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 | Ideal.IsHomogeneous_of_rel_isHomogeneous | [466, 1] | [504, 45] | apply Ideal.sum_mem | case a.refine'_1.intro.intro.intro
R✝ : Type u_1
inst✝¹¹ : CommSemiring R✝
ι✝ : Type u_2
inst✝¹⁰ : DecidableEq ι✝
inst✝⁹ : AddCommMonoid ι✝
A✝ : Type u_3
inst✝⁸ : CommSemiring A✝
inst✝⁷ : DecidableEq A✝
inst✝⁶ : Algebra R✝ A✝
𝒜✝ : ι✝ → Submodule R✝ A✝
rel : A✝ → A✝ → Prop
R : Type u_4
inst✝⁵ : CommRing R
ι : Type u_5
... | case a.refine'_1.intro.intro.intro.a
R✝ : Type u_1
inst✝¹¹ : CommSemiring R✝
ι✝ : Type u_2
inst✝¹⁰ : DecidableEq ι✝
inst✝⁹ : AddCommMonoid ι✝
A✝ : Type u_3
inst✝⁸ : CommSemiring A✝
inst✝⁷ : DecidableEq A✝
inst✝⁶ : Algebra R✝ A✝
𝒜✝ : ι✝ → Submodule R✝ A✝
rel : A✝ → A✝ → Prop
R : Type u_4
inst✝⁵ : CommRing R
ι : Type u_... | Please generate a tactic in lean4 to solve the state.
STATE:
case a.refine'_1.intro.intro.intro
R✝ : Type u_1
inst✝¹¹ : CommSemiring R✝
ι✝ : Type u_2
inst✝¹⁰ : DecidableEq ι✝
inst✝⁹ : AddCommMonoid ι✝
A✝ : Type u_3
inst✝⁸ : CommSemiring A✝
inst✝⁷ : DecidableEq A✝
inst✝⁶ : Algebra R✝ A✝
𝒜✝ : ι✝ → Submodule R✝ A✝
rel : ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | Ideal.IsHomogeneous_of_rel_isHomogeneous | [466, 1] | [504, 45] | intro i _ | case a.refine'_1.intro.intro.intro.a
R✝ : Type u_1
inst✝¹¹ : CommSemiring R✝
ι✝ : Type u_2
inst✝¹⁰ : DecidableEq ι✝
inst✝⁹ : AddCommMonoid ι✝
A✝ : Type u_3
inst✝⁸ : CommSemiring A✝
inst✝⁷ : DecidableEq A✝
inst✝⁶ : Algebra R✝ A✝
𝒜✝ : ι✝ → Submodule R✝ A✝
rel : A✝ → A✝ → Prop
R : Type u_4
inst✝⁵ : CommRing R
ι : Type u_... | case a.refine'_1.intro.intro.intro.a
R✝ : Type u_1
inst✝¹¹ : CommSemiring R✝
ι✝ : Type u_2
inst✝¹⁰ : DecidableEq ι✝
inst✝⁹ : AddCommMonoid ι✝
A✝ : Type u_3
inst✝⁸ : CommSemiring A✝
inst✝⁷ : DecidableEq A✝
inst✝⁶ : Algebra R✝ A✝
𝒜✝ : ι✝ → Submodule R✝ A✝
rel : A✝ → A✝ → Prop
R : Type u_4
inst✝⁵ : CommRing R
ι : Type u_... | Please generate a tactic in lean4 to solve the state.
STATE:
case a.refine'_1.intro.intro.intro.a
R✝ : Type u_1
inst✝¹¹ : CommSemiring R✝
ι✝ : Type u_2
inst✝¹⁰ : DecidableEq ι✝
inst✝⁹ : AddCommMonoid ι✝
A✝ : Type u_3
inst✝⁸ : CommSemiring A✝
inst✝⁷ : DecidableEq A✝
inst✝⁶ : Algebra R✝ A✝
𝒜✝ : ι✝ → Submodule R✝ A✝
rel ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | Ideal.IsHomogeneous_of_rel_isHomogeneous | [466, 1] | [504, 45] | rw [coeAddMonoidHom_of] | case a.refine'_1.intro.intro.intro.a
R✝ : Type u_1
inst✝¹¹ : CommSemiring R✝
ι✝ : Type u_2
inst✝¹⁰ : DecidableEq ι✝
inst✝⁹ : AddCommMonoid ι✝
A✝ : Type u_3
inst✝⁸ : CommSemiring A✝
inst✝⁷ : DecidableEq A✝
inst✝⁶ : Algebra R✝ A✝
𝒜✝ : ι✝ → Submodule R✝ A✝
rel : A✝ → A✝ → Prop
R : Type u_4
inst✝⁵ : CommRing R
ι : Type u_... | case a.refine'_1.intro.intro.intro.a
R✝ : Type u_1
inst✝¹¹ : CommSemiring R✝
ι✝ : Type u_2
inst✝¹⁰ : DecidableEq ι✝
inst✝⁹ : AddCommMonoid ι✝
A✝ : Type u_3
inst✝⁸ : CommSemiring A✝
inst✝⁷ : DecidableEq A✝
inst✝⁶ : Algebra R✝ A✝
𝒜✝ : ι✝ → Submodule R✝ A✝
rel : A✝ → A✝ → Prop
R : Type u_4
inst✝⁵ : CommRing R
ι : Type u_... | Please generate a tactic in lean4 to solve the state.
STATE:
case a.refine'_1.intro.intro.intro.a
R✝ : Type u_1
inst✝¹¹ : CommSemiring R✝
ι✝ : Type u_2
inst✝¹⁰ : DecidableEq ι✝
inst✝⁹ : AddCommMonoid ι✝
A✝ : Type u_3
inst✝⁸ : CommSemiring A✝
inst✝⁷ : DecidableEq A✝
inst✝⁶ : Algebra R✝ A✝
𝒜✝ : ι✝ → Submodule R✝ A✝
rel ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | Ideal.IsHomogeneous_of_rel_isHomogeneous | [466, 1] | [504, 45] | apply Ideal.subset_span | case a.refine'_1.intro.intro.intro.a
R✝ : Type u_1
inst✝¹¹ : CommSemiring R✝
ι✝ : Type u_2
inst✝¹⁰ : DecidableEq ι✝
inst✝⁹ : AddCommMonoid ι✝
A✝ : Type u_3
inst✝⁸ : CommSemiring A✝
inst✝⁷ : DecidableEq A✝
inst✝⁶ : Algebra R✝ A✝
𝒜✝ : ι✝ → Submodule R✝ A✝
rel : A✝ → A✝ → Prop
R : Type u_4
inst✝⁵ : CommRing R
ι : Type u_... | case a.refine'_1.intro.intro.intro.a.a
R✝ : Type u_1
inst✝¹¹ : CommSemiring R✝
ι✝ : Type u_2
inst✝¹⁰ : DecidableEq ι✝
inst✝⁹ : AddCommMonoid ι✝
A✝ : Type u_3
inst✝⁸ : CommSemiring A✝
inst✝⁷ : DecidableEq A✝
inst✝⁶ : Algebra R✝ A✝
𝒜✝ : ι✝ → Submodule R✝ A✝
rel : A✝ → A✝ → Prop
R : Type u_4
inst✝⁵ : CommRing R
ι : Type ... | Please generate a tactic in lean4 to solve the state.
STATE:
case a.refine'_1.intro.intro.intro.a
R✝ : Type u_1
inst✝¹¹ : CommSemiring R✝
ι✝ : Type u_2
inst✝¹⁰ : DecidableEq ι✝
inst✝⁹ : AddCommMonoid ι✝
A✝ : Type u_3
inst✝⁸ : CommSemiring A✝
inst✝⁷ : DecidableEq A✝
inst✝⁶ : Algebra R✝ A✝
𝒜✝ : ι✝ → Submodule R✝ A✝
rel ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | Ideal.IsHomogeneous_of_rel_isHomogeneous | [466, 1] | [504, 45] | use h𝒜.decompose' a i | case a.refine'_1.intro.intro.intro.a.a
R✝ : Type u_1
inst✝¹¹ : CommSemiring R✝
ι✝ : Type u_2
inst✝¹⁰ : DecidableEq ι✝
inst✝⁹ : AddCommMonoid ι✝
A✝ : Type u_3
inst✝⁸ : CommSemiring A✝
inst✝⁷ : DecidableEq A✝
inst✝⁶ : Algebra R✝ A✝
𝒜✝ : ι✝ → Submodule R✝ A✝
rel : A✝ → A✝ → Prop
R : Type u_4
inst✝⁵ : CommRing R
ι : Type ... | case h
R✝ : Type u_1
inst✝¹¹ : CommSemiring R✝
ι✝ : Type u_2
inst✝¹⁰ : DecidableEq ι✝
inst✝⁹ : AddCommMonoid ι✝
A✝ : Type u_3
inst✝⁸ : CommSemiring A✝
inst✝⁷ : DecidableEq A✝
inst✝⁶ : Algebra R✝ A✝
𝒜✝ : ι✝ → Submodule R✝ A✝
rel : A✝ → A✝ → Prop
R : Type u_4
inst✝⁵ : CommRing R
ι : Type u_5
inst✝⁴ : DecidableEq ι
inst✝... | Please generate a tactic in lean4 to solve the state.
STATE:
case a.refine'_1.intro.intro.intro.a.a
R✝ : Type u_1
inst✝¹¹ : CommSemiring R✝
ι✝ : Type u_2
inst✝¹⁰ : DecidableEq ι✝
inst✝⁹ : AddCommMonoid ι✝
A✝ : Type u_3
inst✝⁸ : CommSemiring A✝
inst✝⁷ : DecidableEq A✝
inst✝⁶ : Algebra R✝ A✝
𝒜✝ : ι✝ → Submodule R✝ A✝
re... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | Ideal.IsHomogeneous_of_rel_isHomogeneous | [466, 1] | [504, 45] | use h𝒜.decompose' b i | case h
R✝ : Type u_1
inst✝¹¹ : CommSemiring R✝
ι✝ : Type u_2
inst✝¹⁰ : DecidableEq ι✝
inst✝⁹ : AddCommMonoid ι✝
A✝ : Type u_3
inst✝⁸ : CommSemiring A✝
inst✝⁷ : DecidableEq A✝
inst✝⁶ : Algebra R✝ A✝
𝒜✝ : ι✝ → Submodule R✝ A✝
rel : A✝ → A✝ → Prop
R : Type u_4
inst✝⁵ : CommRing R
ι : Type u_5
inst✝⁴ : DecidableEq ι
inst✝... | case h
R✝ : Type u_1
inst✝¹¹ : CommSemiring R✝
ι✝ : Type u_2
inst✝¹⁰ : DecidableEq ι✝
inst✝⁹ : AddCommMonoid ι✝
A✝ : Type u_3
inst✝⁸ : CommSemiring A✝
inst✝⁷ : DecidableEq A✝
inst✝⁶ : Algebra R✝ A✝
𝒜✝ : ι✝ → Submodule R✝ A✝
rel : A✝ → A✝ → Prop
R : Type u_4
inst✝⁵ : CommRing R
ι : Type u_5
inst✝⁴ : DecidableEq ι
inst✝... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
R✝ : Type u_1
inst✝¹¹ : CommSemiring R✝
ι✝ : Type u_2
inst✝¹⁰ : DecidableEq ι✝
inst✝⁹ : AddCommMonoid ι✝
A✝ : Type u_3
inst✝⁸ : CommSemiring A✝
inst✝⁷ : DecidableEq A✝
inst✝⁶ : Algebra R✝ A✝
𝒜✝ : ι✝ → Submodule R✝ A✝
rel : A✝ → A✝ → Prop
R : Type u_4
... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | Ideal.IsHomogeneous_of_rel_isHomogeneous | [466, 1] | [504, 45] | constructor | case h
R✝ : Type u_1
inst✝¹¹ : CommSemiring R✝
ι✝ : Type u_2
inst✝¹⁰ : DecidableEq ι✝
inst✝⁹ : AddCommMonoid ι✝
A✝ : Type u_3
inst✝⁸ : CommSemiring A✝
inst✝⁷ : DecidableEq A✝
inst✝⁶ : Algebra R✝ A✝
𝒜✝ : ι✝ → Submodule R✝ A✝
rel : A✝ → A✝ → Prop
R : Type u_4
inst✝⁵ : CommRing R
ι : Type u_5
inst✝⁴ : DecidableEq ι
inst✝... | case h.left
R✝ : Type u_1
inst✝¹¹ : CommSemiring R✝
ι✝ : Type u_2
inst✝¹⁰ : DecidableEq ι✝
inst✝⁹ : AddCommMonoid ι✝
A✝ : Type u_3
inst✝⁸ : CommSemiring A✝
inst✝⁷ : DecidableEq A✝
inst✝⁶ : Algebra R✝ A✝
𝒜✝ : ι✝ → Submodule R✝ A✝
rel : A✝ → A✝ → Prop
R : Type u_4
inst✝⁵ : CommRing R
ι : Type u_5
inst✝⁴ : DecidableEq ι
... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
R✝ : Type u_1
inst✝¹¹ : CommSemiring R✝
ι✝ : Type u_2
inst✝¹⁰ : DecidableEq ι✝
inst✝⁹ : AddCommMonoid ι✝
A✝ : Type u_3
inst✝⁸ : CommSemiring A✝
inst✝⁷ : DecidableEq A✝
inst✝⁶ : Algebra R✝ A✝
𝒜✝ : ι✝ → Submodule R✝ A✝
rel : A✝ → A✝ → Prop
R : Type u_4
... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | Ideal.IsHomogeneous_of_rel_isHomogeneous | [466, 1] | [504, 45] | . use i
simp only [Decomposition.decompose'_eq, SetLike.coe_mem, true_and]
simp only [Rel.IsHomogeneous] at hr
exact hr h' i | case h.left
R✝ : Type u_1
inst✝¹¹ : CommSemiring R✝
ι✝ : Type u_2
inst✝¹⁰ : DecidableEq ι✝
inst✝⁹ : AddCommMonoid ι✝
A✝ : Type u_3
inst✝⁸ : CommSemiring A✝
inst✝⁷ : DecidableEq A✝
inst✝⁶ : Algebra R✝ A✝
𝒜✝ : ι✝ → Submodule R✝ A✝
rel : A✝ → A✝ → Prop
R : Type u_4
inst✝⁵ : CommRing R
ι : Type u_5
inst✝⁴ : DecidableEq ι
... | case h.right
R✝ : Type u_1
inst✝¹¹ : CommSemiring R✝
ι✝ : Type u_2
inst✝¹⁰ : DecidableEq ι✝
inst✝⁹ : AddCommMonoid ι✝
A✝ : Type u_3
inst✝⁸ : CommSemiring A✝
inst✝⁷ : DecidableEq A✝
inst✝⁶ : Algebra R✝ A✝
𝒜✝ : ι✝ → Submodule R✝ A✝
rel : A✝ → A✝ → Prop
R : Type u_4
inst✝⁵ : CommRing R
ι : Type u_5
inst✝⁴ : DecidableEq ι... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.left
R✝ : Type u_1
inst✝¹¹ : CommSemiring R✝
ι✝ : Type u_2
inst✝¹⁰ : DecidableEq ι✝
inst✝⁹ : AddCommMonoid ι✝
A✝ : Type u_3
inst✝⁸ : CommSemiring A✝
inst✝⁷ : DecidableEq A✝
inst✝⁶ : Algebra R✝ A✝
𝒜✝ : ι✝ → Submodule R✝ A✝
rel : A✝ → A✝ → Prop
R : Type... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | Ideal.IsHomogeneous_of_rel_isHomogeneous | [466, 1] | [504, 45] | . simp only [← h, Decomposition.decompose'_eq, decompose_add,
add_apply, AddSubmonoid.coe_add, Submodule.coe_toAddSubmonoid] | case h.right
R✝ : Type u_1
inst✝¹¹ : CommSemiring R✝
ι✝ : Type u_2
inst✝¹⁰ : DecidableEq ι✝
inst✝⁹ : AddCommMonoid ι✝
A✝ : Type u_3
inst✝⁸ : CommSemiring A✝
inst✝⁷ : DecidableEq A✝
inst✝⁶ : Algebra R✝ A✝
𝒜✝ : ι✝ → Submodule R✝ A✝
rel : A✝ → A✝ → Prop
R : Type u_4
inst✝⁵ : CommRing R
ι : Type u_5
inst✝⁴ : DecidableEq ι... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.right
R✝ : Type u_1
inst✝¹¹ : CommSemiring R✝
ι✝ : Type u_2
inst✝¹⁰ : DecidableEq ι✝
inst✝⁹ : AddCommMonoid ι✝
A✝ : Type u_3
inst✝⁸ : CommSemiring A✝
inst✝⁷ : DecidableEq A✝
inst✝⁶ : Algebra R✝ A✝
𝒜✝ : ι✝ → Submodule R✝ A✝
rel : A✝ → A✝ → Prop
R : Typ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | Ideal.IsHomogeneous_of_rel_isHomogeneous | [466, 1] | [504, 45] | use i | case h.left
R✝ : Type u_1
inst✝¹¹ : CommSemiring R✝
ι✝ : Type u_2
inst✝¹⁰ : DecidableEq ι✝
inst✝⁹ : AddCommMonoid ι✝
A✝ : Type u_3
inst✝⁸ : CommSemiring A✝
inst✝⁷ : DecidableEq A✝
inst✝⁶ : Algebra R✝ A✝
𝒜✝ : ι✝ → Submodule R✝ A✝
rel : A✝ → A✝ → Prop
R : Type u_4
inst✝⁵ : CommRing R
ι : Type u_5
inst✝⁴ : DecidableEq ι
... | case h
R✝ : Type u_1
inst✝¹¹ : CommSemiring R✝
ι✝ : Type u_2
inst✝¹⁰ : DecidableEq ι✝
inst✝⁹ : AddCommMonoid ι✝
A✝ : Type u_3
inst✝⁸ : CommSemiring A✝
inst✝⁷ : DecidableEq A✝
inst✝⁶ : Algebra R✝ A✝
𝒜✝ : ι✝ → Submodule R✝ A✝
rel : A✝ → A✝ → Prop
R : Type u_4
inst✝⁵ : CommRing R
ι : Type u_5
inst✝⁴ : DecidableEq ι
inst✝... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.left
R✝ : Type u_1
inst✝¹¹ : CommSemiring R✝
ι✝ : Type u_2
inst✝¹⁰ : DecidableEq ι✝
inst✝⁹ : AddCommMonoid ι✝
A✝ : Type u_3
inst✝⁸ : CommSemiring A✝
inst✝⁷ : DecidableEq A✝
inst✝⁶ : Algebra R✝ A✝
𝒜✝ : ι✝ → Submodule R✝ A✝
rel : A✝ → A✝ → Prop
R : Type... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | Ideal.IsHomogeneous_of_rel_isHomogeneous | [466, 1] | [504, 45] | simp only [Decomposition.decompose'_eq, SetLike.coe_mem, true_and] | case h
R✝ : Type u_1
inst✝¹¹ : CommSemiring R✝
ι✝ : Type u_2
inst✝¹⁰ : DecidableEq ι✝
inst✝⁹ : AddCommMonoid ι✝
A✝ : Type u_3
inst✝⁸ : CommSemiring A✝
inst✝⁷ : DecidableEq A✝
inst✝⁶ : Algebra R✝ A✝
𝒜✝ : ι✝ → Submodule R✝ A✝
rel : A✝ → A✝ → Prop
R : Type u_4
inst✝⁵ : CommRing R
ι : Type u_5
inst✝⁴ : DecidableEq ι
inst✝... | case h
R✝ : Type u_1
inst✝¹¹ : CommSemiring R✝
ι✝ : Type u_2
inst✝¹⁰ : DecidableEq ι✝
inst✝⁹ : AddCommMonoid ι✝
A✝ : Type u_3
inst✝⁸ : CommSemiring A✝
inst✝⁷ : DecidableEq A✝
inst✝⁶ : Algebra R✝ A✝
𝒜✝ : ι✝ → Submodule R✝ A✝
rel : A✝ → A✝ → Prop
R : Type u_4
inst✝⁵ : CommRing R
ι : Type u_5
inst✝⁴ : DecidableEq ι
inst✝... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
R✝ : Type u_1
inst✝¹¹ : CommSemiring R✝
ι✝ : Type u_2
inst✝¹⁰ : DecidableEq ι✝
inst✝⁹ : AddCommMonoid ι✝
A✝ : Type u_3
inst✝⁸ : CommSemiring A✝
inst✝⁷ : DecidableEq A✝
inst✝⁶ : Algebra R✝ A✝
𝒜✝ : ι✝ → Submodule R✝ A✝
rel : A✝ → A✝ → Prop
R : Type u_4
... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | Ideal.IsHomogeneous_of_rel_isHomogeneous | [466, 1] | [504, 45] | simp only [Rel.IsHomogeneous] at hr | case h
R✝ : Type u_1
inst✝¹¹ : CommSemiring R✝
ι✝ : Type u_2
inst✝¹⁰ : DecidableEq ι✝
inst✝⁹ : AddCommMonoid ι✝
A✝ : Type u_3
inst✝⁸ : CommSemiring A✝
inst✝⁷ : DecidableEq A✝
inst✝⁶ : Algebra R✝ A✝
𝒜✝ : ι✝ → Submodule R✝ A✝
rel : A✝ → A✝ → Prop
R : Type u_4
inst✝⁵ : CommRing R
ι : Type u_5
inst✝⁴ : DecidableEq ι
inst✝... | case h
R✝ : Type u_1
inst✝¹¹ : CommSemiring R✝
ι✝ : Type u_2
inst✝¹⁰ : DecidableEq ι✝
inst✝⁹ : AddCommMonoid ι✝
A✝ : Type u_3
inst✝⁸ : CommSemiring A✝
inst✝⁷ : DecidableEq A✝
inst✝⁶ : Algebra R✝ A✝
𝒜✝ : ι✝ → Submodule R✝ A✝
rel : A✝ → A✝ → Prop
R : Type u_4
inst✝⁵ : CommRing R
ι : Type u_5
inst✝⁴ : DecidableEq ι
inst✝... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
R✝ : Type u_1
inst✝¹¹ : CommSemiring R✝
ι✝ : Type u_2
inst✝¹⁰ : DecidableEq ι✝
inst✝⁹ : AddCommMonoid ι✝
A✝ : Type u_3
inst✝⁸ : CommSemiring A✝
inst✝⁷ : DecidableEq A✝
inst✝⁶ : Algebra R✝ A✝
𝒜✝ : ι✝ → Submodule R✝ A✝
rel : A✝ → A✝ → Prop
R : Type u_4
... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | Ideal.IsHomogeneous_of_rel_isHomogeneous | [466, 1] | [504, 45] | exact hr h' i | case h
R✝ : Type u_1
inst✝¹¹ : CommSemiring R✝
ι✝ : Type u_2
inst✝¹⁰ : DecidableEq ι✝
inst✝⁹ : AddCommMonoid ι✝
A✝ : Type u_3
inst✝⁸ : CommSemiring A✝
inst✝⁷ : DecidableEq A✝
inst✝⁶ : Algebra R✝ A✝
𝒜✝ : ι✝ → Submodule R✝ A✝
rel : A✝ → A✝ → Prop
R : Type u_4
inst✝⁵ : CommRing R
ι : Type u_5
inst✝⁴ : DecidableEq ι
inst✝... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
R✝ : Type u_1
inst✝¹¹ : CommSemiring R✝
ι✝ : Type u_2
inst✝¹⁰ : DecidableEq ι✝
inst✝⁹ : AddCommMonoid ι✝
A✝ : Type u_3
inst✝⁸ : CommSemiring A✝
inst✝⁷ : DecidableEq A✝
inst✝⁶ : Algebra R✝ A✝
𝒜✝ : ι✝ → Submodule R✝ A✝
rel : A✝ → A✝ → Prop
R : Type u_4
... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | Ideal.IsHomogeneous_of_rel_isHomogeneous | [466, 1] | [504, 45] | simp only [← h, Decomposition.decompose'_eq, decompose_add,
add_apply, AddSubmonoid.coe_add, Submodule.coe_toAddSubmonoid] | case h.right
R✝ : Type u_1
inst✝¹¹ : CommSemiring R✝
ι✝ : Type u_2
inst✝¹⁰ : DecidableEq ι✝
inst✝⁹ : AddCommMonoid ι✝
A✝ : Type u_3
inst✝⁸ : CommSemiring A✝
inst✝⁷ : DecidableEq A✝
inst✝⁶ : Algebra R✝ A✝
𝒜✝ : ι✝ → Submodule R✝ A✝
rel : A✝ → A✝ → Prop
R : Type u_4
inst✝⁵ : CommRing R
ι : Type u_5
inst✝⁴ : DecidableEq ι... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.right
R✝ : Type u_1
inst✝¹¹ : CommSemiring R✝
ι✝ : Type u_2
inst✝¹⁰ : DecidableEq ι✝
inst✝⁹ : AddCommMonoid ι✝
A✝ : Type u_3
inst✝⁸ : CommSemiring A✝
inst✝⁷ : DecidableEq A✝
inst✝⁶ : Algebra R✝ A✝
𝒜✝ : ι✝ → Submodule R✝ A✝
rel : A✝ → A✝ → Prop
R : Typ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | Ideal.IsHomogeneous_of_rel_isHomogeneous | [466, 1] | [504, 45] | simp only [Submodule.zero_mem] | case a.refine'_2
R✝ : Type u_1
inst✝¹¹ : CommSemiring R✝
ι✝ : Type u_2
inst✝¹⁰ : DecidableEq ι✝
inst✝⁹ : AddCommMonoid ι✝
A✝ : Type u_3
inst✝⁸ : CommSemiring A✝
inst✝⁷ : DecidableEq A✝
inst✝⁶ : Algebra R✝ A✝
𝒜✝ : ι✝ → Submodule R✝ A✝
rel : A✝ → A✝ → Prop
R : Type u_4
inst✝⁵ : CommRing R
ι : Type u_5
inst✝⁴ : Decidable... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a.refine'_2
R✝ : Type u_1
inst✝¹¹ : CommSemiring R✝
ι✝ : Type u_2
inst✝¹⁰ : DecidableEq ι✝
inst✝⁹ : AddCommMonoid ι✝
A✝ : Type u_3
inst✝⁸ : CommSemiring A✝
inst✝⁷ : DecidableEq A✝
inst✝⁶ : Algebra R✝ A✝
𝒜✝ : ι✝ → Submodule R✝ A✝
rel : A✝ → A✝ → Prop
R :... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | Ideal.IsHomogeneous_of_rel_isHomogeneous | [466, 1] | [504, 45] | intro x y hx hy | case a.refine'_3
R✝ : Type u_1
inst✝¹¹ : CommSemiring R✝
ι✝ : Type u_2
inst✝¹⁰ : DecidableEq ι✝
inst✝⁹ : AddCommMonoid ι✝
A✝ : Type u_3
inst✝⁸ : CommSemiring A✝
inst✝⁷ : DecidableEq A✝
inst✝⁶ : Algebra R✝ A✝
𝒜✝ : ι✝ → Submodule R✝ A✝
rel : A✝ → A✝ → Prop
R : Type u_4
inst✝⁵ : CommRing R
ι : Type u_5
inst✝⁴ : Decidable... | case a.refine'_3
R✝ : Type u_1
inst✝¹¹ : CommSemiring R✝
ι✝ : Type u_2
inst✝¹⁰ : DecidableEq ι✝
inst✝⁹ : AddCommMonoid ι✝
A✝ : Type u_3
inst✝⁸ : CommSemiring A✝
inst✝⁷ : DecidableEq A✝
inst✝⁶ : Algebra R✝ A✝
𝒜✝ : ι✝ → Submodule R✝ A✝
rel : A✝ → A✝ → Prop
R : Type u_4
inst✝⁵ : CommRing R
ι : Type u_5
inst✝⁴ : Decidable... | Please generate a tactic in lean4 to solve the state.
STATE:
case a.refine'_3
R✝ : Type u_1
inst✝¹¹ : CommSemiring R✝
ι✝ : Type u_2
inst✝¹⁰ : DecidableEq ι✝
inst✝⁹ : AddCommMonoid ι✝
A✝ : Type u_3
inst✝⁸ : CommSemiring A✝
inst✝⁷ : DecidableEq A✝
inst✝⁶ : Algebra R✝ A✝
𝒜✝ : ι✝ → Submodule R✝ A✝
rel : A✝ → A✝ → Prop
R :... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | Ideal.IsHomogeneous_of_rel_isHomogeneous | [466, 1] | [504, 45] | exact Ideal.add_mem _ hx hy | case a.refine'_3
R✝ : Type u_1
inst✝¹¹ : CommSemiring R✝
ι✝ : Type u_2
inst✝¹⁰ : DecidableEq ι✝
inst✝⁹ : AddCommMonoid ι✝
A✝ : Type u_3
inst✝⁸ : CommSemiring A✝
inst✝⁷ : DecidableEq A✝
inst✝⁶ : Algebra R✝ A✝
𝒜✝ : ι✝ → Submodule R✝ A✝
rel : A✝ → A✝ → Prop
R : Type u_4
inst✝⁵ : CommRing R
ι : Type u_5
inst✝⁴ : Decidable... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a.refine'_3
R✝ : Type u_1
inst✝¹¹ : CommSemiring R✝
ι✝ : Type u_2
inst✝¹⁰ : DecidableEq ι✝
inst✝⁹ : AddCommMonoid ι✝
A✝ : Type u_3
inst✝⁸ : CommSemiring A✝
inst✝⁷ : DecidableEq A✝
inst✝⁶ : Algebra R✝ A✝
𝒜✝ : ι✝ → Submodule R✝ A✝
rel : A✝ → A✝ → Prop
R :... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | Ideal.IsHomogeneous_of_rel_isHomogeneous | [466, 1] | [504, 45] | intro a x hx | case a.refine'_4
R✝ : Type u_1
inst✝¹¹ : CommSemiring R✝
ι✝ : Type u_2
inst✝¹⁰ : DecidableEq ι✝
inst✝⁹ : AddCommMonoid ι✝
A✝ : Type u_3
inst✝⁸ : CommSemiring A✝
inst✝⁷ : DecidableEq A✝
inst✝⁶ : Algebra R✝ A✝
𝒜✝ : ι✝ → Submodule R✝ A✝
rel : A✝ → A✝ → Prop
R : Type u_4
inst✝⁵ : CommRing R
ι : Type u_5
inst✝⁴ : Decidable... | case a.refine'_4
R✝ : Type u_1
inst✝¹¹ : CommSemiring R✝
ι✝ : Type u_2
inst✝¹⁰ : DecidableEq ι✝
inst✝⁹ : AddCommMonoid ι✝
A✝ : Type u_3
inst✝⁸ : CommSemiring A✝
inst✝⁷ : DecidableEq A✝
inst✝⁶ : Algebra R✝ A✝
𝒜✝ : ι✝ → Submodule R✝ A✝
rel : A✝ → A✝ → Prop
R : Type u_4
inst✝⁵ : CommRing R
ι : Type u_5
inst✝⁴ : Decidable... | Please generate a tactic in lean4 to solve the state.
STATE:
case a.refine'_4
R✝ : Type u_1
inst✝¹¹ : CommSemiring R✝
ι✝ : Type u_2
inst✝¹⁰ : DecidableEq ι✝
inst✝⁹ : AddCommMonoid ι✝
A✝ : Type u_3
inst✝⁸ : CommSemiring A✝
inst✝⁷ : DecidableEq A✝
inst✝⁶ : Algebra R✝ A✝
𝒜✝ : ι✝ → Submodule R✝ A✝
rel : A✝ → A✝ → Prop
R :... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | Ideal.IsHomogeneous_of_rel_isHomogeneous | [466, 1] | [504, 45] | simp only [smul_eq_mul] | case a.refine'_4
R✝ : Type u_1
inst✝¹¹ : CommSemiring R✝
ι✝ : Type u_2
inst✝¹⁰ : DecidableEq ι✝
inst✝⁹ : AddCommMonoid ι✝
A✝ : Type u_3
inst✝⁸ : CommSemiring A✝
inst✝⁷ : DecidableEq A✝
inst✝⁶ : Algebra R✝ A✝
𝒜✝ : ι✝ → Submodule R✝ A✝
rel : A✝ → A✝ → Prop
R : Type u_4
inst✝⁵ : CommRing R
ι : Type u_5
inst✝⁴ : Decidable... | case a.refine'_4
R✝ : Type u_1
inst✝¹¹ : CommSemiring R✝
ι✝ : Type u_2
inst✝¹⁰ : DecidableEq ι✝
inst✝⁹ : AddCommMonoid ι✝
A✝ : Type u_3
inst✝⁸ : CommSemiring A✝
inst✝⁷ : DecidableEq A✝
inst✝⁶ : Algebra R✝ A✝
𝒜✝ : ι✝ → Submodule R✝ A✝
rel : A✝ → A✝ → Prop
R : Type u_4
inst✝⁵ : CommRing R
ι : Type u_5
inst✝⁴ : Decidable... | Please generate a tactic in lean4 to solve the state.
STATE:
case a.refine'_4
R✝ : Type u_1
inst✝¹¹ : CommSemiring R✝
ι✝ : Type u_2
inst✝¹⁰ : DecidableEq ι✝
inst✝⁹ : AddCommMonoid ι✝
A✝ : Type u_3
inst✝⁸ : CommSemiring A✝
inst✝⁷ : DecidableEq A✝
inst✝⁶ : Algebra R✝ A✝
𝒜✝ : ι✝ → Submodule R✝ A✝
rel : A✝ → A✝ → Prop
R :... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | Ideal.IsHomogeneous_of_rel_isHomogeneous | [466, 1] | [504, 45] | apply Ideal.mul_mem_left _ _ hx | case a.refine'_4
R✝ : Type u_1
inst✝¹¹ : CommSemiring R✝
ι✝ : Type u_2
inst✝¹⁰ : DecidableEq ι✝
inst✝⁹ : AddCommMonoid ι✝
A✝ : Type u_3
inst✝⁸ : CommSemiring A✝
inst✝⁷ : DecidableEq A✝
inst✝⁶ : Algebra R✝ A✝
𝒜✝ : ι✝ → Submodule R✝ A✝
rel : A✝ → A✝ → Prop
R : Type u_4
inst✝⁵ : CommRing R
ι : Type u_5
inst✝⁴ : Decidable... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a.refine'_4
R✝ : Type u_1
inst✝¹¹ : CommSemiring R✝
ι✝ : Type u_2
inst✝¹⁰ : DecidableEq ι✝
inst✝⁹ : AddCommMonoid ι✝
A✝ : Type u_3
inst✝⁸ : CommSemiring A✝
inst✝⁷ : DecidableEq A✝
inst✝⁶ : Algebra R✝ A✝
𝒜✝ : ι✝ → Submodule R✝ A✝
rel : A✝ → A✝ → Prop
R :... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | Ideal.IsHomogeneous_of_rel_isHomogeneous | [466, 1] | [504, 45] | intro x hx' | case a
R✝ : Type u_1
inst✝¹¹ : CommSemiring R✝
ι✝ : Type u_2
inst✝¹⁰ : DecidableEq ι✝
inst✝⁹ : AddCommMonoid ι✝
A✝ : Type u_3
inst✝⁸ : CommSemiring A✝
inst✝⁷ : DecidableEq A✝
inst✝⁶ : Algebra R✝ A✝
𝒜✝ : ι✝ → Submodule R✝ A✝
rel : A✝ → A✝ → Prop
R : Type u_4
inst✝⁵ : CommRing R
ι : Type u_5
inst✝⁴ : DecidableEq ι
inst✝... | case a
R✝ : Type u_1
inst✝¹¹ : CommSemiring R✝
ι✝ : Type u_2
inst✝¹⁰ : DecidableEq ι✝
inst✝⁹ : AddCommMonoid ι✝
A✝ : Type u_3
inst✝⁸ : CommSemiring A✝
inst✝⁷ : DecidableEq A✝
inst✝⁶ : Algebra R✝ A✝
𝒜✝ : ι✝ → Submodule R✝ A✝
rel : A✝ → A✝ → Prop
R : Type u_4
inst✝⁵ : CommRing R
ι : Type u_5
inst✝⁴ : DecidableEq ι
inst✝... | Please generate a tactic in lean4 to solve the state.
STATE:
case a
R✝ : Type u_1
inst✝¹¹ : CommSemiring R✝
ι✝ : Type u_2
inst✝¹⁰ : DecidableEq ι✝
inst✝⁹ : AddCommMonoid ι✝
A✝ : Type u_3
inst✝⁸ : CommSemiring A✝
inst✝⁷ : DecidableEq A✝
inst✝⁶ : Algebra R✝ A✝
𝒜✝ : ι✝ → Submodule R✝ A✝
rel : A✝ → A✝ → Prop
R : Type u_4
... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | Ideal.IsHomogeneous_of_rel_isHomogeneous | [466, 1] | [504, 45] | refine' Submodule.span_induction hx' _ (Submodule.zero_mem _)
(fun _ _ hx hy => Ideal.add_mem _ hx hy) (fun a _ hx => Ideal.mul_mem_left _ a hx) | case a
R✝ : Type u_1
inst✝¹¹ : CommSemiring R✝
ι✝ : Type u_2
inst✝¹⁰ : DecidableEq ι✝
inst✝⁹ : AddCommMonoid ι✝
A✝ : Type u_3
inst✝⁸ : CommSemiring A✝
inst✝⁷ : DecidableEq A✝
inst✝⁶ : Algebra R✝ A✝
𝒜✝ : ι✝ → Submodule R✝ A✝
rel : A✝ → A✝ → Prop
R : Type u_4
inst✝⁵ : CommRing R
ι : Type u_5
inst✝⁴ : DecidableEq ι
inst✝... | case a
R✝ : Type u_1
inst✝¹¹ : CommSemiring R✝
ι✝ : Type u_2
inst✝¹⁰ : DecidableEq ι✝
inst✝⁹ : AddCommMonoid ι✝
A✝ : Type u_3
inst✝⁸ : CommSemiring A✝
inst✝⁷ : DecidableEq A✝
inst✝⁶ : Algebra R✝ A✝
𝒜✝ : ι✝ → Submodule R✝ A✝
rel : A✝ → A✝ → Prop
R : Type u_4
inst✝⁵ : CommRing R
ι : Type u_5
inst✝⁴ : DecidableEq ι
inst✝... | Please generate a tactic in lean4 to solve the state.
STATE:
case a
R✝ : Type u_1
inst✝¹¹ : CommSemiring R✝
ι✝ : Type u_2
inst✝¹⁰ : DecidableEq ι✝
inst✝⁹ : AddCommMonoid ι✝
A✝ : Type u_3
inst✝⁸ : CommSemiring A✝
inst✝⁷ : DecidableEq A✝
inst✝⁶ : Algebra R✝ A✝
𝒜✝ : ι✝ → Submodule R✝ A✝
rel : A✝ → A✝ → Prop
R : Type u_4
... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | Ideal.IsHomogeneous_of_rel_isHomogeneous | [466, 1] | [504, 45] | . rintro x ⟨a, b, ⟨i, _, _, h'⟩, h⟩
exact Ideal.subset_span ⟨a, b, h', h⟩ | case a
R✝ : Type u_1
inst✝¹¹ : CommSemiring R✝
ι✝ : Type u_2
inst✝¹⁰ : DecidableEq ι✝
inst✝⁹ : AddCommMonoid ι✝
A✝ : Type u_3
inst✝⁸ : CommSemiring A✝
inst✝⁷ : DecidableEq A✝
inst✝⁶ : Algebra R✝ A✝
𝒜✝ : ι✝ → Submodule R✝ A✝
rel : A✝ → A✝ → Prop
R : Type u_4
inst✝⁵ : CommRing R
ι : Type u_5
inst✝⁴ : DecidableEq ι
inst✝... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a
R✝ : Type u_1
inst✝¹¹ : CommSemiring R✝
ι✝ : Type u_2
inst✝¹⁰ : DecidableEq ι✝
inst✝⁹ : AddCommMonoid ι✝
A✝ : Type u_3
inst✝⁸ : CommSemiring A✝
inst✝⁷ : DecidableEq A✝
inst✝⁶ : Algebra R✝ A✝
𝒜✝ : ι✝ → Submodule R✝ A✝
rel : A✝ → A✝ → Prop
R : Type u_4
... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | Ideal.IsHomogeneous_of_rel_isHomogeneous | [466, 1] | [504, 45] | rintro x ⟨a, b, ⟨i, _, _, h'⟩, h⟩ | case a
R✝ : Type u_1
inst✝¹¹ : CommSemiring R✝
ι✝ : Type u_2
inst✝¹⁰ : DecidableEq ι✝
inst✝⁹ : AddCommMonoid ι✝
A✝ : Type u_3
inst✝⁸ : CommSemiring A✝
inst✝⁷ : DecidableEq A✝
inst✝⁶ : Algebra R✝ A✝
𝒜✝ : ι✝ → Submodule R✝ A✝
rel : A✝ → A✝ → Prop
R : Type u_4
inst✝⁵ : CommRing R
ι : Type u_5
inst✝⁴ : DecidableEq ι
inst✝... | case a.intro.intro.intro.intro.intro.intro
R✝ : Type u_1
inst✝¹¹ : CommSemiring R✝
ι✝ : Type u_2
inst✝¹⁰ : DecidableEq ι✝
inst✝⁹ : AddCommMonoid ι✝
A✝ : Type u_3
inst✝⁸ : CommSemiring A✝
inst✝⁷ : DecidableEq A✝
inst✝⁶ : Algebra R✝ A✝
𝒜✝ : ι✝ → Submodule R✝ A✝
rel : A✝ → A✝ → Prop
R : Type u_4
inst✝⁵ : CommRing R
ι : T... | Please generate a tactic in lean4 to solve the state.
STATE:
case a
R✝ : Type u_1
inst✝¹¹ : CommSemiring R✝
ι✝ : Type u_2
inst✝¹⁰ : DecidableEq ι✝
inst✝⁹ : AddCommMonoid ι✝
A✝ : Type u_3
inst✝⁸ : CommSemiring A✝
inst✝⁷ : DecidableEq A✝
inst✝⁶ : Algebra R✝ A✝
𝒜✝ : ι✝ → Submodule R✝ A✝
rel : A✝ → A✝ → Prop
R : Type u_4
... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | Ideal.IsHomogeneous_of_rel_isHomogeneous | [466, 1] | [504, 45] | exact Ideal.subset_span ⟨a, b, h', h⟩ | case a.intro.intro.intro.intro.intro.intro
R✝ : Type u_1
inst✝¹¹ : CommSemiring R✝
ι✝ : Type u_2
inst✝¹⁰ : DecidableEq ι✝
inst✝⁹ : AddCommMonoid ι✝
A✝ : Type u_3
inst✝⁸ : CommSemiring A✝
inst✝⁷ : DecidableEq A✝
inst✝⁶ : Algebra R✝ A✝
𝒜✝ : ι✝ → Submodule R✝ A✝
rel : A✝ → A✝ → Prop
R : Type u_4
inst✝⁵ : CommRing R
ι : T... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a.intro.intro.intro.intro.intro.intro
R✝ : Type u_1
inst✝¹¹ : CommSemiring R✝
ι✝ : Type u_2
inst✝¹⁰ : DecidableEq ι✝
inst✝⁹ : AddCommMonoid ι✝
A✝ : Type u_3
inst✝⁸ : CommSemiring A✝
inst✝⁷ : DecidableEq A✝
inst✝⁶ : Algebra R✝ A✝
𝒜✝ : ι✝ → Submodule R✝ A... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | foo_mul | [535, 1] | [541, 39] | obtain ⟨a, ha, rfl⟩ := ha | 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
a b : RingQuot rel
i j : ι
ha : a ∈ quotSubmodule R 𝒜 rel i
hb : b ∈ quotSub... | case 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
b : RingQuot rel
i j : ι
hb : b ∈ quotSubmodule R 𝒜 rel j
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
a b : RingQuot r... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | foo_mul | [535, 1] | [541, 39] | obtain ⟨b, hb, rfl⟩ := hb | case 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
b : RingQuot rel
i j : ι
hb : b ∈ quotSubmodule R 𝒜 rel j
a... | case intro.intro.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
i j : ι
a : A
ha : a ∈ ↑(𝒜 i)
b : A
hb : b ∈ ↑(... | Please generate a tactic in lean4 to solve the state.
STATE:
case 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... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | foo_mul | [535, 1] | [541, 39] | rw [← map_mul] | case intro.intro.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
i j : ι
a : A
ha : a ∈ ↑(𝒜 i)
b : A
hb : b ∈ ↑(... | case intro.intro.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
i j : ι
a : A
ha : a ∈ ↑(𝒜 i)
b : A
hb : b ∈ ↑(... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.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 : Grad... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | foo_mul | [535, 1] | [541, 39] | exact ⟨a * b, h𝒜.mul_mem ha hb, rfl⟩ | case intro.intro.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
i j : ι
a : A
ha : a ∈ ↑(𝒜 i)
b : A
hb : b ∈ ↑(... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.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 : Grad... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | quotDecompose_left_inv'_aux | [550, 1] | [559, 6] | apply linearMap_ext | 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
⊢ (coeLinearMap fun i => Submodule.map (RingQuot.mkAlgHom R rel) (𝒜 i)) ∘ₗ lmap' (quotCompMap R 𝒜 ... | case 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
⊢ ∀ (i : ι),
((coeLinearMap fun i => Submodule.map (RingQuot.mkAlgHom R rel) (𝒜 i)) ∘ₗ l... | 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
⊢ (coeLinearMap fun i => Submodule.map ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | quotDecompose_left_inv'_aux | [550, 1] | [559, 6] | intro i | case 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
⊢ ∀ (i : ι),
((coeLinearMap fun i => Submodule.map (RingQuot.mkAlgHom R rel) (𝒜 i)) ∘ₗ l... | case 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
i : ι
⊢ ((coeLinearMap fun i => Submodule.map (RingQuot.mkAlgHom R rel) (𝒜 i)) ∘ₗ lmap' (quo... | Please generate a tactic in lean4 to solve the state.
STATE:
case 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
⊢ ∀ (i : ι),
((coeLinearMap ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | quotDecompose_left_inv'_aux | [550, 1] | [559, 6] | ext ⟨x, hx⟩ | case 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
i : ι
⊢ ((coeLinearMap fun i => Submodule.map (RingQuot.mkAlgHom R rel) (𝒜 i)) ∘ₗ lmap' (quo... | case H.h.mk
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
i : ι
x : A
hx : x ∈ 𝒜 i
⊢ (((coeLinearMap fun i => Submodule.map (RingQuot.mkAlgHom R ... | Please generate a tactic in lean4 to solve the state.
STATE:
case 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
i : ι
⊢ ((coeLinearMap fun i => ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | quotDecompose_left_inv'_aux | [550, 1] | [559, 6] | simp only [quotCompMap, LinearMap.coe_comp, comp_apply, AlgHom.toLinearMap_apply, lmap'_lof] | case H.h.mk
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
i : ι
x : A
hx : x ∈ 𝒜 i
⊢ (((coeLinearMap fun i => Submodule.map (RingQuot.mkAlgHom R ... | case H.h.mk
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
i : ι
x : A
hx : x ∈ 𝒜 i
⊢ (coeLinearMap fun i => Submodule.map (RingQuot.mkAlgHom R re... | Please generate a tactic in lean4 to solve the state.
STATE:
case H.h.mk
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
i : ι
x : A
hx : x ∈ 𝒜 i
⊢... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | quotDecompose_left_inv'_aux | [550, 1] | [559, 6] | simp only [lof_eq_of, coeLinearMap_of] | case H.h.mk
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
i : ι
x : A
hx : x ∈ 𝒜 i
⊢ (coeLinearMap fun i => Submodule.map (RingQuot.mkAlgHom R re... | case H.h.mk
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
i : ι
x : A
hx : x ∈ 𝒜 i
⊢ ↑({ toFun := fun u => ⟨(RingQuot.mkAlgHom R rel) ↑u, ⋯⟩, map... | Please generate a tactic in lean4 to solve the state.
STATE:
case H.h.mk
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
i : ι
x : A
hx : x ∈ 𝒜 i
⊢... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | quotDecompose_left_inv'_aux | [550, 1] | [559, 6] | rfl | case H.h.mk
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
i : ι
x : A
hx : x ∈ 𝒜 i
⊢ ↑({ toFun := fun u => ⟨(RingQuot.mkAlgHom R rel) ↑u, ⋯⟩, map... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case H.h.mk
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
i : ι
x : A
hx : x ∈ 𝒜 i
⊢... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | quotDecompose_left_inv'_aux_apply | [561, 1] | [567, 10] | let e := quotDecompose_left_inv'_aux R 𝒜 rel | 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
x : ⨁ (i : ι), ↥(𝒜 i)
⊢ (coeLinearMap fun i => Submodule.map (RingQuot.mkAlgHom R rel) (𝒜 i)) ((lm... | 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
x : ⨁ (i : ι), ↥(𝒜 i)
e : (coeLinearMap fun i => Submodule.map (RingQuot.mkAlgHom R rel) (𝒜 i)) ∘ₗ... | 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
x : ⨁ (i : ι), ↥(𝒜 i)
⊢ (coeLinearMap ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | quotDecompose_left_inv'_aux_apply | [561, 1] | [567, 10] | simp only [LinearMap.ext_iff, LinearMap.comp_apply, AlgHom.toLinearMap_apply] at e | 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
x : ⨁ (i : ι), ↥(𝒜 i)
e : (coeLinearMap fun i => Submodule.map (RingQuot.mkAlgHom R rel) (𝒜 i)) ∘ₗ... | 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
x : ⨁ (i : ι), ↥(𝒜 i)
e :
∀ (x : ⨁ (i : ι), ↥(𝒜 i)),
(coeLinearMap fun i => Submodule.map (R... | 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
x : ⨁ (i : ι), ↥(𝒜 i)
e : (coeLinearMa... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | quotDecompose_left_inv'_aux_apply | [561, 1] | [567, 10] | apply e | 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
x : ⨁ (i : ι), ↥(𝒜 i)
e :
∀ (x : ⨁ (i : ι), ↥(𝒜 i)),
(coeLinearMap fun i => Submodule.map (R... | no goals | 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
x : ⨁ (i : ι), ↥(𝒜 i)
e :
∀ (x : ⨁ (... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | quotDecompose'_apply | [569, 1] | [582, 41] | suffices (quotDecompose' R 𝒜 rel).comp (lmap' (quotCompMap R 𝒜 rel)) =
(RingQuot.mkAlgHom R rel).toLinearMap.comp (coeLinearMap 𝒜) by
simp only [LinearMap.ext_iff, LinearMap.comp_apply, AlgHom.toLinearMap_apply] at this
apply this | 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
a : ⨁ (i : ι), ↥(𝒜 i)
⊢ (quotDecompose' R 𝒜 rel) ((lmap' (quotCompMap R 𝒜 rel)) a) = (RingQuot.mk... | 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
a : ⨁ (i : ι), ↥(𝒜 i)
⊢ quotDecompose' R 𝒜 rel ∘ₗ lmap' (quotCompMap R 𝒜 rel) = (RingQuot.mkAlgHo... | 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
a : ⨁ (i : ι), ↥(𝒜 i)
⊢ (quotDecompose... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | quotDecompose'_apply | [569, 1] | [582, 41] | apply linearMap_ext | 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
a : ⨁ (i : ι), ↥(𝒜 i)
⊢ quotDecompose' R 𝒜 rel ∘ₗ lmap' (quotCompMap R 𝒜 rel) = (RingQuot.mkAlgHo... | case 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
a : ⨁ (i : ι), ↥(𝒜 i)
⊢ ∀ (i : ι),
(quotDecompose' R 𝒜 rel ∘ₗ lmap' (quotCompMap R 𝒜 r... | 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
a : ⨁ (i : ι), ↥(𝒜 i)
⊢ quotDecompose'... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | quotDecompose'_apply | [569, 1] | [582, 41] | intro i | case 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
a : ⨁ (i : ι), ↥(𝒜 i)
⊢ ∀ (i : ι),
(quotDecompose' R 𝒜 rel ∘ₗ lmap' (quotCompMap R 𝒜 r... | case 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
a : ⨁ (i : ι), ↥(𝒜 i)
i : ι
⊢ (quotDecompose' R 𝒜 rel ∘ₗ lmap' (quotCompMap R 𝒜 rel)) ∘ₗ l... | Please generate a tactic in lean4 to solve the state.
STATE:
case 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
a : ⨁ (i : ι), ↥(𝒜 i)
⊢ ∀ (i : ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | quotDecompose'_apply | [569, 1] | [582, 41] | ext ⟨x, hx⟩ | case 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
a : ⨁ (i : ι), ↥(𝒜 i)
i : ι
⊢ (quotDecompose' R 𝒜 rel ∘ₗ lmap' (quotCompMap R 𝒜 rel)) ∘ₗ l... | case H.h.mk
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
a : ⨁ (i : ι), ↥(𝒜 i)
i : ι
x : A
hx : x ∈ 𝒜 i
⊢ ((quotDecompose' R 𝒜 rel ∘ₗ lmap' (q... | Please generate a tactic in lean4 to solve the state.
STATE:
case 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
a : ⨁ (i : ι), ↥(𝒜 i)
i : ι
⊢ (... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | quotDecompose'_apply | [569, 1] | [582, 41] | simp only [quotDecompose', LinearMap.coe_comp, comp_apply, AlgHom.toLinearMap_apply,
lmap'_lof, toModule_lof] | case H.h.mk
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
a : ⨁ (i : ι), ↥(𝒜 i)
i : ι
x : A
hx : x ∈ 𝒜 i
⊢ ((quotDecompose' R 𝒜 rel ∘ₗ lmap' (q... | case H.h.mk
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
a : ⨁ (i : ι), ↥(𝒜 i)
i : ι
x : A
hx : x ∈ 𝒜 i
⊢ (quotSubmodule R 𝒜 rel i).subtype ((... | Please generate a tactic in lean4 to solve the state.
STATE:
case H.h.mk
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
a : ⨁ (i : ι), ↥(𝒜 i)
i : ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | quotDecompose'_apply | [569, 1] | [582, 41] | simp only [lof_eq_of, coeLinearMap_of, quotCompMap, LinearMap.coe_mk,
AddHom.coe_mk, Submodule.coeSubtype] | case H.h.mk
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
a : ⨁ (i : ι), ↥(𝒜 i)
i : ι
x : A
hx : x ∈ 𝒜 i
⊢ (quotSubmodule R 𝒜 rel i).subtype ((... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case H.h.mk
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
a : ⨁ (i : ι), ↥(𝒜 i)
i : ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | quotDecompose'_apply | [569, 1] | [582, 41] | simp only [LinearMap.ext_iff, LinearMap.comp_apply, AlgHom.toLinearMap_apply] at this | 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
a : ⨁ (i : ι), ↥(𝒜 i)
this : quotDecompose' R 𝒜 rel ∘ₗ lmap' (quotCompMap R 𝒜 rel) = (RingQuot.mk... | 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
a : ⨁ (i : ι), ↥(𝒜 i)
this :
∀ (x : ⨁ (i : ι), ↥(𝒜 i)),
(quotDecompose' R 𝒜 rel) ((lmap' (q... | 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
a : ⨁ (i : ι), ↥(𝒜 i)
this : quotDecom... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | quotDecompose'_apply | [569, 1] | [582, 41] | apply this | 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
a : ⨁ (i : ι), ↥(𝒜 i)
this :
∀ (x : ⨁ (i : ι), ↥(𝒜 i)),
(quotDecompose' R 𝒜 rel) ((lmap' (q... | no goals | 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
a : ⨁ (i : ι), ↥(𝒜 i)
this :
∀ (x : ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | lmap'_quotCompMap_apply | [584, 1] | [589, 23] | simp only [lmap'_apply] | 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
i : ι
a : ⨁ (i : ι), ↥(𝒜 i)
⊢ (RingQuot.mkAlgHom R rel) ↑(((decompose fun i ... | 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
i : ι
a : ⨁ (i : ι), ↥(𝒜 i)
⊢ (RingQuot.mkAlgHom R rel) ↑(((decompose fun i ... | 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
i : ι
a : ⨁ (i :... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | lmap'_quotCompMap_apply | [584, 1] | [589, 23] | congr | 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
i : ι
a : ⨁ (i : ι), ↥(𝒜 i)
⊢ (RingQuot.mkAlgHom R rel) ↑(((decompose fun i ... | case h.e_6.h.e_self.e_a
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
i : ι
a : ⨁ (i : ι), ↥(𝒜 i)
⊢ (decompose fun i => 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
i : ι
a : ⨁ (i :... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | lmap'_quotCompMap_apply | [584, 1] | [589, 23] | exact h𝒜.right_inv a | case h.e_6.h.e_self.e_a
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
i : ι
a : ⨁ (i : ι), ↥(𝒜 i)
⊢ (decompose fun i => 𝒜... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e_6.h.e_self.e_a
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 : GradedAlg... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | quotDecompose'_surjective | [591, 1] | [599, 42] | intro x | 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
⊢ Surjective ⇑(quotDecompose' R 𝒜 rel) | 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
x : RingQuot rel
⊢ ∃ a, (quotDecompose' R 𝒜 rel) a = x | 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
⊢ Surjective ⇑(q... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | quotDecompose'_surjective | [591, 1] | [599, 42] | obtain ⟨a, rfl⟩ := RingQuot.mkAlgHom_surjective R rel x | 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
x : RingQuot rel
⊢ ∃ a, (quotDecompose' R 𝒜 rel) a = x | case 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
a : A
⊢ ∃ a_1, (quotDecompose' R 𝒜 rel) a_1 = (RingQuot.mkAlgHom ... | 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
x : RingQuot rel... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | quotDecompose'_surjective | [591, 1] | [599, 42] | let e : (coeLinearMap 𝒜) ((decomposeAlgEquiv 𝒜).toLinearMap a) = a :=
h𝒜.left_inv a | case 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
a : A
⊢ ∃ a_1, (quotDecompose' R 𝒜 rel) a_1 = (RingQuot.mkAlgHom ... | case 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
a : A
e : (coeLinearMap 𝒜) ((decomposeAlgEquiv 𝒜).toLinearMap a)... | Please generate a tactic in lean4 to solve the state.
STATE:
case 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
a : A... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | quotDecompose'_surjective | [591, 1] | [599, 42] | use (lmap' (quotCompMap R 𝒜 rel)) ((decomposeAlgEquiv 𝒜).toLinearMap a) | case 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
a : A
e : (coeLinearMap 𝒜) ((decomposeAlgEquiv 𝒜).toLinearMap a)... | case 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
a : A
e : (coeLinearMap 𝒜) ((decomposeAlgEquiv 𝒜).toLinearMap a) = a... | Please generate a tactic in lean4 to solve the state.
STATE:
case 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
a : A... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | quotDecompose'_surjective | [591, 1] | [599, 42] | conv_rhs => rw [← e] | case 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
a : A
e : (coeLinearMap 𝒜) ((decomposeAlgEquiv 𝒜).toLinearMap a) = a... | case 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
a : A
e : (coeLinearMap 𝒜) ((decomposeAlgEquiv 𝒜).toLinearMap a) = a... | Please generate a tactic in lean4 to solve the state.
STATE:
case 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
a : A
e :... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | quotDecompose'_surjective | [591, 1] | [599, 42] | apply quotDecompose_left_inv'_aux_apply | case 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
a : A
e : (coeLinearMap 𝒜) ((decomposeAlgEquiv 𝒜).toLinearMap a) = a... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case 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
a : A
e :... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | obvious_iff | [601, 1] | [617, 49] | constructor | 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
x y : A
⊢ (RingQuot.mkRingHom rel) x = (RingQuot.mkRingHom rel) y ↔ RingConGen.Rel rel x y | case mp
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
x y : A
⊢ (RingQuot.mkRingHom rel) x = (RingQuot.mkRingHom rel) y → RingConGen.Rel rel x y
... | 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
x y : A
⊢ (RingQuot.mkRingHom rel) x = ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | obvious_iff | [601, 1] | [617, 49] | . intro h
suffices ∀ x, Quot.mk (RingQuot.Rel rel) x = ((RingQuot.mkRingHom rel) x).toQuot by
rw [← RingQuot.eqvGen_rel_eq, ← Quot.eq, this x, this y, h]
intro x
simp only [RingQuot.mkRingHom]
rfl | case mp
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
x y : A
⊢ (RingQuot.mkRingHom rel) x = (RingQuot.mkRingHom rel) y → RingConGen.Rel rel x y
... | case mpr
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
x y : A
⊢ RingConGen.Rel rel x y → (RingQuot.mkRingHom rel) x = (RingQuot.mkRingHom rel) y | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
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
x y : A
⊢ (RingQuot.mkRingHom r... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | obvious_iff | [601, 1] | [617, 49] | . intro h
induction h with
| of x y h => exact RingQuot.mkRingHom_rel h
| refl x => exact rfl
| symm _ k => exact k.symm
| trans h h' k k' => rw [k, k']
| add _ _ k k' => simp only [map_add, k, k']
| mul _ _ k k' => simp only [map_mul, k, k'] | case mpr
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
x y : A
⊢ RingConGen.Rel rel x y → (RingQuot.mkRingHom rel) x = (RingQuot.mkRingHom rel) y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
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
x y : A
⊢ RingConGen.Rel rel x... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | obvious_iff | [601, 1] | [617, 49] | intro h | case mp
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
x y : A
⊢ (RingQuot.mkRingHom rel) x = (RingQuot.mkRingHom rel) y → RingConGen.Rel rel x y | case mp
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
x y : A
h : (RingQuot.mkRingHom rel) x = (RingQuot.mkRingHom rel) y
⊢ RingConGen.Rel rel x y | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
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
x y : A
⊢ (RingQuot.mkRingHom r... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | obvious_iff | [601, 1] | [617, 49] | suffices ∀ x, Quot.mk (RingQuot.Rel rel) x = ((RingQuot.mkRingHom rel) x).toQuot by
rw [← RingQuot.eqvGen_rel_eq, ← Quot.eq, this x, this y, h] | case mp
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
x y : A
h : (RingQuot.mkRingHom rel) x = (RingQuot.mkRingHom rel) y
⊢ RingConGen.Rel rel x y | case mp
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
x y : A
h : (RingQuot.mkRingHom rel) x = (RingQuot.mkRingHom rel) y
⊢ ∀ (x : A), Quot.mk (Ri... | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
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
x y : A
h : (RingQuot.mkRingHom... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | obvious_iff | [601, 1] | [617, 49] | intro x | case mp
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
x y : A
h : (RingQuot.mkRingHom rel) x = (RingQuot.mkRingHom rel) y
⊢ ∀ (x : A), Quot.mk (Ri... | case mp
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
x✝ y : A
h : (RingQuot.mkRingHom rel) x✝ = (RingQuot.mkRingHom rel) y
x : A
⊢ Quot.mk (RingQ... | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
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
x y : A
h : (RingQuot.mkRingHom... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | obvious_iff | [601, 1] | [617, 49] | simp only [RingQuot.mkRingHom] | case mp
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
x✝ y : A
h : (RingQuot.mkRingHom rel) x✝ = (RingQuot.mkRingHom rel) y
x : A
⊢ Quot.mk (RingQ... | case mp
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
x✝ y : A
h : (RingQuot.mkRingHom rel) x✝ = (RingQuot.mkRingHom rel) y
x : A
⊢ Quot.mk (RingQ... | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
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
x✝ y : A
h : (RingQuot.mkRingHo... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | obvious_iff | [601, 1] | [617, 49] | rfl | case mp
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
x✝ y : A
h : (RingQuot.mkRingHom rel) x✝ = (RingQuot.mkRingHom rel) y
x : A
⊢ Quot.mk (RingQ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
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
x✝ y : A
h : (RingQuot.mkRingHo... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | obvious_iff | [601, 1] | [617, 49] | rw [← RingQuot.eqvGen_rel_eq, ← Quot.eq, this x, this y, 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
x y : A
h : (RingQuot.mkRingHom rel) x = (RingQuot.mkRingHom rel) y
this : ∀ (x : A), Quot.mk (RingQ... | no goals | 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
x y : A
h : (RingQuot.mkRingHom rel) x ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | obvious_iff | [601, 1] | [617, 49] | intro h | case mpr
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
x y : A
⊢ RingConGen.Rel rel x y → (RingQuot.mkRingHom rel) x = (RingQuot.mkRingHom rel) y | case mpr
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
x y : A
h : RingConGen.Rel rel x y
⊢ (RingQuot.mkRingHom rel) x = (RingQuot.mkRingHom rel) ... | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
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
x y : A
⊢ RingConGen.Rel rel x... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | obvious_iff | [601, 1] | [617, 49] | induction h with
| of x y h => exact RingQuot.mkRingHom_rel h
| refl x => exact rfl
| symm _ k => exact k.symm
| trans h h' k k' => rw [k, k']
| add _ _ k k' => simp only [map_add, k, k']
| mul _ _ k k' => simp only [map_mul, k, k'] | case mpr
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
x y : A
h : RingConGen.Rel rel x y
⊢ (RingQuot.mkRingHom rel) x = (RingQuot.mkRingHom rel) ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
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
x y : A
h : RingConGen.Rel rel... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | obvious_iff | [601, 1] | [617, 49] | exact RingQuot.mkRingHom_rel h | case mpr.of
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
x✝ y✝ x y : A
h : rel x y
⊢ (RingQuot.mkRingHom rel) x = (RingQuot.mkRingHom rel) y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.of
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
x✝ y✝ x y : A
h : rel x y
⊢... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | obvious_iff | [601, 1] | [617, 49] | exact rfl | case mpr.refl
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
x✝ y x : A
⊢ (RingQuot.mkRingHom rel) x = (RingQuot.mkRingHom rel) x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.refl
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
x✝ y x : A
⊢ (RingQuot.mk... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | obvious_iff | [601, 1] | [617, 49] | exact k.symm | case mpr.symm
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
x y x✝ y✝ : A
a✝ : RingConGen.Rel rel x✝ y✝
k : (RingQuot.mkRingHom rel) x✝ = (RingQuo... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.symm
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
x y x✝ y✝ : A
a✝ : RingCo... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | obvious_iff | [601, 1] | [617, 49] | rw [k, k'] | case mpr.trans
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
x y x✝ y✝ z✝ : A
h : RingConGen.Rel rel x✝ y✝
h' : RingConGen.Rel rel y✝ z✝
k : (Ring... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.trans
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
x y x✝ y✝ z✝ : A
h : Rin... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | obvious_iff | [601, 1] | [617, 49] | simp only [map_add, k, k'] | case mpr.add
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
x y w✝ x✝ y✝ z✝ : A
a✝¹ : RingConGen.Rel rel w✝ x✝
a✝ : RingConGen.Rel rel y✝ z✝
k : (R... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.add
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
x y w✝ x✝ y✝ z✝ : A
a✝¹ : ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | obvious_iff | [601, 1] | [617, 49] | simp only [map_mul, k, k'] | case mpr.mul
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
x y w✝ x✝ y✝ z✝ : A
a✝¹ : RingConGen.Rel rel w✝ x✝
a✝ : RingConGen.Rel rel y✝ z✝
k : (R... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.mul
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
x y w✝ x✝ y✝ z✝ : A
a✝¹ : ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | quotDecompose_injective | [619, 1] | [623, 54] | rw [← AlgHom.coe_toRingHom, RingQuot.mkAlgHom_coe R rel, obvious_iff] at hxy ⊢ | 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
x y : A
hxy : (RingQuot.mkAlgHom R rel) x = (... | 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
x y : A
hxy : RingConGen.Rel rel x y
i : ι
⊢ ... | 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 | [619, 1] | [623, 54] | exact RingConGen.RelIsHomogeneous_of 𝒜 _ hrel hxy i | 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
x y : A
hxy : RingConGen.Rel rel x y
i : ι
⊢ ... | no goals | 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_surjective2 | [625, 1] | [628, 23] | apply lmap'_surjective (quotCompMap R 𝒜 rel) | 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
⊢ Surjective ⇑(lmap' (quotCompMap R 𝒜 rel)) | 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
⊢ ∀ (i : ι), Surjective ⇑(quotCompMap R 𝒜 rel i) | 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
⊢ Surjective ⇑(lmap' (quotCompMap R 𝒜 ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | quotDecompose_surjective2 | [625, 1] | [628, 23] | 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
⊢ ∀ (i : ι), Surjective ⇑(quotCompMap R 𝒜 rel i) | 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
i : ι
a : A
ha : a ∈ ↑(𝒜 i)
⊢ ∃ a_1, (quotCompMap R 𝒜 rel i) a_1 = ⟨(RingQuot.... | 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
⊢ ∀ (i : ι), Surjective ⇑(quotCompMap R... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | quotDecompose_surjective2 | [625, 1] | [628, 23] | 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
i : ι
a : A
ha : a ∈ ↑(𝒜 i)
⊢ ∃ a_1, (quotCompMap R 𝒜 rel i) a_1 = ⟨(RingQuot.... | 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
i : ι
a : A
ha : a ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | quotDecompose'_injective | [630, 1] | [647, 23] | intro x y hxy | 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
⊢ Injective ⇑(quotDecompose' R 𝒜 rel) | 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
x y : ⨁ (i : ι), ↥(quotSubmodule R 𝒜 rel i)
... | 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 | [630, 1] | [647, 23] | obtain ⟨a, ha, rfl⟩ := quotDecompose_surjective2 R 𝒜 rel x | 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
x y : ⨁ (i : ι), ↥(quotSubmodule R 𝒜 rel i)
... | case intro.refl
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
y : ⨁ (i : ι), ↥(quotSubmodul... | 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 | [630, 1] | [647, 23] | obtain ⟨b, hb, rfl⟩ := quotDecompose_surjective2 R 𝒜 rel y | case intro.refl
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
y : ⨁ (i : ι), ↥(quotSubmodul... | case intro.refl.intro.refl
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
a b : ⨁ (i : ι), ↥... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.refl
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
... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | quotDecompose'_injective | [630, 1] | [647, 23] | simp only [quotDecompose'_apply] at hxy | case intro.refl.intro.refl
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
a b : ⨁ (i : ι), ↥... | case intro.refl.intro.refl
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
a b : ⨁ (i : ι), ↥... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.refl.intro.refl
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 : Graded... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | quotDecompose'_injective | [630, 1] | [647, 23] | let hxy' := quotDecompose_injective R 𝒜 rel hrel hxy | case intro.refl.intro.refl
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
a b : ⨁ (i : ι), ↥... | case intro.refl.intro.refl
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
a b : ⨁ (i : ι), ↥... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.refl.intro.refl
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 : Graded... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | quotDecompose'_injective | [630, 1] | [647, 23] | apply DFinsupp.ext | case intro.refl.intro.refl
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
a b : ⨁ (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
a b : ⨁ (i : ι),... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.refl.intro.refl
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 : Graded... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | quotDecompose'_injective | [630, 1] | [647, 23] | intro 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
a b : ⨁ (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
a b : ⨁ (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 | [630, 1] | [647, 23] | 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
a b : ⨁ (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
a b : ⨁ (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 | [630, 1] | [647, 23] | 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
a b : ⨁ (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
a b : ⨁ (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 | [630, 1] | [647, 23] | suffices ∀ a, RingQuot.mkAlgHom R rel ↑(((decompose fun i => 𝒜 i)
((coeLinearMap fun i => 𝒜 i) a)) i) = ((lmap' (quotCompMap R 𝒜 rel)) a) i by
simpa only [this, 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
a b : ⨁ (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
a b : ⨁ (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 | [630, 1] | [647, 23] | intro a | 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
a b : ⨁ (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
a✝ b : ⨁ (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 | [630, 1] | [647, 23] | simp only [lmap'_apply] | 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
a✝ b : ⨁ (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
a✝ b : ⨁ (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 | [630, 1] | [647, 23] | congr | 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
a✝ b : ⨁ (i : ι)... | case intro.refl.intro.refl.h.h.e_6.h.e_self.e_a
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 r... | 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 | [630, 1] | [647, 23] | exact h𝒜.right_inv a | case intro.refl.intro.refl.h.h.e_6.h.e_self.e_a
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 r... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.refl.intro.refl.h.h.e_6.h.e_self.e_a
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 → ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | quotDecompose'_injective | [630, 1] | [647, 23] | simpa only [this, SetLike.coe_eq_coe] using hxy' | 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
a b : ⨁ (i : ι), ↥(𝒜 i)
hxy : (RingQuot.mkAl... | no goals | 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] | have hφ : ∀ i, Surjective (quotCompMap R 𝒜 rel i) := by
rintro i ⟨x, ⟨a, ha, rfl⟩ ⟩
exact ⟨⟨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
⊢ Injective ⇑(coeLinearMap fun i => Submodule... | 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), Surjective ⇑(quotCompMap R 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] | intro x y hxy | 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), Surjective ⇑(quotCompMap R 𝒜... | 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), Surjective ⇑(quotCompMap R 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] | obtain ⟨a, ha, rfl⟩ := lmap'_surjective (quotCompMap R 𝒜 rel) hφ x | 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), Surjective ⇑(quotCompMap R 𝒜... | case intro.refl
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), Surjective ⇑(... | 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] | obtain ⟨b, hb, rfl⟩ := lmap'_surjective (quotCompMap R 𝒜 rel) hφ y | case intro.refl
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), Surjective ⇑(... | case intro.refl.intro.refl
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), Su... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.refl
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
... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | quotDecompose_injective' | [650, 1] | [664, 70] | simp only [quotDecompose_left_inv'_aux_apply] at hxy | case intro.refl.intro.refl
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), Su... | case intro.refl.intro.refl
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), Su... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.refl.intro.refl
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 : Graded... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | quotDecompose_injective' | [650, 1] | [664, 70] | let hxy' := quotDecompose_injective R 𝒜 rel hrel hxy | case intro.refl.intro.refl
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), Su... | case intro.refl.intro.refl
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), Su... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.refl.intro.refl
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 : Graded... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | quotDecompose_injective' | [650, 1] | [664, 70] | apply DFinsupp.ext | case intro.refl.intro.refl
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), Su... | 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
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 : Graded... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | quotDecompose_injective' | [650, 1] | [664, 70] | intro 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... |
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