory/AugmentationIdeal.lean | Algebra.TensorProduct.map_includeRight_eq_range_baseChange | [159, 1] | [209, 34] | simp only [_root_.map_zero, baseChange_tmul, coeSubtype, smul_eq_mul, zero_mul] | case h
R✝ : Type u_1
inst✝⁶ : CommRing R✝
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R✝ A
J : Ideal A
S : Type u_3
inst✝³ : CommRing S
inst✝² : Algebra A S
I : Ideal S
R : Type u_4
inst✝¹ : CommRing R
inst✝ : Algebra A R
x : R ⊗[A] S
hx : x ∈ Ideal.map includeRight I
r : R
s : ↥(Submodule.restrictScalars A I)
⊢ ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
R✝ : Type u_1
inst✝⁶ : CommRing R✝
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R✝ A
J : Ideal A
S : Type u_3
inst✝³ : CommRing S
inst✝² : Algebra A S
I : Ideal S
R : Type u_4
inst✝¹ : CommRing R
inst✝ : Algebra A R
x : R ⊗[A] S
hx : x ∈ Ideal.map... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/RingTheory/AugmentationIdeal.lean | Algebra.TensorProduct.map_includeRight_eq_range_baseChange | [159, 1] | [209, 34] | use (u * r) ⊗ₜ[A] (v • s) | case h.mp.smul.intro.tmul.tmul
R✝ : Type u_1
inst✝⁶ : CommRing R✝
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R✝ A
J : Ideal A
S : Type u_3
inst✝³ : CommRing S
inst✝² : Algebra A S
I : Ideal S
R : Type u_4
inst✝¹ : CommRing R
inst✝ : Algebra A R
x : R ⊗[A] S
hx : x ∈ Ideal.map includeRight I
r : R
s : ↥(Submodule... | case h
R✝ : Type u_1
inst✝⁶ : CommRing R✝
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R✝ A
J : Ideal A
S : Type u_3
inst✝³ : CommRing S
inst✝² : Algebra A S
I : Ideal S
R : Type u_4
inst✝¹ : CommRing R
inst✝ : Algebra A R
x : R ⊗[A] S
hx : x ∈ Ideal.map includeRight I
r : R
s : ↥(Submodule.restrictScalars A I)
u ... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp.smul.intro.tmul.tmul
R✝ : Type u_1
inst✝⁶ : CommRing R✝
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R✝ A
J : Ideal A
S : Type u_3
inst✝³ : CommRing S
inst✝² : Algebra A S
I : Ideal S
R : Type u_4
inst✝¹ : CommRing R
inst✝ : Algebra A R
x : R ⊗... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/RingTheory/AugmentationIdeal.lean | Algebra.TensorProduct.map_includeRight_eq_range_baseChange | [159, 1] | [209, 34] | simp only [baseChange_tmul, coeSubtype, smul_eq_mul,
Algebra.TensorProduct.tmul_mul_tmul] | case h
R✝ : Type u_1
inst✝⁶ : CommRing R✝
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R✝ A
J : Ideal A
S : Type u_3
inst✝³ : CommRing S
inst✝² : Algebra A S
I : Ideal S
R : Type u_4
inst✝¹ : CommRing R
inst✝ : Algebra A R
x : R ⊗[A] S
hx : x ∈ Ideal.map includeRight I
r : R
s : ↥(Submodule.restrictScalars A I)
u ... | case h
R✝ : Type u_1
inst✝⁶ : CommRing R✝
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R✝ A
J : Ideal A
S : Type u_3
inst✝³ : CommRing S
inst✝² : Algebra A S
I : Ideal S
R : Type u_4
inst✝¹ : CommRing R
inst✝ : Algebra A R
x : R ⊗[A] S
hx : x ∈ Ideal.map includeRight I
r : R
s : ↥(Submodule.restrictScalars A I)
u ... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
R✝ : Type u_1
inst✝⁶ : CommRing R✝
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R✝ A
J : Ideal A
S : Type u_3
inst✝³ : CommRing S
inst✝² : Algebra A S
I : Ideal S
R : Type u_4
inst✝¹ : CommRing R
inst✝ : Algebra A R
x : R ⊗[A] S
hx : x ∈ Ideal.map... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/RingTheory/AugmentationIdeal.lean | Algebra.TensorProduct.map_includeRight_eq_range_baseChange | [159, 1] | [209, 34] | rw [Submodule.coe_smul, smul_eq_mul] | case h
R✝ : Type u_1
inst✝⁶ : CommRing R✝
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R✝ A
J : Ideal A
S : Type u_3
inst✝³ : CommRing S
inst✝² : Algebra A S
I : Ideal S
R : Type u_4
inst✝¹ : CommRing R
inst✝ : Algebra A R
x : R ⊗[A] S
hx : x ∈ Ideal.map includeRight I
r : R
s : ↥(Submodule.restrictScalars A I)
u ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
R✝ : Type u_1
inst✝⁶ : CommRing R✝
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R✝ A
J : Ideal A
S : Type u_3
inst✝³ : CommRing S
inst✝² : Algebra A S
I : Ideal S
R : Type u_4
inst✝¹ : CommRing R
inst✝ : Algebra A R
x : R ⊗[A] S
hx : x ∈ Ideal.map... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/RingTheory/AugmentationIdeal.lean | Algebra.TensorProduct.map_includeRight_eq_range_baseChange | [159, 1] | [209, 34] | obtain ⟨x, hx⟩ := hu | case h.mp.smul.intro.tmul.add
R✝ : Type u_1
inst✝⁶ : CommRing R✝
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R✝ A
J : Ideal A
S : Type u_3
inst✝³ : CommRing S
inst✝² : Algebra A S
I : Ideal S
R : Type u_4
inst✝¹ : CommRing R
inst✝ : Algebra A R
x : R ⊗[A] S
hx : x ∈ Ideal.map includeRight I
r : R
s : ↥(Submodule.... | case h.mp.smul.intro.tmul.add.intro
R✝ : Type u_1
inst✝⁶ : CommRing R✝
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R✝ A
J : Ideal A
S : Type u_3
inst✝³ : CommRing S
inst✝² : Algebra A S
I : Ideal S
R : Type u_4
inst✝¹ : CommRing R
inst✝ : Algebra A R
x✝ : R ⊗[A] S
hx✝ : x✝ ∈ Ideal.map includeRight I
r : R
s : ↥(S... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp.smul.intro.tmul.add
R✝ : Type u_1
inst✝⁶ : CommRing R✝
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R✝ A
J : Ideal A
S : Type u_3
inst✝³ : CommRing S
inst✝² : Algebra A S
I : Ideal S
R : Type u_4
inst✝¹ : CommRing R
inst✝ : Algebra A R
x : R ⊗[... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/RingTheory/AugmentationIdeal.lean | Algebra.TensorProduct.map_includeRight_eq_range_baseChange | [159, 1] | [209, 34] | obtain ⟨y, hy⟩ := hv | case h.mp.smul.intro.tmul.add.intro
R✝ : Type u_1
inst✝⁶ : CommRing R✝
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R✝ A
J : Ideal A
S : Type u_3
inst✝³ : CommRing S
inst✝² : Algebra A S
I : Ideal S
R : Type u_4
inst✝¹ : CommRing R
inst✝ : Algebra A R
x✝ : R ⊗[A] S
hx✝ : x✝ ∈ Ideal.map includeRight I
r : R
s : ↥(S... | case h.mp.smul.intro.tmul.add.intro.intro
R✝ : Type u_1
inst✝⁶ : CommRing R✝
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R✝ A
J : Ideal A
S : Type u_3
inst✝³ : CommRing S
inst✝² : Algebra A S
I : Ideal S
R : Type u_4
inst✝¹ : CommRing R
inst✝ : Algebra A R
x✝ : R ⊗[A] S
hx✝ : x✝ ∈ Ideal.map includeRight I
r : R
s... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp.smul.intro.tmul.add.intro
R✝ : Type u_1
inst✝⁶ : CommRing R✝
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R✝ A
J : Ideal A
S : Type u_3
inst✝³ : CommRing S
inst✝² : Algebra A S
I : Ideal S
R : Type u_4
inst✝¹ : CommRing R
inst✝ : Algebra A R
x✝... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/RingTheory/AugmentationIdeal.lean | Algebra.TensorProduct.map_includeRight_eq_range_baseChange | [159, 1] | [209, 34] | use x + y | case h.mp.smul.intro.tmul.add.intro.intro
R✝ : Type u_1
inst✝⁶ : CommRing R✝
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R✝ A
J : Ideal A
S : Type u_3
inst✝³ : CommRing S
inst✝² : Algebra A S
I : Ideal S
R : Type u_4
inst✝¹ : CommRing R
inst✝ : Algebra A R
x✝ : R ⊗[A] S
hx✝ : x✝ ∈ Ideal.map includeRight I
r : R
s... | case h
R✝ : Type u_1
inst✝⁶ : CommRing R✝
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R✝ A
J : Ideal A
S : Type u_3
inst✝³ : CommRing S
inst✝² : Algebra A S
I : Ideal S
R : Type u_4
inst✝¹ : CommRing R
inst✝ : Algebra A R
x✝ : R ⊗[A] S
hx✝ : x✝ ∈ Ideal.map includeRight I
r : R
s : ↥(Submodule.restrictScalars A I)... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp.smul.intro.tmul.add.intro.intro
R✝ : Type u_1
inst✝⁶ : CommRing R✝
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R✝ A
J : Ideal A
S : Type u_3
inst✝³ : CommRing S
inst✝² : Algebra A S
I : Ideal S
R : Type u_4
inst✝¹ : CommRing R
inst✝ : Algebra ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/RingTheory/AugmentationIdeal.lean | Algebra.TensorProduct.map_includeRight_eq_range_baseChange | [159, 1] | [209, 34] | rw [LinearMap.map_add, add_smul, hx, hy] | case h
R✝ : Type u_1
inst✝⁶ : CommRing R✝
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R✝ A
J : Ideal A
S : Type u_3
inst✝³ : CommRing S
inst✝² : Algebra A S
I : Ideal S
R : Type u_4
inst✝¹ : CommRing R
inst✝ : Algebra A R
x✝ : R ⊗[A] S
hx✝ : x✝ ∈ Ideal.map includeRight I
r : R
s : ↥(Submodule.restrictScalars A I)... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
R✝ : Type u_1
inst✝⁶ : CommRing R✝
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R✝ A
J : Ideal A
S : Type u_3
inst✝³ : CommRing S
inst✝² : Algebra A S
I : Ideal S
R : Type u_4
inst✝¹ : CommRing R
inst✝ : Algebra A R
x✝ : R ⊗[A] S
hx✝ : x✝ ∈ Ideal.... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/RingTheory/AugmentationIdeal.lean | Algebra.TensorProduct.map_includeRight_eq_range_baseChange | [159, 1] | [209, 34] | obtain ⟨x', hx⟩ := hx | case h.mp.smul.intro.add
R✝ : Type u_1
inst✝⁶ : CommRing R✝
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R✝ A
J : Ideal A
S : Type u_3
inst✝³ : CommRing S
inst✝² : Algebra A S
I : Ideal S
R : Type u_4
inst✝¹ : CommRing R
inst✝ : Algebra A R
x✝ : R ⊗[A] S
hx✝ : x✝ ∈ Ideal.map includeRight I
a : R ⊗[A] S
x y : R ⊗[A... | case h.mp.smul.intro.add.intro
R✝ : Type u_1
inst✝⁶ : CommRing R✝
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R✝ A
J : Ideal A
S : Type u_3
inst✝³ : CommRing S
inst✝² : Algebra A S
I : Ideal S
R : Type u_4
inst✝¹ : CommRing R
inst✝ : Algebra A R
x✝ : R ⊗[A] S
hx✝ : x✝ ∈ Ideal.map includeRight I
a : R ⊗[A] S
x y :... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp.smul.intro.add
R✝ : Type u_1
inst✝⁶ : CommRing R✝
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R✝ A
J : Ideal A
S : Type u_3
inst✝³ : CommRing S
inst✝² : Algebra A S
I : Ideal S
R : Type u_4
inst✝¹ : CommRing R
inst✝ : Algebra A R
x✝ : R ⊗[A] S... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/RingTheory/AugmentationIdeal.lean | Algebra.TensorProduct.map_includeRight_eq_range_baseChange | [159, 1] | [209, 34] | obtain ⟨y', hy⟩ := hy | case h.mp.smul.intro.add.intro
R✝ : Type u_1
inst✝⁶ : CommRing R✝
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R✝ A
J : Ideal A
S : Type u_3
inst✝³ : CommRing S
inst✝² : Algebra A S
I : Ideal S
R : Type u_4
inst✝¹ : CommRing R
inst✝ : Algebra A R
x✝ : R ⊗[A] S
hx✝ : x✝ ∈ Ideal.map includeRight I
a : R ⊗[A] S
x y :... | case h.mp.smul.intro.add.intro.intro
R✝ : Type u_1
inst✝⁶ : CommRing R✝
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R✝ A
J : Ideal A
S : Type u_3
inst✝³ : CommRing S
inst✝² : Algebra A S
I : Ideal S
R : Type u_4
inst✝¹ : CommRing R
inst✝ : Algebra A R
x✝ : R ⊗[A] S
hx✝ : x✝ ∈ Ideal.map includeRight I
a : R ⊗[A] S... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp.smul.intro.add.intro
R✝ : Type u_1
inst✝⁶ : CommRing R✝
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R✝ A
J : Ideal A
S : Type u_3
inst✝³ : CommRing S
inst✝² : Algebra A S
I : Ideal S
R : Type u_4
inst✝¹ : CommRing R
inst✝ : Algebra A R
x✝ : R ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/RingTheory/AugmentationIdeal.lean | Algebra.TensorProduct.map_includeRight_eq_range_baseChange | [159, 1] | [209, 34] | use x' + y' | case h.mp.smul.intro.add.intro.intro
R✝ : Type u_1
inst✝⁶ : CommRing R✝
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R✝ A
J : Ideal A
S : Type u_3
inst✝³ : CommRing S
inst✝² : Algebra A S
I : Ideal S
R : Type u_4
inst✝¹ : CommRing R
inst✝ : Algebra A R
x✝ : R ⊗[A] S
hx✝ : x✝ ∈ Ideal.map includeRight I
a : R ⊗[A] S... | case h
R✝ : Type u_1
inst✝⁶ : CommRing R✝
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R✝ A
J : Ideal A
S : Type u_3
inst✝³ : CommRing S
inst✝² : Algebra A S
I : Ideal S
R : Type u_4
inst✝¹ : CommRing R
inst✝ : Algebra A R
x✝ : R ⊗[A] S
hx✝ : x✝ ∈ Ideal.map includeRight I
a : R ⊗[A] S
x y x' : R ⊗[A] ↥(Submodule.r... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp.smul.intro.add.intro.intro
R✝ : Type u_1
inst✝⁶ : CommRing R✝
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R✝ A
J : Ideal A
S : Type u_3
inst✝³ : CommRing S
inst✝² : Algebra A S
I : Ideal S
R : Type u_4
inst✝¹ : CommRing R
inst✝ : Algebra A R
x... |
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