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 | Rel.IsHomogeneous_of_isPureHomogeneous | [283, 1] | [293, 16] | simp only [AlgEquiv.toLinearMap_apply, decomposeAlgEquiv_coe, decompose_of_mem_ne _ ha hij,
decompose_of_mem_ne _ hb hij] | case neg
R : Type u_1
inst✝⁶ : CommSemiring R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : AddCommMonoid ι
A : Type u_3
inst✝³ : CommSemiring A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
rel : A → A → Prop
inst✝ : GradedAlgebra 𝒜
hrel : IsPureHomogeneous 𝒜 rel
hrel0 : rel 0 0
a b : A
h : rel a... | case neg
R : Type u_1
inst✝⁶ : CommSemiring R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : AddCommMonoid ι
A : Type u_3
inst✝³ : CommSemiring A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
rel : A → A → Prop
inst✝ : GradedAlgebra 𝒜
hrel : IsPureHomogeneous 𝒜 rel
hrel0 : rel 0 0
a b : A
h : rel a... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
R : Type u_1
inst✝⁶ : CommSemiring R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : AddCommMonoid ι
A : Type u_3
inst✝³ : CommSemiring A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
rel : A → A → Prop
inst✝ : GradedAlgebra 𝒜
hrel... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | Rel.IsHomogeneous_of_isPureHomogeneous | [283, 1] | [293, 16] | exact hrel0 | case neg
R : Type u_1
inst✝⁶ : CommSemiring R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : AddCommMonoid ι
A : Type u_3
inst✝³ : CommSemiring A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
rel : A → A → Prop
inst✝ : GradedAlgebra 𝒜
hrel : IsPureHomogeneous 𝒜 rel
hrel0 : rel 0 0
a b : A
h : rel a... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
R : Type u_1
inst✝⁶ : CommSemiring R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : AddCommMonoid ι
A : Type u_3
inst✝³ : CommSemiring A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
rel : A → A → Prop
inst✝ : GradedAlgebra 𝒜
hrel... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | RingConGen.Rel.sum | [297, 1] | [309, 65] | revert hs | R : Type u_1
inst✝⁷ : CommSemiring R
ι✝ : Type u_2
inst✝⁶ : DecidableEq ι✝
inst✝⁵ : AddCommMonoid ι✝
A : Type u_3
inst✝⁴ : CommSemiring A
inst✝³ : DecidableEq A
inst✝² : Algebra R A
𝒜 : ι✝ → Submodule R A
rel : A → A → Prop
α : Type u_4
inst✝¹ : Ring α
r : RingCon α
ι : Type u_5
inst✝ : DecidableEq ι
a b : ι → α
s : F... | R : Type u_1
inst✝⁷ : CommSemiring R
ι✝ : Type u_2
inst✝⁶ : DecidableEq ι✝
inst✝⁵ : AddCommMonoid ι✝
A : Type u_3
inst✝⁴ : CommSemiring A
inst✝³ : DecidableEq A
inst✝² : Algebra R A
𝒜 : ι✝ → Submodule R A
rel : A → A → Prop
α : Type u_4
inst✝¹ : Ring α
r : RingCon α
ι : Type u_5
inst✝ : DecidableEq ι
a b : ι → α
s : F... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝⁷ : CommSemiring R
ι✝ : Type u_2
inst✝⁶ : DecidableEq ι✝
inst✝⁵ : AddCommMonoid ι✝
A : Type u_3
inst✝⁴ : CommSemiring A
inst✝³ : DecidableEq A
inst✝² : Algebra R A
𝒜 : ι✝ → Submodule R A
rel : A → A → Prop
α : Type u_4
inst✝¹ : Ring α
r : R... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | RingConGen.Rel.sum | [297, 1] | [309, 65] | induction' s using Finset.induction_on with j t hj ht hjt | R : Type u_1
inst✝⁷ : CommSemiring R
ι✝ : Type u_2
inst✝⁶ : DecidableEq ι✝
inst✝⁵ : AddCommMonoid ι✝
A : Type u_3
inst✝⁴ : CommSemiring A
inst✝³ : DecidableEq A
inst✝² : Algebra R A
𝒜 : ι✝ → Submodule R A
rel : A → A → Prop
α : Type u_4
inst✝¹ : Ring α
r : RingCon α
ι : Type u_5
inst✝ : DecidableEq ι
a b : ι → α
s : F... | case empty
R : Type u_1
inst✝⁷ : CommSemiring R
ι✝ : Type u_2
inst✝⁶ : DecidableEq ι✝
inst✝⁵ : AddCommMonoid ι✝
A : Type u_3
inst✝⁴ : CommSemiring A
inst✝³ : DecidableEq A
inst✝² : Algebra R A
𝒜 : ι✝ → Submodule R A
rel : A → A → Prop
α : Type u_4
inst✝¹ : Ring α
r : RingCon α
ι : Type u_5
inst✝ : DecidableEq ι
a b : ... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝⁷ : CommSemiring R
ι✝ : Type u_2
inst✝⁶ : DecidableEq ι✝
inst✝⁵ : AddCommMonoid ι✝
A : Type u_3
inst✝⁴ : CommSemiring A
inst✝³ : DecidableEq A
inst✝² : Algebra R A
𝒜 : ι✝ → Submodule R A
rel : A → A → Prop
α : Type u_4
inst✝¹ : Ring α
r : R... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | RingConGen.Rel.sum | [297, 1] | [309, 65] | . exact fun _ => RingConGen.Rel.refl _ | case empty
R : Type u_1
inst✝⁷ : CommSemiring R
ι✝ : Type u_2
inst✝⁶ : DecidableEq ι✝
inst✝⁵ : AddCommMonoid ι✝
A : Type u_3
inst✝⁴ : CommSemiring A
inst✝³ : DecidableEq A
inst✝² : Algebra R A
𝒜 : ι✝ → Submodule R A
rel : A → A → Prop
α : Type u_4
inst✝¹ : Ring α
r : RingCon α
ι : Type u_5
inst✝ : DecidableEq ι
a b : ... | case insert
R : Type u_1
inst✝⁷ : CommSemiring R
ι✝ : Type u_2
inst✝⁶ : DecidableEq ι✝
inst✝⁵ : AddCommMonoid ι✝
A : Type u_3
inst✝⁴ : CommSemiring A
inst✝³ : DecidableEq A
inst✝² : Algebra R A
𝒜 : ι✝ → Submodule R A
rel : A → A → Prop
α : Type u_4
inst✝¹ : Ring α
r : RingCon α
ι : Type u_5
inst✝ : DecidableEq ι
a b :... | Please generate a tactic in lean4 to solve the state.
STATE:
case empty
R : Type u_1
inst✝⁷ : CommSemiring R
ι✝ : Type u_2
inst✝⁶ : DecidableEq ι✝
inst✝⁵ : AddCommMonoid ι✝
A : Type u_3
inst✝⁴ : CommSemiring A
inst✝³ : DecidableEq A
inst✝² : Algebra R A
𝒜 : ι✝ → Submodule R A
rel : A → A → Prop
α : Type u_4
inst✝¹ : R... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | RingConGen.Rel.sum | [297, 1] | [309, 65] | . intro hs
simp only [Finset.sum_insert hj]
apply RingConGen.Rel.add
. exact RingConGen.Rel.of _ _ (hs j (Finset.mem_insert_self j t))
. exact ht (fun i hi => hs i (Finset.mem_insert_of_mem hi)) | case insert
R : Type u_1
inst✝⁷ : CommSemiring R
ι✝ : Type u_2
inst✝⁶ : DecidableEq ι✝
inst✝⁵ : AddCommMonoid ι✝
A : Type u_3
inst✝⁴ : CommSemiring A
inst✝³ : DecidableEq A
inst✝² : Algebra R A
𝒜 : ι✝ → Submodule R A
rel : A → A → Prop
α : Type u_4
inst✝¹ : Ring α
r : RingCon α
ι : Type u_5
inst✝ : DecidableEq ι
a b :... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case insert
R : Type u_1
inst✝⁷ : CommSemiring R
ι✝ : Type u_2
inst✝⁶ : DecidableEq ι✝
inst✝⁵ : AddCommMonoid ι✝
A : Type u_3
inst✝⁴ : CommSemiring A
inst✝³ : DecidableEq A
inst✝² : Algebra R A
𝒜 : ι✝ → Submodule R A
rel : A → A → Prop
α : Type u_4
inst✝¹ : ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | RingConGen.Rel.sum | [297, 1] | [309, 65] | exact fun _ => RingConGen.Rel.refl _ | case empty
R : Type u_1
inst✝⁷ : CommSemiring R
ι✝ : Type u_2
inst✝⁶ : DecidableEq ι✝
inst✝⁵ : AddCommMonoid ι✝
A : Type u_3
inst✝⁴ : CommSemiring A
inst✝³ : DecidableEq A
inst✝² : Algebra R A
𝒜 : ι✝ → Submodule R A
rel : A → A → Prop
α : Type u_4
inst✝¹ : Ring α
r : RingCon α
ι : Type u_5
inst✝ : DecidableEq ι
a b : ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case empty
R : Type u_1
inst✝⁷ : CommSemiring R
ι✝ : Type u_2
inst✝⁶ : DecidableEq ι✝
inst✝⁵ : AddCommMonoid ι✝
A : Type u_3
inst✝⁴ : CommSemiring A
inst✝³ : DecidableEq A
inst✝² : Algebra R A
𝒜 : ι✝ → Submodule R A
rel : A → A → Prop
α : Type u_4
inst✝¹ : R... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | RingConGen.Rel.sum | [297, 1] | [309, 65] | intro hs | case insert
R : Type u_1
inst✝⁷ : CommSemiring R
ι✝ : Type u_2
inst✝⁶ : DecidableEq ι✝
inst✝⁵ : AddCommMonoid ι✝
A : Type u_3
inst✝⁴ : CommSemiring A
inst✝³ : DecidableEq A
inst✝² : Algebra R A
𝒜 : ι✝ → Submodule R A
rel : A → A → Prop
α : Type u_4
inst✝¹ : Ring α
r : RingCon α
ι : Type u_5
inst✝ : DecidableEq ι
a b :... | case insert
R : Type u_1
inst✝⁷ : CommSemiring R
ι✝ : Type u_2
inst✝⁶ : DecidableEq ι✝
inst✝⁵ : AddCommMonoid ι✝
A : Type u_3
inst✝⁴ : CommSemiring A
inst✝³ : DecidableEq A
inst✝² : Algebra R A
𝒜 : ι✝ → Submodule R A
rel : A → A → Prop
α : Type u_4
inst✝¹ : Ring α
r : RingCon α
ι : Type u_5
inst✝ : DecidableEq ι
a b :... | Please generate a tactic in lean4 to solve the state.
STATE:
case insert
R : Type u_1
inst✝⁷ : CommSemiring R
ι✝ : Type u_2
inst✝⁶ : DecidableEq ι✝
inst✝⁵ : AddCommMonoid ι✝
A : Type u_3
inst✝⁴ : CommSemiring A
inst✝³ : DecidableEq A
inst✝² : Algebra R A
𝒜 : ι✝ → Submodule R A
rel : A → A → Prop
α : Type u_4
inst✝¹ : ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | RingConGen.Rel.sum | [297, 1] | [309, 65] | simp only [Finset.sum_insert hj] | case insert
R : Type u_1
inst✝⁷ : CommSemiring R
ι✝ : Type u_2
inst✝⁶ : DecidableEq ι✝
inst✝⁵ : AddCommMonoid ι✝
A : Type u_3
inst✝⁴ : CommSemiring A
inst✝³ : DecidableEq A
inst✝² : Algebra R A
𝒜 : ι✝ → Submodule R A
rel : A → A → Prop
α : Type u_4
inst✝¹ : Ring α
r : RingCon α
ι : Type u_5
inst✝ : DecidableEq ι
a b :... | case insert
R : Type u_1
inst✝⁷ : CommSemiring R
ι✝ : Type u_2
inst✝⁶ : DecidableEq ι✝
inst✝⁵ : AddCommMonoid ι✝
A : Type u_3
inst✝⁴ : CommSemiring A
inst✝³ : DecidableEq A
inst✝² : Algebra R A
𝒜 : ι✝ → Submodule R A
rel : A → A → Prop
α : Type u_4
inst✝¹ : Ring α
r : RingCon α
ι : Type u_5
inst✝ : DecidableEq ι
a b :... | Please generate a tactic in lean4 to solve the state.
STATE:
case insert
R : Type u_1
inst✝⁷ : CommSemiring R
ι✝ : Type u_2
inst✝⁶ : DecidableEq ι✝
inst✝⁵ : AddCommMonoid ι✝
A : Type u_3
inst✝⁴ : CommSemiring A
inst✝³ : DecidableEq A
inst✝² : Algebra R A
𝒜 : ι✝ → Submodule R A
rel : A → A → Prop
α : Type u_4
inst✝¹ : ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | RingConGen.Rel.sum | [297, 1] | [309, 65] | apply RingConGen.Rel.add | case insert
R : Type u_1
inst✝⁷ : CommSemiring R
ι✝ : Type u_2
inst✝⁶ : DecidableEq ι✝
inst✝⁵ : AddCommMonoid ι✝
A : Type u_3
inst✝⁴ : CommSemiring A
inst✝³ : DecidableEq A
inst✝² : Algebra R A
𝒜 : ι✝ → Submodule R A
rel : A → A → Prop
α : Type u_4
inst✝¹ : Ring α
r : RingCon α
ι : Type u_5
inst✝ : DecidableEq ι
a b :... | case insert.a
R : Type u_1
inst✝⁷ : CommSemiring R
ι✝ : Type u_2
inst✝⁶ : DecidableEq ι✝
inst✝⁵ : AddCommMonoid ι✝
A : Type u_3
inst✝⁴ : CommSemiring A
inst✝³ : DecidableEq A
inst✝² : Algebra R A
𝒜 : ι✝ → Submodule R A
rel : A → A → Prop
α : Type u_4
inst✝¹ : Ring α
r : RingCon α
ι : Type u_5
inst✝ : DecidableEq ι
a b... | Please generate a tactic in lean4 to solve the state.
STATE:
case insert
R : Type u_1
inst✝⁷ : CommSemiring R
ι✝ : Type u_2
inst✝⁶ : DecidableEq ι✝
inst✝⁵ : AddCommMonoid ι✝
A : Type u_3
inst✝⁴ : CommSemiring A
inst✝³ : DecidableEq A
inst✝² : Algebra R A
𝒜 : ι✝ → Submodule R A
rel : A → A → Prop
α : Type u_4
inst✝¹ : ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | RingConGen.Rel.sum | [297, 1] | [309, 65] | . exact RingConGen.Rel.of _ _ (hs j (Finset.mem_insert_self j t)) | case insert.a
R : Type u_1
inst✝⁷ : CommSemiring R
ι✝ : Type u_2
inst✝⁶ : DecidableEq ι✝
inst✝⁵ : AddCommMonoid ι✝
A : Type u_3
inst✝⁴ : CommSemiring A
inst✝³ : DecidableEq A
inst✝² : Algebra R A
𝒜 : ι✝ → Submodule R A
rel : A → A → Prop
α : Type u_4
inst✝¹ : Ring α
r : RingCon α
ι : Type u_5
inst✝ : DecidableEq ι
a b... | case insert.a
R : Type u_1
inst✝⁷ : CommSemiring R
ι✝ : Type u_2
inst✝⁶ : DecidableEq ι✝
inst✝⁵ : AddCommMonoid ι✝
A : Type u_3
inst✝⁴ : CommSemiring A
inst✝³ : DecidableEq A
inst✝² : Algebra R A
𝒜 : ι✝ → Submodule R A
rel : A → A → Prop
α : Type u_4
inst✝¹ : Ring α
r : RingCon α
ι : Type u_5
inst✝ : DecidableEq ι
a b... | Please generate a tactic in lean4 to solve the state.
STATE:
case insert.a
R : Type u_1
inst✝⁷ : CommSemiring R
ι✝ : Type u_2
inst✝⁶ : DecidableEq ι✝
inst✝⁵ : AddCommMonoid ι✝
A : Type u_3
inst✝⁴ : CommSemiring A
inst✝³ : DecidableEq A
inst✝² : Algebra R A
𝒜 : ι✝ → Submodule R A
rel : A → A → Prop
α : Type u_4
inst✝¹ ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | RingConGen.Rel.sum | [297, 1] | [309, 65] | . exact ht (fun i hi => hs i (Finset.mem_insert_of_mem hi)) | case insert.a
R : Type u_1
inst✝⁷ : CommSemiring R
ι✝ : Type u_2
inst✝⁶ : DecidableEq ι✝
inst✝⁵ : AddCommMonoid ι✝
A : Type u_3
inst✝⁴ : CommSemiring A
inst✝³ : DecidableEq A
inst✝² : Algebra R A
𝒜 : ι✝ → Submodule R A
rel : A → A → Prop
α : Type u_4
inst✝¹ : Ring α
r : RingCon α
ι : Type u_5
inst✝ : DecidableEq ι
a b... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case insert.a
R : Type u_1
inst✝⁷ : CommSemiring R
ι✝ : Type u_2
inst✝⁶ : DecidableEq ι✝
inst✝⁵ : AddCommMonoid ι✝
A : Type u_3
inst✝⁴ : CommSemiring A
inst✝³ : DecidableEq A
inst✝² : Algebra R A
𝒜 : ι✝ → Submodule R A
rel : A → A → Prop
α : Type u_4
inst✝¹ ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | RingConGen.Rel.sum | [297, 1] | [309, 65] | exact RingConGen.Rel.of _ _ (hs j (Finset.mem_insert_self j t)) | case insert.a
R : Type u_1
inst✝⁷ : CommSemiring R
ι✝ : Type u_2
inst✝⁶ : DecidableEq ι✝
inst✝⁵ : AddCommMonoid ι✝
A : Type u_3
inst✝⁴ : CommSemiring A
inst✝³ : DecidableEq A
inst✝² : Algebra R A
𝒜 : ι✝ → Submodule R A
rel : A → A → Prop
α : Type u_4
inst✝¹ : Ring α
r : RingCon α
ι : Type u_5
inst✝ : DecidableEq ι
a b... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case insert.a
R : Type u_1
inst✝⁷ : CommSemiring R
ι✝ : Type u_2
inst✝⁶ : DecidableEq ι✝
inst✝⁵ : AddCommMonoid ι✝
A : Type u_3
inst✝⁴ : CommSemiring A
inst✝³ : DecidableEq A
inst✝² : Algebra R A
𝒜 : ι✝ → Submodule R A
rel : A → A → Prop
α : Type u_4
inst✝¹ ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | RingConGen.Rel.sum | [297, 1] | [309, 65] | exact ht (fun i hi => hs i (Finset.mem_insert_of_mem hi)) | case insert.a
R : Type u_1
inst✝⁷ : CommSemiring R
ι✝ : Type u_2
inst✝⁶ : DecidableEq ι✝
inst✝⁵ : AddCommMonoid ι✝
A : Type u_3
inst✝⁴ : CommSemiring A
inst✝³ : DecidableEq A
inst✝² : Algebra R A
𝒜 : ι✝ → Submodule R A
rel : A → A → Prop
α : Type u_4
inst✝¹ : Ring α
r : RingCon α
ι : Type u_5
inst✝ : DecidableEq ι
a b... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case insert.a
R : Type u_1
inst✝⁷ : CommSemiring R
ι✝ : Type u_2
inst✝⁶ : DecidableEq ι✝
inst✝⁵ : AddCommMonoid ι✝
A : Type u_3
inst✝⁴ : CommSemiring A
inst✝³ : DecidableEq A
inst✝² : Algebra R A
𝒜 : ι✝ → Submodule R A
rel : A → A → Prop
α : Type u_4
inst✝¹ ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | DFinsupp.sum_of_support_le | [312, 1] | [321, 23] | simp only [DFinsupp.sum] | R : Type u_1
inst✝⁷ : CommSemiring R
ι✝ : Type u_2
inst✝⁶ : DecidableEq ι✝
inst✝⁵ : AddCommMonoid ι✝
A : Type u_3
inst✝⁴ : CommSemiring A
inst✝³ : DecidableEq A
inst✝² : Algebra R A
𝒜 : ι✝ → Submodule R A
rel : A → A → Prop
M : Type u_4
inst✝¹ : AddCommMonoid M
ι : Type u_5
dec_ι : DecidableEq ι
β : ι → Type u_6
inst✝... | R : Type u_1
inst✝⁷ : CommSemiring R
ι✝ : Type u_2
inst✝⁶ : DecidableEq ι✝
inst✝⁵ : AddCommMonoid ι✝
A : Type u_3
inst✝⁴ : CommSemiring A
inst✝³ : DecidableEq A
inst✝² : Algebra R A
𝒜 : ι✝ → Submodule R A
rel : A → A → Prop
M : Type u_4
inst✝¹ : AddCommMonoid M
ι : Type u_5
dec_ι : DecidableEq ι
β : ι → Type u_6
inst✝... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝⁷ : CommSemiring R
ι✝ : Type u_2
inst✝⁶ : DecidableEq ι✝
inst✝⁵ : AddCommMonoid ι✝
A : Type u_3
inst✝⁴ : CommSemiring A
inst✝³ : DecidableEq A
inst✝² : Algebra R A
𝒜 : ι✝ → Submodule R A
rel : A → A → Prop
M : Type u_4
inst✝¹ : AddCommMonoi... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | DFinsupp.sum_of_support_le | [312, 1] | [321, 23] | apply Finset.sum_subset hs | R : Type u_1
inst✝⁷ : CommSemiring R
ι✝ : Type u_2
inst✝⁶ : DecidableEq ι✝
inst✝⁵ : AddCommMonoid ι✝
A : Type u_3
inst✝⁴ : CommSemiring A
inst✝³ : DecidableEq A
inst✝² : Algebra R A
𝒜 : ι✝ → Submodule R A
rel : A → A → Prop
M : Type u_4
inst✝¹ : AddCommMonoid M
ι : Type u_5
dec_ι : DecidableEq ι
β : ι → Type u_6
inst✝... | R : Type u_1
inst✝⁷ : CommSemiring R
ι✝ : Type u_2
inst✝⁶ : DecidableEq ι✝
inst✝⁵ : AddCommMonoid ι✝
A : Type u_3
inst✝⁴ : CommSemiring A
inst✝³ : DecidableEq A
inst✝² : Algebra R A
𝒜 : ι✝ → Submodule R A
rel : A → A → Prop
M : Type u_4
inst✝¹ : AddCommMonoid M
ι : Type u_5
dec_ι : DecidableEq ι
β : ι → Type u_6
inst✝... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝⁷ : CommSemiring R
ι✝ : Type u_2
inst✝⁶ : DecidableEq ι✝
inst✝⁵ : AddCommMonoid ι✝
A : Type u_3
inst✝⁴ : CommSemiring A
inst✝³ : DecidableEq A
inst✝² : Algebra R A
𝒜 : ι✝ → Submodule R A
rel : A → A → Prop
M : Type u_4
inst✝¹ : AddCommMonoi... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | DFinsupp.sum_of_support_le | [312, 1] | [321, 23] | . intro i _ hi'
simp only [DFinsupp.mem_support_toFun, ne_eq, not_not] at hi'
rw [hi', map_zero] | R : Type u_1
inst✝⁷ : CommSemiring R
ι✝ : Type u_2
inst✝⁶ : DecidableEq ι✝
inst✝⁵ : AddCommMonoid ι✝
A : Type u_3
inst✝⁴ : CommSemiring A
inst✝³ : DecidableEq A
inst✝² : Algebra R A
𝒜 : ι✝ → Submodule R A
rel : A → A → Prop
M : Type u_4
inst✝¹ : AddCommMonoid M
ι : Type u_5
dec_ι : DecidableEq ι
β : ι → Type u_6
inst✝... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝⁷ : CommSemiring R
ι✝ : Type u_2
inst✝⁶ : DecidableEq ι✝
inst✝⁵ : AddCommMonoid ι✝
A : Type u_3
inst✝⁴ : CommSemiring A
inst✝³ : DecidableEq A
inst✝² : Algebra R A
𝒜 : ι✝ → Submodule R A
rel : A → A → Prop
M : Type u_4
inst✝¹ : AddCommMonoi... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | DFinsupp.sum_of_support_le | [312, 1] | [321, 23] | intro i _ hi' | R : Type u_1
inst✝⁷ : CommSemiring R
ι✝ : Type u_2
inst✝⁶ : DecidableEq ι✝
inst✝⁵ : AddCommMonoid ι✝
A : Type u_3
inst✝⁴ : CommSemiring A
inst✝³ : DecidableEq A
inst✝² : Algebra R A
𝒜 : ι✝ → Submodule R A
rel : A → A → Prop
M : Type u_4
inst✝¹ : AddCommMonoid M
ι : Type u_5
dec_ι : DecidableEq ι
β : ι → Type u_6
inst✝... | R : Type u_1
inst✝⁷ : CommSemiring R
ι✝ : Type u_2
inst✝⁶ : DecidableEq ι✝
inst✝⁵ : AddCommMonoid ι✝
A : Type u_3
inst✝⁴ : CommSemiring A
inst✝³ : DecidableEq A
inst✝² : Algebra R A
𝒜 : ι✝ → Submodule R A
rel : A → A → Prop
M : Type u_4
inst✝¹ : AddCommMonoid M
ι : Type u_5
dec_ι : DecidableEq ι
β : ι → Type u_6
inst✝... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝⁷ : CommSemiring R
ι✝ : Type u_2
inst✝⁶ : DecidableEq ι✝
inst✝⁵ : AddCommMonoid ι✝
A : Type u_3
inst✝⁴ : CommSemiring A
inst✝³ : DecidableEq A
inst✝² : Algebra R A
𝒜 : ι✝ → Submodule R A
rel : A → A → Prop
M : Type u_4
inst✝¹ : AddCommMonoi... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | DFinsupp.sum_of_support_le | [312, 1] | [321, 23] | simp only [DFinsupp.mem_support_toFun, ne_eq, not_not] at hi' | R : Type u_1
inst✝⁷ : CommSemiring R
ι✝ : Type u_2
inst✝⁶ : DecidableEq ι✝
inst✝⁵ : AddCommMonoid ι✝
A : Type u_3
inst✝⁴ : CommSemiring A
inst✝³ : DecidableEq A
inst✝² : Algebra R A
𝒜 : ι✝ → Submodule R A
rel : A → A → Prop
M : Type u_4
inst✝¹ : AddCommMonoid M
ι : Type u_5
dec_ι : DecidableEq ι
β : ι → Type u_6
inst✝... | R : Type u_1
inst✝⁷ : CommSemiring R
ι✝ : Type u_2
inst✝⁶ : DecidableEq ι✝
inst✝⁵ : AddCommMonoid ι✝
A : Type u_3
inst✝⁴ : CommSemiring A
inst✝³ : DecidableEq A
inst✝² : Algebra R A
𝒜 : ι✝ → Submodule R A
rel : A → A → Prop
M : Type u_4
inst✝¹ : AddCommMonoid M
ι : Type u_5
dec_ι : DecidableEq ι
β : ι → Type u_6
inst✝... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝⁷ : CommSemiring R
ι✝ : Type u_2
inst✝⁶ : DecidableEq ι✝
inst✝⁵ : AddCommMonoid ι✝
A : Type u_3
inst✝⁴ : CommSemiring A
inst✝³ : DecidableEq A
inst✝² : Algebra R A
𝒜 : ι✝ → Submodule R A
rel : A → A → Prop
M : Type u_4
inst✝¹ : AddCommMonoi... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | DFinsupp.sum_of_support_le | [312, 1] | [321, 23] | rw [hi', map_zero] | R : Type u_1
inst✝⁷ : CommSemiring R
ι✝ : Type u_2
inst✝⁶ : DecidableEq ι✝
inst✝⁵ : AddCommMonoid ι✝
A : Type u_3
inst✝⁴ : CommSemiring A
inst✝³ : DecidableEq A
inst✝² : Algebra R A
𝒜 : ι✝ → Submodule R A
rel : A → A → Prop
M : Type u_4
inst✝¹ : AddCommMonoid M
ι : Type u_5
dec_ι : DecidableEq ι
β : ι → Type u_6
inst✝... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝⁷ : CommSemiring R
ι✝ : Type u_2
inst✝⁶ : DecidableEq ι✝
inst✝⁵ : AddCommMonoid ι✝
A : Type u_3
inst✝⁴ : CommSemiring A
inst✝³ : DecidableEq A
inst✝² : Algebra R A
𝒜 : ι✝ → Submodule R A
rel : A → A → Prop
M : Type u_4
inst✝¹ : AddCommMonoi... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | DirectSum.sum_of_support_le | [323, 1] | [332, 23] | conv_rhs => rw [← sum_support_of β x] | R : Type u_1
inst✝⁶ : CommSemiring R
ι✝ : Type u_2
inst✝⁵ : DecidableEq ι✝
inst✝⁴ : AddCommMonoid ι✝
A : Type u_3
inst✝³ : CommSemiring A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι✝ → Submodule R A
rel : A → A → Prop
ι : Type u_4
dec_ι : DecidableEq ι
β : ι → Type u_5
inst✝ : (i : ι) → AddCommMonoid (β i)
inst ... | R : Type u_1
inst✝⁶ : CommSemiring R
ι✝ : Type u_2
inst✝⁵ : DecidableEq ι✝
inst✝⁴ : AddCommMonoid ι✝
A : Type u_3
inst✝³ : CommSemiring A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι✝ → Submodule R A
rel : A → A → Prop
ι : Type u_4
dec_ι : DecidableEq ι
β : ι → Type u_5
inst✝ : (i : ι) → AddCommMonoid (β i)
inst ... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝⁶ : CommSemiring R
ι✝ : Type u_2
inst✝⁵ : DecidableEq ι✝
inst✝⁴ : AddCommMonoid ι✝
A : Type u_3
inst✝³ : CommSemiring A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι✝ → Submodule R A
rel : A → A → Prop
ι : Type u_4
dec_ι : DecidableEq ι... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | DirectSum.sum_of_support_le | [323, 1] | [332, 23] | rw [eq_comm] | R : Type u_1
inst✝⁶ : CommSemiring R
ι✝ : Type u_2
inst✝⁵ : DecidableEq ι✝
inst✝⁴ : AddCommMonoid ι✝
A : Type u_3
inst✝³ : CommSemiring A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι✝ → Submodule R A
rel : A → A → Prop
ι : Type u_4
dec_ι : DecidableEq ι
β : ι → Type u_5
inst✝ : (i : ι) → AddCommMonoid (β i)
inst ... | R : Type u_1
inst✝⁶ : CommSemiring R
ι✝ : Type u_2
inst✝⁵ : DecidableEq ι✝
inst✝⁴ : AddCommMonoid ι✝
A : Type u_3
inst✝³ : CommSemiring A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι✝ → Submodule R A
rel : A → A → Prop
ι : Type u_4
dec_ι : DecidableEq ι
β : ι → Type u_5
inst✝ : (i : ι) → AddCommMonoid (β i)
inst ... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝⁶ : CommSemiring R
ι✝ : Type u_2
inst✝⁵ : DecidableEq ι✝
inst✝⁴ : AddCommMonoid ι✝
A : Type u_3
inst✝³ : CommSemiring A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι✝ → Submodule R A
rel : A → A → Prop
ι : Type u_4
dec_ι : DecidableEq ι... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | DirectSum.sum_of_support_le | [323, 1] | [332, 23] | apply Finset.sum_subset hs | R : Type u_1
inst✝⁶ : CommSemiring R
ι✝ : Type u_2
inst✝⁵ : DecidableEq ι✝
inst✝⁴ : AddCommMonoid ι✝
A : Type u_3
inst✝³ : CommSemiring A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι✝ → Submodule R A
rel : A → A → Prop
ι : Type u_4
dec_ι : DecidableEq ι
β : ι → Type u_5
inst✝ : (i : ι) → AddCommMonoid (β i)
inst ... | R : Type u_1
inst✝⁶ : CommSemiring R
ι✝ : Type u_2
inst✝⁵ : DecidableEq ι✝
inst✝⁴ : AddCommMonoid ι✝
A : Type u_3
inst✝³ : CommSemiring A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι✝ → Submodule R A
rel : A → A → Prop
ι : Type u_4
dec_ι : DecidableEq ι
β : ι → Type u_5
inst✝ : (i : ι) → AddCommMonoid (β i)
inst ... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝⁶ : CommSemiring R
ι✝ : Type u_2
inst✝⁵ : DecidableEq ι✝
inst✝⁴ : AddCommMonoid ι✝
A : Type u_3
inst✝³ : CommSemiring A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι✝ → Submodule R A
rel : A → A → Prop
ι : Type u_4
dec_ι : DecidableEq ι... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | DirectSum.sum_of_support_le | [323, 1] | [332, 23] | . intro i _ hi'
simp only [DFinsupp.mem_support_toFun, ne_eq, not_not] at hi'
rw [hi', map_zero] | R : Type u_1
inst✝⁶ : CommSemiring R
ι✝ : Type u_2
inst✝⁵ : DecidableEq ι✝
inst✝⁴ : AddCommMonoid ι✝
A : Type u_3
inst✝³ : CommSemiring A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι✝ → Submodule R A
rel : A → A → Prop
ι : Type u_4
dec_ι : DecidableEq ι
β : ι → Type u_5
inst✝ : (i : ι) → AddCommMonoid (β i)
inst ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝⁶ : CommSemiring R
ι✝ : Type u_2
inst✝⁵ : DecidableEq ι✝
inst✝⁴ : AddCommMonoid ι✝
A : Type u_3
inst✝³ : CommSemiring A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι✝ → Submodule R A
rel : A → A → Prop
ι : Type u_4
dec_ι : DecidableEq ι... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | DirectSum.sum_of_support_le | [323, 1] | [332, 23] | intro i _ hi' | R : Type u_1
inst✝⁶ : CommSemiring R
ι✝ : Type u_2
inst✝⁵ : DecidableEq ι✝
inst✝⁴ : AddCommMonoid ι✝
A : Type u_3
inst✝³ : CommSemiring A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι✝ → Submodule R A
rel : A → A → Prop
ι : Type u_4
dec_ι : DecidableEq ι
β : ι → Type u_5
inst✝ : (i : ι) → AddCommMonoid (β i)
inst ... | R : Type u_1
inst✝⁶ : CommSemiring R
ι✝ : Type u_2
inst✝⁵ : DecidableEq ι✝
inst✝⁴ : AddCommMonoid ι✝
A : Type u_3
inst✝³ : CommSemiring A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι✝ → Submodule R A
rel : A → A → Prop
ι : Type u_4
dec_ι : DecidableEq ι
β : ι → Type u_5
inst✝ : (i : ι) → AddCommMonoid (β i)
inst ... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝⁶ : CommSemiring R
ι✝ : Type u_2
inst✝⁵ : DecidableEq ι✝
inst✝⁴ : AddCommMonoid ι✝
A : Type u_3
inst✝³ : CommSemiring A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι✝ → Submodule R A
rel : A → A → Prop
ι : Type u_4
dec_ι : DecidableEq ι... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | DirectSum.sum_of_support_le | [323, 1] | [332, 23] | simp only [DFinsupp.mem_support_toFun, ne_eq, not_not] at hi' | R : Type u_1
inst✝⁶ : CommSemiring R
ι✝ : Type u_2
inst✝⁵ : DecidableEq ι✝
inst✝⁴ : AddCommMonoid ι✝
A : Type u_3
inst✝³ : CommSemiring A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι✝ → Submodule R A
rel : A → A → Prop
ι : Type u_4
dec_ι : DecidableEq ι
β : ι → Type u_5
inst✝ : (i : ι) → AddCommMonoid (β i)
inst ... | R : Type u_1
inst✝⁶ : CommSemiring R
ι✝ : Type u_2
inst✝⁵ : DecidableEq ι✝
inst✝⁴ : AddCommMonoid ι✝
A : Type u_3
inst✝³ : CommSemiring A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι✝ → Submodule R A
rel : A → A → Prop
ι : Type u_4
dec_ι : DecidableEq ι
β : ι → Type u_5
inst✝ : (i : ι) → AddCommMonoid (β i)
inst ... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝⁶ : CommSemiring R
ι✝ : Type u_2
inst✝⁵ : DecidableEq ι✝
inst✝⁴ : AddCommMonoid ι✝
A : Type u_3
inst✝³ : CommSemiring A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι✝ → Submodule R A
rel : A → A → Prop
ι : Type u_4
dec_ι : DecidableEq ι... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | DirectSum.sum_of_support_le | [323, 1] | [332, 23] | rw [hi', map_zero] | R : Type u_1
inst✝⁶ : CommSemiring R
ι✝ : Type u_2
inst✝⁵ : DecidableEq ι✝
inst✝⁴ : AddCommMonoid ι✝
A : Type u_3
inst✝³ : CommSemiring A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι✝ → Submodule R A
rel : A → A → Prop
ι : Type u_4
dec_ι : DecidableEq ι
β : ι → Type u_5
inst✝ : (i : ι) → AddCommMonoid (β i)
inst ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝⁶ : CommSemiring R
ι✝ : Type u_2
inst✝⁵ : DecidableEq ι✝
inst✝⁴ : AddCommMonoid ι✝
A : Type u_3
inst✝³ : CommSemiring A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι✝ → Submodule R A
rel : A → A → Prop
ι : Type u_4
dec_ι : DecidableEq ι... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | DirectSum.finsupp_sum_support_decompose' | [334, 1] | [343, 6] | conv_lhs => rw [← sum_support_decompose ℳ r] | R : Type u_1
inst✝¹⁰ : CommSemiring R
ι✝ : Type u_2
inst✝⁹ : DecidableEq ι✝
inst✝⁸ : AddCommMonoid ι✝
A : Type u_3
inst✝⁷ : CommSemiring A
inst✝⁶ : DecidableEq A
inst✝⁵ : Algebra R A
𝒜 : ι✝ → Submodule R A
rel : A → A → Prop
ι : Type u_3
M : Type u_1
σ : Type u_2
inst✝⁴ : DecidableEq ι
inst✝³ : AddCommMonoid M
inst✝² ... | R : Type u_1
inst✝¹⁰ : CommSemiring R
ι✝ : Type u_2
inst✝⁹ : DecidableEq ι✝
inst✝⁸ : AddCommMonoid ι✝
A : Type u_3
inst✝⁷ : CommSemiring A
inst✝⁶ : DecidableEq A
inst✝⁵ : Algebra R A
𝒜 : ι✝ → Submodule R A
rel : A → A → Prop
ι : Type u_3
M : Type u_1
σ : Type u_2
inst✝⁴ : DecidableEq ι
inst✝³ : AddCommMonoid M
inst✝² ... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝¹⁰ : CommSemiring R
ι✝ : Type u_2
inst✝⁹ : DecidableEq ι✝
inst✝⁸ : AddCommMonoid ι✝
A : Type u_3
inst✝⁷ : CommSemiring A
inst✝⁶ : DecidableEq A
inst✝⁵ : Algebra R A
𝒜 : ι✝ → Submodule R A
rel : A → A → Prop
ι : Type u_3
M : Type u_1
σ : Typ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | DirectSum.finsupp_sum_support_decompose' | [334, 1] | [343, 6] | rfl | R : Type u_1
inst✝¹⁰ : CommSemiring R
ι✝ : Type u_2
inst✝⁹ : DecidableEq ι✝
inst✝⁸ : AddCommMonoid ι✝
A : Type u_3
inst✝⁷ : CommSemiring A
inst✝⁶ : DecidableEq A
inst✝⁵ : Algebra R A
𝒜 : ι✝ → Submodule R A
rel : A → A → Prop
ι : Type u_3
M : Type u_1
σ : Type u_2
inst✝⁴ : DecidableEq ι
inst✝³ : AddCommMonoid M
inst✝² ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝¹⁰ : CommSemiring R
ι✝ : Type u_2
inst✝⁹ : DecidableEq ι✝
inst✝⁸ : AddCommMonoid ι✝
A : Type u_3
inst✝⁷ : CommSemiring A
inst✝⁶ : DecidableEq A
inst✝⁵ : Algebra R A
𝒜 : ι✝ → Submodule R A
rel : A → A → Prop
ι : Type u_3
M : Type u_1
σ : Typ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | EqvGenIsHomogeneous_of | [345, 1] | [352, 75] | intro a b h | R : Type u_1
inst✝⁶ : CommSemiring R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : AddCommMonoid ι
A : Type u_3
inst✝³ : CommSemiring A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
rel : A → A → Prop
inst✝ : GradedAlgebra 𝒜
hr : Rel.IsHomogeneous 𝒜 rel
⊢ Rel.IsHomogeneous 𝒜 (EqvGen rel) | R : Type u_1
inst✝⁶ : CommSemiring R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : AddCommMonoid ι
A : Type u_3
inst✝³ : CommSemiring A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
rel : A → A → Prop
inst✝ : GradedAlgebra 𝒜
hr : Rel.IsHomogeneous 𝒜 rel
a b : A
h : EqvGen rel a b
⊢ ∀ (i : ι), EqvG... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝⁶ : CommSemiring R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : AddCommMonoid ι
A : Type u_3
inst✝³ : CommSemiring A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
rel : A → A → Prop
inst✝ : GradedAlgebra 𝒜
hr : Rel.IsHo... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | EqvGenIsHomogeneous_of | [345, 1] | [352, 75] | induction h with
| rel a b h => exact fun i => EqvGen.rel _ _ (hr h i)
| refl a => exact fun i => EqvGen.refl _
| symm a b _ k => exact fun i => EqvGen.symm _ _ (k i)
| trans a b c _ _ k k' => exact fun i => EqvGen.trans _ _ _ (k i) (k' i) | R : Type u_1
inst✝⁶ : CommSemiring R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : AddCommMonoid ι
A : Type u_3
inst✝³ : CommSemiring A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
rel : A → A → Prop
inst✝ : GradedAlgebra 𝒜
hr : Rel.IsHomogeneous 𝒜 rel
a b : A
h : EqvGen rel a b
⊢ ∀ (i : ι), EqvG... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝⁶ : CommSemiring R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : AddCommMonoid ι
A : Type u_3
inst✝³ : CommSemiring A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
rel : A → A → Prop
inst✝ : GradedAlgebra 𝒜
hr : Rel.IsHo... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | EqvGenIsHomogeneous_of | [345, 1] | [352, 75] | exact fun i => EqvGen.rel _ _ (hr h i) | case rel
R : Type u_1
inst✝⁶ : CommSemiring R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : AddCommMonoid ι
A : Type u_3
inst✝³ : CommSemiring A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
rel : A → A → Prop
inst✝ : GradedAlgebra 𝒜
hr : Rel.IsHomogeneous 𝒜 rel
a✝ b✝ a b : A
h : rel a b
⊢ ∀ (i : ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case rel
R : Type u_1
inst✝⁶ : CommSemiring R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : AddCommMonoid ι
A : Type u_3
inst✝³ : CommSemiring A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
rel : A → A → Prop
inst✝ : GradedAlgebra 𝒜
hr :... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | EqvGenIsHomogeneous_of | [345, 1] | [352, 75] | exact fun i => EqvGen.refl _ | case refl
R : Type u_1
inst✝⁶ : CommSemiring R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : AddCommMonoid ι
A : Type u_3
inst✝³ : CommSemiring A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
rel : A → A → Prop
inst✝ : GradedAlgebra 𝒜
hr : Rel.IsHomogeneous 𝒜 rel
a✝ b a : A
⊢ ∀ (i : ι), EqvGen rel... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refl
R : Type u_1
inst✝⁶ : CommSemiring R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : AddCommMonoid ι
A : Type u_3
inst✝³ : CommSemiring A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
rel : A → A → Prop
inst✝ : GradedAlgebra 𝒜
hr ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | EqvGenIsHomogeneous_of | [345, 1] | [352, 75] | exact fun i => EqvGen.symm _ _ (k i) | case symm
R : Type u_1
inst✝⁶ : CommSemiring R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : AddCommMonoid ι
A : Type u_3
inst✝³ : CommSemiring A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
rel : A → A → Prop
inst✝ : GradedAlgebra 𝒜
hr : Rel.IsHomogeneous 𝒜 rel
a✝¹ b✝ a b : A
a✝ : EqvGen rel a b... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case symm
R : Type u_1
inst✝⁶ : CommSemiring R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : AddCommMonoid ι
A : Type u_3
inst✝³ : CommSemiring A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
rel : A → A → Prop
inst✝ : GradedAlgebra 𝒜
hr ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | EqvGenIsHomogeneous_of | [345, 1] | [352, 75] | exact fun i => EqvGen.trans _ _ _ (k i) (k' i) | case trans
R : Type u_1
inst✝⁶ : CommSemiring R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : AddCommMonoid ι
A : Type u_3
inst✝³ : CommSemiring A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
rel : A → A → Prop
inst✝ : GradedAlgebra 𝒜
hr : Rel.IsHomogeneous 𝒜 rel
a✝² b✝ a b c : A
a✝¹ : EqvGen rel... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case trans
R : Type u_1
inst✝⁶ : CommSemiring R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : AddCommMonoid ι
A : Type u_3
inst✝³ : CommSemiring A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
rel : A → A → Prop
inst✝ : GradedAlgebra 𝒜
hr... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | rel_of_sum_of_rel_add | [354, 1] | [364, 59] | revert H | R : Type u_1
inst✝⁷ : CommSemiring R
ι✝ : Type u_2
inst✝⁶ : DecidableEq ι✝
inst✝⁵ : AddCommMonoid ι✝
A✝ : Type u_3
inst✝⁴ : CommSemiring A✝
inst✝³ : DecidableEq A✝
inst✝² : Algebra R A✝
𝒜 : ι✝ → Submodule R A✝
rel : A✝ → A✝ → Prop
A : Type u_4
inst✝¹ : AddCommMonoid A
r : A → A → Prop
hr_zero : r 0 0
hr_add : ∀ {a b c... | R : Type u_1
inst✝⁷ : CommSemiring R
ι✝ : Type u_2
inst✝⁶ : DecidableEq ι✝
inst✝⁵ : AddCommMonoid ι✝
A✝ : Type u_3
inst✝⁴ : CommSemiring A✝
inst✝³ : DecidableEq A✝
inst✝² : Algebra R A✝
𝒜 : ι✝ → Submodule R A✝
rel : A✝ → A✝ → Prop
A : Type u_4
inst✝¹ : AddCommMonoid A
r : A → A → Prop
hr_zero : r 0 0
hr_add : ∀ {a b c... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝⁷ : CommSemiring R
ι✝ : Type u_2
inst✝⁶ : DecidableEq ι✝
inst✝⁵ : AddCommMonoid ι✝
A✝ : Type u_3
inst✝⁴ : CommSemiring A✝
inst✝³ : DecidableEq A✝
inst✝² : Algebra R A✝
𝒜 : ι✝ → Submodule R A✝
rel : A✝ → A✝ → Prop
A : Type u_4
inst✝¹ : AddCo... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | rel_of_sum_of_rel_add | [354, 1] | [364, 59] | induction s using Finset.induction_on with
| empty => exact fun _ => hr_zero
| @insert i s hi hs =>
intro H
simp only [Finset.sum_insert hi]
exact hr_add (H _ (Finset.mem_insert_self _ _))
(hs (fun _ hi => H _ (Finset.mem_insert_of_mem hi))) | R : Type u_1
inst✝⁷ : CommSemiring R
ι✝ : Type u_2
inst✝⁶ : DecidableEq ι✝
inst✝⁵ : AddCommMonoid ι✝
A✝ : Type u_3
inst✝⁴ : CommSemiring A✝
inst✝³ : DecidableEq A✝
inst✝² : Algebra R A✝
𝒜 : ι✝ → Submodule R A✝
rel : A✝ → A✝ → Prop
A : Type u_4
inst✝¹ : AddCommMonoid A
r : A → A → Prop
hr_zero : r 0 0
hr_add : ∀ {a b c... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝⁷ : CommSemiring R
ι✝ : Type u_2
inst✝⁶ : DecidableEq ι✝
inst✝⁵ : AddCommMonoid ι✝
A✝ : Type u_3
inst✝⁴ : CommSemiring A✝
inst✝³ : DecidableEq A✝
inst✝² : Algebra R A✝
𝒜 : ι✝ → Submodule R A✝
rel : A✝ → A✝ → Prop
A : Type u_4
inst✝¹ : AddCo... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | rel_of_sum_of_rel_add | [354, 1] | [364, 59] | exact fun _ => hr_zero | case empty
R : Type u_1
inst✝⁷ : CommSemiring R
ι✝ : Type u_2
inst✝⁶ : DecidableEq ι✝
inst✝⁵ : AddCommMonoid ι✝
A✝ : Type u_3
inst✝⁴ : CommSemiring A✝
inst✝³ : DecidableEq A✝
inst✝² : Algebra R A✝
𝒜 : ι✝ → Submodule R A✝
rel : A✝ → A✝ → Prop
A : Type u_4
inst✝¹ : AddCommMonoid A
r : A → A → Prop
hr_zero : r 0 0
hr_add... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case empty
R : Type u_1
inst✝⁷ : CommSemiring R
ι✝ : Type u_2
inst✝⁶ : DecidableEq ι✝
inst✝⁵ : AddCommMonoid ι✝
A✝ : Type u_3
inst✝⁴ : CommSemiring A✝
inst✝³ : DecidableEq A✝
inst✝² : Algebra R A✝
𝒜 : ι✝ → Submodule R A✝
rel : A✝ → A✝ → Prop
A : Type u_4
ins... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | rel_of_sum_of_rel_add | [354, 1] | [364, 59] | intro H | case insert
R : Type u_1
inst✝⁷ : CommSemiring R
ι✝ : Type u_2
inst✝⁶ : DecidableEq ι✝
inst✝⁵ : AddCommMonoid ι✝
A✝ : Type u_3
inst✝⁴ : CommSemiring A✝
inst✝³ : DecidableEq A✝
inst✝² : Algebra R A✝
𝒜 : ι✝ → Submodule R A✝
rel : A✝ → A✝ → Prop
A : Type u_4
inst✝¹ : AddCommMonoid A
r : A → A → Prop
hr_zero : r 0 0
hr_ad... | case insert
R : Type u_1
inst✝⁷ : CommSemiring R
ι✝ : Type u_2
inst✝⁶ : DecidableEq ι✝
inst✝⁵ : AddCommMonoid ι✝
A✝ : Type u_3
inst✝⁴ : CommSemiring A✝
inst✝³ : DecidableEq A✝
inst✝² : Algebra R A✝
𝒜 : ι✝ → Submodule R A✝
rel : A✝ → A✝ → Prop
A : Type u_4
inst✝¹ : AddCommMonoid A
r : A → A → Prop
hr_zero : r 0 0
hr_ad... | Please generate a tactic in lean4 to solve the state.
STATE:
case insert
R : Type u_1
inst✝⁷ : CommSemiring R
ι✝ : Type u_2
inst✝⁶ : DecidableEq ι✝
inst✝⁵ : AddCommMonoid ι✝
A✝ : Type u_3
inst✝⁴ : CommSemiring A✝
inst✝³ : DecidableEq A✝
inst✝² : Algebra R A✝
𝒜 : ι✝ → Submodule R A✝
rel : A✝ → A✝ → Prop
A : Type u_4
in... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | rel_of_sum_of_rel_add | [354, 1] | [364, 59] | simp only [Finset.sum_insert hi] | case insert
R : Type u_1
inst✝⁷ : CommSemiring R
ι✝ : Type u_2
inst✝⁶ : DecidableEq ι✝
inst✝⁵ : AddCommMonoid ι✝
A✝ : Type u_3
inst✝⁴ : CommSemiring A✝
inst✝³ : DecidableEq A✝
inst✝² : Algebra R A✝
𝒜 : ι✝ → Submodule R A✝
rel : A✝ → A✝ → Prop
A : Type u_4
inst✝¹ : AddCommMonoid A
r : A → A → Prop
hr_zero : r 0 0
hr_ad... | case insert
R : Type u_1
inst✝⁷ : CommSemiring R
ι✝ : Type u_2
inst✝⁶ : DecidableEq ι✝
inst✝⁵ : AddCommMonoid ι✝
A✝ : Type u_3
inst✝⁴ : CommSemiring A✝
inst✝³ : DecidableEq A✝
inst✝² : Algebra R A✝
𝒜 : ι✝ → Submodule R A✝
rel : A✝ → A✝ → Prop
A : Type u_4
inst✝¹ : AddCommMonoid A
r : A → A → Prop
hr_zero : r 0 0
hr_ad... | Please generate a tactic in lean4 to solve the state.
STATE:
case insert
R : Type u_1
inst✝⁷ : CommSemiring R
ι✝ : Type u_2
inst✝⁶ : DecidableEq ι✝
inst✝⁵ : AddCommMonoid ι✝
A✝ : Type u_3
inst✝⁴ : CommSemiring A✝
inst✝³ : DecidableEq A✝
inst✝² : Algebra R A✝
𝒜 : ι✝ → Submodule R A✝
rel : A✝ → A✝ → Prop
A : Type u_4
in... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | rel_of_sum_of_rel_add | [354, 1] | [364, 59] | exact hr_add (H _ (Finset.mem_insert_self _ _))
(hs (fun _ hi => H _ (Finset.mem_insert_of_mem hi))) | case insert
R : Type u_1
inst✝⁷ : CommSemiring R
ι✝ : Type u_2
inst✝⁶ : DecidableEq ι✝
inst✝⁵ : AddCommMonoid ι✝
A✝ : Type u_3
inst✝⁴ : CommSemiring A✝
inst✝³ : DecidableEq A✝
inst✝² : Algebra R A✝
𝒜 : ι✝ → Submodule R A✝
rel : A✝ → A✝ → Prop
A : Type u_4
inst✝¹ : AddCommMonoid A
r : A → A → Prop
hr_zero : r 0 0
hr_ad... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case insert
R : Type u_1
inst✝⁷ : CommSemiring R
ι✝ : Type u_2
inst✝⁶ : DecidableEq ι✝
inst✝⁵ : AddCommMonoid ι✝
A✝ : Type u_3
inst✝⁴ : CommSemiring A✝
inst✝³ : DecidableEq A✝
inst✝² : Algebra R A✝
𝒜 : ι✝ → Submodule R A✝
rel : A✝ → A✝ → Prop
A : Type u_4
in... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | rel_of_finsupp_sum_of_rel_add | [366, 1] | [373, 32] | rw [Finsupp.sum_of_support_subset f (Finset.subset_union_left _ g.support),
Finsupp.sum_of_support_subset g (Finset.subset_union_right f.support _)] | R : Type u_1
inst✝⁷ : CommSemiring R
ι✝ : Type u_2
inst✝⁶ : DecidableEq ι✝
inst✝⁵ : AddCommMonoid ι✝
A✝ : Type u_3
inst✝⁴ : CommSemiring A✝
inst✝³ : DecidableEq A✝
inst✝² : Algebra R A✝
𝒜 : ι✝ → Submodule R A✝
rel : A✝ → A✝ → Prop
A : Type u_4
inst✝¹ : AddCommMonoid A
r : A → A → Prop
hr_zero : r 0 0
hr_add : ∀ {a b c... | R : Type u_1
inst✝⁷ : CommSemiring R
ι✝ : Type u_2
inst✝⁶ : DecidableEq ι✝
inst✝⁵ : AddCommMonoid ι✝
A✝ : Type u_3
inst✝⁴ : CommSemiring A✝
inst✝³ : DecidableEq A✝
inst✝² : Algebra R A✝
𝒜 : ι✝ → Submodule R A✝
rel : A✝ → A✝ → Prop
A : Type u_4
inst✝¹ : AddCommMonoid A
r : A → A → Prop
hr_zero : r 0 0
hr_add : ∀ {a b c... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝⁷ : CommSemiring R
ι✝ : Type u_2
inst✝⁶ : DecidableEq ι✝
inst✝⁵ : AddCommMonoid ι✝
A✝ : Type u_3
inst✝⁴ : CommSemiring A✝
inst✝³ : DecidableEq A✝
inst✝² : Algebra R A✝
𝒜 : ι✝ → Submodule R A✝
rel : A✝ → A✝ → Prop
A : Type u_4
inst✝¹ : AddCo... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | rel_of_finsupp_sum_of_rel_add | [366, 1] | [373, 32] | exact rel_of_sum_of_rel_add hr_zero hr_add (fun i _ => H i) | R : Type u_1
inst✝⁷ : CommSemiring R
ι✝ : Type u_2
inst✝⁶ : DecidableEq ι✝
inst✝⁵ : AddCommMonoid ι✝
A✝ : Type u_3
inst✝⁴ : CommSemiring A✝
inst✝³ : DecidableEq A✝
inst✝² : Algebra R A✝
𝒜 : ι✝ → Submodule R A✝
rel : A✝ → A✝ → Prop
A : Type u_4
inst✝¹ : AddCommMonoid A
r : A → A → Prop
hr_zero : r 0 0
hr_add : ∀ {a b c... | case h
R : Type u_1
inst✝⁷ : CommSemiring R
ι✝ : Type u_2
inst✝⁶ : DecidableEq ι✝
inst✝⁵ : AddCommMonoid ι✝
A✝ : Type u_3
inst✝⁴ : CommSemiring A✝
inst✝³ : DecidableEq A✝
inst✝² : Algebra R A✝
𝒜 : ι✝ → Submodule R A✝
rel : A✝ → A✝ → Prop
A : Type u_4
inst✝¹ : AddCommMonoid A
r : A → A → Prop
hr_zero : r 0 0
hr_add : ∀... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝⁷ : CommSemiring R
ι✝ : Type u_2
inst✝⁶ : DecidableEq ι✝
inst✝⁵ : AddCommMonoid ι✝
A✝ : Type u_3
inst✝⁴ : CommSemiring A✝
inst✝³ : DecidableEq A✝
inst✝² : Algebra R A✝
𝒜 : ι✝ → Submodule R A✝
rel : A✝ → A✝ → Prop
A : Type u_4
inst✝¹ : AddCo... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | rel_of_finsupp_sum_of_rel_add | [366, 1] | [373, 32] | all_goals { intro _ _ ; rfl } | case h
R : Type u_1
inst✝⁷ : CommSemiring R
ι✝ : Type u_2
inst✝⁶ : DecidableEq ι✝
inst✝⁵ : AddCommMonoid ι✝
A✝ : Type u_3
inst✝⁴ : CommSemiring A✝
inst✝³ : DecidableEq A✝
inst✝² : Algebra R A✝
𝒜 : ι✝ → Submodule R A✝
rel : A✝ → A✝ → Prop
A : Type u_4
inst✝¹ : AddCommMonoid A
r : A → A → Prop
hr_zero : r 0 0
hr_add : ∀... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
R : Type u_1
inst✝⁷ : CommSemiring R
ι✝ : Type u_2
inst✝⁶ : DecidableEq ι✝
inst✝⁵ : AddCommMonoid ι✝
A✝ : Type u_3
inst✝⁴ : CommSemiring A✝
inst✝³ : DecidableEq A✝
inst✝² : Algebra R A✝
𝒜 : ι✝ → Submodule R A✝
rel : A✝ → A✝ → Prop
A : Type u_4
inst✝¹ ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | rel_of_finsupp_sum_of_rel_add | [366, 1] | [373, 32] | intro _ _ | case h
R : Type u_1
inst✝⁷ : CommSemiring R
ι✝ : Type u_2
inst✝⁶ : DecidableEq ι✝
inst✝⁵ : AddCommMonoid ι✝
A✝ : Type u_3
inst✝⁴ : CommSemiring A✝
inst✝³ : DecidableEq A✝
inst✝² : Algebra R A✝
𝒜 : ι✝ → Submodule R A✝
rel : A✝ → A✝ → Prop
A : Type u_4
inst✝¹ : AddCommMonoid A
r : A → A → Prop
hr_zero : r 0 0
hr_add : ∀... | case h
R : Type u_1
inst✝⁷ : CommSemiring R
ι✝ : Type u_2
inst✝⁶ : DecidableEq ι✝
inst✝⁵ : AddCommMonoid ι✝
A✝ : Type u_3
inst✝⁴ : CommSemiring A✝
inst✝³ : DecidableEq A✝
inst✝² : Algebra R A✝
𝒜 : ι✝ → Submodule R A✝
rel : A✝ → A✝ → Prop
A : Type u_4
inst✝¹ : AddCommMonoid A
r : A → A → Prop
hr_zero : r 0 0
hr_add : ∀... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
R : Type u_1
inst✝⁷ : CommSemiring R
ι✝ : Type u_2
inst✝⁶ : DecidableEq ι✝
inst✝⁵ : AddCommMonoid ι✝
A✝ : Type u_3
inst✝⁴ : CommSemiring A✝
inst✝³ : DecidableEq A✝
inst✝² : Algebra R A✝
𝒜 : ι✝ → Submodule R A✝
rel : A✝ → A✝ → Prop
A : Type u_4
inst✝¹ ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | rel_of_finsupp_sum_of_rel_add | [366, 1] | [373, 32] | rfl | case h
R : Type u_1
inst✝⁷ : CommSemiring R
ι✝ : Type u_2
inst✝⁶ : DecidableEq ι✝
inst✝⁵ : AddCommMonoid ι✝
A✝ : Type u_3
inst✝⁴ : CommSemiring A✝
inst✝³ : DecidableEq A✝
inst✝² : Algebra R A✝
𝒜 : ι✝ → Submodule R A✝
rel : A✝ → A✝ → Prop
A : Type u_4
inst✝¹ : AddCommMonoid A
r : A → A → Prop
hr_zero : r 0 0
hr_add : ∀... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
R : Type u_1
inst✝⁷ : CommSemiring R
ι✝ : Type u_2
inst✝⁶ : DecidableEq ι✝
inst✝⁵ : AddCommMonoid ι✝
A✝ : Type u_3
inst✝⁴ : CommSemiring A✝
inst✝³ : DecidableEq A✝
inst✝² : Algebra R A✝
𝒜 : ι✝ → Submodule R A✝
rel : A✝ → A✝ → Prop
A : Type u_4
inst✝¹ ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | rel_of_dsum_of_rel_add | [375, 1] | [387, 59] | revert H | R : Type u_1
inst✝⁸ : CommSemiring R
ι✝ : Type u_2
inst✝⁷ : DecidableEq ι✝
inst✝⁶ : AddCommMonoid ι✝
A✝ : Type u_3
inst✝⁵ : CommSemiring A✝
inst✝⁴ : DecidableEq A✝
inst✝³ : Algebra R A✝
𝒜 : ι✝ → Submodule R A✝
rel : A✝ → A✝ → Prop
A : Type u_4
inst✝² : AddCommMonoid A
r : A → A → Prop
hr_zero : r 0 0
hr_add : ∀ {a b c... | R : Type u_1
inst✝⁸ : CommSemiring R
ι✝ : Type u_2
inst✝⁷ : DecidableEq ι✝
inst✝⁶ : AddCommMonoid ι✝
A✝ : Type u_3
inst✝⁵ : CommSemiring A✝
inst✝⁴ : DecidableEq A✝
inst✝³ : Algebra R A✝
𝒜 : ι✝ → Submodule R A✝
rel : A✝ → A✝ → Prop
A : Type u_4
inst✝² : AddCommMonoid A
r : A → A → Prop
hr_zero : r 0 0
hr_add : ∀ {a b c... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝⁸ : CommSemiring R
ι✝ : Type u_2
inst✝⁷ : DecidableEq ι✝
inst✝⁶ : AddCommMonoid ι✝
A✝ : Type u_3
inst✝⁵ : CommSemiring A✝
inst✝⁴ : DecidableEq A✝
inst✝³ : Algebra R A✝
𝒜 : ι✝ → Submodule R A✝
rel : A✝ → A✝ → Prop
A : Type u_4
inst✝² : AddCo... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | rel_of_dsum_of_rel_add | [375, 1] | [387, 59] | induction s using Finset.induction_on with
| empty => exact fun _ => hr_zero
| @insert i s hi hs =>
intro H
simp only [Finset.sum_insert hi]
exact hr_add (H _ (Finset.mem_insert_self _ _))
(hs (fun _ hi => H _ (Finset.mem_insert_of_mem hi))) | R : Type u_1
inst✝⁸ : CommSemiring R
ι✝ : Type u_2
inst✝⁷ : DecidableEq ι✝
inst✝⁶ : AddCommMonoid ι✝
A✝ : Type u_3
inst✝⁵ : CommSemiring A✝
inst✝⁴ : DecidableEq A✝
inst✝³ : Algebra R A✝
𝒜 : ι✝ → Submodule R A✝
rel : A✝ → A✝ → Prop
A : Type u_4
inst✝² : AddCommMonoid A
r : A → A → Prop
hr_zero : r 0 0
hr_add : ∀ {a b c... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝⁸ : CommSemiring R
ι✝ : Type u_2
inst✝⁷ : DecidableEq ι✝
inst✝⁶ : AddCommMonoid ι✝
A✝ : Type u_3
inst✝⁵ : CommSemiring A✝
inst✝⁴ : DecidableEq A✝
inst✝³ : Algebra R A✝
𝒜 : ι✝ → Submodule R A✝
rel : A✝ → A✝ → Prop
A : Type u_4
inst✝² : AddCo... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | rel_of_dsum_of_rel_add | [375, 1] | [387, 59] | exact fun _ => hr_zero | case empty
R : Type u_1
inst✝⁸ : CommSemiring R
ι✝ : Type u_2
inst✝⁷ : DecidableEq ι✝
inst✝⁶ : AddCommMonoid ι✝
A✝ : Type u_3
inst✝⁵ : CommSemiring A✝
inst✝⁴ : DecidableEq A✝
inst✝³ : Algebra R A✝
𝒜 : ι✝ → Submodule R A✝
rel : A✝ → A✝ → Prop
A : Type u_4
inst✝² : AddCommMonoid A
r : A → A → Prop
hr_zero : r 0 0
hr_add... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case empty
R : Type u_1
inst✝⁸ : CommSemiring R
ι✝ : Type u_2
inst✝⁷ : DecidableEq ι✝
inst✝⁶ : AddCommMonoid ι✝
A✝ : Type u_3
inst✝⁵ : CommSemiring A✝
inst✝⁴ : DecidableEq A✝
inst✝³ : Algebra R A✝
𝒜 : ι✝ → Submodule R A✝
rel : A✝ → A✝ → Prop
A : Type u_4
ins... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | rel_of_dsum_of_rel_add | [375, 1] | [387, 59] | intro H | case insert
R : Type u_1
inst✝⁸ : CommSemiring R
ι✝ : Type u_2
inst✝⁷ : DecidableEq ι✝
inst✝⁶ : AddCommMonoid ι✝
A✝ : Type u_3
inst✝⁵ : CommSemiring A✝
inst✝⁴ : DecidableEq A✝
inst✝³ : Algebra R A✝
𝒜 : ι✝ → Submodule R A✝
rel : A✝ → A✝ → Prop
A : Type u_4
inst✝² : AddCommMonoid A
r : A → A → Prop
hr_zero : r 0 0
hr_ad... | case insert
R : Type u_1
inst✝⁸ : CommSemiring R
ι✝ : Type u_2
inst✝⁷ : DecidableEq ι✝
inst✝⁶ : AddCommMonoid ι✝
A✝ : Type u_3
inst✝⁵ : CommSemiring A✝
inst✝⁴ : DecidableEq A✝
inst✝³ : Algebra R A✝
𝒜 : ι✝ → Submodule R A✝
rel : A✝ → A✝ → Prop
A : Type u_4
inst✝² : AddCommMonoid A
r : A → A → Prop
hr_zero : r 0 0
hr_ad... | Please generate a tactic in lean4 to solve the state.
STATE:
case insert
R : Type u_1
inst✝⁸ : CommSemiring R
ι✝ : Type u_2
inst✝⁷ : DecidableEq ι✝
inst✝⁶ : AddCommMonoid ι✝
A✝ : Type u_3
inst✝⁵ : CommSemiring A✝
inst✝⁴ : DecidableEq A✝
inst✝³ : Algebra R A✝
𝒜 : ι✝ → Submodule R A✝
rel : A✝ → A✝ → Prop
A : Type u_4
in... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | rel_of_dsum_of_rel_add | [375, 1] | [387, 59] | simp only [Finset.sum_insert hi] | case insert
R : Type u_1
inst✝⁸ : CommSemiring R
ι✝ : Type u_2
inst✝⁷ : DecidableEq ι✝
inst✝⁶ : AddCommMonoid ι✝
A✝ : Type u_3
inst✝⁵ : CommSemiring A✝
inst✝⁴ : DecidableEq A✝
inst✝³ : Algebra R A✝
𝒜 : ι✝ → Submodule R A✝
rel : A✝ → A✝ → Prop
A : Type u_4
inst✝² : AddCommMonoid A
r : A → A → Prop
hr_zero : r 0 0
hr_ad... | case insert
R : Type u_1
inst✝⁸ : CommSemiring R
ι✝ : Type u_2
inst✝⁷ : DecidableEq ι✝
inst✝⁶ : AddCommMonoid ι✝
A✝ : Type u_3
inst✝⁵ : CommSemiring A✝
inst✝⁴ : DecidableEq A✝
inst✝³ : Algebra R A✝
𝒜 : ι✝ → Submodule R A✝
rel : A✝ → A✝ → Prop
A : Type u_4
inst✝² : AddCommMonoid A
r : A → A → Prop
hr_zero : r 0 0
hr_ad... | Please generate a tactic in lean4 to solve the state.
STATE:
case insert
R : Type u_1
inst✝⁸ : CommSemiring R
ι✝ : Type u_2
inst✝⁷ : DecidableEq ι✝
inst✝⁶ : AddCommMonoid ι✝
A✝ : Type u_3
inst✝⁵ : CommSemiring A✝
inst✝⁴ : DecidableEq A✝
inst✝³ : Algebra R A✝
𝒜 : ι✝ → Submodule R A✝
rel : A✝ → A✝ → Prop
A : Type u_4
in... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | rel_of_dsum_of_rel_add | [375, 1] | [387, 59] | exact hr_add (H _ (Finset.mem_insert_self _ _))
(hs (fun _ hi => H _ (Finset.mem_insert_of_mem hi))) | case insert
R : Type u_1
inst✝⁸ : CommSemiring R
ι✝ : Type u_2
inst✝⁷ : DecidableEq ι✝
inst✝⁶ : AddCommMonoid ι✝
A✝ : Type u_3
inst✝⁵ : CommSemiring A✝
inst✝⁴ : DecidableEq A✝
inst✝³ : Algebra R A✝
𝒜 : ι✝ → Submodule R A✝
rel : A✝ → A✝ → Prop
A : Type u_4
inst✝² : AddCommMonoid A
r : A → A → Prop
hr_zero : r 0 0
hr_ad... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case insert
R : Type u_1
inst✝⁸ : CommSemiring R
ι✝ : Type u_2
inst✝⁷ : DecidableEq ι✝
inst✝⁶ : AddCommMonoid ι✝
A✝ : Type u_3
inst✝⁵ : CommSemiring A✝
inst✝⁴ : DecidableEq A✝
inst✝³ : Algebra R A✝
𝒜 : ι✝ → Submodule R A✝
rel : A✝ → A✝ → Prop
A : Type u_4
in... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | rel_of_dfinsupp_sum_of_rel_add | [389, 1] | [397, 62] | rw [DFinsupp.sum_of_support_le (Finset.subset_union_left f.support g.support),
DFinsupp.sum_of_support_le (Finset.subset_union_right f.support g.support)] | R : Type u_1
inst✝⁹ : CommSemiring R
ι✝ : Type u_2
inst✝⁸ : DecidableEq ι✝
inst✝⁷ : AddCommMonoid ι✝
A✝ : Type u_3
inst✝⁶ : CommSemiring A✝
inst✝⁵ : DecidableEq A✝
inst✝⁴ : Algebra R A✝
𝒜 : ι✝ → Submodule R A✝
rel : A✝ → A✝ → Prop
A : Type u_4
inst✝³ : AddCommMonoid A
r : A → A → Prop
hr_zero : r 0 0
hr_add : ∀ {a b c... | R : Type u_1
inst✝⁹ : CommSemiring R
ι✝ : Type u_2
inst✝⁸ : DecidableEq ι✝
inst✝⁷ : AddCommMonoid ι✝
A✝ : Type u_3
inst✝⁶ : CommSemiring A✝
inst✝⁵ : DecidableEq A✝
inst✝⁴ : Algebra R A✝
𝒜 : ι✝ → Submodule R A✝
rel : A✝ → A✝ → Prop
A : Type u_4
inst✝³ : AddCommMonoid A
r : A → A → Prop
hr_zero : r 0 0
hr_add : ∀ {a b c... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝⁹ : CommSemiring R
ι✝ : Type u_2
inst✝⁸ : DecidableEq ι✝
inst✝⁷ : AddCommMonoid ι✝
A✝ : Type u_3
inst✝⁶ : CommSemiring A✝
inst✝⁵ : DecidableEq A✝
inst✝⁴ : Algebra R A✝
𝒜 : ι✝ → Submodule R A✝
rel : A✝ → A✝ → Prop
A : Type u_4
inst✝³ : AddCo... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | rel_of_dfinsupp_sum_of_rel_add | [389, 1] | [397, 62] | exact rel_of_sum_of_rel_add hr_zero hr_add (fun i _ => H i) | R : Type u_1
inst✝⁹ : CommSemiring R
ι✝ : Type u_2
inst✝⁸ : DecidableEq ι✝
inst✝⁷ : AddCommMonoid ι✝
A✝ : Type u_3
inst✝⁶ : CommSemiring A✝
inst✝⁵ : DecidableEq A✝
inst✝⁴ : Algebra R A✝
𝒜 : ι✝ → Submodule R A✝
rel : A✝ → A✝ → Prop
A : Type u_4
inst✝³ : AddCommMonoid A
r : A → A → Prop
hr_zero : r 0 0
hr_add : ∀ {a b c... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝⁹ : CommSemiring R
ι✝ : Type u_2
inst✝⁸ : DecidableEq ι✝
inst✝⁷ : AddCommMonoid ι✝
A✝ : Type u_3
inst✝⁶ : CommSemiring A✝
inst✝⁵ : DecidableEq A✝
inst✝⁴ : Algebra R A✝
𝒜 : ι✝ → Submodule R A✝
rel : A✝ → A✝ → Prop
A : Type u_4
inst✝³ : AddCo... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | RingConGen.RelIsHomogeneous_of | [423, 1] | [448, 34] | intro a b h | R : Type u_1
inst✝⁶ : CommSemiring R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : AddCommMonoid ι
A : Type u_3
inst✝³ : CommSemiring A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
rel : A → A → Prop
inst✝ : GradedAlgebra 𝒜
hr : Rel.IsHomogeneous 𝒜 rel
⊢ Rel.IsHomogeneous 𝒜 (Rel rel) | R : Type u_1
inst✝⁶ : CommSemiring R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : AddCommMonoid ι
A : Type u_3
inst✝³ : CommSemiring A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
rel : A → A → Prop
inst✝ : GradedAlgebra 𝒜
hr : Rel.IsHomogeneous 𝒜 rel
a b : A
h : Rel rel a b
⊢ ∀ (i : ι), Rel rel... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝⁶ : CommSemiring R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : AddCommMonoid ι
A : Type u_3
inst✝³ : CommSemiring A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
rel : A → A → Prop
inst✝ : GradedAlgebra 𝒜
hr : Rel.IsHo... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | RingConGen.RelIsHomogeneous_of | [423, 1] | [448, 34] | induction h with
| of x y h => exact fun i => RingConGen.Rel.of _ _ (hr h i)
| refl x => exact fun _ => RingConGen.Rel.refl _
| symm _ h' => exact fun i => RingConGen.Rel.symm (h' i)
| trans _ _ k k' => exact fun i => RingConGen.Rel.trans (k i) (k' i)
| add _ _ k k' =>
intro i
simp only [map_add]
exact RingConG... | R : Type u_1
inst✝⁶ : CommSemiring R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : AddCommMonoid ι
A : Type u_3
inst✝³ : CommSemiring A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
rel : A → A → Prop
inst✝ : GradedAlgebra 𝒜
hr : Rel.IsHomogeneous 𝒜 rel
a b : A
h : Rel rel a b
⊢ ∀ (i : ι), Rel rel... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝⁶ : CommSemiring R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : AddCommMonoid ι
A : Type u_3
inst✝³ : CommSemiring A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
rel : A → A → Prop
inst✝ : GradedAlgebra 𝒜
hr : Rel.IsHo... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | RingConGen.RelIsHomogeneous_of | [423, 1] | [448, 34] | exact fun i => RingConGen.Rel.of _ _ (hr h i) | case of
R : Type u_1
inst✝⁶ : CommSemiring R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : AddCommMonoid ι
A : Type u_3
inst✝³ : CommSemiring A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
rel : A → A → Prop
inst✝ : GradedAlgebra 𝒜
hr : Rel.IsHomogeneous 𝒜 rel
a b x y : A
h : rel x y
⊢ ∀ (i : ι),... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case of
R : Type u_1
inst✝⁶ : CommSemiring R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : AddCommMonoid ι
A : Type u_3
inst✝³ : CommSemiring A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
rel : A → A → Prop
inst✝ : GradedAlgebra 𝒜
hr : ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | RingConGen.RelIsHomogeneous_of | [423, 1] | [448, 34] | exact fun _ => RingConGen.Rel.refl _ | case refl
R : Type u_1
inst✝⁶ : CommSemiring R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : AddCommMonoid ι
A : Type u_3
inst✝³ : CommSemiring A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
rel : A → A → Prop
inst✝ : GradedAlgebra 𝒜
hr : Rel.IsHomogeneous 𝒜 rel
a b x : A
⊢ ∀ (i : ι), Rel rel ↑((... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refl
R : Type u_1
inst✝⁶ : CommSemiring R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : AddCommMonoid ι
A : Type u_3
inst✝³ : CommSemiring A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
rel : A → A → Prop
inst✝ : GradedAlgebra 𝒜
hr ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | RingConGen.RelIsHomogeneous_of | [423, 1] | [448, 34] | exact fun i => RingConGen.Rel.symm (h' i) | case symm
R : Type u_1
inst✝⁶ : CommSemiring R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : AddCommMonoid ι
A : Type u_3
inst✝³ : CommSemiring A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
rel : A → A → Prop
inst✝ : GradedAlgebra 𝒜
hr : Rel.IsHomogeneous 𝒜 rel
a b x✝ y✝ : A
a✝ : Rel rel x✝ y✝
h... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case symm
R : Type u_1
inst✝⁶ : CommSemiring R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : AddCommMonoid ι
A : Type u_3
inst✝³ : CommSemiring A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
rel : A → A → Prop
inst✝ : GradedAlgebra 𝒜
hr ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | RingConGen.RelIsHomogeneous_of | [423, 1] | [448, 34] | exact fun i => RingConGen.Rel.trans (k i) (k' i) | case trans
R : Type u_1
inst✝⁶ : CommSemiring R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : AddCommMonoid ι
A : Type u_3
inst✝³ : CommSemiring A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
rel : A → A → Prop
inst✝ : GradedAlgebra 𝒜
hr : Rel.IsHomogeneous 𝒜 rel
a b x✝ y✝ z✝ : A
a✝¹ : Rel rel x✝... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case trans
R : Type u_1
inst✝⁶ : CommSemiring R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : AddCommMonoid ι
A : Type u_3
inst✝³ : CommSemiring A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
rel : A → A → Prop
inst✝ : GradedAlgebra 𝒜
hr... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | RingConGen.RelIsHomogeneous_of | [423, 1] | [448, 34] | intro i | case add
R : Type u_1
inst✝⁶ : CommSemiring R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : AddCommMonoid ι
A : Type u_3
inst✝³ : CommSemiring A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
rel : A → A → Prop
inst✝ : GradedAlgebra 𝒜
hr : Rel.IsHomogeneous 𝒜 rel
a b w✝ x✝ y✝ z✝ : A
a✝¹ : Rel rel w... | case add
R : Type u_1
inst✝⁶ : CommSemiring R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : AddCommMonoid ι
A : Type u_3
inst✝³ : CommSemiring A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
rel : A → A → Prop
inst✝ : GradedAlgebra 𝒜
hr : Rel.IsHomogeneous 𝒜 rel
a b w✝ x✝ y✝ z✝ : A
a✝¹ : Rel rel w... | Please generate a tactic in lean4 to solve the state.
STATE:
case add
R : Type u_1
inst✝⁶ : CommSemiring R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : AddCommMonoid ι
A : Type u_3
inst✝³ : CommSemiring A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
rel : A → A → Prop
inst✝ : GradedAlgebra 𝒜
hr :... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | RingConGen.RelIsHomogeneous_of | [423, 1] | [448, 34] | simp only [map_add] | case add
R : Type u_1
inst✝⁶ : CommSemiring R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : AddCommMonoid ι
A : Type u_3
inst✝³ : CommSemiring A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
rel : A → A → Prop
inst✝ : GradedAlgebra 𝒜
hr : Rel.IsHomogeneous 𝒜 rel
a b w✝ x✝ y✝ z✝ : A
a✝¹ : Rel rel w... | case add
R : Type u_1
inst✝⁶ : CommSemiring R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : AddCommMonoid ι
A : Type u_3
inst✝³ : CommSemiring A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
rel : A → A → Prop
inst✝ : GradedAlgebra 𝒜
hr : Rel.IsHomogeneous 𝒜 rel
a b w✝ x✝ y✝ z✝ : A
a✝¹ : Rel rel w... | Please generate a tactic in lean4 to solve the state.
STATE:
case add
R : Type u_1
inst✝⁶ : CommSemiring R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : AddCommMonoid ι
A : Type u_3
inst✝³ : CommSemiring A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
rel : A → A → Prop
inst✝ : GradedAlgebra 𝒜
hr :... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | RingConGen.RelIsHomogeneous_of | [423, 1] | [448, 34] | exact RingConGen.Rel.add (k i) (k' i) | case add
R : Type u_1
inst✝⁶ : CommSemiring R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : AddCommMonoid ι
A : Type u_3
inst✝³ : CommSemiring A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
rel : A → A → Prop
inst✝ : GradedAlgebra 𝒜
hr : Rel.IsHomogeneous 𝒜 rel
a b w✝ x✝ y✝ z✝ : A
a✝¹ : Rel rel w... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case add
R : Type u_1
inst✝⁶ : CommSemiring R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : AddCommMonoid ι
A : Type u_3
inst✝³ : CommSemiring A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
rel : A → A → Prop
inst✝ : GradedAlgebra 𝒜
hr :... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | RingConGen.RelIsHomogeneous_of | [423, 1] | [448, 34] | intro n | case mul
R : Type u_1
inst✝⁶ : CommSemiring R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : AddCommMonoid ι
A : Type u_3
inst✝³ : CommSemiring A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
rel : A → A → Prop
inst✝ : GradedAlgebra 𝒜
hr : Rel.IsHomogeneous 𝒜 rel
a✝² b✝ a b c d : A
a✝¹ : Rel rel a ... | case mul
R : Type u_1
inst✝⁶ : CommSemiring R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : AddCommMonoid ι
A : Type u_3
inst✝³ : CommSemiring A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
rel : A → A → Prop
inst✝ : GradedAlgebra 𝒜
hr : Rel.IsHomogeneous 𝒜 rel
a✝² b✝ a b c d : A
a✝¹ : Rel rel a ... | Please generate a tactic in lean4 to solve the state.
STATE:
case mul
R : Type u_1
inst✝⁶ : CommSemiring R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : AddCommMonoid ι
A : Type u_3
inst✝³ : CommSemiring A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
rel : A → A → Prop
inst✝ : GradedAlgebra 𝒜
hr :... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | RingConGen.RelIsHomogeneous_of | [423, 1] | [448, 34] | simp only [AlgEquiv.toLinearMap_apply, map_mul, coe_mul_apply_eq_dfinsupp_sum] | case mul
R : Type u_1
inst✝⁶ : CommSemiring R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : AddCommMonoid ι
A : Type u_3
inst✝³ : CommSemiring A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
rel : A → A → Prop
inst✝ : GradedAlgebra 𝒜
hr : Rel.IsHomogeneous 𝒜 rel
a✝² b✝ a b c d : A
a✝¹ : Rel rel a ... | case mul
R : Type u_1
inst✝⁶ : CommSemiring R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : AddCommMonoid ι
A : Type u_3
inst✝³ : CommSemiring A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
rel : A → A → Prop
inst✝ : GradedAlgebra 𝒜
hr : Rel.IsHomogeneous 𝒜 rel
a✝² b✝ a b c d : A
a✝¹ : Rel rel a ... | Please generate a tactic in lean4 to solve the state.
STATE:
case mul
R : Type u_1
inst✝⁶ : CommSemiring R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : AddCommMonoid ι
A : Type u_3
inst✝³ : CommSemiring A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
rel : A → A → Prop
inst✝ : GradedAlgebra 𝒜
hr :... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | RingConGen.RelIsHomogeneous_of | [423, 1] | [448, 34] | apply rel_of_dfinsupp_sum_of_rel_add (RingConGen.Rel.refl 0) (RingConGen.Rel.add)
(Φy 𝒜 n (decomposeAlgEquiv 𝒜 c)) (Φy 𝒜 n (decomposeAlgEquiv 𝒜 d)) | case mul
R : Type u_1
inst✝⁶ : CommSemiring R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : AddCommMonoid ι
A : Type u_3
inst✝³ : CommSemiring A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
rel : A → A → Prop
inst✝ : GradedAlgebra 𝒜
hr : Rel.IsHomogeneous 𝒜 rel
a✝² b✝ a b c d : A
a✝¹ : Rel rel a ... | case mul
R : Type u_1
inst✝⁶ : CommSemiring R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : AddCommMonoid ι
A : Type u_3
inst✝³ : CommSemiring A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
rel : A → A → Prop
inst✝ : GradedAlgebra 𝒜
hr : Rel.IsHomogeneous 𝒜 rel
a✝² b✝ a b c d : A
a✝¹ : Rel rel a ... | Please generate a tactic in lean4 to solve the state.
STATE:
case mul
R : Type u_1
inst✝⁶ : CommSemiring R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : AddCommMonoid ι
A : Type u_3
inst✝³ : CommSemiring A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
rel : A → A → Prop
inst✝ : GradedAlgebra 𝒜
hr :... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | RingConGen.RelIsHomogeneous_of | [423, 1] | [448, 34] | intro i | case mul
R : Type u_1
inst✝⁶ : CommSemiring R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : AddCommMonoid ι
A : Type u_3
inst✝³ : CommSemiring A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
rel : A → A → Prop
inst✝ : GradedAlgebra 𝒜
hr : Rel.IsHomogeneous 𝒜 rel
a✝² b✝ a b c d : A
a✝¹ : Rel rel a ... | case mul
R : Type u_1
inst✝⁶ : CommSemiring R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : AddCommMonoid ι
A : Type u_3
inst✝³ : CommSemiring A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
rel : A → A → Prop
inst✝ : GradedAlgebra 𝒜
hr : Rel.IsHomogeneous 𝒜 rel
a✝² b✝ a b c d : A
a✝¹ : Rel rel a ... | Please generate a tactic in lean4 to solve the state.
STATE:
case mul
R : Type u_1
inst✝⁶ : CommSemiring R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : AddCommMonoid ι
A : Type u_3
inst✝³ : CommSemiring A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
rel : A → A → Prop
inst✝ : GradedAlgebra 𝒜
hr :... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | RingConGen.RelIsHomogeneous_of | [423, 1] | [448, 34] | apply rel_of_dfinsupp_sum_of_rel_add (RingConGen.Rel.refl 0) (RingConGen.Rel.add) | case mul
R : Type u_1
inst✝⁶ : CommSemiring R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : AddCommMonoid ι
A : Type u_3
inst✝³ : CommSemiring A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
rel : A → A → Prop
inst✝ : GradedAlgebra 𝒜
hr : Rel.IsHomogeneous 𝒜 rel
a✝² b✝ a b c d : A
a✝¹ : Rel rel a ... | case mul.H
R : Type u_1
inst✝⁶ : CommSemiring R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : AddCommMonoid ι
A : Type u_3
inst✝³ : CommSemiring A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
rel : A → A → Prop
inst✝ : GradedAlgebra 𝒜
hr : Rel.IsHomogeneous 𝒜 rel
a✝² b✝ a b c d : A
a✝¹ : Rel rel ... | Please generate a tactic in lean4 to solve the state.
STATE:
case mul
R : Type u_1
inst✝⁶ : CommSemiring R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : AddCommMonoid ι
A : Type u_3
inst✝³ : CommSemiring A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
rel : A → A → Prop
inst✝ : GradedAlgebra 𝒜
hr :... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | RingConGen.RelIsHomogeneous_of | [423, 1] | [448, 34] | intro j | case mul.H
R : Type u_1
inst✝⁶ : CommSemiring R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : AddCommMonoid ι
A : Type u_3
inst✝³ : CommSemiring A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
rel : A → A → Prop
inst✝ : GradedAlgebra 𝒜
hr : Rel.IsHomogeneous 𝒜 rel
a✝² b✝ a b c d : A
a✝¹ : Rel rel ... | case mul.H
R : Type u_1
inst✝⁶ : CommSemiring R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : AddCommMonoid ι
A : Type u_3
inst✝³ : CommSemiring A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
rel : A → A → Prop
inst✝ : GradedAlgebra 𝒜
hr : Rel.IsHomogeneous 𝒜 rel
a✝² b✝ a b c d : A
a✝¹ : Rel rel ... | Please generate a tactic in lean4 to solve the state.
STATE:
case mul.H
R : Type u_1
inst✝⁶ : CommSemiring R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : AddCommMonoid ι
A : Type u_3
inst✝³ : CommSemiring A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
rel : A → A → Prop
inst✝ : GradedAlgebra 𝒜
hr... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | RingConGen.RelIsHomogeneous_of | [423, 1] | [448, 34] | dsimp only [Φ] | case mul.H
R : Type u_1
inst✝⁶ : CommSemiring R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : AddCommMonoid ι
A : Type u_3
inst✝³ : CommSemiring A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
rel : A → A → Prop
inst✝ : GradedAlgebra 𝒜
hr : Rel.IsHomogeneous 𝒜 rel
a✝² b✝ a b c d : A
a✝¹ : Rel rel ... | case mul.H
R : Type u_1
inst✝⁶ : CommSemiring R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : AddCommMonoid ι
A : Type u_3
inst✝³ : CommSemiring A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
rel : A → A → Prop
inst✝ : GradedAlgebra 𝒜
hr : Rel.IsHomogeneous 𝒜 rel
a✝² b✝ a b c d : A
a✝¹ : Rel rel ... | Please generate a tactic in lean4 to solve the state.
STATE:
case mul.H
R : Type u_1
inst✝⁶ : CommSemiring R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : AddCommMonoid ι
A : Type u_3
inst✝³ : CommSemiring A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
rel : A → A → Prop
inst✝ : GradedAlgebra 𝒜
hr... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | RingConGen.RelIsHomogeneous_of | [423, 1] | [448, 34] | by_cases hn : i + j = n | case mul.H
R : Type u_1
inst✝⁶ : CommSemiring R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : AddCommMonoid ι
A : Type u_3
inst✝³ : CommSemiring A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
rel : A → A → Prop
inst✝ : GradedAlgebra 𝒜
hr : Rel.IsHomogeneous 𝒜 rel
a✝² b✝ a b c d : A
a✝¹ : Rel rel ... | case pos
R : Type u_1
inst✝⁶ : CommSemiring R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : AddCommMonoid ι
A : Type u_3
inst✝³ : CommSemiring A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
rel : A → A → Prop
inst✝ : GradedAlgebra 𝒜
hr : Rel.IsHomogeneous 𝒜 rel
a✝² b✝ a b c d : A
a✝¹ : Rel rel a ... | Please generate a tactic in lean4 to solve the state.
STATE:
case mul.H
R : Type u_1
inst✝⁶ : CommSemiring R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : AddCommMonoid ι
A : Type u_3
inst✝³ : CommSemiring A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
rel : A → A → Prop
inst✝ : GradedAlgebra 𝒜
hr... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | RingConGen.RelIsHomogeneous_of | [423, 1] | [448, 34] | . simp only [if_pos hn]
exact RingConGen.Rel.mul (k i) (k' j) | case pos
R : Type u_1
inst✝⁶ : CommSemiring R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : AddCommMonoid ι
A : Type u_3
inst✝³ : CommSemiring A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
rel : A → A → Prop
inst✝ : GradedAlgebra 𝒜
hr : Rel.IsHomogeneous 𝒜 rel
a✝² b✝ a b c d : A
a✝¹ : Rel rel a ... | case neg
R : Type u_1
inst✝⁶ : CommSemiring R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : AddCommMonoid ι
A : Type u_3
inst✝³ : CommSemiring A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
rel : A → A → Prop
inst✝ : GradedAlgebra 𝒜
hr : Rel.IsHomogeneous 𝒜 rel
a✝² b✝ a b c d : A
a✝¹ : Rel rel a ... | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
R : Type u_1
inst✝⁶ : CommSemiring R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : AddCommMonoid ι
A : Type u_3
inst✝³ : CommSemiring A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
rel : A → A → Prop
inst✝ : GradedAlgebra 𝒜
hr :... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | RingConGen.RelIsHomogeneous_of | [423, 1] | [448, 34] | . simp only [if_neg hn]
exact RingConGen.Rel.refl _ | case neg
R : Type u_1
inst✝⁶ : CommSemiring R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : AddCommMonoid ι
A : Type u_3
inst✝³ : CommSemiring A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
rel : A → A → Prop
inst✝ : GradedAlgebra 𝒜
hr : Rel.IsHomogeneous 𝒜 rel
a✝² b✝ a b c d : A
a✝¹ : Rel rel a ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
R : Type u_1
inst✝⁶ : CommSemiring R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : AddCommMonoid ι
A : Type u_3
inst✝³ : CommSemiring A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
rel : A → A → Prop
inst✝ : GradedAlgebra 𝒜
hr :... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | RingConGen.RelIsHomogeneous_of | [423, 1] | [448, 34] | simp only [if_pos hn] | case pos
R : Type u_1
inst✝⁶ : CommSemiring R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : AddCommMonoid ι
A : Type u_3
inst✝³ : CommSemiring A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
rel : A → A → Prop
inst✝ : GradedAlgebra 𝒜
hr : Rel.IsHomogeneous 𝒜 rel
a✝² b✝ a b c d : A
a✝¹ : Rel rel a ... | case pos
R : Type u_1
inst✝⁶ : CommSemiring R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : AddCommMonoid ι
A : Type u_3
inst✝³ : CommSemiring A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
rel : A → A → Prop
inst✝ : GradedAlgebra 𝒜
hr : Rel.IsHomogeneous 𝒜 rel
a✝² b✝ a b c d : A
a✝¹ : Rel rel a ... | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
R : Type u_1
inst✝⁶ : CommSemiring R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : AddCommMonoid ι
A : Type u_3
inst✝³ : CommSemiring A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
rel : A → A → Prop
inst✝ : GradedAlgebra 𝒜
hr :... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | RingConGen.RelIsHomogeneous_of | [423, 1] | [448, 34] | exact RingConGen.Rel.mul (k i) (k' j) | case pos
R : Type u_1
inst✝⁶ : CommSemiring R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : AddCommMonoid ι
A : Type u_3
inst✝³ : CommSemiring A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
rel : A → A → Prop
inst✝ : GradedAlgebra 𝒜
hr : Rel.IsHomogeneous 𝒜 rel
a✝² b✝ a b c d : A
a✝¹ : Rel rel a ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
R : Type u_1
inst✝⁶ : CommSemiring R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : AddCommMonoid ι
A : Type u_3
inst✝³ : CommSemiring A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
rel : A → A → Prop
inst✝ : GradedAlgebra 𝒜
hr :... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | RingConGen.RelIsHomogeneous_of | [423, 1] | [448, 34] | simp only [if_neg hn] | case neg
R : Type u_1
inst✝⁶ : CommSemiring R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : AddCommMonoid ι
A : Type u_3
inst✝³ : CommSemiring A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
rel : A → A → Prop
inst✝ : GradedAlgebra 𝒜
hr : Rel.IsHomogeneous 𝒜 rel
a✝² b✝ a b c d : A
a✝¹ : Rel rel a ... | case neg
R : Type u_1
inst✝⁶ : CommSemiring R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : AddCommMonoid ι
A : Type u_3
inst✝³ : CommSemiring A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
rel : A → A → Prop
inst✝ : GradedAlgebra 𝒜
hr : Rel.IsHomogeneous 𝒜 rel
a✝² b✝ a b c d : A
a✝¹ : Rel rel a ... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
R : Type u_1
inst✝⁶ : CommSemiring R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : AddCommMonoid ι
A : Type u_3
inst✝³ : CommSemiring A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
rel : A → A → Prop
inst✝ : GradedAlgebra 𝒜
hr :... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | RingConGen.RelIsHomogeneous_of | [423, 1] | [448, 34] | exact RingConGen.Rel.refl _ | case neg
R : Type u_1
inst✝⁶ : CommSemiring R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : AddCommMonoid ι
A : Type u_3
inst✝³ : CommSemiring A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
rel : A → A → Prop
inst✝ : GradedAlgebra 𝒜
hr : Rel.IsHomogeneous 𝒜 rel
a✝² b✝ a b c d : A
a✝¹ : Rel rel a ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
R : Type u_1
inst✝⁶ : CommSemiring R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : AddCommMonoid ι
A : Type u_3
inst✝³ : CommSemiring A
inst✝² : DecidableEq A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
rel : A → A → Prop
inst✝ : GradedAlgebra 𝒜
hr :... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | Ideal.IsHomogeneous_of_rel_isPureHomogeneous | [456, 1] | [463, 38] | apply Ideal.homogeneous_span | R✝ : Type u_1
inst✝¹² : CommSemiring R✝
ι✝ : Type u_2
inst✝¹¹ : DecidableEq ι✝
inst✝¹⁰ : AddCommMonoid ι✝
A✝ : Type u_3
inst✝⁹ : CommSemiring A✝
inst✝⁸ : DecidableEq A✝
inst✝⁷ : Algebra R✝ A✝
𝒜✝ : ι✝ → Submodule R✝ A✝
rel : A✝ → A✝ → Prop
R : Type u_4
inst✝⁶ : CommRing R
ι : Type u_5
inst✝⁵ : DecidableEq ι
inst✝⁴ : Ad... | case h
R✝ : Type u_1
inst✝¹² : CommSemiring R✝
ι✝ : Type u_2
inst✝¹¹ : DecidableEq ι✝
inst✝¹⁰ : AddCommMonoid ι✝
A✝ : Type u_3
inst✝⁹ : CommSemiring A✝
inst✝⁸ : DecidableEq A✝
inst✝⁷ : Algebra R✝ A✝
𝒜✝ : ι✝ → Submodule R✝ A✝
rel : A✝ → A✝ → Prop
R : Type u_4
inst✝⁶ : CommRing R
ι : Type u_5
inst✝⁵ : DecidableEq ι
inst... | Please generate a tactic in lean4 to solve the state.
STATE:
R✝ : Type u_1
inst✝¹² : CommSemiring R✝
ι✝ : Type u_2
inst✝¹¹ : DecidableEq ι✝
inst✝¹⁰ : AddCommMonoid ι✝
A✝ : Type u_3
inst✝⁹ : CommSemiring A✝
inst✝⁸ : DecidableEq A✝
inst✝⁷ : Algebra R✝ A✝
𝒜✝ : ι✝ → Submodule R✝ A✝
rel : A✝ → A✝ → Prop
R : Type u_4
inst✝⁶... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | Ideal.IsHomogeneous_of_rel_isPureHomogeneous | [456, 1] | [463, 38] | rintro x ⟨a, b, ⟨h, heq⟩⟩ | case h
R✝ : Type u_1
inst✝¹² : CommSemiring R✝
ι✝ : Type u_2
inst✝¹¹ : DecidableEq ι✝
inst✝¹⁰ : AddCommMonoid ι✝
A✝ : Type u_3
inst✝⁹ : CommSemiring A✝
inst✝⁸ : DecidableEq A✝
inst✝⁷ : Algebra R✝ A✝
𝒜✝ : ι✝ → Submodule R✝ A✝
rel : A✝ → A✝ → Prop
R : Type u_4
inst✝⁶ : CommRing R
ι : Type u_5
inst✝⁵ : DecidableEq ι
inst... | case h.intro.intro.intro
R✝ : Type u_1
inst✝¹² : CommSemiring R✝
ι✝ : Type u_2
inst✝¹¹ : DecidableEq ι✝
inst✝¹⁰ : AddCommMonoid ι✝
A✝ : Type u_3
inst✝⁹ : CommSemiring A✝
inst✝⁸ : DecidableEq A✝
inst✝⁷ : Algebra R✝ A✝
𝒜✝ : ι✝ → Submodule R✝ A✝
rel : A✝ → A✝ → Prop
R : Type u_4
inst✝⁶ : CommRing R
ι : Type u_5
inst✝⁵ : ... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
R✝ : Type u_1
inst✝¹² : CommSemiring R✝
ι✝ : Type u_2
inst✝¹¹ : DecidableEq ι✝
inst✝¹⁰ : AddCommMonoid ι✝
A✝ : Type u_3
inst✝⁹ : CommSemiring A✝
inst✝⁸ : DecidableEq A✝
inst✝⁷ : 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_isPureHomogeneous | [456, 1] | [463, 38] | obtain ⟨i, hi⟩ := hr h | case h.intro.intro.intro
R✝ : Type u_1
inst✝¹² : CommSemiring R✝
ι✝ : Type u_2
inst✝¹¹ : DecidableEq ι✝
inst✝¹⁰ : AddCommMonoid ι✝
A✝ : Type u_3
inst✝⁹ : CommSemiring A✝
inst✝⁸ : DecidableEq A✝
inst✝⁷ : Algebra R✝ A✝
𝒜✝ : ι✝ → Submodule R✝ A✝
rel : A✝ → A✝ → Prop
R : Type u_4
inst✝⁶ : CommRing R
ι : Type u_5
inst✝⁵ : ... | case h.intro.intro.intro.intro
R✝ : Type u_1
inst✝¹² : CommSemiring R✝
ι✝ : Type u_2
inst✝¹¹ : DecidableEq ι✝
inst✝¹⁰ : AddCommMonoid ι✝
A✝ : Type u_3
inst✝⁹ : CommSemiring A✝
inst✝⁸ : DecidableEq A✝
inst✝⁷ : Algebra R✝ A✝
𝒜✝ : ι✝ → Submodule R✝ A✝
rel : A✝ → A✝ → Prop
R : Type u_4
inst✝⁶ : CommRing R
ι : Type u_5
ins... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.intro.intro.intro
R✝ : Type u_1
inst✝¹² : CommSemiring R✝
ι✝ : Type u_2
inst✝¹¹ : DecidableEq ι✝
inst✝¹⁰ : AddCommMonoid ι✝
A✝ : Type u_3
inst✝⁹ : CommSemiring A✝
inst✝⁸ : DecidableEq A✝
inst✝⁷ : Algebra R✝ A✝
𝒜✝ : ι✝ → Submodule R✝ A✝
rel : A✝ → A✝ →... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | Ideal.IsHomogeneous_of_rel_isPureHomogeneous | [456, 1] | [463, 38] | use i | case h.intro.intro.intro.intro
R✝ : Type u_1
inst✝¹² : CommSemiring R✝
ι✝ : Type u_2
inst✝¹¹ : DecidableEq ι✝
inst✝¹⁰ : AddCommMonoid ι✝
A✝ : Type u_3
inst✝⁹ : CommSemiring A✝
inst✝⁸ : DecidableEq A✝
inst✝⁷ : Algebra R✝ A✝
𝒜✝ : ι✝ → Submodule R✝ A✝
rel : A✝ → A✝ → Prop
R : Type u_4
inst✝⁶ : CommRing R
ι : Type u_5
ins... | case h
R✝ : Type u_1
inst✝¹² : CommSemiring R✝
ι✝ : Type u_2
inst✝¹¹ : DecidableEq ι✝
inst✝¹⁰ : AddCommMonoid ι✝
A✝ : Type u_3
inst✝⁹ : CommSemiring A✝
inst✝⁸ : DecidableEq A✝
inst✝⁷ : Algebra R✝ A✝
𝒜✝ : ι✝ → Submodule R✝ A✝
rel : A✝ → A✝ → Prop
R : Type u_4
inst✝⁶ : CommRing R
ι : Type u_5
inst✝⁵ : DecidableEq ι
inst... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.intro.intro.intro.intro
R✝ : Type u_1
inst✝¹² : CommSemiring R✝
ι✝ : Type u_2
inst✝¹¹ : DecidableEq ι✝
inst✝¹⁰ : AddCommMonoid ι✝
A✝ : Type u_3
inst✝⁹ : CommSemiring A✝
inst✝⁸ : DecidableEq A✝
inst✝⁷ : Algebra R✝ A✝
𝒜✝ : ι✝ → Submodule R✝ A✝
rel : A✝ ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | Ideal.IsHomogeneous_of_rel_isPureHomogeneous | [456, 1] | [463, 38] | rw [(eq_sub_iff_add_eq).mpr heq] | case h
R✝ : Type u_1
inst✝¹² : CommSemiring R✝
ι✝ : Type u_2
inst✝¹¹ : DecidableEq ι✝
inst✝¹⁰ : AddCommMonoid ι✝
A✝ : Type u_3
inst✝⁹ : CommSemiring A✝
inst✝⁸ : DecidableEq A✝
inst✝⁷ : Algebra R✝ A✝
𝒜✝ : ι✝ → Submodule R✝ A✝
rel : A✝ → A✝ → Prop
R : Type u_4
inst✝⁶ : CommRing R
ι : Type u_5
inst✝⁵ : DecidableEq ι
inst... | case h
R✝ : Type u_1
inst✝¹² : CommSemiring R✝
ι✝ : Type u_2
inst✝¹¹ : DecidableEq ι✝
inst✝¹⁰ : AddCommMonoid ι✝
A✝ : Type u_3
inst✝⁹ : CommSemiring A✝
inst✝⁸ : DecidableEq A✝
inst✝⁷ : Algebra R✝ A✝
𝒜✝ : ι✝ → Submodule R✝ A✝
rel : A✝ → A✝ → Prop
R : Type u_4
inst✝⁶ : CommRing R
ι : Type u_5
inst✝⁵ : DecidableEq ι
inst... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
R✝ : Type u_1
inst✝¹² : CommSemiring R✝
ι✝ : Type u_2
inst✝¹¹ : DecidableEq ι✝
inst✝¹⁰ : AddCommMonoid ι✝
A✝ : Type u_3
inst✝⁹ : CommSemiring A✝
inst✝⁸ : DecidableEq A✝
inst✝⁷ : 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_isPureHomogeneous | [456, 1] | [463, 38] | exact Submodule.sub_mem _ hi.1 hi.2 | case h
R✝ : Type u_1
inst✝¹² : CommSemiring R✝
ι✝ : Type u_2
inst✝¹¹ : DecidableEq ι✝
inst✝¹⁰ : AddCommMonoid ι✝
A✝ : Type u_3
inst✝⁹ : CommSemiring A✝
inst✝⁸ : DecidableEq A✝
inst✝⁷ : Algebra R✝ A✝
𝒜✝ : ι✝ → Submodule R✝ A✝
rel : A✝ → A✝ → Prop
R : Type u_4
inst✝⁶ : CommRing R
ι : Type u_5
inst✝⁵ : DecidableEq ι
inst... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
R✝ : Type u_1
inst✝¹² : CommSemiring R✝
ι✝ : Type u_2
inst✝¹¹ : DecidableEq ι✝
inst✝¹⁰ : AddCommMonoid ι✝
A✝ : Type u_3
inst✝⁹ : CommSemiring A✝
inst✝⁸ : DecidableEq A✝
inst✝⁷ : 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] | let r' : A → A → Prop := fun a b => ∃ i, a ∈ 𝒜 i ∧ b ∈ 𝒜 i ∧ r a b | 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✝³ : Add... | 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✝³ : Add... | Please generate a tactic in lean4 to solve the state.
STATE:
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 ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | Ideal.IsHomogeneous_of_rel_isHomogeneous | [466, 1] | [504, 45] | suffices Ideal.ofRel r = Ideal.ofRel r' by
rw [this]
apply Ideal.IsHomogeneous_of_rel_isPureHomogeneous
rintro a b ⟨i, h⟩
exact ⟨i, h.1, h.2.1⟩ | 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✝³ : Add... | 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✝³ : Add... | Please generate a tactic in lean4 to solve the state.
STATE:
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 ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | Ideal.IsHomogeneous_of_rel_isHomogeneous | [466, 1] | [504, 45] | apply le_antisymm | 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✝³ : Add... | 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:
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 ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | Ideal.IsHomogeneous_of_rel_isHomogeneous | [466, 1] | [504, 45] | . intro x hx
refine' Submodule.span_induction hx _ _ _ _
. rintro x ⟨a, b, h', h⟩
rw [← h𝒜.left_inv x, ← sum_support_of _ (Decomposition.decompose' x),
map_sum]
apply Ideal.sum_mem
intro i _
rw [coeAddMonoidHom_of]
apply Ideal.subset_span
use h𝒜.decompose' a i
use h𝒜.decompose' ... | 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] | . intro x hx'
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)
. 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] | rw [this] | 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✝³ : Add... | 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✝³ : Add... | Please generate a tactic in lean4 to solve the state.
STATE:
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 ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | Ideal.IsHomogeneous_of_rel_isHomogeneous | [466, 1] | [504, 45] | apply Ideal.IsHomogeneous_of_rel_isPureHomogeneous | 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✝³ : Add... | case hr
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:
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 ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedRingQuot.lean | Ideal.IsHomogeneous_of_rel_isHomogeneous | [466, 1] | [504, 45] | rintro a b ⟨i, h⟩ | case hr
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 hr.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
inst✝⁴ : DecidableEq ... | Please generate a tactic in lean4 to solve the state.
STATE:
case hr
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 ⟨i, h.1, h.2.1⟩ | case hr.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
inst✝⁴ : DecidableEq ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hr.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 : Ty... |
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 _ _ _ _ | 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.refine'_1
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
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, h', h⟩
rw [← h𝒜.left_inv x, ← sum_support_of _ (Decomposition.decompose' x),
map_sum]
apply Ideal.sum_mem
intro i _
rw [coeAddMonoidHom_of]
apply Ideal.subset_span
use h𝒜.decompose' a i
use h𝒜.decompose' b i
simp only [exists_prop]
constructor
. use i
simp only [Decompos... | case a.refine'_1
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'_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... | Please generate a tactic in lean4 to solve the state.
STATE:
case a.refine'_1
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 [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... | 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'_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
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... | 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'_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
simp only [smul_eq_mul]
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] | rintro x ⟨a, b, h', h⟩ | case a.refine'_1
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'_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
... | Please generate a tactic in lean4 to solve the state.
STATE:
case a.refine'_1
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] | rw [← h𝒜.left_inv x, ← sum_support_of _ (Decomposition.decompose' x),
map_sum] | 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
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
... | 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 : ... |
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