url stringclasses 147
values | commit stringclasses 147
values | file_path stringlengths 7 101 | full_name stringlengths 1 94 | start stringlengths 6 10 | end stringlengths 6 11 | tactic stringlengths 1 11.2k | state_before stringlengths 3 2.09M | state_after stringlengths 6 2.09M | input stringlengths 73 2.09M |
|---|---|---|---|---|---|---|---|---|---|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Misc.lean | TrivSqZeroExt.mem_kerIdeal_iff_exists | [129, 1] | [131, 62] | rw [mem_kerIdeal_iff_inr] | R : Type u_1
M : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : Module Rᵐᵒᵖ M
inst✝ : IsCentralScalar R M
x : TrivSqZeroExt R M
⊢ x ∈ kerIdeal R M ↔ ∃ m, x = inr m | R : Type u_1
M : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : Module Rᵐᵒᵖ M
inst✝ : IsCentralScalar R M
x : TrivSqZeroExt R M
⊢ x = inr x.snd ↔ ∃ m, x = inr m | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
M : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : Module Rᵐᵒᵖ M
inst✝ : IsCentralScalar R M
x : TrivSqZeroExt R M
⊢ x ∈ kerIdeal R M ↔ ∃ m, x = inr m
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Misc.lean | TrivSqZeroExt.mem_kerIdeal_iff_exists | [129, 1] | [131, 62] | exact ⟨fun h => ⟨x.snd, h⟩, fun ⟨m, hm⟩ => by rw [hm]; rfl⟩ | R : Type u_1
M : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : Module Rᵐᵒᵖ M
inst✝ : IsCentralScalar R M
x : TrivSqZeroExt R M
⊢ x = inr x.snd ↔ ∃ m, x = inr m | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
M : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : Module Rᵐᵒᵖ M
inst✝ : IsCentralScalar R M
x : TrivSqZeroExt R M
⊢ x = inr x.snd ↔ ∃ m, x = inr m
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Misc.lean | TrivSqZeroExt.mem_kerIdeal_iff_exists | [129, 1] | [131, 62] | rw [hm] | R : Type u_1
M : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : Module Rᵐᵒᵖ M
inst✝ : IsCentralScalar R M
x : TrivSqZeroExt R M
x✝ : ∃ m, x = inr m
m : M
hm : x = inr m
⊢ x = inr x.snd | R : Type u_1
M : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : Module Rᵐᵒᵖ M
inst✝ : IsCentralScalar R M
x : TrivSqZeroExt R M
x✝ : ∃ m, x = inr m
m : M
hm : x = inr m
⊢ inr m = inr (inr m).snd | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
M : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : Module Rᵐᵒᵖ M
inst✝ : IsCentralScalar R M
x : TrivSqZeroExt R M
x✝ : ∃ m, x = inr m
m : M
hm : x = inr m
⊢ x = inr x.snd
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Misc.lean | TrivSqZeroExt.mem_kerIdeal_iff_exists | [129, 1] | [131, 62] | rfl | R : Type u_1
M : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : Module Rᵐᵒᵖ M
inst✝ : IsCentralScalar R M
x : TrivSqZeroExt R M
x✝ : ∃ m, x = inr m
m : M
hm : x = inr m
⊢ inr m = inr (inr m).snd | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
M : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : Module Rᵐᵒᵖ M
inst✝ : IsCentralScalar R M
x : TrivSqZeroExt R M
x✝ : ∃ m, x = inr m
m : M
hm : x = inr m
⊢ inr m = inr (inr m).snd
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Misc.lean | TrivSqZeroExt.sqZero | [134, 1] | [138, 36] | simp only [pow_two, zero_eq_bot, eq_bot_iff, mul_le, mem_kerIdeal_iff_inr] | R : Type u_1
M : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : Module Rᵐᵒᵖ M
inst✝ : IsCentralScalar R M
⊢ kerIdeal R M ^ 2 = ⊥ | R : Type u_1
M : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : Module Rᵐᵒᵖ M
inst✝ : IsCentralScalar R M
⊢ ∀ (r : TrivSqZeroExt R M), r = inr r.snd → ∀ (s : TrivSqZeroExt R M), s = inr s.snd → r * s ∈ ⊥ | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
M : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : Module Rᵐᵒᵖ M
inst✝ : IsCentralScalar R M
⊢ kerIdeal R M ^ 2 = ⊥
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Misc.lean | TrivSqZeroExt.sqZero | [134, 1] | [138, 36] | rintro x hx y hy | R : Type u_1
M : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : Module Rᵐᵒᵖ M
inst✝ : IsCentralScalar R M
⊢ ∀ (r : TrivSqZeroExt R M), r = inr r.snd → ∀ (s : TrivSqZeroExt R M), s = inr s.snd → r * s ∈ ⊥ | R : Type u_1
M : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : Module Rᵐᵒᵖ M
inst✝ : IsCentralScalar R M
x : TrivSqZeroExt R M
hx : x = inr x.snd
y : TrivSqZeroExt R M
hy : y = inr y.snd
⊢ x * y ∈ ⊥ | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
M : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : Module Rᵐᵒᵖ M
inst✝ : IsCentralScalar R M
⊢ ∀ (r : TrivSqZeroExt R M), r = inr r.snd → ∀ (s : TrivSqZeroExt R M), s = inr s.snd → r * s ∈ ⊥
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Misc.lean | TrivSqZeroExt.sqZero | [134, 1] | [138, 36] | rw [hx, hy, mem_bot, inr_mul_inr] | R : Type u_1
M : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : Module Rᵐᵒᵖ M
inst✝ : IsCentralScalar R M
x : TrivSqZeroExt R M
hx : x = inr x.snd
y : TrivSqZeroExt R M
hy : y = inr y.snd
⊢ x * y ∈ ⊥ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
M : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : Module Rᵐᵒᵖ M
inst✝ : IsCentralScalar R M
x : TrivSqZeroExt R M
hx : x = inr x.snd
y : TrivSqZeroExt R M
hy : y = inr y.snd
⊢ x * y ∈ ⊥
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Misc.lean | Finsupp.prod.mem_grade | [326, 1] | [343, 21] | rw [Finsupp.prod, Finsupp.sum] | R : Type u_3
inst✝⁷ : CommSemiring R
S : Type ?u.167638
inst✝⁶ : CommSemiring S
inst✝⁵ : Algebra R S
κ : Type u_1
A : Type u_2
inst✝⁴ : AddCommMonoid κ
inst✝³ : DecidableEq κ
inst✝² : CommSemiring A
inst✝¹ : Algebra R A
𝒜 : κ → Submodule R A
inst✝ : GradedAlgebra 𝒜
σ : Type u_4
c : σ →₀ ℕ
f : σ → A
d : σ → κ
hc : ∀ s... | R : Type u_3
inst✝⁷ : CommSemiring R
S : Type ?u.167638
inst✝⁶ : CommSemiring S
inst✝⁵ : Algebra R S
κ : Type u_1
A : Type u_2
inst✝⁴ : AddCommMonoid κ
inst✝³ : DecidableEq κ
inst✝² : CommSemiring A
inst✝¹ : Algebra R A
𝒜 : κ → Submodule R A
inst✝ : GradedAlgebra 𝒜
σ : Type u_4
c : σ →₀ ℕ
f : σ → A
d : σ → κ
hc : ∀ s... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_3
inst✝⁷ : CommSemiring R
S : Type ?u.167638
inst✝⁶ : CommSemiring S
inst✝⁵ : Algebra R S
κ : Type u_1
A : Type u_2
inst✝⁴ : AddCommMonoid κ
inst✝³ : DecidableEq κ
inst✝² : CommSemiring A
inst✝¹ : Algebra R A
𝒜 : κ → Submodule R A
inst✝ : GradedAl... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Misc.lean | Finsupp.prod.mem_grade | [326, 1] | [343, 21] | let p : Finset σ → Prop := fun s =>
s ⊆ c.support → (s.prod fun i => f i ^ c i) ∈ 𝒜 (s.sum fun i => c i • d i) | R : Type u_3
inst✝⁷ : CommSemiring R
S : Type ?u.167638
inst✝⁶ : CommSemiring S
inst✝⁵ : Algebra R S
κ : Type u_1
A : Type u_2
inst✝⁴ : AddCommMonoid κ
inst✝³ : DecidableEq κ
inst✝² : CommSemiring A
inst✝¹ : Algebra R A
𝒜 : κ → Submodule R A
inst✝ : GradedAlgebra 𝒜
σ : Type u_4
c : σ →₀ ℕ
f : σ → A
d : σ → κ
hc : ∀ s... | R : Type u_3
inst✝⁷ : CommSemiring R
S : Type ?u.167638
inst✝⁶ : CommSemiring S
inst✝⁵ : Algebra R S
κ : Type u_1
A : Type u_2
inst✝⁴ : AddCommMonoid κ
inst✝³ : DecidableEq κ
inst✝² : CommSemiring A
inst✝¹ : Algebra R A
𝒜 : κ → Submodule R A
inst✝ : GradedAlgebra 𝒜
σ : Type u_4
c : σ →₀ ℕ
f : σ → A
d : σ → κ
hc : ∀ s... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_3
inst✝⁷ : CommSemiring R
S : Type ?u.167638
inst✝⁶ : CommSemiring S
inst✝⁵ : Algebra R S
κ : Type u_1
A : Type u_2
inst✝⁴ : AddCommMonoid κ
inst✝³ : DecidableEq κ
inst✝² : CommSemiring A
inst✝¹ : Algebra R A
𝒜 : κ → Submodule R A
inst✝ : GradedAl... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Misc.lean | Finsupp.prod.mem_grade | [326, 1] | [343, 21] | apply @Finset.induction_on σ p _ c.support | R : Type u_3
inst✝⁷ : CommSemiring R
S : Type ?u.167638
inst✝⁶ : CommSemiring S
inst✝⁵ : Algebra R S
κ : Type u_1
A : Type u_2
inst✝⁴ : AddCommMonoid κ
inst✝³ : DecidableEq κ
inst✝² : CommSemiring A
inst✝¹ : Algebra R A
𝒜 : κ → Submodule R A
inst✝ : GradedAlgebra 𝒜
σ : Type u_4
c : σ →₀ ℕ
f : σ → A
d : σ → κ
hc : ∀ s... | case empty
R : Type u_3
inst✝⁷ : CommSemiring R
S : Type ?u.167638
inst✝⁶ : CommSemiring S
inst✝⁵ : Algebra R S
κ : Type u_1
A : Type u_2
inst✝⁴ : AddCommMonoid κ
inst✝³ : DecidableEq κ
inst✝² : CommSemiring A
inst✝¹ : Algebra R A
𝒜 : κ → Submodule R A
inst✝ : GradedAlgebra 𝒜
σ : Type u_4
c : σ →₀ ℕ
f : σ → A
d : σ →... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_3
inst✝⁷ : CommSemiring R
S : Type ?u.167638
inst✝⁶ : CommSemiring S
inst✝⁵ : Algebra R S
κ : Type u_1
A : Type u_2
inst✝⁴ : AddCommMonoid κ
inst✝³ : DecidableEq κ
inst✝² : CommSemiring A
inst✝¹ : Algebra R A
𝒜 : κ → Submodule R A
inst✝ : GradedAl... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Misc.lean | Finsupp.prod.mem_grade | [326, 1] | [343, 21] | exact imp_intro (SetLike.one_mem_graded 𝒜) | case empty
R : Type u_3
inst✝⁷ : CommSemiring R
S : Type ?u.167638
inst✝⁶ : CommSemiring S
inst✝⁵ : Algebra R S
κ : Type u_1
A : Type u_2
inst✝⁴ : AddCommMonoid κ
inst✝³ : DecidableEq κ
inst✝² : CommSemiring A
inst✝¹ : Algebra R A
𝒜 : κ → Submodule R A
inst✝ : GradedAlgebra 𝒜
σ : Type u_4
c : σ →₀ ℕ
f : σ → A
d : σ →... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case empty
R : Type u_3
inst✝⁷ : CommSemiring R
S : Type ?u.167638
inst✝⁶ : CommSemiring S
inst✝⁵ : Algebra R S
κ : Type u_1
A : Type u_2
inst✝⁴ : AddCommMonoid κ
inst✝³ : DecidableEq κ
inst✝² : CommSemiring A
inst✝¹ : Algebra R A
𝒜 : κ → Submodule R A
inst✝... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Misc.lean | Finsupp.prod.mem_grade | [326, 1] | [343, 21] | intro a s ha hs hs' | case insert
R : Type u_3
inst✝⁷ : CommSemiring R
S : Type ?u.167638
inst✝⁶ : CommSemiring S
inst✝⁵ : Algebra R S
κ : Type u_1
A : Type u_2
inst✝⁴ : AddCommMonoid κ
inst✝³ : DecidableEq κ
inst✝² : CommSemiring A
inst✝¹ : Algebra R A
𝒜 : κ → Submodule R A
inst✝ : GradedAlgebra 𝒜
σ : Type u_4
c : σ →₀ ℕ
f : σ → A
d : σ ... | case insert
R : Type u_3
inst✝⁷ : CommSemiring R
S : Type ?u.167638
inst✝⁶ : CommSemiring S
inst✝⁵ : Algebra R S
κ : Type u_1
A : Type u_2
inst✝⁴ : AddCommMonoid κ
inst✝³ : DecidableEq κ
inst✝² : CommSemiring A
inst✝¹ : Algebra R A
𝒜 : κ → Submodule R A
inst✝ : GradedAlgebra 𝒜
σ : Type u_4
c : σ →₀ ℕ
f : σ → A
d : σ ... | Please generate a tactic in lean4 to solve the state.
STATE:
case insert
R : Type u_3
inst✝⁷ : CommSemiring R
S : Type ?u.167638
inst✝⁶ : CommSemiring S
inst✝⁵ : Algebra R S
κ : Type u_1
A : Type u_2
inst✝⁴ : AddCommMonoid κ
inst✝³ : DecidableEq κ
inst✝² : CommSemiring A
inst✝¹ : Algebra R A
𝒜 : κ → Submodule R A
inst... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Misc.lean | Finsupp.prod.mem_grade | [326, 1] | [343, 21] | rw [Finset.prod_insert ha, Finset.sum_insert ha] | case insert
R : Type u_3
inst✝⁷ : CommSemiring R
S : Type ?u.167638
inst✝⁶ : CommSemiring S
inst✝⁵ : Algebra R S
κ : Type u_1
A : Type u_2
inst✝⁴ : AddCommMonoid κ
inst✝³ : DecidableEq κ
inst✝² : CommSemiring A
inst✝¹ : Algebra R A
𝒜 : κ → Submodule R A
inst✝ : GradedAlgebra 𝒜
σ : Type u_4
c : σ →₀ ℕ
f : σ → A
d : σ ... | case insert
R : Type u_3
inst✝⁷ : CommSemiring R
S : Type ?u.167638
inst✝⁶ : CommSemiring S
inst✝⁵ : Algebra R S
κ : Type u_1
A : Type u_2
inst✝⁴ : AddCommMonoid κ
inst✝³ : DecidableEq κ
inst✝² : CommSemiring A
inst✝¹ : Algebra R A
𝒜 : κ → Submodule R A
inst✝ : GradedAlgebra 𝒜
σ : Type u_4
c : σ →₀ ℕ
f : σ → A
d : σ ... | Please generate a tactic in lean4 to solve the state.
STATE:
case insert
R : Type u_3
inst✝⁷ : CommSemiring R
S : Type ?u.167638
inst✝⁶ : CommSemiring S
inst✝⁵ : Algebra R S
κ : Type u_1
A : Type u_2
inst✝⁴ : AddCommMonoid κ
inst✝³ : DecidableEq κ
inst✝² : CommSemiring A
inst✝¹ : Algebra R A
𝒜 : κ → Submodule R A
inst... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Misc.lean | Finsupp.prod.mem_grade | [326, 1] | [343, 21] | exact
SetLike.mul_mem_graded (SetLike.pow_mem_graded _ (hc a (hs' (mem_insert_self a s))))
(hs (subset_trans (subset_insert a s) hs')) | case insert
R : Type u_3
inst✝⁷ : CommSemiring R
S : Type ?u.167638
inst✝⁶ : CommSemiring S
inst✝⁵ : Algebra R S
κ : Type u_1
A : Type u_2
inst✝⁴ : AddCommMonoid κ
inst✝³ : DecidableEq κ
inst✝² : CommSemiring A
inst✝¹ : Algebra R A
𝒜 : κ → Submodule R A
inst✝ : GradedAlgebra 𝒜
σ : Type u_4
c : σ →₀ ℕ
f : σ → A
d : σ ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case insert
R : Type u_3
inst✝⁷ : CommSemiring R
S : Type ?u.167638
inst✝⁶ : CommSemiring S
inst✝⁵ : Algebra R S
κ : Type u_1
A : Type u_2
inst✝⁴ : AddCommMonoid κ
inst✝³ : DecidableEq κ
inst✝² : CommSemiring A
inst✝¹ : Algebra R A
𝒜 : κ → Submodule R A
inst... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Misc.lean | Finsupp.prod.mem_grade | [326, 1] | [343, 21] | exact subset_rfl | case a
R : Type u_3
inst✝⁷ : CommSemiring R
S : Type ?u.167638
inst✝⁶ : CommSemiring S
inst✝⁵ : Algebra R S
κ : Type u_1
A : Type u_2
inst✝⁴ : AddCommMonoid κ
inst✝³ : DecidableEq κ
inst✝² : CommSemiring A
inst✝¹ : Algebra R A
𝒜 : κ → Submodule R A
inst✝ : GradedAlgebra 𝒜
σ : Type u_4
c : σ →₀ ℕ
f : σ → A
d : σ → κ
h... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a
R : Type u_3
inst✝⁷ : CommSemiring R
S : Type ?u.167638
inst✝⁶ : CommSemiring S
inst✝⁵ : Algebra R S
κ : Type u_1
A : Type u_2
inst✝⁴ : AddCommMonoid κ
inst✝³ : DecidableEq κ
inst✝² : CommSemiring A
inst✝¹ : Algebra R A
𝒜 : κ → Submodule R A
inst✝ : G... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Misc.lean | GalgHom.isHomogeneous'_aeval | [356, 1] | [375, 38] | intro i p hp | R : Type u_5
inst✝⁸ : CommSemiring R
S : Type ?u.178387
inst✝⁷ : CommSemiring S
inst✝⁶ : Algebra R S
σ : Type u_1
ι : Type u_2
κ : Type u_3
inst✝⁵ : AddCommMonoid ι
inst✝⁴ : AddCommMonoid κ
inst✝³ : DecidableEq κ
A : Type u_4
inst✝² : CommSemiring A
inst✝¹ : Algebra R A
𝒜 : κ → Submodule R A
inst✝ : GradedAlgebra 𝒜
w... | R : Type u_5
inst✝⁸ : CommSemiring R
S : Type ?u.178387
inst✝⁷ : CommSemiring S
inst✝⁶ : Algebra R S
σ : Type u_1
ι : Type u_2
κ : Type u_3
inst✝⁵ : AddCommMonoid ι
inst✝⁴ : AddCommMonoid κ
inst✝³ : DecidableEq κ
A : Type u_4
inst✝² : CommSemiring A
inst✝¹ : Algebra R A
𝒜 : κ → Submodule R A
inst✝ : GradedAlgebra 𝒜
w... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_5
inst✝⁸ : CommSemiring R
S : Type ?u.178387
inst✝⁷ : CommSemiring S
inst✝⁶ : Algebra R S
σ : Type u_1
ι : Type u_2
κ : Type u_3
inst✝⁵ : AddCommMonoid ι
inst✝⁴ : AddCommMonoid κ
inst✝³ : DecidableEq κ
A : Type u_4
inst✝² : CommSemiring A
inst✝¹ : ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Misc.lean | GalgHom.isHomogeneous'_aeval | [356, 1] | [375, 38] | simp only [mem_weightedHomogeneousSubmodule, IsWeightedHomogeneous] at hp | R : Type u_5
inst✝⁸ : CommSemiring R
S : Type ?u.178387
inst✝⁷ : CommSemiring S
inst✝⁶ : Algebra R S
σ : Type u_1
ι : Type u_2
κ : Type u_3
inst✝⁵ : AddCommMonoid ι
inst✝⁴ : AddCommMonoid κ
inst✝³ : DecidableEq κ
A : Type u_4
inst✝² : CommSemiring A
inst✝¹ : Algebra R A
𝒜 : κ → Submodule R A
inst✝ : GradedAlgebra 𝒜
w... | R : Type u_5
inst✝⁸ : CommSemiring R
S : Type ?u.178387
inst✝⁷ : CommSemiring S
inst✝⁶ : Algebra R S
σ : Type u_1
ι : Type u_2
κ : Type u_3
inst✝⁵ : AddCommMonoid ι
inst✝⁴ : AddCommMonoid κ
inst✝³ : DecidableEq κ
A : Type u_4
inst✝² : CommSemiring A
inst✝¹ : Algebra R A
𝒜 : κ → Submodule R A
inst✝ : GradedAlgebra 𝒜
w... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_5
inst✝⁸ : CommSemiring R
S : Type ?u.178387
inst✝⁷ : CommSemiring S
inst✝⁶ : Algebra R S
σ : Type u_1
ι : Type u_2
κ : Type u_3
inst✝⁵ : AddCommMonoid ι
inst✝⁴ : AddCommMonoid κ
inst✝³ : DecidableEq κ
A : Type u_4
inst✝² : CommSemiring A
inst✝¹ : ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Misc.lean | GalgHom.isHomogeneous'_aeval | [356, 1] | [375, 38] | rw [p.as_sum, map_sum] | R : Type u_5
inst✝⁸ : CommSemiring R
S : Type ?u.178387
inst✝⁷ : CommSemiring S
inst✝⁶ : Algebra R S
σ : Type u_1
ι : Type u_2
κ : Type u_3
inst✝⁵ : AddCommMonoid ι
inst✝⁴ : AddCommMonoid κ
inst✝³ : DecidableEq κ
A : Type u_4
inst✝² : CommSemiring A
inst✝¹ : Algebra R A
𝒜 : κ → Submodule R A
inst✝ : GradedAlgebra 𝒜
w... | R : Type u_5
inst✝⁸ : CommSemiring R
S : Type ?u.178387
inst✝⁷ : CommSemiring S
inst✝⁶ : Algebra R S
σ : Type u_1
ι : Type u_2
κ : Type u_3
inst✝⁵ : AddCommMonoid ι
inst✝⁴ : AddCommMonoid κ
inst✝³ : DecidableEq κ
A : Type u_4
inst✝² : CommSemiring A
inst✝¹ : Algebra R A
𝒜 : κ → Submodule R A
inst✝ : GradedAlgebra 𝒜
w... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_5
inst✝⁸ : CommSemiring R
S : Type ?u.178387
inst✝⁷ : CommSemiring S
inst✝⁶ : Algebra R S
σ : Type u_1
ι : Type u_2
κ : Type u_3
inst✝⁵ : AddCommMonoid ι
inst✝⁴ : AddCommMonoid κ
inst✝³ : DecidableEq κ
A : Type u_4
inst✝² : CommSemiring A
inst✝¹ : ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Misc.lean | GalgHom.isHomogeneous'_aeval | [356, 1] | [375, 38] | apply Submodule.sum_mem | R : Type u_5
inst✝⁸ : CommSemiring R
S : Type ?u.178387
inst✝⁷ : CommSemiring S
inst✝⁶ : Algebra R S
σ : Type u_1
ι : Type u_2
κ : Type u_3
inst✝⁵ : AddCommMonoid ι
inst✝⁴ : AddCommMonoid κ
inst✝³ : DecidableEq κ
A : Type u_4
inst✝² : CommSemiring A
inst✝¹ : Algebra R A
𝒜 : κ → Submodule R A
inst✝ : GradedAlgebra 𝒜
w... | case a
R : Type u_5
inst✝⁸ : CommSemiring R
S : Type ?u.178387
inst✝⁷ : CommSemiring S
inst✝⁶ : Algebra R S
σ : Type u_1
ι : Type u_2
κ : Type u_3
inst✝⁵ : AddCommMonoid ι
inst✝⁴ : AddCommMonoid κ
inst✝³ : DecidableEq κ
A : Type u_4
inst✝² : CommSemiring A
inst✝¹ : Algebra R A
𝒜 : κ → Submodule R A
inst✝ : GradedAlgeb... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_5
inst✝⁸ : CommSemiring R
S : Type ?u.178387
inst✝⁷ : CommSemiring S
inst✝⁶ : Algebra R S
σ : Type u_1
ι : Type u_2
κ : Type u_3
inst✝⁵ : AddCommMonoid ι
inst✝⁴ : AddCommMonoid κ
inst✝³ : DecidableEq κ
A : Type u_4
inst✝² : CommSemiring A
inst✝¹ : ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Misc.lean | GalgHom.isHomogeneous'_aeval | [356, 1] | [375, 38] | intro c hc | case a
R : Type u_5
inst✝⁸ : CommSemiring R
S : Type ?u.178387
inst✝⁷ : CommSemiring S
inst✝⁶ : Algebra R S
σ : Type u_1
ι : Type u_2
κ : Type u_3
inst✝⁵ : AddCommMonoid ι
inst✝⁴ : AddCommMonoid κ
inst✝³ : DecidableEq κ
A : Type u_4
inst✝² : CommSemiring A
inst✝¹ : Algebra R A
𝒜 : κ → Submodule R A
inst✝ : GradedAlgeb... | case a
R : Type u_5
inst✝⁸ : CommSemiring R
S : Type ?u.178387
inst✝⁷ : CommSemiring S
inst✝⁶ : Algebra R S
σ : Type u_1
ι : Type u_2
κ : Type u_3
inst✝⁵ : AddCommMonoid ι
inst✝⁴ : AddCommMonoid κ
inst✝³ : DecidableEq κ
A : Type u_4
inst✝² : CommSemiring A
inst✝¹ : Algebra R A
𝒜 : κ → Submodule R A
inst✝ : GradedAlgeb... | Please generate a tactic in lean4 to solve the state.
STATE:
case a
R : Type u_5
inst✝⁸ : CommSemiring R
S : Type ?u.178387
inst✝⁷ : CommSemiring S
inst✝⁶ : Algebra R S
σ : Type u_1
ι : Type u_2
κ : Type u_3
inst✝⁵ : AddCommMonoid ι
inst✝⁴ : AddCommMonoid κ
inst✝³ : DecidableEq κ
A : Type u_4
inst✝² : CommSemiring A
in... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Misc.lean | GalgHom.isHomogeneous'_aeval | [356, 1] | [375, 38] | rw [aeval_monomial, ← smul_eq_mul, algebraMap_smul] | case a
R : Type u_5
inst✝⁸ : CommSemiring R
S : Type ?u.178387
inst✝⁷ : CommSemiring S
inst✝⁶ : Algebra R S
σ : Type u_1
ι : Type u_2
κ : Type u_3
inst✝⁵ : AddCommMonoid ι
inst✝⁴ : AddCommMonoid κ
inst✝³ : DecidableEq κ
A : Type u_4
inst✝² : CommSemiring A
inst✝¹ : Algebra R A
𝒜 : κ → Submodule R A
inst✝ : GradedAlgeb... | case a
R : Type u_5
inst✝⁸ : CommSemiring R
S : Type ?u.178387
inst✝⁷ : CommSemiring S
inst✝⁶ : Algebra R S
σ : Type u_1
ι : Type u_2
κ : Type u_3
inst✝⁵ : AddCommMonoid ι
inst✝⁴ : AddCommMonoid κ
inst✝³ : DecidableEq κ
A : Type u_4
inst✝² : CommSemiring A
inst✝¹ : Algebra R A
𝒜 : κ → Submodule R A
inst✝ : GradedAlgeb... | Please generate a tactic in lean4 to solve the state.
STATE:
case a
R : Type u_5
inst✝⁸ : CommSemiring R
S : Type ?u.178387
inst✝⁷ : CommSemiring S
inst✝⁶ : Algebra R S
σ : Type u_1
ι : Type u_2
κ : Type u_3
inst✝⁵ : AddCommMonoid ι
inst✝⁴ : AddCommMonoid κ
inst✝³ : DecidableEq κ
A : Type u_4
inst✝² : CommSemiring A
in... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Misc.lean | GalgHom.isHomogeneous'_aeval | [356, 1] | [375, 38] | apply Submodule.smul_mem | case a
R : Type u_5
inst✝⁸ : CommSemiring R
S : Type ?u.178387
inst✝⁷ : CommSemiring S
inst✝⁶ : Algebra R S
σ : Type u_1
ι : Type u_2
κ : Type u_3
inst✝⁵ : AddCommMonoid ι
inst✝⁴ : AddCommMonoid κ
inst✝³ : DecidableEq κ
A : Type u_4
inst✝² : CommSemiring A
inst✝¹ : Algebra R A
𝒜 : κ → Submodule R A
inst✝ : GradedAlgeb... | case a.h
R : Type u_5
inst✝⁸ : CommSemiring R
S : Type ?u.178387
inst✝⁷ : CommSemiring S
inst✝⁶ : Algebra R S
σ : Type u_1
ι : Type u_2
κ : Type u_3
inst✝⁵ : AddCommMonoid ι
inst✝⁴ : AddCommMonoid κ
inst✝³ : DecidableEq κ
A : Type u_4
inst✝² : CommSemiring A
inst✝¹ : Algebra R A
𝒜 : κ → Submodule R A
inst✝ : GradedAlg... | Please generate a tactic in lean4 to solve the state.
STATE:
case a
R : Type u_5
inst✝⁸ : CommSemiring R
S : Type ?u.178387
inst✝⁷ : CommSemiring S
inst✝⁶ : Algebra R S
σ : Type u_1
ι : Type u_2
κ : Type u_3
inst✝⁵ : AddCommMonoid ι
inst✝⁴ : AddCommMonoid κ
inst✝³ : DecidableEq κ
A : Type u_4
inst✝² : CommSemiring A
in... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Misc.lean | GalgHom.isHomogeneous'_aeval | [356, 1] | [375, 38] | convert Finsupp.prod.mem_grade 𝒜 c f _ fun s _ => h s | case a.h
R : Type u_5
inst✝⁸ : CommSemiring R
S : Type ?u.178387
inst✝⁷ : CommSemiring S
inst✝⁶ : Algebra R S
σ : Type u_1
ι : Type u_2
κ : Type u_3
inst✝⁵ : AddCommMonoid ι
inst✝⁴ : AddCommMonoid κ
inst✝³ : DecidableEq κ
A : Type u_4
inst✝² : CommSemiring A
inst✝¹ : Algebra R A
𝒜 : κ → Submodule R A
inst✝ : GradedAlg... | case h.e'_5.h.e'_1
R : Type u_5
inst✝⁸ : CommSemiring R
S : Type ?u.178387
inst✝⁷ : CommSemiring S
inst✝⁶ : Algebra R S
σ : Type u_1
ι : Type u_2
κ : Type u_3
inst✝⁵ : AddCommMonoid ι
inst✝⁴ : AddCommMonoid κ
inst✝³ : DecidableEq κ
A : Type u_4
inst✝² : CommSemiring A
inst✝¹ : Algebra R A
𝒜 : κ → Submodule R A
inst✝ :... | Please generate a tactic in lean4 to solve the state.
STATE:
case a.h
R : Type u_5
inst✝⁸ : CommSemiring R
S : Type ?u.178387
inst✝⁷ : CommSemiring S
inst✝⁶ : Algebra R S
σ : Type u_1
ι : Type u_2
κ : Type u_3
inst✝⁵ : AddCommMonoid ι
inst✝⁴ : AddCommMonoid κ
inst✝³ : DecidableEq κ
A : Type u_4
inst✝² : CommSemiring A
... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Misc.lean | GalgHom.isHomogeneous'_aeval | [356, 1] | [375, 38] | rw [← hp (mem_support_iff.mp hc), MvPolynomial.weightedDegree_apply] | case h.e'_5.h.e'_1
R : Type u_5
inst✝⁸ : CommSemiring R
S : Type ?u.178387
inst✝⁷ : CommSemiring S
inst✝⁶ : Algebra R S
σ : Type u_1
ι : Type u_2
κ : Type u_3
inst✝⁵ : AddCommMonoid ι
inst✝⁴ : AddCommMonoid κ
inst✝³ : DecidableEq κ
A : Type u_4
inst✝² : CommSemiring A
inst✝¹ : Algebra R A
𝒜 : κ → Submodule R A
inst✝ :... | case h.e'_5.h.e'_1
R : Type u_5
inst✝⁸ : CommSemiring R
S : Type ?u.178387
inst✝⁷ : CommSemiring S
inst✝⁶ : Algebra R S
σ : Type u_1
ι : Type u_2
κ : Type u_3
inst✝⁵ : AddCommMonoid ι
inst✝⁴ : AddCommMonoid κ
inst✝³ : DecidableEq κ
A : Type u_4
inst✝² : CommSemiring A
inst✝¹ : Algebra R A
𝒜 : κ → Submodule R A
inst✝ :... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_5.h.e'_1
R : Type u_5
inst✝⁸ : CommSemiring R
S : Type ?u.178387
inst✝⁷ : CommSemiring S
inst✝⁶ : Algebra R S
σ : Type u_1
ι : Type u_2
κ : Type u_3
inst✝⁵ : AddCommMonoid ι
inst✝⁴ : AddCommMonoid κ
inst✝³ : DecidableEq κ
A : Type u_4
inst✝² : CommS... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Misc.lean | GalgHom.isHomogeneous'_aeval | [356, 1] | [375, 38] | rw [Finsupp.sum, map_sum, Finsupp.sum_of_support_subset _ le_rfl] | case h.e'_5.h.e'_1
R : Type u_5
inst✝⁸ : CommSemiring R
S : Type ?u.178387
inst✝⁷ : CommSemiring S
inst✝⁶ : Algebra R S
σ : Type u_1
ι : Type u_2
κ : Type u_3
inst✝⁵ : AddCommMonoid ι
inst✝⁴ : AddCommMonoid κ
inst✝³ : DecidableEq κ
A : Type u_4
inst✝² : CommSemiring A
inst✝¹ : Algebra R A
𝒜 : κ → Submodule R A
inst✝ :... | case h.e'_5.h.e'_1
R : Type u_5
inst✝⁸ : CommSemiring R
S : Type ?u.178387
inst✝⁷ : CommSemiring S
inst✝⁶ : Algebra R S
σ : Type u_1
ι : Type u_2
κ : Type u_3
inst✝⁵ : AddCommMonoid ι
inst✝⁴ : AddCommMonoid κ
inst✝³ : DecidableEq κ
A : Type u_4
inst✝² : CommSemiring A
inst✝¹ : Algebra R A
𝒜 : κ → Submodule R A
inst✝ :... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_5.h.e'_1
R : Type u_5
inst✝⁸ : CommSemiring R
S : Type ?u.178387
inst✝⁷ : CommSemiring S
inst✝⁶ : Algebra R S
σ : Type u_1
ι : Type u_2
κ : Type u_3
inst✝⁵ : AddCommMonoid ι
inst✝⁴ : AddCommMonoid κ
inst✝³ : DecidableEq κ
A : Type u_4
inst✝² : CommS... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Misc.lean | GalgHom.isHomogeneous'_aeval | [356, 1] | [375, 38] | apply Finset.sum_congr rfl | case h.e'_5.h.e'_1
R : Type u_5
inst✝⁸ : CommSemiring R
S : Type ?u.178387
inst✝⁷ : CommSemiring S
inst✝⁶ : Algebra R S
σ : Type u_1
ι : Type u_2
κ : Type u_3
inst✝⁵ : AddCommMonoid ι
inst✝⁴ : AddCommMonoid κ
inst✝³ : DecidableEq κ
A : Type u_4
inst✝² : CommSemiring A
inst✝¹ : Algebra R A
𝒜 : κ → Submodule R A
inst✝ :... | case h.e'_5.h.e'_1
R : Type u_5
inst✝⁸ : CommSemiring R
S : Type ?u.178387
inst✝⁷ : CommSemiring S
inst✝⁶ : Algebra R S
σ : Type u_1
ι : Type u_2
κ : Type u_3
inst✝⁵ : AddCommMonoid ι
inst✝⁴ : AddCommMonoid κ
inst✝³ : DecidableEq κ
A : Type u_4
inst✝² : CommSemiring A
inst✝¹ : Algebra R A
𝒜 : κ → Submodule R A
inst✝ :... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_5.h.e'_1
R : Type u_5
inst✝⁸ : CommSemiring R
S : Type ?u.178387
inst✝⁷ : CommSemiring S
inst✝⁶ : Algebra R S
σ : Type u_1
ι : Type u_2
κ : Type u_3
inst✝⁵ : AddCommMonoid ι
inst✝⁴ : AddCommMonoid κ
inst✝³ : DecidableEq κ
A : Type u_4
inst✝² : CommS... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Misc.lean | GalgHom.isHomogeneous'_aeval | [356, 1] | [375, 38] | . intro x _ ; simp only [map_nsmul] | case h.e'_5.h.e'_1
R : Type u_5
inst✝⁸ : CommSemiring R
S : Type ?u.178387
inst✝⁷ : CommSemiring S
inst✝⁶ : Algebra R S
σ : Type u_1
ι : Type u_2
κ : Type u_3
inst✝⁵ : AddCommMonoid ι
inst✝⁴ : AddCommMonoid κ
inst✝³ : DecidableEq κ
A : Type u_4
inst✝² : CommSemiring A
inst✝¹ : Algebra R A
𝒜 : κ → Submodule R A
inst✝ :... | case h.e'_5.h.e'_1.h
R : Type u_5
inst✝⁸ : CommSemiring R
S : Type ?u.178387
inst✝⁷ : CommSemiring S
inst✝⁶ : Algebra R S
σ : Type u_1
ι : Type u_2
κ : Type u_3
inst✝⁵ : AddCommMonoid ι
inst✝⁴ : AddCommMonoid κ
inst✝³ : DecidableEq κ
A : Type u_4
inst✝² : CommSemiring A
inst✝¹ : Algebra R A
𝒜 : κ → Submodule R A
inst✝... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_5.h.e'_1
R : Type u_5
inst✝⁸ : CommSemiring R
S : Type ?u.178387
inst✝⁷ : CommSemiring S
inst✝⁶ : Algebra R S
σ : Type u_1
ι : Type u_2
κ : Type u_3
inst✝⁵ : AddCommMonoid ι
inst✝⁴ : AddCommMonoid κ
inst✝³ : DecidableEq κ
A : Type u_4
inst✝² : CommS... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Misc.lean | GalgHom.isHomogeneous'_aeval | [356, 1] | [375, 38] | . intro s _ ; simp only [zero_smul] | case h.e'_5.h.e'_1.h
R : Type u_5
inst✝⁸ : CommSemiring R
S : Type ?u.178387
inst✝⁷ : CommSemiring S
inst✝⁶ : Algebra R S
σ : Type u_1
ι : Type u_2
κ : Type u_3
inst✝⁵ : AddCommMonoid ι
inst✝⁴ : AddCommMonoid κ
inst✝³ : DecidableEq κ
A : Type u_4
inst✝² : CommSemiring A
inst✝¹ : Algebra R A
𝒜 : κ → Submodule R A
inst✝... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_5.h.e'_1.h
R : Type u_5
inst✝⁸ : CommSemiring R
S : Type ?u.178387
inst✝⁷ : CommSemiring S
inst✝⁶ : Algebra R S
σ : Type u_1
ι : Type u_2
κ : Type u_3
inst✝⁵ : AddCommMonoid ι
inst✝⁴ : AddCommMonoid κ
inst✝³ : DecidableEq κ
A : Type u_4
inst✝² : Com... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Misc.lean | GalgHom.isHomogeneous'_aeval | [356, 1] | [375, 38] | intro x _ | case h.e'_5.h.e'_1
R : Type u_5
inst✝⁸ : CommSemiring R
S : Type ?u.178387
inst✝⁷ : CommSemiring S
inst✝⁶ : Algebra R S
σ : Type u_1
ι : Type u_2
κ : Type u_3
inst✝⁵ : AddCommMonoid ι
inst✝⁴ : AddCommMonoid κ
inst✝³ : DecidableEq κ
A : Type u_4
inst✝² : CommSemiring A
inst✝¹ : Algebra R A
𝒜 : κ → Submodule R A
inst✝ :... | case h.e'_5.h.e'_1
R : Type u_5
inst✝⁸ : CommSemiring R
S : Type ?u.178387
inst✝⁷ : CommSemiring S
inst✝⁶ : Algebra R S
σ : Type u_1
ι : Type u_2
κ : Type u_3
inst✝⁵ : AddCommMonoid ι
inst✝⁴ : AddCommMonoid κ
inst✝³ : DecidableEq κ
A : Type u_4
inst✝² : CommSemiring A
inst✝¹ : Algebra R A
𝒜 : κ → Submodule R A
inst✝ :... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_5.h.e'_1
R : Type u_5
inst✝⁸ : CommSemiring R
S : Type ?u.178387
inst✝⁷ : CommSemiring S
inst✝⁶ : Algebra R S
σ : Type u_1
ι : Type u_2
κ : Type u_3
inst✝⁵ : AddCommMonoid ι
inst✝⁴ : AddCommMonoid κ
inst✝³ : DecidableEq κ
A : Type u_4
inst✝² : CommS... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Misc.lean | GalgHom.isHomogeneous'_aeval | [356, 1] | [375, 38] | simp only [map_nsmul] | case h.e'_5.h.e'_1
R : Type u_5
inst✝⁸ : CommSemiring R
S : Type ?u.178387
inst✝⁷ : CommSemiring S
inst✝⁶ : Algebra R S
σ : Type u_1
ι : Type u_2
κ : Type u_3
inst✝⁵ : AddCommMonoid ι
inst✝⁴ : AddCommMonoid κ
inst✝³ : DecidableEq κ
A : Type u_4
inst✝² : CommSemiring A
inst✝¹ : Algebra R A
𝒜 : κ → Submodule R A
inst✝ :... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_5.h.e'_1
R : Type u_5
inst✝⁸ : CommSemiring R
S : Type ?u.178387
inst✝⁷ : CommSemiring S
inst✝⁶ : Algebra R S
σ : Type u_1
ι : Type u_2
κ : Type u_3
inst✝⁵ : AddCommMonoid ι
inst✝⁴ : AddCommMonoid κ
inst✝³ : DecidableEq κ
A : Type u_4
inst✝² : CommS... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Misc.lean | GalgHom.isHomogeneous'_aeval | [356, 1] | [375, 38] | intro s _ | case h.e'_5.h.e'_1.h
R : Type u_5
inst✝⁸ : CommSemiring R
S : Type ?u.178387
inst✝⁷ : CommSemiring S
inst✝⁶ : Algebra R S
σ : Type u_1
ι : Type u_2
κ : Type u_3
inst✝⁵ : AddCommMonoid ι
inst✝⁴ : AddCommMonoid κ
inst✝³ : DecidableEq κ
A : Type u_4
inst✝² : CommSemiring A
inst✝¹ : Algebra R A
𝒜 : κ → Submodule R A
inst✝... | case h.e'_5.h.e'_1.h
R : Type u_5
inst✝⁸ : CommSemiring R
S : Type ?u.178387
inst✝⁷ : CommSemiring S
inst✝⁶ : Algebra R S
σ : Type u_1
ι : Type u_2
κ : Type u_3
inst✝⁵ : AddCommMonoid ι
inst✝⁴ : AddCommMonoid κ
inst✝³ : DecidableEq κ
A : Type u_4
inst✝² : CommSemiring A
inst✝¹ : Algebra R A
𝒜 : κ → Submodule R A
inst✝... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_5.h.e'_1.h
R : Type u_5
inst✝⁸ : CommSemiring R
S : Type ?u.178387
inst✝⁷ : CommSemiring S
inst✝⁶ : Algebra R S
σ : Type u_1
ι : Type u_2
κ : Type u_3
inst✝⁵ : AddCommMonoid ι
inst✝⁴ : AddCommMonoid κ
inst✝³ : DecidableEq κ
A : Type u_4
inst✝² : Com... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Misc.lean | GalgHom.isHomogeneous'_aeval | [356, 1] | [375, 38] | simp only [zero_smul] | case h.e'_5.h.e'_1.h
R : Type u_5
inst✝⁸ : CommSemiring R
S : Type ?u.178387
inst✝⁷ : CommSemiring S
inst✝⁶ : Algebra R S
σ : Type u_1
ι : Type u_2
κ : Type u_3
inst✝⁵ : AddCommMonoid ι
inst✝⁴ : AddCommMonoid κ
inst✝³ : DecidableEq κ
A : Type u_4
inst✝² : CommSemiring A
inst✝¹ : Algebra R A
𝒜 : κ → Submodule R A
inst✝... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_5.h.e'_1.h
R : Type u_5
inst✝⁸ : CommSemiring R
S : Type ?u.178387
inst✝⁷ : CommSemiring S
inst✝⁶ : Algebra R S
σ : Type u_1
ι : Type u_2
κ : Type u_3
inst✝⁵ : AddCommMonoid ι
inst✝⁴ : AddCommMonoid κ
inst✝³ : DecidableEq κ
A : Type u_4
inst✝² : Com... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Misc.lean | MvPolynomial.vars_X_subset | [380, 1] | [391, 28] | intro u | R : Type u_1
σ : Type u_2
n : σ
inst✝ : CommSemiring R
⊢ (X n).vars ⊆ {n} | R : Type u_1
σ : Type u_2
n : σ
inst✝ : CommSemiring R
u : σ
⊢ u ∈ (X n).vars → u ∈ {n} | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
σ : Type u_2
n : σ
inst✝ : CommSemiring R
⊢ (X n).vars ⊆ {n}
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Misc.lean | MvPolynomial.vars_X_subset | [380, 1] | [391, 28] | rw [X, mem_vars, mem_singleton] | R : Type u_1
σ : Type u_2
n : σ
inst✝ : CommSemiring R
u : σ
⊢ u ∈ (X n).vars → u ∈ {n} | R : Type u_1
σ : Type u_2
n : σ
inst✝ : CommSemiring R
u : σ
⊢ (∃ d ∈ ((monomial (Finsupp.single n 1)) 1).support, u ∈ d.support) → u = n | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
σ : Type u_2
n : σ
inst✝ : CommSemiring R
u : σ
⊢ u ∈ (X n).vars → u ∈ {n}
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Misc.lean | MvPolynomial.vars_X_subset | [380, 1] | [391, 28] | rintro ⟨c, hc, hc'⟩ | R : Type u_1
σ : Type u_2
n : σ
inst✝ : CommSemiring R
u : σ
⊢ (∃ d ∈ ((monomial (Finsupp.single n 1)) 1).support, u ∈ d.support) → u = n | case intro.intro
R : Type u_1
σ : Type u_2
n : σ
inst✝ : CommSemiring R
u : σ
c : σ →₀ ℕ
hc : c ∈ ((monomial (Finsupp.single n 1)) 1).support
hc' : u ∈ c.support
⊢ u = n | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
σ : Type u_2
n : σ
inst✝ : CommSemiring R
u : σ
⊢ (∃ d ∈ ((monomial (Finsupp.single n 1)) 1).support, u ∈ d.support) → u = n
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Misc.lean | MvPolynomial.vars_X_subset | [380, 1] | [391, 28] | by_contra h' | case intro.intro
R : Type u_1
σ : Type u_2
n : σ
inst✝ : CommSemiring R
u : σ
c : σ →₀ ℕ
hc : c ∈ ((monomial (Finsupp.single n 1)) 1).support
hc' : u ∈ c.support
⊢ u = n | case intro.intro
R : Type u_1
σ : Type u_2
n : σ
inst✝ : CommSemiring R
u : σ
c : σ →₀ ℕ
hc : c ∈ ((monomial (Finsupp.single n 1)) 1).support
hc' : u ∈ c.support
h' : ¬u = n
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
R : Type u_1
σ : Type u_2
n : σ
inst✝ : CommSemiring R
u : σ
c : σ →₀ ℕ
hc : c ∈ ((monomial (Finsupp.single n 1)) 1).support
hc' : u ∈ c.support
⊢ u = n
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Misc.lean | MvPolynomial.vars_X_subset | [380, 1] | [391, 28] | rw [mem_support_iff, coeff_monomial, ne_eq] at hc | case intro.intro
R : Type u_1
σ : Type u_2
n : σ
inst✝ : CommSemiring R
u : σ
c : σ →₀ ℕ
hc : c ∈ ((monomial (Finsupp.single n 1)) 1).support
hc' : u ∈ c.support
h' : ¬u = n
⊢ False | case intro.intro
R : Type u_1
σ : Type u_2
n : σ
inst✝ : CommSemiring R
u : σ
c : σ →₀ ℕ
hc : ¬(if Finsupp.single n 1 = c then 1 else 0) = 0
hc' : u ∈ c.support
h' : ¬u = n
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
R : Type u_1
σ : Type u_2
n : σ
inst✝ : CommSemiring R
u : σ
c : σ →₀ ℕ
hc : c ∈ ((monomial (Finsupp.single n 1)) 1).support
hc' : u ∈ c.support
h' : ¬u = n
⊢ False
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Misc.lean | MvPolynomial.vars_X_subset | [380, 1] | [391, 28] | by_cases h : Finsupp.single n 1 = c | case intro.intro
R : Type u_1
σ : Type u_2
n : σ
inst✝ : CommSemiring R
u : σ
c : σ →₀ ℕ
hc : ¬(if Finsupp.single n 1 = c then 1 else 0) = 0
hc' : u ∈ c.support
h' : ¬u = n
⊢ False | case pos
R : Type u_1
σ : Type u_2
n : σ
inst✝ : CommSemiring R
u : σ
c : σ →₀ ℕ
hc : ¬(if Finsupp.single n 1 = c then 1 else 0) = 0
hc' : u ∈ c.support
h' : ¬u = n
h : Finsupp.single n 1 = c
⊢ False
case neg
R : Type u_1
σ : Type u_2
n : σ
inst✝ : CommSemiring R
u : σ
c : σ →₀ ℕ
hc : ¬(if Finsupp.single n 1 = c then ... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
R : Type u_1
σ : Type u_2
n : σ
inst✝ : CommSemiring R
u : σ
c : σ →₀ ℕ
hc : ¬(if Finsupp.single n 1 = c then 1 else 0) = 0
hc' : u ∈ c.support
h' : ¬u = n
⊢ False
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Misc.lean | MvPolynomial.vars_X_subset | [380, 1] | [391, 28] | rw [← h, Finsupp.mem_support_iff, ne_eq, Finsupp.single_apply] at hc' | case pos
R : Type u_1
σ : Type u_2
n : σ
inst✝ : CommSemiring R
u : σ
c : σ →₀ ℕ
hc : ¬(if Finsupp.single n 1 = c then 1 else 0) = 0
hc' : u ∈ c.support
h' : ¬u = n
h : Finsupp.single n 1 = c
⊢ False | case pos
R : Type u_1
σ : Type u_2
n : σ
inst✝ : CommSemiring R
u : σ
c : σ →₀ ℕ
hc : ¬(if Finsupp.single n 1 = c then 1 else 0) = 0
hc' : ¬(if n = u then 1 else 0) = 0
h' : ¬u = n
h : Finsupp.single n 1 = c
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
R : Type u_1
σ : Type u_2
n : σ
inst✝ : CommSemiring R
u : σ
c : σ →₀ ℕ
hc : ¬(if Finsupp.single n 1 = c then 1 else 0) = 0
hc' : u ∈ c.support
h' : ¬u = n
h : Finsupp.single n 1 = c
⊢ False
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Misc.lean | MvPolynomial.vars_X_subset | [380, 1] | [391, 28] | apply hc' | case pos
R : Type u_1
σ : Type u_2
n : σ
inst✝ : CommSemiring R
u : σ
c : σ →₀ ℕ
hc : ¬(if Finsupp.single n 1 = c then 1 else 0) = 0
hc' : ¬(if n = u then 1 else 0) = 0
h' : ¬u = n
h : Finsupp.single n 1 = c
⊢ False | case pos
R : Type u_1
σ : Type u_2
n : σ
inst✝ : CommSemiring R
u : σ
c : σ →₀ ℕ
hc : ¬(if Finsupp.single n 1 = c then 1 else 0) = 0
hc' : ¬(if n = u then 1 else 0) = 0
h' : ¬u = n
h : Finsupp.single n 1 = c
⊢ (if n = u then 1 else 0) = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
R : Type u_1
σ : Type u_2
n : σ
inst✝ : CommSemiring R
u : σ
c : σ →₀ ℕ
hc : ¬(if Finsupp.single n 1 = c then 1 else 0) = 0
hc' : ¬(if n = u then 1 else 0) = 0
h' : ¬u = n
h : Finsupp.single n 1 = c
⊢ False
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Misc.lean | MvPolynomial.vars_X_subset | [380, 1] | [391, 28] | rw [if_neg (Ne.symm h')] | case pos
R : Type u_1
σ : Type u_2
n : σ
inst✝ : CommSemiring R
u : σ
c : σ →₀ ℕ
hc : ¬(if Finsupp.single n 1 = c then 1 else 0) = 0
hc' : ¬(if n = u then 1 else 0) = 0
h' : ¬u = n
h : Finsupp.single n 1 = c
⊢ (if n = u then 1 else 0) = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
R : Type u_1
σ : Type u_2
n : σ
inst✝ : CommSemiring R
u : σ
c : σ →₀ ℕ
hc : ¬(if Finsupp.single n 1 = c then 1 else 0) = 0
hc' : ¬(if n = u then 1 else 0) = 0
h' : ¬u = n
h : Finsupp.single n 1 = c
⊢ (if n = u then 1 else 0) = 0
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Misc.lean | MvPolynomial.vars_X_subset | [380, 1] | [391, 28] | apply hc | case neg
R : Type u_1
σ : Type u_2
n : σ
inst✝ : CommSemiring R
u : σ
c : σ →₀ ℕ
hc : ¬(if Finsupp.single n 1 = c then 1 else 0) = 0
hc' : u ∈ c.support
h' : ¬u = n
h : ¬Finsupp.single n 1 = c
⊢ False | case neg
R : Type u_1
σ : Type u_2
n : σ
inst✝ : CommSemiring R
u : σ
c : σ →₀ ℕ
hc : ¬(if Finsupp.single n 1 = c then 1 else 0) = 0
hc' : u ∈ c.support
h' : ¬u = n
h : ¬Finsupp.single n 1 = c
⊢ (if Finsupp.single n 1 = c then 1 else 0) = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
R : Type u_1
σ : Type u_2
n : σ
inst✝ : CommSemiring R
u : σ
c : σ →₀ ℕ
hc : ¬(if Finsupp.single n 1 = c then 1 else 0) = 0
hc' : u ∈ c.support
h' : ¬u = n
h : ¬Finsupp.single n 1 = c
⊢ False
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Misc.lean | MvPolynomial.vars_X_subset | [380, 1] | [391, 28] | rw [if_neg h] | case neg
R : Type u_1
σ : Type u_2
n : σ
inst✝ : CommSemiring R
u : σ
c : σ →₀ ℕ
hc : ¬(if Finsupp.single n 1 = c then 1 else 0) = 0
hc' : u ∈ c.support
h' : ¬u = n
h : ¬Finsupp.single n 1 = c
⊢ (if Finsupp.single n 1 = c then 1 else 0) = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
R : Type u_1
σ : Type u_2
n : σ
inst✝ : CommSemiring R
u : σ
c : σ →₀ ℕ
hc : ¬(if Finsupp.single n 1 = c then 1 else 0) = 0
hc' : u ∈ c.support
h' : ¬u = n
h : ¬Finsupp.single n 1 = c
⊢ (if Finsupp.single n 1 = c then 1 else 0) = 0
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Misc.lean | variable_mem_supported | [416, 1] | [422, 11] | rw [mem_supported] | R : Type u_2
M : Type u_1
inst✝ : CommSemiring R
nm : ℕ × M
hn : 0 < nm.1
⊢ X nm ∈ supported R {nm | 0 < nm.1} | R : Type u_2
M : Type u_1
inst✝ : CommSemiring R
nm : ℕ × M
hn : 0 < nm.1
⊢ ↑(X nm).vars ⊆ {nm | 0 < nm.1} | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
M : Type u_1
inst✝ : CommSemiring R
nm : ℕ × M
hn : 0 < nm.1
⊢ X nm ∈ supported R {nm | 0 < nm.1}
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Misc.lean | variable_mem_supported | [416, 1] | [422, 11] | refine' Set.Subset.trans (Finset.coe_subset.mpr (vars_X_subset nm)) _ | R : Type u_2
M : Type u_1
inst✝ : CommSemiring R
nm : ℕ × M
hn : 0 < nm.1
⊢ ↑(X nm).vars ⊆ {nm | 0 < nm.1} | R : Type u_2
M : Type u_1
inst✝ : CommSemiring R
nm : ℕ × M
hn : 0 < nm.1
⊢ ↑{nm} ⊆ {nm | 0 < nm.1} | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
M : Type u_1
inst✝ : CommSemiring R
nm : ℕ × M
hn : 0 < nm.1
⊢ ↑(X nm).vars ⊆ {nm | 0 < nm.1}
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Misc.lean | variable_mem_supported | [416, 1] | [422, 11] | rw [coe_singleton, Set.singleton_subset_iff, Set.mem_setOf_eq] | R : Type u_2
M : Type u_1
inst✝ : CommSemiring R
nm : ℕ × M
hn : 0 < nm.1
⊢ ↑{nm} ⊆ {nm | 0 < nm.1} | R : Type u_2
M : Type u_1
inst✝ : CommSemiring R
nm : ℕ × M
hn : 0 < nm.1
⊢ 0 < nm.1 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
M : Type u_1
inst✝ : CommSemiring R
nm : ℕ × M
hn : 0 < nm.1
⊢ ↑{nm} ⊆ {nm | 0 < nm.1}
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Misc.lean | variable_mem_supported | [416, 1] | [422, 11] | exact hn | R : Type u_2
M : Type u_1
inst✝ : CommSemiring R
nm : ℕ × M
hn : 0 < nm.1
⊢ 0 < nm.1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
M : Type u_1
inst✝ : CommSemiring R
nm : ℕ × M
hn : 0 < nm.1
⊢ 0 < nm.1
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Misc.lean | toSupported_isHomogeneous' | [430, 1] | [457, 38] | have heq : aeval
((supported R {nm : ℕ × M | 0 < nm.fst}).val.toFun ∘
fun nm : ℕ × M =>
if h : 0 < nm.fst
then ⟨X nm, variable_mem_supported R nm h⟩
else 1) =
(supported R {nm : ℕ × M | 0 < nm.fst}).val.comp (toSupported R) :=
by
apply MvPolynomial.algHom_ext
intro nm
simp on... | R : Type u_2
M : Type u_1
inst✝² : CommSemiring R
inst✝¹ : DecidableEq M
inst✝ : DecidableEq R
⊢ GalgHom.isHomogeneous' (MvPolynomial (ℕ × M) R) (weightedHomogeneousSubmodule R Prod.fst) (MvPolynomial (ℕ × M) R)
(weightedHomogeneousSubmodule R Prod.fst) id ((supported R {nm | 0 < nm.1}).val.comp (toSupported R)) | R : Type u_2
M : Type u_1
inst✝² : CommSemiring R
inst✝¹ : DecidableEq M
inst✝ : DecidableEq R
heq :
aeval ((↑↑(supported R {nm | 0 < nm.1}).val.toRingHom).toFun ∘ fun nm => if h : 0 < nm.1 then ⟨X nm, ⋯⟩ else 1) =
(supported R {nm | 0 < nm.1}).val.comp (toSupported R)
⊢ GalgHom.isHomogeneous' (MvPolynomial (ℕ × ... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
M : Type u_1
inst✝² : CommSemiring R
inst✝¹ : DecidableEq M
inst✝ : DecidableEq R
⊢ GalgHom.isHomogeneous' (MvPolynomial (ℕ × M) R) (weightedHomogeneousSubmodule R Prod.fst) (MvPolynomial (ℕ × M) R)
(weightedHomogeneousSubmodule R Prod.fst) i... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Misc.lean | toSupported_isHomogeneous' | [430, 1] | [457, 38] | rw [← heq] | R : Type u_2
M : Type u_1
inst✝² : CommSemiring R
inst✝¹ : DecidableEq M
inst✝ : DecidableEq R
heq :
aeval ((↑↑(supported R {nm | 0 < nm.1}).val.toRingHom).toFun ∘ fun nm => if h : 0 < nm.1 then ⟨X nm, ⋯⟩ else 1) =
(supported R {nm | 0 < nm.1}).val.comp (toSupported R)
⊢ GalgHom.isHomogeneous' (MvPolynomial (ℕ × ... | R : Type u_2
M : Type u_1
inst✝² : CommSemiring R
inst✝¹ : DecidableEq M
inst✝ : DecidableEq R
heq :
aeval ((↑↑(supported R {nm | 0 < nm.1}).val.toRingHom).toFun ∘ fun nm => if h : 0 < nm.1 then ⟨X nm, ⋯⟩ else 1) =
(supported R {nm | 0 < nm.1}).val.comp (toSupported R)
⊢ GalgHom.isHomogeneous' (MvPolynomial (ℕ × ... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
M : Type u_1
inst✝² : CommSemiring R
inst✝¹ : DecidableEq M
inst✝ : DecidableEq R
heq :
aeval ((↑↑(supported R {nm | 0 < nm.1}).val.toRingHom).toFun ∘ fun nm => if h : 0 < nm.1 then ⟨X nm, ⋯⟩ else 1) =
(supported R {nm | 0 < nm.1}).val.comp... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Misc.lean | toSupported_isHomogeneous' | [430, 1] | [457, 38] | apply GalgHom.isHomogeneous'_aeval (ℕ × M) (MvPolynomial (ℕ × M) R)
(weightedHomogeneousSubmodule R Prod.fst) Prod.fst (AddMonoidHom.id ℕ) | R : Type u_2
M : Type u_1
inst✝² : CommSemiring R
inst✝¹ : DecidableEq M
inst✝ : DecidableEq R
heq :
aeval ((↑↑(supported R {nm | 0 < nm.1}).val.toRingHom).toFun ∘ fun nm => if h : 0 < nm.1 then ⟨X nm, ⋯⟩ else 1) =
(supported R {nm | 0 < nm.1}).val.comp (toSupported R)
⊢ GalgHom.isHomogeneous' (MvPolynomial (ℕ × ... | case h
R : Type u_2
M : Type u_1
inst✝² : CommSemiring R
inst✝¹ : DecidableEq M
inst✝ : DecidableEq R
heq :
aeval ((↑↑(supported R {nm | 0 < nm.1}).val.toRingHom).toFun ∘ fun nm => if h : 0 < nm.1 then ⟨X nm, ⋯⟩ else 1) =
(supported R {nm | 0 < nm.1}).val.comp (toSupported R)
⊢ ∀ (s : ℕ × M),
((↑↑(supported R... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
M : Type u_1
inst✝² : CommSemiring R
inst✝¹ : DecidableEq M
inst✝ : DecidableEq R
heq :
aeval ((↑↑(supported R {nm | 0 < nm.1}).val.toRingHom).toFun ∘ fun nm => if h : 0 < nm.1 then ⟨X nm, ⋯⟩ else 1) =
(supported R {nm | 0 < nm.1}).val.comp... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Misc.lean | toSupported_isHomogeneous' | [430, 1] | [457, 38] | apply MvPolynomial.algHom_ext | R : Type u_2
M : Type u_1
inst✝² : CommSemiring R
inst✝¹ : DecidableEq M
inst✝ : DecidableEq R
⊢ aeval ((↑↑(supported R {nm | 0 < nm.1}).val.toRingHom).toFun ∘ fun nm => if h : 0 < nm.1 then ⟨X nm, ⋯⟩ else 1) =
(supported R {nm | 0 < nm.1}).val.comp (toSupported R) | case hf
R : Type u_2
M : Type u_1
inst✝² : CommSemiring R
inst✝¹ : DecidableEq M
inst✝ : DecidableEq R
⊢ ∀ (i : ℕ × M),
(aeval ((↑↑(supported R {nm | 0 < nm.1}).val.toRingHom).toFun ∘ fun nm => if h : 0 < nm.1 then ⟨X nm, ⋯⟩ else 1))
(X i) =
((supported R {nm | 0 < nm.1}).val.comp (toSupported R)) (X ... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
M : Type u_1
inst✝² : CommSemiring R
inst✝¹ : DecidableEq M
inst✝ : DecidableEq R
⊢ aeval ((↑↑(supported R {nm | 0 < nm.1}).val.toRingHom).toFun ∘ fun nm => if h : 0 < nm.1 then ⟨X nm, ⋯⟩ else 1) =
(supported R {nm | 0 < nm.1}).val.comp (toSu... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Misc.lean | toSupported_isHomogeneous' | [430, 1] | [457, 38] | intro nm | case hf
R : Type u_2
M : Type u_1
inst✝² : CommSemiring R
inst✝¹ : DecidableEq M
inst✝ : DecidableEq R
⊢ ∀ (i : ℕ × M),
(aeval ((↑↑(supported R {nm | 0 < nm.1}).val.toRingHom).toFun ∘ fun nm => if h : 0 < nm.1 then ⟨X nm, ⋯⟩ else 1))
(X i) =
((supported R {nm | 0 < nm.1}).val.comp (toSupported R)) (X ... | case hf
R : Type u_2
M : Type u_1
inst✝² : CommSemiring R
inst✝¹ : DecidableEq M
inst✝ : DecidableEq R
nm : ℕ × M
⊢ (aeval ((↑↑(supported R {nm | 0 < nm.1}).val.toRingHom).toFun ∘ fun nm => if h : 0 < nm.1 then ⟨X nm, ⋯⟩ else 1))
(X nm) =
((supported R {nm | 0 < nm.1}).val.comp (toSupported R)) (X nm) | Please generate a tactic in lean4 to solve the state.
STATE:
case hf
R : Type u_2
M : Type u_1
inst✝² : CommSemiring R
inst✝¹ : DecidableEq M
inst✝ : DecidableEq R
⊢ ∀ (i : ℕ × M),
(aeval ((↑↑(supported R {nm | 0 < nm.1}).val.toRingHom).toFun ∘ fun nm => if h : 0 < nm.1 then ⟨X nm, ⋯⟩ else 1))
(X i) =
... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Misc.lean | toSupported_isHomogeneous' | [430, 1] | [457, 38] | simp only [toSupported, AlgHom.toFun_eq_coe, Function.comp_apply, AlgHom.coe_comp, aeval_X] | case hf
R : Type u_2
M : Type u_1
inst✝² : CommSemiring R
inst✝¹ : DecidableEq M
inst✝ : DecidableEq R
nm : ℕ × M
⊢ (aeval ((↑↑(supported R {nm | 0 < nm.1}).val.toRingHom).toFun ∘ fun nm => if h : 0 < nm.1 then ⟨X nm, ⋯⟩ else 1))
(X nm) =
((supported R {nm | 0 < nm.1}).val.comp (toSupported R)) (X nm) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hf
R : Type u_2
M : Type u_1
inst✝² : CommSemiring R
inst✝¹ : DecidableEq M
inst✝ : DecidableEq R
nm : ℕ × M
⊢ (aeval ((↑↑(supported R {nm | 0 < nm.1}).val.toRingHom).toFun ∘ fun nm => if h : 0 < nm.1 then ⟨X nm, ⋯⟩ else 1))
(X nm) =
((supporte... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Misc.lean | toSupported_isHomogeneous' | [430, 1] | [457, 38] | intro nm | case h
R : Type u_2
M : Type u_1
inst✝² : CommSemiring R
inst✝¹ : DecidableEq M
inst✝ : DecidableEq R
heq :
aeval ((↑↑(supported R {nm | 0 < nm.1}).val.toRingHom).toFun ∘ fun nm => if h : 0 < nm.1 then ⟨X nm, ⋯⟩ else 1) =
(supported R {nm | 0 < nm.1}).val.comp (toSupported R)
⊢ ∀ (s : ℕ × M),
((↑↑(supported R... | case h
R : Type u_2
M : Type u_1
inst✝² : CommSemiring R
inst✝¹ : DecidableEq M
inst✝ : DecidableEq R
heq :
aeval ((↑↑(supported R {nm | 0 < nm.1}).val.toRingHom).toFun ∘ fun nm => if h : 0 < nm.1 then ⟨X nm, ⋯⟩ else 1) =
(supported R {nm | 0 < nm.1}).val.comp (toSupported R)
nm : ℕ × M
⊢ ((↑↑(supported R {nm | 0... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
R : Type u_2
M : Type u_1
inst✝² : CommSemiring R
inst✝¹ : DecidableEq M
inst✝ : DecidableEq R
heq :
aeval ((↑↑(supported R {nm | 0 < nm.1}).val.toRingHom).toFun ∘ fun nm => if h : 0 < nm.1 then ⟨X nm, ⋯⟩ else 1) =
(supported R {nm | 0 < nm.1}).v... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Misc.lean | toSupported_isHomogeneous' | [430, 1] | [457, 38] | simp only [mem_weightedHomogeneousSubmodule, AlgHom.toFun_eq_coe, Subalgebra.coe_val,
Function.comp_apply, AddMonoidHom.id_apply] | case h
R : Type u_2
M : Type u_1
inst✝² : CommSemiring R
inst✝¹ : DecidableEq M
inst✝ : DecidableEq R
heq :
aeval ((↑↑(supported R {nm | 0 < nm.1}).val.toRingHom).toFun ∘ fun nm => if h : 0 < nm.1 then ⟨X nm, ⋯⟩ else 1) =
(supported R {nm | 0 < nm.1}).val.comp (toSupported R)
nm : ℕ × M
⊢ ((↑↑(supported R {nm | 0... | case h
R : Type u_2
M : Type u_1
inst✝² : CommSemiring R
inst✝¹ : DecidableEq M
inst✝ : DecidableEq R
heq :
aeval ((↑↑(supported R {nm | 0 < nm.1}).val.toRingHom).toFun ∘ fun nm => if h : 0 < nm.1 then ⟨X nm, ⋯⟩ else 1) =
(supported R {nm | 0 < nm.1}).val.comp (toSupported R)
nm : ℕ × M
⊢ IsWeightedHomogeneous Pr... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
R : Type u_2
M : Type u_1
inst✝² : CommSemiring R
inst✝¹ : DecidableEq M
inst✝ : DecidableEq R
heq :
aeval ((↑↑(supported R {nm | 0 < nm.1}).val.toRingHom).toFun ∘ fun nm => if h : 0 < nm.1 then ⟨X nm, ⋯⟩ else 1) =
(supported R {nm | 0 < nm.1}).v... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Misc.lean | toSupported_isHomogeneous' | [430, 1] | [457, 38] | split_ifs with h | case h
R : Type u_2
M : Type u_1
inst✝² : CommSemiring R
inst✝¹ : DecidableEq M
inst✝ : DecidableEq R
heq :
aeval ((↑↑(supported R {nm | 0 < nm.1}).val.toRingHom).toFun ∘ fun nm => if h : 0 < nm.1 then ⟨X nm, ⋯⟩ else 1) =
(supported R {nm | 0 < nm.1}).val.comp (toSupported R)
nm : ℕ × M
⊢ IsWeightedHomogeneous Pr... | case pos
R : Type u_2
M : Type u_1
inst✝² : CommSemiring R
inst✝¹ : DecidableEq M
inst✝ : DecidableEq R
heq :
aeval ((↑↑(supported R {nm | 0 < nm.1}).val.toRingHom).toFun ∘ fun nm => if h : 0 < nm.1 then ⟨X nm, ⋯⟩ else 1) =
(supported R {nm | 0 < nm.1}).val.comp (toSupported R)
nm : ℕ × M
h : 0 < nm.1
⊢ IsWeighte... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
R : Type u_2
M : Type u_1
inst✝² : CommSemiring R
inst✝¹ : DecidableEq M
inst✝ : DecidableEq R
heq :
aeval ((↑↑(supported R {nm | 0 < nm.1}).val.toRingHom).toFun ∘ fun nm => if h : 0 < nm.1 then ⟨X nm, ⋯⟩ else 1) =
(supported R {nm | 0 < nm.1}).v... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Misc.lean | toSupported_isHomogeneous' | [430, 1] | [457, 38] | apply isWeightedHomogeneous_X | case pos
R : Type u_2
M : Type u_1
inst✝² : CommSemiring R
inst✝¹ : DecidableEq M
inst✝ : DecidableEq R
heq :
aeval ((↑↑(supported R {nm | 0 < nm.1}).val.toRingHom).toFun ∘ fun nm => if h : 0 < nm.1 then ⟨X nm, ⋯⟩ else 1) =
(supported R {nm | 0 < nm.1}).val.comp (toSupported R)
nm : ℕ × M
h : 0 < nm.1
⊢ IsWeighte... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
R : Type u_2
M : Type u_1
inst✝² : CommSemiring R
inst✝¹ : DecidableEq M
inst✝ : DecidableEq R
heq :
aeval ((↑↑(supported R {nm | 0 < nm.1}).val.toRingHom).toFun ∘ fun nm => if h : 0 < nm.1 then ⟨X nm, ⋯⟩ else 1) =
(supported R {nm | 0 < nm.1})... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Misc.lean | toSupported_isHomogeneous' | [430, 1] | [457, 38] | simp only [not_lt, le_zero_iff] at h | case neg
R : Type u_2
M : Type u_1
inst✝² : CommSemiring R
inst✝¹ : DecidableEq M
inst✝ : DecidableEq R
heq :
aeval ((↑↑(supported R {nm | 0 < nm.1}).val.toRingHom).toFun ∘ fun nm => if h : 0 < nm.1 then ⟨X nm, ⋯⟩ else 1) =
(supported R {nm | 0 < nm.1}).val.comp (toSupported R)
nm : ℕ × M
h : ¬0 < nm.1
⊢ IsWeight... | case neg
R : Type u_2
M : Type u_1
inst✝² : CommSemiring R
inst✝¹ : DecidableEq M
inst✝ : DecidableEq R
heq :
aeval ((↑↑(supported R {nm | 0 < nm.1}).val.toRingHom).toFun ∘ fun nm => if h : 0 < nm.1 then ⟨X nm, ⋯⟩ else 1) =
(supported R {nm | 0 < nm.1}).val.comp (toSupported R)
nm : ℕ × M
h : nm.1 = 0
⊢ IsWeighte... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
R : Type u_2
M : Type u_1
inst✝² : CommSemiring R
inst✝¹ : DecidableEq M
inst✝ : DecidableEq R
heq :
aeval ((↑↑(supported R {nm | 0 < nm.1}).val.toRingHom).toFun ∘ fun nm => if h : 0 < nm.1 then ⟨X nm, ⋯⟩ else 1) =
(supported R {nm | 0 < nm.1})... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Misc.lean | toSupported_isHomogeneous' | [430, 1] | [457, 38] | rw [h, OneMemClass.coe_one] | case neg
R : Type u_2
M : Type u_1
inst✝² : CommSemiring R
inst✝¹ : DecidableEq M
inst✝ : DecidableEq R
heq :
aeval ((↑↑(supported R {nm | 0 < nm.1}).val.toRingHom).toFun ∘ fun nm => if h : 0 < nm.1 then ⟨X nm, ⋯⟩ else 1) =
(supported R {nm | 0 < nm.1}).val.comp (toSupported R)
nm : ℕ × M
h : nm.1 = 0
⊢ IsWeighte... | case neg
R : Type u_2
M : Type u_1
inst✝² : CommSemiring R
inst✝¹ : DecidableEq M
inst✝ : DecidableEq R
heq :
aeval ((↑↑(supported R {nm | 0 < nm.1}).val.toRingHom).toFun ∘ fun nm => if h : 0 < nm.1 then ⟨X nm, ⋯⟩ else 1) =
(supported R {nm | 0 < nm.1}).val.comp (toSupported R)
nm : ℕ × M
h : nm.1 = 0
⊢ IsWeighte... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
R : Type u_2
M : Type u_1
inst✝² : CommSemiring R
inst✝¹ : DecidableEq M
inst✝ : DecidableEq R
heq :
aeval ((↑↑(supported R {nm | 0 < nm.1}).val.toRingHom).toFun ∘ fun nm => if h : 0 < nm.1 then ⟨X nm, ⋯⟩ else 1) =
(supported R {nm | 0 < nm.1})... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Misc.lean | toSupported_isHomogeneous' | [430, 1] | [457, 38] | apply isWeightedHomogeneous_one | case neg
R : Type u_2
M : Type u_1
inst✝² : CommSemiring R
inst✝¹ : DecidableEq M
inst✝ : DecidableEq R
heq :
aeval ((↑↑(supported R {nm | 0 < nm.1}).val.toRingHom).toFun ∘ fun nm => if h : 0 < nm.1 then ⟨X nm, ⋯⟩ else 1) =
(supported R {nm | 0 < nm.1}).val.comp (toSupported R)
nm : ℕ × M
h : nm.1 = 0
⊢ IsWeighte... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
R : Type u_2
M : Type u_1
inst✝² : CommSemiring R
inst✝¹ : DecidableEq M
inst✝ : DecidableEq R
heq :
aeval ((↑↑(supported R {nm | 0 < nm.1}).val.toRingHom).toFun ∘ fun nm => if h : 0 < nm.1 then ⟨X nm, ⋯⟩ else 1) =
(supported R {nm | 0 < nm.1})... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Misc.lean | eq_finsupp_single_of_degree_one | [463, 1] | [496, 35] | rw [weightedDegree_apply, Finsupp.sum] at hd | R : Type ?u.209305
M : Type u_1
inst✝¹ : CommSemiring R
inst✝ : DecidableEq M
d : ℕ × M →₀ ℕ
hd : (weightedDegree Prod.fst) d = 1
hsupp : ∀ nm ∈ d.support, 0 < nm.1
⊢ ∃ m, Finsupp.single (1, m) 1 = d | R : Type ?u.209305
M : Type u_1
inst✝¹ : CommSemiring R
inst✝ : DecidableEq M
d : ℕ × M →₀ ℕ
hd : ∑ a ∈ d.support, d a • a.1 = 1
hsupp : ∀ nm ∈ d.support, 0 < nm.1
⊢ ∃ m, Finsupp.single (1, m) 1 = d | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type ?u.209305
M : Type u_1
inst✝¹ : CommSemiring R
inst✝ : DecidableEq M
d : ℕ × M →₀ ℕ
hd : (weightedDegree Prod.fst) d = 1
hsupp : ∀ nm ∈ d.support, 0 < nm.1
⊢ ∃ m, Finsupp.single (1, m) 1 = d
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Misc.lean | eq_finsupp_single_of_degree_one | [463, 1] | [496, 35] | have hnm : ∃ nm : ℕ × M, d nm • nm.fst = 1 := by
by_contra h0
rw [not_exists] at h0
have hd0 : (d.support.sum fun a : ℕ × M => d a • a.fst) = 0 := by
rw [Finset.sum_eq_zero]
intro nm hnm
rw [← Nat.lt_one_iff]
apply lt_of_le_of_ne _ (h0 nm)
rw [← hd]
exact Finset.single_le_sum (fun x _ => z... | R : Type ?u.209305
M : Type u_1
inst✝¹ : CommSemiring R
inst✝ : DecidableEq M
d : ℕ × M →₀ ℕ
hd : ∑ a ∈ d.support, d a • a.1 = 1
hsupp : ∀ nm ∈ d.support, 0 < nm.1
⊢ ∃ m, Finsupp.single (1, m) 1 = d | R : Type ?u.209305
M : Type u_1
inst✝¹ : CommSemiring R
inst✝ : DecidableEq M
d : ℕ × M →₀ ℕ
hd : ∑ a ∈ d.support, d a • a.1 = 1
hsupp : ∀ nm ∈ d.support, 0 < nm.1
hnm : ∃ nm, d nm • nm.1 = 1
⊢ ∃ m, Finsupp.single (1, m) 1 = d | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type ?u.209305
M : Type u_1
inst✝¹ : CommSemiring R
inst✝ : DecidableEq M
d : ℕ × M →₀ ℕ
hd : ∑ a ∈ d.support, d a • a.1 = 1
hsupp : ∀ nm ∈ d.support, 0 < nm.1
⊢ ∃ m, Finsupp.single (1, m) 1 = d
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Misc.lean | eq_finsupp_single_of_degree_one | [463, 1] | [496, 35] | obtain ⟨nm, hnm⟩ := hnm | R : Type ?u.209305
M : Type u_1
inst✝¹ : CommSemiring R
inst✝ : DecidableEq M
d : ℕ × M →₀ ℕ
hd : ∑ a ∈ d.support, d a • a.1 = 1
hsupp : ∀ nm ∈ d.support, 0 < nm.1
hnm : ∃ nm, d nm • nm.1 = 1
⊢ ∃ m, Finsupp.single (1, m) 1 = d | case intro
R : Type ?u.209305
M : Type u_1
inst✝¹ : CommSemiring R
inst✝ : DecidableEq M
d : ℕ × M →₀ ℕ
hd : ∑ a ∈ d.support, d a • a.1 = 1
hsupp : ∀ nm ∈ d.support, 0 < nm.1
nm : ℕ × M
hnm : d nm • nm.1 = 1
⊢ ∃ m, Finsupp.single (1, m) 1 = d | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type ?u.209305
M : Type u_1
inst✝¹ : CommSemiring R
inst✝ : DecidableEq M
d : ℕ × M →₀ ℕ
hd : ∑ a ∈ d.support, d a • a.1 = 1
hsupp : ∀ nm ∈ d.support, 0 < nm.1
hnm : ∃ nm, d nm • nm.1 = 1
⊢ ∃ m, Finsupp.single (1, m) 1 = d
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Misc.lean | eq_finsupp_single_of_degree_one | [463, 1] | [496, 35] | rw [← hnm] at hd | case intro
R : Type ?u.209305
M : Type u_1
inst✝¹ : CommSemiring R
inst✝ : DecidableEq M
d : ℕ × M →₀ ℕ
hd : ∑ a ∈ d.support, d a • a.1 = 1
hsupp : ∀ nm ∈ d.support, 0 < nm.1
nm : ℕ × M
hnm : d nm • nm.1 = 1
⊢ ∃ m, Finsupp.single (1, m) 1 = d | case intro
R : Type ?u.209305
M : Type u_1
inst✝¹ : CommSemiring R
inst✝ : DecidableEq M
d : ℕ × M →₀ ℕ
hsupp : ∀ nm ∈ d.support, 0 < nm.1
nm : ℕ × M
hd : ∑ a ∈ d.support, d a • a.1 = d nm • nm.1
hnm : d nm • nm.1 = 1
⊢ ∃ m, Finsupp.single (1, m) 1 = d | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
R : Type ?u.209305
M : Type u_1
inst✝¹ : CommSemiring R
inst✝ : DecidableEq M
d : ℕ × M →₀ ℕ
hd : ∑ a ∈ d.support, d a • a.1 = 1
hsupp : ∀ nm ∈ d.support, 0 < nm.1
nm : ℕ × M
hnm : d nm • nm.1 = 1
⊢ ∃ m, Finsupp.single (1, m) 1 = d
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Misc.lean | eq_finsupp_single_of_degree_one | [463, 1] | [496, 35] | simp only [Algebra.id.smul_eq_mul, mul_eq_one] at hnm | case intro
R : Type ?u.209305
M : Type u_1
inst✝¹ : CommSemiring R
inst✝ : DecidableEq M
d : ℕ × M →₀ ℕ
hsupp : ∀ nm ∈ d.support, 0 < nm.1
nm : ℕ × M
hd : ∑ a ∈ d.support, d a • a.1 = d nm • nm.1
hnm : d nm • nm.1 = 1
⊢ ∃ m, Finsupp.single (1, m) 1 = d | case intro
R : Type ?u.209305
M : Type u_1
inst✝¹ : CommSemiring R
inst✝ : DecidableEq M
d : ℕ × M →₀ ℕ
hsupp : ∀ nm ∈ d.support, 0 < nm.1
nm : ℕ × M
hd : ∑ a ∈ d.support, d a • a.1 = d nm • nm.1
hnm : d nm = 1 ∧ nm.1 = 1
⊢ ∃ m, Finsupp.single (1, m) 1 = d | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
R : Type ?u.209305
M : Type u_1
inst✝¹ : CommSemiring R
inst✝ : DecidableEq M
d : ℕ × M →₀ ℕ
hsupp : ∀ nm ∈ d.support, 0 < nm.1
nm : ℕ × M
hd : ∑ a ∈ d.support, d a • a.1 = d nm • nm.1
hnm : d nm • nm.1 = 1
⊢ ∃ m, Finsupp.single (1, m) 1 = d
TACTIC... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Misc.lean | eq_finsupp_single_of_degree_one | [463, 1] | [496, 35] | use nm.snd | case intro
R : Type ?u.209305
M : Type u_1
inst✝¹ : CommSemiring R
inst✝ : DecidableEq M
d : ℕ × M →₀ ℕ
hsupp : ∀ nm ∈ d.support, 0 < nm.1
nm : ℕ × M
hd : ∑ a ∈ d.support, d a • a.1 = d nm • nm.1
hnm : d nm = 1 ∧ nm.1 = 1
⊢ ∃ m, Finsupp.single (1, m) 1 = d | case h
R : Type ?u.209305
M : Type u_1
inst✝¹ : CommSemiring R
inst✝ : DecidableEq M
d : ℕ × M →₀ ℕ
hsupp : ∀ nm ∈ d.support, 0 < nm.1
nm : ℕ × M
hd : ∑ a ∈ d.support, d a • a.1 = d nm • nm.1
hnm : d nm = 1 ∧ nm.1 = 1
⊢ Finsupp.single (1, nm.2) 1 = d | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
R : Type ?u.209305
M : Type u_1
inst✝¹ : CommSemiring R
inst✝ : DecidableEq M
d : ℕ × M →₀ ℕ
hsupp : ∀ nm ∈ d.support, 0 < nm.1
nm : ℕ × M
hd : ∑ a ∈ d.support, d a • a.1 = d nm • nm.1
hnm : d nm = 1 ∧ nm.1 = 1
⊢ ∃ m, Finsupp.single (1, m) 1 = d
TA... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Misc.lean | eq_finsupp_single_of_degree_one | [463, 1] | [496, 35] | ext ab | case h
R : Type ?u.209305
M : Type u_1
inst✝¹ : CommSemiring R
inst✝ : DecidableEq M
d : ℕ × M →₀ ℕ
hsupp : ∀ nm ∈ d.support, 0 < nm.1
nm : ℕ × M
hd : ∑ a ∈ d.support, d a • a.1 = d nm • nm.1
hnm : d nm = 1 ∧ nm.1 = 1
⊢ Finsupp.single (1, nm.2) 1 = d | case h.h
R : Type ?u.209305
M : Type u_1
inst✝¹ : CommSemiring R
inst✝ : DecidableEq M
d : ℕ × M →₀ ℕ
hsupp : ∀ nm ∈ d.support, 0 < nm.1
nm : ℕ × M
hd : ∑ a ∈ d.support, d a • a.1 = d nm • nm.1
hnm : d nm = 1 ∧ nm.1 = 1
ab : ℕ × M
⊢ (Finsupp.single (1, nm.2) 1) ab = d ab | Please generate a tactic in lean4 to solve the state.
STATE:
case h
R : Type ?u.209305
M : Type u_1
inst✝¹ : CommSemiring R
inst✝ : DecidableEq M
d : ℕ × M →₀ ℕ
hsupp : ∀ nm ∈ d.support, 0 < nm.1
nm : ℕ × M
hd : ∑ a ∈ d.support, d a • a.1 = d nm • nm.1
hnm : d nm = 1 ∧ nm.1 = 1
⊢ Finsupp.single (1, nm.2) 1 = d
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Misc.lean | eq_finsupp_single_of_degree_one | [463, 1] | [496, 35] | rw [Finsupp.single_apply] | case h.h
R : Type ?u.209305
M : Type u_1
inst✝¹ : CommSemiring R
inst✝ : DecidableEq M
d : ℕ × M →₀ ℕ
hsupp : ∀ nm ∈ d.support, 0 < nm.1
nm : ℕ × M
hd : ∑ a ∈ d.support, d a • a.1 = d nm • nm.1
hnm : d nm = 1 ∧ nm.1 = 1
ab : ℕ × M
⊢ (Finsupp.single (1, nm.2) 1) ab = d ab | case h.h
R : Type ?u.209305
M : Type u_1
inst✝¹ : CommSemiring R
inst✝ : DecidableEq M
d : ℕ × M →₀ ℕ
hsupp : ∀ nm ∈ d.support, 0 < nm.1
nm : ℕ × M
hd : ∑ a ∈ d.support, d a • a.1 = d nm • nm.1
hnm : d nm = 1 ∧ nm.1 = 1
ab : ℕ × M
⊢ (if (1, nm.2) = ab then 1 else 0) = d ab | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h
R : Type ?u.209305
M : Type u_1
inst✝¹ : CommSemiring R
inst✝ : DecidableEq M
d : ℕ × M →₀ ℕ
hsupp : ∀ nm ∈ d.support, 0 < nm.1
nm : ℕ × M
hd : ∑ a ∈ d.support, d a • a.1 = d nm • nm.1
hnm : d nm = 1 ∧ nm.1 = 1
ab : ℕ × M
⊢ (Finsupp.single (1, nm.2) ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Misc.lean | eq_finsupp_single_of_degree_one | [463, 1] | [496, 35] | split_ifs with hab <;> rw [← hnm.2, eq_comm, Prod.mk.eta] at hab | case h.h
R : Type ?u.209305
M : Type u_1
inst✝¹ : CommSemiring R
inst✝ : DecidableEq M
d : ℕ × M →₀ ℕ
hsupp : ∀ nm ∈ d.support, 0 < nm.1
nm : ℕ × M
hd : ∑ a ∈ d.support, d a • a.1 = d nm • nm.1
hnm : d nm = 1 ∧ nm.1 = 1
ab : ℕ × M
⊢ (if (1, nm.2) = ab then 1 else 0) = d ab | case pos
R : Type ?u.209305
M : Type u_1
inst✝¹ : CommSemiring R
inst✝ : DecidableEq M
d : ℕ × M →₀ ℕ
hsupp : ∀ nm ∈ d.support, 0 < nm.1
nm : ℕ × M
hd : ∑ a ∈ d.support, d a • a.1 = d nm • nm.1
hnm : d nm = 1 ∧ nm.1 = 1
ab : ℕ × M
hab : ab = nm
⊢ 1 = d ab
case neg
R : Type ?u.209305
M : Type u_1
inst✝¹ : CommSemiring ... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h
R : Type ?u.209305
M : Type u_1
inst✝¹ : CommSemiring R
inst✝ : DecidableEq M
d : ℕ × M →₀ ℕ
hsupp : ∀ nm ∈ d.support, 0 < nm.1
nm : ℕ × M
hd : ∑ a ∈ d.support, d a • a.1 = d nm • nm.1
hnm : d nm = 1 ∧ nm.1 = 1
ab : ℕ × M
⊢ (if (1, nm.2) = ab then 1 ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Misc.lean | eq_finsupp_single_of_degree_one | [463, 1] | [496, 35] | by_contra h0 | R : Type ?u.209305
M : Type u_1
inst✝¹ : CommSemiring R
inst✝ : DecidableEq M
d : ℕ × M →₀ ℕ
hd : ∑ a ∈ d.support, d a • a.1 = 1
hsupp : ∀ nm ∈ d.support, 0 < nm.1
⊢ ∃ nm, d nm • nm.1 = 1 | R : Type ?u.209305
M : Type u_1
inst✝¹ : CommSemiring R
inst✝ : DecidableEq M
d : ℕ × M →₀ ℕ
hd : ∑ a ∈ d.support, d a • a.1 = 1
hsupp : ∀ nm ∈ d.support, 0 < nm.1
h0 : ¬∃ nm, d nm • nm.1 = 1
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type ?u.209305
M : Type u_1
inst✝¹ : CommSemiring R
inst✝ : DecidableEq M
d : ℕ × M →₀ ℕ
hd : ∑ a ∈ d.support, d a • a.1 = 1
hsupp : ∀ nm ∈ d.support, 0 < nm.1
⊢ ∃ nm, d nm • nm.1 = 1
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Misc.lean | eq_finsupp_single_of_degree_one | [463, 1] | [496, 35] | rw [not_exists] at h0 | R : Type ?u.209305
M : Type u_1
inst✝¹ : CommSemiring R
inst✝ : DecidableEq M
d : ℕ × M →₀ ℕ
hd : ∑ a ∈ d.support, d a • a.1 = 1
hsupp : ∀ nm ∈ d.support, 0 < nm.1
h0 : ¬∃ nm, d nm • nm.1 = 1
⊢ False | R : Type ?u.209305
M : Type u_1
inst✝¹ : CommSemiring R
inst✝ : DecidableEq M
d : ℕ × M →₀ ℕ
hd : ∑ a ∈ d.support, d a • a.1 = 1
hsupp : ∀ nm ∈ d.support, 0 < nm.1
h0 : ∀ (x : ℕ × M), ¬d x • x.1 = 1
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type ?u.209305
M : Type u_1
inst✝¹ : CommSemiring R
inst✝ : DecidableEq M
d : ℕ × M →₀ ℕ
hd : ∑ a ∈ d.support, d a • a.1 = 1
hsupp : ∀ nm ∈ d.support, 0 < nm.1
h0 : ¬∃ nm, d nm • nm.1 = 1
⊢ False
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Misc.lean | eq_finsupp_single_of_degree_one | [463, 1] | [496, 35] | have hd0 : (d.support.sum fun a : ℕ × M => d a • a.fst) = 0 := by
rw [Finset.sum_eq_zero]
intro nm hnm
rw [← Nat.lt_one_iff]
apply lt_of_le_of_ne _ (h0 nm)
rw [← hd]
exact Finset.single_le_sum (fun x _ => zero_le (d x • x.fst)) hnm | R : Type ?u.209305
M : Type u_1
inst✝¹ : CommSemiring R
inst✝ : DecidableEq M
d : ℕ × M →₀ ℕ
hd : ∑ a ∈ d.support, d a • a.1 = 1
hsupp : ∀ nm ∈ d.support, 0 < nm.1
h0 : ∀ (x : ℕ × M), ¬d x • x.1 = 1
⊢ False | R : Type ?u.209305
M : Type u_1
inst✝¹ : CommSemiring R
inst✝ : DecidableEq M
d : ℕ × M →₀ ℕ
hd : ∑ a ∈ d.support, d a • a.1 = 1
hsupp : ∀ nm ∈ d.support, 0 < nm.1
h0 : ∀ (x : ℕ × M), ¬d x • x.1 = 1
hd0 : ∑ a ∈ d.support, d a • a.1 = 0
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type ?u.209305
M : Type u_1
inst✝¹ : CommSemiring R
inst✝ : DecidableEq M
d : ℕ × M →₀ ℕ
hd : ∑ a ∈ d.support, d a • a.1 = 1
hsupp : ∀ nm ∈ d.support, 0 < nm.1
h0 : ∀ (x : ℕ × M), ¬d x • x.1 = 1
⊢ False
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Misc.lean | eq_finsupp_single_of_degree_one | [463, 1] | [496, 35] | rw [hd0] at hd | R : Type ?u.209305
M : Type u_1
inst✝¹ : CommSemiring R
inst✝ : DecidableEq M
d : ℕ × M →₀ ℕ
hd : ∑ a ∈ d.support, d a • a.1 = 1
hsupp : ∀ nm ∈ d.support, 0 < nm.1
h0 : ∀ (x : ℕ × M), ¬d x • x.1 = 1
hd0 : ∑ a ∈ d.support, d a • a.1 = 0
⊢ False | R : Type ?u.209305
M : Type u_1
inst✝¹ : CommSemiring R
inst✝ : DecidableEq M
d : ℕ × M →₀ ℕ
hd : 0 = 1
hsupp : ∀ nm ∈ d.support, 0 < nm.1
h0 : ∀ (x : ℕ × M), ¬d x • x.1 = 1
hd0 : ∑ a ∈ d.support, d a • a.1 = 0
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type ?u.209305
M : Type u_1
inst✝¹ : CommSemiring R
inst✝ : DecidableEq M
d : ℕ × M →₀ ℕ
hd : ∑ a ∈ d.support, d a • a.1 = 1
hsupp : ∀ nm ∈ d.support, 0 < nm.1
h0 : ∀ (x : ℕ × M), ¬d x • x.1 = 1
hd0 : ∑ a ∈ d.support, d a • a.1 = 0
⊢ False
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Misc.lean | eq_finsupp_single_of_degree_one | [463, 1] | [496, 35] | exact zero_ne_one hd | R : Type ?u.209305
M : Type u_1
inst✝¹ : CommSemiring R
inst✝ : DecidableEq M
d : ℕ × M →₀ ℕ
hd : 0 = 1
hsupp : ∀ nm ∈ d.support, 0 < nm.1
h0 : ∀ (x : ℕ × M), ¬d x • x.1 = 1
hd0 : ∑ a ∈ d.support, d a • a.1 = 0
⊢ False | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type ?u.209305
M : Type u_1
inst✝¹ : CommSemiring R
inst✝ : DecidableEq M
d : ℕ × M →₀ ℕ
hd : 0 = 1
hsupp : ∀ nm ∈ d.support, 0 < nm.1
h0 : ∀ (x : ℕ × M), ¬d x • x.1 = 1
hd0 : ∑ a ∈ d.support, d a • a.1 = 0
⊢ False
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Misc.lean | eq_finsupp_single_of_degree_one | [463, 1] | [496, 35] | rw [Finset.sum_eq_zero] | R : Type ?u.209305
M : Type u_1
inst✝¹ : CommSemiring R
inst✝ : DecidableEq M
d : ℕ × M →₀ ℕ
hd : ∑ a ∈ d.support, d a • a.1 = 1
hsupp : ∀ nm ∈ d.support, 0 < nm.1
h0 : ∀ (x : ℕ × M), ¬d x • x.1 = 1
⊢ ∑ a ∈ d.support, d a • a.1 = 0 | R : Type ?u.209305
M : Type u_1
inst✝¹ : CommSemiring R
inst✝ : DecidableEq M
d : ℕ × M →₀ ℕ
hd : ∑ a ∈ d.support, d a • a.1 = 1
hsupp : ∀ nm ∈ d.support, 0 < nm.1
h0 : ∀ (x : ℕ × M), ¬d x • x.1 = 1
⊢ ∀ x ∈ d.support, d x • x.1 = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type ?u.209305
M : Type u_1
inst✝¹ : CommSemiring R
inst✝ : DecidableEq M
d : ℕ × M →₀ ℕ
hd : ∑ a ∈ d.support, d a • a.1 = 1
hsupp : ∀ nm ∈ d.support, 0 < nm.1
h0 : ∀ (x : ℕ × M), ¬d x • x.1 = 1
⊢ ∑ a ∈ d.support, d a • a.1 = 0
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Misc.lean | eq_finsupp_single_of_degree_one | [463, 1] | [496, 35] | intro nm hnm | R : Type ?u.209305
M : Type u_1
inst✝¹ : CommSemiring R
inst✝ : DecidableEq M
d : ℕ × M →₀ ℕ
hd : ∑ a ∈ d.support, d a • a.1 = 1
hsupp : ∀ nm ∈ d.support, 0 < nm.1
h0 : ∀ (x : ℕ × M), ¬d x • x.1 = 1
⊢ ∀ x ∈ d.support, d x • x.1 = 0 | R : Type ?u.209305
M : Type u_1
inst✝¹ : CommSemiring R
inst✝ : DecidableEq M
d : ℕ × M →₀ ℕ
hd : ∑ a ∈ d.support, d a • a.1 = 1
hsupp : ∀ nm ∈ d.support, 0 < nm.1
h0 : ∀ (x : ℕ × M), ¬d x • x.1 = 1
nm : ℕ × M
hnm : nm ∈ d.support
⊢ d nm • nm.1 = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type ?u.209305
M : Type u_1
inst✝¹ : CommSemiring R
inst✝ : DecidableEq M
d : ℕ × M →₀ ℕ
hd : ∑ a ∈ d.support, d a • a.1 = 1
hsupp : ∀ nm ∈ d.support, 0 < nm.1
h0 : ∀ (x : ℕ × M), ¬d x • x.1 = 1
⊢ ∀ x ∈ d.support, d x • x.1 = 0
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Misc.lean | eq_finsupp_single_of_degree_one | [463, 1] | [496, 35] | rw [← Nat.lt_one_iff] | R : Type ?u.209305
M : Type u_1
inst✝¹ : CommSemiring R
inst✝ : DecidableEq M
d : ℕ × M →₀ ℕ
hd : ∑ a ∈ d.support, d a • a.1 = 1
hsupp : ∀ nm ∈ d.support, 0 < nm.1
h0 : ∀ (x : ℕ × M), ¬d x • x.1 = 1
nm : ℕ × M
hnm : nm ∈ d.support
⊢ d nm • nm.1 = 0 | R : Type ?u.209305
M : Type u_1
inst✝¹ : CommSemiring R
inst✝ : DecidableEq M
d : ℕ × M →₀ ℕ
hd : ∑ a ∈ d.support, d a • a.1 = 1
hsupp : ∀ nm ∈ d.support, 0 < nm.1
h0 : ∀ (x : ℕ × M), ¬d x • x.1 = 1
nm : ℕ × M
hnm : nm ∈ d.support
⊢ d nm • nm.1 < 1 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type ?u.209305
M : Type u_1
inst✝¹ : CommSemiring R
inst✝ : DecidableEq M
d : ℕ × M →₀ ℕ
hd : ∑ a ∈ d.support, d a • a.1 = 1
hsupp : ∀ nm ∈ d.support, 0 < nm.1
h0 : ∀ (x : ℕ × M), ¬d x • x.1 = 1
nm : ℕ × M
hnm : nm ∈ d.support
⊢ d nm • nm.1 = 0
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Misc.lean | eq_finsupp_single_of_degree_one | [463, 1] | [496, 35] | apply lt_of_le_of_ne _ (h0 nm) | R : Type ?u.209305
M : Type u_1
inst✝¹ : CommSemiring R
inst✝ : DecidableEq M
d : ℕ × M →₀ ℕ
hd : ∑ a ∈ d.support, d a • a.1 = 1
hsupp : ∀ nm ∈ d.support, 0 < nm.1
h0 : ∀ (x : ℕ × M), ¬d x • x.1 = 1
nm : ℕ × M
hnm : nm ∈ d.support
⊢ d nm • nm.1 < 1 | R : Type ?u.209305
M : Type u_1
inst✝¹ : CommSemiring R
inst✝ : DecidableEq M
d : ℕ × M →₀ ℕ
hd : ∑ a ∈ d.support, d a • a.1 = 1
hsupp : ∀ nm ∈ d.support, 0 < nm.1
h0 : ∀ (x : ℕ × M), ¬d x • x.1 = 1
nm : ℕ × M
hnm : nm ∈ d.support
⊢ d nm • nm.1 ≤ 1 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type ?u.209305
M : Type u_1
inst✝¹ : CommSemiring R
inst✝ : DecidableEq M
d : ℕ × M →₀ ℕ
hd : ∑ a ∈ d.support, d a • a.1 = 1
hsupp : ∀ nm ∈ d.support, 0 < nm.1
h0 : ∀ (x : ℕ × M), ¬d x • x.1 = 1
nm : ℕ × M
hnm : nm ∈ d.support
⊢ d nm • nm.1 < 1
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Misc.lean | eq_finsupp_single_of_degree_one | [463, 1] | [496, 35] | rw [← hd] | R : Type ?u.209305
M : Type u_1
inst✝¹ : CommSemiring R
inst✝ : DecidableEq M
d : ℕ × M →₀ ℕ
hd : ∑ a ∈ d.support, d a • a.1 = 1
hsupp : ∀ nm ∈ d.support, 0 < nm.1
h0 : ∀ (x : ℕ × M), ¬d x • x.1 = 1
nm : ℕ × M
hnm : nm ∈ d.support
⊢ d nm • nm.1 ≤ 1 | R : Type ?u.209305
M : Type u_1
inst✝¹ : CommSemiring R
inst✝ : DecidableEq M
d : ℕ × M →₀ ℕ
hd : ∑ a ∈ d.support, d a • a.1 = 1
hsupp : ∀ nm ∈ d.support, 0 < nm.1
h0 : ∀ (x : ℕ × M), ¬d x • x.1 = 1
nm : ℕ × M
hnm : nm ∈ d.support
⊢ d nm • nm.1 ≤ ∑ a ∈ d.support, d a • a.1 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type ?u.209305
M : Type u_1
inst✝¹ : CommSemiring R
inst✝ : DecidableEq M
d : ℕ × M →₀ ℕ
hd : ∑ a ∈ d.support, d a • a.1 = 1
hsupp : ∀ nm ∈ d.support, 0 < nm.1
h0 : ∀ (x : ℕ × M), ¬d x • x.1 = 1
nm : ℕ × M
hnm : nm ∈ d.support
⊢ d nm • nm.1 ≤ 1
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Misc.lean | eq_finsupp_single_of_degree_one | [463, 1] | [496, 35] | exact Finset.single_le_sum (fun x _ => zero_le (d x • x.fst)) hnm | R : Type ?u.209305
M : Type u_1
inst✝¹ : CommSemiring R
inst✝ : DecidableEq M
d : ℕ × M →₀ ℕ
hd : ∑ a ∈ d.support, d a • a.1 = 1
hsupp : ∀ nm ∈ d.support, 0 < nm.1
h0 : ∀ (x : ℕ × M), ¬d x • x.1 = 1
nm : ℕ × M
hnm : nm ∈ d.support
⊢ d nm • nm.1 ≤ ∑ a ∈ d.support, d a • a.1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type ?u.209305
M : Type u_1
inst✝¹ : CommSemiring R
inst✝ : DecidableEq M
d : ℕ × M →₀ ℕ
hd : ∑ a ∈ d.support, d a • a.1 = 1
hsupp : ∀ nm ∈ d.support, 0 < nm.1
h0 : ∀ (x : ℕ × M), ¬d x • x.1 = 1
nm : ℕ × M
hnm : nm ∈ d.support
⊢ d nm • nm.1 ≤ ∑ a ∈ d.supp... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Misc.lean | eq_finsupp_single_of_degree_one | [463, 1] | [496, 35] | rw [hab, hnm.1] | case pos
R : Type ?u.209305
M : Type u_1
inst✝¹ : CommSemiring R
inst✝ : DecidableEq M
d : ℕ × M →₀ ℕ
hsupp : ∀ nm ∈ d.support, 0 < nm.1
nm : ℕ × M
hd : ∑ a ∈ d.support, d a • a.1 = d nm • nm.1
hnm : d nm = 1 ∧ nm.1 = 1
ab : ℕ × M
hab : ab = nm
⊢ 1 = d ab | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
R : Type ?u.209305
M : Type u_1
inst✝¹ : CommSemiring R
inst✝ : DecidableEq M
d : ℕ × M →₀ ℕ
hsupp : ∀ nm ∈ d.support, 0 < nm.1
nm : ℕ × M
hd : ∑ a ∈ d.support, d a • a.1 = d nm • nm.1
hnm : d nm = 1 ∧ nm.1 = 1
ab : ℕ × M
hab : ab = nm
⊢ 1 = d ab
TAC... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Misc.lean | eq_finsupp_single_of_degree_one | [463, 1] | [496, 35] | rw [eq_comm] | case neg
R : Type ?u.209305
M : Type u_1
inst✝¹ : CommSemiring R
inst✝ : DecidableEq M
d : ℕ × M →₀ ℕ
hsupp : ∀ nm ∈ d.support, 0 < nm.1
nm : ℕ × M
hd : ∑ a ∈ d.support, d a • a.1 = d nm • nm.1
hnm : d nm = 1 ∧ nm.1 = 1
ab : ℕ × M
hab : ¬ab = nm
⊢ 0 = d ab | case neg
R : Type ?u.209305
M : Type u_1
inst✝¹ : CommSemiring R
inst✝ : DecidableEq M
d : ℕ × M →₀ ℕ
hsupp : ∀ nm ∈ d.support, 0 < nm.1
nm : ℕ × M
hd : ∑ a ∈ d.support, d a • a.1 = d nm • nm.1
hnm : d nm = 1 ∧ nm.1 = 1
ab : ℕ × M
hab : ¬ab = nm
⊢ d ab = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
R : Type ?u.209305
M : Type u_1
inst✝¹ : CommSemiring R
inst✝ : DecidableEq M
d : ℕ × M →₀ ℕ
hsupp : ∀ nm ∈ d.support, 0 < nm.1
nm : ℕ × M
hd : ∑ a ∈ d.support, d a • a.1 = d nm • nm.1
hnm : d nm = 1 ∧ nm.1 = 1
ab : ℕ × M
hab : ¬ab = nm
⊢ 0 = d ab
TA... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Misc.lean | eq_finsupp_single_of_degree_one | [463, 1] | [496, 35] | by_contra hab' | case neg
R : Type ?u.209305
M : Type u_1
inst✝¹ : CommSemiring R
inst✝ : DecidableEq M
d : ℕ × M →₀ ℕ
hsupp : ∀ nm ∈ d.support, 0 < nm.1
nm : ℕ × M
hd : ∑ a ∈ d.support, d a • a.1 = d nm • nm.1
hnm : d nm = 1 ∧ nm.1 = 1
ab : ℕ × M
hab : ¬ab = nm
⊢ d ab = 0 | case neg
R : Type ?u.209305
M : Type u_1
inst✝¹ : CommSemiring R
inst✝ : DecidableEq M
d : ℕ × M →₀ ℕ
hsupp : ∀ nm ∈ d.support, 0 < nm.1
nm : ℕ × M
hd : ∑ a ∈ d.support, d a • a.1 = d nm • nm.1
hnm : d nm = 1 ∧ nm.1 = 1
ab : ℕ × M
hab : ¬ab = nm
hab' : ¬d ab = 0
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
R : Type ?u.209305
M : Type u_1
inst✝¹ : CommSemiring R
inst✝ : DecidableEq M
d : ℕ × M →₀ ℕ
hsupp : ∀ nm ∈ d.support, 0 < nm.1
nm : ℕ × M
hd : ∑ a ∈ d.support, d a • a.1 = d nm • nm.1
hnm : d nm = 1 ∧ nm.1 = 1
ab : ℕ × M
hab : ¬ab = nm
⊢ d ab = 0
TA... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Misc.lean | eq_finsupp_single_of_degree_one | [463, 1] | [496, 35] | have hne0 : d ab * ab.fst ≠ 0 :=
mul_ne_zero hab' (ne_of_gt (hsupp ab (Finsupp.mem_support_iff.mpr hab'))) | case neg
R : Type ?u.209305
M : Type u_1
inst✝¹ : CommSemiring R
inst✝ : DecidableEq M
d : ℕ × M →₀ ℕ
hsupp : ∀ nm ∈ d.support, 0 < nm.1
nm : ℕ × M
hd : ∑ a ∈ d.support, d a • a.1 = d nm • nm.1
hnm : d nm = 1 ∧ nm.1 = 1
ab : ℕ × M
hab : ¬ab = nm
hab' : ¬d ab = 0
⊢ False | case neg
R : Type ?u.209305
M : Type u_1
inst✝¹ : CommSemiring R
inst✝ : DecidableEq M
d : ℕ × M →₀ ℕ
hsupp : ∀ nm ∈ d.support, 0 < nm.1
nm : ℕ × M
hd : ∑ a ∈ d.support, d a • a.1 = d nm • nm.1
hnm : d nm = 1 ∧ nm.1 = 1
ab : ℕ × M
hab : ¬ab = nm
hab' : ¬d ab = 0
hne0 : d ab * ab.1 ≠ 0
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
R : Type ?u.209305
M : Type u_1
inst✝¹ : CommSemiring R
inst✝ : DecidableEq M
d : ℕ × M →₀ ℕ
hsupp : ∀ nm ∈ d.support, 0 < nm.1
nm : ℕ × M
hd : ∑ a ∈ d.support, d a • a.1 = d nm • nm.1
hnm : d nm = 1 ∧ nm.1 = 1
ab : ℕ × M
hab : ¬ab = nm
hab' : ¬d ab ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Misc.lean | eq_finsupp_single_of_degree_one | [463, 1] | [496, 35] | have hnm_mem : nm ∈ d.support := by rw [Finsupp.mem_support_iff, hnm.1]; exact one_ne_zero | case neg
R : Type ?u.209305
M : Type u_1
inst✝¹ : CommSemiring R
inst✝ : DecidableEq M
d : ℕ × M →₀ ℕ
hsupp : ∀ nm ∈ d.support, 0 < nm.1
nm : ℕ × M
hd : ∑ a ∈ d.support, d a • a.1 = d nm • nm.1
hnm : d nm = 1 ∧ nm.1 = 1
ab : ℕ × M
hab : ¬ab = nm
hab' : ¬d ab = 0
hne0 : d ab * ab.1 ≠ 0
⊢ False | case neg
R : Type ?u.209305
M : Type u_1
inst✝¹ : CommSemiring R
inst✝ : DecidableEq M
d : ℕ × M →₀ ℕ
hsupp : ∀ nm ∈ d.support, 0 < nm.1
nm : ℕ × M
hd : ∑ a ∈ d.support, d a • a.1 = d nm • nm.1
hnm : d nm = 1 ∧ nm.1 = 1
ab : ℕ × M
hab : ¬ab = nm
hab' : ¬d ab = 0
hne0 : d ab * ab.1 ≠ 0
hnm_mem : nm ∈ d.support
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
R : Type ?u.209305
M : Type u_1
inst✝¹ : CommSemiring R
inst✝ : DecidableEq M
d : ℕ × M →₀ ℕ
hsupp : ∀ nm ∈ d.support, 0 < nm.1
nm : ℕ × M
hd : ∑ a ∈ d.support, d a • a.1 = d nm • nm.1
hnm : d nm = 1 ∧ nm.1 = 1
ab : ℕ × M
hab : ¬ab = nm
hab' : ¬d ab ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Misc.lean | eq_finsupp_single_of_degree_one | [463, 1] | [496, 35] | simp only [Finset.sum_eq_sum_diff_singleton_add hnm_mem, add_left_eq_self,
Algebra.id.smul_eq_mul, sum_eq_zero_iff, mem_sdiff,
Finsupp.mem_support_iff, mem_singleton] at hd | case neg
R : Type ?u.209305
M : Type u_1
inst✝¹ : CommSemiring R
inst✝ : DecidableEq M
d : ℕ × M →₀ ℕ
hsupp : ∀ nm ∈ d.support, 0 < nm.1
nm : ℕ × M
hd : ∑ a ∈ d.support, d a • a.1 = d nm • nm.1
hnm : d nm = 1 ∧ nm.1 = 1
ab : ℕ × M
hab : ¬ab = nm
hab' : ¬d ab = 0
hne0 : d ab * ab.1 ≠ 0
hnm_mem : nm ∈ d.support
⊢ False | case neg
R : Type ?u.209305
M : Type u_1
inst✝¹ : CommSemiring R
inst✝ : DecidableEq M
d : ℕ × M →₀ ℕ
hsupp : ∀ nm ∈ d.support, 0 < nm.1
nm : ℕ × M
hnm : d nm = 1 ∧ nm.1 = 1
ab : ℕ × M
hab : ¬ab = nm
hab' : ¬d ab = 0
hne0 : d ab * ab.1 ≠ 0
hnm_mem : nm ∈ d.support
hd : ∀ (x : ℕ × M), d x ≠ 0 ∧ ¬x = nm → d x * x.1 = 0
⊢... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
R : Type ?u.209305
M : Type u_1
inst✝¹ : CommSemiring R
inst✝ : DecidableEq M
d : ℕ × M →₀ ℕ
hsupp : ∀ nm ∈ d.support, 0 < nm.1
nm : ℕ × M
hd : ∑ a ∈ d.support, d a • a.1 = d nm • nm.1
hnm : d nm = 1 ∧ nm.1 = 1
ab : ℕ × M
hab : ¬ab = nm
hab' : ¬d ab ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Misc.lean | eq_finsupp_single_of_degree_one | [463, 1] | [496, 35] | exact hne0 (hd ab ⟨hab', hab⟩) | case neg
R : Type ?u.209305
M : Type u_1
inst✝¹ : CommSemiring R
inst✝ : DecidableEq M
d : ℕ × M →₀ ℕ
hsupp : ∀ nm ∈ d.support, 0 < nm.1
nm : ℕ × M
hnm : d nm = 1 ∧ nm.1 = 1
ab : ℕ × M
hab : ¬ab = nm
hab' : ¬d ab = 0
hne0 : d ab * ab.1 ≠ 0
hnm_mem : nm ∈ d.support
hd : ∀ (x : ℕ × M), d x ≠ 0 ∧ ¬x = nm → d x * x.1 = 0
⊢... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
R : Type ?u.209305
M : Type u_1
inst✝¹ : CommSemiring R
inst✝ : DecidableEq M
d : ℕ × M →₀ ℕ
hsupp : ∀ nm ∈ d.support, 0 < nm.1
nm : ℕ × M
hnm : d nm = 1 ∧ nm.1 = 1
ab : ℕ × M
hab : ¬ab = nm
hab' : ¬d ab = 0
hne0 : d ab * ab.1 ≠ 0
hnm_mem : nm ∈ d.su... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Misc.lean | eq_finsupp_single_of_degree_one | [463, 1] | [496, 35] | rw [Finsupp.mem_support_iff, hnm.1] | R : Type ?u.209305
M : Type u_1
inst✝¹ : CommSemiring R
inst✝ : DecidableEq M
d : ℕ × M →₀ ℕ
hsupp : ∀ nm ∈ d.support, 0 < nm.1
nm : ℕ × M
hd : ∑ a ∈ d.support, d a • a.1 = d nm • nm.1
hnm : d nm = 1 ∧ nm.1 = 1
ab : ℕ × M
hab : ¬ab = nm
hab' : ¬d ab = 0
hne0 : d ab * ab.1 ≠ 0
⊢ nm ∈ d.support | R : Type ?u.209305
M : Type u_1
inst✝¹ : CommSemiring R
inst✝ : DecidableEq M
d : ℕ × M →₀ ℕ
hsupp : ∀ nm ∈ d.support, 0 < nm.1
nm : ℕ × M
hd : ∑ a ∈ d.support, d a • a.1 = d nm • nm.1
hnm : d nm = 1 ∧ nm.1 = 1
ab : ℕ × M
hab : ¬ab = nm
hab' : ¬d ab = 0
hne0 : d ab * ab.1 ≠ 0
⊢ 1 ≠ 0 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type ?u.209305
M : Type u_1
inst✝¹ : CommSemiring R
inst✝ : DecidableEq M
d : ℕ × M →₀ ℕ
hsupp : ∀ nm ∈ d.support, 0 < nm.1
nm : ℕ × M
hd : ∑ a ∈ d.support, d a • a.1 = d nm • nm.1
hnm : d nm = 1 ∧ nm.1 = 1
ab : ℕ × M
hab : ¬ab = nm
hab' : ¬d ab = 0
hne0 ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Misc.lean | eq_finsupp_single_of_degree_one | [463, 1] | [496, 35] | exact one_ne_zero | R : Type ?u.209305
M : Type u_1
inst✝¹ : CommSemiring R
inst✝ : DecidableEq M
d : ℕ × M →₀ ℕ
hsupp : ∀ nm ∈ d.support, 0 < nm.1
nm : ℕ × M
hd : ∑ a ∈ d.support, d a • a.1 = d nm • nm.1
hnm : d nm = 1 ∧ nm.1 = 1
ab : ℕ × M
hab : ¬ab = nm
hab' : ¬d ab = 0
hne0 : d ab * ab.1 ≠ 0
⊢ 1 ≠ 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type ?u.209305
M : Type u_1
inst✝¹ : CommSemiring R
inst✝ : DecidableEq M
d : ℕ × M →₀ ℕ
hsupp : ∀ nm ∈ d.support, 0 < nm.1
nm : ℕ × M
hd : ∑ a ∈ d.support, d a • a.1 = d nm • nm.1
hnm : d nm = 1 ∧ nm.1 = 1
ab : ℕ × M
hab : ¬ab = nm
hab' : ¬d ab = 0
hne0 ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/WeightedHomogeneous.lean | MvPolynomial.weightedHomogeneousComponent_of_weighted_homogeneous_polynomial_same | [590, 1] | [600, 36] | ext x | R : Type u_2
M : Type u_1
inst✝² : CommSemiring R
σ : Type u_3
inst✝¹ : AddCommMonoid M
w : σ → M
n : M
φ ψ : MvPolynomial σ R
inst✝ : DecidableEq M
m : M
p : MvPolynomial σ R
hp : IsWeightedHomogeneous w p m
⊢ (weightedHomogeneousComponent w m) p = p | case a
R : Type u_2
M : Type u_1
inst✝² : CommSemiring R
σ : Type u_3
inst✝¹ : AddCommMonoid M
w : σ → M
n : M
φ ψ : MvPolynomial σ R
inst✝ : DecidableEq M
m : M
p : MvPolynomial σ R
hp : IsWeightedHomogeneous w p m
x : σ →₀ ℕ
⊢ coeff x ((weightedHomogeneousComponent w m) p) = coeff x p | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
M : Type u_1
inst✝² : CommSemiring R
σ : Type u_3
inst✝¹ : AddCommMonoid M
w : σ → M
n : M
φ ψ : MvPolynomial σ R
inst✝ : DecidableEq M
m : M
p : MvPolynomial σ R
hp : IsWeightedHomogeneous w p m
⊢ (weightedHomogeneousComponent w m) p = p
TACTIC:... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/WeightedHomogeneous.lean | MvPolynomial.weightedHomogeneousComponent_of_weighted_homogeneous_polynomial_same | [590, 1] | [600, 36] | rw [coeff_weightedHomogeneousComponent] | case a
R : Type u_2
M : Type u_1
inst✝² : CommSemiring R
σ : Type u_3
inst✝¹ : AddCommMonoid M
w : σ → M
n : M
φ ψ : MvPolynomial σ R
inst✝ : DecidableEq M
m : M
p : MvPolynomial σ R
hp : IsWeightedHomogeneous w p m
x : σ →₀ ℕ
⊢ coeff x ((weightedHomogeneousComponent w m) p) = coeff x p | case a
R : Type u_2
M : Type u_1
inst✝² : CommSemiring R
σ : Type u_3
inst✝¹ : AddCommMonoid M
w : σ → M
n : M
φ ψ : MvPolynomial σ R
inst✝ : DecidableEq M
m : M
p : MvPolynomial σ R
hp : IsWeightedHomogeneous w p m
x : σ →₀ ℕ
⊢ (if (weightedDegree w) x = m then coeff x p else 0) = coeff x p | Please generate a tactic in lean4 to solve the state.
STATE:
case a
R : Type u_2
M : Type u_1
inst✝² : CommSemiring R
σ : Type u_3
inst✝¹ : AddCommMonoid M
w : σ → M
n : M
φ ψ : MvPolynomial σ R
inst✝ : DecidableEq M
m : M
p : MvPolynomial σ R
hp : IsWeightedHomogeneous w p m
x : σ →₀ ℕ
⊢ coeff x ((weightedHomogeneousC... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/WeightedHomogeneous.lean | MvPolynomial.weightedHomogeneousComponent_of_weighted_homogeneous_polynomial_same | [590, 1] | [600, 36] | by_cases zero_coeff : coeff x p = 0 | case a
R : Type u_2
M : Type u_1
inst✝² : CommSemiring R
σ : Type u_3
inst✝¹ : AddCommMonoid M
w : σ → M
n : M
φ ψ : MvPolynomial σ R
inst✝ : DecidableEq M
m : M
p : MvPolynomial σ R
hp : IsWeightedHomogeneous w p m
x : σ →₀ ℕ
⊢ (if (weightedDegree w) x = m then coeff x p else 0) = coeff x p | case pos
R : Type u_2
M : Type u_1
inst✝² : CommSemiring R
σ : Type u_3
inst✝¹ : AddCommMonoid M
w : σ → M
n : M
φ ψ : MvPolynomial σ R
inst✝ : DecidableEq M
m : M
p : MvPolynomial σ R
hp : IsWeightedHomogeneous w p m
x : σ →₀ ℕ
zero_coeff : coeff x p = 0
⊢ (if (weightedDegree w) x = m then coeff x p else 0) = coeff x ... | Please generate a tactic in lean4 to solve the state.
STATE:
case a
R : Type u_2
M : Type u_1
inst✝² : CommSemiring R
σ : Type u_3
inst✝¹ : AddCommMonoid M
w : σ → M
n : M
φ ψ : MvPolynomial σ R
inst✝ : DecidableEq M
m : M
p : MvPolynomial σ R
hp : IsWeightedHomogeneous w p m
x : σ →₀ ℕ
⊢ (if (weightedDegree w) x = m t... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/WeightedHomogeneous.lean | MvPolynomial.weightedHomogeneousComponent_of_weighted_homogeneous_polynomial_same | [590, 1] | [600, 36] | split_ifs | case pos
R : Type u_2
M : Type u_1
inst✝² : CommSemiring R
σ : Type u_3
inst✝¹ : AddCommMonoid M
w : σ → M
n : M
φ ψ : MvPolynomial σ R
inst✝ : DecidableEq M
m : M
p : MvPolynomial σ R
hp : IsWeightedHomogeneous w p m
x : σ →₀ ℕ
zero_coeff : coeff x p = 0
⊢ (if (weightedDegree w) x = m then coeff x p else 0) = coeff x ... | case pos
R : Type u_2
M : Type u_1
inst✝² : CommSemiring R
σ : Type u_3
inst✝¹ : AddCommMonoid M
w : σ → M
n : M
φ ψ : MvPolynomial σ R
inst✝ : DecidableEq M
m : M
p : MvPolynomial σ R
hp : IsWeightedHomogeneous w p m
x : σ →₀ ℕ
zero_coeff : coeff x p = 0
h✝ : (weightedDegree w) x = m
⊢ coeff x p = coeff x p
case neg
... | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
R : Type u_2
M : Type u_1
inst✝² : CommSemiring R
σ : Type u_3
inst✝¹ : AddCommMonoid M
w : σ → M
n : M
φ ψ : MvPolynomial σ R
inst✝ : DecidableEq M
m : M
p : MvPolynomial σ R
hp : IsWeightedHomogeneous w p m
x : σ →₀ ℕ
zero_coeff : coeff x p = 0
⊢ (... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/WeightedHomogeneous.lean | MvPolynomial.weightedHomogeneousComponent_of_weighted_homogeneous_polynomial_same | [590, 1] | [600, 36] | rfl | case pos
R : Type u_2
M : Type u_1
inst✝² : CommSemiring R
σ : Type u_3
inst✝¹ : AddCommMonoid M
w : σ → M
n : M
φ ψ : MvPolynomial σ R
inst✝ : DecidableEq M
m : M
p : MvPolynomial σ R
hp : IsWeightedHomogeneous w p m
x : σ →₀ ℕ
zero_coeff : coeff x p = 0
h✝ : (weightedDegree w) x = m
⊢ coeff x p = coeff x p
case neg
... | case neg
R : Type u_2
M : Type u_1
inst✝² : CommSemiring R
σ : Type u_3
inst✝¹ : AddCommMonoid M
w : σ → M
n : M
φ ψ : MvPolynomial σ R
inst✝ : DecidableEq M
m : M
p : MvPolynomial σ R
hp : IsWeightedHomogeneous w p m
x : σ →₀ ℕ
zero_coeff : coeff x p = 0
h✝ : ¬(weightedDegree w) x = m
⊢ 0 = coeff x p | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
R : Type u_2
M : Type u_1
inst✝² : CommSemiring R
σ : Type u_3
inst✝¹ : AddCommMonoid M
w : σ → M
n : M
φ ψ : MvPolynomial σ R
inst✝ : DecidableEq M
m : M
p : MvPolynomial σ R
hp : IsWeightedHomogeneous w p m
x : σ →₀ ℕ
zero_coeff : coeff x p = 0
h✝ ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/WeightedHomogeneous.lean | MvPolynomial.weightedHomogeneousComponent_of_weighted_homogeneous_polynomial_same | [590, 1] | [600, 36] | rw [zero_coeff] | case neg
R : Type u_2
M : Type u_1
inst✝² : CommSemiring R
σ : Type u_3
inst✝¹ : AddCommMonoid M
w : σ → M
n : M
φ ψ : MvPolynomial σ R
inst✝ : DecidableEq M
m : M
p : MvPolynomial σ R
hp : IsWeightedHomogeneous w p m
x : σ →₀ ℕ
zero_coeff : coeff x p = 0
h✝ : ¬(weightedDegree w) x = m
⊢ 0 = coeff x p | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
R : Type u_2
M : Type u_1
inst✝² : CommSemiring R
σ : Type u_3
inst✝¹ : AddCommMonoid M
w : σ → M
n : M
φ ψ : MvPolynomial σ R
inst✝ : DecidableEq M
m : M
p : MvPolynomial σ R
hp : IsWeightedHomogeneous w p m
x : σ →₀ ℕ
zero_coeff : coeff x p = 0
h✝ ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/WeightedHomogeneous.lean | MvPolynomial.weightedHomogeneousComponent_of_weighted_homogeneous_polynomial_same | [590, 1] | [600, 36] | rw [hp zero_coeff, if_pos] | case neg
R : Type u_2
M : Type u_1
inst✝² : CommSemiring R
σ : Type u_3
inst✝¹ : AddCommMonoid M
w : σ → M
n : M
φ ψ : MvPolynomial σ R
inst✝ : DecidableEq M
m : M
p : MvPolynomial σ R
hp : IsWeightedHomogeneous w p m
x : σ →₀ ℕ
zero_coeff : ¬coeff x p = 0
⊢ (if (weightedDegree w) x = m then coeff x p else 0) = coeff x... | case neg.hc
R : Type u_2
M : Type u_1
inst✝² : CommSemiring R
σ : Type u_3
inst✝¹ : AddCommMonoid M
w : σ → M
n : M
φ ψ : MvPolynomial σ R
inst✝ : DecidableEq M
m : M
p : MvPolynomial σ R
hp : IsWeightedHomogeneous w p m
x : σ →₀ ℕ
zero_coeff : ¬coeff x p = 0
⊢ m = m | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
R : Type u_2
M : Type u_1
inst✝² : CommSemiring R
σ : Type u_3
inst✝¹ : AddCommMonoid M
w : σ → M
n : M
φ ψ : MvPolynomial σ R
inst✝ : DecidableEq M
m : M
p : MvPolynomial σ R
hp : IsWeightedHomogeneous w p m
x : σ →₀ ℕ
zero_coeff : ¬coeff x p = 0
⊢ ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/WeightedHomogeneous.lean | MvPolynomial.weightedHomogeneousComponent_of_weighted_homogeneous_polynomial_same | [590, 1] | [600, 36] | rfl | case neg.hc
R : Type u_2
M : Type u_1
inst✝² : CommSemiring R
σ : Type u_3
inst✝¹ : AddCommMonoid M
w : σ → M
n : M
φ ψ : MvPolynomial σ R
inst✝ : DecidableEq M
m : M
p : MvPolynomial σ R
hp : IsWeightedHomogeneous w p m
x : σ →₀ ℕ
zero_coeff : ¬coeff x p = 0
⊢ m = m | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.hc
R : Type u_2
M : Type u_1
inst✝² : CommSemiring R
σ : Type u_3
inst✝¹ : AddCommMonoid M
w : σ → M
n : M
φ ψ : MvPolynomial σ R
inst✝ : DecidableEq M
m : M
p : MvPolynomial σ R
hp : IsWeightedHomogeneous w p m
x : σ →₀ ℕ
zero_coeff : ¬coeff x p = 0... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/WeightedHomogeneous.lean | MvPolynomial.weightedHomogeneousComponent_of_weightedHomogeneous_ne | [604, 1] | [615, 71] | intro hn | R : Type u_2
M : Type u_1
inst✝² : CommSemiring R
σ : Type u_3
inst✝¹ : AddCommMonoid M
w : σ → M
n✝ : M
φ ψ : MvPolynomial σ R
inst✝ : DecidableEq M
m n : M
p : MvPolynomial σ R
hp : IsWeightedHomogeneous w p m
⊢ n ≠ m → (weightedHomogeneousComponent w n) p = 0 | R : Type u_2
M : Type u_1
inst✝² : CommSemiring R
σ : Type u_3
inst✝¹ : AddCommMonoid M
w : σ → M
n✝ : M
φ ψ : MvPolynomial σ R
inst✝ : DecidableEq M
m n : M
p : MvPolynomial σ R
hp : IsWeightedHomogeneous w p m
hn : n ≠ m
⊢ (weightedHomogeneousComponent w n) p = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
M : Type u_1
inst✝² : CommSemiring R
σ : Type u_3
inst✝¹ : AddCommMonoid M
w : σ → M
n✝ : M
φ ψ : MvPolynomial σ R
inst✝ : DecidableEq M
m n : M
p : MvPolynomial σ R
hp : IsWeightedHomogeneous w p m
⊢ n ≠ m → (weightedHomogeneousComponent w n) p ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/WeightedHomogeneous.lean | MvPolynomial.weightedHomogeneousComponent_of_weightedHomogeneous_ne | [604, 1] | [615, 71] | ext x | R : Type u_2
M : Type u_1
inst✝² : CommSemiring R
σ : Type u_3
inst✝¹ : AddCommMonoid M
w : σ → M
n✝ : M
φ ψ : MvPolynomial σ R
inst✝ : DecidableEq M
m n : M
p : MvPolynomial σ R
hp : IsWeightedHomogeneous w p m
hn : n ≠ m
⊢ (weightedHomogeneousComponent w n) p = 0 | case a
R : Type u_2
M : Type u_1
inst✝² : CommSemiring R
σ : Type u_3
inst✝¹ : AddCommMonoid M
w : σ → M
n✝ : M
φ ψ : MvPolynomial σ R
inst✝ : DecidableEq M
m n : M
p : MvPolynomial σ R
hp : IsWeightedHomogeneous w p m
hn : n ≠ m
x : σ →₀ ℕ
⊢ coeff x ((weightedHomogeneousComponent w n) p) = coeff x 0 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
M : Type u_1
inst✝² : CommSemiring R
σ : Type u_3
inst✝¹ : AddCommMonoid M
w : σ → M
n✝ : M
φ ψ : MvPolynomial σ R
inst✝ : DecidableEq M
m n : M
p : MvPolynomial σ R
hp : IsWeightedHomogeneous w p m
hn : n ≠ m
⊢ (weightedHomogeneousComponent w n)... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/WeightedHomogeneous.lean | MvPolynomial.weightedHomogeneousComponent_of_weightedHomogeneous_ne | [604, 1] | [615, 71] | rw [coeff_weightedHomogeneousComponent] | case a
R : Type u_2
M : Type u_1
inst✝² : CommSemiring R
σ : Type u_3
inst✝¹ : AddCommMonoid M
w : σ → M
n✝ : M
φ ψ : MvPolynomial σ R
inst✝ : DecidableEq M
m n : M
p : MvPolynomial σ R
hp : IsWeightedHomogeneous w p m
hn : n ≠ m
x : σ →₀ ℕ
⊢ coeff x ((weightedHomogeneousComponent w n) p) = coeff x 0 | case a
R : Type u_2
M : Type u_1
inst✝² : CommSemiring R
σ : Type u_3
inst✝¹ : AddCommMonoid M
w : σ → M
n✝ : M
φ ψ : MvPolynomial σ R
inst✝ : DecidableEq M
m n : M
p : MvPolynomial σ R
hp : IsWeightedHomogeneous w p m
hn : n ≠ m
x : σ →₀ ℕ
⊢ (if (weightedDegree w) x = n then coeff x p else 0) = coeff x 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case a
R : Type u_2
M : Type u_1
inst✝² : CommSemiring R
σ : Type u_3
inst✝¹ : AddCommMonoid M
w : σ → M
n✝ : M
φ ψ : MvPolynomial σ R
inst✝ : DecidableEq M
m n : M
p : MvPolynomial σ R
hp : IsWeightedHomogeneous w p m
hn : n ≠ m
x : σ →₀ ℕ
⊢ coeff x ((weight... |
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