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/ForMathlib/GradedRingQuot.lean | quotDecompose_injective' | [650, 1] | [664, 70] | specialize hxy' i | case intro.refl.intro.refl.h
R : Type u_1
inst✝⁵ : CommSemiring R
ι : Type u_2
inst✝⁴ : DecidableEq ι
inst✝³ : AddCommMonoid ι
A : Type u_3
inst✝² : CommSemiring A
inst✝¹ : DecidableEq A
inst✝ : Algebra R A
𝒜 : ι → Submodule R A
rel : A → A → Prop
h𝒜 : GradedAlgebra 𝒜
hrel : Rel.IsHomogeneous 𝒜 rel
hφ : ∀ (i : ι), ... | case intro.refl.intro.refl.h
R : Type u_1
inst✝⁵ : CommSemiring R
ι : Type u_2
inst✝⁴ : DecidableEq ι
inst✝³ : AddCommMonoid ι
A : Type u_3
inst✝² : CommSemiring A
inst✝¹ : DecidableEq A
inst✝ : Algebra R A
𝒜 : ι → Submodule R A
rel : A → A → Prop
h𝒜 : GradedAlgebra 𝒜
hrel : Rel.IsHomogeneous 𝒜 rel
hφ : ∀ (i : ι), ... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.refl.intro.refl.h
R : Type u_1
inst✝⁵ : CommSemiring R
ι : Type u_2
inst✝⁴ : DecidableEq ι
inst✝³ : AddCommMonoid ι
A : Type u_3
inst✝² : CommSemiring A
inst✝¹ : DecidableEq A
inst✝ : Algebra R A
𝒜 : ι → Submodule R A
rel : A → A → Prop
h𝒜 : Grad... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | quotDecompose_injective' | [650, 1] | [664, 70] | simp only [Decomposition.decompose'_eq] at hxy' | case intro.refl.intro.refl.h
R : Type u_1
inst✝⁵ : CommSemiring R
ι : Type u_2
inst✝⁴ : DecidableEq ι
inst✝³ : AddCommMonoid ι
A : Type u_3
inst✝² : CommSemiring A
inst✝¹ : DecidableEq A
inst✝ : Algebra R A
𝒜 : ι → Submodule R A
rel : A → A → Prop
h𝒜 : GradedAlgebra 𝒜
hrel : Rel.IsHomogeneous 𝒜 rel
hφ : ∀ (i : ι), ... | case intro.refl.intro.refl.h
R : Type u_1
inst✝⁵ : CommSemiring R
ι : Type u_2
inst✝⁴ : DecidableEq ι
inst✝³ : AddCommMonoid ι
A : Type u_3
inst✝² : CommSemiring A
inst✝¹ : DecidableEq A
inst✝ : Algebra R A
𝒜 : ι → Submodule R A
rel : A → A → Prop
h𝒜 : GradedAlgebra 𝒜
hrel : Rel.IsHomogeneous 𝒜 rel
hφ : ∀ (i : ι), ... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.refl.intro.refl.h
R : Type u_1
inst✝⁵ : CommSemiring R
ι : Type u_2
inst✝⁴ : DecidableEq ι
inst✝³ : AddCommMonoid ι
A : Type u_3
inst✝² : CommSemiring A
inst✝¹ : DecidableEq A
inst✝ : Algebra R A
𝒜 : ι → Submodule R A
rel : A → A → Prop
h𝒜 : Grad... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | quotDecompose_injective' | [650, 1] | [664, 70] | simpa only [lmap'_quotCompMap_apply, SetLike.coe_eq_coe] using hxy' | case intro.refl.intro.refl.h
R : Type u_1
inst✝⁵ : CommSemiring R
ι : Type u_2
inst✝⁴ : DecidableEq ι
inst✝³ : AddCommMonoid ι
A : Type u_3
inst✝² : CommSemiring A
inst✝¹ : DecidableEq A
inst✝ : Algebra R A
𝒜 : ι → Submodule R A
rel : A → A → Prop
h𝒜 : GradedAlgebra 𝒜
hrel : Rel.IsHomogeneous 𝒜 rel
hφ : ∀ (i : ι), ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.refl.intro.refl.h
R : Type u_1
inst✝⁵ : CommSemiring R
ι : Type u_2
inst✝⁴ : DecidableEq ι
inst✝³ : AddCommMonoid ι
A : Type u_3
inst✝² : CommSemiring A
inst✝¹ : DecidableEq A
inst✝ : Algebra R A
𝒜 : ι → Submodule R A
rel : A → A → Prop
h𝒜 : Grad... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | quotDecompose_injective' | [650, 1] | [664, 70] | rintro i ⟨x, ⟨a, ha, rfl⟩ ⟩ | R : Type u_1
inst✝⁵ : CommSemiring R
ι : Type u_2
inst✝⁴ : DecidableEq ι
inst✝³ : AddCommMonoid ι
A : Type u_3
inst✝² : CommSemiring A
inst✝¹ : DecidableEq A
inst✝ : Algebra R A
𝒜 : ι → Submodule R A
rel : A → A → Prop
h𝒜 : GradedAlgebra 𝒜
hrel : Rel.IsHomogeneous 𝒜 rel
⊢ ∀ (i : ι), Surjective ⇑(quotCompMap R 𝒜 re... | case mk.intro.intro
R : Type u_1
inst✝⁵ : CommSemiring R
ι : Type u_2
inst✝⁴ : DecidableEq ι
inst✝³ : AddCommMonoid ι
A : Type u_3
inst✝² : CommSemiring A
inst✝¹ : DecidableEq A
inst✝ : Algebra R A
𝒜 : ι → Submodule R A
rel : A → A → Prop
h𝒜 : GradedAlgebra 𝒜
hrel : Rel.IsHomogeneous 𝒜 rel
i : ι
a : A
ha : a ∈ ↑(𝒜... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝⁵ : CommSemiring R
ι : Type u_2
inst✝⁴ : DecidableEq ι
inst✝³ : AddCommMonoid ι
A : Type u_3
inst✝² : CommSemiring A
inst✝¹ : DecidableEq A
inst✝ : Algebra R A
𝒜 : ι → Submodule R A
rel : A → A → Prop
h𝒜 : GradedAlgebra 𝒜
hrel : Rel.IsHom... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | quotDecompose_injective' | [650, 1] | [664, 70] | exact ⟨⟨a, ha⟩, rfl⟩ | case mk.intro.intro
R : Type u_1
inst✝⁵ : CommSemiring R
ι : Type u_2
inst✝⁴ : DecidableEq ι
inst✝³ : AddCommMonoid ι
A : Type u_3
inst✝² : CommSemiring A
inst✝¹ : DecidableEq A
inst✝ : Algebra R A
𝒜 : ι → Submodule R A
rel : A → A → Prop
h𝒜 : GradedAlgebra 𝒜
hrel : Rel.IsHomogeneous 𝒜 rel
i : ι
a : A
ha : a ∈ ↑(𝒜... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mk.intro.intro
R : Type u_1
inst✝⁵ : CommSemiring R
ι : Type u_2
inst✝⁴ : DecidableEq ι
inst✝³ : AddCommMonoid ι
A : Type u_3
inst✝² : CommSemiring A
inst✝¹ : DecidableEq A
inst✝ : Algebra R A
𝒜 : ι → Submodule R A
rel : A → A → Prop
h𝒜 : GradedAlgebra... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | Ideal.quotDecomposeLaux_of_mem_eq_zero | [720, 1] | [726, 16] | rw [Ideal.quotDecomposeLaux, LinearMap.comp_apply, lmap'_apply, quotCompMap] | R : Type u_1
inst✝⁶ : CommRing R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : AddCommMonoid ι
A : Type u_3
inst✝³ : CommRing A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
I : Ideal A
rel : A → A → Prop
inst✝ : GradedAlgebra 𝒜
hI : IsHomogeneous 𝒜 I
x : A
hx : x ∈ I
i : ι
⊢ ((quotDecomposeLaux R... | R : Type u_1
inst✝⁶ : CommRing R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : AddCommMonoid ι
A : Type u_3
inst✝³ : CommRing A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
I : Ideal A
rel : A → A → Prop
inst✝ : GradedAlgebra 𝒜
hI : IsHomogeneous 𝒜 I
x : A
hx : x ∈ I
i : ι
⊢ { toFun := fun u => ⟨... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝⁶ : CommRing R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : AddCommMonoid ι
A : Type u_3
inst✝³ : CommRing A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
I : Ideal A
rel : A → A → Prop
inst✝ : GradedAlgebra 𝒜
hI : IsHo... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | Ideal.quotDecomposeLaux_of_mem_eq_zero | [720, 1] | [726, 16] | simp only [Ideal.Quotient.mkₐ_eq_mk, AlgEquiv.toLinearMap_apply, decomposeAlgEquiv_apply,
LinearMap.coe_mk, AddHom.coe_mk, Submodule.mk_eq_zero] | R : Type u_1
inst✝⁶ : CommRing R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : AddCommMonoid ι
A : Type u_3
inst✝³ : CommRing A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
I : Ideal A
rel : A → A → Prop
inst✝ : GradedAlgebra 𝒜
hI : IsHomogeneous 𝒜 I
x : A
hx : x ∈ I
i : ι
⊢ { toFun := fun u => ⟨... | R : Type u_1
inst✝⁶ : CommRing R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : AddCommMonoid ι
A : Type u_3
inst✝³ : CommRing A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
I : Ideal A
rel : A → A → Prop
inst✝ : GradedAlgebra 𝒜
hI : IsHomogeneous 𝒜 I
x : A
hx : x ∈ I
i : ι
⊢ (Quotient.mk I) ↑(((d... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝⁶ : CommRing R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : AddCommMonoid ι
A : Type u_3
inst✝³ : CommRing A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
I : Ideal A
rel : A → A → Prop
inst✝ : GradedAlgebra 𝒜
hI : IsHo... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | Ideal.quotDecomposeLaux_of_mem_eq_zero | [720, 1] | [726, 16] | rw [Ideal.Quotient.eq_zero_iff_mem] | R : Type u_1
inst✝⁶ : CommRing R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : AddCommMonoid ι
A : Type u_3
inst✝³ : CommRing A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
I : Ideal A
rel : A → A → Prop
inst✝ : GradedAlgebra 𝒜
hI : IsHomogeneous 𝒜 I
x : A
hx : x ∈ I
i : ι
⊢ (Quotient.mk I) ↑(((d... | R : Type u_1
inst✝⁶ : CommRing R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : AddCommMonoid ι
A : Type u_3
inst✝³ : CommRing A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
I : Ideal A
rel : A → A → Prop
inst✝ : GradedAlgebra 𝒜
hI : IsHomogeneous 𝒜 I
x : A
hx : x ∈ I
i : ι
⊢ ↑(((decompose 𝒜) x) ... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝⁶ : CommRing R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : AddCommMonoid ι
A : Type u_3
inst✝³ : CommRing A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
I : Ideal A
rel : A → A → Prop
inst✝ : GradedAlgebra 𝒜
hI : IsHo... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | Ideal.quotDecomposeLaux_of_mem_eq_zero | [720, 1] | [726, 16] | exact hI i hx | R : Type u_1
inst✝⁶ : CommRing R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : AddCommMonoid ι
A : Type u_3
inst✝³ : CommRing A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
I : Ideal A
rel : A → A → Prop
inst✝ : GradedAlgebra 𝒜
hI : IsHomogeneous 𝒜 I
x : A
hx : x ∈ I
i : ι
⊢ ↑(((decompose 𝒜) x) ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝⁶ : CommRing R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : AddCommMonoid ι
A : Type u_3
inst✝³ : CommRing A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
I : Ideal A
rel : A → A → Prop
inst✝ : GradedAlgebra 𝒜
hI : IsHo... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | Ideal.quotDecomposition_left_inv'_aux | [748, 9] | [758, 6] | apply linearMap_ext | R : Type u_1
inst✝⁶ : CommRing R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : AddCommMonoid ι
A : Type u_3
inst✝³ : CommRing A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
I : Ideal A
rel : A → A → Prop
inst✝ : GradedAlgebra 𝒜
⊢ coeLinearMap (quotSubmodule R 𝒜 I) ∘ₗ lmap' (quotCompMap R 𝒜 I) =
... | case H
R : Type u_1
inst✝⁶ : CommRing R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : AddCommMonoid ι
A : Type u_3
inst✝³ : CommRing A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
I : Ideal A
rel : A → A → Prop
inst✝ : GradedAlgebra 𝒜
⊢ ∀ (i : ι),
(coeLinearMap (quotSubmodule R 𝒜 I) ∘ₗ lmap' ... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝⁶ : CommRing R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : AddCommMonoid ι
A : Type u_3
inst✝³ : CommRing A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
I : Ideal A
rel : A → A → Prop
inst✝ : GradedAlgebra 𝒜
⊢ coeLine... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | Ideal.quotDecomposition_left_inv'_aux | [748, 9] | [758, 6] | intro i | case H
R : Type u_1
inst✝⁶ : CommRing R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : AddCommMonoid ι
A : Type u_3
inst✝³ : CommRing A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
I : Ideal A
rel : A → A → Prop
inst✝ : GradedAlgebra 𝒜
⊢ ∀ (i : ι),
(coeLinearMap (quotSubmodule R 𝒜 I) ∘ₗ lmap' ... | case H
R : Type u_1
inst✝⁶ : CommRing R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : AddCommMonoid ι
A : Type u_3
inst✝³ : CommRing A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
I : Ideal A
rel : A → A → Prop
inst✝ : GradedAlgebra 𝒜
i : ι
⊢ (coeLinearMap (quotSubmodule R 𝒜 I) ∘ₗ lmap' (quotComp... | Please generate a tactic in lean4 to solve the state.
STATE:
case H
R : Type u_1
inst✝⁶ : CommRing R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : AddCommMonoid ι
A : Type u_3
inst✝³ : CommRing A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
I : Ideal A
rel : A → A → Prop
inst✝ : GradedAlgebra 𝒜
⊢ ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | Ideal.quotDecomposition_left_inv'_aux | [748, 9] | [758, 6] | ext x | case H
R : Type u_1
inst✝⁶ : CommRing R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : AddCommMonoid ι
A : Type u_3
inst✝³ : CommRing A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
I : Ideal A
rel : A → A → Prop
inst✝ : GradedAlgebra 𝒜
i : ι
⊢ (coeLinearMap (quotSubmodule R 𝒜 I) ∘ₗ lmap' (quotComp... | case H.h
R : Type u_1
inst✝⁶ : CommRing R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : AddCommMonoid ι
A : Type u_3
inst✝³ : CommRing A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
I : Ideal A
rel : A → A → Prop
inst✝ : GradedAlgebra 𝒜
i : ι
x : ↥(𝒜 i)
⊢ ((coeLinearMap (quotSubmodule R 𝒜 I) ∘ₗ ... | Please generate a tactic in lean4 to solve the state.
STATE:
case H
R : Type u_1
inst✝⁶ : CommRing R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : AddCommMonoid ι
A : Type u_3
inst✝³ : CommRing A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
I : Ideal A
rel : A → A → Prop
inst✝ : GradedAlgebra 𝒜
i ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | Ideal.quotDecomposition_left_inv'_aux | [748, 9] | [758, 6] | dsimp only [LinearMap.coe_comp, comp_apply] | case H.h
R : Type u_1
inst✝⁶ : CommRing R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : AddCommMonoid ι
A : Type u_3
inst✝³ : CommRing A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
I : Ideal A
rel : A → A → Prop
inst✝ : GradedAlgebra 𝒜
i : ι
x : ↥(𝒜 i)
⊢ ((coeLinearMap (quotSubmodule R 𝒜 I) ∘ₗ ... | case H.h
R : Type u_1
inst✝⁶ : CommRing R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : AddCommMonoid ι
A : Type u_3
inst✝³ : CommRing A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
I : Ideal A
rel : A → A → Prop
inst✝ : GradedAlgebra 𝒜
i : ι
x : ↥(𝒜 i)
⊢ (coeLinearMap (quotSubmodule R 𝒜 I)) ((l... | Please generate a tactic in lean4 to solve the state.
STATE:
case H.h
R : Type u_1
inst✝⁶ : CommRing R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : AddCommMonoid ι
A : Type u_3
inst✝³ : CommRing A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
I : Ideal A
rel : A → A → Prop
inst✝ : GradedAlgebra 𝒜
... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | Ideal.quotDecomposition_left_inv'_aux | [748, 9] | [758, 6] | change _ = (Submodule.mkQ (Submodule.restrictScalars R I)) (_) | case H.h
R : Type u_1
inst✝⁶ : CommRing R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : AddCommMonoid ι
A : Type u_3
inst✝³ : CommRing A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
I : Ideal A
rel : A → A → Prop
inst✝ : GradedAlgebra 𝒜
i : ι
x : ↥(𝒜 i)
⊢ (coeLinearMap (quotSubmodule R 𝒜 I)) ((l... | case H.h
R : Type u_1
inst✝⁶ : CommRing R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : AddCommMonoid ι
A : Type u_3
inst✝³ : CommRing A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
I : Ideal A
rel : A → A → Prop
inst✝ : GradedAlgebra 𝒜
i : ι
x : ↥(𝒜 i)
⊢ (coeLinearMap (quotSubmodule R 𝒜 I)) ((l... | Please generate a tactic in lean4 to solve the state.
STATE:
case H.h
R : Type u_1
inst✝⁶ : CommRing R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : AddCommMonoid ι
A : Type u_3
inst✝³ : CommRing A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
I : Ideal A
rel : A → A → Prop
inst✝ : GradedAlgebra 𝒜
... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | Ideal.quotDecomposition_left_inv'_aux | [748, 9] | [758, 6] | rw [lmap'_lof] | case H.h
R : Type u_1
inst✝⁶ : CommRing R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : AddCommMonoid ι
A : Type u_3
inst✝³ : CommRing A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
I : Ideal A
rel : A → A → Prop
inst✝ : GradedAlgebra 𝒜
i : ι
x : ↥(𝒜 i)
⊢ (coeLinearMap (quotSubmodule R 𝒜 I)) ((l... | case H.h
R : Type u_1
inst✝⁶ : CommRing R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : AddCommMonoid ι
A : Type u_3
inst✝³ : CommRing A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
I : Ideal A
rel : A → A → Prop
inst✝ : GradedAlgebra 𝒜
i : ι
x : ↥(𝒜 i)
⊢ (coeLinearMap (quotSubmodule R 𝒜 I)) ((l... | Please generate a tactic in lean4 to solve the state.
STATE:
case H.h
R : Type u_1
inst✝⁶ : CommRing R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : AddCommMonoid ι
A : Type u_3
inst✝³ : CommRing A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
I : Ideal A
rel : A → A → Prop
inst✝ : GradedAlgebra 𝒜
... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | Ideal.quotDecomposition_left_inv'_aux | [748, 9] | [758, 6] | simp only [lof_eq_of, coeLinearMap_of, Submodule.mkQ_apply] | case H.h
R : Type u_1
inst✝⁶ : CommRing R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : AddCommMonoid ι
A : Type u_3
inst✝³ : CommRing A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
I : Ideal A
rel : A → A → Prop
inst✝ : GradedAlgebra 𝒜
i : ι
x : ↥(𝒜 i)
⊢ (coeLinearMap (quotSubmodule R 𝒜 I)) ((l... | case H.h
R : Type u_1
inst✝⁶ : CommRing R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : AddCommMonoid ι
A : Type u_3
inst✝³ : CommRing A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
I : Ideal A
rel : A → A → Prop
inst✝ : GradedAlgebra 𝒜
i : ι
x : ↥(𝒜 i)
⊢ ↑((quotCompMap R 𝒜 I i) x) = Submodule.Q... | Please generate a tactic in lean4 to solve the state.
STATE:
case H.h
R : Type u_1
inst✝⁶ : CommRing R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : AddCommMonoid ι
A : Type u_3
inst✝³ : CommRing A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
I : Ideal A
rel : A → A → Prop
inst✝ : GradedAlgebra 𝒜
... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | Ideal.quotDecomposition_left_inv'_aux | [748, 9] | [758, 6] | rfl | case H.h
R : Type u_1
inst✝⁶ : CommRing R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : AddCommMonoid ι
A : Type u_3
inst✝³ : CommRing A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
I : Ideal A
rel : A → A → Prop
inst✝ : GradedAlgebra 𝒜
i : ι
x : ↥(𝒜 i)
⊢ ↑((quotCompMap R 𝒜 I i) x) = Submodule.Q... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case H.h
R : Type u_1
inst✝⁶ : CommRing R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : AddCommMonoid ι
A : Type u_3
inst✝³ : CommRing A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
I : Ideal A
rel : A → A → Prop
inst✝ : GradedAlgebra 𝒜
... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | Ideal.quotDecomposition_left_inv' | [760, 1] | [771, 6] | intro x | R : Type u_1
inst✝⁵ : CommRing R
ι : Type u_2
inst✝⁴ : DecidableEq ι
inst✝³ : AddCommMonoid ι
A : Type u_3
inst✝² : CommRing A
inst✝¹ : DecidableEq A
inst✝ : Algebra R A
𝒜 : ι → Submodule R A
I : Ideal A
rel : A → A → Prop
h𝒜 : GradedAlgebra 𝒜
hI : IsHomogeneous 𝒜 I
⊢ LeftInverse ⇑(coeLinearMap (quotSubmodule R 𝒜 ... | R : Type u_1
inst✝⁵ : CommRing R
ι : Type u_2
inst✝⁴ : DecidableEq ι
inst✝³ : AddCommMonoid ι
A : Type u_3
inst✝² : CommRing A
inst✝¹ : DecidableEq A
inst✝ : Algebra R A
𝒜 : ι → Submodule R A
I : Ideal A
rel : A → A → Prop
h𝒜 : GradedAlgebra 𝒜
hI : IsHomogeneous 𝒜 I
x : A ⧸ I
⊢ (coeLinearMap (quotSubmodule R 𝒜 I))... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝⁵ : CommRing R
ι : Type u_2
inst✝⁴ : DecidableEq ι
inst✝³ : AddCommMonoid ι
A : Type u_3
inst✝² : CommRing A
inst✝¹ : DecidableEq A
inst✝ : Algebra R A
𝒜 : ι → Submodule R A
I : Ideal A
rel : A → A → Prop
h𝒜 : GradedAlgebra 𝒜
hI : IsHomog... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | Ideal.quotDecomposition_left_inv' | [760, 1] | [771, 6] | obtain ⟨(a : A), rfl⟩ := Ideal.Quotient.mk_surjective x | R : Type u_1
inst✝⁵ : CommRing R
ι : Type u_2
inst✝⁴ : DecidableEq ι
inst✝³ : AddCommMonoid ι
A : Type u_3
inst✝² : CommRing A
inst✝¹ : DecidableEq A
inst✝ : Algebra R A
𝒜 : ι → Submodule R A
I : Ideal A
rel : A → A → Prop
h𝒜 : GradedAlgebra 𝒜
hI : IsHomogeneous 𝒜 I
x : A ⧸ I
⊢ (coeLinearMap (quotSubmodule R 𝒜 I))... | case intro
R : Type u_1
inst✝⁵ : CommRing R
ι : Type u_2
inst✝⁴ : DecidableEq ι
inst✝³ : AddCommMonoid ι
A : Type u_3
inst✝² : CommRing A
inst✝¹ : DecidableEq A
inst✝ : Algebra R A
𝒜 : ι → Submodule R A
I : Ideal A
rel : A → A → Prop
h𝒜 : GradedAlgebra 𝒜
hI : IsHomogeneous 𝒜 I
a : A
⊢ (coeLinearMap (quotSubmodule R... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝⁵ : CommRing R
ι : Type u_2
inst✝⁴ : DecidableEq ι
inst✝³ : AddCommMonoid ι
A : Type u_3
inst✝² : CommRing A
inst✝¹ : DecidableEq A
inst✝ : Algebra R A
𝒜 : ι → Submodule R A
I : Ideal A
rel : A → A → Prop
h𝒜 : GradedAlgebra 𝒜
hI : IsHomog... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | Ideal.quotDecomposition_left_inv' | [760, 1] | [771, 6] | conv_lhs =>
rw [quotDecomposeLaux_apply_mk, quotDecomposeLaux, LinearMap.comp_apply]
simp only [AlgEquiv.toLinearMap_apply, ← LinearMap.comp_apply] | case intro
R : Type u_1
inst✝⁵ : CommRing R
ι : Type u_2
inst✝⁴ : DecidableEq ι
inst✝³ : AddCommMonoid ι
A : Type u_3
inst✝² : CommRing A
inst✝¹ : DecidableEq A
inst✝ : Algebra R A
𝒜 : ι → Submodule R A
I : Ideal A
rel : A → A → Prop
h𝒜 : GradedAlgebra 𝒜
hI : IsHomogeneous 𝒜 I
a : A
⊢ (coeLinearMap (quotSubmodule R... | case intro
R : Type u_1
inst✝⁵ : CommRing R
ι : Type u_2
inst✝⁴ : DecidableEq ι
inst✝³ : AddCommMonoid ι
A : Type u_3
inst✝² : CommRing A
inst✝¹ : DecidableEq A
inst✝ : Algebra R A
𝒜 : ι → Submodule R A
I : Ideal A
rel : A → A → Prop
h𝒜 : GradedAlgebra 𝒜
hI : IsHomogeneous 𝒜 I
a : A
⊢ (coeLinearMap (quotSubmodule R... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
R : Type u_1
inst✝⁵ : CommRing R
ι : Type u_2
inst✝⁴ : DecidableEq ι
inst✝³ : AddCommMonoid ι
A : Type u_3
inst✝² : CommRing A
inst✝¹ : DecidableEq A
inst✝ : Algebra R A
𝒜 : ι → Submodule R A
I : Ideal A
rel : A → A → Prop
h𝒜 : GradedAlgebra 𝒜
h... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | Ideal.quotDecomposition_left_inv' | [760, 1] | [771, 6] | rw [Ideal.quotDecomposition_left_inv'_aux] | case intro
R : Type u_1
inst✝⁵ : CommRing R
ι : Type u_2
inst✝⁴ : DecidableEq ι
inst✝³ : AddCommMonoid ι
A : Type u_3
inst✝² : CommRing A
inst✝¹ : DecidableEq A
inst✝ : Algebra R A
𝒜 : ι → Submodule R A
I : Ideal A
rel : A → A → Prop
h𝒜 : GradedAlgebra 𝒜
hI : IsHomogeneous 𝒜 I
a : A
⊢ (coeLinearMap (quotSubmodule R... | case intro
R : Type u_1
inst✝⁵ : CommRing R
ι : Type u_2
inst✝⁴ : DecidableEq ι
inst✝³ : AddCommMonoid ι
A : Type u_3
inst✝² : CommRing A
inst✝¹ : DecidableEq A
inst✝ : Algebra R A
𝒜 : ι → Submodule R A
I : Ideal A
rel : A → A → Prop
h𝒜 : GradedAlgebra 𝒜
hI : IsHomogeneous 𝒜 I
a : A
⊢ ((Submodule.restrictScalars R ... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
R : Type u_1
inst✝⁵ : CommRing R
ι : Type u_2
inst✝⁴ : DecidableEq ι
inst✝³ : AddCommMonoid ι
A : Type u_3
inst✝² : CommRing A
inst✝¹ : DecidableEq A
inst✝ : Algebra R A
𝒜 : ι → Submodule R A
I : Ideal A
rel : A → A → Prop
h𝒜 : GradedAlgebra 𝒜
h... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | Ideal.quotDecomposition_left_inv' | [760, 1] | [771, 6] | conv_rhs =>
rw [← h𝒜.left_inv a]
simp only [← LinearMap.comp_apply] | case intro
R : Type u_1
inst✝⁵ : CommRing R
ι : Type u_2
inst✝⁴ : DecidableEq ι
inst✝³ : AddCommMonoid ι
A : Type u_3
inst✝² : CommRing A
inst✝¹ : DecidableEq A
inst✝ : Algebra R A
𝒜 : ι → Submodule R A
I : Ideal A
rel : A → A → Prop
h𝒜 : GradedAlgebra 𝒜
hI : IsHomogeneous 𝒜 I
a : A
⊢ ((Submodule.restrictScalars R ... | case intro
R : Type u_1
inst✝⁵ : CommRing R
ι : Type u_2
inst✝⁴ : DecidableEq ι
inst✝³ : AddCommMonoid ι
A : Type u_3
inst✝² : CommRing A
inst✝¹ : DecidableEq A
inst✝ : Algebra R A
𝒜 : ι → Submodule R A
I : Ideal A
rel : A → A → Prop
h𝒜 : GradedAlgebra 𝒜
hI : IsHomogeneous 𝒜 I
a : A
⊢ ((Submodule.restrictScalars R ... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
R : Type u_1
inst✝⁵ : CommRing R
ι : Type u_2
inst✝⁴ : DecidableEq ι
inst✝³ : AddCommMonoid ι
A : Type u_3
inst✝² : CommRing A
inst✝¹ : DecidableEq A
inst✝ : Algebra R A
𝒜 : ι → Submodule R A
I : Ideal A
rel : A → A → Prop
h𝒜 : GradedAlgebra 𝒜
h... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | Ideal.quotDecomposition_left_inv' | [760, 1] | [771, 6] | rfl | case intro
R : Type u_1
inst✝⁵ : CommRing R
ι : Type u_2
inst✝⁴ : DecidableEq ι
inst✝³ : AddCommMonoid ι
A : Type u_3
inst✝² : CommRing A
inst✝¹ : DecidableEq A
inst✝ : Algebra R A
𝒜 : ι → Submodule R A
I : Ideal A
rel : A → A → Prop
h𝒜 : GradedAlgebra 𝒜
hI : IsHomogeneous 𝒜 I
a : A
⊢ ((Submodule.restrictScalars R ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
R : Type u_1
inst✝⁵ : CommRing R
ι : Type u_2
inst✝⁴ : DecidableEq ι
inst✝³ : AddCommMonoid ι
A : Type u_3
inst✝² : CommRing A
inst✝¹ : DecidableEq A
inst✝ : Algebra R A
𝒜 : ι → Submodule R A
I : Ideal A
rel : A → A → Prop
h𝒜 : GradedAlgebra 𝒜
h... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | Ideal.quotDecomposition_right_inv' | [777, 1] | [798, 32] | rw [rightInverse_iff_comp, ← LinearMap.coe_comp, ← @LinearMap.id_coe R] | R : Type u_1
inst✝⁶ : CommRing R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : AddCommMonoid ι
A : Type u_3
inst✝³ : CommRing A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
I : Ideal A
rel : A → A → Prop
inst✝ : GradedAlgebra 𝒜
hI : IsHomogeneous 𝒜 I
⊢ Function.RightInverse ⇑(coeLinearMap (quotSu... | R : Type u_1
inst✝⁶ : CommRing R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : AddCommMonoid ι
A : Type u_3
inst✝³ : CommRing A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
I : Ideal A
rel : A → A → Prop
inst✝ : GradedAlgebra 𝒜
hI : IsHomogeneous 𝒜 I
⊢ ⇑(quotDecompose R 𝒜 hI ∘ₗ coeLinearMap (quo... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝⁶ : CommRing R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : AddCommMonoid ι
A : Type u_3
inst✝³ : CommRing A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
I : Ideal A
rel : A → A → Prop
inst✝ : GradedAlgebra 𝒜
hI : IsHo... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | Ideal.quotDecomposition_right_inv' | [777, 1] | [798, 32] | apply congr_arg | R : Type u_1
inst✝⁶ : CommRing R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : AddCommMonoid ι
A : Type u_3
inst✝³ : CommRing A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
I : Ideal A
rel : A → A → Prop
inst✝ : GradedAlgebra 𝒜
hI : IsHomogeneous 𝒜 I
⊢ ⇑(quotDecompose R 𝒜 hI ∘ₗ coeLinearMap (quo... | case h
R : Type u_1
inst✝⁶ : CommRing R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : AddCommMonoid ι
A : Type u_3
inst✝³ : CommRing A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
I : Ideal A
rel : A → A → Prop
inst✝ : GradedAlgebra 𝒜
hI : IsHomogeneous 𝒜 I
⊢ quotDecompose R 𝒜 hI ∘ₗ coeLinearMap... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝⁶ : CommRing R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : AddCommMonoid ι
A : Type u_3
inst✝³ : CommRing A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
I : Ideal A
rel : A → A → Prop
inst✝ : GradedAlgebra 𝒜
hI : IsHo... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | Ideal.quotDecomposition_right_inv' | [777, 1] | [798, 32] | apply linearMap_ext | case h
R : Type u_1
inst✝⁶ : CommRing R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : AddCommMonoid ι
A : Type u_3
inst✝³ : CommRing A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
I : Ideal A
rel : A → A → Prop
inst✝ : GradedAlgebra 𝒜
hI : IsHomogeneous 𝒜 I
⊢ quotDecompose R 𝒜 hI ∘ₗ coeLinearMap... | case h.H
R : Type u_1
inst✝⁶ : CommRing R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : AddCommMonoid ι
A : Type u_3
inst✝³ : CommRing A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
I : Ideal A
rel : A → A → Prop
inst✝ : GradedAlgebra 𝒜
hI : IsHomogeneous 𝒜 I
⊢ ∀ (i : ι),
(quotDecompose R 𝒜 ... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
R : Type u_1
inst✝⁶ : CommRing R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : AddCommMonoid ι
A : Type u_3
inst✝³ : CommRing A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
I : Ideal A
rel : A → A → Prop
inst✝ : GradedAlgebra 𝒜
hI... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | Ideal.quotDecomposition_right_inv' | [777, 1] | [798, 32] | intro i | case h.H
R : Type u_1
inst✝⁶ : CommRing R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : AddCommMonoid ι
A : Type u_3
inst✝³ : CommRing A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
I : Ideal A
rel : A → A → Prop
inst✝ : GradedAlgebra 𝒜
hI : IsHomogeneous 𝒜 I
⊢ ∀ (i : ι),
(quotDecompose R 𝒜 ... | case h.H
R : Type u_1
inst✝⁶ : CommRing R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : AddCommMonoid ι
A : Type u_3
inst✝³ : CommRing A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
I : Ideal A
rel : A → A → Prop
inst✝ : GradedAlgebra 𝒜
hI : IsHomogeneous 𝒜 I
i : ι
⊢ (quotDecompose R 𝒜 hI ∘ₗ coe... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.H
R : Type u_1
inst✝⁶ : CommRing R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : AddCommMonoid ι
A : Type u_3
inst✝³ : CommRing A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
I : Ideal A
rel : A → A → Prop
inst✝ : GradedAlgebra 𝒜
... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | Ideal.quotDecomposition_right_inv' | [777, 1] | [798, 32] | ext y | case h.H
R : Type u_1
inst✝⁶ : CommRing R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : AddCommMonoid ι
A : Type u_3
inst✝³ : CommRing A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
I : Ideal A
rel : A → A → Prop
inst✝ : GradedAlgebra 𝒜
hI : IsHomogeneous 𝒜 I
i : ι
⊢ (quotDecompose R 𝒜 hI ∘ₗ coe... | case h.H.h
R : Type u_1
inst✝⁶ : CommRing R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : AddCommMonoid ι
A : Type u_3
inst✝³ : CommRing A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
I : Ideal A
rel : A → A → Prop
inst✝ : GradedAlgebra 𝒜
hI : IsHomogeneous 𝒜 I
i : ι
y : ↥(quotSubmodule R 𝒜 I i)... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.H
R : Type u_1
inst✝⁶ : CommRing R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : AddCommMonoid ι
A : Type u_3
inst✝³ : CommRing A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
I : Ideal A
rel : A → A → Prop
inst✝ : GradedAlgebra 𝒜
... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | Ideal.quotDecomposition_right_inv' | [777, 1] | [798, 32] | obtain ⟨x, hx, hxy⟩ := y.prop | case h.H.h
R : Type u_1
inst✝⁶ : CommRing R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : AddCommMonoid ι
A : Type u_3
inst✝³ : CommRing A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
I : Ideal A
rel : A → A → Prop
inst✝ : GradedAlgebra 𝒜
hI : IsHomogeneous 𝒜 I
i : ι
y : ↥(quotSubmodule R 𝒜 I i)... | case h.H.h.intro.intro
R : Type u_1
inst✝⁶ : CommRing R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : AddCommMonoid ι
A : Type u_3
inst✝³ : CommRing A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
I : Ideal A
rel : A → A → Prop
inst✝ : GradedAlgebra 𝒜
hI : IsHomogeneous 𝒜 I
i : ι
y : ↥(quotSubmodu... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.H.h
R : Type u_1
inst✝⁶ : CommRing R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : AddCommMonoid ι
A : Type u_3
inst✝³ : CommRing A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
I : Ideal A
rel : A → A → Prop
inst✝ : GradedAlgebra �... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | Ideal.quotDecomposition_right_inv' | [777, 1] | [798, 32] | simp only [LinearMap.coe_comp, comp_apply, LinearMap.id_comp, lof_eq_of, coeLinearMap_of] | case h.H.h.intro.intro
R : Type u_1
inst✝⁶ : CommRing R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : AddCommMonoid ι
A : Type u_3
inst✝³ : CommRing A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
I : Ideal A
rel : A → A → Prop
inst✝ : GradedAlgebra 𝒜
hI : IsHomogeneous 𝒜 I
i : ι
y : ↥(quotSubmodu... | case h.H.h.intro.intro
R : Type u_1
inst✝⁶ : CommRing R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : AddCommMonoid ι
A : Type u_3
inst✝³ : CommRing A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
I : Ideal A
rel : A → A → Prop
inst✝ : GradedAlgebra 𝒜
hI : IsHomogeneous 𝒜 I
i : ι
y : ↥(quotSubmodu... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.H.h.intro.intro
R : Type u_1
inst✝⁶ : CommRing R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : AddCommMonoid ι
A : Type u_3
inst✝³ : CommRing A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
I : Ideal A
rel : A → A → Prop
inst✝ : Gra... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | Ideal.quotDecomposition_right_inv' | [777, 1] | [798, 32] | rw [← hxy, Ideal.Quotient.mkₐ_eq_mk, Ideal.quotDecomposeLaux_apply_mk, Ideal.quotDecomposeLaux] | case h.H.h.intro.intro
R : Type u_1
inst✝⁶ : CommRing R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : AddCommMonoid ι
A : Type u_3
inst✝³ : CommRing A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
I : Ideal A
rel : A → A → Prop
inst✝ : GradedAlgebra 𝒜
hI : IsHomogeneous 𝒜 I
i : ι
y : ↥(quotSubmodu... | case h.H.h.intro.intro
R : Type u_1
inst✝⁶ : CommRing R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : AddCommMonoid ι
A : Type u_3
inst✝³ : CommRing A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
I : Ideal A
rel : A → A → Prop
inst✝ : GradedAlgebra 𝒜
hI : IsHomogeneous 𝒜 I
i : ι
y : ↥(quotSubmodu... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.H.h.intro.intro
R : Type u_1
inst✝⁶ : CommRing R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : AddCommMonoid ι
A : Type u_3
inst✝³ : CommRing A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
I : Ideal A
rel : A → A → Prop
inst✝ : Gra... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | Ideal.quotDecomposition_right_inv' | [777, 1] | [798, 32] | simp only [LinearMap.coe_comp, comp_apply] | case h.H.h.intro.intro
R : Type u_1
inst✝⁶ : CommRing R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : AddCommMonoid ι
A : Type u_3
inst✝³ : CommRing A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
I : Ideal A
rel : A → A → Prop
inst✝ : GradedAlgebra 𝒜
hI : IsHomogeneous 𝒜 I
i : ι
y : ↥(quotSubmodu... | case h.H.h.intro.intro
R : Type u_1
inst✝⁶ : CommRing R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : AddCommMonoid ι
A : Type u_3
inst✝³ : CommRing A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
I : Ideal A
rel : A → A → Prop
inst✝ : GradedAlgebra 𝒜
hI : IsHomogeneous 𝒜 I
i : ι
y : ↥(quotSubmodu... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.H.h.intro.intro
R : Type u_1
inst✝⁶ : CommRing R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : AddCommMonoid ι
A : Type u_3
inst✝³ : CommRing A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
I : Ideal A
rel : A → A → Prop
inst✝ : Gra... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | Ideal.quotDecomposition_right_inv' | [777, 1] | [798, 32] | change lmap' _ (decompose 𝒜 x) = _ | case h.H.h.intro.intro
R : Type u_1
inst✝⁶ : CommRing R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : AddCommMonoid ι
A : Type u_3
inst✝³ : CommRing A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
I : Ideal A
rel : A → A → Prop
inst✝ : GradedAlgebra 𝒜
hI : IsHomogeneous 𝒜 I
i : ι
y : ↥(quotSubmodu... | case h.H.h.intro.intro
R : Type u_1
inst✝⁶ : CommRing R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : AddCommMonoid ι
A : Type u_3
inst✝³ : CommRing A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
I : Ideal A
rel : A → A → Prop
inst✝ : GradedAlgebra 𝒜
hI : IsHomogeneous 𝒜 I
i : ι
y : ↥(quotSubmodu... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.H.h.intro.intro
R : Type u_1
inst✝⁶ : CommRing R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : AddCommMonoid ι
A : Type u_3
inst✝³ : CommRing A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
I : Ideal A
rel : A → A → Prop
inst✝ : Gra... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | Ideal.quotDecomposition_right_inv' | [777, 1] | [798, 32] | suffices decompose 𝒜 x = lof R ι (fun i => 𝒜 i) i (⟨x, hx⟩ : 𝒜 i) by
rw [this, lmap'_lof, lof_eq_of]
apply congr_arg₂ _ rfl
rw [quotCompMap]
simp only [Ideal.Quotient.mkₐ_eq_mk, Submodule.coe_mk, LinearMap.coe_mk]
rw [← Subtype.coe_inj, Subtype.coe_mk, ← hxy]
simp only [Ideal.Quotient.mkₐ_eq_mk]
rfl | case h.H.h.intro.intro
R : Type u_1
inst✝⁶ : CommRing R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : AddCommMonoid ι
A : Type u_3
inst✝³ : CommRing A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
I : Ideal A
rel : A → A → Prop
inst✝ : GradedAlgebra 𝒜
hI : IsHomogeneous 𝒜 I
i : ι
y : ↥(quotSubmodu... | case h.H.h.intro.intro
R : Type u_1
inst✝⁶ : CommRing R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : AddCommMonoid ι
A : Type u_3
inst✝³ : CommRing A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
I : Ideal A
rel : A → A → Prop
inst✝ : GradedAlgebra 𝒜
hI : IsHomogeneous 𝒜 I
i : ι
y : ↥(quotSubmodu... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.H.h.intro.intro
R : Type u_1
inst✝⁶ : CommRing R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : AddCommMonoid ι
A : Type u_3
inst✝³ : CommRing A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
I : Ideal A
rel : A → A → Prop
inst✝ : Gra... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | Ideal.quotDecomposition_right_inv' | [777, 1] | [798, 32] | conv_lhs => rw [← Subtype.coe_mk x hx] | case h.H.h.intro.intro
R : Type u_1
inst✝⁶ : CommRing R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : AddCommMonoid ι
A : Type u_3
inst✝³ : CommRing A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
I : Ideal A
rel : A → A → Prop
inst✝ : GradedAlgebra 𝒜
hI : IsHomogeneous 𝒜 I
i : ι
y : ↥(quotSubmodu... | case h.H.h.intro.intro
R : Type u_1
inst✝⁶ : CommRing R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : AddCommMonoid ι
A : Type u_3
inst✝³ : CommRing A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
I : Ideal A
rel : A → A → Prop
inst✝ : GradedAlgebra 𝒜
hI : IsHomogeneous 𝒜 I
i : ι
y : ↥(quotSubmodu... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.H.h.intro.intro
R : Type u_1
inst✝⁶ : CommRing R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : AddCommMonoid ι
A : Type u_3
inst✝³ : CommRing A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
I : Ideal A
rel : A → A → Prop
inst✝ : Gra... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | Ideal.quotDecomposition_right_inv' | [777, 1] | [798, 32] | rw [decompose_coe, lof_eq_of] | case h.H.h.intro.intro
R : Type u_1
inst✝⁶ : CommRing R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : AddCommMonoid ι
A : Type u_3
inst✝³ : CommRing A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
I : Ideal A
rel : A → A → Prop
inst✝ : GradedAlgebra 𝒜
hI : IsHomogeneous 𝒜 I
i : ι
y : ↥(quotSubmodu... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.H.h.intro.intro
R : Type u_1
inst✝⁶ : CommRing R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : AddCommMonoid ι
A : Type u_3
inst✝³ : CommRing A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
I : Ideal A
rel : A → A → Prop
inst✝ : Gra... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | Ideal.quotDecomposition_right_inv' | [777, 1] | [798, 32] | rw [this, lmap'_lof, lof_eq_of] | R : Type u_1
inst✝⁶ : CommRing R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : AddCommMonoid ι
A : Type u_3
inst✝³ : CommRing A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
I : Ideal A
rel : A → A → Prop
inst✝ : GradedAlgebra 𝒜
hI : IsHomogeneous 𝒜 I
i : ι
y : ↥(quotSubmodule R 𝒜 I i)
x : A
hx :... | R : Type u_1
inst✝⁶ : CommRing R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : AddCommMonoid ι
A : Type u_3
inst✝³ : CommRing A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
I : Ideal A
rel : A → A → Prop
inst✝ : GradedAlgebra 𝒜
hI : IsHomogeneous 𝒜 I
i : ι
y : ↥(quotSubmodule R 𝒜 I i)
x : A
hx :... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝⁶ : CommRing R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : AddCommMonoid ι
A : Type u_3
inst✝³ : CommRing A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
I : Ideal A
rel : A → A → Prop
inst✝ : GradedAlgebra 𝒜
hI : IsHo... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | Ideal.quotDecomposition_right_inv' | [777, 1] | [798, 32] | apply congr_arg₂ _ rfl | R : Type u_1
inst✝⁶ : CommRing R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : AddCommMonoid ι
A : Type u_3
inst✝³ : CommRing A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
I : Ideal A
rel : A → A → Prop
inst✝ : GradedAlgebra 𝒜
hI : IsHomogeneous 𝒜 I
i : ι
y : ↥(quotSubmodule R 𝒜 I i)
x : A
hx :... | R : Type u_1
inst✝⁶ : CommRing R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : AddCommMonoid ι
A : Type u_3
inst✝³ : CommRing A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
I : Ideal A
rel : A → A → Prop
inst✝ : GradedAlgebra 𝒜
hI : IsHomogeneous 𝒜 I
i : ι
y : ↥(quotSubmodule R 𝒜 I i)
x : A
hx :... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝⁶ : CommRing R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : AddCommMonoid ι
A : Type u_3
inst✝³ : CommRing A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
I : Ideal A
rel : A → A → Prop
inst✝ : GradedAlgebra 𝒜
hI : IsHo... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | Ideal.quotDecomposition_right_inv' | [777, 1] | [798, 32] | rw [quotCompMap] | R : Type u_1
inst✝⁶ : CommRing R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : AddCommMonoid ι
A : Type u_3
inst✝³ : CommRing A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
I : Ideal A
rel : A → A → Prop
inst✝ : GradedAlgebra 𝒜
hI : IsHomogeneous 𝒜 I
i : ι
y : ↥(quotSubmodule R 𝒜 I i)
x : A
hx :... | R : Type u_1
inst✝⁶ : CommRing R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : AddCommMonoid ι
A : Type u_3
inst✝³ : CommRing A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
I : Ideal A
rel : A → A → Prop
inst✝ : GradedAlgebra 𝒜
hI : IsHomogeneous 𝒜 I
i : ι
y : ↥(quotSubmodule R 𝒜 I i)
x : A
hx :... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝⁶ : CommRing R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : AddCommMonoid ι
A : Type u_3
inst✝³ : CommRing A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
I : Ideal A
rel : A → A → Prop
inst✝ : GradedAlgebra 𝒜
hI : IsHo... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | Ideal.quotDecomposition_right_inv' | [777, 1] | [798, 32] | simp only [Ideal.Quotient.mkₐ_eq_mk, Submodule.coe_mk, LinearMap.coe_mk] | R : Type u_1
inst✝⁶ : CommRing R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : AddCommMonoid ι
A : Type u_3
inst✝³ : CommRing A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
I : Ideal A
rel : A → A → Prop
inst✝ : GradedAlgebra 𝒜
hI : IsHomogeneous 𝒜 I
i : ι
y : ↥(quotSubmodule R 𝒜 I i)
x : A
hx :... | R : Type u_1
inst✝⁶ : CommRing R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : AddCommMonoid ι
A : Type u_3
inst✝³ : CommRing A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
I : Ideal A
rel : A → A → Prop
inst✝ : GradedAlgebra 𝒜
hI : IsHomogeneous 𝒜 I
i : ι
y : ↥(quotSubmodule R 𝒜 I i)
x : A
hx :... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝⁶ : CommRing R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : AddCommMonoid ι
A : Type u_3
inst✝³ : CommRing A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
I : Ideal A
rel : A → A → Prop
inst✝ : GradedAlgebra 𝒜
hI : IsHo... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | Ideal.quotDecomposition_right_inv' | [777, 1] | [798, 32] | rw [← Subtype.coe_inj, Subtype.coe_mk, ← hxy] | R : Type u_1
inst✝⁶ : CommRing R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : AddCommMonoid ι
A : Type u_3
inst✝³ : CommRing A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
I : Ideal A
rel : A → A → Prop
inst✝ : GradedAlgebra 𝒜
hI : IsHomogeneous 𝒜 I
i : ι
y : ↥(quotSubmodule R 𝒜 I i)
x : A
hx :... | R : Type u_1
inst✝⁶ : CommRing R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : AddCommMonoid ι
A : Type u_3
inst✝³ : CommRing A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
I : Ideal A
rel : A → A → Prop
inst✝ : GradedAlgebra 𝒜
hI : IsHomogeneous 𝒜 I
i : ι
y : ↥(quotSubmodule R 𝒜 I i)
x : A
hx :... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝⁶ : CommRing R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : AddCommMonoid ι
A : Type u_3
inst✝³ : CommRing A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
I : Ideal A
rel : A → A → Prop
inst✝ : GradedAlgebra 𝒜
hI : IsHo... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | Ideal.quotDecomposition_right_inv' | [777, 1] | [798, 32] | simp only [Ideal.Quotient.mkₐ_eq_mk] | R : Type u_1
inst✝⁶ : CommRing R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : AddCommMonoid ι
A : Type u_3
inst✝³ : CommRing A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
I : Ideal A
rel : A → A → Prop
inst✝ : GradedAlgebra 𝒜
hI : IsHomogeneous 𝒜 I
i : ι
y : ↥(quotSubmodule R 𝒜 I i)
x : A
hx :... | R : Type u_1
inst✝⁶ : CommRing R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : AddCommMonoid ι
A : Type u_3
inst✝³ : CommRing A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
I : Ideal A
rel : A → A → Prop
inst✝ : GradedAlgebra 𝒜
hI : IsHomogeneous 𝒜 I
i : ι
y : ↥(quotSubmodule R 𝒜 I i)
x : A
hx :... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝⁶ : CommRing R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : AddCommMonoid ι
A : Type u_3
inst✝³ : CommRing A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
I : Ideal A
rel : A → A → Prop
inst✝ : GradedAlgebra 𝒜
hI : IsHo... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | Ideal.quotDecomposition_right_inv' | [777, 1] | [798, 32] | rfl | R : Type u_1
inst✝⁶ : CommRing R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : AddCommMonoid ι
A : Type u_3
inst✝³ : CommRing A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
I : Ideal A
rel : A → A → Prop
inst✝ : GradedAlgebra 𝒜
hI : IsHomogeneous 𝒜 I
i : ι
y : ↥(quotSubmodule R 𝒜 I i)
x : A
hx :... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝⁶ : CommRing R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : AddCommMonoid ι
A : Type u_3
inst✝³ : CommRing A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
I : Ideal A
rel : A → A → Prop
inst✝ : GradedAlgebra 𝒜
hI : IsHo... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | Ideal.mem_quotSubmodule_iff | [810, 1] | [812, 72] | rw [Ideal.quotSubmodule, Submodule.mem_map, Ideal.Quotient.mkₐ_eq_mk] | R : Type u_1
inst✝⁵ : CommRing R
ι : Type u_2
inst✝⁴ : DecidableEq ι
inst✝³ : AddCommMonoid ι
A : Type u_3
inst✝² : CommRing A
inst✝¹ : DecidableEq A
inst✝ : Algebra R A
𝒜 : ι → Submodule R A
I : Ideal A
rel : A → A → Prop
i : ι
g : A ⧸ I
⊢ g ∈ quotSubmodule R 𝒜 I i ↔ ∃ a ∈ 𝒜 i, (Quotient.mk I) a = g | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝⁵ : CommRing R
ι : Type u_2
inst✝⁴ : DecidableEq ι
inst✝³ : AddCommMonoid ι
A : Type u_3
inst✝² : CommRing A
inst✝¹ : DecidableEq A
inst✝ : Algebra R A
𝒜 : ι → Submodule R A
I : Ideal A
rel : A → A → Prop
i : ι
g : A ⧸ I
⊢ g ∈ quotSubmodule... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | unused_files/Ideal.lean | Ideal.image_eq_map_of_surjective | [6, 1] | [13, 45] | symm | A : Type u_1
B : Type u_2
inst✝¹ : Semiring A
inst✝ : Semiring B
f : A →+* B
I : Ideal A
hf : Function.Surjective ⇑f
⊢ ⇑f '' ↑I = ↑(map f I) | A : Type u_1
B : Type u_2
inst✝¹ : Semiring A
inst✝ : Semiring B
f : A →+* B
I : Ideal A
hf : Function.Surjective ⇑f
⊢ ↑(map f I) = ⇑f '' ↑I | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
B : Type u_2
inst✝¹ : Semiring A
inst✝ : Semiring B
f : A →+* B
I : Ideal A
hf : Function.Surjective ⇑f
⊢ ⇑f '' ↑I = ↑(map f I)
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | unused_files/Ideal.lean | Ideal.image_eq_map_of_surjective | [6, 1] | [13, 45] | ext x | A : Type u_1
B : Type u_2
inst✝¹ : Semiring A
inst✝ : Semiring B
f : A →+* B
I : Ideal A
hf : Function.Surjective ⇑f
⊢ ↑(map f I) = ⇑f '' ↑I | case h
A : Type u_1
B : Type u_2
inst✝¹ : Semiring A
inst✝ : Semiring B
f : A →+* B
I : Ideal A
hf : Function.Surjective ⇑f
x : B
⊢ x ∈ ↑(map f I) ↔ x ∈ ⇑f '' ↑I | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
B : Type u_2
inst✝¹ : Semiring A
inst✝ : Semiring B
f : A →+* B
I : Ideal A
hf : Function.Surjective ⇑f
⊢ ↑(map f I) = ⇑f '' ↑I
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | unused_files/Ideal.lean | Ideal.image_eq_map_of_surjective | [6, 1] | [13, 45] | simp only [Set.mem_image, SetLike.mem_coe] | case h
A : Type u_1
B : Type u_2
inst✝¹ : Semiring A
inst✝ : Semiring B
f : A →+* B
I : Ideal A
hf : Function.Surjective ⇑f
x : B
⊢ x ∈ ↑(map f I) ↔ x ∈ ⇑f '' ↑I | case h
A : Type u_1
B : Type u_2
inst✝¹ : Semiring A
inst✝ : Semiring B
f : A →+* B
I : Ideal A
hf : Function.Surjective ⇑f
x : B
⊢ x ∈ map f I ↔ ∃ x_1 ∈ I, f x_1 = x | Please generate a tactic in lean4 to solve the state.
STATE:
case h
A : Type u_1
B : Type u_2
inst✝¹ : Semiring A
inst✝ : Semiring B
f : A →+* B
I : Ideal A
hf : Function.Surjective ⇑f
x : B
⊢ x ∈ ↑(map f I) ↔ x ∈ ⇑f '' ↑I
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | unused_files/Ideal.lean | Ideal.image_eq_map_of_surjective | [6, 1] | [13, 45] | apply Ideal.mem_map_iff_of_surjective _ hf | case h
A : Type u_1
B : Type u_2
inst✝¹ : Semiring A
inst✝ : Semiring B
f : A →+* B
I : Ideal A
hf : Function.Surjective ⇑f
x : B
⊢ x ∈ map f I ↔ ∃ x_1 ∈ I, f x_1 = x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
A : Type u_1
B : Type u_2
inst✝¹ : Semiring A
inst✝ : Semiring B
f : A →+* B
I : Ideal A
hf : Function.Surjective ⇑f
x : B
⊢ x ∈ map f I ↔ ∃ x_1 ∈ I, f x_1 = x
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/RingTheory/SubmoduleMem.lean | Submodule.mem_span_iff_exists_sum | [8, 1] | [13, 80] | rw [← top_smul (span R (Set.range f)), mem_ideal_smul_span_iff_exists_sum] | R : Type u_1
inst✝² : CommSemiring R
M : Type u_2
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
ι : Type u_3
f : ι → M
x : M
⊢ x ∈ span R (Set.range f) ↔ ∃ a, (a.sum fun i c => c • f i) = x | R : Type u_1
inst✝² : CommSemiring R
M : Type u_2
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
ι : Type u_3
f : ι → M
x : M
⊢ (∃ a, ∃ (_ : ∀ (i : ι), a i ∈ ⊤), (a.sum fun i c => c • f i) = x) ↔ ∃ a, (a.sum fun i c => c • f i) = x | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝² : CommSemiring R
M : Type u_2
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
ι : Type u_3
f : ι → M
x : M
⊢ x ∈ span R (Set.range f) ↔ ∃ a, (a.sum fun i c => c • f i) = x
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/RingTheory/SubmoduleMem.lean | Submodule.mem_span_iff_exists_sum | [8, 1] | [13, 80] | exact exists_congr fun a => ⟨fun ⟨_, h⟩ => h, fun h => ⟨fun i => mem_top, h⟩⟩ | R : Type u_1
inst✝² : CommSemiring R
M : Type u_2
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
ι : Type u_3
f : ι → M
x : M
⊢ (∃ a, ∃ (_ : ∀ (i : ι), a i ∈ ⊤), (a.sum fun i c => c • f i) = x) ↔ ∃ a, (a.sum fun i c => c • f i) = x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝² : CommSemiring R
M : Type u_2
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
ι : Type u_3
f : ι → M
x : M
⊢ (∃ a, ∃ (_ : ∀ (i : ι), a i ∈ ⊤), (a.sum fun i c => c • f i) = x) ↔ ∃ a, (a.sum fun i c => c • f i) = x
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/RingTheory/SubmoduleMem.lean | Submodule.mem_span_iff_exists_sum' | [16, 1] | [28, 12] | rw [← top_smul (span R (f '' s)), mem_ideal_smul_span_iff_exists_sum'] | R : Type u_1
inst✝² : CommSemiring R
M : Type u_2
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
ι : Type u_3
s : Set ι
f : ι → M
x : M
⊢ x ∈ span R (f '' s) ↔ ∃ a, (a.sum fun i c => c • f ↑i) = x | R : Type u_1
inst✝² : CommSemiring R
M : Type u_2
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
ι : Type u_3
s : Set ι
f : ι → M
x : M
⊢ (∃ a, ∃ (_ : ∀ (i : ↑s), a i ∈ ⊤), (a.sum fun i c => c • f ↑i) = x) ↔ ∃ a, (a.sum fun i c => c • f ↑i) = x | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝² : CommSemiring R
M : Type u_2
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
ι : Type u_3
s : Set ι
f : ι → M
x : M
⊢ x ∈ span R (f '' s) ↔ ∃ a, (a.sum fun i c => c • f ↑i) = x
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/RingTheory/SubmoduleMem.lean | Submodule.mem_span_iff_exists_sum' | [16, 1] | [28, 12] | apply exists_congr | R : Type u_1
inst✝² : CommSemiring R
M : Type u_2
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
ι : Type u_3
s : Set ι
f : ι → M
x : M
⊢ (∃ a, ∃ (_ : ∀ (i : ↑s), a i ∈ ⊤), (a.sum fun i c => c • f ↑i) = x) ↔ ∃ a, (a.sum fun i c => c • f ↑i) = x | case h
R : Type u_1
inst✝² : CommSemiring R
M : Type u_2
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
ι : Type u_3
s : Set ι
f : ι → M
x : M
⊢ ∀ (a : ↑s →₀ R), (∃ (_ : ∀ (i : ↑s), a i ∈ ⊤), (a.sum fun i c => c • f ↑i) = x) ↔ (a.sum fun i c => c • f ↑i) = x | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝² : CommSemiring R
M : Type u_2
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
ι : Type u_3
s : Set ι
f : ι → M
x : M
⊢ (∃ a, ∃ (_ : ∀ (i : ↑s), a i ∈ ⊤), (a.sum fun i c => c • f ↑i) = x) ↔ ∃ a, (a.sum fun i c => c • f ↑i) = x
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/RingTheory/SubmoduleMem.lean | Submodule.mem_span_iff_exists_sum' | [16, 1] | [28, 12] | intro a | case h
R : Type u_1
inst✝² : CommSemiring R
M : Type u_2
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
ι : Type u_3
s : Set ι
f : ι → M
x : M
⊢ ∀ (a : ↑s →₀ R), (∃ (_ : ∀ (i : ↑s), a i ∈ ⊤), (a.sum fun i c => c • f ↑i) = x) ↔ (a.sum fun i c => c • f ↑i) = x | case h
R : Type u_1
inst✝² : CommSemiring R
M : Type u_2
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
ι : Type u_3
s : Set ι
f : ι → M
x : M
a : ↑s →₀ R
⊢ (∃ (_ : ∀ (i : ↑s), a i ∈ ⊤), (a.sum fun i c => c • f ↑i) = x) ↔ (a.sum fun i c => c • f ↑i) = x | Please generate a tactic in lean4 to solve the state.
STATE:
case h
R : Type u_1
inst✝² : CommSemiring R
M : Type u_2
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
ι : Type u_3
s : Set ι
f : ι → M
x : M
⊢ ∀ (a : ↑s →₀ R), (∃ (_ : ∀ (i : ↑s), a i ∈ ⊤), (a.sum fun i c => c • f ↑i) = x) ↔ (a.sum fun i c => c • f ↑i) = x
TAC... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/RingTheory/SubmoduleMem.lean | Submodule.mem_span_iff_exists_sum' | [16, 1] | [28, 12] | constructor | case h
R : Type u_1
inst✝² : CommSemiring R
M : Type u_2
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
ι : Type u_3
s : Set ι
f : ι → M
x : M
a : ↑s →₀ R
⊢ (∃ (_ : ∀ (i : ↑s), a i ∈ ⊤), (a.sum fun i c => c • f ↑i) = x) ↔ (a.sum fun i c => c • f ↑i) = x | case h.mp
R : Type u_1
inst✝² : CommSemiring R
M : Type u_2
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
ι : Type u_3
s : Set ι
f : ι → M
x : M
a : ↑s →₀ R
⊢ (∃ (_ : ∀ (i : ↑s), a i ∈ ⊤), (a.sum fun i c => c • f ↑i) = x) → (a.sum fun i c => c • f ↑i) = x
case h.mpr
R : Type u_1
inst✝² : CommSemiring R
M : Type u_2
inst... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
R : Type u_1
inst✝² : CommSemiring R
M : Type u_2
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
ι : Type u_3
s : Set ι
f : ι → M
x : M
a : ↑s →₀ R
⊢ (∃ (_ : ∀ (i : ↑s), a i ∈ ⊤), (a.sum fun i c => c • f ↑i) = x) ↔ (a.sum fun i c => c • f ↑i) = x
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/RingTheory/SubmoduleMem.lean | Submodule.mem_span_iff_exists_sum' | [16, 1] | [28, 12] | . rintro ⟨_, h⟩
exact h | case h.mp
R : Type u_1
inst✝² : CommSemiring R
M : Type u_2
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
ι : Type u_3
s : Set ι
f : ι → M
x : M
a : ↑s →₀ R
⊢ (∃ (_ : ∀ (i : ↑s), a i ∈ ⊤), (a.sum fun i c => c • f ↑i) = x) → (a.sum fun i c => c • f ↑i) = x
case h.mpr
R : Type u_1
inst✝² : CommSemiring R
M : Type u_2
inst... | case h.mpr
R : Type u_1
inst✝² : CommSemiring R
M : Type u_2
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
ι : Type u_3
s : Set ι
f : ι → M
x : M
a : ↑s →₀ R
⊢ (a.sum fun i c => c • f ↑i) = x → ∃ (_ : ∀ (i : ↑s), a i ∈ ⊤), (a.sum fun i c => c • f ↑i) = x | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp
R : Type u_1
inst✝² : CommSemiring R
M : Type u_2
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
ι : Type u_3
s : Set ι
f : ι → M
x : M
a : ↑s →₀ R
⊢ (∃ (_ : ∀ (i : ↑s), a i ∈ ⊤), (a.sum fun i c => c • f ↑i) = x) → (a.sum fun i c => c • f ↑i) = x
case... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/RingTheory/SubmoduleMem.lean | Submodule.mem_span_iff_exists_sum' | [16, 1] | [28, 12] | . intro h
simp only [mem_top, Subtype.forall, implies_true, exists_prop, true_and]
exact h | case h.mpr
R : Type u_1
inst✝² : CommSemiring R
M : Type u_2
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
ι : Type u_3
s : Set ι
f : ι → M
x : M
a : ↑s →₀ R
⊢ (a.sum fun i c => c • f ↑i) = x → ∃ (_ : ∀ (i : ↑s), a i ∈ ⊤), (a.sum fun i c => c • f ↑i) = x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr
R : Type u_1
inst✝² : CommSemiring R
M : Type u_2
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
ι : Type u_3
s : Set ι
f : ι → M
x : M
a : ↑s →₀ R
⊢ (a.sum fun i c => c • f ↑i) = x → ∃ (_ : ∀ (i : ↑s), a i ∈ ⊤), (a.sum fun i c => c • f ↑i) = x
TACTIC... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/RingTheory/SubmoduleMem.lean | Submodule.mem_span_iff_exists_sum' | [16, 1] | [28, 12] | rintro ⟨_, h⟩ | case h.mp
R : Type u_1
inst✝² : CommSemiring R
M : Type u_2
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
ι : Type u_3
s : Set ι
f : ι → M
x : M
a : ↑s →₀ R
⊢ (∃ (_ : ∀ (i : ↑s), a i ∈ ⊤), (a.sum fun i c => c • f ↑i) = x) → (a.sum fun i c => c • f ↑i) = x | case h.mp.intro
R : Type u_1
inst✝² : CommSemiring R
M : Type u_2
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
ι : Type u_3
s : Set ι
f : ι → M
x : M
a : ↑s →₀ R
w✝ : ∀ (i : ↑s), a i ∈ ⊤
h : (a.sum fun i c => c • f ↑i) = x
⊢ (a.sum fun i c => c • f ↑i) = x | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp
R : Type u_1
inst✝² : CommSemiring R
M : Type u_2
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
ι : Type u_3
s : Set ι
f : ι → M
x : M
a : ↑s →₀ R
⊢ (∃ (_ : ∀ (i : ↑s), a i ∈ ⊤), (a.sum fun i c => c • f ↑i) = x) → (a.sum fun i c => c • f ↑i) = x
TACTI... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/RingTheory/SubmoduleMem.lean | Submodule.mem_span_iff_exists_sum' | [16, 1] | [28, 12] | exact h | case h.mp.intro
R : Type u_1
inst✝² : CommSemiring R
M : Type u_2
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
ι : Type u_3
s : Set ι
f : ι → M
x : M
a : ↑s →₀ R
w✝ : ∀ (i : ↑s), a i ∈ ⊤
h : (a.sum fun i c => c • f ↑i) = x
⊢ (a.sum fun i c => c • f ↑i) = x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp.intro
R : Type u_1
inst✝² : CommSemiring R
M : Type u_2
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
ι : Type u_3
s : Set ι
f : ι → M
x : M
a : ↑s →₀ R
w✝ : ∀ (i : ↑s), a i ∈ ⊤
h : (a.sum fun i c => c • f ↑i) = x
⊢ (a.sum fun i c => c • f ↑i) = x
TAC... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/RingTheory/SubmoduleMem.lean | Submodule.mem_span_iff_exists_sum' | [16, 1] | [28, 12] | intro h | case h.mpr
R : Type u_1
inst✝² : CommSemiring R
M : Type u_2
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
ι : Type u_3
s : Set ι
f : ι → M
x : M
a : ↑s →₀ R
⊢ (a.sum fun i c => c • f ↑i) = x → ∃ (_ : ∀ (i : ↑s), a i ∈ ⊤), (a.sum fun i c => c • f ↑i) = x | case h.mpr
R : Type u_1
inst✝² : CommSemiring R
M : Type u_2
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
ι : Type u_3
s : Set ι
f : ι → M
x : M
a : ↑s →₀ R
h : (a.sum fun i c => c • f ↑i) = x
⊢ ∃ (_ : ∀ (i : ↑s), a i ∈ ⊤), (a.sum fun i c => c • f ↑i) = x | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr
R : Type u_1
inst✝² : CommSemiring R
M : Type u_2
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
ι : Type u_3
s : Set ι
f : ι → M
x : M
a : ↑s →₀ R
⊢ (a.sum fun i c => c • f ↑i) = x → ∃ (_ : ∀ (i : ↑s), a i ∈ ⊤), (a.sum fun i c => c • f ↑i) = x
TACTIC... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/RingTheory/SubmoduleMem.lean | Submodule.mem_span_iff_exists_sum' | [16, 1] | [28, 12] | simp only [mem_top, Subtype.forall, implies_true, exists_prop, true_and] | case h.mpr
R : Type u_1
inst✝² : CommSemiring R
M : Type u_2
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
ι : Type u_3
s : Set ι
f : ι → M
x : M
a : ↑s →₀ R
h : (a.sum fun i c => c • f ↑i) = x
⊢ ∃ (_ : ∀ (i : ↑s), a i ∈ ⊤), (a.sum fun i c => c • f ↑i) = x | case h.mpr
R : Type u_1
inst✝² : CommSemiring R
M : Type u_2
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
ι : Type u_3
s : Set ι
f : ι → M
x : M
a : ↑s →₀ R
h : (a.sum fun i c => c • f ↑i) = x
⊢ (a.sum fun i c => c • f ↑i) = x | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr
R : Type u_1
inst✝² : CommSemiring R
M : Type u_2
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
ι : Type u_3
s : Set ι
f : ι → M
x : M
a : ↑s →₀ R
h : (a.sum fun i c => c • f ↑i) = x
⊢ ∃ (_ : ∀ (i : ↑s), a i ∈ ⊤), (a.sum fun i c => c • f ↑i) = x
TACT... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/RingTheory/SubmoduleMem.lean | Submodule.mem_span_iff_exists_sum' | [16, 1] | [28, 12] | exact h | case h.mpr
R : Type u_1
inst✝² : CommSemiring R
M : Type u_2
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
ι : Type u_3
s : Set ι
f : ι → M
x : M
a : ↑s →₀ R
h : (a.sum fun i c => c • f ↑i) = x
⊢ (a.sum fun i c => c • f ↑i) = x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr
R : Type u_1
inst✝² : CommSemiring R
M : Type u_2
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
ι : Type u_3
s : Set ι
f : ι → M
x : M
a : ↑s →₀ R
h : (a.sum fun i c => c • f ↑i) = x
⊢ (a.sum fun i c => c • f ↑i) = x
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/RingTheory/SubmoduleMem.lean | Submodule.mem_span_set_iff_exists_sum | [31, 1] | [37, 80] | conv_lhs => rw [← Set.image_id s] | R : Type u_1
inst✝² : CommSemiring R
M : Type u_2
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
s : Set M
x : M
⊢ x ∈ span R s ↔ ∃ a, (a.sum fun i c => c • ↑i) = x | R : Type u_1
inst✝² : CommSemiring R
M : Type u_2
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
s : Set M
x : M
⊢ x ∈ span R (id '' s) ↔ ∃ a, (a.sum fun i c => c • ↑i) = x | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝² : CommSemiring R
M : Type u_2
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
s : Set M
x : M
⊢ x ∈ span R s ↔ ∃ a, (a.sum fun i c => c • ↑i) = x
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/RingTheory/SubmoduleMem.lean | Submodule.mem_span_set_iff_exists_sum | [31, 1] | [37, 80] | rw [← top_smul (span R (id '' s)), mem_ideal_smul_span_iff_exists_sum'] | R : Type u_1
inst✝² : CommSemiring R
M : Type u_2
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
s : Set M
x : M
⊢ x ∈ span R (id '' s) ↔ ∃ a, (a.sum fun i c => c • ↑i) = x | R : Type u_1
inst✝² : CommSemiring R
M : Type u_2
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
s : Set M
x : M
⊢ (∃ a, ∃ (_ : ∀ (i : ↑s), a i ∈ ⊤), (a.sum fun i c => c • id ↑i) = x) ↔ ∃ a, (a.sum fun i c => c • ↑i) = x | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝² : CommSemiring R
M : Type u_2
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
s : Set M
x : M
⊢ x ∈ span R (id '' s) ↔ ∃ a, (a.sum fun i c => c • ↑i) = x
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/RingTheory/SubmoduleMem.lean | Submodule.mem_span_set_iff_exists_sum | [31, 1] | [37, 80] | exact exists_congr fun a => ⟨fun ⟨_, h⟩ => h, fun h => ⟨fun i => mem_top, h⟩⟩ | R : Type u_1
inst✝² : CommSemiring R
M : Type u_2
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
s : Set M
x : M
⊢ (∃ a, ∃ (_ : ∀ (i : ↑s), a i ∈ ⊤), (a.sum fun i c => c • id ↑i) = x) ↔ ∃ a, (a.sum fun i c => c • ↑i) = x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝² : CommSemiring R
M : Type u_2
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
s : Set M
x : M
⊢ (∃ a, ∃ (_ : ∀ (i : ↑s), a i ∈ ⊤), (a.sum fun i c => c • id ↑i) = x) ↔ ∃ a, (a.sum fun i c => c • ↑i) = x
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Exponential.lean | DividedPowerAlgebra.coeff_exp' | [35, 1] | [37, 29] | simp only [coeff_mk, exp'] | R : Type u_1
inst✝⁶ : CommRing R
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R A
M : Type u_3
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : Module A M
inst✝ : IsScalarTower R A M
m : M
n : ℕ
⊢ (coeff (DividedPowerAlgebra R M) n) (exp' R m) = dp R n m | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝⁶ : CommRing R
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R A
M : Type u_3
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : Module A M
inst✝ : IsScalarTower R A M
m : M
n : ℕ
⊢ (coeff (DividedPowerAlgebra R M) n) (exp' R m) = dp ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Exponential.lean | DividedPowerAlgebra.isExponential_exp' | [39, 1] | [44, 49] | rw [isExponential_iff] | R : Type u_1
inst✝⁶ : CommRing R
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R A
M : Type u_3
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : Module A M
inst✝ : IsScalarTower R A M
m : M
⊢ (exp' R m).IsExponential | R : Type u_1
inst✝⁶ : CommRing R
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R A
M : Type u_3
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : Module A M
inst✝ : IsScalarTower R A M
m : M
⊢ (constantCoeff (DividedPowerAlgebra R M)) (exp' R m) = 1 ∧
∀ (p q : ℕ),
↑((p + q).choose p) * (coeff (Divided... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝⁶ : CommRing R
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R A
M : Type u_3
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : Module A M
inst✝ : IsScalarTower R A M
m : M
⊢ (exp' R m).IsExponential
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Exponential.lean | DividedPowerAlgebra.isExponential_exp' | [39, 1] | [44, 49] | constructor | R : Type u_1
inst✝⁶ : CommRing R
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R A
M : Type u_3
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : Module A M
inst✝ : IsScalarTower R A M
m : M
⊢ (constantCoeff (DividedPowerAlgebra R M)) (exp' R m) = 1 ∧
∀ (p q : ℕ),
↑((p + q).choose p) * (coeff (Divided... | case left
R : Type u_1
inst✝⁶ : CommRing R
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R A
M : Type u_3
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : Module A M
inst✝ : IsScalarTower R A M
m : M
⊢ (constantCoeff (DividedPowerAlgebra R M)) (exp' R m) = 1
case right
R : Type u_1
inst✝⁶ : CommRing R
A : Typ... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝⁶ : CommRing R
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R A
M : Type u_3
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : Module A M
inst✝ : IsScalarTower R A M
m : M
⊢ (constantCoeff (DividedPowerAlgebra R M)) (exp' R m) = 1 ∧... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Exponential.lean | DividedPowerAlgebra.isExponential_exp' | [39, 1] | [44, 49] | rw [← coeff_zero_eq_constantCoeff, coeff_exp', dp_zero] | case left
R : Type u_1
inst✝⁶ : CommRing R
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R A
M : Type u_3
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : Module A M
inst✝ : IsScalarTower R A M
m : M
⊢ (constantCoeff (DividedPowerAlgebra R M)) (exp' R m) = 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case left
R : Type u_1
inst✝⁶ : CommRing R
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R A
M : Type u_3
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : Module A M
inst✝ : IsScalarTower R A M
m : M
⊢ (constantCoeff (DividedPowerAlgebra R M)) (exp' ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Exponential.lean | DividedPowerAlgebra.isExponential_exp' | [39, 1] | [44, 49] | intro p q | case right
R : Type u_1
inst✝⁶ : CommRing R
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R A
M : Type u_3
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : Module A M
inst✝ : IsScalarTower R A M
m : M
⊢ ∀ (p q : ℕ),
↑((p + q).choose p) * (coeff (DividedPowerAlgebra R M) (p + q)) (exp' R m) =
(coeff (... | case right
R : Type u_1
inst✝⁶ : CommRing R
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R A
M : Type u_3
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : Module A M
inst✝ : IsScalarTower R A M
m : M
p q : ℕ
⊢ ↑((p + q).choose p) * (coeff (DividedPowerAlgebra R M) (p + q)) (exp' R m) =
(coeff (DividedPowe... | Please generate a tactic in lean4 to solve the state.
STATE:
case right
R : Type u_1
inst✝⁶ : CommRing R
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R A
M : Type u_3
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : Module A M
inst✝ : IsScalarTower R A M
m : M
⊢ ∀ (p q : ℕ),
↑((p + q).choose p) * (coeff (... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Exponential.lean | DividedPowerAlgebra.isExponential_exp' | [39, 1] | [44, 49] | simp only [coeff_exp', dp_mul, nsmul_eq_mul] | case right
R : Type u_1
inst✝⁶ : CommRing R
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R A
M : Type u_3
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : Module A M
inst✝ : IsScalarTower R A M
m : M
p q : ℕ
⊢ ↑((p + q).choose p) * (coeff (DividedPowerAlgebra R M) (p + q)) (exp' R m) =
(coeff (DividedPowe... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case right
R : Type u_1
inst✝⁶ : CommRing R
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R A
M : Type u_3
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : Module A M
inst✝ : IsScalarTower R A M
m : M
p q : ℕ
⊢ ↑((p + q).choose p) * (coeff (DividedPo... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Exponential.lean | DividedPowerAlgebra.coeff_exp | [53, 1] | [54, 34] | simp only [coe_exp, coeff_exp'] | R : Type u_1
inst✝⁶ : CommRing R
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R A
M : Type u_3
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : Module A M
inst✝ : IsScalarTower R A M
m : M
n : ℕ
⊢ (coeff (DividedPowerAlgebra R M) n) ↑(exp R m) = dp R n m | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝⁶ : CommRing R
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R A
M : Type u_3
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : Module A M
inst✝ : IsScalarTower R A M
m : M
n : ℕ
⊢ (coeff (DividedPowerAlgebra R M) n) ↑(exp R m) = dp ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Exponential.lean | DividedPowerAlgebra.coeff_exp_LinearMap | [81, 1] | [83, 36] | rw [coe_exp_LinearMap, coeff_exp] | R : Type u_1
inst✝⁶ : CommRing R
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R A
M : Type u_3
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : Module A M
inst✝ : IsScalarTower R A M
n : ℕ
m : M
⊢ (coeff (DividedPowerAlgebra R M) n) ↑((exp_LinearMap R M) m) = dp R n m | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝⁶ : CommRing R
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R A
M : Type u_3
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : Module A M
inst✝ : IsScalarTower R A M
n : ℕ
m : M
⊢ (coeff (DividedPowerAlgebra R M) n) ↑((exp_LinearMap... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Exponential.lean | DividedPowerAlgebra.dividedPowerAlgebra_exponentialModule_equiv_symm_apply | [132, 1] | [137, 57] | unfold dividedPowerAlgebra_exponentialModule_equiv | R : Type u_1
inst✝⁸ : CommRing R
A : Type u_2
inst✝⁷ : CommRing A
inst✝⁶ : Algebra R A
M : Type u_3
inst✝⁵ : AddCommMonoid M
inst✝⁴ : Module R M
inst✝³ : Module A M
inst✝² : IsScalarTower R A M
S : Type u_4
inst✝¹ : CommRing S
inst✝ : Algebra R S
β : M →ₗ[R] ↥(ExponentialModule S)
n : ℕ
m : M
⊢ ((dividedPowerAlgebra_ex... | R : Type u_1
inst✝⁸ : CommRing R
A : Type u_2
inst✝⁷ : CommRing A
inst✝⁶ : Algebra R A
M : Type u_3
inst✝⁵ : AddCommMonoid M
inst✝⁴ : Module R M
inst✝³ : Module A M
inst✝² : IsScalarTower R A M
S : Type u_4
inst✝¹ : CommRing S
inst✝ : Algebra R S
β : M →ₗ[R] ↥(ExponentialModule S)
n : ℕ
m : M
⊢ ({ toFun := fun α => lin... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝⁸ : CommRing R
A : Type u_2
inst✝⁷ : CommRing A
inst✝⁶ : Algebra R A
M : Type u_3
inst✝⁵ : AddCommMonoid M
inst✝⁴ : Module R M
inst✝³ : Module A M
inst✝² : IsScalarTower R A M
S : Type u_4
inst✝¹ : CommRing S
inst✝ : Algebra R S
β : M →ₗ[R] ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Exponential.lean | DividedPowerAlgebra.dividedPowerAlgebra_exponentialModule_equiv_symm_apply | [132, 1] | [137, 57] | simp only [Equiv.coe_fn_symm_mk, lift'AlgHom_apply_dp] | R : Type u_1
inst✝⁸ : CommRing R
A : Type u_2
inst✝⁷ : CommRing A
inst✝⁶ : Algebra R A
M : Type u_3
inst✝⁵ : AddCommMonoid M
inst✝⁴ : Module R M
inst✝³ : Module A M
inst✝² : IsScalarTower R A M
S : Type u_4
inst✝¹ : CommRing S
inst✝ : Algebra R S
β : M →ₗ[R] ↥(ExponentialModule S)
n : ℕ
m : M
⊢ ({ toFun := fun α => lin... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝⁸ : CommRing R
A : Type u_2
inst✝⁷ : CommRing A
inst✝⁶ : Algebra R A
M : Type u_3
inst✝⁵ : AddCommMonoid M
inst✝⁴ : Module R M
inst✝³ : Module A M
inst✝² : IsScalarTower R A M
S : Type u_4
inst✝¹ : CommRing S
inst✝ : Algebra R S
β : M →ₗ[R] ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/StronglySummable/Topology.lean | MvPowerSeries.StronglySummable.sum_eq_tsum | [29, 1] | [39, 37] | ext d | σ : Type u_1
α : Type u_2
inst✝³ : DecidableEq σ
ι : Type u_3
inst✝² : Semiring α
inst✝¹ : TopologicalSpace α
inst✝ : T2Space α
f : ι → MvPowerSeries σ α
hf : StronglySummable f
⊢ hf.sum = tsum f | case h
σ : Type u_1
α : Type u_2
inst✝³ : DecidableEq σ
ι : Type u_3
inst✝² : Semiring α
inst✝¹ : TopologicalSpace α
inst✝ : T2Space α
f : ι → MvPowerSeries σ α
hf : StronglySummable f
d : σ →₀ ℕ
⊢ (coeff α d) hf.sum = (coeff α d) (tsum f) | Please generate a tactic in lean4 to solve the state.
STATE:
σ : Type u_1
α : Type u_2
inst✝³ : DecidableEq σ
ι : Type u_3
inst✝² : Semiring α
inst✝¹ : TopologicalSpace α
inst✝ : T2Space α
f : ι → MvPowerSeries σ α
hf : StronglySummable f
⊢ hf.sum = tsum f
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/StronglySummable/Topology.lean | MvPowerSeries.StronglySummable.sum_eq_tsum | [29, 1] | [39, 37] | rw [tsum_def, dif_pos hf.summable] | case h
σ : Type u_1
α : Type u_2
inst✝³ : DecidableEq σ
ι : Type u_3
inst✝² : Semiring α
inst✝¹ : TopologicalSpace α
inst✝ : T2Space α
f : ι → MvPowerSeries σ α
hf : StronglySummable f
d : σ →₀ ℕ
⊢ (coeff α d) hf.sum = (coeff α d) (tsum f) | case h
σ : Type u_1
α : Type u_2
inst✝³ : DecidableEq σ
ι : Type u_3
inst✝² : Semiring α
inst✝¹ : TopologicalSpace α
inst✝ : T2Space α
f : ι → MvPowerSeries σ α
hf : StronglySummable f
d : σ →₀ ℕ
⊢ (coeff α d) hf.sum = (coeff α d) (if (support f).Finite then finsum f else Exists.choose ⋯) | Please generate a tactic in lean4 to solve the state.
STATE:
case h
σ : Type u_1
α : Type u_2
inst✝³ : DecidableEq σ
ι : Type u_3
inst✝² : Semiring α
inst✝¹ : TopologicalSpace α
inst✝ : T2Space α
f : ι → MvPowerSeries σ α
hf : StronglySummable f
d : σ →₀ ℕ
⊢ (coeff α d) hf.sum = (coeff α d) (tsum f)
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/StronglySummable/Topology.lean | MvPowerSeries.StronglySummable.sum_eq_tsum | [29, 1] | [39, 37] | apply HasSum.unique (hf.hasSum_coeff d) | case h
σ : Type u_1
α : Type u_2
inst✝³ : DecidableEq σ
ι : Type u_3
inst✝² : Semiring α
inst✝¹ : TopologicalSpace α
inst✝ : T2Space α
f : ι → MvPowerSeries σ α
hf : StronglySummable f
d : σ →₀ ℕ
⊢ (coeff α d) hf.sum = (coeff α d) (if (support f).Finite then finsum f else Exists.choose ⋯) | case h
σ : Type u_1
α : Type u_2
inst✝³ : DecidableEq σ
ι : Type u_3
inst✝² : Semiring α
inst✝¹ : TopologicalSpace α
inst✝ : T2Space α
f : ι → MvPowerSeries σ α
hf : StronglySummable f
d : σ →₀ ℕ
⊢ HasSum (fun i => (coeff α d) (f i)) ((coeff α d) (if (support f).Finite then finsum f else Exists.choose ⋯)) | Please generate a tactic in lean4 to solve the state.
STATE:
case h
σ : Type u_1
α : Type u_2
inst✝³ : DecidableEq σ
ι : Type u_3
inst✝² : Semiring α
inst✝¹ : TopologicalSpace α
inst✝ : T2Space α
f : ι → MvPowerSeries σ α
hf : StronglySummable f
d : σ →₀ ℕ
⊢ (coeff α d) hf.sum = (coeff α d) (if (support f).Finite then ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/StronglySummable/Topology.lean | MvPowerSeries.StronglySummable.sum_eq_tsum | [29, 1] | [39, 37] | apply HasSum.map | case h
σ : Type u_1
α : Type u_2
inst✝³ : DecidableEq σ
ι : Type u_3
inst✝² : Semiring α
inst✝¹ : TopologicalSpace α
inst✝ : T2Space α
f : ι → MvPowerSeries σ α
hf : StronglySummable f
d : σ →₀ ℕ
⊢ HasSum (fun i => (coeff α d) (f i)) ((coeff α d) (if (support f).Finite then finsum f else Exists.choose ⋯)) | case h.hf
σ : Type u_1
α : Type u_2
inst✝³ : DecidableEq σ
ι : Type u_3
inst✝² : Semiring α
inst✝¹ : TopologicalSpace α
inst✝ : T2Space α
f : ι → MvPowerSeries σ α
hf : StronglySummable f
d : σ →₀ ℕ
⊢ HasSum (fun i => f i) (if (support f).Finite then finsum f else Exists.choose ⋯)
case h.hg
σ : Type u_1
α : Type u_2
i... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
σ : Type u_1
α : Type u_2
inst✝³ : DecidableEq σ
ι : Type u_3
inst✝² : Semiring α
inst✝¹ : TopologicalSpace α
inst✝ : T2Space α
f : ι → MvPowerSeries σ α
hf : StronglySummable f
d : σ →₀ ℕ
⊢ HasSum (fun i => (coeff α d) (f i)) ((coeff α d) (if (support... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/StronglySummable/Topology.lean | MvPowerSeries.StronglySummable.sum_eq_tsum | [29, 1] | [39, 37] | . split_ifs with h
. rw [← tsum_eq_finsum h]
exact hf.summable.hasSum
. exact (Classical.choose_spec hf.summable) | case h.hf
σ : Type u_1
α : Type u_2
inst✝³ : DecidableEq σ
ι : Type u_3
inst✝² : Semiring α
inst✝¹ : TopologicalSpace α
inst✝ : T2Space α
f : ι → MvPowerSeries σ α
hf : StronglySummable f
d : σ →₀ ℕ
⊢ HasSum (fun i => f i) (if (support f).Finite then finsum f else Exists.choose ⋯)
case h.hg
σ : Type u_1
α : Type u_2
i... | case h.hg
σ : Type u_1
α : Type u_2
inst✝³ : DecidableEq σ
ι : Type u_3
inst✝² : Semiring α
inst✝¹ : TopologicalSpace α
inst✝ : T2Space α
f : ι → MvPowerSeries σ α
hf : StronglySummable f
d : σ →₀ ℕ
⊢ Continuous ⇑(coeff α d) | Please generate a tactic in lean4 to solve the state.
STATE:
case h.hf
σ : Type u_1
α : Type u_2
inst✝³ : DecidableEq σ
ι : Type u_3
inst✝² : Semiring α
inst✝¹ : TopologicalSpace α
inst✝ : T2Space α
f : ι → MvPowerSeries σ α
hf : StronglySummable f
d : σ →₀ ℕ
⊢ HasSum (fun i => f i) (if (support f).Finite then finsum f... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/StronglySummable/Topology.lean | MvPowerSeries.StronglySummable.sum_eq_tsum | [29, 1] | [39, 37] | . exact continuous_component σ α d | case h.hg
σ : Type u_1
α : Type u_2
inst✝³ : DecidableEq σ
ι : Type u_3
inst✝² : Semiring α
inst✝¹ : TopologicalSpace α
inst✝ : T2Space α
f : ι → MvPowerSeries σ α
hf : StronglySummable f
d : σ →₀ ℕ
⊢ Continuous ⇑(coeff α d) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.hg
σ : Type u_1
α : Type u_2
inst✝³ : DecidableEq σ
ι : Type u_3
inst✝² : Semiring α
inst✝¹ : TopologicalSpace α
inst✝ : T2Space α
f : ι → MvPowerSeries σ α
hf : StronglySummable f
d : σ →₀ ℕ
⊢ Continuous ⇑(coeff α d)
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/StronglySummable/Topology.lean | MvPowerSeries.StronglySummable.sum_eq_tsum | [29, 1] | [39, 37] | split_ifs with h | case h.hf
σ : Type u_1
α : Type u_2
inst✝³ : DecidableEq σ
ι : Type u_3
inst✝² : Semiring α
inst✝¹ : TopologicalSpace α
inst✝ : T2Space α
f : ι → MvPowerSeries σ α
hf : StronglySummable f
d : σ →₀ ℕ
⊢ HasSum (fun i => f i) (if (support f).Finite then finsum f else Exists.choose ⋯) | case pos
σ : Type u_1
α : Type u_2
inst✝³ : DecidableEq σ
ι : Type u_3
inst✝² : Semiring α
inst✝¹ : TopologicalSpace α
inst✝ : T2Space α
f : ι → MvPowerSeries σ α
hf : StronglySummable f
d : σ →₀ ℕ
h : (support f).Finite
⊢ HasSum (fun i => f i) (finsum f)
case neg
σ : Type u_1
α : Type u_2
inst✝³ : DecidableEq σ
ι : T... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.hf
σ : Type u_1
α : Type u_2
inst✝³ : DecidableEq σ
ι : Type u_3
inst✝² : Semiring α
inst✝¹ : TopologicalSpace α
inst✝ : T2Space α
f : ι → MvPowerSeries σ α
hf : StronglySummable f
d : σ →₀ ℕ
⊢ HasSum (fun i => f i) (if (support f).Finite then finsum f... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/StronglySummable/Topology.lean | MvPowerSeries.StronglySummable.sum_eq_tsum | [29, 1] | [39, 37] | . rw [← tsum_eq_finsum h]
exact hf.summable.hasSum | case pos
σ : Type u_1
α : Type u_2
inst✝³ : DecidableEq σ
ι : Type u_3
inst✝² : Semiring α
inst✝¹ : TopologicalSpace α
inst✝ : T2Space α
f : ι → MvPowerSeries σ α
hf : StronglySummable f
d : σ →₀ ℕ
h : (support f).Finite
⊢ HasSum (fun i => f i) (finsum f)
case neg
σ : Type u_1
α : Type u_2
inst✝³ : DecidableEq σ
ι : T... | case neg
σ : Type u_1
α : Type u_2
inst✝³ : DecidableEq σ
ι : Type u_3
inst✝² : Semiring α
inst✝¹ : TopologicalSpace α
inst✝ : T2Space α
f : ι → MvPowerSeries σ α
hf : StronglySummable f
d : σ →₀ ℕ
h : ¬(support f).Finite
⊢ HasSum (fun i => f i) (Exists.choose ⋯) | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
σ : Type u_1
α : Type u_2
inst✝³ : DecidableEq σ
ι : Type u_3
inst✝² : Semiring α
inst✝¹ : TopologicalSpace α
inst✝ : T2Space α
f : ι → MvPowerSeries σ α
hf : StronglySummable f
d : σ →₀ ℕ
h : (support f).Finite
⊢ HasSum (fun i => f i) (finsum f)
ca... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/StronglySummable/Topology.lean | MvPowerSeries.StronglySummable.sum_eq_tsum | [29, 1] | [39, 37] | . exact (Classical.choose_spec hf.summable) | case neg
σ : Type u_1
α : Type u_2
inst✝³ : DecidableEq σ
ι : Type u_3
inst✝² : Semiring α
inst✝¹ : TopologicalSpace α
inst✝ : T2Space α
f : ι → MvPowerSeries σ α
hf : StronglySummable f
d : σ →₀ ℕ
h : ¬(support f).Finite
⊢ HasSum (fun i => f i) (Exists.choose ⋯) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
σ : Type u_1
α : Type u_2
inst✝³ : DecidableEq σ
ι : Type u_3
inst✝² : Semiring α
inst✝¹ : TopologicalSpace α
inst✝ : T2Space α
f : ι → MvPowerSeries σ α
hf : StronglySummable f
d : σ →₀ ℕ
h : ¬(support f).Finite
⊢ HasSum (fun i => f i) (Exists.choos... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/StronglySummable/Topology.lean | MvPowerSeries.StronglySummable.sum_eq_tsum | [29, 1] | [39, 37] | rw [← tsum_eq_finsum h] | case pos
σ : Type u_1
α : Type u_2
inst✝³ : DecidableEq σ
ι : Type u_3
inst✝² : Semiring α
inst✝¹ : TopologicalSpace α
inst✝ : T2Space α
f : ι → MvPowerSeries σ α
hf : StronglySummable f
d : σ →₀ ℕ
h : (support f).Finite
⊢ HasSum (fun i => f i) (finsum f) | case pos
σ : Type u_1
α : Type u_2
inst✝³ : DecidableEq σ
ι : Type u_3
inst✝² : Semiring α
inst✝¹ : TopologicalSpace α
inst✝ : T2Space α
f : ι → MvPowerSeries σ α
hf : StronglySummable f
d : σ →₀ ℕ
h : (support f).Finite
⊢ HasSum (fun i => f i) (∑' (b : ι), f b) | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
σ : Type u_1
α : Type u_2
inst✝³ : DecidableEq σ
ι : Type u_3
inst✝² : Semiring α
inst✝¹ : TopologicalSpace α
inst✝ : T2Space α
f : ι → MvPowerSeries σ α
hf : StronglySummable f
d : σ →₀ ℕ
h : (support f).Finite
⊢ HasSum (fun i => f i) (finsum f)
TAC... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/StronglySummable/Topology.lean | MvPowerSeries.StronglySummable.sum_eq_tsum | [29, 1] | [39, 37] | exact hf.summable.hasSum | case pos
σ : Type u_1
α : Type u_2
inst✝³ : DecidableEq σ
ι : Type u_3
inst✝² : Semiring α
inst✝¹ : TopologicalSpace α
inst✝ : T2Space α
f : ι → MvPowerSeries σ α
hf : StronglySummable f
d : σ →₀ ℕ
h : (support f).Finite
⊢ HasSum (fun i => f i) (∑' (b : ι), f b) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
σ : Type u_1
α : Type u_2
inst✝³ : DecidableEq σ
ι : Type u_3
inst✝² : Semiring α
inst✝¹ : TopologicalSpace α
inst✝ : T2Space α
f : ι → MvPowerSeries σ α
hf : StronglySummable f
d : σ →₀ ℕ
h : (support f).Finite
⊢ HasSum (fun i => f i) (∑' (b : ι), f... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/StronglySummable/Topology.lean | MvPowerSeries.StronglySummable.sum_eq_tsum | [29, 1] | [39, 37] | exact (Classical.choose_spec hf.summable) | case neg
σ : Type u_1
α : Type u_2
inst✝³ : DecidableEq σ
ι : Type u_3
inst✝² : Semiring α
inst✝¹ : TopologicalSpace α
inst✝ : T2Space α
f : ι → MvPowerSeries σ α
hf : StronglySummable f
d : σ →₀ ℕ
h : ¬(support f).Finite
⊢ HasSum (fun i => f i) (Exists.choose ⋯) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
σ : Type u_1
α : Type u_2
inst✝³ : DecidableEq σ
ι : Type u_3
inst✝² : Semiring α
inst✝¹ : TopologicalSpace α
inst✝ : T2Space α
f : ι → MvPowerSeries σ α
hf : StronglySummable f
d : σ →₀ ℕ
h : ¬(support f).Finite
⊢ HasSum (fun i => f i) (Exists.choos... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/StronglySummable/Topology.lean | MvPowerSeries.StronglySummable.sum_eq_tsum | [29, 1] | [39, 37] | exact continuous_component σ α d | case h.hg
σ : Type u_1
α : Type u_2
inst✝³ : DecidableEq σ
ι : Type u_3
inst✝² : Semiring α
inst✝¹ : TopologicalSpace α
inst✝ : T2Space α
f : ι → MvPowerSeries σ α
hf : StronglySummable f
d : σ →₀ ℕ
⊢ Continuous ⇑(coeff α d) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.hg
σ : Type u_1
α : Type u_2
inst✝³ : DecidableEq σ
ι : Type u_3
inst✝² : Semiring α
inst✝¹ : TopologicalSpace α
inst✝ : T2Space α
f : ι → MvPowerSeries σ α
hf : StronglySummable f
d : σ →₀ ℕ
⊢ Continuous ⇑(coeff α d)
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/StronglySummable/Topology.lean | MvPowerSeries.StronglySummable.iff_summable' | [74, 1] | [79, 59] | haveI := topologicalRing σ α | σ : Type u_1
α : Type u_2
inst✝³ : DecidableEq σ
ι : Type u_3
inst✝² : Ring α
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
f : ι → MvPowerSeries σ α
⊢ StronglySummable f ↔ Filter.Tendsto f Filter.cofinite (nhds 0) | σ : Type u_1
α : Type u_2
inst✝³ : DecidableEq σ
ι : Type u_3
inst✝² : Ring α
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
f : ι → MvPowerSeries σ α
this : TopologicalRing (MvPowerSeries σ α)
⊢ StronglySummable f ↔ Filter.Tendsto f Filter.cofinite (nhds 0) | Please generate a tactic in lean4 to solve the state.
STATE:
σ : Type u_1
α : Type u_2
inst✝³ : DecidableEq σ
ι : Type u_3
inst✝² : Ring α
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
f : ι → MvPowerSeries σ α
⊢ StronglySummable f ↔ Filter.Tendsto f Filter.cofinite (nhds 0)
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/StronglySummable/Topology.lean | MvPowerSeries.StronglySummable.iff_summable' | [74, 1] | [79, 59] | refine' ⟨fun hf => hf.summable.tendsto_cofinite_zero, _⟩ | σ : Type u_1
α : Type u_2
inst✝³ : DecidableEq σ
ι : Type u_3
inst✝² : Ring α
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
f : ι → MvPowerSeries σ α
this : TopologicalRing (MvPowerSeries σ α)
⊢ StronglySummable f ↔ Filter.Tendsto f Filter.cofinite (nhds 0) | σ : Type u_1
α : Type u_2
inst✝³ : DecidableEq σ
ι : Type u_3
inst✝² : Ring α
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
f : ι → MvPowerSeries σ α
this : TopologicalRing (MvPowerSeries σ α)
⊢ Filter.Tendsto f Filter.cofinite (nhds 0) → StronglySummable f | Please generate a tactic in lean4 to solve the state.
STATE:
σ : Type u_1
α : Type u_2
inst✝³ : DecidableEq σ
ι : Type u_3
inst✝² : Ring α
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
f : ι → MvPowerSeries σ α
this : TopologicalRing (MvPowerSeries σ α)
⊢ StronglySummable f ↔ Filter.Tendsto f Filter.cofinite (... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/StronglySummable/Topology.lean | MvPowerSeries.StronglySummable.iff_summable' | [74, 1] | [79, 59] | rw [StronglySummable, nhds_pi, Filter.tendsto_pi] | σ : Type u_1
α : Type u_2
inst✝³ : DecidableEq σ
ι : Type u_3
inst✝² : Ring α
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
f : ι → MvPowerSeries σ α
this : TopologicalRing (MvPowerSeries σ α)
⊢ Filter.Tendsto f Filter.cofinite (nhds 0) → StronglySummable f | σ : Type u_1
α : Type u_2
inst✝³ : DecidableEq σ
ι : Type u_3
inst✝² : Ring α
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
f : ι → MvPowerSeries σ α
this : TopologicalRing (MvPowerSeries σ α)
⊢ (∀ (i : σ →₀ ℕ), Filter.Tendsto (fun x => f x i) Filter.cofinite (nhds (0 i))) →
∀ (d : σ →₀ ℕ), (support fun i ... | Please generate a tactic in lean4 to solve the state.
STATE:
σ : Type u_1
α : Type u_2
inst✝³ : DecidableEq σ
ι : Type u_3
inst✝² : Ring α
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
f : ι → MvPowerSeries σ α
this : TopologicalRing (MvPowerSeries σ α)
⊢ Filter.Tendsto f Filter.cofinite (nhds 0) → StronglySum... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/StronglySummable/Topology.lean | MvPowerSeries.StronglySummable.iff_summable' | [74, 1] | [79, 59] | exact forall_imp fun d => finite_support_of_tendsto_zero | σ : Type u_1
α : Type u_2
inst✝³ : DecidableEq σ
ι : Type u_3
inst✝² : Ring α
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
f : ι → MvPowerSeries σ α
this : TopologicalRing (MvPowerSeries σ α)
⊢ (∀ (i : σ →₀ ℕ), Filter.Tendsto (fun x => f x i) Filter.cofinite (nhds (0 i))) →
∀ (d : σ →₀ ℕ), (support fun i ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
σ : Type u_1
α : Type u_2
inst✝³ : DecidableEq σ
ι : Type u_3
inst✝² : Ring α
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
f : ι → MvPowerSeries σ α
this : TopologicalRing (MvPowerSeries σ α)
⊢ (∀ (i : σ →₀ ℕ), Filter.Tendsto (fun x => f x i) Filter... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/StronglySummable/Topology.lean | MvPowerSeries.StronglySummable.hasProd_of_one_add | [113, 1] | [152, 62] | haveI := uniformAddGroup σ α | σ : Type u_1
α : Type u_2
inst✝³ : DecidableEq σ
ι : Type u_3
f : ι → MvPowerSeries σ α
inst✝² : CommRing α
inst✝¹ : UniformSpace α
inst✝ : UniformAddGroup α
hf : StronglySummable f
⊢ HasProd (fun i => 1 + f i) ⋯.prod | σ : Type u_1
α : Type u_2
inst✝³ : DecidableEq σ
ι : Type u_3
f : ι → MvPowerSeries σ α
inst✝² : CommRing α
inst✝¹ : UniformSpace α
inst✝ : UniformAddGroup α
hf : StronglySummable f
this : UniformAddGroup (MvPowerSeries σ α)
⊢ HasProd (fun i => 1 + f i) ⋯.prod | Please generate a tactic in lean4 to solve the state.
STATE:
σ : Type u_1
α : Type u_2
inst✝³ : DecidableEq σ
ι : Type u_3
f : ι → MvPowerSeries σ α
inst✝² : CommRing α
inst✝¹ : UniformSpace α
inst✝ : UniformAddGroup α
hf : StronglySummable f
⊢ HasProd (fun i => 1 + f i) ⋯.prod
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/StronglySummable/Topology.lean | MvPowerSeries.StronglySummable.hasProd_of_one_add | [113, 1] | [152, 62] | intro V hV | σ : Type u_1
α : Type u_2
inst✝³ : DecidableEq σ
ι : Type u_3
f : ι → MvPowerSeries σ α
inst✝² : CommRing α
inst✝¹ : UniformSpace α
inst✝ : UniformAddGroup α
hf : StronglySummable f
this : UniformAddGroup (MvPowerSeries σ α)
⊢ HasProd (fun i => 1 + f i) ⋯.prod | σ : Type u_1
α : Type u_2
inst✝³ : DecidableEq σ
ι : Type u_3
f : ι → MvPowerSeries σ α
inst✝² : CommRing α
inst✝¹ : UniformSpace α
inst✝ : UniformAddGroup α
hf : StronglySummable f
this : UniformAddGroup (MvPowerSeries σ α)
V : Set (MvPowerSeries σ α)
hV : V ∈ nhds ⋯.prod
⊢ V ∈ Filter.map (fun s => ∏ b ∈ s, (fun i => ... | Please generate a tactic in lean4 to solve the state.
STATE:
σ : Type u_1
α : Type u_2
inst✝³ : DecidableEq σ
ι : Type u_3
f : ι → MvPowerSeries σ α
inst✝² : CommRing α
inst✝¹ : UniformSpace α
inst✝ : UniformAddGroup α
hf : StronglySummable f
this : UniformAddGroup (MvPowerSeries σ α)
⊢ HasProd (fun i => 1 + f i) ⋯.pro... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/StronglySummable/Topology.lean | MvPowerSeries.StronglySummable.hasProd_of_one_add | [113, 1] | [152, 62] | simp only [Filter.mem_map, Filter.mem_atTop_sets, ge_iff_le, Finset.le_eq_subset,
Set.mem_preimage] | σ : Type u_1
α : Type u_2
inst✝³ : DecidableEq σ
ι : Type u_3
f : ι → MvPowerSeries σ α
inst✝² : CommRing α
inst✝¹ : UniformSpace α
inst✝ : UniformAddGroup α
hf : StronglySummable f
this : UniformAddGroup (MvPowerSeries σ α)
V : Set (MvPowerSeries σ α)
hV : V ∈ nhds ⋯.prod
⊢ V ∈ Filter.map (fun s => ∏ b ∈ s, (fun i => ... | σ : Type u_1
α : Type u_2
inst✝³ : DecidableEq σ
ι : Type u_3
f : ι → MvPowerSeries σ α
inst✝² : CommRing α
inst✝¹ : UniformSpace α
inst✝ : UniformAddGroup α
hf : StronglySummable f
this : UniformAddGroup (MvPowerSeries σ α)
V : Set (MvPowerSeries σ α)
hV : V ∈ nhds ⋯.prod
⊢ ∃ a, ∀ (b : Finset ι), a ⊆ b → ∏ x ∈ b, (1 +... | Please generate a tactic in lean4 to solve the state.
STATE:
σ : Type u_1
α : Type u_2
inst✝³ : DecidableEq σ
ι : Type u_3
f : ι → MvPowerSeries σ α
inst✝² : CommRing α
inst✝¹ : UniformSpace α
inst✝ : UniformAddGroup α
hf : StronglySummable f
this : UniformAddGroup (MvPowerSeries σ α)
V : Set (MvPowerSeries σ α)
hV : V... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/StronglySummable/Topology.lean | MvPowerSeries.StronglySummable.hasProd_of_one_add | [113, 1] | [152, 62] | let V₀ := Add.add hf.toStronglyMultipliable.prod ⁻¹' V | σ : Type u_1
α : Type u_2
inst✝³ : DecidableEq σ
ι : Type u_3
f : ι → MvPowerSeries σ α
inst✝² : CommRing α
inst✝¹ : UniformSpace α
inst✝ : UniformAddGroup α
hf : StronglySummable f
this : UniformAddGroup (MvPowerSeries σ α)
V : Set (MvPowerSeries σ α)
hV : V ∈ nhds ⋯.prod
⊢ ∃ a, ∀ (b : Finset ι), a ⊆ b → ∏ x ∈ b, (1 +... | σ : Type u_1
α : Type u_2
inst✝³ : DecidableEq σ
ι : Type u_3
f : ι → MvPowerSeries σ α
inst✝² : CommRing α
inst✝¹ : UniformSpace α
inst✝ : UniformAddGroup α
hf : StronglySummable f
this : UniformAddGroup (MvPowerSeries σ α)
V : Set (MvPowerSeries σ α)
hV : V ∈ nhds ⋯.prod
V₀ : Set (MvPowerSeries σ α) := Add.add ⋯.prod... | Please generate a tactic in lean4 to solve the state.
STATE:
σ : Type u_1
α : Type u_2
inst✝³ : DecidableEq σ
ι : Type u_3
f : ι → MvPowerSeries σ α
inst✝² : CommRing α
inst✝¹ : UniformSpace α
inst✝ : UniformAddGroup α
hf : StronglySummable f
this : UniformAddGroup (MvPowerSeries σ α)
V : Set (MvPowerSeries σ α)
hV : V... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/StronglySummable/Topology.lean | MvPowerSeries.StronglySummable.hasProd_of_one_add | [113, 1] | [152, 62] | have hV'₀ : V = Add.add (-hf.toStronglyMultipliable.prod) ⁻¹' V₀ := by
rw [← Set.preimage_comp, eq_comm]
convert Set.preimage_id
rw [Function.funext_iff]
intro f
simp only [comp_apply, id_eq]
change _ + (_ + f) = f
simp_rw [← add_assoc, add_right_neg, zero_add] | σ : Type u_1
α : Type u_2
inst✝³ : DecidableEq σ
ι : Type u_3
f : ι → MvPowerSeries σ α
inst✝² : CommRing α
inst✝¹ : UniformSpace α
inst✝ : UniformAddGroup α
hf : StronglySummable f
this : UniformAddGroup (MvPowerSeries σ α)
V : Set (MvPowerSeries σ α)
hV : V ∈ nhds ⋯.prod
V₀ : Set (MvPowerSeries σ α) := Add.add ⋯.prod... | σ : Type u_1
α : Type u_2
inst✝³ : DecidableEq σ
ι : Type u_3
f : ι → MvPowerSeries σ α
inst✝² : CommRing α
inst✝¹ : UniformSpace α
inst✝ : UniformAddGroup α
hf : StronglySummable f
this : UniformAddGroup (MvPowerSeries σ α)
V : Set (MvPowerSeries σ α)
hV : V ∈ nhds ⋯.prod
V₀ : Set (MvPowerSeries σ α) := Add.add ⋯.prod... | Please generate a tactic in lean4 to solve the state.
STATE:
σ : Type u_1
α : Type u_2
inst✝³ : DecidableEq σ
ι : Type u_3
f : ι → MvPowerSeries σ α
inst✝² : CommRing α
inst✝¹ : UniformSpace α
inst✝ : UniformAddGroup α
hf : StronglySummable f
this : UniformAddGroup (MvPowerSeries σ α)
V : Set (MvPowerSeries σ α)
hV : V... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/StronglySummable/Topology.lean | MvPowerSeries.StronglySummable.hasProd_of_one_add | [113, 1] | [152, 62] | have hV₀ : V₀ ∈ nhds (0 : MvPowerSeries σ α) := by
apply continuousAt_def.mp (Continuous.continuousAt (continuous_add_left _))
rw [add_zero]
exact hV | σ : Type u_1
α : Type u_2
inst✝³ : DecidableEq σ
ι : Type u_3
f : ι → MvPowerSeries σ α
inst✝² : CommRing α
inst✝¹ : UniformSpace α
inst✝ : UniformAddGroup α
hf : StronglySummable f
this : UniformAddGroup (MvPowerSeries σ α)
V : Set (MvPowerSeries σ α)
hV : V ∈ nhds ⋯.prod
V₀ : Set (MvPowerSeries σ α) := Add.add ⋯.prod... | σ : Type u_1
α : Type u_2
inst✝³ : DecidableEq σ
ι : Type u_3
f : ι → MvPowerSeries σ α
inst✝² : CommRing α
inst✝¹ : UniformSpace α
inst✝ : UniformAddGroup α
hf : StronglySummable f
this : UniformAddGroup (MvPowerSeries σ α)
V : Set (MvPowerSeries σ α)
hV : V ∈ nhds ⋯.prod
V₀ : Set (MvPowerSeries σ α) := Add.add ⋯.prod... | Please generate a tactic in lean4 to solve the state.
STATE:
σ : Type u_1
α : Type u_2
inst✝³ : DecidableEq σ
ι : Type u_3
f : ι → MvPowerSeries σ α
inst✝² : CommRing α
inst✝¹ : UniformSpace α
inst✝ : UniformAddGroup α
hf : StronglySummable f
this : UniformAddGroup (MvPowerSeries σ α)
V : Set (MvPowerSeries σ α)
hV : V... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/StronglySummable/Topology.lean | MvPowerSeries.StronglySummable.hasProd_of_one_add | [113, 1] | [152, 62] | rw [nhds_pi, Filter.mem_pi] at hV₀ | σ : Type u_1
α : Type u_2
inst✝³ : DecidableEq σ
ι : Type u_3
f : ι → MvPowerSeries σ α
inst✝² : CommRing α
inst✝¹ : UniformSpace α
inst✝ : UniformAddGroup α
hf : StronglySummable f
this : UniformAddGroup (MvPowerSeries σ α)
V : Set (MvPowerSeries σ α)
hV : V ∈ nhds ⋯.prod
V₀ : Set (MvPowerSeries σ α) := Add.add ⋯.prod... | σ : Type u_1
α : Type u_2
inst✝³ : DecidableEq σ
ι : Type u_3
f : ι → MvPowerSeries σ α
inst✝² : CommRing α
inst✝¹ : UniformSpace α
inst✝ : UniformAddGroup α
hf : StronglySummable f
this : UniformAddGroup (MvPowerSeries σ α)
V : Set (MvPowerSeries σ α)
hV : V ∈ nhds ⋯.prod
V₀ : Set (MvPowerSeries σ α) := Add.add ⋯.prod... | Please generate a tactic in lean4 to solve the state.
STATE:
σ : Type u_1
α : Type u_2
inst✝³ : DecidableEq σ
ι : Type u_3
f : ι → MvPowerSeries σ α
inst✝² : CommRing α
inst✝¹ : UniformSpace α
inst✝ : UniformAddGroup α
hf : StronglySummable f
this : UniformAddGroup (MvPowerSeries σ α)
V : Set (MvPowerSeries σ α)
hV : V... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/StronglySummable/Topology.lean | MvPowerSeries.StronglySummable.hasProd_of_one_add | [113, 1] | [152, 62] | obtain ⟨D, hD, t, ht, htV₀⟩ := hV₀ | σ : Type u_1
α : Type u_2
inst✝³ : DecidableEq σ
ι : Type u_3
f : ι → MvPowerSeries σ α
inst✝² : CommRing α
inst✝¹ : UniformSpace α
inst✝ : UniformAddGroup α
hf : StronglySummable f
this : UniformAddGroup (MvPowerSeries σ α)
V : Set (MvPowerSeries σ α)
hV : V ∈ nhds ⋯.prod
V₀ : Set (MvPowerSeries σ α) := Add.add ⋯.prod... | case intro.intro.intro.intro
σ : Type u_1
α : Type u_2
inst✝³ : DecidableEq σ
ι : Type u_3
f : ι → MvPowerSeries σ α
inst✝² : CommRing α
inst✝¹ : UniformSpace α
inst✝ : UniformAddGroup α
hf : StronglySummable f
this : UniformAddGroup (MvPowerSeries σ α)
V : Set (MvPowerSeries σ α)
hV : V ∈ nhds ⋯.prod
V₀ : Set (MvPower... | Please generate a tactic in lean4 to solve the state.
STATE:
σ : Type u_1
α : Type u_2
inst✝³ : DecidableEq σ
ι : Type u_3
f : ι → MvPowerSeries σ α
inst✝² : CommRing α
inst✝¹ : UniformSpace α
inst✝ : UniformAddGroup α
hf : StronglySummable f
this : UniformAddGroup (MvPowerSeries σ α)
V : Set (MvPowerSeries σ α)
hV : V... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/StronglySummable/Topology.lean | MvPowerSeries.StronglySummable.hasProd_of_one_add | [113, 1] | [152, 62] | use hf.unionOfSupportOfCoeffLe (hD.toFinset.sup id) | case intro.intro.intro.intro
σ : Type u_1
α : Type u_2
inst✝³ : DecidableEq σ
ι : Type u_3
f : ι → MvPowerSeries σ α
inst✝² : CommRing α
inst✝¹ : UniformSpace α
inst✝ : UniformAddGroup α
hf : StronglySummable f
this : UniformAddGroup (MvPowerSeries σ α)
V : Set (MvPowerSeries σ α)
hV : V ∈ nhds ⋯.prod
V₀ : Set (MvPower... | case h
σ : Type u_1
α : Type u_2
inst✝³ : DecidableEq σ
ι : Type u_3
f : ι → MvPowerSeries σ α
inst✝² : CommRing α
inst✝¹ : UniformSpace α
inst✝ : UniformAddGroup α
hf : StronglySummable f
this : UniformAddGroup (MvPowerSeries σ α)
V : Set (MvPowerSeries σ α)
hV : V ∈ nhds ⋯.prod
V₀ : Set (MvPowerSeries σ α) := Add.add... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro
σ : Type u_1
α : Type u_2
inst✝³ : DecidableEq σ
ι : Type u_3
f : ι → MvPowerSeries σ α
inst✝² : CommRing α
inst✝¹ : UniformSpace α
inst✝ : UniformAddGroup α
hf : StronglySummable f
this : UniformAddGroup (MvPowerSeries σ α)
V : S... |
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