url stringclasses 147
values | commit stringclasses 147
values | file_path stringlengths 7 101 | full_name stringlengths 1 94 | start stringlengths 6 10 | end stringlengths 6 11 | tactic stringlengths 1 11.2k | state_before stringlengths 3 2.09M | state_after stringlengths 6 2.09M | input stringlengths 73 2.09M |
|---|---|---|---|---|---|---|---|---|---|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Init.lean | DividedPowerAlgebra.LinearMap.dp_smul | [509, 1] | [513, 34] | rw [f.map_smul, algebra_compatible_smul S r (f a)] | R : Type u_1
M : Type u_2
inst✝⁸ : CommSemiring R
inst✝⁷ : AddCommMonoid M
inst✝⁶ : Module R M
S : Type u_3
inst✝⁵ : CommSemiring S
inst✝⁴ : Algebra R S
N : Type u_4
inst✝³ : AddCommMonoid N
inst✝² : Module R N
inst✝¹ : Module S N
inst✝ : IsScalarTower R S N
f : M →ₗ[R] N
r : R
n : ℕ
a : M
⊢ dp S n (f (r • a)) = r ^ n ... | R : Type u_1
M : Type u_2
inst✝⁸ : CommSemiring R
inst✝⁷ : AddCommMonoid M
inst✝⁶ : Module R M
S : Type u_3
inst✝⁵ : CommSemiring S
inst✝⁴ : Algebra R S
N : Type u_4
inst✝³ : AddCommMonoid N
inst✝² : Module R N
inst✝¹ : Module S N
inst✝ : IsScalarTower R S N
f : M →ₗ[R] N
r : R
n : ℕ
a : M
⊢ dp S n ((algebraMap R S) r ... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
M : Type u_2
inst✝⁸ : CommSemiring R
inst✝⁷ : AddCommMonoid M
inst✝⁶ : Module R M
S : Type u_3
inst✝⁵ : CommSemiring S
inst✝⁴ : Algebra R S
N : Type u_4
inst✝³ : AddCommMonoid N
inst✝² : Module R N
inst✝¹ : Module S N
inst✝ : IsScalarTower R S N
... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Init.lean | DividedPowerAlgebra.LinearMap.dp_smul | [509, 1] | [513, 34] | rw [DividedPowerAlgebra.dp_smul S ((algebraMap R S) r) n (f a)] | R : Type u_1
M : Type u_2
inst✝⁸ : CommSemiring R
inst✝⁷ : AddCommMonoid M
inst✝⁶ : Module R M
S : Type u_3
inst✝⁵ : CommSemiring S
inst✝⁴ : Algebra R S
N : Type u_4
inst✝³ : AddCommMonoid N
inst✝² : Module R N
inst✝¹ : Module S N
inst✝ : IsScalarTower R S N
f : M →ₗ[R] N
r : R
n : ℕ
a : M
⊢ dp S n ((algebraMap R S) r ... | R : Type u_1
M : Type u_2
inst✝⁸ : CommSemiring R
inst✝⁷ : AddCommMonoid M
inst✝⁶ : Module R M
S : Type u_3
inst✝⁵ : CommSemiring S
inst✝⁴ : Algebra R S
N : Type u_4
inst✝³ : AddCommMonoid N
inst✝² : Module R N
inst✝¹ : Module S N
inst✝ : IsScalarTower R S N
f : M →ₗ[R] N
r : R
n : ℕ
a : M
⊢ (algebraMap R S) r ^ n • dp... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
M : Type u_2
inst✝⁸ : CommSemiring R
inst✝⁷ : AddCommMonoid M
inst✝⁶ : Module R M
S : Type u_3
inst✝⁵ : CommSemiring S
inst✝⁴ : Algebra R S
N : Type u_4
inst✝³ : AddCommMonoid N
inst✝² : Module R N
inst✝¹ : Module S N
inst✝ : IsScalarTower R S N
... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Init.lean | DividedPowerAlgebra.LinearMap.dp_smul | [509, 1] | [513, 34] | rw [← map_pow, algebraMap_smul] | R : Type u_1
M : Type u_2
inst✝⁸ : CommSemiring R
inst✝⁷ : AddCommMonoid M
inst✝⁶ : Module R M
S : Type u_3
inst✝⁵ : CommSemiring S
inst✝⁴ : Algebra R S
N : Type u_4
inst✝³ : AddCommMonoid N
inst✝² : Module R N
inst✝¹ : Module S N
inst✝ : IsScalarTower R S N
f : M →ₗ[R] N
r : R
n : ℕ
a : M
⊢ (algebraMap R S) r ^ n • dp... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
M : Type u_2
inst✝⁸ : CommSemiring R
inst✝⁷ : AddCommMonoid M
inst✝⁶ : Module R M
S : Type u_3
inst✝⁵ : CommSemiring S
inst✝⁴ : Algebra R S
N : Type u_4
inst✝³ : AddCommMonoid N
inst✝² : Module R N
inst✝¹ : Module S N
inst✝ : IsScalarTower R S N
... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Init.lean | DividedPowerAlgebra.LinearMap.dp_add | [519, 1] | [522, 43] | rw [map_add, DividedPowerAlgebra.dp_add] | R : Type u_1
M : Type u_2
inst✝⁸ : CommSemiring R
inst✝⁷ : AddCommMonoid M
inst✝⁶ : Module R M
S : Type u_3
inst✝⁵ : CommSemiring S
inst✝⁴ : Algebra R S
N : Type u_4
inst✝³ : AddCommMonoid N
inst✝² : Module R N
inst✝¹ : Module S N
inst✝ : IsScalarTower R S N
f : M →ₗ[R] N
n : ℕ
a b : M
⊢ dp S n (f (a + b)) = ∑ k ∈ anti... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
M : Type u_2
inst✝⁸ : CommSemiring R
inst✝⁷ : AddCommMonoid M
inst✝⁶ : Module R M
S : Type u_3
inst✝⁵ : CommSemiring S
inst✝⁴ : Algebra R S
N : Type u_4
inst✝³ : AddCommMonoid N
inst✝² : Module R N
inst✝¹ : Module S N
inst✝ : IsScalarTower R S N
... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Init.lean | DividedPowerAlgebra.LinearMap.liftAlgHom_apply | [538, 1] | [541, 41] | rw [LinearMap.lift, lift'AlgHom_apply] | R : Type u_1
M : Type u_2
inst✝⁸ : CommSemiring R
inst✝⁷ : AddCommMonoid M
inst✝⁶ : Module R M
S : Type u_3
inst✝⁵ : CommSemiring S
inst✝⁴ : Algebra R S
N : Type u_4
inst✝³ : AddCommMonoid N
inst✝² : Module R N
inst✝¹ : Module S N
inst✝ : IsScalarTower R S N
f : M →ₗ[R] N
p : MvPolynomial (ℕ × M) R
⊢ (lift R S f) (mk p... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
M : Type u_2
inst✝⁸ : CommSemiring R
inst✝⁷ : AddCommMonoid M
inst✝⁶ : Module R M
S : Type u_3
inst✝⁵ : CommSemiring S
inst✝⁴ : Algebra R S
N : Type u_4
inst✝³ : AddCommMonoid N
inst✝² : Module R N
inst✝¹ : Module S N
inst✝ : IsScalarTower R S N
... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Init.lean | DividedPowerAlgebra.LinearMap.liftAlgHom_dp | [543, 1] | [545, 44] | rw [LinearMap.lift, lift'AlgHom_apply_dp] | R : Type u_1
M : Type u_2
inst✝⁸ : CommSemiring R
inst✝⁷ : AddCommMonoid M
inst✝⁶ : Module R M
S : Type u_3
inst✝⁵ : CommSemiring S
inst✝⁴ : Algebra R S
N : Type u_4
inst✝³ : AddCommMonoid N
inst✝² : Module R N
inst✝¹ : Module S N
inst✝ : IsScalarTower R S N
f : M →ₗ[R] N
n : ℕ
a : M
⊢ (lift R S f) (dp R n a) = dp S n ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
M : Type u_2
inst✝⁸ : CommSemiring R
inst✝⁷ : AddCommMonoid M
inst✝⁶ : Module R M
S : Type u_3
inst✝⁵ : CommSemiring S
inst✝⁴ : Algebra R S
N : Type u_4
inst✝³ : AddCommMonoid N
inst✝² : Module R N
inst✝¹ : Module S N
inst✝ : IsScalarTower R S N
... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | DirectSum.map_of | [102, 1] | [106, 6] | simp only [map, toAddMonoid_of, AddMonoidHom.coe_comp, AddMonoidHom.coe_coe] | ι : Type u_1
inst✝⁵ : DecidableEq ι
R : Type u_2
inst✝⁴ : Semiring R
β : ι → Type u_3
γ : ι → Type u_4
inst✝³ : (i : ι) → AddCommMonoid (β i)
inst✝² : (i : ι) → AddCommMonoid (γ i)
F : ι → Type u_5
inst✝¹ : (i : ι) → FunLike (F i) (β i) (γ i)
inst✝ : ∀ (i : ι), AddMonoidHomClass (F i) (β i) (γ i)
h : (i : ι) → F i
i : ... | ι : Type u_1
inst✝⁵ : DecidableEq ι
R : Type u_2
inst✝⁴ : Semiring R
β : ι → Type u_3
γ : ι → Type u_4
inst✝³ : (i : ι) → AddCommMonoid (β i)
inst✝² : (i : ι) → AddCommMonoid (γ i)
F : ι → Type u_5
inst✝¹ : (i : ι) → FunLike (F i) (β i) (γ i)
inst✝ : ∀ (i : ι), AddMonoidHomClass (F i) (β i) (γ i)
h : (i : ι) → F i
i : ... | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
inst✝⁵ : DecidableEq ι
R : Type u_2
inst✝⁴ : Semiring R
β : ι → Type u_3
γ : ι → Type u_4
inst✝³ : (i : ι) → AddCommMonoid (β i)
inst✝² : (i : ι) → AddCommMonoid (γ i)
F : ι → Type u_5
inst✝¹ : (i : ι) → FunLike (F i) (β i) (γ i)
inst✝ : ∀ (i : ι... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | DirectSum.map_of | [102, 1] | [106, 6] | rfl | ι : Type u_1
inst✝⁵ : DecidableEq ι
R : Type u_2
inst✝⁴ : Semiring R
β : ι → Type u_3
γ : ι → Type u_4
inst✝³ : (i : ι) → AddCommMonoid (β i)
inst✝² : (i : ι) → AddCommMonoid (γ i)
F : ι → Type u_5
inst✝¹ : (i : ι) → FunLike (F i) (β i) (γ i)
inst✝ : ∀ (i : ι), AddMonoidHomClass (F i) (β i) (γ i)
h : (i : ι) → F i
i : ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
inst✝⁵ : DecidableEq ι
R : Type u_2
inst✝⁴ : Semiring R
β : ι → Type u_3
γ : ι → Type u_4
inst✝³ : (i : ι) → AddCommMonoid (β i)
inst✝² : (i : ι) → AddCommMonoid (γ i)
F : ι → Type u_5
inst✝¹ : (i : ι) → FunLike (F i) (β i) (γ i)
inst✝ : ∀ (i : ι... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | DirectSum.map_apply | [108, 1] | [127, 49] | let f : DirectSum ι β →+ γ i :=
{ toFun := fun x => map h x i
map_zero' := by simp only [map_zero, zero_apply]
map_add' := by simp only [map_add, add_apply, forall_const] } | ι : Type u_1
inst✝⁵ : DecidableEq ι
R : Type u_2
inst✝⁴ : Semiring R
β : ι → Type u_3
γ : ι → Type u_4
inst✝³ : (i : ι) → AddCommMonoid (β i)
inst✝² : (i : ι) → AddCommMonoid (γ i)
F : ι → Type u_5
inst✝¹ : (i : ι) → FunLike (F i) (β i) (γ i)
inst✝ : ∀ (i : ι), AddMonoidHomClass (F i) (β i) (γ i)
h : (i : ι) → F i
x : ... | ι : Type u_1
inst✝⁵ : DecidableEq ι
R : Type u_2
inst✝⁴ : Semiring R
β : ι → Type u_3
γ : ι → Type u_4
inst✝³ : (i : ι) → AddCommMonoid (β i)
inst✝² : (i : ι) → AddCommMonoid (γ i)
F : ι → Type u_5
inst✝¹ : (i : ι) → FunLike (F i) (β i) (γ i)
inst✝ : ∀ (i : ι), AddMonoidHomClass (F i) (β i) (γ i)
h : (i : ι) → F i
x : ... | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
inst✝⁵ : DecidableEq ι
R : Type u_2
inst✝⁴ : Semiring R
β : ι → Type u_3
γ : ι → Type u_4
inst✝³ : (i : ι) → AddCommMonoid (β i)
inst✝² : (i : ι) → AddCommMonoid (γ i)
F : ι → Type u_5
inst✝¹ : (i : ι) → FunLike (F i) (β i) (γ i)
inst✝ : ∀ (i : ι... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | DirectSum.map_apply | [108, 1] | [127, 49] | let g : DirectSum ι β →+ γ i :=
{ toFun := fun y => h i (y i)
map_zero' := by simp only [zero_apply, map_zero]
map_add' := by simp only [add_apply, map_add, forall_const] } | ι : Type u_1
inst✝⁵ : DecidableEq ι
R : Type u_2
inst✝⁴ : Semiring R
β : ι → Type u_3
γ : ι → Type u_4
inst✝³ : (i : ι) → AddCommMonoid (β i)
inst✝² : (i : ι) → AddCommMonoid (γ i)
F : ι → Type u_5
inst✝¹ : (i : ι) → FunLike (F i) (β i) (γ i)
inst✝ : ∀ (i : ι), AddMonoidHomClass (F i) (β i) (γ i)
h : (i : ι) → F i
x : ... | ι : Type u_1
inst✝⁵ : DecidableEq ι
R : Type u_2
inst✝⁴ : Semiring R
β : ι → Type u_3
γ : ι → Type u_4
inst✝³ : (i : ι) → AddCommMonoid (β i)
inst✝² : (i : ι) → AddCommMonoid (γ i)
F : ι → Type u_5
inst✝¹ : (i : ι) → FunLike (F i) (β i) (γ i)
inst✝ : ∀ (i : ι), AddMonoidHomClass (F i) (β i) (γ i)
h : (i : ι) → F i
x : ... | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
inst✝⁵ : DecidableEq ι
R : Type u_2
inst✝⁴ : Semiring R
β : ι → Type u_3
γ : ι → Type u_4
inst✝³ : (i : ι) → AddCommMonoid (β i)
inst✝² : (i : ι) → AddCommMonoid (γ i)
F : ι → Type u_5
inst✝¹ : (i : ι) → FunLike (F i) (β i) (γ i)
inst✝ : ∀ (i : ι... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | DirectSum.map_apply | [108, 1] | [127, 49] | change f x = g x | ι : Type u_1
inst✝⁵ : DecidableEq ι
R : Type u_2
inst✝⁴ : Semiring R
β : ι → Type u_3
γ : ι → Type u_4
inst✝³ : (i : ι) → AddCommMonoid (β i)
inst✝² : (i : ι) → AddCommMonoid (γ i)
F : ι → Type u_5
inst✝¹ : (i : ι) → FunLike (F i) (β i) (γ i)
inst✝ : ∀ (i : ι), AddMonoidHomClass (F i) (β i) (γ i)
h : (i : ι) → F i
x : ... | ι : Type u_1
inst✝⁵ : DecidableEq ι
R : Type u_2
inst✝⁴ : Semiring R
β : ι → Type u_3
γ : ι → Type u_4
inst✝³ : (i : ι) → AddCommMonoid (β i)
inst✝² : (i : ι) → AddCommMonoid (γ i)
F : ι → Type u_5
inst✝¹ : (i : ι) → FunLike (F i) (β i) (γ i)
inst✝ : ∀ (i : ι), AddMonoidHomClass (F i) (β i) (γ i)
h : (i : ι) → F i
x : ... | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
inst✝⁵ : DecidableEq ι
R : Type u_2
inst✝⁴ : Semiring R
β : ι → Type u_3
γ : ι → Type u_4
inst✝³ : (i : ι) → AddCommMonoid (β i)
inst✝² : (i : ι) → AddCommMonoid (γ i)
F : ι → Type u_5
inst✝¹ : (i : ι) → FunLike (F i) (β i) (γ i)
inst✝ : ∀ (i : ι... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | DirectSum.map_apply | [108, 1] | [127, 49] | suffices f = g by
rw [this] | ι : Type u_1
inst✝⁵ : DecidableEq ι
R : Type u_2
inst✝⁴ : Semiring R
β : ι → Type u_3
γ : ι → Type u_4
inst✝³ : (i : ι) → AddCommMonoid (β i)
inst✝² : (i : ι) → AddCommMonoid (γ i)
F : ι → Type u_5
inst✝¹ : (i : ι) → FunLike (F i) (β i) (γ i)
inst✝ : ∀ (i : ι), AddMonoidHomClass (F i) (β i) (γ i)
h : (i : ι) → F i
x : ... | ι : Type u_1
inst✝⁵ : DecidableEq ι
R : Type u_2
inst✝⁴ : Semiring R
β : ι → Type u_3
γ : ι → Type u_4
inst✝³ : (i : ι) → AddCommMonoid (β i)
inst✝² : (i : ι) → AddCommMonoid (γ i)
F : ι → Type u_5
inst✝¹ : (i : ι) → FunLike (F i) (β i) (γ i)
inst✝ : ∀ (i : ι), AddMonoidHomClass (F i) (β i) (γ i)
h : (i : ι) → F i
x : ... | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
inst✝⁵ : DecidableEq ι
R : Type u_2
inst✝⁴ : Semiring R
β : ι → Type u_3
γ : ι → Type u_4
inst✝³ : (i : ι) → AddCommMonoid (β i)
inst✝² : (i : ι) → AddCommMonoid (γ i)
F : ι → Type u_5
inst✝¹ : (i : ι) → FunLike (F i) (β i) (γ i)
inst✝ : ∀ (i : ι... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | DirectSum.map_apply | [108, 1] | [127, 49] | apply addHom_ext | ι : Type u_1
inst✝⁵ : DecidableEq ι
R : Type u_2
inst✝⁴ : Semiring R
β : ι → Type u_3
γ : ι → Type u_4
inst✝³ : (i : ι) → AddCommMonoid (β i)
inst✝² : (i : ι) → AddCommMonoid (γ i)
F : ι → Type u_5
inst✝¹ : (i : ι) → FunLike (F i) (β i) (γ i)
inst✝ : ∀ (i : ι), AddMonoidHomClass (F i) (β i) (γ i)
h : (i : ι) → F i
x : ... | case H
ι : Type u_1
inst✝⁵ : DecidableEq ι
R : Type u_2
inst✝⁴ : Semiring R
β : ι → Type u_3
γ : ι → Type u_4
inst✝³ : (i : ι) → AddCommMonoid (β i)
inst✝² : (i : ι) → AddCommMonoid (γ i)
F : ι → Type u_5
inst✝¹ : (i : ι) → FunLike (F i) (β i) (γ i)
inst✝ : ∀ (i : ι), AddMonoidHomClass (F i) (β i) (γ i)
h : (i : ι) → F... | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
inst✝⁵ : DecidableEq ι
R : Type u_2
inst✝⁴ : Semiring R
β : ι → Type u_3
γ : ι → Type u_4
inst✝³ : (i : ι) → AddCommMonoid (β i)
inst✝² : (i : ι) → AddCommMonoid (γ i)
F : ι → Type u_5
inst✝¹ : (i : ι) → FunLike (F i) (β i) (γ i)
inst✝ : ∀ (i : ι... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | DirectSum.map_apply | [108, 1] | [127, 49] | intros j y | case H
ι : Type u_1
inst✝⁵ : DecidableEq ι
R : Type u_2
inst✝⁴ : Semiring R
β : ι → Type u_3
γ : ι → Type u_4
inst✝³ : (i : ι) → AddCommMonoid (β i)
inst✝² : (i : ι) → AddCommMonoid (γ i)
F : ι → Type u_5
inst✝¹ : (i : ι) → FunLike (F i) (β i) (γ i)
inst✝ : ∀ (i : ι), AddMonoidHomClass (F i) (β i) (γ i)
h : (i : ι) → F... | case H
ι : Type u_1
inst✝⁵ : DecidableEq ι
R : Type u_2
inst✝⁴ : Semiring R
β : ι → Type u_3
γ : ι → Type u_4
inst✝³ : (i : ι) → AddCommMonoid (β i)
inst✝² : (i : ι) → AddCommMonoid (γ i)
F : ι → Type u_5
inst✝¹ : (i : ι) → FunLike (F i) (β i) (γ i)
inst✝ : ∀ (i : ι), AddMonoidHomClass (F i) (β i) (γ i)
h : (i : ι) → F... | Please generate a tactic in lean4 to solve the state.
STATE:
case H
ι : Type u_1
inst✝⁵ : DecidableEq ι
R : Type u_2
inst✝⁴ : Semiring R
β : ι → Type u_3
γ : ι → Type u_4
inst✝³ : (i : ι) → AddCommMonoid (β i)
inst✝² : (i : ι) → AddCommMonoid (γ i)
F : ι → Type u_5
inst✝¹ : (i : ι) → FunLike (F i) (β i) (γ i)
inst✝ : ∀... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | DirectSum.map_apply | [108, 1] | [127, 49] | simp only [AddMonoidHom.coe_mk, ZeroHom.coe_mk, map_of, f, g] | case H
ι : Type u_1
inst✝⁵ : DecidableEq ι
R : Type u_2
inst✝⁴ : Semiring R
β : ι → Type u_3
γ : ι → Type u_4
inst✝³ : (i : ι) → AddCommMonoid (β i)
inst✝² : (i : ι) → AddCommMonoid (γ i)
F : ι → Type u_5
inst✝¹ : (i : ι) → FunLike (F i) (β i) (γ i)
inst✝ : ∀ (i : ι), AddMonoidHomClass (F i) (β i) (γ i)
h : (i : ι) → F... | case H
ι : Type u_1
inst✝⁵ : DecidableEq ι
R : Type u_2
inst✝⁴ : Semiring R
β : ι → Type u_3
γ : ι → Type u_4
inst✝³ : (i : ι) → AddCommMonoid (β i)
inst✝² : (i : ι) → AddCommMonoid (γ i)
F : ι → Type u_5
inst✝¹ : (i : ι) → FunLike (F i) (β i) (γ i)
inst✝ : ∀ (i : ι), AddMonoidHomClass (F i) (β i) (γ i)
h : (i : ι) → F... | Please generate a tactic in lean4 to solve the state.
STATE:
case H
ι : Type u_1
inst✝⁵ : DecidableEq ι
R : Type u_2
inst✝⁴ : Semiring R
β : ι → Type u_3
γ : ι → Type u_4
inst✝³ : (i : ι) → AddCommMonoid (β i)
inst✝² : (i : ι) → AddCommMonoid (γ i)
F : ι → Type u_5
inst✝¹ : (i : ι) → FunLike (F i) (β i) (γ i)
inst✝ : ∀... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | DirectSum.map_apply | [108, 1] | [127, 49] | by_cases hj : j = i | case H
ι : Type u_1
inst✝⁵ : DecidableEq ι
R : Type u_2
inst✝⁴ : Semiring R
β : ι → Type u_3
γ : ι → Type u_4
inst✝³ : (i : ι) → AddCommMonoid (β i)
inst✝² : (i : ι) → AddCommMonoid (γ i)
F : ι → Type u_5
inst✝¹ : (i : ι) → FunLike (F i) (β i) (γ i)
inst✝ : ∀ (i : ι), AddMonoidHomClass (F i) (β i) (γ i)
h : (i : ι) → F... | case pos
ι : Type u_1
inst✝⁵ : DecidableEq ι
R : Type u_2
inst✝⁴ : Semiring R
β : ι → Type u_3
γ : ι → Type u_4
inst✝³ : (i : ι) → AddCommMonoid (β i)
inst✝² : (i : ι) → AddCommMonoid (γ i)
F : ι → Type u_5
inst✝¹ : (i : ι) → FunLike (F i) (β i) (γ i)
inst✝ : ∀ (i : ι), AddMonoidHomClass (F i) (β i) (γ i)
h : (i : ι) →... | Please generate a tactic in lean4 to solve the state.
STATE:
case H
ι : Type u_1
inst✝⁵ : DecidableEq ι
R : Type u_2
inst✝⁴ : Semiring R
β : ι → Type u_3
γ : ι → Type u_4
inst✝³ : (i : ι) → AddCommMonoid (β i)
inst✝² : (i : ι) → AddCommMonoid (γ i)
F : ι → Type u_5
inst✝¹ : (i : ι) → FunLike (F i) (β i) (γ i)
inst✝ : ∀... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | DirectSum.map_apply | [108, 1] | [127, 49] | simp only [map_zero, zero_apply] | ι : Type u_1
inst✝⁵ : DecidableEq ι
R : Type u_2
inst✝⁴ : Semiring R
β : ι → Type u_3
γ : ι → Type u_4
inst✝³ : (i : ι) → AddCommMonoid (β i)
inst✝² : (i : ι) → AddCommMonoid (γ i)
F : ι → Type u_5
inst✝¹ : (i : ι) → FunLike (F i) (β i) (γ i)
inst✝ : ∀ (i : ι), AddMonoidHomClass (F i) (β i) (γ i)
h : (i : ι) → F i
x : ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
inst✝⁵ : DecidableEq ι
R : Type u_2
inst✝⁴ : Semiring R
β : ι → Type u_3
γ : ι → Type u_4
inst✝³ : (i : ι) → AddCommMonoid (β i)
inst✝² : (i : ι) → AddCommMonoid (γ i)
F : ι → Type u_5
inst✝¹ : (i : ι) → FunLike (F i) (β i) (γ i)
inst✝ : ∀ (i : ι... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | DirectSum.map_apply | [108, 1] | [127, 49] | simp only [map_add, add_apply, forall_const] | ι : Type u_1
inst✝⁵ : DecidableEq ι
R : Type u_2
inst✝⁴ : Semiring R
β : ι → Type u_3
γ : ι → Type u_4
inst✝³ : (i : ι) → AddCommMonoid (β i)
inst✝² : (i : ι) → AddCommMonoid (γ i)
F : ι → Type u_5
inst✝¹ : (i : ι) → FunLike (F i) (β i) (γ i)
inst✝ : ∀ (i : ι), AddMonoidHomClass (F i) (β i) (γ i)
h : (i : ι) → F i
x : ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
inst✝⁵ : DecidableEq ι
R : Type u_2
inst✝⁴ : Semiring R
β : ι → Type u_3
γ : ι → Type u_4
inst✝³ : (i : ι) → AddCommMonoid (β i)
inst✝² : (i : ι) → AddCommMonoid (γ i)
F : ι → Type u_5
inst✝¹ : (i : ι) → FunLike (F i) (β i) (γ i)
inst✝ : ∀ (i : ι... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | DirectSum.map_apply | [108, 1] | [127, 49] | simp only [zero_apply, map_zero] | ι : Type u_1
inst✝⁵ : DecidableEq ι
R : Type u_2
inst✝⁴ : Semiring R
β : ι → Type u_3
γ : ι → Type u_4
inst✝³ : (i : ι) → AddCommMonoid (β i)
inst✝² : (i : ι) → AddCommMonoid (γ i)
F : ι → Type u_5
inst✝¹ : (i : ι) → FunLike (F i) (β i) (γ i)
inst✝ : ∀ (i : ι), AddMonoidHomClass (F i) (β i) (γ i)
h : (i : ι) → F i
x : ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
inst✝⁵ : DecidableEq ι
R : Type u_2
inst✝⁴ : Semiring R
β : ι → Type u_3
γ : ι → Type u_4
inst✝³ : (i : ι) → AddCommMonoid (β i)
inst✝² : (i : ι) → AddCommMonoid (γ i)
F : ι → Type u_5
inst✝¹ : (i : ι) → FunLike (F i) (β i) (γ i)
inst✝ : ∀ (i : ι... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | DirectSum.map_apply | [108, 1] | [127, 49] | simp only [add_apply, map_add, forall_const] | ι : Type u_1
inst✝⁵ : DecidableEq ι
R : Type u_2
inst✝⁴ : Semiring R
β : ι → Type u_3
γ : ι → Type u_4
inst✝³ : (i : ι) → AddCommMonoid (β i)
inst✝² : (i : ι) → AddCommMonoid (γ i)
F : ι → Type u_5
inst✝¹ : (i : ι) → FunLike (F i) (β i) (γ i)
inst✝ : ∀ (i : ι), AddMonoidHomClass (F i) (β i) (γ i)
h : (i : ι) → F i
x : ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
inst✝⁵ : DecidableEq ι
R : Type u_2
inst✝⁴ : Semiring R
β : ι → Type u_3
γ : ι → Type u_4
inst✝³ : (i : ι) → AddCommMonoid (β i)
inst✝² : (i : ι) → AddCommMonoid (γ i)
F : ι → Type u_5
inst✝¹ : (i : ι) → FunLike (F i) (β i) (γ i)
inst✝ : ∀ (i : ι... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | DirectSum.map_apply | [108, 1] | [127, 49] | rw [this] | ι : Type u_1
inst✝⁵ : DecidableEq ι
R : Type u_2
inst✝⁴ : Semiring R
β : ι → Type u_3
γ : ι → Type u_4
inst✝³ : (i : ι) → AddCommMonoid (β i)
inst✝² : (i : ι) → AddCommMonoid (γ i)
F : ι → Type u_5
inst✝¹ : (i : ι) → FunLike (F i) (β i) (γ i)
inst✝ : ∀ (i : ι), AddMonoidHomClass (F i) (β i) (γ i)
h : (i : ι) → F i
x : ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
inst✝⁵ : DecidableEq ι
R : Type u_2
inst✝⁴ : Semiring R
β : ι → Type u_3
γ : ι → Type u_4
inst✝³ : (i : ι) → AddCommMonoid (β i)
inst✝² : (i : ι) → AddCommMonoid (γ i)
F : ι → Type u_5
inst✝¹ : (i : ι) → FunLike (F i) (β i) (γ i)
inst✝ : ∀ (i : ι... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | DirectSum.map_apply | [108, 1] | [127, 49] | rw [← hj] | case pos
ι : Type u_1
inst✝⁵ : DecidableEq ι
R : Type u_2
inst✝⁴ : Semiring R
β : ι → Type u_3
γ : ι → Type u_4
inst✝³ : (i : ι) → AddCommMonoid (β i)
inst✝² : (i : ι) → AddCommMonoid (γ i)
F : ι → Type u_5
inst✝¹ : (i : ι) → FunLike (F i) (β i) (γ i)
inst✝ : ∀ (i : ι), AddMonoidHomClass (F i) (β i) (γ i)
h : (i : ι) →... | case pos
ι : Type u_1
inst✝⁵ : DecidableEq ι
R : Type u_2
inst✝⁴ : Semiring R
β : ι → Type u_3
γ : ι → Type u_4
inst✝³ : (i : ι) → AddCommMonoid (β i)
inst✝² : (i : ι) → AddCommMonoid (γ i)
F : ι → Type u_5
inst✝¹ : (i : ι) → FunLike (F i) (β i) (γ i)
inst✝ : ∀ (i : ι), AddMonoidHomClass (F i) (β i) (γ i)
h : (i : ι) →... | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
ι : Type u_1
inst✝⁵ : DecidableEq ι
R : Type u_2
inst✝⁴ : Semiring R
β : ι → Type u_3
γ : ι → Type u_4
inst✝³ : (i : ι) → AddCommMonoid (β i)
inst✝² : (i : ι) → AddCommMonoid (γ i)
F : ι → Type u_5
inst✝¹ : (i : ι) → FunLike (F i) (β i) (γ i)
inst✝ :... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | DirectSum.map_apply | [108, 1] | [127, 49] | simp only [of_eq_same] | case pos
ι : Type u_1
inst✝⁵ : DecidableEq ι
R : Type u_2
inst✝⁴ : Semiring R
β : ι → Type u_3
γ : ι → Type u_4
inst✝³ : (i : ι) → AddCommMonoid (β i)
inst✝² : (i : ι) → AddCommMonoid (γ i)
F : ι → Type u_5
inst✝¹ : (i : ι) → FunLike (F i) (β i) (γ i)
inst✝ : ∀ (i : ι), AddMonoidHomClass (F i) (β i) (γ i)
h : (i : ι) →... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
ι : Type u_1
inst✝⁵ : DecidableEq ι
R : Type u_2
inst✝⁴ : Semiring R
β : ι → Type u_3
γ : ι → Type u_4
inst✝³ : (i : ι) → AddCommMonoid (β i)
inst✝² : (i : ι) → AddCommMonoid (γ i)
F : ι → Type u_5
inst✝¹ : (i : ι) → FunLike (F i) (β i) (γ i)
inst✝ :... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | DirectSum.map_apply | [108, 1] | [127, 49] | simp only [of_eq_of_ne _ j i _ hj, map_zero] | case neg
ι : Type u_1
inst✝⁵ : DecidableEq ι
R : Type u_2
inst✝⁴ : Semiring R
β : ι → Type u_3
γ : ι → Type u_4
inst✝³ : (i : ι) → AddCommMonoid (β i)
inst✝² : (i : ι) → AddCommMonoid (γ i)
F : ι → Type u_5
inst✝¹ : (i : ι) → FunLike (F i) (β i) (γ i)
inst✝ : ∀ (i : ι), AddMonoidHomClass (F i) (β i) (γ i)
h : (i : ι) →... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
ι : Type u_1
inst✝⁵ : DecidableEq ι
R : Type u_2
inst✝⁴ : Semiring R
β : ι → Type u_3
γ : ι → Type u_4
inst✝³ : (i : ι) → AddCommMonoid (β i)
inst✝² : (i : ι) → AddCommMonoid (γ i)
F : ι → Type u_5
inst✝¹ : (i : ι) → FunLike (F i) (β i) (γ i)
inst✝ :... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | DirectSum.map'_of | [129, 1] | [132, 63] | simp only [map', toAddMonoid_of, AddMonoidHom.coe_comp] | ι : Type u_1
inst✝³ : DecidableEq ι
R : Type u_2
inst✝² : Semiring R
β : ι → Type u_3
γ : ι → Type u_4
inst✝¹ : (i : ι) → AddCommMonoid (β i)
inst✝ : (i : ι) → AddCommMonoid (γ i)
h : (i : ι) → β i →+ γ i
i : ι
x : β i
⊢ (map' h) ((of β i) x) = (of γ i) ((h i) x) | ι : Type u_1
inst✝³ : DecidableEq ι
R : Type u_2
inst✝² : Semiring R
β : ι → Type u_3
γ : ι → Type u_4
inst✝¹ : (i : ι) → AddCommMonoid (β i)
inst✝ : (i : ι) → AddCommMonoid (γ i)
h : (i : ι) → β i →+ γ i
i : ι
x : β i
⊢ (⇑(of (fun i => γ i) i) ∘ ⇑(h i)) x = (of γ i) ((h i) x) | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
inst✝³ : DecidableEq ι
R : Type u_2
inst✝² : Semiring R
β : ι → Type u_3
γ : ι → Type u_4
inst✝¹ : (i : ι) → AddCommMonoid (β i)
inst✝ : (i : ι) → AddCommMonoid (γ i)
h : (i : ι) → β i →+ γ i
i : ι
x : β i
⊢ (map' h) ((of β i) x) = (of γ i) ((h i... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | DirectSum.map'_of | [129, 1] | [132, 63] | rfl | ι : Type u_1
inst✝³ : DecidableEq ι
R : Type u_2
inst✝² : Semiring R
β : ι → Type u_3
γ : ι → Type u_4
inst✝¹ : (i : ι) → AddCommMonoid (β i)
inst✝ : (i : ι) → AddCommMonoid (γ i)
h : (i : ι) → β i →+ γ i
i : ι
x : β i
⊢ (⇑(of (fun i => γ i) i) ∘ ⇑(h i)) x = (of γ i) ((h i) x) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
inst✝³ : DecidableEq ι
R : Type u_2
inst✝² : Semiring R
β : ι → Type u_3
γ : ι → Type u_4
inst✝¹ : (i : ι) → AddCommMonoid (β i)
inst✝ : (i : ι) → AddCommMonoid (γ i)
h : (i : ι) → β i →+ γ i
i : ι
x : β i
⊢ (⇑(of (fun i => γ i) i) ∘ ⇑(h i)) x = ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | DirectSum.lmap'_lof | [134, 1] | [137, 59] | simp only [lmap', toModule_lof, LinearMap.coe_comp] | ι : Type u_1
inst✝⁵ : DecidableEq ι
R : Type u_2
inst✝⁴ : Semiring R
β : ι → Type u_3
γ : ι → Type u_4
inst✝³ : (i : ι) → AddCommMonoid (β i)
inst✝² : (i : ι) → AddCommMonoid (γ i)
inst✝¹ : (i : ι) → Module R (β i)
inst✝ : (i : ι) → Module R (γ i)
h : (i : ι) → β i →ₗ[R] γ i
i : ι
x : β i
⊢ (lmap' h) ((lof R ι β i) x) ... | ι : Type u_1
inst✝⁵ : DecidableEq ι
R : Type u_2
inst✝⁴ : Semiring R
β : ι → Type u_3
γ : ι → Type u_4
inst✝³ : (i : ι) → AddCommMonoid (β i)
inst✝² : (i : ι) → AddCommMonoid (γ i)
inst✝¹ : (i : ι) → Module R (β i)
inst✝ : (i : ι) → Module R (γ i)
h : (i : ι) → β i →ₗ[R] γ i
i : ι
x : β i
⊢ (⇑(lof R ι (fun i => γ i) i)... | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
inst✝⁵ : DecidableEq ι
R : Type u_2
inst✝⁴ : Semiring R
β : ι → Type u_3
γ : ι → Type u_4
inst✝³ : (i : ι) → AddCommMonoid (β i)
inst✝² : (i : ι) → AddCommMonoid (γ i)
inst✝¹ : (i : ι) → Module R (β i)
inst✝ : (i : ι) → Module R (γ i)
h : (i : ι)... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | DirectSum.lmap'_lof | [134, 1] | [137, 59] | rfl | ι : Type u_1
inst✝⁵ : DecidableEq ι
R : Type u_2
inst✝⁴ : Semiring R
β : ι → Type u_3
γ : ι → Type u_4
inst✝³ : (i : ι) → AddCommMonoid (β i)
inst✝² : (i : ι) → AddCommMonoid (γ i)
inst✝¹ : (i : ι) → Module R (β i)
inst✝ : (i : ι) → Module R (γ i)
h : (i : ι) → β i →ₗ[R] γ i
i : ι
x : β i
⊢ (⇑(lof R ι (fun i => γ i) i)... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
inst✝⁵ : DecidableEq ι
R : Type u_2
inst✝⁴ : Semiring R
β : ι → Type u_3
γ : ι → Type u_4
inst✝³ : (i : ι) → AddCommMonoid (β i)
inst✝² : (i : ι) → AddCommMonoid (γ i)
inst✝¹ : (i : ι) → Module R (β i)
inst✝ : (i : ι) → Module R (γ i)
h : (i : ι)... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | DirectSum.lmap'_surjective | [139, 1] | [149, 33] | intro c | ι : Type u_1
inst✝⁵ : DecidableEq ι
R : Type u_2
inst✝⁴ : Semiring R
β : ι → Type u_3
γ : ι → Type u_4
inst✝³ : (i : ι) → AddCommMonoid (β i)
inst✝² : (i : ι) → Module R (β i)
inst✝¹ : (i : ι) → AddCommMonoid (γ i)
inst✝ : (i : ι) → Module R (γ i)
f : (i : ι) → β i →ₗ[R] γ i
h : ∀ (i : ι), Surjective ⇑(f i)
⊢ Surjectiv... | ι : Type u_1
inst✝⁵ : DecidableEq ι
R : Type u_2
inst✝⁴ : Semiring R
β : ι → Type u_3
γ : ι → Type u_4
inst✝³ : (i : ι) → AddCommMonoid (β i)
inst✝² : (i : ι) → Module R (β i)
inst✝¹ : (i : ι) → AddCommMonoid (γ i)
inst✝ : (i : ι) → Module R (γ i)
f : (i : ι) → β i →ₗ[R] γ i
h : ∀ (i : ι), Surjective ⇑(f i)
c : ⨁ (i : ... | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
inst✝⁵ : DecidableEq ι
R : Type u_2
inst✝⁴ : Semiring R
β : ι → Type u_3
γ : ι → Type u_4
inst✝³ : (i : ι) → AddCommMonoid (β i)
inst✝² : (i : ι) → Module R (β i)
inst✝¹ : (i : ι) → AddCommMonoid (γ i)
inst✝ : (i : ι) → Module R (γ i)
f : (i : ι)... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | DirectSum.lmap'_surjective | [139, 1] | [149, 33] | induction' c using DirectSum.induction_on with i xi x y hx hy | ι : Type u_1
inst✝⁵ : DecidableEq ι
R : Type u_2
inst✝⁴ : Semiring R
β : ι → Type u_3
γ : ι → Type u_4
inst✝³ : (i : ι) → AddCommMonoid (β i)
inst✝² : (i : ι) → Module R (β i)
inst✝¹ : (i : ι) → AddCommMonoid (γ i)
inst✝ : (i : ι) → Module R (γ i)
f : (i : ι) → β i →ₗ[R] γ i
h : ∀ (i : ι), Surjective ⇑(f i)
c : ⨁ (i : ... | case H_zero
ι : Type u_1
inst✝⁵ : DecidableEq ι
R : Type u_2
inst✝⁴ : Semiring R
β : ι → Type u_3
γ : ι → Type u_4
inst✝³ : (i : ι) → AddCommMonoid (β i)
inst✝² : (i : ι) → Module R (β i)
inst✝¹ : (i : ι) → AddCommMonoid (γ i)
inst✝ : (i : ι) → Module R (γ i)
f : (i : ι) → β i →ₗ[R] γ i
h : ∀ (i : ι), Surjective ⇑(f i)... | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
inst✝⁵ : DecidableEq ι
R : Type u_2
inst✝⁴ : Semiring R
β : ι → Type u_3
γ : ι → Type u_4
inst✝³ : (i : ι) → AddCommMonoid (β i)
inst✝² : (i : ι) → Module R (β i)
inst✝¹ : (i : ι) → AddCommMonoid (γ i)
inst✝ : (i : ι) → Module R (γ i)
f : (i : ι)... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | DirectSum.lmap'_surjective | [139, 1] | [149, 33] | . exact ⟨0, map_zero _⟩ | case H_zero
ι : Type u_1
inst✝⁵ : DecidableEq ι
R : Type u_2
inst✝⁴ : Semiring R
β : ι → Type u_3
γ : ι → Type u_4
inst✝³ : (i : ι) → AddCommMonoid (β i)
inst✝² : (i : ι) → Module R (β i)
inst✝¹ : (i : ι) → AddCommMonoid (γ i)
inst✝ : (i : ι) → Module R (γ i)
f : (i : ι) → β i →ₗ[R] γ i
h : ∀ (i : ι), Surjective ⇑(f i)... | case H_basic
ι : Type u_1
inst✝⁵ : DecidableEq ι
R : Type u_2
inst✝⁴ : Semiring R
β : ι → Type u_3
γ : ι → Type u_4
inst✝³ : (i : ι) → AddCommMonoid (β i)
inst✝² : (i : ι) → Module R (β i)
inst✝¹ : (i : ι) → AddCommMonoid (γ i)
inst✝ : (i : ι) → Module R (γ i)
f : (i : ι) → β i →ₗ[R] γ i
h : ∀ (i : ι), Surjective ⇑(f i... | Please generate a tactic in lean4 to solve the state.
STATE:
case H_zero
ι : Type u_1
inst✝⁵ : DecidableEq ι
R : Type u_2
inst✝⁴ : Semiring R
β : ι → Type u_3
γ : ι → Type u_4
inst✝³ : (i : ι) → AddCommMonoid (β i)
inst✝² : (i : ι) → Module R (β i)
inst✝¹ : (i : ι) → AddCommMonoid (γ i)
inst✝ : (i : ι) → Module R (γ i)... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | DirectSum.lmap'_surjective | [139, 1] | [149, 33] | . use of _ i (h i xi).choose
rw [← lof_eq_of R, lmap'_lof, lof_eq_of, (h i xi).choose_spec] | case H_basic
ι : Type u_1
inst✝⁵ : DecidableEq ι
R : Type u_2
inst✝⁴ : Semiring R
β : ι → Type u_3
γ : ι → Type u_4
inst✝³ : (i : ι) → AddCommMonoid (β i)
inst✝² : (i : ι) → Module R (β i)
inst✝¹ : (i : ι) → AddCommMonoid (γ i)
inst✝ : (i : ι) → Module R (γ i)
f : (i : ι) → β i →ₗ[R] γ i
h : ∀ (i : ι), Surjective ⇑(f i... | case H_plus
ι : Type u_1
inst✝⁵ : DecidableEq ι
R : Type u_2
inst✝⁴ : Semiring R
β : ι → Type u_3
γ : ι → Type u_4
inst✝³ : (i : ι) → AddCommMonoid (β i)
inst✝² : (i : ι) → Module R (β i)
inst✝¹ : (i : ι) → AddCommMonoid (γ i)
inst✝ : (i : ι) → Module R (γ i)
f : (i : ι) → β i →ₗ[R] γ i
h : ∀ (i : ι), Surjective ⇑(f i)... | Please generate a tactic in lean4 to solve the state.
STATE:
case H_basic
ι : Type u_1
inst✝⁵ : DecidableEq ι
R : Type u_2
inst✝⁴ : Semiring R
β : ι → Type u_3
γ : ι → Type u_4
inst✝³ : (i : ι) → AddCommMonoid (β i)
inst✝² : (i : ι) → Module R (β i)
inst✝¹ : (i : ι) → AddCommMonoid (γ i)
inst✝ : (i : ι) → Module R (γ i... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | DirectSum.lmap'_surjective | [139, 1] | [149, 33] | . obtain ⟨a, ha, rfl⟩ := hx
obtain ⟨b, hb, rfl⟩ := hy
exact ⟨a + b, map_add _ _ _⟩ | case H_plus
ι : Type u_1
inst✝⁵ : DecidableEq ι
R : Type u_2
inst✝⁴ : Semiring R
β : ι → Type u_3
γ : ι → Type u_4
inst✝³ : (i : ι) → AddCommMonoid (β i)
inst✝² : (i : ι) → Module R (β i)
inst✝¹ : (i : ι) → AddCommMonoid (γ i)
inst✝ : (i : ι) → Module R (γ i)
f : (i : ι) → β i →ₗ[R] γ i
h : ∀ (i : ι), Surjective ⇑(f i)... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case H_plus
ι : Type u_1
inst✝⁵ : DecidableEq ι
R : Type u_2
inst✝⁴ : Semiring R
β : ι → Type u_3
γ : ι → Type u_4
inst✝³ : (i : ι) → AddCommMonoid (β i)
inst✝² : (i : ι) → Module R (β i)
inst✝¹ : (i : ι) → AddCommMonoid (γ i)
inst✝ : (i : ι) → Module R (γ i)... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | DirectSum.lmap'_surjective | [139, 1] | [149, 33] | exact ⟨0, map_zero _⟩ | case H_zero
ι : Type u_1
inst✝⁵ : DecidableEq ι
R : Type u_2
inst✝⁴ : Semiring R
β : ι → Type u_3
γ : ι → Type u_4
inst✝³ : (i : ι) → AddCommMonoid (β i)
inst✝² : (i : ι) → Module R (β i)
inst✝¹ : (i : ι) → AddCommMonoid (γ i)
inst✝ : (i : ι) → Module R (γ i)
f : (i : ι) → β i →ₗ[R] γ i
h : ∀ (i : ι), Surjective ⇑(f i)... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case H_zero
ι : Type u_1
inst✝⁵ : DecidableEq ι
R : Type u_2
inst✝⁴ : Semiring R
β : ι → Type u_3
γ : ι → Type u_4
inst✝³ : (i : ι) → AddCommMonoid (β i)
inst✝² : (i : ι) → Module R (β i)
inst✝¹ : (i : ι) → AddCommMonoid (γ i)
inst✝ : (i : ι) → Module R (γ i)... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | DirectSum.lmap'_surjective | [139, 1] | [149, 33] | use of _ i (h i xi).choose | case H_basic
ι : Type u_1
inst✝⁵ : DecidableEq ι
R : Type u_2
inst✝⁴ : Semiring R
β : ι → Type u_3
γ : ι → Type u_4
inst✝³ : (i : ι) → AddCommMonoid (β i)
inst✝² : (i : ι) → Module R (β i)
inst✝¹ : (i : ι) → AddCommMonoid (γ i)
inst✝ : (i : ι) → Module R (γ i)
f : (i : ι) → β i →ₗ[R] γ i
h : ∀ (i : ι), Surjective ⇑(f i... | case h
ι : Type u_1
inst✝⁵ : DecidableEq ι
R : Type u_2
inst✝⁴ : Semiring R
β : ι → Type u_3
γ : ι → Type u_4
inst✝³ : (i : ι) → AddCommMonoid (β i)
inst✝² : (i : ι) → Module R (β i)
inst✝¹ : (i : ι) → AddCommMonoid (γ i)
inst✝ : (i : ι) → Module R (γ i)
f : (i : ι) → β i →ₗ[R] γ i
h : ∀ (i : ι), Surjective ⇑(f i)
i : ... | Please generate a tactic in lean4 to solve the state.
STATE:
case H_basic
ι : Type u_1
inst✝⁵ : DecidableEq ι
R : Type u_2
inst✝⁴ : Semiring R
β : ι → Type u_3
γ : ι → Type u_4
inst✝³ : (i : ι) → AddCommMonoid (β i)
inst✝² : (i : ι) → Module R (β i)
inst✝¹ : (i : ι) → AddCommMonoid (γ i)
inst✝ : (i : ι) → Module R (γ i... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | DirectSum.lmap'_surjective | [139, 1] | [149, 33] | rw [← lof_eq_of R, lmap'_lof, lof_eq_of, (h i xi).choose_spec] | case h
ι : Type u_1
inst✝⁵ : DecidableEq ι
R : Type u_2
inst✝⁴ : Semiring R
β : ι → Type u_3
γ : ι → Type u_4
inst✝³ : (i : ι) → AddCommMonoid (β i)
inst✝² : (i : ι) → Module R (β i)
inst✝¹ : (i : ι) → AddCommMonoid (γ i)
inst✝ : (i : ι) → Module R (γ i)
f : (i : ι) → β i →ₗ[R] γ i
h : ∀ (i : ι), Surjective ⇑(f i)
i : ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
ι : Type u_1
inst✝⁵ : DecidableEq ι
R : Type u_2
inst✝⁴ : Semiring R
β : ι → Type u_3
γ : ι → Type u_4
inst✝³ : (i : ι) → AddCommMonoid (β i)
inst✝² : (i : ι) → Module R (β i)
inst✝¹ : (i : ι) → AddCommMonoid (γ i)
inst✝ : (i : ι) → Module R (γ i)
f : ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | DirectSum.lmap'_surjective | [139, 1] | [149, 33] | obtain ⟨a, ha, rfl⟩ := hx | case H_plus
ι : Type u_1
inst✝⁵ : DecidableEq ι
R : Type u_2
inst✝⁴ : Semiring R
β : ι → Type u_3
γ : ι → Type u_4
inst✝³ : (i : ι) → AddCommMonoid (β i)
inst✝² : (i : ι) → Module R (β i)
inst✝¹ : (i : ι) → AddCommMonoid (γ i)
inst✝ : (i : ι) → Module R (γ i)
f : (i : ι) → β i →ₗ[R] γ i
h : ∀ (i : ι), Surjective ⇑(f i)... | case H_plus.intro.refl
ι : Type u_1
inst✝⁵ : DecidableEq ι
R : Type u_2
inst✝⁴ : Semiring R
β : ι → Type u_3
γ : ι → Type u_4
inst✝³ : (i : ι) → AddCommMonoid (β i)
inst✝² : (i : ι) → Module R (β i)
inst✝¹ : (i : ι) → AddCommMonoid (γ i)
inst✝ : (i : ι) → Module R (γ i)
f : (i : ι) → β i →ₗ[R] γ i
h : ∀ (i : ι), Surjec... | Please generate a tactic in lean4 to solve the state.
STATE:
case H_plus
ι : Type u_1
inst✝⁵ : DecidableEq ι
R : Type u_2
inst✝⁴ : Semiring R
β : ι → Type u_3
γ : ι → Type u_4
inst✝³ : (i : ι) → AddCommMonoid (β i)
inst✝² : (i : ι) → Module R (β i)
inst✝¹ : (i : ι) → AddCommMonoid (γ i)
inst✝ : (i : ι) → Module R (γ i)... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | DirectSum.lmap'_surjective | [139, 1] | [149, 33] | obtain ⟨b, hb, rfl⟩ := hy | case H_plus.intro.refl
ι : Type u_1
inst✝⁵ : DecidableEq ι
R : Type u_2
inst✝⁴ : Semiring R
β : ι → Type u_3
γ : ι → Type u_4
inst✝³ : (i : ι) → AddCommMonoid (β i)
inst✝² : (i : ι) → Module R (β i)
inst✝¹ : (i : ι) → AddCommMonoid (γ i)
inst✝ : (i : ι) → Module R (γ i)
f : (i : ι) → β i →ₗ[R] γ i
h : ∀ (i : ι), Surjec... | case H_plus.intro.refl.intro.refl
ι : Type u_1
inst✝⁵ : DecidableEq ι
R : Type u_2
inst✝⁴ : Semiring R
β : ι → Type u_3
γ : ι → Type u_4
inst✝³ : (i : ι) → AddCommMonoid (β i)
inst✝² : (i : ι) → Module R (β i)
inst✝¹ : (i : ι) → AddCommMonoid (γ i)
inst✝ : (i : ι) → Module R (γ i)
f : (i : ι) → β i →ₗ[R] γ i
h : ∀ (i :... | Please generate a tactic in lean4 to solve the state.
STATE:
case H_plus.intro.refl
ι : Type u_1
inst✝⁵ : DecidableEq ι
R : Type u_2
inst✝⁴ : Semiring R
β : ι → Type u_3
γ : ι → Type u_4
inst✝³ : (i : ι) → AddCommMonoid (β i)
inst✝² : (i : ι) → Module R (β i)
inst✝¹ : (i : ι) → AddCommMonoid (γ i)
inst✝ : (i : ι) → Mod... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | DirectSum.lmap'_surjective | [139, 1] | [149, 33] | exact ⟨a + b, map_add _ _ _⟩ | case H_plus.intro.refl.intro.refl
ι : Type u_1
inst✝⁵ : DecidableEq ι
R : Type u_2
inst✝⁴ : Semiring R
β : ι → Type u_3
γ : ι → Type u_4
inst✝³ : (i : ι) → AddCommMonoid (β i)
inst✝² : (i : ι) → Module R (β i)
inst✝¹ : (i : ι) → AddCommMonoid (γ i)
inst✝ : (i : ι) → Module R (γ i)
f : (i : ι) → β i →ₗ[R] γ i
h : ∀ (i :... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case H_plus.intro.refl.intro.refl
ι : Type u_1
inst✝⁵ : DecidableEq ι
R : Type u_2
inst✝⁴ : Semiring R
β : ι → Type u_3
γ : ι → Type u_4
inst✝³ : (i : ι) → AddCommMonoid (β i)
inst✝² : (i : ι) → Module R (β i)
inst✝¹ : (i : ι) → AddCommMonoid (γ i)
inst✝ : (i... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | DirectSum.lmap'_apply | [151, 1] | [161, 50] | simp only [apply_eq_component R, ← LinearMap.comp_apply] | ι : Type u_1
inst✝⁵ : DecidableEq ι
R : Type u_2
inst✝⁴ : Semiring R
β : ι → Type u_3
γ : ι → Type u_4
inst✝³ : (i : ι) → AddCommMonoid (β i)
inst✝² : (i : ι) → AddCommMonoid (γ i)
inst✝¹ : (i : ι) → Module R (β i)
inst✝ : (i : ι) → Module R (γ i)
h : (i : ι) → β i →ₗ[R] γ i
x : ⨁ (i : ι), β i
i : ι
⊢ ((lmap' h) x) i =... | ι : Type u_1
inst✝⁵ : DecidableEq ι
R : Type u_2
inst✝⁴ : Semiring R
β : ι → Type u_3
γ : ι → Type u_4
inst✝³ : (i : ι) → AddCommMonoid (β i)
inst✝² : (i : ι) → AddCommMonoid (γ i)
inst✝¹ : (i : ι) → Module R (β i)
inst✝ : (i : ι) → Module R (γ i)
h : (i : ι) → β i →ₗ[R] γ i
x : ⨁ (i : ι), β i
i : ι
⊢ (component R ι (f... | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
inst✝⁵ : DecidableEq ι
R : Type u_2
inst✝⁴ : Semiring R
β : ι → Type u_3
γ : ι → Type u_4
inst✝³ : (i : ι) → AddCommMonoid (β i)
inst✝² : (i : ι) → AddCommMonoid (γ i)
inst✝¹ : (i : ι) → Module R (β i)
inst✝ : (i : ι) → Module R (γ i)
h : (i : ι)... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | DirectSum.lmap'_apply | [151, 1] | [161, 50] | apply LinearMap.congr_fun | ι : Type u_1
inst✝⁵ : DecidableEq ι
R : Type u_2
inst✝⁴ : Semiring R
β : ι → Type u_3
γ : ι → Type u_4
inst✝³ : (i : ι) → AddCommMonoid (β i)
inst✝² : (i : ι) → AddCommMonoid (γ i)
inst✝¹ : (i : ι) → Module R (β i)
inst✝ : (i : ι) → Module R (γ i)
h : (i : ι) → β i →ₗ[R] γ i
x : ⨁ (i : ι), β i
i : ι
⊢ (component R ι (f... | case h
ι : Type u_1
inst✝⁵ : DecidableEq ι
R : Type u_2
inst✝⁴ : Semiring R
β : ι → Type u_3
γ : ι → Type u_4
inst✝³ : (i : ι) → AddCommMonoid (β i)
inst✝² : (i : ι) → AddCommMonoid (γ i)
inst✝¹ : (i : ι) → Module R (β i)
inst✝ : (i : ι) → Module R (γ i)
h : (i : ι) → β i →ₗ[R] γ i
x : ⨁ (i : ι), β i
i : ι
⊢ component ... | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
inst✝⁵ : DecidableEq ι
R : Type u_2
inst✝⁴ : Semiring R
β : ι → Type u_3
γ : ι → Type u_4
inst✝³ : (i : ι) → AddCommMonoid (β i)
inst✝² : (i : ι) → AddCommMonoid (γ i)
inst✝¹ : (i : ι) → Module R (β i)
inst✝ : (i : ι) → Module R (γ i)
h : (i : ι)... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | DirectSum.lmap'_apply | [151, 1] | [161, 50] | ext j y | case h
ι : Type u_1
inst✝⁵ : DecidableEq ι
R : Type u_2
inst✝⁴ : Semiring R
β : ι → Type u_3
γ : ι → Type u_4
inst✝³ : (i : ι) → AddCommMonoid (β i)
inst✝² : (i : ι) → AddCommMonoid (γ i)
inst✝¹ : (i : ι) → Module R (β i)
inst✝ : (i : ι) → Module R (γ i)
h : (i : ι) → β i →ₗ[R] γ i
x : ⨁ (i : ι), β i
i : ι
⊢ component ... | case h.H.h
ι : Type u_1
inst✝⁵ : DecidableEq ι
R : Type u_2
inst✝⁴ : Semiring R
β : ι → Type u_3
γ : ι → Type u_4
inst✝³ : (i : ι) → AddCommMonoid (β i)
inst✝² : (i : ι) → AddCommMonoid (γ i)
inst✝¹ : (i : ι) → Module R (β i)
inst✝ : (i : ι) → Module R (γ i)
h : (i : ι) → β i →ₗ[R] γ i
x : ⨁ (i : ι), β i
i j : ι
y : β ... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
ι : Type u_1
inst✝⁵ : DecidableEq ι
R : Type u_2
inst✝⁴ : Semiring R
β : ι → Type u_3
γ : ι → Type u_4
inst✝³ : (i : ι) → AddCommMonoid (β i)
inst✝² : (i : ι) → AddCommMonoid (γ i)
inst✝¹ : (i : ι) → Module R (β i)
inst✝ : (i : ι) → Module R (γ i)
h : ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | DirectSum.lmap'_apply | [151, 1] | [161, 50] | simp only [LinearMap.comp_apply, lmap'_lof] | case h.H.h
ι : Type u_1
inst✝⁵ : DecidableEq ι
R : Type u_2
inst✝⁴ : Semiring R
β : ι → Type u_3
γ : ι → Type u_4
inst✝³ : (i : ι) → AddCommMonoid (β i)
inst✝² : (i : ι) → AddCommMonoid (γ i)
inst✝¹ : (i : ι) → Module R (β i)
inst✝ : (i : ι) → Module R (γ i)
h : (i : ι) → β i →ₗ[R] γ i
x : ⨁ (i : ι), β i
i j : ι
y : β ... | case h.H.h
ι : Type u_1
inst✝⁵ : DecidableEq ι
R : Type u_2
inst✝⁴ : Semiring R
β : ι → Type u_3
γ : ι → Type u_4
inst✝³ : (i : ι) → AddCommMonoid (β i)
inst✝² : (i : ι) → AddCommMonoid (γ i)
inst✝¹ : (i : ι) → Module R (β i)
inst✝ : (i : ι) → Module R (γ i)
h : (i : ι) → β i →ₗ[R] γ i
x : ⨁ (i : ι), β i
i j : ι
y : β ... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.H.h
ι : Type u_1
inst✝⁵ : DecidableEq ι
R : Type u_2
inst✝⁴ : Semiring R
β : ι → Type u_3
γ : ι → Type u_4
inst✝³ : (i : ι) → AddCommMonoid (β i)
inst✝² : (i : ι) → AddCommMonoid (γ i)
inst✝¹ : (i : ι) → Module R (β i)
inst✝ : (i : ι) → Module R (γ i)
... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | DirectSum.lmap'_apply | [151, 1] | [161, 50] | simp only [lof_eq_of, ← apply_eq_component] | case h.H.h
ι : Type u_1
inst✝⁵ : DecidableEq ι
R : Type u_2
inst✝⁴ : Semiring R
β : ι → Type u_3
γ : ι → Type u_4
inst✝³ : (i : ι) → AddCommMonoid (β i)
inst✝² : (i : ι) → AddCommMonoid (γ i)
inst✝¹ : (i : ι) → Module R (β i)
inst✝ : (i : ι) → Module R (γ i)
h : (i : ι) → β i →ₗ[R] γ i
x : ⨁ (i : ι), β i
i j : ι
y : β ... | case h.H.h
ι : Type u_1
inst✝⁵ : DecidableEq ι
R : Type u_2
inst✝⁴ : Semiring R
β : ι → Type u_3
γ : ι → Type u_4
inst✝³ : (i : ι) → AddCommMonoid (β i)
inst✝² : (i : ι) → AddCommMonoid (γ i)
inst✝¹ : (i : ι) → Module R (β i)
inst✝ : (i : ι) → Module R (γ i)
h : (i : ι) → β i →ₗ[R] γ i
x : ⨁ (i : ι), β i
i j : ι
y : β ... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.H.h
ι : Type u_1
inst✝⁵ : DecidableEq ι
R : Type u_2
inst✝⁴ : Semiring R
β : ι → Type u_3
γ : ι → Type u_4
inst✝³ : (i : ι) → AddCommMonoid (β i)
inst✝² : (i : ι) → AddCommMonoid (γ i)
inst✝¹ : (i : ι) → Module R (β i)
inst✝ : (i : ι) → Module R (γ i)
... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | DirectSum.lmap'_apply | [151, 1] | [161, 50] | by_cases hji : j = i | case h.H.h
ι : Type u_1
inst✝⁵ : DecidableEq ι
R : Type u_2
inst✝⁴ : Semiring R
β : ι → Type u_3
γ : ι → Type u_4
inst✝³ : (i : ι) → AddCommMonoid (β i)
inst✝² : (i : ι) → AddCommMonoid (γ i)
inst✝¹ : (i : ι) → Module R (β i)
inst✝ : (i : ι) → Module R (γ i)
h : (i : ι) → β i →ₗ[R] γ i
x : ⨁ (i : ι), β i
i j : ι
y : β ... | case pos
ι : Type u_1
inst✝⁵ : DecidableEq ι
R : Type u_2
inst✝⁴ : Semiring R
β : ι → Type u_3
γ : ι → Type u_4
inst✝³ : (i : ι) → AddCommMonoid (β i)
inst✝² : (i : ι) → AddCommMonoid (γ i)
inst✝¹ : (i : ι) → Module R (β i)
inst✝ : (i : ι) → Module R (γ i)
h : (i : ι) → β i →ₗ[R] γ i
x : ⨁ (i : ι), β i
i j : ι
y : β j
... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.H.h
ι : Type u_1
inst✝⁵ : DecidableEq ι
R : Type u_2
inst✝⁴ : Semiring R
β : ι → Type u_3
γ : ι → Type u_4
inst✝³ : (i : ι) → AddCommMonoid (β i)
inst✝² : (i : ι) → AddCommMonoid (γ i)
inst✝¹ : (i : ι) → Module R (β i)
inst✝ : (i : ι) → Module R (γ i)
... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | DirectSum.lmap'_apply | [151, 1] | [161, 50] | rw [← hji] | case pos
ι : Type u_1
inst✝⁵ : DecidableEq ι
R : Type u_2
inst✝⁴ : Semiring R
β : ι → Type u_3
γ : ι → Type u_4
inst✝³ : (i : ι) → AddCommMonoid (β i)
inst✝² : (i : ι) → AddCommMonoid (γ i)
inst✝¹ : (i : ι) → Module R (β i)
inst✝ : (i : ι) → Module R (γ i)
h : (i : ι) → β i →ₗ[R] γ i
x : ⨁ (i : ι), β i
i j : ι
y : β j
... | case pos
ι : Type u_1
inst✝⁵ : DecidableEq ι
R : Type u_2
inst✝⁴ : Semiring R
β : ι → Type u_3
γ : ι → Type u_4
inst✝³ : (i : ι) → AddCommMonoid (β i)
inst✝² : (i : ι) → AddCommMonoid (γ i)
inst✝¹ : (i : ι) → Module R (β i)
inst✝ : (i : ι) → Module R (γ i)
h : (i : ι) → β i →ₗ[R] γ i
x : ⨁ (i : ι), β i
i j : ι
y : β j
... | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
ι : Type u_1
inst✝⁵ : DecidableEq ι
R : Type u_2
inst✝⁴ : Semiring R
β : ι → Type u_3
γ : ι → Type u_4
inst✝³ : (i : ι) → AddCommMonoid (β i)
inst✝² : (i : ι) → AddCommMonoid (γ i)
inst✝¹ : (i : ι) → Module R (β i)
inst✝ : (i : ι) → Module R (γ i)
h ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | DirectSum.lmap'_apply | [151, 1] | [161, 50] | simp only [of_eq_same] | case pos
ι : Type u_1
inst✝⁵ : DecidableEq ι
R : Type u_2
inst✝⁴ : Semiring R
β : ι → Type u_3
γ : ι → Type u_4
inst✝³ : (i : ι) → AddCommMonoid (β i)
inst✝² : (i : ι) → AddCommMonoid (γ i)
inst✝¹ : (i : ι) → Module R (β i)
inst✝ : (i : ι) → Module R (γ i)
h : (i : ι) → β i →ₗ[R] γ i
x : ⨁ (i : ι), β i
i j : ι
y : β j
... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
ι : Type u_1
inst✝⁵ : DecidableEq ι
R : Type u_2
inst✝⁴ : Semiring R
β : ι → Type u_3
γ : ι → Type u_4
inst✝³ : (i : ι) → AddCommMonoid (β i)
inst✝² : (i : ι) → AddCommMonoid (γ i)
inst✝¹ : (i : ι) → Module R (β i)
inst✝ : (i : ι) → Module R (γ i)
h ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | DirectSum.lmap'_apply | [151, 1] | [161, 50] | simp only [of_eq_of_ne _ j i _ hji, map_zero] | case neg
ι : Type u_1
inst✝⁵ : DecidableEq ι
R : Type u_2
inst✝⁴ : Semiring R
β : ι → Type u_3
γ : ι → Type u_4
inst✝³ : (i : ι) → AddCommMonoid (β i)
inst✝² : (i : ι) → AddCommMonoid (γ i)
inst✝¹ : (i : ι) → Module R (β i)
inst✝ : (i : ι) → Module R (γ i)
h : (i : ι) → β i →ₗ[R] γ i
x : ⨁ (i : ι), β i
i j : ι
y : β j
... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
ι : Type u_1
inst✝⁵ : DecidableEq ι
R : Type u_2
inst✝⁴ : Semiring R
β : ι → Type u_3
γ : ι → Type u_4
inst✝³ : (i : ι) → AddCommMonoid (β i)
inst✝² : (i : ι) → AddCommMonoid (γ i)
inst✝¹ : (i : ι) → Module R (β i)
inst✝ : (i : ι) → Module R (γ i)
h ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | DirectSum.toModule_comp_lmap'_eq | [163, 1] | [170, 70] | rw [← LinearMap.comp_apply] | ι : Type u_1
inst✝⁷ : DecidableEq ι
R : Type u_2
inst✝⁶ : Semiring R
β : ι → Type u_3
γ : ι → Type u_4
δ : Type u_5
inst✝⁵ : (i : ι) → AddCommMonoid (β i)
inst✝⁴ : (i : ι) → AddCommMonoid (γ i)
inst✝³ : AddCommMonoid δ
inst✝² : (i : ι) → Module R (β i)
inst✝¹ : (i : ι) → Module R (γ i)
inst✝ : Module R δ
h : (i : ι) → ... | ι : Type u_1
inst✝⁷ : DecidableEq ι
R : Type u_2
inst✝⁶ : Semiring R
β : ι → Type u_3
γ : ι → Type u_4
δ : Type u_5
inst✝⁵ : (i : ι) → AddCommMonoid (β i)
inst✝⁴ : (i : ι) → AddCommMonoid (γ i)
inst✝³ : AddCommMonoid δ
inst✝² : (i : ι) → Module R (β i)
inst✝¹ : (i : ι) → Module R (γ i)
inst✝ : Module R δ
h : (i : ι) → ... | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
inst✝⁷ : DecidableEq ι
R : Type u_2
inst✝⁶ : Semiring R
β : ι → Type u_3
γ : ι → Type u_4
δ : Type u_5
inst✝⁵ : (i : ι) → AddCommMonoid (β i)
inst✝⁴ : (i : ι) → AddCommMonoid (γ i)
inst✝³ : AddCommMonoid δ
inst✝² : (i : ι) → Module R (β i)
inst✝¹... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | DirectSum.toModule_comp_lmap'_eq | [163, 1] | [170, 70] | apply LinearMap.congr_fun | ι : Type u_1
inst✝⁷ : DecidableEq ι
R : Type u_2
inst✝⁶ : Semiring R
β : ι → Type u_3
γ : ι → Type u_4
δ : Type u_5
inst✝⁵ : (i : ι) → AddCommMonoid (β i)
inst✝⁴ : (i : ι) → AddCommMonoid (γ i)
inst✝³ : AddCommMonoid δ
inst✝² : (i : ι) → Module R (β i)
inst✝¹ : (i : ι) → Module R (γ i)
inst✝ : Module R δ
h : (i : ι) → ... | case h
ι : Type u_1
inst✝⁷ : DecidableEq ι
R : Type u_2
inst✝⁶ : Semiring R
β : ι → Type u_3
γ : ι → Type u_4
δ : Type u_5
inst✝⁵ : (i : ι) → AddCommMonoid (β i)
inst✝⁴ : (i : ι) → AddCommMonoid (γ i)
inst✝³ : AddCommMonoid δ
inst✝² : (i : ι) → Module R (β i)
inst✝¹ : (i : ι) → Module R (γ i)
inst✝ : Module R δ
h : (i ... | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
inst✝⁷ : DecidableEq ι
R : Type u_2
inst✝⁶ : Semiring R
β : ι → Type u_3
γ : ι → Type u_4
δ : Type u_5
inst✝⁵ : (i : ι) → AddCommMonoid (β i)
inst✝⁴ : (i : ι) → AddCommMonoid (γ i)
inst✝³ : AddCommMonoid δ
inst✝² : (i : ι) → Module R (β i)
inst✝¹... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | DirectSum.toModule_comp_lmap'_eq | [163, 1] | [170, 70] | ext i y | case h
ι : Type u_1
inst✝⁷ : DecidableEq ι
R : Type u_2
inst✝⁶ : Semiring R
β : ι → Type u_3
γ : ι → Type u_4
δ : Type u_5
inst✝⁵ : (i : ι) → AddCommMonoid (β i)
inst✝⁴ : (i : ι) → AddCommMonoid (γ i)
inst✝³ : AddCommMonoid δ
inst✝² : (i : ι) → Module R (β i)
inst✝¹ : (i : ι) → Module R (γ i)
inst✝ : Module R δ
h : (i ... | case h.H.h
ι : Type u_1
inst✝⁷ : DecidableEq ι
R : Type u_2
inst✝⁶ : Semiring R
β : ι → Type u_3
γ : ι → Type u_4
δ : Type u_5
inst✝⁵ : (i : ι) → AddCommMonoid (β i)
inst✝⁴ : (i : ι) → AddCommMonoid (γ i)
inst✝³ : AddCommMonoid δ
inst✝² : (i : ι) → Module R (β i)
inst✝¹ : (i : ι) → Module R (γ i)
inst✝ : Module R δ
h :... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
ι : Type u_1
inst✝⁷ : DecidableEq ι
R : Type u_2
inst✝⁶ : Semiring R
β : ι → Type u_3
γ : ι → Type u_4
δ : Type u_5
inst✝⁵ : (i : ι) → AddCommMonoid (β i)
inst✝⁴ : (i : ι) → AddCommMonoid (γ i)
inst✝³ : AddCommMonoid δ
inst✝² : (i : ι) → Module R (β i)... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | DirectSum.toModule_comp_lmap'_eq | [163, 1] | [170, 70] | simp only [LinearMap.coe_comp, comp_apply, lmap'_lof, toModule_lof] | case h.H.h
ι : Type u_1
inst✝⁷ : DecidableEq ι
R : Type u_2
inst✝⁶ : Semiring R
β : ι → Type u_3
γ : ι → Type u_4
δ : Type u_5
inst✝⁵ : (i : ι) → AddCommMonoid (β i)
inst✝⁴ : (i : ι) → AddCommMonoid (γ i)
inst✝³ : AddCommMonoid δ
inst✝² : (i : ι) → Module R (β i)
inst✝¹ : (i : ι) → Module R (γ i)
inst✝ : Module R δ
h :... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.H.h
ι : Type u_1
inst✝⁷ : DecidableEq ι
R : Type u_2
inst✝⁶ : Semiring R
β : ι → Type u_3
γ : ι → Type u_4
δ : Type u_5
inst✝⁵ : (i : ι) → AddCommMonoid (β i)
inst✝⁴ : (i : ι) → AddCommMonoid (γ i)
inst✝³ : AddCommMonoid δ
inst✝² : (i : ι) → Module R (... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | DirectSum.map'_apply | [172, 1] | [189, 49] | let f : (⨁ i, β i) →+ γ i :=
{ toFun := fun x => map' h x i
map_zero' := by simp only [map_zero, zero_apply]
map_add' := by simp only [map_add, add_apply, eq_self_iff_true, forall_const] } | ι : Type u_1
inst✝³ : DecidableEq ι
R : Type u_2
inst✝² : Semiring R
β : ι → Type u_3
γ : ι → Type u_4
inst✝¹ : (i : ι) → AddCommMonoid (β i)
inst✝ : (i : ι) → AddCommMonoid (γ i)
h : (i : ι) → β i →+ γ i
x : ⨁ (i : ι), β i
i : ι
⊢ ((map' h) x) i = (h i) (x i) | ι : Type u_1
inst✝³ : DecidableEq ι
R : Type u_2
inst✝² : Semiring R
β : ι → Type u_3
γ : ι → Type u_4
inst✝¹ : (i : ι) → AddCommMonoid (β i)
inst✝ : (i : ι) → AddCommMonoid (γ i)
h : (i : ι) → β i →+ γ i
x : ⨁ (i : ι), β i
i : ι
f : (⨁ (i : ι), β i) →+ γ i := { toFun := fun x => ((map' h) x) i, map_zero' := ⋯, map_add... | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
inst✝³ : DecidableEq ι
R : Type u_2
inst✝² : Semiring R
β : ι → Type u_3
γ : ι → Type u_4
inst✝¹ : (i : ι) → AddCommMonoid (β i)
inst✝ : (i : ι) → AddCommMonoid (γ i)
h : (i : ι) → β i →+ γ i
x : ⨁ (i : ι), β i
i : ι
⊢ ((map' h) x) i = (h i) (x i... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | DirectSum.map'_apply | [172, 1] | [189, 49] | let g : (⨁ i, β i) →+ γ i :=
{ toFun := fun y => h i (y i)
map_zero' := by simp only [zero_apply, map_zero]
map_add' := by simp only [add_apply, map_add, eq_self_iff_true, forall_const] } | ι : Type u_1
inst✝³ : DecidableEq ι
R : Type u_2
inst✝² : Semiring R
β : ι → Type u_3
γ : ι → Type u_4
inst✝¹ : (i : ι) → AddCommMonoid (β i)
inst✝ : (i : ι) → AddCommMonoid (γ i)
h : (i : ι) → β i →+ γ i
x : ⨁ (i : ι), β i
i : ι
f : (⨁ (i : ι), β i) →+ γ i := { toFun := fun x => ((map' h) x) i, map_zero' := ⋯, map_add... | ι : Type u_1
inst✝³ : DecidableEq ι
R : Type u_2
inst✝² : Semiring R
β : ι → Type u_3
γ : ι → Type u_4
inst✝¹ : (i : ι) → AddCommMonoid (β i)
inst✝ : (i : ι) → AddCommMonoid (γ i)
h : (i : ι) → β i →+ γ i
x : ⨁ (i : ι), β i
i : ι
f : (⨁ (i : ι), β i) →+ γ i := { toFun := fun x => ((map' h) x) i, map_zero' := ⋯, map_add... | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
inst✝³ : DecidableEq ι
R : Type u_2
inst✝² : Semiring R
β : ι → Type u_3
γ : ι → Type u_4
inst✝¹ : (i : ι) → AddCommMonoid (β i)
inst✝ : (i : ι) → AddCommMonoid (γ i)
h : (i : ι) → β i →+ γ i
x : ⨁ (i : ι), β i
i : ι
f : (⨁ (i : ι), β i) →+ γ i :... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | DirectSum.map'_apply | [172, 1] | [189, 49] | change f x = g x | ι : Type u_1
inst✝³ : DecidableEq ι
R : Type u_2
inst✝² : Semiring R
β : ι → Type u_3
γ : ι → Type u_4
inst✝¹ : (i : ι) → AddCommMonoid (β i)
inst✝ : (i : ι) → AddCommMonoid (γ i)
h : (i : ι) → β i →+ γ i
x : ⨁ (i : ι), β i
i : ι
f : (⨁ (i : ι), β i) →+ γ i := { toFun := fun x => ((map' h) x) i, map_zero' := ⋯, map_add... | ι : Type u_1
inst✝³ : DecidableEq ι
R : Type u_2
inst✝² : Semiring R
β : ι → Type u_3
γ : ι → Type u_4
inst✝¹ : (i : ι) → AddCommMonoid (β i)
inst✝ : (i : ι) → AddCommMonoid (γ i)
h : (i : ι) → β i →+ γ i
x : ⨁ (i : ι), β i
i : ι
f : (⨁ (i : ι), β i) →+ γ i := { toFun := fun x => ((map' h) x) i, map_zero' := ⋯, map_add... | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
inst✝³ : DecidableEq ι
R : Type u_2
inst✝² : Semiring R
β : ι → Type u_3
γ : ι → Type u_4
inst✝¹ : (i : ι) → AddCommMonoid (β i)
inst✝ : (i : ι) → AddCommMonoid (γ i)
h : (i : ι) → β i →+ γ i
x : ⨁ (i : ι), β i
i : ι
f : (⨁ (i : ι), β i) →+ γ i :... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | DirectSum.map'_apply | [172, 1] | [189, 49] | apply DFunLike.congr_fun | ι : Type u_1
inst✝³ : DecidableEq ι
R : Type u_2
inst✝² : Semiring R
β : ι → Type u_3
γ : ι → Type u_4
inst✝¹ : (i : ι) → AddCommMonoid (β i)
inst✝ : (i : ι) → AddCommMonoid (γ i)
h : (i : ι) → β i →+ γ i
x : ⨁ (i : ι), β i
i : ι
f : (⨁ (i : ι), β i) →+ γ i := { toFun := fun x => ((map' h) x) i, map_zero' := ⋯, map_add... | case h₁
ι : Type u_1
inst✝³ : DecidableEq ι
R : Type u_2
inst✝² : Semiring R
β : ι → Type u_3
γ : ι → Type u_4
inst✝¹ : (i : ι) → AddCommMonoid (β i)
inst✝ : (i : ι) → AddCommMonoid (γ i)
h : (i : ι) → β i →+ γ i
x : ⨁ (i : ι), β i
i : ι
f : (⨁ (i : ι), β i) →+ γ i := { toFun := fun x => ((map' h) x) i, map_zero' := ⋯,... | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
inst✝³ : DecidableEq ι
R : Type u_2
inst✝² : Semiring R
β : ι → Type u_3
γ : ι → Type u_4
inst✝¹ : (i : ι) → AddCommMonoid (β i)
inst✝ : (i : ι) → AddCommMonoid (γ i)
h : (i : ι) → β i →+ γ i
x : ⨁ (i : ι), β i
i : ι
f : (⨁ (i : ι), β i) →+ γ i :... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | DirectSum.map'_apply | [172, 1] | [189, 49] | ext j y | case h₁
ι : Type u_1
inst✝³ : DecidableEq ι
R : Type u_2
inst✝² : Semiring R
β : ι → Type u_3
γ : ι → Type u_4
inst✝¹ : (i : ι) → AddCommMonoid (β i)
inst✝ : (i : ι) → AddCommMonoid (γ i)
h : (i : ι) → β i →+ γ i
x : ⨁ (i : ι), β i
i : ι
f : (⨁ (i : ι), β i) →+ γ i := { toFun := fun x => ((map' h) x) i, map_zero' := ⋯,... | case h₁.H.h
ι : Type u_1
inst✝³ : DecidableEq ι
R : Type u_2
inst✝² : Semiring R
β : ι → Type u_3
γ : ι → Type u_4
inst✝¹ : (i : ι) → AddCommMonoid (β i)
inst✝ : (i : ι) → AddCommMonoid (γ i)
h : (i : ι) → β i →+ γ i
x : ⨁ (i : ι), β i
i : ι
f : (⨁ (i : ι), β i) →+ γ i := { toFun := fun x => ((map' h) x) i, map_zero' :... | Please generate a tactic in lean4 to solve the state.
STATE:
case h₁
ι : Type u_1
inst✝³ : DecidableEq ι
R : Type u_2
inst✝² : Semiring R
β : ι → Type u_3
γ : ι → Type u_4
inst✝¹ : (i : ι) → AddCommMonoid (β i)
inst✝ : (i : ι) → AddCommMonoid (γ i)
h : (i : ι) → β i →+ γ i
x : ⨁ (i : ι), β i
i : ι
f : (⨁ (i : ι), β i) ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | DirectSum.map'_apply | [172, 1] | [189, 49] | simp only [f, g, AddMonoidHom.coe_comp, AddMonoidHom.coe_mk, ZeroHom.coe_mk, comp_apply,
map'_of] | case h₁.H.h
ι : Type u_1
inst✝³ : DecidableEq ι
R : Type u_2
inst✝² : Semiring R
β : ι → Type u_3
γ : ι → Type u_4
inst✝¹ : (i : ι) → AddCommMonoid (β i)
inst✝ : (i : ι) → AddCommMonoid (γ i)
h : (i : ι) → β i →+ γ i
x : ⨁ (i : ι), β i
i : ι
f : (⨁ (i : ι), β i) →+ γ i := { toFun := fun x => ((map' h) x) i, map_zero' :... | case h₁.H.h
ι : Type u_1
inst✝³ : DecidableEq ι
R : Type u_2
inst✝² : Semiring R
β : ι → Type u_3
γ : ι → Type u_4
inst✝¹ : (i : ι) → AddCommMonoid (β i)
inst✝ : (i : ι) → AddCommMonoid (γ i)
h : (i : ι) → β i →+ γ i
x : ⨁ (i : ι), β i
i : ι
f : (⨁ (i : ι), β i) →+ γ i := { toFun := fun x => ((map' h) x) i, map_zero' :... | Please generate a tactic in lean4 to solve the state.
STATE:
case h₁.H.h
ι : Type u_1
inst✝³ : DecidableEq ι
R : Type u_2
inst✝² : Semiring R
β : ι → Type u_3
γ : ι → Type u_4
inst✝¹ : (i : ι) → AddCommMonoid (β i)
inst✝ : (i : ι) → AddCommMonoid (γ i)
h : (i : ι) → β i →+ γ i
x : ⨁ (i : ι), β i
i : ι
f : (⨁ (i : ι), β... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | DirectSum.map'_apply | [172, 1] | [189, 49] | by_cases hj : j = i | case h₁.H.h
ι : Type u_1
inst✝³ : DecidableEq ι
R : Type u_2
inst✝² : Semiring R
β : ι → Type u_3
γ : ι → Type u_4
inst✝¹ : (i : ι) → AddCommMonoid (β i)
inst✝ : (i : ι) → AddCommMonoid (γ i)
h : (i : ι) → β i →+ γ i
x : ⨁ (i : ι), β i
i : ι
f : (⨁ (i : ι), β i) →+ γ i := { toFun := fun x => ((map' h) x) i, map_zero' :... | case pos
ι : Type u_1
inst✝³ : DecidableEq ι
R : Type u_2
inst✝² : Semiring R
β : ι → Type u_3
γ : ι → Type u_4
inst✝¹ : (i : ι) → AddCommMonoid (β i)
inst✝ : (i : ι) → AddCommMonoid (γ i)
h : (i : ι) → β i →+ γ i
x : ⨁ (i : ι), β i
i : ι
f : (⨁ (i : ι), β i) →+ γ i := { toFun := fun x => ((map' h) x) i, map_zero' := ⋯... | Please generate a tactic in lean4 to solve the state.
STATE:
case h₁.H.h
ι : Type u_1
inst✝³ : DecidableEq ι
R : Type u_2
inst✝² : Semiring R
β : ι → Type u_3
γ : ι → Type u_4
inst✝¹ : (i : ι) → AddCommMonoid (β i)
inst✝ : (i : ι) → AddCommMonoid (γ i)
h : (i : ι) → β i →+ γ i
x : ⨁ (i : ι), β i
i : ι
f : (⨁ (i : ι), β... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | DirectSum.map'_apply | [172, 1] | [189, 49] | simp only [map_zero, zero_apply] | ι : Type u_1
inst✝³ : DecidableEq ι
R : Type u_2
inst✝² : Semiring R
β : ι → Type u_3
γ : ι → Type u_4
inst✝¹ : (i : ι) → AddCommMonoid (β i)
inst✝ : (i : ι) → AddCommMonoid (γ i)
h : (i : ι) → β i →+ γ i
x : ⨁ (i : ι), β i
i : ι
⊢ (fun x => ((map' h) x) i) 0 = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
inst✝³ : DecidableEq ι
R : Type u_2
inst✝² : Semiring R
β : ι → Type u_3
γ : ι → Type u_4
inst✝¹ : (i : ι) → AddCommMonoid (β i)
inst✝ : (i : ι) → AddCommMonoid (γ i)
h : (i : ι) → β i →+ γ i
x : ⨁ (i : ι), β i
i : ι
⊢ (fun x => ((map' h) x) i) 0... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | DirectSum.map'_apply | [172, 1] | [189, 49] | simp only [map_add, add_apply, eq_self_iff_true, forall_const] | ι : Type u_1
inst✝³ : DecidableEq ι
R : Type u_2
inst✝² : Semiring R
β : ι → Type u_3
γ : ι → Type u_4
inst✝¹ : (i : ι) → AddCommMonoid (β i)
inst✝ : (i : ι) → AddCommMonoid (γ i)
h : (i : ι) → β i →+ γ i
x : ⨁ (i : ι), β i
i : ι
⊢ ∀ (x y : ⨁ (i : ι), β i),
{ toFun := fun x => ((map' h) x) i, map_zero' := ⋯ }.toFun... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
inst✝³ : DecidableEq ι
R : Type u_2
inst✝² : Semiring R
β : ι → Type u_3
γ : ι → Type u_4
inst✝¹ : (i : ι) → AddCommMonoid (β i)
inst✝ : (i : ι) → AddCommMonoid (γ i)
h : (i : ι) → β i →+ γ i
x : ⨁ (i : ι), β i
i : ι
⊢ ∀ (x y : ⨁ (i : ι), β i),
... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | DirectSum.map'_apply | [172, 1] | [189, 49] | simp only [zero_apply, map_zero] | ι : Type u_1
inst✝³ : DecidableEq ι
R : Type u_2
inst✝² : Semiring R
β : ι → Type u_3
γ : ι → Type u_4
inst✝¹ : (i : ι) → AddCommMonoid (β i)
inst✝ : (i : ι) → AddCommMonoid (γ i)
h : (i : ι) → β i →+ γ i
x : ⨁ (i : ι), β i
i : ι
f : (⨁ (i : ι), β i) →+ γ i := { toFun := fun x => ((map' h) x) i, map_zero' := ⋯, map_add... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
inst✝³ : DecidableEq ι
R : Type u_2
inst✝² : Semiring R
β : ι → Type u_3
γ : ι → Type u_4
inst✝¹ : (i : ι) → AddCommMonoid (β i)
inst✝ : (i : ι) → AddCommMonoid (γ i)
h : (i : ι) → β i →+ γ i
x : ⨁ (i : ι), β i
i : ι
f : (⨁ (i : ι), β i) →+ γ i :... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | DirectSum.map'_apply | [172, 1] | [189, 49] | simp only [add_apply, map_add, eq_self_iff_true, forall_const] | ι : Type u_1
inst✝³ : DecidableEq ι
R : Type u_2
inst✝² : Semiring R
β : ι → Type u_3
γ : ι → Type u_4
inst✝¹ : (i : ι) → AddCommMonoid (β i)
inst✝ : (i : ι) → AddCommMonoid (γ i)
h : (i : ι) → β i →+ γ i
x : ⨁ (i : ι), β i
i : ι
f : (⨁ (i : ι), β i) →+ γ i := { toFun := fun x => ((map' h) x) i, map_zero' := ⋯, map_add... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
inst✝³ : DecidableEq ι
R : Type u_2
inst✝² : Semiring R
β : ι → Type u_3
γ : ι → Type u_4
inst✝¹ : (i : ι) → AddCommMonoid (β i)
inst✝ : (i : ι) → AddCommMonoid (γ i)
h : (i : ι) → β i →+ γ i
x : ⨁ (i : ι), β i
i : ι
f : (⨁ (i : ι), β i) →+ γ i :... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | DirectSum.map'_apply | [172, 1] | [189, 49] | rw [← hj] | case pos
ι : Type u_1
inst✝³ : DecidableEq ι
R : Type u_2
inst✝² : Semiring R
β : ι → Type u_3
γ : ι → Type u_4
inst✝¹ : (i : ι) → AddCommMonoid (β i)
inst✝ : (i : ι) → AddCommMonoid (γ i)
h : (i : ι) → β i →+ γ i
x : ⨁ (i : ι), β i
i : ι
f : (⨁ (i : ι), β i) →+ γ i := { toFun := fun x => ((map' h) x) i, map_zero' := ⋯... | case pos
ι : Type u_1
inst✝³ : DecidableEq ι
R : Type u_2
inst✝² : Semiring R
β : ι → Type u_3
γ : ι → Type u_4
inst✝¹ : (i : ι) → AddCommMonoid (β i)
inst✝ : (i : ι) → AddCommMonoid (γ i)
h : (i : ι) → β i →+ γ i
x : ⨁ (i : ι), β i
i : ι
f : (⨁ (i : ι), β i) →+ γ i := { toFun := fun x => ((map' h) x) i, map_zero' := ⋯... | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
ι : Type u_1
inst✝³ : DecidableEq ι
R : Type u_2
inst✝² : Semiring R
β : ι → Type u_3
γ : ι → Type u_4
inst✝¹ : (i : ι) → AddCommMonoid (β i)
inst✝ : (i : ι) → AddCommMonoid (γ i)
h : (i : ι) → β i →+ γ i
x : ⨁ (i : ι), β i
i : ι
f : (⨁ (i : ι), β i)... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | DirectSum.map'_apply | [172, 1] | [189, 49] | simp only [of_eq_same] | case pos
ι : Type u_1
inst✝³ : DecidableEq ι
R : Type u_2
inst✝² : Semiring R
β : ι → Type u_3
γ : ι → Type u_4
inst✝¹ : (i : ι) → AddCommMonoid (β i)
inst✝ : (i : ι) → AddCommMonoid (γ i)
h : (i : ι) → β i →+ γ i
x : ⨁ (i : ι), β i
i : ι
f : (⨁ (i : ι), β i) →+ γ i := { toFun := fun x => ((map' h) x) i, map_zero' := ⋯... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
ι : Type u_1
inst✝³ : DecidableEq ι
R : Type u_2
inst✝² : Semiring R
β : ι → Type u_3
γ : ι → Type u_4
inst✝¹ : (i : ι) → AddCommMonoid (β i)
inst✝ : (i : ι) → AddCommMonoid (γ i)
h : (i : ι) → β i →+ γ i
x : ⨁ (i : ι), β i
i : ι
f : (⨁ (i : ι), β i)... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | DirectSum.map'_apply | [172, 1] | [189, 49] | simp only [of_eq_of_ne _ j i _ hj, map_zero] | case neg
ι : Type u_1
inst✝³ : DecidableEq ι
R : Type u_2
inst✝² : Semiring R
β : ι → Type u_3
γ : ι → Type u_4
inst✝¹ : (i : ι) → AddCommMonoid (β i)
inst✝ : (i : ι) → AddCommMonoid (γ i)
h : (i : ι) → β i →+ γ i
x : ⨁ (i : ι), β i
i : ι
f : (⨁ (i : ι), β i) →+ γ i := { toFun := fun x => ((map' h) x) i, map_zero' := ⋯... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
ι : Type u_1
inst✝³ : DecidableEq ι
R : Type u_2
inst✝² : Semiring R
β : ι → Type u_3
γ : ι → Type u_4
inst✝¹ : (i : ι) → AddCommMonoid (β i)
inst✝ : (i : ι) → AddCommMonoid (γ i)
h : (i : ι) → β i →+ γ i
x : ⨁ (i : ι), β i
i : ι
f : (⨁ (i : ι), β i)... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | DirectSum.mk_eq_sum' | [196, 1] | [205, 92] | apply DFinsupp.ext | ι : Type u_1
inst✝² : DecidableEq ι
R : Type u_2
inst✝¹ : Semiring R
β : ι → Type u_3
inst✝ : (i : ι) → AddCommMonoid (β i)
s : Finset ι
f : (i : ι) → β i
⊢ ((mk β s) fun i => f ↑i) = ∑ i ∈ s, (of β i) (f i) | case h
ι : Type u_1
inst✝² : DecidableEq ι
R : Type u_2
inst✝¹ : Semiring R
β : ι → Type u_3
inst✝ : (i : ι) → AddCommMonoid (β i)
s : Finset ι
f : (i : ι) → β i
⊢ ∀ (i : ι), ((mk β s) fun i => f ↑i) i = (∑ i ∈ s, (of β i) (f i)) i | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
inst✝² : DecidableEq ι
R : Type u_2
inst✝¹ : Semiring R
β : ι → Type u_3
inst✝ : (i : ι) → AddCommMonoid (β i)
s : Finset ι
f : (i : ι) → β i
⊢ ((mk β s) fun i => f ↑i) = ∑ i ∈ s, (of β i) (f i)
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | DirectSum.mk_eq_sum' | [196, 1] | [205, 92] | intro i | case h
ι : Type u_1
inst✝² : DecidableEq ι
R : Type u_2
inst✝¹ : Semiring R
β : ι → Type u_3
inst✝ : (i : ι) → AddCommMonoid (β i)
s : Finset ι
f : (i : ι) → β i
⊢ ∀ (i : ι), ((mk β s) fun i => f ↑i) i = (∑ i ∈ s, (of β i) (f i)) i | case h
ι : Type u_1
inst✝² : DecidableEq ι
R : Type u_2
inst✝¹ : Semiring R
β : ι → Type u_3
inst✝ : (i : ι) → AddCommMonoid (β i)
s : Finset ι
f : (i : ι) → β i
i : ι
⊢ ((mk β s) fun i => f ↑i) i = (∑ i ∈ s, (of β i) (f i)) i | Please generate a tactic in lean4 to solve the state.
STATE:
case h
ι : Type u_1
inst✝² : DecidableEq ι
R : Type u_2
inst✝¹ : Semiring R
β : ι → Type u_3
inst✝ : (i : ι) → AddCommMonoid (β i)
s : Finset ι
f : (i : ι) → β i
⊢ ∀ (i : ι), ((mk β s) fun i => f ↑i) i = (∑ i ∈ s, (of β i) (f i)) i
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | DirectSum.mk_eq_sum' | [196, 1] | [205, 92] | convert mk_apply s _ i | case h
ι : Type u_1
inst✝² : DecidableEq ι
R : Type u_2
inst✝¹ : Semiring R
β : ι → Type u_3
inst✝ : (i : ι) → AddCommMonoid (β i)
s : Finset ι
f : (i : ι) → β i
i : ι
⊢ ((mk β s) fun i => f ↑i) i = (∑ i ∈ s, (of β i) (f i)) i | case h.e'_3
ι : Type u_1
inst✝² : DecidableEq ι
R : Type u_2
inst✝¹ : Semiring R
β : ι → Type u_3
inst✝ : (i : ι) → AddCommMonoid (β i)
s : Finset ι
f : (i : ι) → β i
i : ι
⊢ (∑ i ∈ s, (of β i) (f i)) i = if h : i ∈ s then f ↑⟨i, h⟩ else 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case h
ι : Type u_1
inst✝² : DecidableEq ι
R : Type u_2
inst✝¹ : Semiring R
β : ι → Type u_3
inst✝ : (i : ι) → AddCommMonoid (β i)
s : Finset ι
f : (i : ι) → β i
i : ι
⊢ ((mk β s) fun i => f ↑i) i = (∑ i ∈ s, (of β i) (f i)) i
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | DirectSum.mk_eq_sum' | [196, 1] | [205, 92] | rw [DFinsupp.finset_sum_apply] | case h.e'_3
ι : Type u_1
inst✝² : DecidableEq ι
R : Type u_2
inst✝¹ : Semiring R
β : ι → Type u_3
inst✝ : (i : ι) → AddCommMonoid (β i)
s : Finset ι
f : (i : ι) → β i
i : ι
⊢ (∑ i ∈ s, (of β i) (f i)) i = if h : i ∈ s then f ↑⟨i, h⟩ else 0 | case h.e'_3
ι : Type u_1
inst✝² : DecidableEq ι
R : Type u_2
inst✝¹ : Semiring R
β : ι → Type u_3
inst✝ : (i : ι) → AddCommMonoid (β i)
s : Finset ι
f : (i : ι) → β i
i : ι
⊢ ∑ a ∈ s, ((of β a) (f a)) i = if h : i ∈ s then f ↑⟨i, h⟩ else 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_3
ι : Type u_1
inst✝² : DecidableEq ι
R : Type u_2
inst✝¹ : Semiring R
β : ι → Type u_3
inst✝ : (i : ι) → AddCommMonoid (β i)
s : Finset ι
f : (i : ι) → β i
i : ι
⊢ (∑ i ∈ s, (of β i) (f i)) i = if h : i ∈ s then f ↑⟨i, h⟩ else 0
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | DirectSum.mk_eq_sum' | [196, 1] | [205, 92] | split_ifs with hi | case h.e'_3
ι : Type u_1
inst✝² : DecidableEq ι
R : Type u_2
inst✝¹ : Semiring R
β : ι → Type u_3
inst✝ : (i : ι) → AddCommMonoid (β i)
s : Finset ι
f : (i : ι) → β i
i : ι
⊢ ∑ a ∈ s, ((of β a) (f a)) i = if h : i ∈ s then f ↑⟨i, h⟩ else 0 | case pos
ι : Type u_1
inst✝² : DecidableEq ι
R : Type u_2
inst✝¹ : Semiring R
β : ι → Type u_3
inst✝ : (i : ι) → AddCommMonoid (β i)
s : Finset ι
f : (i : ι) → β i
i : ι
hi : i ∈ s
⊢ ∑ a ∈ s, ((of β a) (f a)) i = f ↑⟨i, hi⟩
case neg
ι : Type u_1
inst✝² : DecidableEq ι
R : Type u_2
inst✝¹ : Semiring R
β : ι → Type u_3
... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_3
ι : Type u_1
inst✝² : DecidableEq ι
R : Type u_2
inst✝¹ : Semiring R
β : ι → Type u_3
inst✝ : (i : ι) → AddCommMonoid (β i)
s : Finset ι
f : (i : ι) → β i
i : ι
⊢ ∑ a ∈ s, ((of β a) (f a)) i = if h : i ∈ s then f ↑⟨i, h⟩ else 0
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | DirectSum.mk_eq_sum' | [196, 1] | [205, 92] | rw [Finset.sum_eq_single_of_mem i hi (fun j _ hij => of_eq_of_ne _ _ _ _ hij),
← lof_eq_of ℕ, lof_apply] | case pos
ι : Type u_1
inst✝² : DecidableEq ι
R : Type u_2
inst✝¹ : Semiring R
β : ι → Type u_3
inst✝ : (i : ι) → AddCommMonoid (β i)
s : Finset ι
f : (i : ι) → β i
i : ι
hi : i ∈ s
⊢ ∑ a ∈ s, ((of β a) (f a)) i = f ↑⟨i, hi⟩ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
ι : Type u_1
inst✝² : DecidableEq ι
R : Type u_2
inst✝¹ : Semiring R
β : ι → Type u_3
inst✝ : (i : ι) → AddCommMonoid (β i)
s : Finset ι
f : (i : ι) → β i
i : ι
hi : i ∈ s
⊢ ∑ a ∈ s, ((of β a) (f a)) i = f ↑⟨i, hi⟩
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | DirectSum.mk_eq_sum' | [196, 1] | [205, 92] | exact Finset.sum_eq_zero (fun j hj => of_eq_of_ne _ _ _ _ (ne_of_mem_of_not_mem hj hi)) | case neg
ι : Type u_1
inst✝² : DecidableEq ι
R : Type u_2
inst✝¹ : Semiring R
β : ι → Type u_3
inst✝ : (i : ι) → AddCommMonoid (β i)
s : Finset ι
f : (i : ι) → β i
i : ι
hi : i ∉ s
⊢ ∑ a ∈ s, ((of β a) (f a)) i = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
ι : Type u_1
inst✝² : DecidableEq ι
R : Type u_2
inst✝¹ : Semiring R
β : ι → Type u_3
inst✝ : (i : ι) → AddCommMonoid (β i)
s : Finset ι
f : (i : ι) → β i
i : ι
hi : i ∉ s
⊢ ∑ a ∈ s, ((of β a) (f a)) i = 0
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | DFinsupp.mk_eq_sum | [208, 1] | [216, 70] | ext i | ι : Type u_1
inst✝² : DecidableEq ι
R : Type u_2
inst✝¹ : Semiring R
β : ι → Type u_3
inst✝ : (i : ι) → AddCommMonoid (β i)
s : Finset ι
f : (i : ι) → β i
⊢ (DFinsupp.mk s fun i => f ↑i) = ∑ i ∈ s, (of β i) (f i) | case h
ι : Type u_1
inst✝² : DecidableEq ι
R : Type u_2
inst✝¹ : Semiring R
β : ι → Type u_3
inst✝ : (i : ι) → AddCommMonoid (β i)
s : Finset ι
f : (i : ι) → β i
i : ι
⊢ (DFinsupp.mk s fun i => f ↑i) i = (∑ i ∈ s, (of β i) (f i)) i | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
inst✝² : DecidableEq ι
R : Type u_2
inst✝¹ : Semiring R
β : ι → Type u_3
inst✝ : (i : ι) → AddCommMonoid (β i)
s : Finset ι
f : (i : ι) → β i
⊢ (DFinsupp.mk s fun i => f ↑i) = ∑ i ∈ s, (of β i) (f i)
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | DFinsupp.mk_eq_sum | [208, 1] | [216, 70] | simp only [DFinsupp.mk_apply, DFinsupp.finset_sum_apply] | case h
ι : Type u_1
inst✝² : DecidableEq ι
R : Type u_2
inst✝¹ : Semiring R
β : ι → Type u_3
inst✝ : (i : ι) → AddCommMonoid (β i)
s : Finset ι
f : (i : ι) → β i
i : ι
⊢ (DFinsupp.mk s fun i => f ↑i) i = (∑ i ∈ s, (of β i) (f i)) i | case h
ι : Type u_1
inst✝² : DecidableEq ι
R : Type u_2
inst✝¹ : Semiring R
β : ι → Type u_3
inst✝ : (i : ι) → AddCommMonoid (β i)
s : Finset ι
f : (i : ι) → β i
i : ι
⊢ (if h : i ∈ s then f i else 0) = (∑ i ∈ s, (of β i) (f i)) i | Please generate a tactic in lean4 to solve the state.
STATE:
case h
ι : Type u_1
inst✝² : DecidableEq ι
R : Type u_2
inst✝¹ : Semiring R
β : ι → Type u_3
inst✝ : (i : ι) → AddCommMonoid (β i)
s : Finset ι
f : (i : ι) → β i
i : ι
⊢ (DFinsupp.mk s fun i => f ↑i) i = (∑ i ∈ s, (of β i) (f i)) i
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | DFinsupp.mk_eq_sum | [208, 1] | [216, 70] | split_ifs with hi | case h
ι : Type u_1
inst✝² : DecidableEq ι
R : Type u_2
inst✝¹ : Semiring R
β : ι → Type u_3
inst✝ : (i : ι) → AddCommMonoid (β i)
s : Finset ι
f : (i : ι) → β i
i : ι
⊢ (if h : i ∈ s then f i else 0) = (∑ i ∈ s, (of β i) (f i)) i | case pos
ι : Type u_1
inst✝² : DecidableEq ι
R : Type u_2
inst✝¹ : Semiring R
β : ι → Type u_3
inst✝ : (i : ι) → AddCommMonoid (β i)
s : Finset ι
f : (i : ι) → β i
i : ι
hi : i ∈ s
⊢ f i = (∑ i ∈ s, (of β i) (f i)) i
case neg
ι : Type u_1
inst✝² : DecidableEq ι
R : Type u_2
inst✝¹ : Semiring R
β : ι → Type u_3
inst✝ :... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
ι : Type u_1
inst✝² : DecidableEq ι
R : Type u_2
inst✝¹ : Semiring R
β : ι → Type u_3
inst✝ : (i : ι) → AddCommMonoid (β i)
s : Finset ι
f : (i : ι) → β i
i : ι
⊢ (if h : i ∈ s then f i else 0) = (∑ i ∈ s, (of β i) (f i)) i
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | DFinsupp.mk_eq_sum | [208, 1] | [216, 70] | rw [DFinsupp.finset_sum_apply, Finset.sum_eq_single_of_mem i hi
(fun j _ hij => of_eq_of_ne _ _ _ _ hij), of_eq_same] | case pos
ι : Type u_1
inst✝² : DecidableEq ι
R : Type u_2
inst✝¹ : Semiring R
β : ι → Type u_3
inst✝ : (i : ι) → AddCommMonoid (β i)
s : Finset ι
f : (i : ι) → β i
i : ι
hi : i ∈ s
⊢ f i = (∑ i ∈ s, (of β i) (f i)) i | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
ι : Type u_1
inst✝² : DecidableEq ι
R : Type u_2
inst✝¹ : Semiring R
β : ι → Type u_3
inst✝ : (i : ι) → AddCommMonoid (β i)
s : Finset ι
f : (i : ι) → β i
i : ι
hi : i ∈ s
⊢ f i = (∑ i ∈ s, (of β i) (f i)) i
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | DFinsupp.mk_eq_sum | [208, 1] | [216, 70] | rw [DFinsupp.finset_sum_apply, Finset.sum_eq_zero
(fun j hj => of_eq_of_ne _ _ _ _ (ne_of_mem_of_not_mem hj hi))] | case neg
ι : Type u_1
inst✝² : DecidableEq ι
R : Type u_2
inst✝¹ : Semiring R
β : ι → Type u_3
inst✝ : (i : ι) → AddCommMonoid (β i)
s : Finset ι
f : (i : ι) → β i
i : ι
hi : i ∉ s
⊢ 0 = (∑ i ∈ s, (of β i) (f i)) i | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
ι : Type u_1
inst✝² : DecidableEq ι
R : Type u_2
inst✝¹ : Semiring R
β : ι → Type u_3
inst✝ : (i : ι) → AddCommMonoid (β i)
s : Finset ι
f : (i : ι) → β i
i : ι
hi : i ∉ s
⊢ 0 = (∑ i ∈ s, (of β i) (f i)) i
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | DirectSum.mk_eq_sum | [218, 1] | [229, 70] | apply DFinsupp.ext | ι : Type u_1
inst✝² : DecidableEq ι
R : Type u_2
inst✝¹ : Semiring R
β : ι → Type u_3
inst✝ : (i : ι) → AddCommMonoid (β i)
s : Finset ι
x : (i : { x // x ∈ s }) → β ↑i
⊢ (mk β s) x = ∑ i ∈ s, (of β i) (if h : i ∈ s then x ⟨i, h⟩ else 0) | case h
ι : Type u_1
inst✝² : DecidableEq ι
R : Type u_2
inst✝¹ : Semiring R
β : ι → Type u_3
inst✝ : (i : ι) → AddCommMonoid (β i)
s : Finset ι
x : (i : { x // x ∈ s }) → β ↑i
⊢ ∀ (i : ι), ((mk β s) x) i = (∑ i ∈ s, (of β i) (if h : i ∈ s then x ⟨i, h⟩ else 0)) i | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
inst✝² : DecidableEq ι
R : Type u_2
inst✝¹ : Semiring R
β : ι → Type u_3
inst✝ : (i : ι) → AddCommMonoid (β i)
s : Finset ι
x : (i : { x // x ∈ s }) → β ↑i
⊢ (mk β s) x = ∑ i ∈ s, (of β i) (if h : i ∈ s then x ⟨i, h⟩ else 0)
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | DirectSum.mk_eq_sum | [218, 1] | [229, 70] | intro i | case h
ι : Type u_1
inst✝² : DecidableEq ι
R : Type u_2
inst✝¹ : Semiring R
β : ι → Type u_3
inst✝ : (i : ι) → AddCommMonoid (β i)
s : Finset ι
x : (i : { x // x ∈ s }) → β ↑i
⊢ ∀ (i : ι), ((mk β s) x) i = (∑ i ∈ s, (of β i) (if h : i ∈ s then x ⟨i, h⟩ else 0)) i | case h
ι : Type u_1
inst✝² : DecidableEq ι
R : Type u_2
inst✝¹ : Semiring R
β : ι → Type u_3
inst✝ : (i : ι) → AddCommMonoid (β i)
s : Finset ι
x : (i : { x // x ∈ s }) → β ↑i
i : ι
⊢ ((mk β s) x) i = (∑ i ∈ s, (of β i) (if h : i ∈ s then x ⟨i, h⟩ else 0)) i | Please generate a tactic in lean4 to solve the state.
STATE:
case h
ι : Type u_1
inst✝² : DecidableEq ι
R : Type u_2
inst✝¹ : Semiring R
β : ι → Type u_3
inst✝ : (i : ι) → AddCommMonoid (β i)
s : Finset ι
x : (i : { x // x ∈ s }) → β ↑i
⊢ ∀ (i : ι), ((mk β s) x) i = (∑ i ∈ s, (of β i) (if h : i ∈ s then x ⟨i, h⟩ else 0... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | DirectSum.mk_eq_sum | [218, 1] | [229, 70] | rw [mk_apply] | case h
ι : Type u_1
inst✝² : DecidableEq ι
R : Type u_2
inst✝¹ : Semiring R
β : ι → Type u_3
inst✝ : (i : ι) → AddCommMonoid (β i)
s : Finset ι
x : (i : { x // x ∈ s }) → β ↑i
i : ι
⊢ ((mk β s) x) i = (∑ i ∈ s, (of β i) (if h : i ∈ s then x ⟨i, h⟩ else 0)) i | case h
ι : Type u_1
inst✝² : DecidableEq ι
R : Type u_2
inst✝¹ : Semiring R
β : ι → Type u_3
inst✝ : (i : ι) → AddCommMonoid (β i)
s : Finset ι
x : (i : { x // x ∈ s }) → β ↑i
i : ι
⊢ (if h : i ∈ s then x ⟨i, h⟩ else 0) = (∑ i ∈ s, (of β i) (if h : i ∈ s then x ⟨i, h⟩ else 0)) i | Please generate a tactic in lean4 to solve the state.
STATE:
case h
ι : Type u_1
inst✝² : DecidableEq ι
R : Type u_2
inst✝¹ : Semiring R
β : ι → Type u_3
inst✝ : (i : ι) → AddCommMonoid (β i)
s : Finset ι
x : (i : { x // x ∈ s }) → β ↑i
i : ι
⊢ ((mk β s) x) i = (∑ i ∈ s, (of β i) (if h : i ∈ s then x ⟨i, h⟩ else 0)) i
... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | DirectSum.mk_eq_sum | [218, 1] | [229, 70] | split_ifs with hi | case h
ι : Type u_1
inst✝² : DecidableEq ι
R : Type u_2
inst✝¹ : Semiring R
β : ι → Type u_3
inst✝ : (i : ι) → AddCommMonoid (β i)
s : Finset ι
x : (i : { x // x ∈ s }) → β ↑i
i : ι
⊢ (if h : i ∈ s then x ⟨i, h⟩ else 0) = (∑ i ∈ s, (of β i) (if h : i ∈ s then x ⟨i, h⟩ else 0)) i | case pos
ι : Type u_1
inst✝² : DecidableEq ι
R : Type u_2
inst✝¹ : Semiring R
β : ι → Type u_3
inst✝ : (i : ι) → AddCommMonoid (β i)
s : Finset ι
x : (i : { x // x ∈ s }) → β ↑i
i : ι
hi : i ∈ s
⊢ x ⟨i, hi⟩ = (∑ i ∈ s, (of β i) (if h : i ∈ s then x ⟨i, h⟩ else 0)) i
case neg
ι : Type u_1
inst✝² : DecidableEq ι
R : Typ... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
ι : Type u_1
inst✝² : DecidableEq ι
R : Type u_2
inst✝¹ : Semiring R
β : ι → Type u_3
inst✝ : (i : ι) → AddCommMonoid (β i)
s : Finset ι
x : (i : { x // x ∈ s }) → β ↑i
i : ι
⊢ (if h : i ∈ s then x ⟨i, h⟩ else 0) = (∑ i ∈ s, (of β i) (if h : i ∈ s then... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | DirectSum.mk_eq_sum | [218, 1] | [229, 70] | rw [DFinsupp.finset_sum_apply, Finset.sum_eq_single i (fun j _ hji => of_eq_of_ne _ _ _ _ hji),
of_eq_same, dif_pos hi] | case pos
ι : Type u_1
inst✝² : DecidableEq ι
R : Type u_2
inst✝¹ : Semiring R
β : ι → Type u_3
inst✝ : (i : ι) → AddCommMonoid (β i)
s : Finset ι
x : (i : { x // x ∈ s }) → β ↑i
i : ι
hi : i ∈ s
⊢ x ⟨i, hi⟩ = (∑ i ∈ s, (of β i) (if h : i ∈ s then x ⟨i, h⟩ else 0)) i | case pos
ι : Type u_1
inst✝² : DecidableEq ι
R : Type u_2
inst✝¹ : Semiring R
β : ι → Type u_3
inst✝ : (i : ι) → AddCommMonoid (β i)
s : Finset ι
x : (i : { x // x ∈ s }) → β ↑i
i : ι
hi : i ∈ s
⊢ i ∉ s → ((of β i) (if h : i ∈ s then x ⟨i, h⟩ else 0)) i = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
ι : Type u_1
inst✝² : DecidableEq ι
R : Type u_2
inst✝¹ : Semiring R
β : ι → Type u_3
inst✝ : (i : ι) → AddCommMonoid (β i)
s : Finset ι
x : (i : { x // x ∈ s }) → β ↑i
i : ι
hi : i ∈ s
⊢ x ⟨i, hi⟩ = (∑ i ∈ s, (of β i) (if h : i ∈ s then x ⟨i, h⟩ els... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | DirectSum.mk_eq_sum | [218, 1] | [229, 70] | intro his | case pos
ι : Type u_1
inst✝² : DecidableEq ι
R : Type u_2
inst✝¹ : Semiring R
β : ι → Type u_3
inst✝ : (i : ι) → AddCommMonoid (β i)
s : Finset ι
x : (i : { x // x ∈ s }) → β ↑i
i : ι
hi : i ∈ s
⊢ i ∉ s → ((of β i) (if h : i ∈ s then x ⟨i, h⟩ else 0)) i = 0 | case pos
ι : Type u_1
inst✝² : DecidableEq ι
R : Type u_2
inst✝¹ : Semiring R
β : ι → Type u_3
inst✝ : (i : ι) → AddCommMonoid (β i)
s : Finset ι
x : (i : { x // x ∈ s }) → β ↑i
i : ι
hi : i ∈ s
his : i ∉ s
⊢ ((of β i) (if h : i ∈ s then x ⟨i, h⟩ else 0)) i = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
ι : Type u_1
inst✝² : DecidableEq ι
R : Type u_2
inst✝¹ : Semiring R
β : ι → Type u_3
inst✝ : (i : ι) → AddCommMonoid (β i)
s : Finset ι
x : (i : { x // x ∈ s }) → β ↑i
i : ι
hi : i ∈ s
⊢ i ∉ s → ((of β i) (if h : i ∈ s then x ⟨i, h⟩ else 0)) i = 0
T... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | DirectSum.mk_eq_sum | [218, 1] | [229, 70] | rw [of_eq_same, dif_neg his] | case pos
ι : Type u_1
inst✝² : DecidableEq ι
R : Type u_2
inst✝¹ : Semiring R
β : ι → Type u_3
inst✝ : (i : ι) → AddCommMonoid (β i)
s : Finset ι
x : (i : { x // x ∈ s }) → β ↑i
i : ι
hi : i ∈ s
his : i ∉ s
⊢ ((of β i) (if h : i ∈ s then x ⟨i, h⟩ else 0)) i = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
ι : Type u_1
inst✝² : DecidableEq ι
R : Type u_2
inst✝¹ : Semiring R
β : ι → Type u_3
inst✝ : (i : ι) → AddCommMonoid (β i)
s : Finset ι
x : (i : { x // x ∈ s }) → β ↑i
i : ι
hi : i ∈ s
his : i ∉ s
⊢ ((of β i) (if h : i ∈ s then x ⟨i, h⟩ else 0)) i =... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | DirectSum.mk_eq_sum | [218, 1] | [229, 70] | rw [DFinsupp.finset_sum_apply, Finset.sum_eq_zero
(fun j hj => of_eq_of_ne _ _ _ _ (ne_of_mem_of_not_mem hj hi))] | case neg
ι : Type u_1
inst✝² : DecidableEq ι
R : Type u_2
inst✝¹ : Semiring R
β : ι → Type u_3
inst✝ : (i : ι) → AddCommMonoid (β i)
s : Finset ι
x : (i : { x // x ∈ s }) → β ↑i
i : ι
hi : i ∉ s
⊢ 0 = (∑ i ∈ s, (of β i) (if h : i ∈ s then x ⟨i, h⟩ else 0)) i | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
ι : Type u_1
inst✝² : DecidableEq ι
R : Type u_2
inst✝¹ : Semiring R
β : ι → Type u_3
inst✝ : (i : ι) → AddCommMonoid (β i)
s : Finset ι
x : (i : { x // x ∈ s }) → β ↑i
i : ι
hi : i ∉ s
⊢ 0 = (∑ i ∈ s, (of β i) (if h : i ∈ s then x ⟨i, h⟩ else 0)) i
... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | DirectSum.toAddMonoid_mk | [231, 1] | [235, 82] | rw [mk_eq_sum, map_sum, Finset.sum_congr rfl (fun i _ => toAddMonoid_of _ _ _)] | ι : Type u_1
inst✝³ : DecidableEq ι
R : Type u_2
inst✝² : Semiring R
β : ι → Type u_3
inst✝¹ : (i : ι) → AddCommMonoid (β i)
γ : Type u_4
inst✝ : AddCommMonoid γ
ψ : (i : ι) → β i →+ γ
s : Finset ι
x : (i : { x // x ∈ s }) → β ↑i
⊢ (toAddMonoid ψ) ((mk β s) x) = ∑ i ∈ s, (ψ i) (if h : i ∈ s then x ⟨i, h⟩ else 0) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
inst✝³ : DecidableEq ι
R : Type u_2
inst✝² : Semiring R
β : ι → Type u_3
inst✝¹ : (i : ι) → AddCommMonoid (β i)
γ : Type u_4
inst✝ : AddCommMonoid γ
ψ : (i : ι) → β i →+ γ
s : Finset ι
x : (i : { x // x ∈ s }) → β ↑i
⊢ (toAddMonoid ψ) ((mk β s) x... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | DirectSum.map_apply' | [237, 1] | [245, 16] | conv_lhs => rw [← sum_support_of β x] | ι : Type u_1
inst✝⁶ : DecidableEq ι
R : Type u_2
inst✝⁵ : Semiring R
β : ι → Type u_3
γ : ι → Type u_4
inst✝⁴ : (i : ι) → AddCommMonoid (β i)
inst✝³ : (i : ι) → AddCommMonoid (γ i)
inst✝² : (i : ι) → (x : β i) → Decidable (x ≠ 0)
F : ι → Type u_5
inst✝¹ : (i : ι) → FunLike (F i) (β i) (γ i)
inst✝ : ∀ (i : ι), AddMonoid... | ι : Type u_1
inst✝⁶ : DecidableEq ι
R : Type u_2
inst✝⁵ : Semiring R
β : ι → Type u_3
γ : ι → Type u_4
inst✝⁴ : (i : ι) → AddCommMonoid (β i)
inst✝³ : (i : ι) → AddCommMonoid (γ i)
inst✝² : (i : ι) → (x : β i) → Decidable (x ≠ 0)
F : ι → Type u_5
inst✝¹ : (i : ι) → FunLike (F i) (β i) (γ i)
inst✝ : ∀ (i : ι), AddMonoid... | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
inst✝⁶ : DecidableEq ι
R : Type u_2
inst✝⁵ : Semiring R
β : ι → Type u_3
γ : ι → Type u_4
inst✝⁴ : (i : ι) → AddCommMonoid (β i)
inst✝³ : (i : ι) → AddCommMonoid (γ i)
inst✝² : (i : ι) → (x : β i) → Decidable (x ≠ 0)
F : ι → Type u_5
inst✝¹ : (i ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | DirectSum.map_apply' | [237, 1] | [245, 16] | simp_rw [map_sum, map_of] | ι : Type u_1
inst✝⁶ : DecidableEq ι
R : Type u_2
inst✝⁵ : Semiring R
β : ι → Type u_3
γ : ι → Type u_4
inst✝⁴ : (i : ι) → AddCommMonoid (β i)
inst✝³ : (i : ι) → AddCommMonoid (γ i)
inst✝² : (i : ι) → (x : β i) → Decidable (x ≠ 0)
F : ι → Type u_5
inst✝¹ : (i : ι) → FunLike (F i) (β i) (γ i)
inst✝ : ∀ (i : ι), AddMonoid... | ι : Type u_1
inst✝⁶ : DecidableEq ι
R : Type u_2
inst✝⁵ : Semiring R
β : ι → Type u_3
γ : ι → Type u_4
inst✝⁴ : (i : ι) → AddCommMonoid (β i)
inst✝³ : (i : ι) → AddCommMonoid (γ i)
inst✝² : (i : ι) → (x : β i) → Decidable (x ≠ 0)
F : ι → Type u_5
inst✝¹ : (i : ι) → FunLike (F i) (β i) (γ i)
inst✝ : ∀ (i : ι), AddMonoid... | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
inst✝⁶ : DecidableEq ι
R : Type u_2
inst✝⁵ : Semiring R
β : ι → Type u_3
γ : ι → Type u_4
inst✝⁴ : (i : ι) → AddCommMonoid (β i)
inst✝³ : (i : ι) → AddCommMonoid (γ i)
inst✝² : (i : ι) → (x : β i) → Decidable (x ≠ 0)
F : ι → Type u_5
inst✝¹ : (i ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | DirectSum.map_apply' | [237, 1] | [245, 16] | rw [eq_comm] | ι : Type u_1
inst✝⁶ : DecidableEq ι
R : Type u_2
inst✝⁵ : Semiring R
β : ι → Type u_3
γ : ι → Type u_4
inst✝⁴ : (i : ι) → AddCommMonoid (β i)
inst✝³ : (i : ι) → AddCommMonoid (γ i)
inst✝² : (i : ι) → (x : β i) → Decidable (x ≠ 0)
F : ι → Type u_5
inst✝¹ : (i : ι) → FunLike (F i) (β i) (γ i)
inst✝ : ∀ (i : ι), AddMonoid... | ι : Type u_1
inst✝⁶ : DecidableEq ι
R : Type u_2
inst✝⁵ : Semiring R
β : ι → Type u_3
γ : ι → Type u_4
inst✝⁴ : (i : ι) → AddCommMonoid (β i)
inst✝³ : (i : ι) → AddCommMonoid (γ i)
inst✝² : (i : ι) → (x : β i) → Decidable (x ≠ 0)
F : ι → Type u_5
inst✝¹ : (i : ι) → FunLike (F i) (β i) (γ i)
inst✝ : ∀ (i : ι), AddMonoid... | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
inst✝⁶ : DecidableEq ι
R : Type u_2
inst✝⁵ : Semiring R
β : ι → Type u_3
γ : ι → Type u_4
inst✝⁴ : (i : ι) → AddCommMonoid (β i)
inst✝³ : (i : ι) → AddCommMonoid (γ i)
inst✝² : (i : ι) → (x : β i) → Decidable (x ≠ 0)
F : ι → Type u_5
inst✝¹ : (i ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | DirectSum.map_apply' | [237, 1] | [245, 16] | convert mk_eq_sum x.support fun i => (h i) (x i) | ι : Type u_1
inst✝⁶ : DecidableEq ι
R : Type u_2
inst✝⁵ : Semiring R
β : ι → Type u_3
γ : ι → Type u_4
inst✝⁴ : (i : ι) → AddCommMonoid (β i)
inst✝³ : (i : ι) → AddCommMonoid (γ i)
inst✝² : (i : ι) → (x : β i) → Decidable (x ≠ 0)
F : ι → Type u_5
inst✝¹ : (i : ι) → FunLike (F i) (β i) (γ i)
inst✝ : ∀ (i : ι), AddMonoid... | case h.e'_3.a.h.e'_6
ι : Type u_1
inst✝⁶ : DecidableEq ι
R : Type u_2
inst✝⁵ : Semiring R
β : ι → Type u_3
γ : ι → Type u_4
inst✝⁴ : (i : ι) → AddCommMonoid (β i)
inst✝³ : (i : ι) → AddCommMonoid (γ i)
inst✝² : (i : ι) → (x : β i) → Decidable (x ≠ 0)
F : ι → Type u_5
inst✝¹ : (i : ι) → FunLike (F i) (β i) (γ i)
inst✝ :... | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
inst✝⁶ : DecidableEq ι
R : Type u_2
inst✝⁵ : Semiring R
β : ι → Type u_3
γ : ι → Type u_4
inst✝⁴ : (i : ι) → AddCommMonoid (β i)
inst✝³ : (i : ι) → AddCommMonoid (γ i)
inst✝² : (i : ι) → (x : β i) → Decidable (x ≠ 0)
F : ι → Type u_5
inst✝¹ : (i ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | DirectSum.map_apply' | [237, 1] | [245, 16] | rwa [dif_pos] | case h.e'_3.a.h.e'_6
ι : Type u_1
inst✝⁶ : DecidableEq ι
R : Type u_2
inst✝⁵ : Semiring R
β : ι → Type u_3
γ : ι → Type u_4
inst✝⁴ : (i : ι) → AddCommMonoid (β i)
inst✝³ : (i : ι) → AddCommMonoid (γ i)
inst✝² : (i : ι) → (x : β i) → Decidable (x ≠ 0)
F : ι → Type u_5
inst✝¹ : (i : ι) → FunLike (F i) (β i) (γ i)
inst✝ :... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_3.a.h.e'_6
ι : Type u_1
inst✝⁶ : DecidableEq ι
R : Type u_2
inst✝⁵ : Semiring R
β : ι → Type u_3
γ : ι → Type u_4
inst✝⁴ : (i : ι) → AddCommMonoid (β i)
inst✝³ : (i : ι) → AddCommMonoid (γ i)
inst✝² : (i : ι) → (x : β i) → Decidable (x ≠ 0)
F : ι → ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | DirectSum.decomposeAlgEquiv_coe | [266, 1] | [267, 52] | rfl | R : Type u_1
inst✝⁶ : CommSemiring R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : AddCommMonoid ι
A : Type u_3
inst✝³ : CommSemiring A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
rel : A → A → Prop
inst✝ : GradedAlgebra 𝒜
a : A
⊢ (decomposeAlgEquiv 𝒜) a = (decompose 𝒜) a | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝⁶ : CommSemiring R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : AddCommMonoid ι
A : Type u_3
inst✝³ : CommSemiring A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
rel : A → A → Prop
inst✝ : GradedAlgebra 𝒜
a : A
⊢ (deco... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | Rel.IsHomogeneous_of_isPureHomogeneous | [283, 1] | [293, 16] | intro a b h i | R : Type u_1
inst✝⁶ : CommSemiring R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : AddCommMonoid ι
A : Type u_3
inst✝³ : CommSemiring A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
rel : A → A → Prop
inst✝ : GradedAlgebra 𝒜
hrel : IsPureHomogeneous 𝒜 rel
hrel0 : rel 0 0
⊢ IsHomogeneous 𝒜 rel | R : Type u_1
inst✝⁶ : CommSemiring R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : AddCommMonoid ι
A : Type u_3
inst✝³ : CommSemiring A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
rel : A → A → Prop
inst✝ : GradedAlgebra 𝒜
hrel : IsPureHomogeneous 𝒜 rel
hrel0 : rel 0 0
a b : A
h : rel a b
i : ι
... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝⁶ : CommSemiring R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : AddCommMonoid ι
A : Type u_3
inst✝³ : CommSemiring A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
rel : A → A → Prop
inst✝ : GradedAlgebra 𝒜
hrel : IsPure... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | Rel.IsHomogeneous_of_isPureHomogeneous | [283, 1] | [293, 16] | obtain ⟨j, ha, hb⟩ := hrel h | R : Type u_1
inst✝⁶ : CommSemiring R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : AddCommMonoid ι
A : Type u_3
inst✝³ : CommSemiring A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
rel : A → A → Prop
inst✝ : GradedAlgebra 𝒜
hrel : IsPureHomogeneous 𝒜 rel
hrel0 : rel 0 0
a b : A
h : rel a b
i : ι
... | case intro.intro
R : Type u_1
inst✝⁶ : CommSemiring R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : AddCommMonoid ι
A : Type u_3
inst✝³ : CommSemiring A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
rel : A → A → Prop
inst✝ : GradedAlgebra 𝒜
hrel : IsPureHomogeneous 𝒜 rel
hrel0 : rel 0 0
a b : A
h... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝⁶ : CommSemiring R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : AddCommMonoid ι
A : Type u_3
inst✝³ : CommSemiring A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
rel : A → A → Prop
inst✝ : GradedAlgebra 𝒜
hrel : IsPure... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | Rel.IsHomogeneous_of_isPureHomogeneous | [283, 1] | [293, 16] | by_cases hij : j = i | case intro.intro
R : Type u_1
inst✝⁶ : CommSemiring R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : AddCommMonoid ι
A : Type u_3
inst✝³ : CommSemiring A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
rel : A → A → Prop
inst✝ : GradedAlgebra 𝒜
hrel : IsPureHomogeneous 𝒜 rel
hrel0 : rel 0 0
a b : A
h... | case pos
R : Type u_1
inst✝⁶ : CommSemiring R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : AddCommMonoid ι
A : Type u_3
inst✝³ : CommSemiring A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
rel : A → A → Prop
inst✝ : GradedAlgebra 𝒜
hrel : IsPureHomogeneous 𝒜 rel
hrel0 : rel 0 0
a b : A
h : rel a... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
R : Type u_1
inst✝⁶ : CommSemiring R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : AddCommMonoid ι
A : Type u_3
inst✝³ : CommSemiring A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
rel : A → A → Prop
inst✝ : GradedAlgebra... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | Rel.IsHomogeneous_of_isPureHomogeneous | [283, 1] | [293, 16] | . rw [← hij]
simp only [AlgEquiv.toLinearMap_apply, decomposeAlgEquiv_coe, decompose_of_mem_same, ha, hb, h] | case pos
R : Type u_1
inst✝⁶ : CommSemiring R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : AddCommMonoid ι
A : Type u_3
inst✝³ : CommSemiring A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
rel : A → A → Prop
inst✝ : GradedAlgebra 𝒜
hrel : IsPureHomogeneous 𝒜 rel
hrel0 : rel 0 0
a b : A
h : rel a... | case neg
R : Type u_1
inst✝⁶ : CommSemiring R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : AddCommMonoid ι
A : Type u_3
inst✝³ : CommSemiring A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
rel : A → A → Prop
inst✝ : GradedAlgebra 𝒜
hrel : IsPureHomogeneous 𝒜 rel
hrel0 : rel 0 0
a b : A
h : rel a... | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
R : Type u_1
inst✝⁶ : CommSemiring R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : AddCommMonoid ι
A : Type u_3
inst✝³ : CommSemiring A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
rel : A → A → Prop
inst✝ : GradedAlgebra 𝒜
hrel... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | Rel.IsHomogeneous_of_isPureHomogeneous | [283, 1] | [293, 16] | . simp only [AlgEquiv.toLinearMap_apply, decomposeAlgEquiv_coe, decompose_of_mem_ne _ ha hij,
decompose_of_mem_ne _ hb hij]
exact hrel0 | case neg
R : Type u_1
inst✝⁶ : CommSemiring R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : AddCommMonoid ι
A : Type u_3
inst✝³ : CommSemiring A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
rel : A → A → Prop
inst✝ : GradedAlgebra 𝒜
hrel : IsPureHomogeneous 𝒜 rel
hrel0 : rel 0 0
a b : A
h : rel a... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
R : Type u_1
inst✝⁶ : CommSemiring R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : AddCommMonoid ι
A : Type u_3
inst✝³ : CommSemiring A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
rel : A → A → Prop
inst✝ : GradedAlgebra 𝒜
hrel... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | Rel.IsHomogeneous_of_isPureHomogeneous | [283, 1] | [293, 16] | rw [← hij] | case pos
R : Type u_1
inst✝⁶ : CommSemiring R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : AddCommMonoid ι
A : Type u_3
inst✝³ : CommSemiring A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
rel : A → A → Prop
inst✝ : GradedAlgebra 𝒜
hrel : IsPureHomogeneous 𝒜 rel
hrel0 : rel 0 0
a b : A
h : rel a... | case pos
R : Type u_1
inst✝⁶ : CommSemiring R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : AddCommMonoid ι
A : Type u_3
inst✝³ : CommSemiring A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
rel : A → A → Prop
inst✝ : GradedAlgebra 𝒜
hrel : IsPureHomogeneous 𝒜 rel
hrel0 : rel 0 0
a b : A
h : rel a... | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
R : Type u_1
inst✝⁶ : CommSemiring R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : AddCommMonoid ι
A : Type u_3
inst✝³ : CommSemiring A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
rel : A → A → Prop
inst✝ : GradedAlgebra 𝒜
hrel... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | Rel.IsHomogeneous_of_isPureHomogeneous | [283, 1] | [293, 16] | simp only [AlgEquiv.toLinearMap_apply, decomposeAlgEquiv_coe, decompose_of_mem_same, ha, hb, h] | case pos
R : Type u_1
inst✝⁶ : CommSemiring R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : AddCommMonoid ι
A : Type u_3
inst✝³ : CommSemiring A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
rel : A → A → Prop
inst✝ : GradedAlgebra 𝒜
hrel : IsPureHomogeneous 𝒜 rel
hrel0 : rel 0 0
a b : A
h : rel a... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
R : Type u_1
inst✝⁶ : CommSemiring R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : AddCommMonoid ι
A : Type u_3
inst✝³ : CommSemiring A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
rel : A → A → Prop
inst✝ : GradedAlgebra 𝒜
hrel... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.