url stringclasses 147
values | commit stringclasses 147
values | file_path stringlengths 7 101 | full_name stringlengths 1 94 | start stringlengths 6 10 | end stringlengths 6 11 | tactic stringlengths 1 11.2k | state_before stringlengths 3 2.09M | state_after stringlengths 6 2.09M | input stringlengths 73 2.09M |
|---|---|---|---|---|---|---|---|---|---|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/LinearTopology.lean | MvPowerSeries.toSubmodulesBasis | [158, 1] | [168, 57] | simp_rw [le_inf_iff] | σ : Type u_1
α : Type u_2
inst✝² : CommRing α
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
d e : σ →₀ ℕ
⊢ ∃ k, basis σ α k ≤ basis σ α d ⊓ basis σ α e | σ : Type u_1
α : Type u_2
inst✝² : CommRing α
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
d e : σ →₀ ℕ
⊢ ∃ k, basis σ α k ≤ basis σ α d ∧ basis σ α k ≤ basis σ α e | Please generate a tactic in lean4 to solve the state.
STATE:
σ : Type u_1
α : Type u_2
inst✝² : CommRing α
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
d e : σ →₀ ℕ
⊢ ∃ k, basis σ α k ≤ basis σ α d ⊓ basis σ α e
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/LinearTopology.lean | MvPowerSeries.toSubmodulesBasis | [158, 1] | [168, 57] | exact ⟨d + e, basis_antitone _ _ (le_self_add), basis_antitone _ _ (le_add_self)⟩ | σ : Type u_1
α : Type u_2
inst✝² : CommRing α
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
d e : σ →₀ ℕ
⊢ ∃ k, basis σ α k ≤ basis σ α d ∧ basis σ α k ≤ basis σ α e | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
σ : Type u_1
α : Type u_2
inst✝² : CommRing α
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
d e : σ →₀ ℕ
⊢ ∃ k, basis σ α k ≤ basis σ α d ∧ basis σ α k ≤ basis σ α e
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/LinearTopology.lean | MvPowerSeries.toSubmodulesBasis | [158, 1] | [168, 57] | rw [Filter.eventually_iff_exists_mem] | σ : Type u_1
α : Type u_2
inst✝² : CommRing α
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
f : MvPowerSeries σ α
d : σ →₀ ℕ
⊢ ∀ᶠ (a : MvPowerSeries σ α) in nhds 0, a • f ∈ basis σ α d | σ : Type u_1
α : Type u_2
inst✝² : CommRing α
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
f : MvPowerSeries σ α
d : σ →₀ ℕ
⊢ ∃ v ∈ nhds 0, ∀ y ∈ v, y • f ∈ basis σ α d | Please generate a tactic in lean4 to solve the state.
STATE:
σ : Type u_1
α : Type u_2
inst✝² : CommRing α
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
f : MvPowerSeries σ α
d : σ →₀ ℕ
⊢ ∀ᶠ (a : MvPowerSeries σ α) in nhds 0, a • f ∈ basis σ α d
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/LinearTopology.lean | MvPowerSeries.toSubmodulesBasis | [158, 1] | [168, 57] | refine ⟨↑(basis σ α d), (basis_mem_nhds_zero σ α d), ?_⟩ | σ : Type u_1
α : Type u_2
inst✝² : CommRing α
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
f : MvPowerSeries σ α
d : σ →₀ ℕ
⊢ ∃ v ∈ nhds 0, ∀ y ∈ v, y • f ∈ basis σ α d | σ : Type u_1
α : Type u_2
inst✝² : CommRing α
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
f : MvPowerSeries σ α
d : σ →₀ ℕ
⊢ ∀ y ∈ ↑(basis σ α d), y • f ∈ basis σ α d | Please generate a tactic in lean4 to solve the state.
STATE:
σ : Type u_1
α : Type u_2
inst✝² : CommRing α
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
f : MvPowerSeries σ α
d : σ →₀ ℕ
⊢ ∃ v ∈ nhds 0, ∀ y ∈ v, y • f ∈ basis σ α d
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/LinearTopology.lean | MvPowerSeries.toSubmodulesBasis | [158, 1] | [168, 57] | intros g hg | σ : Type u_1
α : Type u_2
inst✝² : CommRing α
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
f : MvPowerSeries σ α
d : σ →₀ ℕ
⊢ ∀ y ∈ ↑(basis σ α d), y • f ∈ basis σ α d | σ : Type u_1
α : Type u_2
inst✝² : CommRing α
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
f : MvPowerSeries σ α
d : σ →₀ ℕ
g : MvPowerSeries σ α
hg : g ∈ ↑(basis σ α d)
⊢ g • f ∈ basis σ α d | Please generate a tactic in lean4 to solve the state.
STATE:
σ : Type u_1
α : Type u_2
inst✝² : CommRing α
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
f : MvPowerSeries σ α
d : σ →₀ ℕ
⊢ ∀ y ∈ ↑(basis σ α d), y • f ∈ basis σ α d
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/LinearTopology.lean | MvPowerSeries.toSubmodulesBasis | [158, 1] | [168, 57] | rw [smul_eq_mul, mul_comm] | σ : Type u_1
α : Type u_2
inst✝² : CommRing α
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
f : MvPowerSeries σ α
d : σ →₀ ℕ
g : MvPowerSeries σ α
hg : g ∈ ↑(basis σ α d)
⊢ g • f ∈ basis σ α d | σ : Type u_1
α : Type u_2
inst✝² : CommRing α
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
f : MvPowerSeries σ α
d : σ →₀ ℕ
g : MvPowerSeries σ α
hg : g ∈ ↑(basis σ α d)
⊢ f * g ∈ basis σ α d | Please generate a tactic in lean4 to solve the state.
STATE:
σ : Type u_1
α : Type u_2
inst✝² : CommRing α
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
f : MvPowerSeries σ α
d : σ →₀ ℕ
g : MvPowerSeries σ α
hg : g ∈ ↑(basis σ α d)
⊢ g • f ∈ basis σ α d
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/LinearTopology.lean | MvPowerSeries.toSubmodulesBasis | [158, 1] | [168, 57] | exact Ideal.mul_mem_left _ f (SetLike.mem_coe.mp hg) | σ : Type u_1
α : Type u_2
inst✝² : CommRing α
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
f : MvPowerSeries σ α
d : σ →₀ ℕ
g : MvPowerSeries σ α
hg : g ∈ ↑(basis σ α d)
⊢ f * g ∈ basis σ α d | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
σ : Type u_1
α : Type u_2
inst✝² : CommRing α
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
f : MvPowerSeries σ α
d : σ →₀ ℕ
g : MvPowerSeries σ α
hg : g ∈ ↑(basis σ α d)
⊢ f * g ∈ basis σ α d
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/LinearTopology.lean | MvPowerSeries.mem_nhds_zero_iff | [209, 1] | [224, 72] | let τ := MvPowerSeries.WithPiTopology.topologicalSpace σ α | σ : Type u_1
α : Type u_2
inst✝² : CommRing α
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
U : Set (MvPowerSeries σ α)
⊢ U ∈ nhds 0 ↔ ∃ i, {b | b ∈ basis σ α i} ⊆ U | σ : Type u_1
α : Type u_2
inst✝² : CommRing α
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
U : Set (MvPowerSeries σ α)
τ : TopologicalSpace (MvPowerSeries σ α) := topologicalSpace σ α
⊢ U ∈ nhds 0 ↔ ∃ i, {b | b ∈ basis σ α i} ⊆ U | Please generate a tactic in lean4 to solve the state.
STATE:
σ : Type u_1
α : Type u_2
inst✝² : CommRing α
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
U : Set (MvPowerSeries σ α)
⊢ U ∈ nhds 0 ↔ ∃ i, {b | b ∈ basis σ α i} ⊆ U
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/LinearTopology.lean | MvPowerSeries.mem_nhds_zero_iff | [209, 1] | [224, 72] | constructor | σ : Type u_1
α : Type u_2
inst✝² : CommRing α
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
U : Set (MvPowerSeries σ α)
τ : TopologicalSpace (MvPowerSeries σ α) := topologicalSpace σ α
⊢ U ∈ nhds 0 ↔ ∃ i, {b | b ∈ basis σ α i} ⊆ U | case mp
σ : Type u_1
α : Type u_2
inst✝² : CommRing α
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
U : Set (MvPowerSeries σ α)
τ : TopologicalSpace (MvPowerSeries σ α) := topologicalSpace σ α
⊢ U ∈ nhds 0 → ∃ i, {b | b ∈ basis σ α i} ⊆ U
case mpr
σ : Type u_1
α : Type u_2
inst✝² : CommRing α
inst✝¹ : Topolog... | Please generate a tactic in lean4 to solve the state.
STATE:
σ : Type u_1
α : Type u_2
inst✝² : CommRing α
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
U : Set (MvPowerSeries σ α)
τ : TopologicalSpace (MvPowerSeries σ α) := topologicalSpace σ α
⊢ U ∈ nhds 0 ↔ ∃ i, {b | b ∈ basis σ α i} ⊆ U
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/LinearTopology.lean | MvPowerSeries.mem_nhds_zero_iff | [209, 1] | [224, 72] | rw [nhds_pi, Filter.mem_pi] | case mp
σ : Type u_1
α : Type u_2
inst✝² : CommRing α
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
U : Set (MvPowerSeries σ α)
τ : TopologicalSpace (MvPowerSeries σ α) := topologicalSpace σ α
⊢ U ∈ nhds 0 → ∃ i, {b | b ∈ basis σ α i} ⊆ U | case mp
σ : Type u_1
α : Type u_2
inst✝² : CommRing α
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
U : Set (MvPowerSeries σ α)
τ : TopologicalSpace (MvPowerSeries σ α) := topologicalSpace σ α
⊢ (∃ I, I.Finite ∧ ∃ t, (∀ (i : σ →₀ ℕ), t i ∈ nhds (0 i)) ∧ I.pi t ⊆ U) → ∃ i, {b | b ∈ basis σ α i} ⊆ U | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
σ : Type u_1
α : Type u_2
inst✝² : CommRing α
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
U : Set (MvPowerSeries σ α)
τ : TopologicalSpace (MvPowerSeries σ α) := topologicalSpace σ α
⊢ U ∈ nhds 0 → ∃ i, {b | b ∈ basis σ α i} ⊆ U
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/LinearTopology.lean | MvPowerSeries.mem_nhds_zero_iff | [209, 1] | [224, 72] | rintro ⟨D, hD, t, ht, ht'⟩ | case mp
σ : Type u_1
α : Type u_2
inst✝² : CommRing α
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
U : Set (MvPowerSeries σ α)
τ : TopologicalSpace (MvPowerSeries σ α) := topologicalSpace σ α
⊢ (∃ I, I.Finite ∧ ∃ t, (∀ (i : σ →₀ ℕ), t i ∈ nhds (0 i)) ∧ I.pi t ⊆ U) → ∃ i, {b | b ∈ basis σ α i} ⊆ U | case mp.intro.intro.intro.intro
σ : Type u_1
α : Type u_2
inst✝² : CommRing α
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
U : Set (MvPowerSeries σ α)
τ : TopologicalSpace (MvPowerSeries σ α) := topologicalSpace σ α
D : Set (σ →₀ ℕ)
hD : D.Finite
t : (σ →₀ ℕ) → Set α
ht : ∀ (i : σ →₀ ℕ), t i ∈ nhds (0 i)
ht' ... | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
σ : Type u_1
α : Type u_2
inst✝² : CommRing α
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
U : Set (MvPowerSeries σ α)
τ : TopologicalSpace (MvPowerSeries σ α) := topologicalSpace σ α
⊢ (∃ I, I.Finite ∧ ∃ t, (∀ (i : σ →₀ ℕ), t i ∈ nhds (0 i)... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/LinearTopology.lean | MvPowerSeries.mem_nhds_zero_iff | [209, 1] | [224, 72] | use Finset.sup hD.toFinset id | case mp.intro.intro.intro.intro
σ : Type u_1
α : Type u_2
inst✝² : CommRing α
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
U : Set (MvPowerSeries σ α)
τ : TopologicalSpace (MvPowerSeries σ α) := topologicalSpace σ α
D : Set (σ →₀ ℕ)
hD : D.Finite
t : (σ →₀ ℕ) → Set α
ht : ∀ (i : σ →₀ ℕ), t i ∈ nhds (0 i)
ht' ... | case h
σ : Type u_1
α : Type u_2
inst✝² : CommRing α
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
U : Set (MvPowerSeries σ α)
τ : TopologicalSpace (MvPowerSeries σ α) := topologicalSpace σ α
D : Set (σ →₀ ℕ)
hD : D.Finite
t : (σ →₀ ℕ) → Set α
ht : ∀ (i : σ →₀ ℕ), t i ∈ nhds (0 i)
ht' : D.pi t ⊆ U
⊢ {b | b ∈ b... | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.intro.intro.intro.intro
σ : Type u_1
α : Type u_2
inst✝² : CommRing α
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
U : Set (MvPowerSeries σ α)
τ : TopologicalSpace (MvPowerSeries σ α) := topologicalSpace σ α
D : Set (σ →₀ ℕ)
hD : D.Finite
t ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/LinearTopology.lean | MvPowerSeries.mem_nhds_zero_iff | [209, 1] | [224, 72] | apply subset_trans _ ht' | case h
σ : Type u_1
α : Type u_2
inst✝² : CommRing α
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
U : Set (MvPowerSeries σ α)
τ : TopologicalSpace (MvPowerSeries σ α) := topologicalSpace σ α
D : Set (σ →₀ ℕ)
hD : D.Finite
t : (σ →₀ ℕ) → Set α
ht : ∀ (i : σ →₀ ℕ), t i ∈ nhds (0 i)
ht' : D.pi t ⊆ U
⊢ {b | b ∈ b... | σ : Type u_1
α : Type u_2
inst✝² : CommRing α
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
U : Set (MvPowerSeries σ α)
τ : TopologicalSpace (MvPowerSeries σ α) := topologicalSpace σ α
D : Set (σ →₀ ℕ)
hD : D.Finite
t : (σ →₀ ℕ) → Set α
ht : ∀ (i : σ →₀ ℕ), t i ∈ nhds (0 i)
ht' : D.pi t ⊆ U
⊢ {b | b ∈ basis σ ... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
σ : Type u_1
α : Type u_2
inst✝² : CommRing α
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
U : Set (MvPowerSeries σ α)
τ : TopologicalSpace (MvPowerSeries σ α) := topologicalSpace σ α
D : Set (σ →₀ ℕ)
hD : D.Finite
t : (σ →₀ ℕ) → Set α
ht : ∀... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/LinearTopology.lean | MvPowerSeries.mem_nhds_zero_iff | [209, 1] | [224, 72] | intros f hf e he | σ : Type u_1
α : Type u_2
inst✝² : CommRing α
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
U : Set (MvPowerSeries σ α)
τ : TopologicalSpace (MvPowerSeries σ α) := topologicalSpace σ α
D : Set (σ →₀ ℕ)
hD : D.Finite
t : (σ →₀ ℕ) → Set α
ht : ∀ (i : σ →₀ ℕ), t i ∈ nhds (0 i)
ht' : D.pi t ⊆ U
⊢ {b | b ∈ basis σ ... | σ : Type u_1
α : Type u_2
inst✝² : CommRing α
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
U : Set (MvPowerSeries σ α)
τ : TopologicalSpace (MvPowerSeries σ α) := topologicalSpace σ α
D : Set (σ →₀ ℕ)
hD : D.Finite
t : (σ →₀ ℕ) → Set α
ht : ∀ (i : σ →₀ ℕ), t i ∈ nhds (0 i)
ht' : D.pi t ⊆ U
f : (σ →₀ ℕ) → α
hf... | Please generate a tactic in lean4 to solve the state.
STATE:
σ : Type u_1
α : Type u_2
inst✝² : CommRing α
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
U : Set (MvPowerSeries σ α)
τ : TopologicalSpace (MvPowerSeries σ α) := topologicalSpace σ α
D : Set (σ →₀ ℕ)
hD : D.Finite
t : (σ →₀ ℕ) → Set α
ht : ∀ (i : σ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/LinearTopology.lean | MvPowerSeries.mem_nhds_zero_iff | [209, 1] | [224, 72] | rw [← coeff_eq_apply f e, hf e] | σ : Type u_1
α : Type u_2
inst✝² : CommRing α
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
U : Set (MvPowerSeries σ α)
τ : TopologicalSpace (MvPowerSeries σ α) := topologicalSpace σ α
D : Set (σ →₀ ℕ)
hD : D.Finite
t : (σ →₀ ℕ) → Set α
ht : ∀ (i : σ →₀ ℕ), t i ∈ nhds (0 i)
ht' : D.pi t ⊆ U
f : (σ →₀ ℕ) → α
hf... | σ : Type u_1
α : Type u_2
inst✝² : CommRing α
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
U : Set (MvPowerSeries σ α)
τ : TopologicalSpace (MvPowerSeries σ α) := topologicalSpace σ α
D : Set (σ →₀ ℕ)
hD : D.Finite
t : (σ →₀ ℕ) → Set α
ht : ∀ (i : σ →₀ ℕ), t i ∈ nhds (0 i)
ht' : D.pi t ⊆ U
f : (σ →₀ ℕ) → α
hf... | Please generate a tactic in lean4 to solve the state.
STATE:
σ : Type u_1
α : Type u_2
inst✝² : CommRing α
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
U : Set (MvPowerSeries σ α)
τ : TopologicalSpace (MvPowerSeries σ α) := topologicalSpace σ α
D : Set (σ →₀ ℕ)
hD : D.Finite
t : (σ →₀ ℕ) → Set α
ht : ∀ (i : σ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/LinearTopology.lean | MvPowerSeries.mem_nhds_zero_iff | [209, 1] | [224, 72] | exact mem_of_mem_nhds (ht e) | σ : Type u_1
α : Type u_2
inst✝² : CommRing α
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
U : Set (MvPowerSeries σ α)
τ : TopologicalSpace (MvPowerSeries σ α) := topologicalSpace σ α
D : Set (σ →₀ ℕ)
hD : D.Finite
t : (σ →₀ ℕ) → Set α
ht : ∀ (i : σ →₀ ℕ), t i ∈ nhds (0 i)
ht' : D.pi t ⊆ U
f : (σ →₀ ℕ) → α
hf... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
σ : Type u_1
α : Type u_2
inst✝² : CommRing α
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
U : Set (MvPowerSeries σ α)
τ : TopologicalSpace (MvPowerSeries σ α) := topologicalSpace σ α
D : Set (σ →₀ ℕ)
hD : D.Finite
t : (σ →₀ ℕ) → Set α
ht : ∀ (i : σ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/LinearTopology.lean | MvPowerSeries.mem_nhds_zero_iff | [209, 1] | [224, 72] | rw [← id_eq e] | σ : Type u_1
α : Type u_2
inst✝² : CommRing α
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
U : Set (MvPowerSeries σ α)
τ : TopologicalSpace (MvPowerSeries σ α) := topologicalSpace σ α
D : Set (σ →₀ ℕ)
hD : D.Finite
t : (σ →₀ ℕ) → Set α
ht : ∀ (i : σ →₀ ℕ), t i ∈ nhds (0 i)
ht' : D.pi t ⊆ U
f : (σ →₀ ℕ) → α
hf... | σ : Type u_1
α : Type u_2
inst✝² : CommRing α
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
U : Set (MvPowerSeries σ α)
τ : TopologicalSpace (MvPowerSeries σ α) := topologicalSpace σ α
D : Set (σ →₀ ℕ)
hD : D.Finite
t : (σ →₀ ℕ) → Set α
ht : ∀ (i : σ →₀ ℕ), t i ∈ nhds (0 i)
ht' : D.pi t ⊆ U
f : (σ →₀ ℕ) → α
hf... | Please generate a tactic in lean4 to solve the state.
STATE:
σ : Type u_1
α : Type u_2
inst✝² : CommRing α
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
U : Set (MvPowerSeries σ α)
τ : TopologicalSpace (MvPowerSeries σ α) := topologicalSpace σ α
D : Set (σ →₀ ℕ)
hD : D.Finite
t : (σ →₀ ℕ) → Set α
ht : ∀ (i : σ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/LinearTopology.lean | MvPowerSeries.mem_nhds_zero_iff | [209, 1] | [224, 72] | exact Finset.le_sup ((Set.Finite.mem_toFinset _).mpr he) | σ : Type u_1
α : Type u_2
inst✝² : CommRing α
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
U : Set (MvPowerSeries σ α)
τ : TopologicalSpace (MvPowerSeries σ α) := topologicalSpace σ α
D : Set (σ →₀ ℕ)
hD : D.Finite
t : (σ →₀ ℕ) → Set α
ht : ∀ (i : σ →₀ ℕ), t i ∈ nhds (0 i)
ht' : D.pi t ⊆ U
f : (σ →₀ ℕ) → α
hf... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
σ : Type u_1
α : Type u_2
inst✝² : CommRing α
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
U : Set (MvPowerSeries σ α)
τ : TopologicalSpace (MvPowerSeries σ α) := topologicalSpace σ α
D : Set (σ →₀ ℕ)
hD : D.Finite
t : (σ →₀ ℕ) → Set α
ht : ∀ (i : σ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/LinearTopology.lean | MvPowerSeries.mem_nhds_zero_iff | [209, 1] | [224, 72] | rintro ⟨d, hd⟩ | case mpr
σ : Type u_1
α : Type u_2
inst✝² : CommRing α
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
U : Set (MvPowerSeries σ α)
τ : TopologicalSpace (MvPowerSeries σ α) := topologicalSpace σ α
⊢ (∃ i, {b | b ∈ basis σ α i} ⊆ U) → U ∈ nhds 0 | case mpr.intro
σ : Type u_1
α : Type u_2
inst✝² : CommRing α
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
U : Set (MvPowerSeries σ α)
τ : TopologicalSpace (MvPowerSeries σ α) := topologicalSpace σ α
d : σ →₀ ℕ
hd : {b | b ∈ basis σ α d} ⊆ U
⊢ U ∈ nhds 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
σ : Type u_1
α : Type u_2
inst✝² : CommRing α
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
U : Set (MvPowerSeries σ α)
τ : TopologicalSpace (MvPowerSeries σ α) := topologicalSpace σ α
⊢ (∃ i, {b | b ∈ basis σ α i} ⊆ U) → U ∈ nhds 0
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/LinearTopology.lean | MvPowerSeries.mem_nhds_zero_iff | [209, 1] | [224, 72] | exact (@nhds _ τ 0).sets_of_superset (basis_mem_nhds_zero σ α d) hd | case mpr.intro
σ : Type u_1
α : Type u_2
inst✝² : CommRing α
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
U : Set (MvPowerSeries σ α)
τ : TopologicalSpace (MvPowerSeries σ α) := topologicalSpace σ α
d : σ →₀ ℕ
hd : {b | b ∈ basis σ α d} ⊆ U
⊢ U ∈ nhds 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.intro
σ : Type u_1
α : Type u_2
inst✝² : CommRing α
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
U : Set (MvPowerSeries σ α)
τ : TopologicalSpace (MvPowerSeries σ α) := topologicalSpace σ α
d : σ →₀ ℕ
hd : {b | b ∈ basis σ α d} ⊆ U
⊢ U ∈ nh... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/LinearTopology.lean | MvPowerSeries.has_submodules_basis_topology | [227, 1] | [236, 32] | let τ := MvPowerSeries.WithPiTopology.topologicalSpace σ α | σ : Type u_1
α : Type u_2
inst✝³ : CommRing α
inst✝² : TopologicalSpace α
inst✝¹ inst✝ : DiscreteTopology α
⊢ topologicalSpace σ α = ⋯.topology | σ : Type u_1
α : Type u_2
inst✝³ : CommRing α
inst✝² : TopologicalSpace α
inst✝¹ inst✝ : DiscreteTopology α
τ : TopologicalSpace (MvPowerSeries σ α) := topologicalSpace σ α
⊢ topologicalSpace σ α = ⋯.topology | Please generate a tactic in lean4 to solve the state.
STATE:
σ : Type u_1
α : Type u_2
inst✝³ : CommRing α
inst✝² : TopologicalSpace α
inst✝¹ inst✝ : DiscreteTopology α
⊢ topologicalSpace σ α = ⋯.topology
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/LinearTopology.lean | MvPowerSeries.has_submodules_basis_topology | [227, 1] | [236, 32] | let τ' := (toSubmodulesBasis σ α).topology | σ : Type u_1
α : Type u_2
inst✝³ : CommRing α
inst✝² : TopologicalSpace α
inst✝¹ inst✝ : DiscreteTopology α
τ : TopologicalSpace (MvPowerSeries σ α) := topologicalSpace σ α
⊢ topologicalSpace σ α = ⋯.topology | σ : Type u_1
α : Type u_2
inst✝³ : CommRing α
inst✝² : TopologicalSpace α
inst✝¹ inst✝ : DiscreteTopology α
τ : TopologicalSpace (MvPowerSeries σ α) := topologicalSpace σ α
τ' : TopologicalSpace (MvPowerSeries σ α) := ⋯.topology
⊢ topologicalSpace σ α = ⋯.topology | Please generate a tactic in lean4 to solve the state.
STATE:
σ : Type u_1
α : Type u_2
inst✝³ : CommRing α
inst✝² : TopologicalSpace α
inst✝¹ inst✝ : DiscreteTopology α
τ : TopologicalSpace (MvPowerSeries σ α) := topologicalSpace σ α
⊢ topologicalSpace σ α = ⋯.topology
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/LinearTopology.lean | MvPowerSeries.has_submodules_basis_topology | [227, 1] | [236, 32] | rw [TopologicalAddGroup.ext_iff_nhds_zero τ τ'] | σ : Type u_1
α : Type u_2
inst✝³ : CommRing α
inst✝² : TopologicalSpace α
inst✝¹ inst✝ : DiscreteTopology α
τ : TopologicalSpace (MvPowerSeries σ α) := topologicalSpace σ α
τ' : TopologicalSpace (MvPowerSeries σ α) := ⋯.topology
⊢ topologicalSpace σ α = ⋯.topology | σ : Type u_1
α : Type u_2
inst✝³ : CommRing α
inst✝² : TopologicalSpace α
inst✝¹ inst✝ : DiscreteTopology α
τ : TopologicalSpace (MvPowerSeries σ α) := topologicalSpace σ α
τ' : TopologicalSpace (MvPowerSeries σ α) := ⋯.topology
⊢ nhds 0 = nhds 0 | Please generate a tactic in lean4 to solve the state.
STATE:
σ : Type u_1
α : Type u_2
inst✝³ : CommRing α
inst✝² : TopologicalSpace α
inst✝¹ inst✝ : DiscreteTopology α
τ : TopologicalSpace (MvPowerSeries σ α) := topologicalSpace σ α
τ' : TopologicalSpace (MvPowerSeries σ α) := ⋯.topology
⊢ topologicalSpace σ α = ⋯.top... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/LinearTopology.lean | MvPowerSeries.has_submodules_basis_topology | [227, 1] | [236, 32] | ext s | σ : Type u_1
α : Type u_2
inst✝³ : CommRing α
inst✝² : TopologicalSpace α
inst✝¹ inst✝ : DiscreteTopology α
τ : TopologicalSpace (MvPowerSeries σ α) := topologicalSpace σ α
τ' : TopologicalSpace (MvPowerSeries σ α) := ⋯.topology
⊢ nhds 0 = nhds 0 | case a
σ : Type u_1
α : Type u_2
inst✝³ : CommRing α
inst✝² : TopologicalSpace α
inst✝¹ inst✝ : DiscreteTopology α
τ : TopologicalSpace (MvPowerSeries σ α) := topologicalSpace σ α
τ' : TopologicalSpace (MvPowerSeries σ α) := ⋯.topology
s : Set (MvPowerSeries σ α)
⊢ s ∈ nhds 0 ↔ s ∈ nhds 0 | Please generate a tactic in lean4 to solve the state.
STATE:
σ : Type u_1
α : Type u_2
inst✝³ : CommRing α
inst✝² : TopologicalSpace α
inst✝¹ inst✝ : DiscreteTopology α
τ : TopologicalSpace (MvPowerSeries σ α) := topologicalSpace σ α
τ' : TopologicalSpace (MvPowerSeries σ α) := ⋯.topology
⊢ nhds 0 = nhds 0
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/LinearTopology.lean | MvPowerSeries.has_submodules_basis_topology | [227, 1] | [236, 32] | rw [(RingSubgroupsBasis.hasBasis_nhds (toRingSubgroupsBasis σ α) 0).mem_iff] | case a
σ : Type u_1
α : Type u_2
inst✝³ : CommRing α
inst✝² : TopologicalSpace α
inst✝¹ inst✝ : DiscreteTopology α
τ : TopologicalSpace (MvPowerSeries σ α) := topologicalSpace σ α
τ' : TopologicalSpace (MvPowerSeries σ α) := ⋯.topology
s : Set (MvPowerSeries σ α)
⊢ s ∈ nhds 0 ↔ s ∈ nhds 0 | case a
σ : Type u_1
α : Type u_2
inst✝³ : CommRing α
inst✝² : TopologicalSpace α
inst✝¹ inst✝ : DiscreteTopology α
τ : TopologicalSpace (MvPowerSeries σ α) := topologicalSpace σ α
τ' : TopologicalSpace (MvPowerSeries σ α) := ⋯.topology
s : Set (MvPowerSeries σ α)
⊢ s ∈ nhds 0 ↔ ∃ i, True ∧ {b | b - 0 ∈ Submodule.toAddS... | Please generate a tactic in lean4 to solve the state.
STATE:
case a
σ : Type u_1
α : Type u_2
inst✝³ : CommRing α
inst✝² : TopologicalSpace α
inst✝¹ inst✝ : DiscreteTopology α
τ : TopologicalSpace (MvPowerSeries σ α) := topologicalSpace σ α
τ' : TopologicalSpace (MvPowerSeries σ α) := ⋯.topology
s : Set (MvPowerSeries ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/LinearTopology.lean | MvPowerSeries.has_submodules_basis_topology | [227, 1] | [236, 32] | simp only [sub_zero, Submodule.mem_toAddSubgroup, true_and] | case a
σ : Type u_1
α : Type u_2
inst✝³ : CommRing α
inst✝² : TopologicalSpace α
inst✝¹ inst✝ : DiscreteTopology α
τ : TopologicalSpace (MvPowerSeries σ α) := topologicalSpace σ α
τ' : TopologicalSpace (MvPowerSeries σ α) := ⋯.topology
s : Set (MvPowerSeries σ α)
⊢ s ∈ nhds 0 ↔ ∃ i, True ∧ {b | b - 0 ∈ Submodule.toAddS... | case a
σ : Type u_1
α : Type u_2
inst✝³ : CommRing α
inst✝² : TopologicalSpace α
inst✝¹ inst✝ : DiscreteTopology α
τ : TopologicalSpace (MvPowerSeries σ α) := topologicalSpace σ α
τ' : TopologicalSpace (MvPowerSeries σ α) := ⋯.topology
s : Set (MvPowerSeries σ α)
⊢ s ∈ nhds 0 ↔ ∃ i, {b | b ∈ basis σ α i} ⊆ s | Please generate a tactic in lean4 to solve the state.
STATE:
case a
σ : Type u_1
α : Type u_2
inst✝³ : CommRing α
inst✝² : TopologicalSpace α
inst✝¹ inst✝ : DiscreteTopology α
τ : TopologicalSpace (MvPowerSeries σ α) := topologicalSpace σ α
τ' : TopologicalSpace (MvPowerSeries σ α) := ⋯.topology
s : Set (MvPowerSeries ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/LinearTopology.lean | MvPowerSeries.has_submodules_basis_topology | [227, 1] | [236, 32] | exact mem_nhds_zero_iff σ α s | case a
σ : Type u_1
α : Type u_2
inst✝³ : CommRing α
inst✝² : TopologicalSpace α
inst✝¹ inst✝ : DiscreteTopology α
τ : TopologicalSpace (MvPowerSeries σ α) := topologicalSpace σ α
τ' : TopologicalSpace (MvPowerSeries σ α) := ⋯.topology
s : Set (MvPowerSeries σ α)
⊢ s ∈ nhds 0 ↔ ∃ i, {b | b ∈ basis σ α i} ⊆ s | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a
σ : Type u_1
α : Type u_2
inst✝³ : CommRing α
inst✝² : TopologicalSpace α
inst✝¹ inst✝ : DiscreteTopology α
τ : TopologicalSpace (MvPowerSeries σ α) := topologicalSpace σ α
τ' : TopologicalSpace (MvPowerSeries σ α) := ⋯.topology
s : Set (MvPowerSeries ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeOne.lean | DividedPowerAlgebra.toTrivSqZeroExt_apply_dp_of_two_le | [36, 1] | [39, 11] | rw [toTrivSqZeroExt, liftAlgHom_apply_dp, DividedPowers.OfSquareZero.dpow_of_two_le] | R : Type u_1
inst✝⁶ : CommRing R
M : Type u_2
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : DecidableEq R
inst✝² : DecidableEq M
inst✝¹ : Module Rᵐᵒᵖ M
inst✝ : IsCentralScalar R M
n : ℕ
m : M
hn : 2 ≤ n
⊢ (toTrivSqZeroExt R M) (dp R n m) = 0 | case hn
R : Type u_1
inst✝⁶ : CommRing R
M : Type u_2
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : DecidableEq R
inst✝² : DecidableEq M
inst✝¹ : Module Rᵐᵒᵖ M
inst✝ : IsCentralScalar R M
n : ℕ
m : M
hn : 2 ≤ n
⊢ 2 ≤ n | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝⁶ : CommRing R
M : Type u_2
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : DecidableEq R
inst✝² : DecidableEq M
inst✝¹ : Module Rᵐᵒᵖ M
inst✝ : IsCentralScalar R M
n : ℕ
m : M
hn : 2 ≤ n
⊢ (toTrivSqZeroExt R M) (dp R n m) = 0
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeOne.lean | DividedPowerAlgebra.toTrivSqZeroExt_apply_dp_of_two_le | [36, 1] | [39, 11] | exact hn | case hn
R : Type u_1
inst✝⁶ : CommRing R
M : Type u_2
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : DecidableEq R
inst✝² : DecidableEq M
inst✝¹ : Module Rᵐᵒᵖ M
inst✝ : IsCentralScalar R M
n : ℕ
m : M
hn : 2 ≤ n
⊢ 2 ≤ n | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hn
R : Type u_1
inst✝⁶ : CommRing R
M : Type u_2
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : DecidableEq R
inst✝² : DecidableEq M
inst✝¹ : Module Rᵐᵒᵖ M
inst✝ : IsCentralScalar R M
n : ℕ
m : M
hn : 2 ≤ n
⊢ 2 ≤ n
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeOne.lean | DividedPowerAlgebra.grade_one_eq_span | [44, 1] | [68, 31] | apply le_antisymm | R : Type u_1
inst✝⁶ : CommRing R
M : Type u_2
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : DecidableEq R
inst✝² : DecidableEq M
inst✝¹ : Module Rᵐᵒᵖ M
inst✝ : IsCentralScalar R M
⊢ grade R M 1 = Submodule.span R (Set.range (dp R 1)) | case a
R : Type u_1
inst✝⁶ : CommRing R
M : Type u_2
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : DecidableEq R
inst✝² : DecidableEq M
inst✝¹ : Module Rᵐᵒᵖ M
inst✝ : IsCentralScalar R M
⊢ grade R M 1 ≤ Submodule.span R (Set.range (dp R 1))
case a
R : Type u_1
inst✝⁶ : CommRing R
M : Type u_2
inst✝⁵ : AddCommGr... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝⁶ : CommRing R
M : Type u_2
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : DecidableEq R
inst✝² : DecidableEq M
inst✝¹ : Module Rᵐᵒᵖ M
inst✝ : IsCentralScalar R M
⊢ grade R M 1 = Submodule.span R (Set.range (dp R 1))
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeOne.lean | DividedPowerAlgebra.grade_one_eq_span | [44, 1] | [68, 31] | intro p hp | case a
R : Type u_1
inst✝⁶ : CommRing R
M : Type u_2
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : DecidableEq R
inst✝² : DecidableEq M
inst✝¹ : Module Rᵐᵒᵖ M
inst✝ : IsCentralScalar R M
⊢ grade R M 1 ≤ Submodule.span R (Set.range (dp R 1)) | case a
R : Type u_1
inst✝⁶ : CommRing R
M : Type u_2
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : DecidableEq R
inst✝² : DecidableEq M
inst✝¹ : Module Rᵐᵒᵖ M
inst✝ : IsCentralScalar R M
p : DividedPowerAlgebra R M
hp : p ∈ grade R M 1
⊢ p ∈ Submodule.span R (Set.range (dp R 1)) | Please generate a tactic in lean4 to solve the state.
STATE:
case a
R : Type u_1
inst✝⁶ : CommRing R
M : Type u_2
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : DecidableEq R
inst✝² : DecidableEq M
inst✝¹ : Module Rᵐᵒᵖ M
inst✝ : IsCentralScalar R M
⊢ grade R M 1 ≤ Submodule.span R (Set.range (dp R 1))
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeOne.lean | DividedPowerAlgebra.grade_one_eq_span | [44, 1] | [68, 31] | obtain ⟨q, hq1, hqp⟩ := surjective_of_supported' R M ⟨p, hp⟩ | case a
R : Type u_1
inst✝⁶ : CommRing R
M : Type u_2
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : DecidableEq R
inst✝² : DecidableEq M
inst✝¹ : Module Rᵐᵒᵖ M
inst✝ : IsCentralScalar R M
p : DividedPowerAlgebra R M
hp : p ∈ grade R M 1
⊢ p ∈ Submodule.span R (Set.range (dp R 1)) | case a.intro.intro
R : Type u_1
inst✝⁶ : CommRing R
M : Type u_2
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : DecidableEq R
inst✝² : DecidableEq M
inst✝¹ : Module Rᵐᵒᵖ M
inst✝ : IsCentralScalar R M
p : DividedPowerAlgebra R M
hp : p ∈ grade R M 1
q : ↥(supported R {nm | 0 < nm.1})
hq1 : IsWeightedHomogeneous Pr... | Please generate a tactic in lean4 to solve the state.
STATE:
case a
R : Type u_1
inst✝⁶ : CommRing R
M : Type u_2
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : DecidableEq R
inst✝² : DecidableEq M
inst✝¹ : Module Rᵐᵒᵖ M
inst✝ : IsCentralScalar R M
p : DividedPowerAlgebra R M
hp : p ∈ grade R M 1
⊢ p ∈ Submodule.... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeOne.lean | DividedPowerAlgebra.grade_one_eq_span | [44, 1] | [68, 31] | simp only at hqp | case a.intro.intro
R : Type u_1
inst✝⁶ : CommRing R
M : Type u_2
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : DecidableEq R
inst✝² : DecidableEq M
inst✝¹ : Module Rᵐᵒᵖ M
inst✝ : IsCentralScalar R M
p : DividedPowerAlgebra R M
hp : p ∈ grade R M 1
q : ↥(supported R {nm | 0 < nm.1})
hq1 : IsWeightedHomogeneous Pr... | case a.intro.intro
R : Type u_1
inst✝⁶ : CommRing R
M : Type u_2
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : DecidableEq R
inst✝² : DecidableEq M
inst✝¹ : Module Rᵐᵒᵖ M
inst✝ : IsCentralScalar R M
p : DividedPowerAlgebra R M
hp : p ∈ grade R M 1
q : ↥(supported R {nm | 0 < nm.1})
hq1 : IsWeightedHomogeneous Pr... | Please generate a tactic in lean4 to solve the state.
STATE:
case a.intro.intro
R : Type u_1
inst✝⁶ : CommRing R
M : Type u_2
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : DecidableEq R
inst✝² : DecidableEq M
inst✝¹ : Module Rᵐᵒᵖ M
inst✝ : IsCentralScalar R M
p : DividedPowerAlgebra R M
hp : p ∈ grade R M 1
q : ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeOne.lean | DividedPowerAlgebra.grade_one_eq_span | [44, 1] | [68, 31] | simp only [IsWeightedHomogeneous, ne_eq] at hq1 | case a.intro.intro
R : Type u_1
inst✝⁶ : CommRing R
M : Type u_2
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : DecidableEq R
inst✝² : DecidableEq M
inst✝¹ : Module Rᵐᵒᵖ M
inst✝ : IsCentralScalar R M
p : DividedPowerAlgebra R M
hp : p ∈ grade R M 1
q : ↥(supported R {nm | 0 < nm.1})
hq1 : IsWeightedHomogeneous Pr... | case a.intro.intro
R : Type u_1
inst✝⁶ : CommRing R
M : Type u_2
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : DecidableEq R
inst✝² : DecidableEq M
inst✝¹ : Module Rᵐᵒᵖ M
inst✝ : IsCentralScalar R M
p : DividedPowerAlgebra R M
hp : p ∈ grade R M 1
q : ↥(supported R {nm | 0 < nm.1})
hq1 : ∀ ⦃d : ℕ × M →₀ ℕ⦄, ¬coe... | Please generate a tactic in lean4 to solve the state.
STATE:
case a.intro.intro
R : Type u_1
inst✝⁶ : CommRing R
M : Type u_2
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : DecidableEq R
inst✝² : DecidableEq M
inst✝¹ : Module Rᵐᵒᵖ M
inst✝ : IsCentralScalar R M
p : DividedPowerAlgebra R M
hp : p ∈ grade R M 1
q : ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeOne.lean | DividedPowerAlgebra.grade_one_eq_span | [44, 1] | [68, 31] | rw [← hqp, (q : MvPolynomial (ℕ × M) R).as_sum, map_sum] | case a.intro.intro
R : Type u_1
inst✝⁶ : CommRing R
M : Type u_2
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : DecidableEq R
inst✝² : DecidableEq M
inst✝¹ : Module Rᵐᵒᵖ M
inst✝ : IsCentralScalar R M
p : DividedPowerAlgebra R M
hp : p ∈ grade R M 1
q : ↥(supported R {nm | 0 < nm.1})
hq1 : ∀ ⦃d : ℕ × M →₀ ℕ⦄, ¬coe... | case a.intro.intro
R : Type u_1
inst✝⁶ : CommRing R
M : Type u_2
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : DecidableEq R
inst✝² : DecidableEq M
inst✝¹ : Module Rᵐᵒᵖ M
inst✝ : IsCentralScalar R M
p : DividedPowerAlgebra R M
hp : p ∈ grade R M 1
q : ↥(supported R {nm | 0 < nm.1})
hq1 : ∀ ⦃d : ℕ × M →₀ ℕ⦄, ¬coe... | Please generate a tactic in lean4 to solve the state.
STATE:
case a.intro.intro
R : Type u_1
inst✝⁶ : CommRing R
M : Type u_2
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : DecidableEq R
inst✝² : DecidableEq M
inst✝¹ : Module Rᵐᵒᵖ M
inst✝ : IsCentralScalar R M
p : DividedPowerAlgebra R M
hp : p ∈ grade R M 1
q : ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeOne.lean | DividedPowerAlgebra.grade_one_eq_span | [44, 1] | [68, 31] | apply Submodule.sum_mem (Submodule.span R (Set.range (dp R 1))) | case a.intro.intro
R : Type u_1
inst✝⁶ : CommRing R
M : Type u_2
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : DecidableEq R
inst✝² : DecidableEq M
inst✝¹ : Module Rᵐᵒᵖ M
inst✝ : IsCentralScalar R M
p : DividedPowerAlgebra R M
hp : p ∈ grade R M 1
q : ↥(supported R {nm | 0 < nm.1})
hq1 : ∀ ⦃d : ℕ × M →₀ ℕ⦄, ¬coe... | case a.intro.intro
R : Type u_1
inst✝⁶ : CommRing R
M : Type u_2
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : DecidableEq R
inst✝² : DecidableEq M
inst✝¹ : Module Rᵐᵒᵖ M
inst✝ : IsCentralScalar R M
p : DividedPowerAlgebra R M
hp : p ∈ grade R M 1
q : ↥(supported R {nm | 0 < nm.1})
hq1 : ∀ ⦃d : ℕ × M →₀ ℕ⦄, ¬coe... | Please generate a tactic in lean4 to solve the state.
STATE:
case a.intro.intro
R : Type u_1
inst✝⁶ : CommRing R
M : Type u_2
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : DecidableEq R
inst✝² : DecidableEq M
inst✝¹ : Module Rᵐᵒᵖ M
inst✝ : IsCentralScalar R M
p : DividedPowerAlgebra R M
hp : p ∈ grade R M 1
q : ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeOne.lean | DividedPowerAlgebra.grade_one_eq_span | [44, 1] | [68, 31] | intro d hd | case a.intro.intro
R : Type u_1
inst✝⁶ : CommRing R
M : Type u_2
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : DecidableEq R
inst✝² : DecidableEq M
inst✝¹ : Module Rᵐᵒᵖ M
inst✝ : IsCentralScalar R M
p : DividedPowerAlgebra R M
hp : p ∈ grade R M 1
q : ↥(supported R {nm | 0 < nm.1})
hq1 : ∀ ⦃d : ℕ × M →₀ ℕ⦄, ¬coe... | case a.intro.intro
R : Type u_1
inst✝⁶ : CommRing R
M : Type u_2
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : DecidableEq R
inst✝² : DecidableEq M
inst✝¹ : Module Rᵐᵒᵖ M
inst✝ : IsCentralScalar R M
p : DividedPowerAlgebra R M
hp : p ∈ grade R M 1
q : ↥(supported R {nm | 0 < nm.1})
hq1 : ∀ ⦃d : ℕ × M →₀ ℕ⦄, ¬coe... | Please generate a tactic in lean4 to solve the state.
STATE:
case a.intro.intro
R : Type u_1
inst✝⁶ : CommRing R
M : Type u_2
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : DecidableEq R
inst✝² : DecidableEq M
inst✝¹ : Module Rᵐᵒᵖ M
inst✝ : IsCentralScalar R M
p : DividedPowerAlgebra R M
hp : p ∈ grade R M 1
q : ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeOne.lean | DividedPowerAlgebra.grade_one_eq_span | [44, 1] | [68, 31] | have hsupp : ∀ nm : ℕ × M, nm ∈ d.support → 0 < nm.fst := by
intro nm hnm
apply mem_supported.mp q.2
rw [mem_coe, mem_vars]
exact ⟨d, hd, hnm⟩ | case a.intro.intro
R : Type u_1
inst✝⁶ : CommRing R
M : Type u_2
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : DecidableEq R
inst✝² : DecidableEq M
inst✝¹ : Module Rᵐᵒᵖ M
inst✝ : IsCentralScalar R M
p : DividedPowerAlgebra R M
hp : p ∈ grade R M 1
q : ↥(supported R {nm | 0 < nm.1})
hq1 : ∀ ⦃d : ℕ × M →₀ ℕ⦄, ¬coe... | case a.intro.intro
R : Type u_1
inst✝⁶ : CommRing R
M : Type u_2
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : DecidableEq R
inst✝² : DecidableEq M
inst✝¹ : Module Rᵐᵒᵖ M
inst✝ : IsCentralScalar R M
p : DividedPowerAlgebra R M
hp : p ∈ grade R M 1
q : ↥(supported R {nm | 0 < nm.1})
hq1 : ∀ ⦃d : ℕ × M →₀ ℕ⦄, ¬coe... | Please generate a tactic in lean4 to solve the state.
STATE:
case a.intro.intro
R : Type u_1
inst✝⁶ : CommRing R
M : Type u_2
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : DecidableEq R
inst✝² : DecidableEq M
inst✝¹ : Module Rᵐᵒᵖ M
inst✝ : IsCentralScalar R M
p : DividedPowerAlgebra R M
hp : p ∈ grade R M 1
q : ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeOne.lean | DividedPowerAlgebra.grade_one_eq_span | [44, 1] | [68, 31] | obtain ⟨m, hm⟩ := eq_finsupp_single_of_degree_one M (hq1 (mem_support_iff.mp hd)) hsupp | case a.intro.intro
R : Type u_1
inst✝⁶ : CommRing R
M : Type u_2
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : DecidableEq R
inst✝² : DecidableEq M
inst✝¹ : Module Rᵐᵒᵖ M
inst✝ : IsCentralScalar R M
p : DividedPowerAlgebra R M
hp : p ∈ grade R M 1
q : ↥(supported R {nm | 0 < nm.1})
hq1 : ∀ ⦃d : ℕ × M →₀ ℕ⦄, ¬coe... | case a.intro.intro.intro
R : Type u_1
inst✝⁶ : CommRing R
M : Type u_2
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : DecidableEq R
inst✝² : DecidableEq M
inst✝¹ : Module Rᵐᵒᵖ M
inst✝ : IsCentralScalar R M
p : DividedPowerAlgebra R M
hp : p ∈ grade R M 1
q : ↥(supported R {nm | 0 < nm.1})
hq1 : ∀ ⦃d : ℕ × M →₀ ℕ⦄... | Please generate a tactic in lean4 to solve the state.
STATE:
case a.intro.intro
R : Type u_1
inst✝⁶ : CommRing R
M : Type u_2
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : DecidableEq R
inst✝² : DecidableEq M
inst✝¹ : Module Rᵐᵒᵖ M
inst✝ : IsCentralScalar R M
p : DividedPowerAlgebra R M
hp : p ∈ grade R M 1
q : ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeOne.lean | DividedPowerAlgebra.grade_one_eq_span | [44, 1] | [68, 31] | rw [← hm, monomial_eq, C_mul', map_smul, Finsupp.prod_single_index, pow_one] | case a.intro.intro.intro
R : Type u_1
inst✝⁶ : CommRing R
M : Type u_2
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : DecidableEq R
inst✝² : DecidableEq M
inst✝¹ : Module Rᵐᵒᵖ M
inst✝ : IsCentralScalar R M
p : DividedPowerAlgebra R M
hp : p ∈ grade R M 1
q : ↥(supported R {nm | 0 < nm.1})
hq1 : ∀ ⦃d : ℕ × M →₀ ℕ⦄... | case a.intro.intro.intro
R : Type u_1
inst✝⁶ : CommRing R
M : Type u_2
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : DecidableEq R
inst✝² : DecidableEq M
inst✝¹ : Module Rᵐᵒᵖ M
inst✝ : IsCentralScalar R M
p : DividedPowerAlgebra R M
hp : p ∈ grade R M 1
q : ↥(supported R {nm | 0 < nm.1})
hq1 : ∀ ⦃d : ℕ × M →₀ ℕ⦄... | Please generate a tactic in lean4 to solve the state.
STATE:
case a.intro.intro.intro
R : Type u_1
inst✝⁶ : CommRing R
M : Type u_2
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : DecidableEq R
inst✝² : DecidableEq M
inst✝¹ : Module Rᵐᵒᵖ M
inst✝ : IsCentralScalar R M
p : DividedPowerAlgebra R M
hp : p ∈ grade R M ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeOne.lean | DividedPowerAlgebra.grade_one_eq_span | [44, 1] | [68, 31] | exact Submodule.smul_mem (Submodule.span R (Set.range (dp R 1))) _
(Submodule.subset_span (Set.mem_range.mpr ⟨m, rfl⟩)) | case a.intro.intro.intro
R : Type u_1
inst✝⁶ : CommRing R
M : Type u_2
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : DecidableEq R
inst✝² : DecidableEq M
inst✝¹ : Module Rᵐᵒᵖ M
inst✝ : IsCentralScalar R M
p : DividedPowerAlgebra R M
hp : p ∈ grade R M 1
q : ↥(supported R {nm | 0 < nm.1})
hq1 : ∀ ⦃d : ℕ × M →₀ ℕ⦄... | case a.intro.intro.intro
R : Type u_1
inst✝⁶ : CommRing R
M : Type u_2
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : DecidableEq R
inst✝² : DecidableEq M
inst✝¹ : Module Rᵐᵒᵖ M
inst✝ : IsCentralScalar R M
p : DividedPowerAlgebra R M
hp : p ∈ grade R M 1
q : ↥(supported R {nm | 0 < nm.1})
hq1 : ∀ ⦃d : ℕ × M →₀ ℕ⦄... | Please generate a tactic in lean4 to solve the state.
STATE:
case a.intro.intro.intro
R : Type u_1
inst✝⁶ : CommRing R
M : Type u_2
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : DecidableEq R
inst✝² : DecidableEq M
inst✝¹ : Module Rᵐᵒᵖ M
inst✝ : IsCentralScalar R M
p : DividedPowerAlgebra R M
hp : p ∈ grade R M ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeOne.lean | DividedPowerAlgebra.grade_one_eq_span | [44, 1] | [68, 31] | intro nm hnm | R : Type u_1
inst✝⁶ : CommRing R
M : Type u_2
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : DecidableEq R
inst✝² : DecidableEq M
inst✝¹ : Module Rᵐᵒᵖ M
inst✝ : IsCentralScalar R M
p : DividedPowerAlgebra R M
hp : p ∈ grade R M 1
q : ↥(supported R {nm | 0 < nm.1})
hq1 : ∀ ⦃d : ℕ × M →₀ ℕ⦄, ¬coeff d ↑q = 0 → (weig... | R : Type u_1
inst✝⁶ : CommRing R
M : Type u_2
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : DecidableEq R
inst✝² : DecidableEq M
inst✝¹ : Module Rᵐᵒᵖ M
inst✝ : IsCentralScalar R M
p : DividedPowerAlgebra R M
hp : p ∈ grade R M 1
q : ↥(supported R {nm | 0 < nm.1})
hq1 : ∀ ⦃d : ℕ × M →₀ ℕ⦄, ¬coeff d ↑q = 0 → (weig... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝⁶ : CommRing R
M : Type u_2
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : DecidableEq R
inst✝² : DecidableEq M
inst✝¹ : Module Rᵐᵒᵖ M
inst✝ : IsCentralScalar R M
p : DividedPowerAlgebra R M
hp : p ∈ grade R M 1
q : ↥(supported R {nm |... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeOne.lean | DividedPowerAlgebra.grade_one_eq_span | [44, 1] | [68, 31] | apply mem_supported.mp q.2 | R : Type u_1
inst✝⁶ : CommRing R
M : Type u_2
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : DecidableEq R
inst✝² : DecidableEq M
inst✝¹ : Module Rᵐᵒᵖ M
inst✝ : IsCentralScalar R M
p : DividedPowerAlgebra R M
hp : p ∈ grade R M 1
q : ↥(supported R {nm | 0 < nm.1})
hq1 : ∀ ⦃d : ℕ × M →₀ ℕ⦄, ¬coeff d ↑q = 0 → (weig... | case a
R : Type u_1
inst✝⁶ : CommRing R
M : Type u_2
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : DecidableEq R
inst✝² : DecidableEq M
inst✝¹ : Module Rᵐᵒᵖ M
inst✝ : IsCentralScalar R M
p : DividedPowerAlgebra R M
hp : p ∈ grade R M 1
q : ↥(supported R {nm | 0 < nm.1})
hq1 : ∀ ⦃d : ℕ × M →₀ ℕ⦄, ¬coeff d ↑q = 0 ... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝⁶ : CommRing R
M : Type u_2
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : DecidableEq R
inst✝² : DecidableEq M
inst✝¹ : Module Rᵐᵒᵖ M
inst✝ : IsCentralScalar R M
p : DividedPowerAlgebra R M
hp : p ∈ grade R M 1
q : ↥(supported R {nm |... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeOne.lean | DividedPowerAlgebra.grade_one_eq_span | [44, 1] | [68, 31] | rw [mem_coe, mem_vars] | case a
R : Type u_1
inst✝⁶ : CommRing R
M : Type u_2
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : DecidableEq R
inst✝² : DecidableEq M
inst✝¹ : Module Rᵐᵒᵖ M
inst✝ : IsCentralScalar R M
p : DividedPowerAlgebra R M
hp : p ∈ grade R M 1
q : ↥(supported R {nm | 0 < nm.1})
hq1 : ∀ ⦃d : ℕ × M →₀ ℕ⦄, ¬coeff d ↑q = 0 ... | case a
R : Type u_1
inst✝⁶ : CommRing R
M : Type u_2
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : DecidableEq R
inst✝² : DecidableEq M
inst✝¹ : Module Rᵐᵒᵖ M
inst✝ : IsCentralScalar R M
p : DividedPowerAlgebra R M
hp : p ∈ grade R M 1
q : ↥(supported R {nm | 0 < nm.1})
hq1 : ∀ ⦃d : ℕ × M →₀ ℕ⦄, ¬coeff d ↑q = 0 ... | Please generate a tactic in lean4 to solve the state.
STATE:
case a
R : Type u_1
inst✝⁶ : CommRing R
M : Type u_2
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : DecidableEq R
inst✝² : DecidableEq M
inst✝¹ : Module Rᵐᵒᵖ M
inst✝ : IsCentralScalar R M
p : DividedPowerAlgebra R M
hp : p ∈ grade R M 1
q : ↥(supported ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeOne.lean | DividedPowerAlgebra.grade_one_eq_span | [44, 1] | [68, 31] | exact ⟨d, hd, hnm⟩ | case a
R : Type u_1
inst✝⁶ : CommRing R
M : Type u_2
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : DecidableEq R
inst✝² : DecidableEq M
inst✝¹ : Module Rᵐᵒᵖ M
inst✝ : IsCentralScalar R M
p : DividedPowerAlgebra R M
hp : p ∈ grade R M 1
q : ↥(supported R {nm | 0 < nm.1})
hq1 : ∀ ⦃d : ℕ × M →₀ ℕ⦄, ¬coeff d ↑q = 0 ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a
R : Type u_1
inst✝⁶ : CommRing R
M : Type u_2
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : DecidableEq R
inst✝² : DecidableEq M
inst✝¹ : Module Rᵐᵒᵖ M
inst✝ : IsCentralScalar R M
p : DividedPowerAlgebra R M
hp : p ∈ grade R M 1
q : ↥(supported ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeOne.lean | DividedPowerAlgebra.grade_one_eq_span | [44, 1] | [68, 31] | rw [pow_zero] | case a.intro.intro.intro
R : Type u_1
inst✝⁶ : CommRing R
M : Type u_2
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : DecidableEq R
inst✝² : DecidableEq M
inst✝¹ : Module Rᵐᵒᵖ M
inst✝ : IsCentralScalar R M
p : DividedPowerAlgebra R M
hp : p ∈ grade R M 1
q : ↥(supported R {nm | 0 < nm.1})
hq1 : ∀ ⦃d : ℕ × M →₀ ℕ⦄... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a.intro.intro.intro
R : Type u_1
inst✝⁶ : CommRing R
M : Type u_2
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : DecidableEq R
inst✝² : DecidableEq M
inst✝¹ : Module Rᵐᵒᵖ M
inst✝ : IsCentralScalar R M
p : DividedPowerAlgebra R M
hp : p ∈ grade R M ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeOne.lean | DividedPowerAlgebra.grade_one_eq_span | [44, 1] | [68, 31] | rw [Submodule.span_le] | case a
R : Type u_1
inst✝⁶ : CommRing R
M : Type u_2
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : DecidableEq R
inst✝² : DecidableEq M
inst✝¹ : Module Rᵐᵒᵖ M
inst✝ : IsCentralScalar R M
⊢ Submodule.span R (Set.range (dp R 1)) ≤ grade R M 1 | case a
R : Type u_1
inst✝⁶ : CommRing R
M : Type u_2
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : DecidableEq R
inst✝² : DecidableEq M
inst✝¹ : Module Rᵐᵒᵖ M
inst✝ : IsCentralScalar R M
⊢ Set.range (dp R 1) ⊆ ↑(grade R M 1) | Please generate a tactic in lean4 to solve the state.
STATE:
case a
R : Type u_1
inst✝⁶ : CommRing R
M : Type u_2
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : DecidableEq R
inst✝² : DecidableEq M
inst✝¹ : Module Rᵐᵒᵖ M
inst✝ : IsCentralScalar R M
⊢ Submodule.span R (Set.range (dp R 1)) ≤ grade R M 1
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeOne.lean | DividedPowerAlgebra.grade_one_eq_span | [44, 1] | [68, 31] | intro p hp | case a
R : Type u_1
inst✝⁶ : CommRing R
M : Type u_2
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : DecidableEq R
inst✝² : DecidableEq M
inst✝¹ : Module Rᵐᵒᵖ M
inst✝ : IsCentralScalar R M
⊢ Set.range (dp R 1) ⊆ ↑(grade R M 1) | case a
R : Type u_1
inst✝⁶ : CommRing R
M : Type u_2
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : DecidableEq R
inst✝² : DecidableEq M
inst✝¹ : Module Rᵐᵒᵖ M
inst✝ : IsCentralScalar R M
p : DividedPowerAlgebra R M
hp : p ∈ Set.range (dp R 1)
⊢ p ∈ ↑(grade R M 1) | Please generate a tactic in lean4 to solve the state.
STATE:
case a
R : Type u_1
inst✝⁶ : CommRing R
M : Type u_2
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : DecidableEq R
inst✝² : DecidableEq M
inst✝¹ : Module Rᵐᵒᵖ M
inst✝ : IsCentralScalar R M
⊢ Set.range (dp R 1) ⊆ ↑(grade R M 1)
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeOne.lean | DividedPowerAlgebra.grade_one_eq_span | [44, 1] | [68, 31] | obtain ⟨m, hm⟩ := Set.mem_range.mp hp | case a
R : Type u_1
inst✝⁶ : CommRing R
M : Type u_2
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : DecidableEq R
inst✝² : DecidableEq M
inst✝¹ : Module Rᵐᵒᵖ M
inst✝ : IsCentralScalar R M
p : DividedPowerAlgebra R M
hp : p ∈ Set.range (dp R 1)
⊢ p ∈ ↑(grade R M 1) | case a.intro
R : Type u_1
inst✝⁶ : CommRing R
M : Type u_2
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : DecidableEq R
inst✝² : DecidableEq M
inst✝¹ : Module Rᵐᵒᵖ M
inst✝ : IsCentralScalar R M
p : DividedPowerAlgebra R M
hp : p ∈ Set.range (dp R 1)
m : M
hm : dp R 1 m = p
⊢ p ∈ ↑(grade R M 1) | Please generate a tactic in lean4 to solve the state.
STATE:
case a
R : Type u_1
inst✝⁶ : CommRing R
M : Type u_2
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : DecidableEq R
inst✝² : DecidableEq M
inst✝¹ : Module Rᵐᵒᵖ M
inst✝ : IsCentralScalar R M
p : DividedPowerAlgebra R M
hp : p ∈ Set.range (dp R 1)
⊢ p ∈ ↑(g... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeOne.lean | DividedPowerAlgebra.grade_one_eq_span | [44, 1] | [68, 31] | rw [← hm] | case a.intro
R : Type u_1
inst✝⁶ : CommRing R
M : Type u_2
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : DecidableEq R
inst✝² : DecidableEq M
inst✝¹ : Module Rᵐᵒᵖ M
inst✝ : IsCentralScalar R M
p : DividedPowerAlgebra R M
hp : p ∈ Set.range (dp R 1)
m : M
hm : dp R 1 m = p
⊢ p ∈ ↑(grade R M 1) | case a.intro
R : Type u_1
inst✝⁶ : CommRing R
M : Type u_2
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : DecidableEq R
inst✝² : DecidableEq M
inst✝¹ : Module Rᵐᵒᵖ M
inst✝ : IsCentralScalar R M
p : DividedPowerAlgebra R M
hp : p ∈ Set.range (dp R 1)
m : M
hm : dp R 1 m = p
⊢ dp R 1 m ∈ ↑(grade R M 1) | Please generate a tactic in lean4 to solve the state.
STATE:
case a.intro
R : Type u_1
inst✝⁶ : CommRing R
M : Type u_2
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : DecidableEq R
inst✝² : DecidableEq M
inst✝¹ : Module Rᵐᵒᵖ M
inst✝ : IsCentralScalar R M
p : DividedPowerAlgebra R M
hp : p ∈ Set.range (dp R 1)
m :... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeOne.lean | DividedPowerAlgebra.grade_one_eq_span | [44, 1] | [68, 31] | exact dp_mem_grade R M 1 m | case a.intro
R : Type u_1
inst✝⁶ : CommRing R
M : Type u_2
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : DecidableEq R
inst✝² : DecidableEq M
inst✝¹ : Module Rᵐᵒᵖ M
inst✝ : IsCentralScalar R M
p : DividedPowerAlgebra R M
hp : p ∈ Set.range (dp R 1)
m : M
hm : dp R 1 m = p
⊢ dp R 1 m ∈ ↑(grade R M 1) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a.intro
R : Type u_1
inst✝⁶ : CommRing R
M : Type u_2
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : DecidableEq R
inst✝² : DecidableEq M
inst✝¹ : Module Rᵐᵒᵖ M
inst✝ : IsCentralScalar R M
p : DividedPowerAlgebra R M
hp : p ∈ Set.range (dp R 1)
m :... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeOne.lean | DividedPowerAlgebra.grade_one_eq_span' | [70, 1] | [76, 29] | apply Submodule.map_injective_of_injective (grade R M 1).injective_subtype | R : Type u_1
inst✝⁶ : CommRing R
M : Type u_2
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : DecidableEq R
inst✝² : DecidableEq M
inst✝¹ : Module Rᵐᵒᵖ M
inst✝ : IsCentralScalar R M
⊢ ⊤ = Submodule.span R (Set.range fun m => ⟨dp R 1 m, ⋯⟩) | case a
R : Type u_1
inst✝⁶ : CommRing R
M : Type u_2
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : DecidableEq R
inst✝² : DecidableEq M
inst✝¹ : Module Rᵐᵒᵖ M
inst✝ : IsCentralScalar R M
⊢ Submodule.map (grade R M 1).subtype ⊤ =
Submodule.map (grade R M 1).subtype (Submodule.span R (Set.range fun m => ⟨dp R ... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝⁶ : CommRing R
M : Type u_2
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : DecidableEq R
inst✝² : DecidableEq M
inst✝¹ : Module Rᵐᵒᵖ M
inst✝ : IsCentralScalar R M
⊢ ⊤ = Submodule.span R (Set.range fun m => ⟨dp R 1 m, ⋯⟩)
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeOne.lean | DividedPowerAlgebra.grade_one_eq_span' | [70, 1] | [76, 29] | rw [Submodule.map_subtype_top, Submodule.map_span] | case a
R : Type u_1
inst✝⁶ : CommRing R
M : Type u_2
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : DecidableEq R
inst✝² : DecidableEq M
inst✝¹ : Module Rᵐᵒᵖ M
inst✝ : IsCentralScalar R M
⊢ Submodule.map (grade R M 1).subtype ⊤ =
Submodule.map (grade R M 1).subtype (Submodule.span R (Set.range fun m => ⟨dp R ... | case a
R : Type u_1
inst✝⁶ : CommRing R
M : Type u_2
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : DecidableEq R
inst✝² : DecidableEq M
inst✝¹ : Module Rᵐᵒᵖ M
inst✝ : IsCentralScalar R M
⊢ grade R M 1 = Submodule.span R (⇑(grade R M 1).subtype '' Set.range fun m => ⟨dp R 1 m, ⋯⟩) | Please generate a tactic in lean4 to solve the state.
STATE:
case a
R : Type u_1
inst✝⁶ : CommRing R
M : Type u_2
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : DecidableEq R
inst✝² : DecidableEq M
inst✝¹ : Module Rᵐᵒᵖ M
inst✝ : IsCentralScalar R M
⊢ Submodule.map (grade R M 1).subtype ⊤ =
Submodule.map (grad... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeOne.lean | DividedPowerAlgebra.grade_one_eq_span' | [70, 1] | [76, 29] | simp_rw [grade_one_eq_span R M] | case a
R : Type u_1
inst✝⁶ : CommRing R
M : Type u_2
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : DecidableEq R
inst✝² : DecidableEq M
inst✝¹ : Module Rᵐᵒᵖ M
inst✝ : IsCentralScalar R M
⊢ grade R M 1 = Submodule.span R (⇑(grade R M 1).subtype '' Set.range fun m => ⟨dp R 1 m, ⋯⟩) | case a
R : Type u_1
inst✝⁶ : CommRing R
M : Type u_2
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : DecidableEq R
inst✝² : DecidableEq M
inst✝¹ : Module Rᵐᵒᵖ M
inst✝ : IsCentralScalar R M
⊢ Submodule.span R (Set.range (dp R 1)) = Submodule.span R (⇑(grade R M 1).subtype '' Set.range fun m => ⟨dp R 1 m, ⋯⟩) | Please generate a tactic in lean4 to solve the state.
STATE:
case a
R : Type u_1
inst✝⁶ : CommRing R
M : Type u_2
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : DecidableEq R
inst✝² : DecidableEq M
inst✝¹ : Module Rᵐᵒᵖ M
inst✝ : IsCentralScalar R M
⊢ grade R M 1 = Submodule.span R (⇑(grade R M 1).subtype '' Set.r... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeOne.lean | DividedPowerAlgebra.grade_one_eq_span' | [70, 1] | [76, 29] | rw [← Set.range_comp] | case a
R : Type u_1
inst✝⁶ : CommRing R
M : Type u_2
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : DecidableEq R
inst✝² : DecidableEq M
inst✝¹ : Module Rᵐᵒᵖ M
inst✝ : IsCentralScalar R M
⊢ Submodule.span R (Set.range (dp R 1)) = Submodule.span R (⇑(grade R M 1).subtype '' Set.range fun m => ⟨dp R 1 m, ⋯⟩) | case a
R : Type u_1
inst✝⁶ : CommRing R
M : Type u_2
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : DecidableEq R
inst✝² : DecidableEq M
inst✝¹ : Module Rᵐᵒᵖ M
inst✝ : IsCentralScalar R M
⊢ Submodule.span R (Set.range (dp R 1)) = Submodule.span R (Set.range (⇑(grade R M 1).subtype ∘ fun m => ⟨dp R 1 m, ⋯⟩)) | Please generate a tactic in lean4 to solve the state.
STATE:
case a
R : Type u_1
inst✝⁶ : CommRing R
M : Type u_2
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : DecidableEq R
inst✝² : DecidableEq M
inst✝¹ : Module Rᵐᵒᵖ M
inst✝ : IsCentralScalar R M
⊢ Submodule.span R (Set.range (dp R 1)) = Submodule.span R (⇑(gra... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeOne.lean | DividedPowerAlgebra.grade_one_eq_span' | [70, 1] | [76, 29] | rfl | case a
R : Type u_1
inst✝⁶ : CommRing R
M : Type u_2
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : DecidableEq R
inst✝² : DecidableEq M
inst✝¹ : Module Rᵐᵒᵖ M
inst✝ : IsCentralScalar R M
⊢ Submodule.span R (Set.range (dp R 1)) = Submodule.span R (Set.range (⇑(grade R M 1).subtype ∘ fun m => ⟨dp R 1 m, ⋯⟩)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a
R : Type u_1
inst✝⁶ : CommRing R
M : Type u_2
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : DecidableEq R
inst✝² : DecidableEq M
inst✝¹ : Module Rᵐᵒᵖ M
inst✝ : IsCentralScalar R M
⊢ Submodule.span R (Set.range (dp R 1)) = Submodule.span R (Set.r... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeOne.lean | DividedPowerAlgebra.deg_one_right_inv | [78, 1] | [93, 24] | simp only [Function.rightInverse_iff_comp, ← LinearMap.coe_comp, ← @LinearMap.id_coe R] | R : Type u_1
inst✝⁶ : CommRing R
M : Type u_2
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : DecidableEq R
inst✝² : DecidableEq M
inst✝¹ : Module Rᵐᵒᵖ M
inst✝ : IsCentralScalar R M
⊢ Function.RightInverse (⇑(sndHom R M) ∘ ⇑(toTrivSqZeroExt R M).toLinearMap ∘ ⇑(grade R M 1).subtype)
(⇑(proj' R M 1) ∘ ⇑(ι R M)) | R : Type u_1
inst✝⁶ : CommRing R
M : Type u_2
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : DecidableEq R
inst✝² : DecidableEq M
inst✝¹ : Module Rᵐᵒᵖ M
inst✝ : IsCentralScalar R M
⊢ ⇑((proj' R M 1 ∘ₗ ι R M) ∘ₗ sndHom R M ∘ₗ (toTrivSqZeroExt R M).toLinearMap ∘ₗ (grade R M 1).subtype) = ⇑LinearMap.id | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝⁶ : CommRing R
M : Type u_2
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : DecidableEq R
inst✝² : DecidableEq M
inst✝¹ : Module Rᵐᵒᵖ M
inst✝ : IsCentralScalar R M
⊢ Function.RightInverse (⇑(sndHom R M) ∘ ⇑(toTrivSqZeroExt R M).toLinear... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeOne.lean | DividedPowerAlgebra.deg_one_right_inv | [78, 1] | [93, 24] | rw [DFunLike.coe_injective.eq_iff] | R : Type u_1
inst✝⁶ : CommRing R
M : Type u_2
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : DecidableEq R
inst✝² : DecidableEq M
inst✝¹ : Module Rᵐᵒᵖ M
inst✝ : IsCentralScalar R M
⊢ ⇑((proj' R M 1 ∘ₗ ι R M) ∘ₗ sndHom R M ∘ₗ (toTrivSqZeroExt R M).toLinearMap ∘ₗ (grade R M 1).subtype) = ⇑LinearMap.id | R : Type u_1
inst✝⁶ : CommRing R
M : Type u_2
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : DecidableEq R
inst✝² : DecidableEq M
inst✝¹ : Module Rᵐᵒᵖ M
inst✝ : IsCentralScalar R M
⊢ (proj' R M 1 ∘ₗ ι R M) ∘ₗ sndHom R M ∘ₗ (toTrivSqZeroExt R M).toLinearMap ∘ₗ (grade R M 1).subtype = LinearMap.id | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝⁶ : CommRing R
M : Type u_2
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : DecidableEq R
inst✝² : DecidableEq M
inst✝¹ : Module Rᵐᵒᵖ M
inst✝ : IsCentralScalar R M
⊢ ⇑((proj' R M 1 ∘ₗ ι R M) ∘ₗ sndHom R M ∘ₗ (toTrivSqZeroExt R M).toLine... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeOne.lean | DividedPowerAlgebra.deg_one_right_inv | [78, 1] | [93, 24] | apply LinearMap.ext_on_range (grade_one_eq_span' R M).symm | R : Type u_1
inst✝⁶ : CommRing R
M : Type u_2
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : DecidableEq R
inst✝² : DecidableEq M
inst✝¹ : Module Rᵐᵒᵖ M
inst✝ : IsCentralScalar R M
⊢ (proj' R M 1 ∘ₗ ι R M) ∘ₗ sndHom R M ∘ₗ (toTrivSqZeroExt R M).toLinearMap ∘ₗ (grade R M 1).subtype = LinearMap.id | R : Type u_1
inst✝⁶ : CommRing R
M : Type u_2
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : DecidableEq R
inst✝² : DecidableEq M
inst✝¹ : Module Rᵐᵒᵖ M
inst✝ : IsCentralScalar R M
⊢ ∀ (i : M),
((proj' R M 1 ∘ₗ ι R M) ∘ₗ sndHom R M ∘ₗ (toTrivSqZeroExt R M).toLinearMap ∘ₗ (grade R M 1).subtype) ⟨dp R 1 i, ⋯⟩ =... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝⁶ : CommRing R
M : Type u_2
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : DecidableEq R
inst✝² : DecidableEq M
inst✝¹ : Module Rᵐᵒᵖ M
inst✝ : IsCentralScalar R M
⊢ (proj' R M 1 ∘ₗ ι R M) ∘ₗ sndHom R M ∘ₗ (toTrivSqZeroExt R M).toLinear... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeOne.lean | DividedPowerAlgebra.deg_one_right_inv | [78, 1] | [93, 24] | intro m | R : Type u_1
inst✝⁶ : CommRing R
M : Type u_2
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : DecidableEq R
inst✝² : DecidableEq M
inst✝¹ : Module Rᵐᵒᵖ M
inst✝ : IsCentralScalar R M
⊢ ∀ (i : M),
((proj' R M 1 ∘ₗ ι R M) ∘ₗ sndHom R M ∘ₗ (toTrivSqZeroExt R M).toLinearMap ∘ₗ (grade R M 1).subtype) ⟨dp R 1 i, ⋯⟩ =... | R : Type u_1
inst✝⁶ : CommRing R
M : Type u_2
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : DecidableEq R
inst✝² : DecidableEq M
inst✝¹ : Module Rᵐᵒᵖ M
inst✝ : IsCentralScalar R M
m : M
⊢ ((proj' R M 1 ∘ₗ ι R M) ∘ₗ sndHom R M ∘ₗ (toTrivSqZeroExt R M).toLinearMap ∘ₗ (grade R M 1).subtype) ⟨dp R 1 m, ⋯⟩ =
Line... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝⁶ : CommRing R
M : Type u_2
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : DecidableEq R
inst✝² : DecidableEq M
inst✝¹ : Module Rᵐᵒᵖ M
inst✝ : IsCentralScalar R M
⊢ ∀ (i : M),
((proj' R M 1 ∘ₗ ι R M) ∘ₗ sndHom R M ∘ₗ (toTrivSqZeroE... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeOne.lean | DividedPowerAlgebra.deg_one_right_inv | [78, 1] | [93, 24] | simp only [proj', proj, ι, LinearMap.coe_comp, LinearMap.coe_mk, AddHom.coe_mk,
Submodule.coeSubtype, comp_apply, AlgHom.toLinearMap_apply, sndHom_apply,
LinearMap.id_coe, id_eq] | R : Type u_1
inst✝⁶ : CommRing R
M : Type u_2
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : DecidableEq R
inst✝² : DecidableEq M
inst✝¹ : Module Rᵐᵒᵖ M
inst✝ : IsCentralScalar R M
m : M
⊢ ((proj' R M 1 ∘ₗ ι R M) ∘ₗ sndHom R M ∘ₗ (toTrivSqZeroExt R M).toLinearMap ∘ₗ (grade R M 1).subtype) ⟨dp R 1 m, ⋯⟩ =
Line... | R : Type u_1
inst✝⁶ : CommRing R
M : Type u_2
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : DecidableEq R
inst✝² : DecidableEq M
inst✝¹ : Module Rᵐᵒᵖ M
inst✝ : IsCentralScalar R M
m : M
⊢ ((DirectSum.decompose (grade R M)) (dp R 1 ((toTrivSqZeroExt R M) (dp R 1 m)).snd)) 1 = ⟨dp R 1 m, ⋯⟩ | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝⁶ : CommRing R
M : Type u_2
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : DecidableEq R
inst✝² : DecidableEq M
inst✝¹ : Module Rᵐᵒᵖ M
inst✝ : IsCentralScalar R M
m : M
⊢ ((proj' R M 1 ∘ₗ ι R M) ∘ₗ sndHom R M ∘ₗ (toTrivSqZeroExt R M).t... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeOne.lean | DividedPowerAlgebra.deg_one_right_inv | [78, 1] | [93, 24] | ext | R : Type u_1
inst✝⁶ : CommRing R
M : Type u_2
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : DecidableEq R
inst✝² : DecidableEq M
inst✝¹ : Module Rᵐᵒᵖ M
inst✝ : IsCentralScalar R M
m : M
⊢ ((DirectSum.decompose (grade R M)) (dp R 1 ((toTrivSqZeroExt R M) (dp R 1 m)).snd)) 1 = ⟨dp R 1 m, ⋯⟩ | case a
R : Type u_1
inst✝⁶ : CommRing R
M : Type u_2
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : DecidableEq R
inst✝² : DecidableEq M
inst✝¹ : Module Rᵐᵒᵖ M
inst✝ : IsCentralScalar R M
m : M
⊢ ↑(((DirectSum.decompose (grade R M)) (dp R 1 ((toTrivSqZeroExt R M) (dp R 1 m)).snd)) 1) = ↑⟨dp R 1 m, ⋯⟩ | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝⁶ : CommRing R
M : Type u_2
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : DecidableEq R
inst✝² : DecidableEq M
inst✝¹ : Module Rᵐᵒᵖ M
inst✝ : IsCentralScalar R M
m : M
⊢ ((DirectSum.decompose (grade R M)) (dp R 1 ((toTrivSqZeroExt R M... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeOne.lean | DividedPowerAlgebra.deg_one_right_inv | [78, 1] | [93, 24] | dsimp only | case a
R : Type u_1
inst✝⁶ : CommRing R
M : Type u_2
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : DecidableEq R
inst✝² : DecidableEq M
inst✝¹ : Module Rᵐᵒᵖ M
inst✝ : IsCentralScalar R M
m : M
⊢ ↑(((DirectSum.decompose (grade R M)) (dp R 1 ((toTrivSqZeroExt R M) (dp R 1 m)).snd)) 1) = ↑⟨dp R 1 m, ⋯⟩ | case a
R : Type u_1
inst✝⁶ : CommRing R
M : Type u_2
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : DecidableEq R
inst✝² : DecidableEq M
inst✝¹ : Module Rᵐᵒᵖ M
inst✝ : IsCentralScalar R M
m : M
⊢ ↑(((DirectSum.decompose (grade R M)) (dp R 1 ((toTrivSqZeroExt R M) (dp R 1 m)).snd)) 1) = dp R 1 m | Please generate a tactic in lean4 to solve the state.
STATE:
case a
R : Type u_1
inst✝⁶ : CommRing R
M : Type u_2
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : DecidableEq R
inst✝² : DecidableEq M
inst✝¹ : Module Rᵐᵒᵖ M
inst✝ : IsCentralScalar R M
m : M
⊢ ↑(((DirectSum.decompose (grade R M)) (dp R 1 ((toTrivSqZe... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeOne.lean | DividedPowerAlgebra.deg_one_right_inv | [78, 1] | [93, 24] | rw [← ι_def R M m, toTrivSqZeroExt_ι, ← ι_def, snd_inr, decompose_of_mem_same] | case a
R : Type u_1
inst✝⁶ : CommRing R
M : Type u_2
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : DecidableEq R
inst✝² : DecidableEq M
inst✝¹ : Module Rᵐᵒᵖ M
inst✝ : IsCentralScalar R M
m : M
⊢ ↑(((DirectSum.decompose (grade R M)) (dp R 1 ((toTrivSqZeroExt R M) (dp R 1 m)).snd)) 1) = dp R 1 m | case a.hx
R : Type u_1
inst✝⁶ : CommRing R
M : Type u_2
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : DecidableEq R
inst✝² : DecidableEq M
inst✝¹ : Module Rᵐᵒᵖ M
inst✝ : IsCentralScalar R M
m : M
⊢ (ι R M) m ∈ grade R M 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case a
R : Type u_1
inst✝⁶ : CommRing R
M : Type u_2
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : DecidableEq R
inst✝² : DecidableEq M
inst✝¹ : Module Rᵐᵒᵖ M
inst✝ : IsCentralScalar R M
m : M
⊢ ↑(((DirectSum.decompose (grade R M)) (dp R 1 ((toTrivSqZe... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeOne.lean | DividedPowerAlgebra.deg_one_right_inv | [78, 1] | [93, 24] | apply ι_mem_grade_one | case a.hx
R : Type u_1
inst✝⁶ : CommRing R
M : Type u_2
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : DecidableEq R
inst✝² : DecidableEq M
inst✝¹ : Module Rᵐᵒᵖ M
inst✝ : IsCentralScalar R M
m : M
⊢ (ι R M) m ∈ grade R M 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a.hx
R : Type u_1
inst✝⁶ : CommRing R
M : Type u_2
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : DecidableEq R
inst✝² : DecidableEq M
inst✝¹ : Module Rᵐᵒᵖ M
inst✝ : IsCentralScalar R M
m : M
⊢ (ι R M) m ∈ grade R M 1
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeOne.lean | DividedPowerAlgebra.deg_one_left_inv | [95, 1] | [102, 24] | intro m | R : Type u_1
inst✝⁶ : CommRing R
M : Type u_2
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : DecidableEq R
inst✝² : DecidableEq M
inst✝¹ : Module Rᵐᵒᵖ M
inst✝ : IsCentralScalar R M
⊢ LeftInverse (fun x => ((toTrivSqZeroExt R M) ↑x).snd) (⇑(proj' R M 1) ∘ ⇑(ι R M)) | R : Type u_1
inst✝⁶ : CommRing R
M : Type u_2
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : DecidableEq R
inst✝² : DecidableEq M
inst✝¹ : Module Rᵐᵒᵖ M
inst✝ : IsCentralScalar R M
m : M
⊢ (fun x => ((toTrivSqZeroExt R M) ↑x).snd) ((⇑(proj' R M 1) ∘ ⇑(ι R M)) m) = m | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝⁶ : CommRing R
M : Type u_2
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : DecidableEq R
inst✝² : DecidableEq M
inst✝¹ : Module Rᵐᵒᵖ M
inst✝ : IsCentralScalar R M
⊢ LeftInverse (fun x => ((toTrivSqZeroExt R M) ↑x).snd) (⇑(proj' R M 1) ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeOne.lean | DividedPowerAlgebra.deg_one_left_inv | [95, 1] | [102, 24] | simp only [proj', proj, LinearMap.coe_mk, AddHom.coe_mk, ι, Function.comp_apply] | R : Type u_1
inst✝⁶ : CommRing R
M : Type u_2
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : DecidableEq R
inst✝² : DecidableEq M
inst✝¹ : Module Rᵐᵒᵖ M
inst✝ : IsCentralScalar R M
m : M
⊢ (fun x => ((toTrivSqZeroExt R M) ↑x).snd) ((⇑(proj' R M 1) ∘ ⇑(ι R M)) m) = m | R : Type u_1
inst✝⁶ : CommRing R
M : Type u_2
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : DecidableEq R
inst✝² : DecidableEq M
inst✝¹ : Module Rᵐᵒᵖ M
inst✝ : IsCentralScalar R M
m : M
⊢ ((toTrivSqZeroExt R M) ↑(((DirectSum.decompose (grade R M)) (dp R 1 m)) 1)).snd = m | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝⁶ : CommRing R
M : Type u_2
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : DecidableEq R
inst✝² : DecidableEq M
inst✝¹ : Module Rᵐᵒᵖ M
inst✝ : IsCentralScalar R M
m : M
⊢ (fun x => ((toTrivSqZeroExt R M) ↑x).snd) ((⇑(proj' R M 1) ∘ ⇑(ι... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeOne.lean | DividedPowerAlgebra.deg_one_left_inv | [95, 1] | [102, 24] | rw [← TrivSqZeroExt.snd_inr R m, ← ι_def] | R : Type u_1
inst✝⁶ : CommRing R
M : Type u_2
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : DecidableEq R
inst✝² : DecidableEq M
inst✝¹ : Module Rᵐᵒᵖ M
inst✝ : IsCentralScalar R M
m : M
⊢ ((toTrivSqZeroExt R M) ↑(((DirectSum.decompose (grade R M)) (dp R 1 m)) 1)).snd = m | R : Type u_1
inst✝⁶ : CommRing R
M : Type u_2
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : DecidableEq R
inst✝² : DecidableEq M
inst✝¹ : Module Rᵐᵒᵖ M
inst✝ : IsCentralScalar R M
m : M
⊢ ((toTrivSqZeroExt R M) ↑(((DirectSum.decompose (grade R M)) ((ι R M) (inr m).snd)) 1)).snd = (inr m).snd | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝⁶ : CommRing R
M : Type u_2
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : DecidableEq R
inst✝² : DecidableEq M
inst✝¹ : Module Rᵐᵒᵖ M
inst✝ : IsCentralScalar R M
m : M
⊢ ((toTrivSqZeroExt R M) ↑(((DirectSum.decompose (grade R M)) (dp ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeOne.lean | DividedPowerAlgebra.deg_one_left_inv | [95, 1] | [102, 24] | apply congr_arg | R : Type u_1
inst✝⁶ : CommRing R
M : Type u_2
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : DecidableEq R
inst✝² : DecidableEq M
inst✝¹ : Module Rᵐᵒᵖ M
inst✝ : IsCentralScalar R M
m : M
⊢ ((toTrivSqZeroExt R M) ↑(((DirectSum.decompose (grade R M)) ((ι R M) (inr m).snd)) 1)).snd = (inr m).snd | case h
R : Type u_1
inst✝⁶ : CommRing R
M : Type u_2
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : DecidableEq R
inst✝² : DecidableEq M
inst✝¹ : Module Rᵐᵒᵖ M
inst✝ : IsCentralScalar R M
m : M
⊢ (toTrivSqZeroExt R M) ↑(((DirectSum.decompose (grade R M)) ((ι R M) (inr m).snd)) 1) = inr m | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝⁶ : CommRing R
M : Type u_2
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : DecidableEq R
inst✝² : DecidableEq M
inst✝¹ : Module Rᵐᵒᵖ M
inst✝ : IsCentralScalar R M
m : M
⊢ ((toTrivSqZeroExt R M) ↑(((DirectSum.decompose (grade R M)) ((ι ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeOne.lean | DividedPowerAlgebra.deg_one_left_inv | [95, 1] | [102, 24] | rw [snd_inr, decompose_of_mem_same, toTrivSqZeroExt_ι] | case h
R : Type u_1
inst✝⁶ : CommRing R
M : Type u_2
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : DecidableEq R
inst✝² : DecidableEq M
inst✝¹ : Module Rᵐᵒᵖ M
inst✝ : IsCentralScalar R M
m : M
⊢ (toTrivSqZeroExt R M) ↑(((DirectSum.decompose (grade R M)) ((ι R M) (inr m).snd)) 1) = inr m | case h.hx
R : Type u_1
inst✝⁶ : CommRing R
M : Type u_2
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : DecidableEq R
inst✝² : DecidableEq M
inst✝¹ : Module Rᵐᵒᵖ M
inst✝ : IsCentralScalar R M
m : M
⊢ (ι R M) m ∈ grade R M 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case h
R : Type u_1
inst✝⁶ : CommRing R
M : Type u_2
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : DecidableEq R
inst✝² : DecidableEq M
inst✝¹ : Module Rᵐᵒᵖ M
inst✝ : IsCentralScalar R M
m : M
⊢ (toTrivSqZeroExt R M) ↑(((DirectSum.decompose (grade R M)... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeOne.lean | DividedPowerAlgebra.deg_one_left_inv | [95, 1] | [102, 24] | apply ι_mem_grade_one | case h.hx
R : Type u_1
inst✝⁶ : CommRing R
M : Type u_2
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : DecidableEq R
inst✝² : DecidableEq M
inst✝¹ : Module Rᵐᵒᵖ M
inst✝ : IsCentralScalar R M
m : M
⊢ (ι R M) m ∈ grade R M 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.hx
R : Type u_1
inst✝⁶ : CommRing R
M : Type u_2
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : DecidableEq R
inst✝² : DecidableEq M
inst✝¹ : Module Rᵐᵒᵖ M
inst✝ : IsCentralScalar R M
m : M
⊢ (ι R M) m ∈ grade R M 1
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeOne.lean | DividedPowerAlgebra.ι_toTrivSqZeroExt_of_mem_grade_one | [114, 1] | [121, 77] | suffices ⟨(ι R M) ((sndHom R M) ((toTrivSqZeroExt R M) a)), ι_mem_grade_one R _⟩ =
(⟨a, ha⟩ : grade R M 1) by
simpa only [sndHom_apply, Subtype.mk.injEq] using this | R : Type u_1
inst✝⁶ : CommRing R
M : Type u_2
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : DecidableEq R
inst✝² : DecidableEq M
inst✝¹ : Module Rᵐᵒᵖ M
inst✝ : IsCentralScalar R M
a : DividedPowerAlgebra R M
ha : a ∈ grade R M 1
⊢ (ι R M) ((sndHom R M) ((toTrivSqZeroExt R M) a)) = a | R : Type u_1
inst✝⁶ : CommRing R
M : Type u_2
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : DecidableEq R
inst✝² : DecidableEq M
inst✝¹ : Module Rᵐᵒᵖ M
inst✝ : IsCentralScalar R M
a : DividedPowerAlgebra R M
ha : a ∈ grade R M 1
⊢ ⟨(ι R M) ((sndHom R M) ((toTrivSqZeroExt R M) a)), ⋯⟩ = ⟨a, ha⟩ | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝⁶ : CommRing R
M : Type u_2
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : DecidableEq R
inst✝² : DecidableEq M
inst✝¹ : Module Rᵐᵒᵖ M
inst✝ : IsCentralScalar R M
a : DividedPowerAlgebra R M
ha : a ∈ grade R M 1
⊢ (ι R M) ((sndHom R M)... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeOne.lean | DividedPowerAlgebra.ι_toTrivSqZeroExt_of_mem_grade_one | [114, 1] | [121, 77] | apply (linearEquivDegreeOne R M).symm.injective | R : Type u_1
inst✝⁶ : CommRing R
M : Type u_2
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : DecidableEq R
inst✝² : DecidableEq M
inst✝¹ : Module Rᵐᵒᵖ M
inst✝ : IsCentralScalar R M
a : DividedPowerAlgebra R M
ha : a ∈ grade R M 1
⊢ ⟨(ι R M) ((sndHom R M) ((toTrivSqZeroExt R M) a)), ⋯⟩ = ⟨a, ha⟩ | case a
R : Type u_1
inst✝⁶ : CommRing R
M : Type u_2
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : DecidableEq R
inst✝² : DecidableEq M
inst✝¹ : Module Rᵐᵒᵖ M
inst✝ : IsCentralScalar R M
a : DividedPowerAlgebra R M
ha : a ∈ grade R M 1
⊢ (linearEquivDegreeOne R M).symm ⟨(ι R M) ((sndHom R M) ((toTrivSqZeroExt R ... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝⁶ : CommRing R
M : Type u_2
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : DecidableEq R
inst✝² : DecidableEq M
inst✝¹ : Module Rᵐᵒᵖ M
inst✝ : IsCentralScalar R M
a : DividedPowerAlgebra R M
ha : a ∈ grade R M 1
⊢ ⟨(ι R M) ((sndHom R M... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeOne.lean | DividedPowerAlgebra.ι_toTrivSqZeroExt_of_mem_grade_one | [114, 1] | [121, 77] | rw [← LinearEquiv.invFun_eq_symm] | case a
R : Type u_1
inst✝⁶ : CommRing R
M : Type u_2
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : DecidableEq R
inst✝² : DecidableEq M
inst✝¹ : Module Rᵐᵒᵖ M
inst✝ : IsCentralScalar R M
a : DividedPowerAlgebra R M
ha : a ∈ grade R M 1
⊢ (linearEquivDegreeOne R M).symm ⟨(ι R M) ((sndHom R M) ((toTrivSqZeroExt R ... | case a
R : Type u_1
inst✝⁶ : CommRing R
M : Type u_2
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : DecidableEq R
inst✝² : DecidableEq M
inst✝¹ : Module Rᵐᵒᵖ M
inst✝ : IsCentralScalar R M
a : DividedPowerAlgebra R M
ha : a ∈ grade R M 1
⊢ (linearEquivDegreeOne R M).invFun ⟨(ι R M) ((sndHom R M) ((toTrivSqZeroExt ... | Please generate a tactic in lean4 to solve the state.
STATE:
case a
R : Type u_1
inst✝⁶ : CommRing R
M : Type u_2
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : DecidableEq R
inst✝² : DecidableEq M
inst✝¹ : Module Rᵐᵒᵖ M
inst✝ : IsCentralScalar R M
a : DividedPowerAlgebra R M
ha : a ∈ grade R M 1
⊢ (linearEquivDe... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeOne.lean | DividedPowerAlgebra.ι_toTrivSqZeroExt_of_mem_grade_one | [114, 1] | [121, 77] | simp only [linearEquivDegreeOne, toTrivSqZeroExt_ι, sndHom_apply, snd_inr] | case a
R : Type u_1
inst✝⁶ : CommRing R
M : Type u_2
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : DecidableEq R
inst✝² : DecidableEq M
inst✝¹ : Module Rᵐᵒᵖ M
inst✝ : IsCentralScalar R M
a : DividedPowerAlgebra R M
ha : a ∈ grade R M 1
⊢ (linearEquivDegreeOne R M).invFun ⟨(ι R M) ((sndHom R M) ((toTrivSqZeroExt ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a
R : Type u_1
inst✝⁶ : CommRing R
M : Type u_2
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : DecidableEq R
inst✝² : DecidableEq M
inst✝¹ : Module Rᵐᵒᵖ M
inst✝ : IsCentralScalar R M
a : DividedPowerAlgebra R M
ha : a ∈ grade R M 1
⊢ (linearEquivDe... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeOne.lean | DividedPowerAlgebra.ι_toTrivSqZeroExt_of_mem_grade_one | [114, 1] | [121, 77] | simpa only [sndHom_apply, Subtype.mk.injEq] using this | R : Type u_1
inst✝⁶ : CommRing R
M : Type u_2
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : DecidableEq R
inst✝² : DecidableEq M
inst✝¹ : Module Rᵐᵒᵖ M
inst✝ : IsCentralScalar R M
a : DividedPowerAlgebra R M
ha : a ∈ grade R M 1
this : ⟨(ι R M) ((sndHom R M) ((toTrivSqZeroExt R M) a)), ⋯⟩ = ⟨a, ha⟩
⊢ (ι R M) ((s... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝⁶ : CommRing R
M : Type u_2
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : DecidableEq R
inst✝² : DecidableEq M
inst✝¹ : Module Rᵐᵒᵖ M
inst✝ : IsCentralScalar R M
a : DividedPowerAlgebra R M
ha : a ∈ grade R M 1
this : ⟨(ι R M) ((sndHo... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeOne.lean | DividedPowerAlgebra.mem_grade_one_iff | [123, 1] | [130, 26] | constructor | R : Type u_1
inst✝⁶ : CommRing R
M : Type u_2
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : DecidableEq R
inst✝² : DecidableEq M
inst✝¹ : Module Rᵐᵒᵖ M
inst✝ : IsCentralScalar R M
a : DividedPowerAlgebra R M
⊢ a ∈ grade R M 1 ↔ ∃ m, a = (ι R M) m | case mp
R : Type u_1
inst✝⁶ : CommRing R
M : Type u_2
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : DecidableEq R
inst✝² : DecidableEq M
inst✝¹ : Module Rᵐᵒᵖ M
inst✝ : IsCentralScalar R M
a : DividedPowerAlgebra R M
⊢ a ∈ grade R M 1 → ∃ m, a = (ι R M) m
case mpr
R : Type u_1
inst✝⁶ : CommRing R
M : Type u_2
in... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝⁶ : CommRing R
M : Type u_2
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : DecidableEq R
inst✝² : DecidableEq M
inst✝¹ : Module Rᵐᵒᵖ M
inst✝ : IsCentralScalar R M
a : DividedPowerAlgebra R M
⊢ a ∈ grade R M 1 ↔ ∃ m, a = (ι R M) m
TACTI... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeOne.lean | DividedPowerAlgebra.mem_grade_one_iff | [123, 1] | [130, 26] | . intro ha
use ((sndHom R M) ((toTrivSqZeroExt R M) a))
rw [ι_toTrivSqZeroExt_of_mem_grade_one R M ha] | case mp
R : Type u_1
inst✝⁶ : CommRing R
M : Type u_2
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : DecidableEq R
inst✝² : DecidableEq M
inst✝¹ : Module Rᵐᵒᵖ M
inst✝ : IsCentralScalar R M
a : DividedPowerAlgebra R M
⊢ a ∈ grade R M 1 → ∃ m, a = (ι R M) m
case mpr
R : Type u_1
inst✝⁶ : CommRing R
M : Type u_2
in... | case mpr
R : Type u_1
inst✝⁶ : CommRing R
M : Type u_2
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : DecidableEq R
inst✝² : DecidableEq M
inst✝¹ : Module Rᵐᵒᵖ M
inst✝ : IsCentralScalar R M
a : DividedPowerAlgebra R M
⊢ (∃ m, a = (ι R M) m) → a ∈ grade R M 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
R : Type u_1
inst✝⁶ : CommRing R
M : Type u_2
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : DecidableEq R
inst✝² : DecidableEq M
inst✝¹ : Module Rᵐᵒᵖ M
inst✝ : IsCentralScalar R M
a : DividedPowerAlgebra R M
⊢ a ∈ grade R M 1 → ∃ m, a = (ι R M)... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeOne.lean | DividedPowerAlgebra.mem_grade_one_iff | [123, 1] | [130, 26] | . rintro ⟨m, rfl⟩
apply ι_mem_grade_one | case mpr
R : Type u_1
inst✝⁶ : CommRing R
M : Type u_2
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : DecidableEq R
inst✝² : DecidableEq M
inst✝¹ : Module Rᵐᵒᵖ M
inst✝ : IsCentralScalar R M
a : DividedPowerAlgebra R M
⊢ (∃ m, a = (ι R M) m) → a ∈ grade R M 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
R : Type u_1
inst✝⁶ : CommRing R
M : Type u_2
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : DecidableEq R
inst✝² : DecidableEq M
inst✝¹ : Module Rᵐᵒᵖ M
inst✝ : IsCentralScalar R M
a : DividedPowerAlgebra R M
⊢ (∃ m, a = (ι R M) m) → a ∈ grade ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeOne.lean | DividedPowerAlgebra.mem_grade_one_iff | [123, 1] | [130, 26] | intro ha | case mp
R : Type u_1
inst✝⁶ : CommRing R
M : Type u_2
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : DecidableEq R
inst✝² : DecidableEq M
inst✝¹ : Module Rᵐᵒᵖ M
inst✝ : IsCentralScalar R M
a : DividedPowerAlgebra R M
⊢ a ∈ grade R M 1 → ∃ m, a = (ι R M) m | case mp
R : Type u_1
inst✝⁶ : CommRing R
M : Type u_2
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : DecidableEq R
inst✝² : DecidableEq M
inst✝¹ : Module Rᵐᵒᵖ M
inst✝ : IsCentralScalar R M
a : DividedPowerAlgebra R M
ha : a ∈ grade R M 1
⊢ ∃ m, a = (ι R M) m | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
R : Type u_1
inst✝⁶ : CommRing R
M : Type u_2
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : DecidableEq R
inst✝² : DecidableEq M
inst✝¹ : Module Rᵐᵒᵖ M
inst✝ : IsCentralScalar R M
a : DividedPowerAlgebra R M
⊢ a ∈ grade R M 1 → ∃ m, a = (ι R M)... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeOne.lean | DividedPowerAlgebra.mem_grade_one_iff | [123, 1] | [130, 26] | use ((sndHom R M) ((toTrivSqZeroExt R M) a)) | case mp
R : Type u_1
inst✝⁶ : CommRing R
M : Type u_2
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : DecidableEq R
inst✝² : DecidableEq M
inst✝¹ : Module Rᵐᵒᵖ M
inst✝ : IsCentralScalar R M
a : DividedPowerAlgebra R M
ha : a ∈ grade R M 1
⊢ ∃ m, a = (ι R M) m | case h
R : Type u_1
inst✝⁶ : CommRing R
M : Type u_2
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : DecidableEq R
inst✝² : DecidableEq M
inst✝¹ : Module Rᵐᵒᵖ M
inst✝ : IsCentralScalar R M
a : DividedPowerAlgebra R M
ha : a ∈ grade R M 1
⊢ a = (ι R M) ((sndHom R M) ((toTrivSqZeroExt R M) a)) | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
R : Type u_1
inst✝⁶ : CommRing R
M : Type u_2
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : DecidableEq R
inst✝² : DecidableEq M
inst✝¹ : Module Rᵐᵒᵖ M
inst✝ : IsCentralScalar R M
a : DividedPowerAlgebra R M
ha : a ∈ grade R M 1
⊢ ∃ m, a = (ι R... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeOne.lean | DividedPowerAlgebra.mem_grade_one_iff | [123, 1] | [130, 26] | rw [ι_toTrivSqZeroExt_of_mem_grade_one R M ha] | case h
R : Type u_1
inst✝⁶ : CommRing R
M : Type u_2
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : DecidableEq R
inst✝² : DecidableEq M
inst✝¹ : Module Rᵐᵒᵖ M
inst✝ : IsCentralScalar R M
a : DividedPowerAlgebra R M
ha : a ∈ grade R M 1
⊢ a = (ι R M) ((sndHom R M) ((toTrivSqZeroExt R M) a)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
R : Type u_1
inst✝⁶ : CommRing R
M : Type u_2
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : DecidableEq R
inst✝² : DecidableEq M
inst✝¹ : Module Rᵐᵒᵖ M
inst✝ : IsCentralScalar R M
a : DividedPowerAlgebra R M
ha : a ∈ grade R M 1
⊢ a = (ι R M) ((... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeOne.lean | DividedPowerAlgebra.mem_grade_one_iff | [123, 1] | [130, 26] | rintro ⟨m, rfl⟩ | case mpr
R : Type u_1
inst✝⁶ : CommRing R
M : Type u_2
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : DecidableEq R
inst✝² : DecidableEq M
inst✝¹ : Module Rᵐᵒᵖ M
inst✝ : IsCentralScalar R M
a : DividedPowerAlgebra R M
⊢ (∃ m, a = (ι R M) m) → a ∈ grade R M 1 | case mpr.intro
R : Type u_1
inst✝⁶ : CommRing R
M : Type u_2
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : DecidableEq R
inst✝² : DecidableEq M
inst✝¹ : Module Rᵐᵒᵖ M
inst✝ : IsCentralScalar R M
m : M
⊢ (ι R M) m ∈ grade R M 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
R : Type u_1
inst✝⁶ : CommRing R
M : Type u_2
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : DecidableEq R
inst✝² : DecidableEq M
inst✝¹ : Module Rᵐᵒᵖ M
inst✝ : IsCentralScalar R M
a : DividedPowerAlgebra R M
⊢ (∃ m, a = (ι R M) m) → a ∈ grade ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Graded/GradeOne.lean | DividedPowerAlgebra.mem_grade_one_iff | [123, 1] | [130, 26] | apply ι_mem_grade_one | case mpr.intro
R : Type u_1
inst✝⁶ : CommRing R
M : Type u_2
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : DecidableEq R
inst✝² : DecidableEq M
inst✝¹ : Module Rᵐᵒᵖ M
inst✝ : IsCentralScalar R M
m : M
⊢ (ι R M) m ∈ grade R M 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.intro
R : Type u_1
inst✝⁶ : CommRing R
M : Type u_2
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : DecidableEq R
inst✝² : DecidableEq M
inst✝¹ : Module Rᵐᵒᵖ M
inst✝ : IsCentralScalar R M
m : M
⊢ (ι R M) m ∈ grade R M 1
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/Topology.lean | Finset.prod_one_add | [15, 1] | [21, 44] | simp_rw [add_comm, Finset.prod_add] | ι : Type u_1
α : Type u_2
inst✝¹ : DecidableEq ι
inst✝ : CommRing α
f : ι → α
s : Finset ι
⊢ ∏ i ∈ s, (1 + f i) = ∑ t ∈ s.powerset, t.prod f | ι : Type u_1
α : Type u_2
inst✝¹ : DecidableEq ι
inst✝ : CommRing α
f : ι → α
s : Finset ι
⊢ ∑ t ∈ s.powerset, (∏ i ∈ t, f i) * ∏ i ∈ s \ t, 1 = ∑ t ∈ s.powerset, t.prod f | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
α : Type u_2
inst✝¹ : DecidableEq ι
inst✝ : CommRing α
f : ι → α
s : Finset ι
⊢ ∏ i ∈ s, (1 + f i) = ∑ t ∈ s.powerset, t.prod f
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/Topology.lean | Finset.prod_one_add | [15, 1] | [21, 44] | congr | ι : Type u_1
α : Type u_2
inst✝¹ : DecidableEq ι
inst✝ : CommRing α
f : ι → α
s : Finset ι
⊢ ∑ t ∈ s.powerset, (∏ i ∈ t, f i) * ∏ i ∈ s \ t, 1 = ∑ t ∈ s.powerset, t.prod f | case e_f
ι : Type u_1
α : Type u_2
inst✝¹ : DecidableEq ι
inst✝ : CommRing α
f : ι → α
s : Finset ι
⊢ (fun t => (∏ i ∈ t, f i) * ∏ i ∈ s \ t, 1) = fun t => t.prod f | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
α : Type u_2
inst✝¹ : DecidableEq ι
inst✝ : CommRing α
f : ι → α
s : Finset ι
⊢ ∑ t ∈ s.powerset, (∏ i ∈ t, f i) * ∏ i ∈ s \ t, 1 = ∑ t ∈ s.powerset, t.prod f
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/Topology.lean | Finset.prod_one_add | [15, 1] | [21, 44] | ext t | case e_f
ι : Type u_1
α : Type u_2
inst✝¹ : DecidableEq ι
inst✝ : CommRing α
f : ι → α
s : Finset ι
⊢ (fun t => (∏ i ∈ t, f i) * ∏ i ∈ s \ t, 1) = fun t => t.prod f | case e_f.h
ι : Type u_1
α : Type u_2
inst✝¹ : DecidableEq ι
inst✝ : CommRing α
f : ι → α
s t : Finset ι
⊢ (∏ i ∈ t, f i) * ∏ i ∈ s \ t, 1 = t.prod f | Please generate a tactic in lean4 to solve the state.
STATE:
case e_f
ι : Type u_1
α : Type u_2
inst✝¹ : DecidableEq ι
inst✝ : CommRing α
f : ι → α
s : Finset ι
⊢ (fun t => (∏ i ∈ t, f i) * ∏ i ∈ s \ t, 1) = fun t => t.prod f
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/Topology.lean | Finset.prod_one_add | [15, 1] | [21, 44] | convert mul_one (Finset.prod t fun a => f a) | case e_f.h
ι : Type u_1
α : Type u_2
inst✝¹ : DecidableEq ι
inst✝ : CommRing α
f : ι → α
s t : Finset ι
⊢ (∏ i ∈ t, f i) * ∏ i ∈ s \ t, 1 = t.prod f | case h.e'_2.h.e'_6
ι : Type u_1
α : Type u_2
inst✝¹ : DecidableEq ι
inst✝ : CommRing α
f : ι → α
s t : Finset ι
⊢ ∏ i ∈ s \ t, 1 = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case e_f.h
ι : Type u_1
α : Type u_2
inst✝¹ : DecidableEq ι
inst✝ : CommRing α
f : ι → α
s t : Finset ι
⊢ (∏ i ∈ t, f i) * ∏ i ∈ s \ t, 1 = t.prod f
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/Topology.lean | Finset.prod_one_add | [15, 1] | [21, 44] | exact Finset.prod_eq_one (fun i _ => rfl) | case h.e'_2.h.e'_6
ι : Type u_1
α : Type u_2
inst✝¹ : DecidableEq ι
inst✝ : CommRing α
f : ι → α
s t : Finset ι
⊢ ∏ i ∈ s \ t, 1 = 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_2.h.e'_6
ι : Type u_1
α : Type u_2
inst✝¹ : DecidableEq ι
inst✝ : CommRing α
f : ι → α
s t : Finset ι
⊢ ∏ i ∈ s \ t, 1 = 1
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/Topology.lean | MvPowerSeries.WithPiTopology.tendsto_iff_coeff_tendsto | [61, 1] | [66, 41] | rw [nhds_pi, Filter.tendsto_pi] | σ : Type u_3
α : Type u_1
inst✝¹ : TopologicalSpace α
inst✝ : Semiring α
ι : Type u_2
f : ι → MvPowerSeries σ α
u : Filter ι
g : MvPowerSeries σ α
⊢ Filter.Tendsto f u (nhds g) ↔ ∀ (d : σ →₀ ℕ), Filter.Tendsto (fun i => (coeff α d) (f i)) u (nhds ((coeff α d) g)) | σ : Type u_3
α : Type u_1
inst✝¹ : TopologicalSpace α
inst✝ : Semiring α
ι : Type u_2
f : ι → MvPowerSeries σ α
u : Filter ι
g : MvPowerSeries σ α
⊢ (∀ (i : σ →₀ ℕ), Filter.Tendsto (fun x => f x i) u (nhds (g i))) ↔
∀ (d : σ →₀ ℕ), Filter.Tendsto (fun i => (coeff α d) (f i)) u (nhds ((coeff α d) g)) | Please generate a tactic in lean4 to solve the state.
STATE:
σ : Type u_3
α : Type u_1
inst✝¹ : TopologicalSpace α
inst✝ : Semiring α
ι : Type u_2
f : ι → MvPowerSeries σ α
u : Filter ι
g : MvPowerSeries σ α
⊢ Filter.Tendsto f u (nhds g) ↔ ∀ (d : σ →₀ ℕ), Filter.Tendsto (fun i => (coeff α d) (f i)) u (nhds ((coeff α d)... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/Topology.lean | MvPowerSeries.WithPiTopology.tendsto_iff_coeff_tendsto | [61, 1] | [66, 41] | exact forall_congr' (fun d => Iff.rfl) | σ : Type u_3
α : Type u_1
inst✝¹ : TopologicalSpace α
inst✝ : Semiring α
ι : Type u_2
f : ι → MvPowerSeries σ α
u : Filter ι
g : MvPowerSeries σ α
⊢ (∀ (i : σ →₀ ℕ), Filter.Tendsto (fun x => f x i) u (nhds (g i))) ↔
∀ (d : σ →₀ ℕ), Filter.Tendsto (fun i => (coeff α d) (f i)) u (nhds ((coeff α d) g)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
σ : Type u_3
α : Type u_1
inst✝¹ : TopologicalSpace α
inst✝ : Semiring α
ι : Type u_2
f : ι → MvPowerSeries σ α
u : Filter ι
g : MvPowerSeries σ α
⊢ (∀ (i : σ →₀ ℕ), Filter.Tendsto (fun x => f x i) u (nhds (g i))) ↔
∀ (d : σ →₀ ℕ), Filter.Tendsto (fun i =... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/Topology.lean | MvPowerSeries.continuous_C | [174, 1] | [184, 28] | apply continuous_of_continuousAt_zero | σ : Type u_2
α : Type u_1
inst✝³ : DecidableEq σ
inst✝² : TopologicalSpace α
inst✝¹ : CommRing α
inst✝ : TopologicalRing α
⊢ Continuous ⇑(C σ α) | case hf
σ : Type u_2
α : Type u_1
inst✝³ : DecidableEq σ
inst✝² : TopologicalSpace α
inst✝¹ : CommRing α
inst✝ : TopologicalRing α
⊢ ContinuousAt (⇑(C σ α)) 0 | Please generate a tactic in lean4 to solve the state.
STATE:
σ : Type u_2
α : Type u_1
inst✝³ : DecidableEq σ
inst✝² : TopologicalSpace α
inst✝¹ : CommRing α
inst✝ : TopologicalRing α
⊢ Continuous ⇑(C σ α)
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/Topology.lean | MvPowerSeries.continuous_C | [174, 1] | [184, 28] | rw [continuousAt_pi] | case hf
σ : Type u_2
α : Type u_1
inst✝³ : DecidableEq σ
inst✝² : TopologicalSpace α
inst✝¹ : CommRing α
inst✝ : TopologicalRing α
⊢ ContinuousAt (⇑(C σ α)) 0 | case hf
σ : Type u_2
α : Type u_1
inst✝³ : DecidableEq σ
inst✝² : TopologicalSpace α
inst✝¹ : CommRing α
inst✝ : TopologicalRing α
⊢ ∀ (i : σ →₀ ℕ), ContinuousAt (fun y => (C σ α) y i) 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case hf
σ : Type u_2
α : Type u_1
inst✝³ : DecidableEq σ
inst✝² : TopologicalSpace α
inst✝¹ : CommRing α
inst✝ : TopologicalRing α
⊢ ContinuousAt (⇑(C σ α)) 0
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/Topology.lean | MvPowerSeries.continuous_C | [174, 1] | [184, 28] | intro d | case hf
σ : Type u_2
α : Type u_1
inst✝³ : DecidableEq σ
inst✝² : TopologicalSpace α
inst✝¹ : CommRing α
inst✝ : TopologicalRing α
⊢ ∀ (i : σ →₀ ℕ), ContinuousAt (fun y => (C σ α) y i) 0 | case hf
σ : Type u_2
α : Type u_1
inst✝³ : DecidableEq σ
inst✝² : TopologicalSpace α
inst✝¹ : CommRing α
inst✝ : TopologicalRing α
d : σ →₀ ℕ
⊢ ContinuousAt (fun y => (C σ α) y d) 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case hf
σ : Type u_2
α : Type u_1
inst✝³ : DecidableEq σ
inst✝² : TopologicalSpace α
inst✝¹ : CommRing α
inst✝ : TopologicalRing α
⊢ ∀ (i : σ →₀ ℕ), ContinuousAt (fun y => (C σ α) y i) 0
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/Topology.lean | MvPowerSeries.continuous_C | [174, 1] | [184, 28] | change ContinuousAt (fun y => coeff α d ((C σ α) y)) 0 | case hf
σ : Type u_2
α : Type u_1
inst✝³ : DecidableEq σ
inst✝² : TopologicalSpace α
inst✝¹ : CommRing α
inst✝ : TopologicalRing α
d : σ →₀ ℕ
⊢ ContinuousAt (fun y => (C σ α) y d) 0 | case hf
σ : Type u_2
α : Type u_1
inst✝³ : DecidableEq σ
inst✝² : TopologicalSpace α
inst✝¹ : CommRing α
inst✝ : TopologicalRing α
d : σ →₀ ℕ
⊢ ContinuousAt (fun y => (coeff α d) ((C σ α) y)) 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case hf
σ : Type u_2
α : Type u_1
inst✝³ : DecidableEq σ
inst✝² : TopologicalSpace α
inst✝¹ : CommRing α
inst✝ : TopologicalRing α
d : σ →₀ ℕ
⊢ ContinuousAt (fun y => (C σ α) y d) 0
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/Topology.lean | MvPowerSeries.continuous_C | [174, 1] | [184, 28] | by_cases hd : d = 0 | case hf
σ : Type u_2
α : Type u_1
inst✝³ : DecidableEq σ
inst✝² : TopologicalSpace α
inst✝¹ : CommRing α
inst✝ : TopologicalRing α
d : σ →₀ ℕ
⊢ ContinuousAt (fun y => (coeff α d) ((C σ α) y)) 0 | case pos
σ : Type u_2
α : Type u_1
inst✝³ : DecidableEq σ
inst✝² : TopologicalSpace α
inst✝¹ : CommRing α
inst✝ : TopologicalRing α
d : σ →₀ ℕ
hd : d = 0
⊢ ContinuousAt (fun y => (coeff α d) ((C σ α) y)) 0
case neg
σ : Type u_2
α : Type u_1
inst✝³ : DecidableEq σ
inst✝² : TopologicalSpace α
inst✝¹ : CommRing α
inst✝ :... | Please generate a tactic in lean4 to solve the state.
STATE:
case hf
σ : Type u_2
α : Type u_1
inst✝³ : DecidableEq σ
inst✝² : TopologicalSpace α
inst✝¹ : CommRing α
inst✝ : TopologicalRing α
d : σ →₀ ℕ
⊢ ContinuousAt (fun y => (coeff α d) ((C σ α) y)) 0
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/Topology.lean | MvPowerSeries.continuous_C | [174, 1] | [184, 28] | convert continuousAt_id | case pos
σ : Type u_2
α : Type u_1
inst✝³ : DecidableEq σ
inst✝² : TopologicalSpace α
inst✝¹ : CommRing α
inst✝ : TopologicalRing α
d : σ →₀ ℕ
hd : d = 0
⊢ ContinuousAt (fun y => (coeff α d) ((C σ α) y)) 0 | case h.e'_5.h
σ : Type u_2
α : Type u_1
inst✝³ : DecidableEq σ
inst✝² : TopologicalSpace α
inst✝¹ : CommRing α
inst✝ : TopologicalRing α
d : σ →₀ ℕ
hd : d = 0
x✝ : α
⊢ (coeff α d) ((C σ α) x✝) = id x✝ | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
σ : Type u_2
α : Type u_1
inst✝³ : DecidableEq σ
inst✝² : TopologicalSpace α
inst✝¹ : CommRing α
inst✝ : TopologicalRing α
d : σ →₀ ℕ
hd : d = 0
⊢ ContinuousAt (fun y => (coeff α d) ((C σ α) y)) 0
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/Topology.lean | MvPowerSeries.continuous_C | [174, 1] | [184, 28] | rw [hd, coeff_zero_C, id_eq] | case h.e'_5.h
σ : Type u_2
α : Type u_1
inst✝³ : DecidableEq σ
inst✝² : TopologicalSpace α
inst✝¹ : CommRing α
inst✝ : TopologicalRing α
d : σ →₀ ℕ
hd : d = 0
x✝ : α
⊢ (coeff α d) ((C σ α) x✝) = id x✝ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_5.h
σ : Type u_2
α : Type u_1
inst✝³ : DecidableEq σ
inst✝² : TopologicalSpace α
inst✝¹ : CommRing α
inst✝ : TopologicalRing α
d : σ →₀ ℕ
hd : d = 0
x✝ : α
⊢ (coeff α d) ((C σ α) x✝) = id x✝
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/Topology.lean | MvPowerSeries.continuous_C | [174, 1] | [184, 28] | convert continuousAt_const | case neg
σ : Type u_2
α : Type u_1
inst✝³ : DecidableEq σ
inst✝² : TopologicalSpace α
inst✝¹ : CommRing α
inst✝ : TopologicalRing α
d : σ →₀ ℕ
hd : ¬d = 0
⊢ ContinuousAt (fun y => (coeff α d) ((C σ α) y)) 0 | case h.e'_5.h
σ : Type u_2
α : Type u_1
inst✝³ : DecidableEq σ
inst✝² : TopologicalSpace α
inst✝¹ : CommRing α
inst✝ : TopologicalRing α
d : σ →₀ ℕ
hd : ¬d = 0
x✝ : α
⊢ (coeff α d) ((C σ α) x✝) = ?neg.convert_6✝
case neg.convert_6
σ : Type u_2
α : Type u_1
inst✝³ : DecidableEq σ
inst✝² : TopologicalSpace α
inst✝¹ : Co... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
σ : Type u_2
α : Type u_1
inst✝³ : DecidableEq σ
inst✝² : TopologicalSpace α
inst✝¹ : CommRing α
inst✝ : TopologicalRing α
d : σ →₀ ℕ
hd : ¬d = 0
⊢ ContinuousAt (fun y => (coeff α d) ((C σ α) y)) 0
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/Topology.lean | MvPowerSeries.continuous_C | [174, 1] | [184, 28] | rw [coeff_C, if_neg hd] | case h.e'_5.h
σ : Type u_2
α : Type u_1
inst✝³ : DecidableEq σ
inst✝² : TopologicalSpace α
inst✝¹ : CommRing α
inst✝ : TopologicalRing α
d : σ →₀ ℕ
hd : ¬d = 0
x✝ : α
⊢ (coeff α d) ((C σ α) x✝) = ?neg.convert_6✝
case neg.convert_6
σ : Type u_2
α : Type u_1
inst✝³ : DecidableEq σ
inst✝² : TopologicalSpace α
inst✝¹ : Co... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_5.h
σ : Type u_2
α : Type u_1
inst✝³ : DecidableEq σ
inst✝² : TopologicalSpace α
inst✝¹ : CommRing α
inst✝ : TopologicalRing α
d : σ →₀ ℕ
hd : ¬d = 0
x✝ : α
⊢ (coeff α d) ((C σ α) x✝) = ?neg.convert_6✝
case neg.convert_6
σ : Type u_2
α : Type u_1
i... |
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