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/Topology.lean | MvPowerSeries.variables_tendsto_zero | [186, 1] | [213, 20] | rw [tendsto_pi_nhds] | σ : Type u_1
α : Type u_2
inst✝³ : DecidableEq σ
inst✝² : TopologicalSpace α
inst✝¹ : CommRing α
inst✝ : TopologicalRing α
⊢ Filter.Tendsto (fun s => X s) Filter.cofinite (nhds 0) | σ : Type u_1
α : Type u_2
inst✝³ : DecidableEq σ
inst✝² : TopologicalSpace α
inst✝¹ : CommRing α
inst✝ : TopologicalRing α
⊢ ∀ (x : σ →₀ ℕ), Filter.Tendsto (fun i => X i x) Filter.cofinite (nhds (0 x)) | Please generate a tactic in lean4 to solve the state.
STATE:
σ : Type u_1
α : Type u_2
inst✝³ : DecidableEq σ
inst✝² : TopologicalSpace α
inst✝¹ : CommRing α
inst✝ : TopologicalRing α
⊢ Filter.Tendsto (fun s => X s) Filter.cofinite (nhds 0)
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/Topology.lean | MvPowerSeries.variables_tendsto_zero | [186, 1] | [213, 20] | intro d s hs | σ : Type u_1
α : Type u_2
inst✝³ : DecidableEq σ
inst✝² : TopologicalSpace α
inst✝¹ : CommRing α
inst✝ : TopologicalRing α
⊢ ∀ (x : σ →₀ ℕ), Filter.Tendsto (fun i => X i x) Filter.cofinite (nhds (0 x)) | σ : Type u_1
α : Type u_2
inst✝³ : DecidableEq σ
inst✝² : TopologicalSpace α
inst✝¹ : CommRing α
inst✝ : TopologicalRing α
d : σ →₀ ℕ
s : Set α
hs : s ∈ nhds (0 d)
⊢ s ∈ Filter.map (fun i => X i d) Filter.cofinite | Please generate a tactic in lean4 to solve the state.
STATE:
σ : Type u_1
α : Type u_2
inst✝³ : DecidableEq σ
inst✝² : TopologicalSpace α
inst✝¹ : CommRing α
inst✝ : TopologicalRing α
⊢ ∀ (x : σ →₀ ℕ), Filter.Tendsto (fun i => X i x) Filter.cofinite (nhds (0 x))
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/Topology.lean | MvPowerSeries.variables_tendsto_zero | [186, 1] | [213, 20] | rw [Filter.mem_map, Filter.mem_cofinite, ← Set.preimage_compl] | σ : Type u_1
α : Type u_2
inst✝³ : DecidableEq σ
inst✝² : TopologicalSpace α
inst✝¹ : CommRing α
inst✝ : TopologicalRing α
d : σ →₀ ℕ
s : Set α
hs : s ∈ nhds (0 d)
⊢ s ∈ Filter.map (fun i => X i d) Filter.cofinite | σ : Type u_1
α : Type u_2
inst✝³ : DecidableEq σ
inst✝² : TopologicalSpace α
inst✝¹ : CommRing α
inst✝ : TopologicalRing α
d : σ →₀ ℕ
s : Set α
hs : s ∈ nhds (0 d)
⊢ ((fun i => X i d) ⁻¹' sᶜ).Finite | Please generate a tactic in lean4 to solve the state.
STATE:
σ : Type u_1
α : Type u_2
inst✝³ : DecidableEq σ
inst✝² : TopologicalSpace α
inst✝¹ : CommRing α
inst✝ : TopologicalRing α
d : σ →₀ ℕ
s : Set α
hs : s ∈ nhds (0 d)
⊢ s ∈ Filter.map (fun i => X i d) Filter.cofinite
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/Topology.lean | MvPowerSeries.variables_tendsto_zero | [186, 1] | [213, 20] | by_cases h : ∃ i, d = Finsupp.single i 1 | σ : Type u_1
α : Type u_2
inst✝³ : DecidableEq σ
inst✝² : TopologicalSpace α
inst✝¹ : CommRing α
inst✝ : TopologicalRing α
d : σ →₀ ℕ
s : Set α
hs : s ∈ nhds (0 d)
⊢ ((fun i => X i d) ⁻¹' sᶜ).Finite | case pos
σ : Type u_1
α : Type u_2
inst✝³ : DecidableEq σ
inst✝² : TopologicalSpace α
inst✝¹ : CommRing α
inst✝ : TopologicalRing α
d : σ →₀ ℕ
s : Set α
hs : s ∈ nhds (0 d)
h : ∃ i, d = Finsupp.single i 1
⊢ ((fun i => X i d) ⁻¹' sᶜ).Finite
case neg
σ : Type u_1
α : Type u_2
inst✝³ : DecidableEq σ
inst✝² : TopologicalS... | Please generate a tactic in lean4 to solve the state.
STATE:
σ : Type u_1
α : Type u_2
inst✝³ : DecidableEq σ
inst✝² : TopologicalSpace α
inst✝¹ : CommRing α
inst✝ : TopologicalRing α
d : σ →₀ ℕ
s : Set α
hs : s ∈ nhds (0 d)
⊢ ((fun i => X i d) ⁻¹' sᶜ).Finite
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/Topology.lean | MvPowerSeries.variables_tendsto_zero | [186, 1] | [213, 20] | . convert Set.finite_empty
rw [Set.eq_empty_iff_forall_not_mem]
intro x
rw [Set.mem_preimage, Set.not_mem_compl_iff]
convert mem_of_mem_nhds hs using 1
rw [← coeff_eq_apply (X x) d, coeff_X, if_neg]
rfl
. intro h'
apply h
exact ⟨x, h'⟩ | case neg
σ : Type u_1
α : Type u_2
inst✝³ : DecidableEq σ
inst✝² : TopologicalSpace α
inst✝¹ : CommRing α
inst✝ : TopologicalRing α
d : σ →₀ ℕ
s : Set α
hs : s ∈ nhds (0 d)
h : ¬∃ i, d = Finsupp.single i 1
⊢ ((fun i => X i d) ⁻¹' sᶜ).Finite | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
σ : Type u_1
α : Type u_2
inst✝³ : DecidableEq σ
inst✝² : TopologicalSpace α
inst✝¹ : CommRing α
inst✝ : TopologicalRing α
d : σ →₀ ℕ
s : Set α
hs : s ∈ nhds (0 d)
h : ¬∃ i, d = Finsupp.single i 1
⊢ ((fun i => X i d) ⁻¹' sᶜ).Finite
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/Topology.lean | MvPowerSeries.variables_tendsto_zero | [186, 1] | [213, 20] | obtain ⟨i, rfl⟩ := h | case pos
σ : Type u_1
α : Type u_2
inst✝³ : DecidableEq σ
inst✝² : TopologicalSpace α
inst✝¹ : CommRing α
inst✝ : TopologicalRing α
d : σ →₀ ℕ
s : Set α
hs : s ∈ nhds (0 d)
h : ∃ i, d = Finsupp.single i 1
⊢ ((fun i => X i d) ⁻¹' sᶜ).Finite | case pos.intro
σ : Type u_1
α : Type u_2
inst✝³ : DecidableEq σ
inst✝² : TopologicalSpace α
inst✝¹ : CommRing α
inst✝ : TopologicalRing α
s : Set α
i : σ
hs : s ∈ nhds (0 (Finsupp.single i 1))
⊢ ((fun i_1 => X i_1 (Finsupp.single i 1)) ⁻¹' sᶜ).Finite | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
σ : Type u_1
α : Type u_2
inst✝³ : DecidableEq σ
inst✝² : TopologicalSpace α
inst✝¹ : CommRing α
inst✝ : TopologicalRing α
d : σ →₀ ℕ
s : Set α
hs : s ∈ nhds (0 d)
h : ∃ i, d = Finsupp.single i 1
⊢ ((fun i => X i d) ⁻¹' sᶜ).Finite
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/Topology.lean | MvPowerSeries.variables_tendsto_zero | [186, 1] | [213, 20] | apply Set.Finite.subset (Set.finite_singleton i) | case pos.intro
σ : Type u_1
α : Type u_2
inst✝³ : DecidableEq σ
inst✝² : TopologicalSpace α
inst✝¹ : CommRing α
inst✝ : TopologicalRing α
s : Set α
i : σ
hs : s ∈ nhds (0 (Finsupp.single i 1))
⊢ ((fun i_1 => X i_1 (Finsupp.single i 1)) ⁻¹' sᶜ).Finite | case pos.intro
σ : Type u_1
α : Type u_2
inst✝³ : DecidableEq σ
inst✝² : TopologicalSpace α
inst✝¹ : CommRing α
inst✝ : TopologicalRing α
s : Set α
i : σ
hs : s ∈ nhds (0 (Finsupp.single i 1))
⊢ (fun i_1 => X i_1 (Finsupp.single i 1)) ⁻¹' sᶜ ⊆ {i} | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.intro
σ : Type u_1
α : Type u_2
inst✝³ : DecidableEq σ
inst✝² : TopologicalSpace α
inst✝¹ : CommRing α
inst✝ : TopologicalRing α
s : Set α
i : σ
hs : s ∈ nhds (0 (Finsupp.single i 1))
⊢ ((fun i_1 => X i_1 (Finsupp.single i 1)) ⁻¹' sᶜ).Finite
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/Topology.lean | MvPowerSeries.variables_tendsto_zero | [186, 1] | [213, 20] | intro x | case pos.intro
σ : Type u_1
α : Type u_2
inst✝³ : DecidableEq σ
inst✝² : TopologicalSpace α
inst✝¹ : CommRing α
inst✝ : TopologicalRing α
s : Set α
i : σ
hs : s ∈ nhds (0 (Finsupp.single i 1))
⊢ (fun i_1 => X i_1 (Finsupp.single i 1)) ⁻¹' sᶜ ⊆ {i} | case pos.intro
σ : Type u_1
α : Type u_2
inst✝³ : DecidableEq σ
inst✝² : TopologicalSpace α
inst✝¹ : CommRing α
inst✝ : TopologicalRing α
s : Set α
i : σ
hs : s ∈ nhds (0 (Finsupp.single i 1))
x : σ
⊢ x ∈ (fun i_1 => X i_1 (Finsupp.single i 1)) ⁻¹' sᶜ → x ∈ {i} | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.intro
σ : Type u_1
α : Type u_2
inst✝³ : DecidableEq σ
inst✝² : TopologicalSpace α
inst✝¹ : CommRing α
inst✝ : TopologicalRing α
s : Set α
i : σ
hs : s ∈ nhds (0 (Finsupp.single i 1))
⊢ (fun i_1 => X i_1 (Finsupp.single i 1)) ⁻¹' sᶜ ⊆ {i}
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/Topology.lean | MvPowerSeries.variables_tendsto_zero | [186, 1] | [213, 20] | simp only [OfNat.ofNat, Zero.zero] at hs | case pos.intro
σ : Type u_1
α : Type u_2
inst✝³ : DecidableEq σ
inst✝² : TopologicalSpace α
inst✝¹ : CommRing α
inst✝ : TopologicalRing α
s : Set α
i : σ
hs : s ∈ nhds (0 (Finsupp.single i 1))
x : σ
⊢ x ∈ (fun i_1 => X i_1 (Finsupp.single i 1)) ⁻¹' sᶜ → x ∈ {i} | case pos.intro
σ : Type u_1
α : Type u_2
inst✝³ : DecidableEq σ
inst✝² : TopologicalSpace α
inst✝¹ : CommRing α
inst✝ : TopologicalRing α
s : Set α
i : σ
hs : s ∈ nhds Zero.zero
x : σ
⊢ x ∈ (fun i_1 => X i_1 (Finsupp.single i 1)) ⁻¹' sᶜ → x ∈ {i} | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.intro
σ : Type u_1
α : Type u_2
inst✝³ : DecidableEq σ
inst✝² : TopologicalSpace α
inst✝¹ : CommRing α
inst✝ : TopologicalRing α
s : Set α
i : σ
hs : s ∈ nhds (0 (Finsupp.single i 1))
x : σ
⊢ x ∈ (fun i_1 => X i_1 (Finsupp.single i 1)) ⁻¹' sᶜ → x ∈ {... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/Topology.lean | MvPowerSeries.variables_tendsto_zero | [186, 1] | [213, 20] | rw [Set.mem_preimage, Set.mem_compl_iff, Set.mem_singleton_iff, not_imp_comm] | case pos.intro
σ : Type u_1
α : Type u_2
inst✝³ : DecidableEq σ
inst✝² : TopologicalSpace α
inst✝¹ : CommRing α
inst✝ : TopologicalRing α
s : Set α
i : σ
hs : s ∈ nhds Zero.zero
x : σ
⊢ x ∈ (fun i_1 => X i_1 (Finsupp.single i 1)) ⁻¹' sᶜ → x ∈ {i} | case pos.intro
σ : Type u_1
α : Type u_2
inst✝³ : DecidableEq σ
inst✝² : TopologicalSpace α
inst✝¹ : CommRing α
inst✝ : TopologicalRing α
s : Set α
i : σ
hs : s ∈ nhds Zero.zero
x : σ
⊢ ¬x = i → X x (Finsupp.single i 1) ∈ s | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.intro
σ : Type u_1
α : Type u_2
inst✝³ : DecidableEq σ
inst✝² : TopologicalSpace α
inst✝¹ : CommRing α
inst✝ : TopologicalRing α
s : Set α
i : σ
hs : s ∈ nhds Zero.zero
x : σ
⊢ x ∈ (fun i_1 => X i_1 (Finsupp.single i 1)) ⁻¹' sᶜ → x ∈ {i}
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/Topology.lean | MvPowerSeries.variables_tendsto_zero | [186, 1] | [213, 20] | intro hx | case pos.intro
σ : Type u_1
α : Type u_2
inst✝³ : DecidableEq σ
inst✝² : TopologicalSpace α
inst✝¹ : CommRing α
inst✝ : TopologicalRing α
s : Set α
i : σ
hs : s ∈ nhds Zero.zero
x : σ
⊢ ¬x = i → X x (Finsupp.single i 1) ∈ s | case pos.intro
σ : Type u_1
α : Type u_2
inst✝³ : DecidableEq σ
inst✝² : TopologicalSpace α
inst✝¹ : CommRing α
inst✝ : TopologicalRing α
s : Set α
i : σ
hs : s ∈ nhds Zero.zero
x : σ
hx : ¬x = i
⊢ X x (Finsupp.single i 1) ∈ s | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.intro
σ : Type u_1
α : Type u_2
inst✝³ : DecidableEq σ
inst✝² : TopologicalSpace α
inst✝¹ : CommRing α
inst✝ : TopologicalRing α
s : Set α
i : σ
hs : s ∈ nhds Zero.zero
x : σ
⊢ ¬x = i → X x (Finsupp.single i 1) ∈ s
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/Topology.lean | MvPowerSeries.variables_tendsto_zero | [186, 1] | [213, 20] | convert mem_of_mem_nhds hs | case pos.intro
σ : Type u_1
α : Type u_2
inst✝³ : DecidableEq σ
inst✝² : TopologicalSpace α
inst✝¹ : CommRing α
inst✝ : TopologicalRing α
s : Set α
i : σ
hs : s ∈ nhds Zero.zero
x : σ
hx : ¬x = i
⊢ X x (Finsupp.single i 1) ∈ s | case h.e'_4
σ : Type u_1
α : Type u_2
inst✝³ : DecidableEq σ
inst✝² : TopologicalSpace α
inst✝¹ : CommRing α
inst✝ : TopologicalRing α
s : Set α
i : σ
hs : s ∈ nhds Zero.zero
x : σ
hx : ¬x = i
⊢ X x (Finsupp.single i 1) = Zero.zero | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.intro
σ : Type u_1
α : Type u_2
inst✝³ : DecidableEq σ
inst✝² : TopologicalSpace α
inst✝¹ : CommRing α
inst✝ : TopologicalRing α
s : Set α
i : σ
hs : s ∈ nhds Zero.zero
x : σ
hx : ¬x = i
⊢ X x (Finsupp.single i 1) ∈ s
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/Topology.lean | MvPowerSeries.variables_tendsto_zero | [186, 1] | [213, 20] | rw [← coeff_eq_apply (X x) (Finsupp.single i 1), coeff_X, if_neg] | case h.e'_4
σ : Type u_1
α : Type u_2
inst✝³ : DecidableEq σ
inst✝² : TopologicalSpace α
inst✝¹ : CommRing α
inst✝ : TopologicalRing α
s : Set α
i : σ
hs : s ∈ nhds Zero.zero
x : σ
hx : ¬x = i
⊢ X x (Finsupp.single i 1) = Zero.zero | case h.e'_4
σ : Type u_1
α : Type u_2
inst✝³ : DecidableEq σ
inst✝² : TopologicalSpace α
inst✝¹ : CommRing α
inst✝ : TopologicalRing α
s : Set α
i : σ
hs : s ∈ nhds Zero.zero
x : σ
hx : ¬x = i
⊢ 0 = Zero.zero
case h.e'_4.hnc
σ : Type u_1
α : Type u_2
inst✝³ : DecidableEq σ
inst✝² : TopologicalSpace α
inst✝¹ : CommRing... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_4
σ : Type u_1
α : Type u_2
inst✝³ : DecidableEq σ
inst✝² : TopologicalSpace α
inst✝¹ : CommRing α
inst✝ : TopologicalRing α
s : Set α
i : σ
hs : s ∈ nhds Zero.zero
x : σ
hx : ¬x = i
⊢ X x (Finsupp.single i 1) = Zero.zero
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/Topology.lean | MvPowerSeries.variables_tendsto_zero | [186, 1] | [213, 20] | rfl | case h.e'_4
σ : Type u_1
α : Type u_2
inst✝³ : DecidableEq σ
inst✝² : TopologicalSpace α
inst✝¹ : CommRing α
inst✝ : TopologicalRing α
s : Set α
i : σ
hs : s ∈ nhds Zero.zero
x : σ
hx : ¬x = i
⊢ 0 = Zero.zero
case h.e'_4.hnc
σ : Type u_1
α : Type u_2
inst✝³ : DecidableEq σ
inst✝² : TopologicalSpace α
inst✝¹ : CommRing... | case h.e'_4.hnc
σ : Type u_1
α : Type u_2
inst✝³ : DecidableEq σ
inst✝² : TopologicalSpace α
inst✝¹ : CommRing α
inst✝ : TopologicalRing α
s : Set α
i : σ
hs : s ∈ nhds Zero.zero
x : σ
hx : ¬x = i
⊢ ¬Finsupp.single i 1 = Finsupp.single x 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_4
σ : Type u_1
α : Type u_2
inst✝³ : DecidableEq σ
inst✝² : TopologicalSpace α
inst✝¹ : CommRing α
inst✝ : TopologicalRing α
s : Set α
i : σ
hs : s ∈ nhds Zero.zero
x : σ
hx : ¬x = i
⊢ 0 = Zero.zero
case h.e'_4.hnc
σ : Type u_1
α : Type u_2
inst✝³ ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/Topology.lean | MvPowerSeries.variables_tendsto_zero | [186, 1] | [213, 20] | simp only [Finsupp.single_eq_single_iff, Ne.symm hx, and_true, one_ne_zero, and_self,
or_self, not_false_eq_true] | case h.e'_4.hnc
σ : Type u_1
α : Type u_2
inst✝³ : DecidableEq σ
inst✝² : TopologicalSpace α
inst✝¹ : CommRing α
inst✝ : TopologicalRing α
s : Set α
i : σ
hs : s ∈ nhds Zero.zero
x : σ
hx : ¬x = i
⊢ ¬Finsupp.single i 1 = Finsupp.single x 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_4.hnc
σ : Type u_1
α : Type u_2
inst✝³ : DecidableEq σ
inst✝² : TopologicalSpace α
inst✝¹ : CommRing α
inst✝ : TopologicalRing α
s : Set α
i : σ
hs : s ∈ nhds Zero.zero
x : σ
hx : ¬x = i
⊢ ¬Finsupp.single i 1 = Finsupp.single x 1
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/Topology.lean | MvPowerSeries.variables_tendsto_zero | [186, 1] | [213, 20] | convert Set.finite_empty | case neg
σ : Type u_1
α : Type u_2
inst✝³ : DecidableEq σ
inst✝² : TopologicalSpace α
inst✝¹ : CommRing α
inst✝ : TopologicalRing α
d : σ →₀ ℕ
s : Set α
hs : s ∈ nhds (0 d)
h : ¬∃ i, d = Finsupp.single i 1
⊢ ((fun i => X i d) ⁻¹' sᶜ).Finite | case h.e'_2
σ : Type u_1
α : Type u_2
inst✝³ : DecidableEq σ
inst✝² : TopologicalSpace α
inst✝¹ : CommRing α
inst✝ : TopologicalRing α
d : σ →₀ ℕ
s : Set α
hs : s ∈ nhds (0 d)
h : ¬∃ i, d = Finsupp.single i 1
⊢ (fun i => X i d) ⁻¹' sᶜ = ∅ | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
σ : Type u_1
α : Type u_2
inst✝³ : DecidableEq σ
inst✝² : TopologicalSpace α
inst✝¹ : CommRing α
inst✝ : TopologicalRing α
d : σ →₀ ℕ
s : Set α
hs : s ∈ nhds (0 d)
h : ¬∃ i, d = Finsupp.single i 1
⊢ ((fun i => X i d) ⁻¹' sᶜ).Finite
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/Topology.lean | MvPowerSeries.variables_tendsto_zero | [186, 1] | [213, 20] | rw [Set.eq_empty_iff_forall_not_mem] | case h.e'_2
σ : Type u_1
α : Type u_2
inst✝³ : DecidableEq σ
inst✝² : TopologicalSpace α
inst✝¹ : CommRing α
inst✝ : TopologicalRing α
d : σ →₀ ℕ
s : Set α
hs : s ∈ nhds (0 d)
h : ¬∃ i, d = Finsupp.single i 1
⊢ (fun i => X i d) ⁻¹' sᶜ = ∅ | case h.e'_2
σ : Type u_1
α : Type u_2
inst✝³ : DecidableEq σ
inst✝² : TopologicalSpace α
inst✝¹ : CommRing α
inst✝ : TopologicalRing α
d : σ →₀ ℕ
s : Set α
hs : s ∈ nhds (0 d)
h : ¬∃ i, d = Finsupp.single i 1
⊢ ∀ (x : σ), x ∉ (fun i => X i d) ⁻¹' sᶜ | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_2
σ : Type u_1
α : Type u_2
inst✝³ : DecidableEq σ
inst✝² : TopologicalSpace α
inst✝¹ : CommRing α
inst✝ : TopologicalRing α
d : σ →₀ ℕ
s : Set α
hs : s ∈ nhds (0 d)
h : ¬∃ i, d = Finsupp.single i 1
⊢ (fun i => X i d) ⁻¹' sᶜ = ∅
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/Topology.lean | MvPowerSeries.variables_tendsto_zero | [186, 1] | [213, 20] | intro x | case h.e'_2
σ : Type u_1
α : Type u_2
inst✝³ : DecidableEq σ
inst✝² : TopologicalSpace α
inst✝¹ : CommRing α
inst✝ : TopologicalRing α
d : σ →₀ ℕ
s : Set α
hs : s ∈ nhds (0 d)
h : ¬∃ i, d = Finsupp.single i 1
⊢ ∀ (x : σ), x ∉ (fun i => X i d) ⁻¹' sᶜ | case h.e'_2
σ : Type u_1
α : Type u_2
inst✝³ : DecidableEq σ
inst✝² : TopologicalSpace α
inst✝¹ : CommRing α
inst✝ : TopologicalRing α
d : σ →₀ ℕ
s : Set α
hs : s ∈ nhds (0 d)
h : ¬∃ i, d = Finsupp.single i 1
x : σ
⊢ x ∉ (fun i => X i d) ⁻¹' sᶜ | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_2
σ : Type u_1
α : Type u_2
inst✝³ : DecidableEq σ
inst✝² : TopologicalSpace α
inst✝¹ : CommRing α
inst✝ : TopologicalRing α
d : σ →₀ ℕ
s : Set α
hs : s ∈ nhds (0 d)
h : ¬∃ i, d = Finsupp.single i 1
⊢ ∀ (x : σ), x ∉ (fun i => X i d) ⁻¹' sᶜ
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/Topology.lean | MvPowerSeries.variables_tendsto_zero | [186, 1] | [213, 20] | rw [Set.mem_preimage, Set.not_mem_compl_iff] | case h.e'_2
σ : Type u_1
α : Type u_2
inst✝³ : DecidableEq σ
inst✝² : TopologicalSpace α
inst✝¹ : CommRing α
inst✝ : TopologicalRing α
d : σ →₀ ℕ
s : Set α
hs : s ∈ nhds (0 d)
h : ¬∃ i, d = Finsupp.single i 1
x : σ
⊢ x ∉ (fun i => X i d) ⁻¹' sᶜ | case h.e'_2
σ : Type u_1
α : Type u_2
inst✝³ : DecidableEq σ
inst✝² : TopologicalSpace α
inst✝¹ : CommRing α
inst✝ : TopologicalRing α
d : σ →₀ ℕ
s : Set α
hs : s ∈ nhds (0 d)
h : ¬∃ i, d = Finsupp.single i 1
x : σ
⊢ X x d ∈ s | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_2
σ : Type u_1
α : Type u_2
inst✝³ : DecidableEq σ
inst✝² : TopologicalSpace α
inst✝¹ : CommRing α
inst✝ : TopologicalRing α
d : σ →₀ ℕ
s : Set α
hs : s ∈ nhds (0 d)
h : ¬∃ i, d = Finsupp.single i 1
x : σ
⊢ x ∉ (fun i => X i d) ⁻¹' sᶜ
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/Topology.lean | MvPowerSeries.variables_tendsto_zero | [186, 1] | [213, 20] | convert mem_of_mem_nhds hs using 1 | case h.e'_2
σ : Type u_1
α : Type u_2
inst✝³ : DecidableEq σ
inst✝² : TopologicalSpace α
inst✝¹ : CommRing α
inst✝ : TopologicalRing α
d : σ →₀ ℕ
s : Set α
hs : s ∈ nhds (0 d)
h : ¬∃ i, d = Finsupp.single i 1
x : σ
⊢ X x d ∈ s | case h.e'_4
σ : Type u_1
α : Type u_2
inst✝³ : DecidableEq σ
inst✝² : TopologicalSpace α
inst✝¹ : CommRing α
inst✝ : TopologicalRing α
d : σ →₀ ℕ
s : Set α
hs : s ∈ nhds (0 d)
h : ¬∃ i, d = Finsupp.single i 1
x : σ
⊢ X x d = 0 d | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_2
σ : Type u_1
α : Type u_2
inst✝³ : DecidableEq σ
inst✝² : TopologicalSpace α
inst✝¹ : CommRing α
inst✝ : TopologicalRing α
d : σ →₀ ℕ
s : Set α
hs : s ∈ nhds (0 d)
h : ¬∃ i, d = Finsupp.single i 1
x : σ
⊢ X x d ∈ s
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/Topology.lean | MvPowerSeries.variables_tendsto_zero | [186, 1] | [213, 20] | rw [← coeff_eq_apply (X x) d, coeff_X, if_neg] | case h.e'_4
σ : Type u_1
α : Type u_2
inst✝³ : DecidableEq σ
inst✝² : TopologicalSpace α
inst✝¹ : CommRing α
inst✝ : TopologicalRing α
d : σ →₀ ℕ
s : Set α
hs : s ∈ nhds (0 d)
h : ¬∃ i, d = Finsupp.single i 1
x : σ
⊢ X x d = 0 d | case h.e'_4
σ : Type u_1
α : Type u_2
inst✝³ : DecidableEq σ
inst✝² : TopologicalSpace α
inst✝¹ : CommRing α
inst✝ : TopologicalRing α
d : σ →₀ ℕ
s : Set α
hs : s ∈ nhds (0 d)
h : ¬∃ i, d = Finsupp.single i 1
x : σ
⊢ 0 = 0 d
case h.e'_4.hnc
σ : Type u_1
α : Type u_2
inst✝³ : DecidableEq σ
inst✝² : TopologicalSpace α
i... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_4
σ : Type u_1
α : Type u_2
inst✝³ : DecidableEq σ
inst✝² : TopologicalSpace α
inst✝¹ : CommRing α
inst✝ : TopologicalRing α
d : σ →₀ ℕ
s : Set α
hs : s ∈ nhds (0 d)
h : ¬∃ i, d = Finsupp.single i 1
x : σ
⊢ X x d = 0 d
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/Topology.lean | MvPowerSeries.variables_tendsto_zero | [186, 1] | [213, 20] | rfl | case h.e'_4
σ : Type u_1
α : Type u_2
inst✝³ : DecidableEq σ
inst✝² : TopologicalSpace α
inst✝¹ : CommRing α
inst✝ : TopologicalRing α
d : σ →₀ ℕ
s : Set α
hs : s ∈ nhds (0 d)
h : ¬∃ i, d = Finsupp.single i 1
x : σ
⊢ 0 = 0 d
case h.e'_4.hnc
σ : Type u_1
α : Type u_2
inst✝³ : DecidableEq σ
inst✝² : TopologicalSpace α
i... | case h.e'_4.hnc
σ : Type u_1
α : Type u_2
inst✝³ : DecidableEq σ
inst✝² : TopologicalSpace α
inst✝¹ : CommRing α
inst✝ : TopologicalRing α
d : σ →₀ ℕ
s : Set α
hs : s ∈ nhds (0 d)
h : ¬∃ i, d = Finsupp.single i 1
x : σ
⊢ ¬d = Finsupp.single x 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_4
σ : Type u_1
α : Type u_2
inst✝³ : DecidableEq σ
inst✝² : TopologicalSpace α
inst✝¹ : CommRing α
inst✝ : TopologicalRing α
d : σ →₀ ℕ
s : Set α
hs : s ∈ nhds (0 d)
h : ¬∃ i, d = Finsupp.single i 1
x : σ
⊢ 0 = 0 d
case h.e'_4.hnc
σ : Type u_1
α : ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/Topology.lean | MvPowerSeries.variables_tendsto_zero | [186, 1] | [213, 20] | . intro h'
apply h
exact ⟨x, h'⟩ | case h.e'_4.hnc
σ : Type u_1
α : Type u_2
inst✝³ : DecidableEq σ
inst✝² : TopologicalSpace α
inst✝¹ : CommRing α
inst✝ : TopologicalRing α
d : σ →₀ ℕ
s : Set α
hs : s ∈ nhds (0 d)
h : ¬∃ i, d = Finsupp.single i 1
x : σ
⊢ ¬d = Finsupp.single x 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_4.hnc
σ : Type u_1
α : Type u_2
inst✝³ : DecidableEq σ
inst✝² : TopologicalSpace α
inst✝¹ : CommRing α
inst✝ : TopologicalRing α
d : σ →₀ ℕ
s : Set α
hs : s ∈ nhds (0 d)
h : ¬∃ i, d = Finsupp.single i 1
x : σ
⊢ ¬d = Finsupp.single x 1
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/Topology.lean | MvPowerSeries.variables_tendsto_zero | [186, 1] | [213, 20] | intro h' | case h.e'_4.hnc
σ : Type u_1
α : Type u_2
inst✝³ : DecidableEq σ
inst✝² : TopologicalSpace α
inst✝¹ : CommRing α
inst✝ : TopologicalRing α
d : σ →₀ ℕ
s : Set α
hs : s ∈ nhds (0 d)
h : ¬∃ i, d = Finsupp.single i 1
x : σ
⊢ ¬d = Finsupp.single x 1 | case h.e'_4.hnc
σ : Type u_1
α : Type u_2
inst✝³ : DecidableEq σ
inst✝² : TopologicalSpace α
inst✝¹ : CommRing α
inst✝ : TopologicalRing α
d : σ →₀ ℕ
s : Set α
hs : s ∈ nhds (0 d)
h : ¬∃ i, d = Finsupp.single i 1
x : σ
h' : d = Finsupp.single x 1
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_4.hnc
σ : Type u_1
α : Type u_2
inst✝³ : DecidableEq σ
inst✝² : TopologicalSpace α
inst✝¹ : CommRing α
inst✝ : TopologicalRing α
d : σ →₀ ℕ
s : Set α
hs : s ∈ nhds (0 d)
h : ¬∃ i, d = Finsupp.single i 1
x : σ
⊢ ¬d = Finsupp.single x 1
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/Topology.lean | MvPowerSeries.variables_tendsto_zero | [186, 1] | [213, 20] | apply h | case h.e'_4.hnc
σ : Type u_1
α : Type u_2
inst✝³ : DecidableEq σ
inst✝² : TopologicalSpace α
inst✝¹ : CommRing α
inst✝ : TopologicalRing α
d : σ →₀ ℕ
s : Set α
hs : s ∈ nhds (0 d)
h : ¬∃ i, d = Finsupp.single i 1
x : σ
h' : d = Finsupp.single x 1
⊢ False | case h.e'_4.hnc
σ : Type u_1
α : Type u_2
inst✝³ : DecidableEq σ
inst✝² : TopologicalSpace α
inst✝¹ : CommRing α
inst✝ : TopologicalRing α
d : σ →₀ ℕ
s : Set α
hs : s ∈ nhds (0 d)
h : ¬∃ i, d = Finsupp.single i 1
x : σ
h' : d = Finsupp.single x 1
⊢ ∃ i, d = Finsupp.single i 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_4.hnc
σ : Type u_1
α : Type u_2
inst✝³ : DecidableEq σ
inst✝² : TopologicalSpace α
inst✝¹ : CommRing α
inst✝ : TopologicalRing α
d : σ →₀ ℕ
s : Set α
hs : s ∈ nhds (0 d)
h : ¬∃ i, d = Finsupp.single i 1
x : σ
h' : d = Finsupp.single x 1
⊢ False
TACT... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/Topology.lean | MvPowerSeries.variables_tendsto_zero | [186, 1] | [213, 20] | exact ⟨x, h'⟩ | case h.e'_4.hnc
σ : Type u_1
α : Type u_2
inst✝³ : DecidableEq σ
inst✝² : TopologicalSpace α
inst✝¹ : CommRing α
inst✝ : TopologicalRing α
d : σ →₀ ℕ
s : Set α
hs : s ∈ nhds (0 d)
h : ¬∃ i, d = Finsupp.single i 1
x : σ
h' : d = Finsupp.single x 1
⊢ ∃ i, d = Finsupp.single i 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_4.hnc
σ : Type u_1
α : Type u_2
inst✝³ : DecidableEq σ
inst✝² : TopologicalSpace α
inst✝¹ : CommRing α
inst✝ : TopologicalRing α
d : σ →₀ ℕ
s : Set α
hs : s ∈ nhds (0 d)
h : ¬∃ i, d = Finsupp.single i 1
x : σ
h' : d = Finsupp.single x 1
⊢ ∃ i, d = F... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/Topology.lean | MvPowerSeries.tendsto_pow_zero_of_constantCoeff_nilpotent | [215, 1] | [222, 59] | classical
obtain ⟨m, hm⟩ := hf
simp_rw [tendsto_iff_coeff_tendsto, coeff_zero]
exact fun d => tendsto_atTop_of_eventually_const fun n hn =>
coeff_eq_zero_of_constantCoeff_nilpotent f m hm d n hn | σ : Type u_2
α : Type u_1
inst✝³ : DecidableEq σ
inst✝² : TopologicalSpace α
inst✝¹ : CommRing α
inst✝ : TopologicalRing α
f : MvPowerSeries σ α
hf : IsNilpotent ((constantCoeff σ α) f)
⊢ Filter.Tendsto (fun n => f ^ n) Filter.atTop (nhds 0) | no goals | 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
f : MvPowerSeries σ α
hf : IsNilpotent ((constantCoeff σ α) f)
⊢ Filter.Tendsto (fun n => f ^ n) Filter.atTop (nhds 0)
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/Topology.lean | MvPowerSeries.tendsto_pow_zero_of_constantCoeff_nilpotent | [215, 1] | [222, 59] | obtain ⟨m, hm⟩ := hf | σ : Type u_2
α : Type u_1
inst✝³ : DecidableEq σ
inst✝² : TopologicalSpace α
inst✝¹ : CommRing α
inst✝ : TopologicalRing α
f : MvPowerSeries σ α
hf : IsNilpotent ((constantCoeff σ α) f)
⊢ Filter.Tendsto (fun n => f ^ n) Filter.atTop (nhds 0) | case intro
σ : Type u_2
α : Type u_1
inst✝³ : DecidableEq σ
inst✝² : TopologicalSpace α
inst✝¹ : CommRing α
inst✝ : TopologicalRing α
f : MvPowerSeries σ α
m : ℕ
hm : (constantCoeff σ α) f ^ m = 0
⊢ Filter.Tendsto (fun n => f ^ n) Filter.atTop (nhds 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
f : MvPowerSeries σ α
hf : IsNilpotent ((constantCoeff σ α) f)
⊢ Filter.Tendsto (fun n => f ^ n) Filter.atTop (nhds 0)
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/Topology.lean | MvPowerSeries.tendsto_pow_zero_of_constantCoeff_nilpotent | [215, 1] | [222, 59] | simp_rw [tendsto_iff_coeff_tendsto, coeff_zero] | case intro
σ : Type u_2
α : Type u_1
inst✝³ : DecidableEq σ
inst✝² : TopologicalSpace α
inst✝¹ : CommRing α
inst✝ : TopologicalRing α
f : MvPowerSeries σ α
m : ℕ
hm : (constantCoeff σ α) f ^ m = 0
⊢ Filter.Tendsto (fun n => f ^ n) Filter.atTop (nhds 0) | case intro
σ : Type u_2
α : Type u_1
inst✝³ : DecidableEq σ
inst✝² : TopologicalSpace α
inst✝¹ : CommRing α
inst✝ : TopologicalRing α
f : MvPowerSeries σ α
m : ℕ
hm : (constantCoeff σ α) f ^ m = 0
⊢ ∀ (d : σ →₀ ℕ), Filter.Tendsto (fun i => (coeff α d) (f ^ i)) Filter.atTop (nhds 0) | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
σ : Type u_2
α : Type u_1
inst✝³ : DecidableEq σ
inst✝² : TopologicalSpace α
inst✝¹ : CommRing α
inst✝ : TopologicalRing α
f : MvPowerSeries σ α
m : ℕ
hm : (constantCoeff σ α) f ^ m = 0
⊢ Filter.Tendsto (fun n => f ^ n) Filter.atTop (nhds 0)
TACTIC... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/Topology.lean | MvPowerSeries.tendsto_pow_zero_of_constantCoeff_nilpotent | [215, 1] | [222, 59] | exact fun d => tendsto_atTop_of_eventually_const fun n hn =>
coeff_eq_zero_of_constantCoeff_nilpotent f m hm d n hn | case intro
σ : Type u_2
α : Type u_1
inst✝³ : DecidableEq σ
inst✝² : TopologicalSpace α
inst✝¹ : CommRing α
inst✝ : TopologicalRing α
f : MvPowerSeries σ α
m : ℕ
hm : (constantCoeff σ α) f ^ m = 0
⊢ ∀ (d : σ →₀ ℕ), Filter.Tendsto (fun i => (coeff α d) (f ^ i)) Filter.atTop (nhds 0) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
σ : Type u_2
α : Type u_1
inst✝³ : DecidableEq σ
inst✝² : TopologicalSpace α
inst✝¹ : CommRing α
inst✝ : TopologicalRing α
f : MvPowerSeries σ α
m : ℕ
hm : (constantCoeff σ α) f ^ m = 0
⊢ ∀ (d : σ →₀ ℕ), Filter.Tendsto (fun i => (coeff α d) (f ^ i)... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/Topology.lean | MvPowerSeries.tendsto_pow_zero_of_constantCoeff_zero | [224, 1] | [228, 25] | apply tendsto_pow_zero_of_constantCoeff_nilpotent | σ : Type u_2
α : Type u_1
inst✝³ : DecidableEq σ
inst✝² : TopologicalSpace α
inst✝¹ : CommRing α
inst✝ : TopologicalRing α
f : MvPowerSeries σ α
hf : (constantCoeff σ α) f = 0
⊢ Filter.Tendsto (fun n => f ^ n) Filter.atTop (nhds 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
f : MvPowerSeries σ α
hf : (constantCoeff σ α) f = 0
⊢ IsNilpotent ((constantCoeff σ α) f) | 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
f : MvPowerSeries σ α
hf : (constantCoeff σ α) f = 0
⊢ Filter.Tendsto (fun n => f ^ n) Filter.atTop (nhds 0)
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/Topology.lean | MvPowerSeries.tendsto_pow_zero_of_constantCoeff_zero | [224, 1] | [228, 25] | rw [hf] | 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
f : MvPowerSeries σ α
hf : (constantCoeff σ α) f = 0
⊢ IsNilpotent ((constantCoeff σ α) f) | 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
f : MvPowerSeries σ α
hf : (constantCoeff σ α) f = 0
⊢ IsNilpotent 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
f : MvPowerSeries σ α
hf : (constantCoeff σ α) f = 0
⊢ IsNilpotent ((constantCoeff σ α) f)
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/Topology.lean | MvPowerSeries.tendsto_pow_zero_of_constantCoeff_zero | [224, 1] | [228, 25] | exact IsNilpotent.zero | 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
f : MvPowerSeries σ α
hf : (constantCoeff σ α) f = 0
⊢ IsNilpotent 0 | no goals | 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
f : MvPowerSeries σ α
hf : (constantCoeff σ α) f = 0
⊢ IsNilpotent 0
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/Topology.lean | MvPowerSeries.tendsto_pow_of_constantCoeff_nilpotent_iff | [231, 1] | [252, 48] | refine' ⟨_, tendsto_pow_zero_of_constantCoeff_nilpotent ⟩ | σ : Type u_2
α : Type u_1
inst✝⁴ : DecidableEq σ
inst✝³ : TopologicalSpace α
inst✝² : CommRing α
inst✝¹ : TopologicalRing α
inst✝ : DiscreteTopology α
f : MvPowerSeries σ α
⊢ Filter.Tendsto (fun n => f ^ n) Filter.atTop (nhds 0) ↔ IsNilpotent ((constantCoeff σ α) f) | σ : Type u_2
α : Type u_1
inst✝⁴ : DecidableEq σ
inst✝³ : TopologicalSpace α
inst✝² : CommRing α
inst✝¹ : TopologicalRing α
inst✝ : DiscreteTopology α
f : MvPowerSeries σ α
⊢ Filter.Tendsto (fun n => f ^ n) Filter.atTop (nhds 0) → IsNilpotent ((constantCoeff σ α) f) | Please generate a tactic in lean4 to solve the state.
STATE:
σ : Type u_2
α : Type u_1
inst✝⁴ : DecidableEq σ
inst✝³ : TopologicalSpace α
inst✝² : CommRing α
inst✝¹ : TopologicalRing α
inst✝ : DiscreteTopology α
f : MvPowerSeries σ α
⊢ Filter.Tendsto (fun n => f ^ n) Filter.atTop (nhds 0) ↔ IsNilpotent ((constantCoeff ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/Topology.lean | MvPowerSeries.tendsto_pow_of_constantCoeff_nilpotent_iff | [231, 1] | [252, 48] | intro h | σ : Type u_2
α : Type u_1
inst✝⁴ : DecidableEq σ
inst✝³ : TopologicalSpace α
inst✝² : CommRing α
inst✝¹ : TopologicalRing α
inst✝ : DiscreteTopology α
f : MvPowerSeries σ α
⊢ Filter.Tendsto (fun n => f ^ n) Filter.atTop (nhds 0) → IsNilpotent ((constantCoeff σ α) f) | σ : Type u_2
α : Type u_1
inst✝⁴ : DecidableEq σ
inst✝³ : TopologicalSpace α
inst✝² : CommRing α
inst✝¹ : TopologicalRing α
inst✝ : DiscreteTopology α
f : MvPowerSeries σ α
h : Filter.Tendsto (fun n => f ^ n) Filter.atTop (nhds 0)
⊢ IsNilpotent ((constantCoeff σ α) f) | Please generate a tactic in lean4 to solve the state.
STATE:
σ : Type u_2
α : Type u_1
inst✝⁴ : DecidableEq σ
inst✝³ : TopologicalSpace α
inst✝² : CommRing α
inst✝¹ : TopologicalRing α
inst✝ : DiscreteTopology α
f : MvPowerSeries σ α
⊢ Filter.Tendsto (fun n => f ^ n) Filter.atTop (nhds 0) → IsNilpotent ((constantCoeff ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/Topology.lean | MvPowerSeries.tendsto_pow_of_constantCoeff_nilpotent_iff | [231, 1] | [252, 48] | suffices Filter.Tendsto (fun n : ℕ => constantCoeff σ α (f ^ n)) Filter.atTop (nhds 0) by
simp only [Filter.tendsto_def] at this
specialize this {0} _
suffices ∀ x : α, {x} ∈ nhds x by exact this 0
rw [← discreteTopology_iff_singleton_mem_nhds]; infer_instance
simp only [map_pow, Filter.mem_atTop_sets, ge_if... | σ : Type u_2
α : Type u_1
inst✝⁴ : DecidableEq σ
inst✝³ : TopologicalSpace α
inst✝² : CommRing α
inst✝¹ : TopologicalRing α
inst✝ : DiscreteTopology α
f : MvPowerSeries σ α
h : Filter.Tendsto (fun n => f ^ n) Filter.atTop (nhds 0)
⊢ IsNilpotent ((constantCoeff σ α) f) | σ : Type u_2
α : Type u_1
inst✝⁴ : DecidableEq σ
inst✝³ : TopologicalSpace α
inst✝² : CommRing α
inst✝¹ : TopologicalRing α
inst✝ : DiscreteTopology α
f : MvPowerSeries σ α
h : Filter.Tendsto (fun n => f ^ n) Filter.atTop (nhds 0)
⊢ Filter.Tendsto (fun n => (constantCoeff σ α) (f ^ n)) Filter.atTop (nhds 0) | Please generate a tactic in lean4 to solve the state.
STATE:
σ : Type u_2
α : Type u_1
inst✝⁴ : DecidableEq σ
inst✝³ : TopologicalSpace α
inst✝² : CommRing α
inst✝¹ : TopologicalRing α
inst✝ : DiscreteTopology α
f : MvPowerSeries σ α
h : Filter.Tendsto (fun n => f ^ n) Filter.atTop (nhds 0)
⊢ IsNilpotent ((constantCoef... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/Topology.lean | MvPowerSeries.tendsto_pow_of_constantCoeff_nilpotent_iff | [231, 1] | [252, 48] | simp only [← @comp_apply _ α ℕ] | σ : Type u_2
α : Type u_1
inst✝⁴ : DecidableEq σ
inst✝³ : TopologicalSpace α
inst✝² : CommRing α
inst✝¹ : TopologicalRing α
inst✝ : DiscreteTopology α
f : MvPowerSeries σ α
h : Filter.Tendsto (fun n => f ^ n) Filter.atTop (nhds 0)
⊢ Filter.Tendsto (fun n => (constantCoeff σ α) (f ^ n)) Filter.atTop (nhds 0) | σ : Type u_2
α : Type u_1
inst✝⁴ : DecidableEq σ
inst✝³ : TopologicalSpace α
inst✝² : CommRing α
inst✝¹ : TopologicalRing α
inst✝ : DiscreteTopology α
f : MvPowerSeries σ α
h : Filter.Tendsto (fun n => f ^ n) Filter.atTop (nhds 0)
⊢ Filter.Tendsto (fun n => (⇑(constantCoeff σ α) ∘ HPow.hPow f) n) Filter.atTop (nhds 0) | Please generate a tactic in lean4 to solve the state.
STATE:
σ : Type u_2
α : Type u_1
inst✝⁴ : DecidableEq σ
inst✝³ : TopologicalSpace α
inst✝² : CommRing α
inst✝¹ : TopologicalRing α
inst✝ : DiscreteTopology α
f : MvPowerSeries σ α
h : Filter.Tendsto (fun n => f ^ n) Filter.atTop (nhds 0)
⊢ Filter.Tendsto (fun n => (... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/Topology.lean | MvPowerSeries.tendsto_pow_of_constantCoeff_nilpotent_iff | [231, 1] | [252, 48] | rw [← Filter.tendsto_map'_iff] | σ : Type u_2
α : Type u_1
inst✝⁴ : DecidableEq σ
inst✝³ : TopologicalSpace α
inst✝² : CommRing α
inst✝¹ : TopologicalRing α
inst✝ : DiscreteTopology α
f : MvPowerSeries σ α
h : Filter.Tendsto (fun n => f ^ n) Filter.atTop (nhds 0)
⊢ Filter.Tendsto (fun n => (⇑(constantCoeff σ α) ∘ HPow.hPow f) n) Filter.atTop (nhds 0) | σ : Type u_2
α : Type u_1
inst✝⁴ : DecidableEq σ
inst✝³ : TopologicalSpace α
inst✝² : CommRing α
inst✝¹ : TopologicalRing α
inst✝ : DiscreteTopology α
f : MvPowerSeries σ α
h : Filter.Tendsto (fun n => f ^ n) Filter.atTop (nhds 0)
⊢ Filter.Tendsto (⇑(constantCoeff σ α)) (Filter.map (HPow.hPow f) Filter.atTop) (nhds 0) | Please generate a tactic in lean4 to solve the state.
STATE:
σ : Type u_2
α : Type u_1
inst✝⁴ : DecidableEq σ
inst✝³ : TopologicalSpace α
inst✝² : CommRing α
inst✝¹ : TopologicalRing α
inst✝ : DiscreteTopology α
f : MvPowerSeries σ α
h : Filter.Tendsto (fun n => f ^ n) Filter.atTop (nhds 0)
⊢ Filter.Tendsto (fun n => (... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/Topology.lean | MvPowerSeries.tendsto_pow_of_constantCoeff_nilpotent_iff | [231, 1] | [252, 48] | simp only [Filter.Tendsto, Filter.map_le_iff_le_comap] at h ⊢ | σ : Type u_2
α : Type u_1
inst✝⁴ : DecidableEq σ
inst✝³ : TopologicalSpace α
inst✝² : CommRing α
inst✝¹ : TopologicalRing α
inst✝ : DiscreteTopology α
f : MvPowerSeries σ α
h : Filter.Tendsto (fun n => f ^ n) Filter.atTop (nhds 0)
⊢ Filter.Tendsto (⇑(constantCoeff σ α)) (Filter.map (HPow.hPow f) Filter.atTop) (nhds 0) | σ : Type u_2
α : Type u_1
inst✝⁴ : DecidableEq σ
inst✝³ : TopologicalSpace α
inst✝² : CommRing α
inst✝¹ : TopologicalRing α
inst✝ : DiscreteTopology α
f : MvPowerSeries σ α
h : Filter.atTop ≤ Filter.comap (fun n => f ^ n) (nhds 0)
⊢ Filter.atTop ≤ Filter.comap (HPow.hPow f) (Filter.comap (⇑(constantCoeff σ α)) (nhds 0)... | Please generate a tactic in lean4 to solve the state.
STATE:
σ : Type u_2
α : Type u_1
inst✝⁴ : DecidableEq σ
inst✝³ : TopologicalSpace α
inst✝² : CommRing α
inst✝¹ : TopologicalRing α
inst✝ : DiscreteTopology α
f : MvPowerSeries σ α
h : Filter.Tendsto (fun n => f ^ n) Filter.atTop (nhds 0)
⊢ Filter.Tendsto (⇑(constant... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/Topology.lean | MvPowerSeries.tendsto_pow_of_constantCoeff_nilpotent_iff | [231, 1] | [252, 48] | apply le_trans h | σ : Type u_2
α : Type u_1
inst✝⁴ : DecidableEq σ
inst✝³ : TopologicalSpace α
inst✝² : CommRing α
inst✝¹ : TopologicalRing α
inst✝ : DiscreteTopology α
f : MvPowerSeries σ α
h : Filter.atTop ≤ Filter.comap (fun n => f ^ n) (nhds 0)
⊢ Filter.atTop ≤ Filter.comap (HPow.hPow f) (Filter.comap (⇑(constantCoeff σ α)) (nhds 0)... | σ : Type u_2
α : Type u_1
inst✝⁴ : DecidableEq σ
inst✝³ : TopologicalSpace α
inst✝² : CommRing α
inst✝¹ : TopologicalRing α
inst✝ : DiscreteTopology α
f : MvPowerSeries σ α
h : Filter.atTop ≤ Filter.comap (fun n => f ^ n) (nhds 0)
⊢ Filter.comap (fun n => f ^ n) (nhds 0) ≤ Filter.comap (HPow.hPow f) (Filter.comap (⇑(co... | Please generate a tactic in lean4 to solve the state.
STATE:
σ : Type u_2
α : Type u_1
inst✝⁴ : DecidableEq σ
inst✝³ : TopologicalSpace α
inst✝² : CommRing α
inst✝¹ : TopologicalRing α
inst✝ : DiscreteTopology α
f : MvPowerSeries σ α
h : Filter.atTop ≤ Filter.comap (fun n => f ^ n) (nhds 0)
⊢ Filter.atTop ≤ Filter.coma... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/Topology.lean | MvPowerSeries.tendsto_pow_of_constantCoeff_nilpotent_iff | [231, 1] | [252, 48] | apply Filter.comap_mono | σ : Type u_2
α : Type u_1
inst✝⁴ : DecidableEq σ
inst✝³ : TopologicalSpace α
inst✝² : CommRing α
inst✝¹ : TopologicalRing α
inst✝ : DiscreteTopology α
f : MvPowerSeries σ α
h : Filter.atTop ≤ Filter.comap (fun n => f ^ n) (nhds 0)
⊢ Filter.comap (fun n => f ^ n) (nhds 0) ≤ Filter.comap (HPow.hPow f) (Filter.comap (⇑(co... | case a
σ : Type u_2
α : Type u_1
inst✝⁴ : DecidableEq σ
inst✝³ : TopologicalSpace α
inst✝² : CommRing α
inst✝¹ : TopologicalRing α
inst✝ : DiscreteTopology α
f : MvPowerSeries σ α
h : Filter.atTop ≤ Filter.comap (fun n => f ^ n) (nhds 0)
⊢ nhds 0 ≤ Filter.comap (⇑(constantCoeff σ α)) (nhds 0) | Please generate a tactic in lean4 to solve the state.
STATE:
σ : Type u_2
α : Type u_1
inst✝⁴ : DecidableEq σ
inst✝³ : TopologicalSpace α
inst✝² : CommRing α
inst✝¹ : TopologicalRing α
inst✝ : DiscreteTopology α
f : MvPowerSeries σ α
h : Filter.atTop ≤ Filter.comap (fun n => f ^ n) (nhds 0)
⊢ Filter.comap (fun n => f ^... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/Topology.lean | MvPowerSeries.tendsto_pow_of_constantCoeff_nilpotent_iff | [231, 1] | [252, 48] | rw [← Filter.map_le_iff_le_comap] | case a
σ : Type u_2
α : Type u_1
inst✝⁴ : DecidableEq σ
inst✝³ : TopologicalSpace α
inst✝² : CommRing α
inst✝¹ : TopologicalRing α
inst✝ : DiscreteTopology α
f : MvPowerSeries σ α
h : Filter.atTop ≤ Filter.comap (fun n => f ^ n) (nhds 0)
⊢ nhds 0 ≤ Filter.comap (⇑(constantCoeff σ α)) (nhds 0) | case a
σ : Type u_2
α : Type u_1
inst✝⁴ : DecidableEq σ
inst✝³ : TopologicalSpace α
inst✝² : CommRing α
inst✝¹ : TopologicalRing α
inst✝ : DiscreteTopology α
f : MvPowerSeries σ α
h : Filter.atTop ≤ Filter.comap (fun n => f ^ n) (nhds 0)
⊢ Filter.map (⇑(constantCoeff σ α)) (nhds 0) ≤ nhds 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case a
σ : Type u_2
α : Type u_1
inst✝⁴ : DecidableEq σ
inst✝³ : TopologicalSpace α
inst✝² : CommRing α
inst✝¹ : TopologicalRing α
inst✝ : DiscreteTopology α
f : MvPowerSeries σ α
h : Filter.atTop ≤ Filter.comap (fun n => f ^ n) (nhds 0)
⊢ nhds 0 ≤ Filter.com... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/Topology.lean | MvPowerSeries.tendsto_pow_of_constantCoeff_nilpotent_iff | [231, 1] | [252, 48] | apply continuous_constantCoeff.continuousAt | case a
σ : Type u_2
α : Type u_1
inst✝⁴ : DecidableEq σ
inst✝³ : TopologicalSpace α
inst✝² : CommRing α
inst✝¹ : TopologicalRing α
inst✝ : DiscreteTopology α
f : MvPowerSeries σ α
h : Filter.atTop ≤ Filter.comap (fun n => f ^ n) (nhds 0)
⊢ Filter.map (⇑(constantCoeff σ α)) (nhds 0) ≤ nhds 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a
σ : Type u_2
α : Type u_1
inst✝⁴ : DecidableEq σ
inst✝³ : TopologicalSpace α
inst✝² : CommRing α
inst✝¹ : TopologicalRing α
inst✝ : DiscreteTopology α
f : MvPowerSeries σ α
h : Filter.atTop ≤ Filter.comap (fun n => f ^ n) (nhds 0)
⊢ Filter.map (⇑(const... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/Topology.lean | MvPowerSeries.tendsto_pow_of_constantCoeff_nilpotent_iff | [231, 1] | [252, 48] | simp only [Filter.tendsto_def] at this | σ : Type u_2
α : Type u_1
inst✝⁴ : DecidableEq σ
inst✝³ : TopologicalSpace α
inst✝² : CommRing α
inst✝¹ : TopologicalRing α
inst✝ : DiscreteTopology α
f : MvPowerSeries σ α
h : Filter.Tendsto (fun n => f ^ n) Filter.atTop (nhds 0)
this : Filter.Tendsto (fun n => (constantCoeff σ α) (f ^ n)) Filter.atTop (nhds 0)
⊢ IsNi... | σ : Type u_2
α : Type u_1
inst✝⁴ : DecidableEq σ
inst✝³ : TopologicalSpace α
inst✝² : CommRing α
inst✝¹ : TopologicalRing α
inst✝ : DiscreteTopology α
f : MvPowerSeries σ α
h : Filter.Tendsto (fun n => f ^ n) Filter.atTop (nhds 0)
this : ∀ s ∈ nhds 0, (fun n => (constantCoeff σ α) (f ^ n)) ⁻¹' s ∈ Filter.atTop
⊢ IsNilp... | Please generate a tactic in lean4 to solve the state.
STATE:
σ : Type u_2
α : Type u_1
inst✝⁴ : DecidableEq σ
inst✝³ : TopologicalSpace α
inst✝² : CommRing α
inst✝¹ : TopologicalRing α
inst✝ : DiscreteTopology α
f : MvPowerSeries σ α
h : Filter.Tendsto (fun n => f ^ n) Filter.atTop (nhds 0)
this : Filter.Tendsto (fun n... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/Topology.lean | MvPowerSeries.tendsto_pow_of_constantCoeff_nilpotent_iff | [231, 1] | [252, 48] | specialize this {0} _ | σ : Type u_2
α : Type u_1
inst✝⁴ : DecidableEq σ
inst✝³ : TopologicalSpace α
inst✝² : CommRing α
inst✝¹ : TopologicalRing α
inst✝ : DiscreteTopology α
f : MvPowerSeries σ α
h : Filter.Tendsto (fun n => f ^ n) Filter.atTop (nhds 0)
this : ∀ s ∈ nhds 0, (fun n => (constantCoeff σ α) (f ^ n)) ⁻¹' s ∈ Filter.atTop
⊢ IsNilp... | σ : Type u_2
α : Type u_1
inst✝⁴ : DecidableEq σ
inst✝³ : TopologicalSpace α
inst✝² : CommRing α
inst✝¹ : TopologicalRing α
inst✝ : DiscreteTopology α
f : MvPowerSeries σ α
h : Filter.Tendsto (fun n => f ^ n) Filter.atTop (nhds 0)
this : ∀ s ∈ nhds 0, (fun n => (constantCoeff σ α) (f ^ n)) ⁻¹' s ∈ Filter.atTop
⊢ {0} ∈ ... | Please generate a tactic in lean4 to solve the state.
STATE:
σ : Type u_2
α : Type u_1
inst✝⁴ : DecidableEq σ
inst✝³ : TopologicalSpace α
inst✝² : CommRing α
inst✝¹ : TopologicalRing α
inst✝ : DiscreteTopology α
f : MvPowerSeries σ α
h : Filter.Tendsto (fun n => f ^ n) Filter.atTop (nhds 0)
this : ∀ s ∈ nhds 0, (fun n ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/Topology.lean | MvPowerSeries.tendsto_pow_of_constantCoeff_nilpotent_iff | [231, 1] | [252, 48] | suffices ∀ x : α, {x} ∈ nhds x by exact this 0 | σ : Type u_2
α : Type u_1
inst✝⁴ : DecidableEq σ
inst✝³ : TopologicalSpace α
inst✝² : CommRing α
inst✝¹ : TopologicalRing α
inst✝ : DiscreteTopology α
f : MvPowerSeries σ α
h : Filter.Tendsto (fun n => f ^ n) Filter.atTop (nhds 0)
this : ∀ s ∈ nhds 0, (fun n => (constantCoeff σ α) (f ^ n)) ⁻¹' s ∈ Filter.atTop
⊢ {0} ∈ ... | σ : Type u_2
α : Type u_1
inst✝⁴ : DecidableEq σ
inst✝³ : TopologicalSpace α
inst✝² : CommRing α
inst✝¹ : TopologicalRing α
inst✝ : DiscreteTopology α
f : MvPowerSeries σ α
h : Filter.Tendsto (fun n => f ^ n) Filter.atTop (nhds 0)
this : ∀ s ∈ nhds 0, (fun n => (constantCoeff σ α) (f ^ n)) ⁻¹' s ∈ Filter.atTop
⊢ ∀ (x :... | Please generate a tactic in lean4 to solve the state.
STATE:
σ : Type u_2
α : Type u_1
inst✝⁴ : DecidableEq σ
inst✝³ : TopologicalSpace α
inst✝² : CommRing α
inst✝¹ : TopologicalRing α
inst✝ : DiscreteTopology α
f : MvPowerSeries σ α
h : Filter.Tendsto (fun n => f ^ n) Filter.atTop (nhds 0)
this : ∀ s ∈ nhds 0, (fun n ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/Topology.lean | MvPowerSeries.tendsto_pow_of_constantCoeff_nilpotent_iff | [231, 1] | [252, 48] | rw [← discreteTopology_iff_singleton_mem_nhds] | σ : Type u_2
α : Type u_1
inst✝⁴ : DecidableEq σ
inst✝³ : TopologicalSpace α
inst✝² : CommRing α
inst✝¹ : TopologicalRing α
inst✝ : DiscreteTopology α
f : MvPowerSeries σ α
h : Filter.Tendsto (fun n => f ^ n) Filter.atTop (nhds 0)
this : ∀ s ∈ nhds 0, (fun n => (constantCoeff σ α) (f ^ n)) ⁻¹' s ∈ Filter.atTop
⊢ ∀ (x :... | σ : Type u_2
α : Type u_1
inst✝⁴ : DecidableEq σ
inst✝³ : TopologicalSpace α
inst✝² : CommRing α
inst✝¹ : TopologicalRing α
inst✝ : DiscreteTopology α
f : MvPowerSeries σ α
h : Filter.Tendsto (fun n => f ^ n) Filter.atTop (nhds 0)
this : ∀ s ∈ nhds 0, (fun n => (constantCoeff σ α) (f ^ n)) ⁻¹' s ∈ Filter.atTop
⊢ Discre... | Please generate a tactic in lean4 to solve the state.
STATE:
σ : Type u_2
α : Type u_1
inst✝⁴ : DecidableEq σ
inst✝³ : TopologicalSpace α
inst✝² : CommRing α
inst✝¹ : TopologicalRing α
inst✝ : DiscreteTopology α
f : MvPowerSeries σ α
h : Filter.Tendsto (fun n => f ^ n) Filter.atTop (nhds 0)
this : ∀ s ∈ nhds 0, (fun n ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/Topology.lean | MvPowerSeries.tendsto_pow_of_constantCoeff_nilpotent_iff | [231, 1] | [252, 48] | infer_instance | σ : Type u_2
α : Type u_1
inst✝⁴ : DecidableEq σ
inst✝³ : TopologicalSpace α
inst✝² : CommRing α
inst✝¹ : TopologicalRing α
inst✝ : DiscreteTopology α
f : MvPowerSeries σ α
h : Filter.Tendsto (fun n => f ^ n) Filter.atTop (nhds 0)
this : ∀ s ∈ nhds 0, (fun n => (constantCoeff σ α) (f ^ n)) ⁻¹' s ∈ Filter.atTop
⊢ Discre... | σ : Type u_2
α : Type u_1
inst✝⁴ : DecidableEq σ
inst✝³ : TopologicalSpace α
inst✝² : CommRing α
inst✝¹ : TopologicalRing α
inst✝ : DiscreteTopology α
f : MvPowerSeries σ α
h : Filter.Tendsto (fun n => f ^ n) Filter.atTop (nhds 0)
this : (fun n => (constantCoeff σ α) (f ^ n)) ⁻¹' {0} ∈ Filter.atTop
⊢ IsNilpotent ((cons... | Please generate a tactic in lean4 to solve the state.
STATE:
σ : Type u_2
α : Type u_1
inst✝⁴ : DecidableEq σ
inst✝³ : TopologicalSpace α
inst✝² : CommRing α
inst✝¹ : TopologicalRing α
inst✝ : DiscreteTopology α
f : MvPowerSeries σ α
h : Filter.Tendsto (fun n => f ^ n) Filter.atTop (nhds 0)
this : ∀ s ∈ nhds 0, (fun n ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/Topology.lean | MvPowerSeries.tendsto_pow_of_constantCoeff_nilpotent_iff | [231, 1] | [252, 48] | simp only [map_pow, Filter.mem_atTop_sets, ge_iff_le, Set.mem_preimage,
Set.mem_singleton_iff] at this | σ : Type u_2
α : Type u_1
inst✝⁴ : DecidableEq σ
inst✝³ : TopologicalSpace α
inst✝² : CommRing α
inst✝¹ : TopologicalRing α
inst✝ : DiscreteTopology α
f : MvPowerSeries σ α
h : Filter.Tendsto (fun n => f ^ n) Filter.atTop (nhds 0)
this : (fun n => (constantCoeff σ α) (f ^ n)) ⁻¹' {0} ∈ Filter.atTop
⊢ IsNilpotent ((cons... | σ : Type u_2
α : Type u_1
inst✝⁴ : DecidableEq σ
inst✝³ : TopologicalSpace α
inst✝² : CommRing α
inst✝¹ : TopologicalRing α
inst✝ : DiscreteTopology α
f : MvPowerSeries σ α
h : Filter.Tendsto (fun n => f ^ n) Filter.atTop (nhds 0)
this : ∃ a, ∀ (b : ℕ), a ≤ b → (constantCoeff σ α) f ^ b = 0
⊢ IsNilpotent ((constantCoef... | Please generate a tactic in lean4 to solve the state.
STATE:
σ : Type u_2
α : Type u_1
inst✝⁴ : DecidableEq σ
inst✝³ : TopologicalSpace α
inst✝² : CommRing α
inst✝¹ : TopologicalRing α
inst✝ : DiscreteTopology α
f : MvPowerSeries σ α
h : Filter.Tendsto (fun n => f ^ n) Filter.atTop (nhds 0)
this : (fun n => (constantCo... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/Topology.lean | MvPowerSeries.tendsto_pow_of_constantCoeff_nilpotent_iff | [231, 1] | [252, 48] | obtain ⟨m, hm⟩ := this | σ : Type u_2
α : Type u_1
inst✝⁴ : DecidableEq σ
inst✝³ : TopologicalSpace α
inst✝² : CommRing α
inst✝¹ : TopologicalRing α
inst✝ : DiscreteTopology α
f : MvPowerSeries σ α
h : Filter.Tendsto (fun n => f ^ n) Filter.atTop (nhds 0)
this : ∃ a, ∀ (b : ℕ), a ≤ b → (constantCoeff σ α) f ^ b = 0
⊢ IsNilpotent ((constantCoef... | case intro
σ : Type u_2
α : Type u_1
inst✝⁴ : DecidableEq σ
inst✝³ : TopologicalSpace α
inst✝² : CommRing α
inst✝¹ : TopologicalRing α
inst✝ : DiscreteTopology α
f : MvPowerSeries σ α
h : Filter.Tendsto (fun n => f ^ n) Filter.atTop (nhds 0)
m : ℕ
hm : ∀ (b : ℕ), m ≤ b → (constantCoeff σ α) f ^ b = 0
⊢ IsNilpotent ((co... | Please generate a tactic in lean4 to solve the state.
STATE:
σ : Type u_2
α : Type u_1
inst✝⁴ : DecidableEq σ
inst✝³ : TopologicalSpace α
inst✝² : CommRing α
inst✝¹ : TopologicalRing α
inst✝ : DiscreteTopology α
f : MvPowerSeries σ α
h : Filter.Tendsto (fun n => f ^ n) Filter.atTop (nhds 0)
this : ∃ a, ∀ (b : ℕ), a ≤ b... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/Topology.lean | MvPowerSeries.tendsto_pow_of_constantCoeff_nilpotent_iff | [231, 1] | [252, 48] | use m | case intro
σ : Type u_2
α : Type u_1
inst✝⁴ : DecidableEq σ
inst✝³ : TopologicalSpace α
inst✝² : CommRing α
inst✝¹ : TopologicalRing α
inst✝ : DiscreteTopology α
f : MvPowerSeries σ α
h : Filter.Tendsto (fun n => f ^ n) Filter.atTop (nhds 0)
m : ℕ
hm : ∀ (b : ℕ), m ≤ b → (constantCoeff σ α) f ^ b = 0
⊢ IsNilpotent ((co... | case h
σ : Type u_2
α : Type u_1
inst✝⁴ : DecidableEq σ
inst✝³ : TopologicalSpace α
inst✝² : CommRing α
inst✝¹ : TopologicalRing α
inst✝ : DiscreteTopology α
f : MvPowerSeries σ α
h : Filter.Tendsto (fun n => f ^ n) Filter.atTop (nhds 0)
m : ℕ
hm : ∀ (b : ℕ), m ≤ b → (constantCoeff σ α) f ^ b = 0
⊢ (constantCoeff σ α) ... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
σ : Type u_2
α : Type u_1
inst✝⁴ : DecidableEq σ
inst✝³ : TopologicalSpace α
inst✝² : CommRing α
inst✝¹ : TopologicalRing α
inst✝ : DiscreteTopology α
f : MvPowerSeries σ α
h : Filter.Tendsto (fun n => f ^ n) Filter.atTop (nhds 0)
m : ℕ
hm : ∀ (b :... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/Topology.lean | MvPowerSeries.tendsto_pow_of_constantCoeff_nilpotent_iff | [231, 1] | [252, 48] | apply hm m (le_refl m) | case h
σ : Type u_2
α : Type u_1
inst✝⁴ : DecidableEq σ
inst✝³ : TopologicalSpace α
inst✝² : CommRing α
inst✝¹ : TopologicalRing α
inst✝ : DiscreteTopology α
f : MvPowerSeries σ α
h : Filter.Tendsto (fun n => f ^ n) Filter.atTop (nhds 0)
m : ℕ
hm : ∀ (b : ℕ), m ≤ b → (constantCoeff σ α) f ^ b = 0
⊢ (constantCoeff σ α) ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
σ : Type u_2
α : Type u_1
inst✝⁴ : DecidableEq σ
inst✝³ : TopologicalSpace α
inst✝² : CommRing α
inst✝¹ : TopologicalRing α
inst✝ : DiscreteTopology α
f : MvPowerSeries σ α
h : Filter.Tendsto (fun n => f ^ n) Filter.atTop (nhds 0)
m : ℕ
hm : ∀ (b : ℕ),... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/Topology.lean | MvPowerSeries.tendsto_pow_of_constantCoeff_nilpotent_iff | [231, 1] | [252, 48] | exact this 0 | σ : Type u_2
α : Type u_1
inst✝⁴ : DecidableEq σ
inst✝³ : TopologicalSpace α
inst✝² : CommRing α
inst✝¹ : TopologicalRing α
inst✝ : DiscreteTopology α
f : MvPowerSeries σ α
h : Filter.Tendsto (fun n => f ^ n) Filter.atTop (nhds 0)
this✝ : ∀ s ∈ nhds 0, (fun n => (constantCoeff σ α) (f ^ n)) ⁻¹' s ∈ Filter.atTop
this : ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
σ : Type u_2
α : Type u_1
inst✝⁴ : DecidableEq σ
inst✝³ : TopologicalSpace α
inst✝² : CommRing α
inst✝¹ : TopologicalRing α
inst✝ : DiscreteTopology α
f : MvPowerSeries σ α
h : Filter.Tendsto (fun n => f ^ n) Filter.atTop (nhds 0)
this✝ : ∀ s ∈ nhds 0, (fun n... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/Topology.lean | MvPowerSeries.hasSum_of_monomials_self | [265, 1] | [275, 27] | rw [Pi.hasSum] | σ : Type u_2
α : Type u_1
inst✝¹ : Semiring α
inst✝ : TopologicalSpace α
f : MvPowerSeries σ α
⊢ HasSum (fun d => (monomial α d) ((coeff α d) f)) f | σ : Type u_2
α : Type u_1
inst✝¹ : Semiring α
inst✝ : TopologicalSpace α
f : MvPowerSeries σ α
⊢ ∀ (x : σ →₀ ℕ), HasSum (fun i => (monomial α i) ((coeff α i) f) x) (f x) | Please generate a tactic in lean4 to solve the state.
STATE:
σ : Type u_2
α : Type u_1
inst✝¹ : Semiring α
inst✝ : TopologicalSpace α
f : MvPowerSeries σ α
⊢ HasSum (fun d => (monomial α d) ((coeff α d) f)) f
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/Topology.lean | MvPowerSeries.hasSum_of_monomials_self | [265, 1] | [275, 27] | intro d | σ : Type u_2
α : Type u_1
inst✝¹ : Semiring α
inst✝ : TopologicalSpace α
f : MvPowerSeries σ α
⊢ ∀ (x : σ →₀ ℕ), HasSum (fun i => (monomial α i) ((coeff α i) f) x) (f x) | σ : Type u_2
α : Type u_1
inst✝¹ : Semiring α
inst✝ : TopologicalSpace α
f : MvPowerSeries σ α
d : σ →₀ ℕ
⊢ HasSum (fun i => (monomial α i) ((coeff α i) f) d) (f d) | Please generate a tactic in lean4 to solve the state.
STATE:
σ : Type u_2
α : Type u_1
inst✝¹ : Semiring α
inst✝ : TopologicalSpace α
f : MvPowerSeries σ α
⊢ ∀ (x : σ →₀ ℕ), HasSum (fun i => (monomial α i) ((coeff α i) f) x) (f x)
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/Topology.lean | MvPowerSeries.hasSum_of_monomials_self | [265, 1] | [275, 27] | have hd : ∀ (d' : σ →₀ ℕ), d' ≠ d → (monomial α d') ((coeff α d') f) d = 0 := by
intro d' h
change coeff α d ((monomial α d') ((coeff α d') f)) = 0
rw [coeff_monomial_ne (Ne.symm h)] | σ : Type u_2
α : Type u_1
inst✝¹ : Semiring α
inst✝ : TopologicalSpace α
f : MvPowerSeries σ α
d : σ →₀ ℕ
⊢ HasSum (fun i => (monomial α i) ((coeff α i) f) d) (f d) | σ : Type u_2
α : Type u_1
inst✝¹ : Semiring α
inst✝ : TopologicalSpace α
f : MvPowerSeries σ α
d : σ →₀ ℕ
hd : ∀ (d' : σ →₀ ℕ), d' ≠ d → (monomial α d') ((coeff α d') f) d = 0
⊢ HasSum (fun i => (monomial α i) ((coeff α i) f) d) (f d) | Please generate a tactic in lean4 to solve the state.
STATE:
σ : Type u_2
α : Type u_1
inst✝¹ : Semiring α
inst✝ : TopologicalSpace α
f : MvPowerSeries σ α
d : σ →₀ ℕ
⊢ HasSum (fun i => (monomial α i) ((coeff α i) f) d) (f d)
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/Topology.lean | MvPowerSeries.hasSum_of_monomials_self | [265, 1] | [275, 27] | convert hasSum_single d hd using 1 | σ : Type u_2
α : Type u_1
inst✝¹ : Semiring α
inst✝ : TopologicalSpace α
f : MvPowerSeries σ α
d : σ →₀ ℕ
hd : ∀ (d' : σ →₀ ℕ), d' ≠ d → (monomial α d') ((coeff α d') f) d = 0
⊢ HasSum (fun i => (monomial α i) ((coeff α i) f) d) (f d) | case h.e'_6
σ : Type u_2
α : Type u_1
inst✝¹ : Semiring α
inst✝ : TopologicalSpace α
f : MvPowerSeries σ α
d : σ →₀ ℕ
hd : ∀ (d' : σ →₀ ℕ), d' ≠ d → (monomial α d') ((coeff α d') f) d = 0
⊢ f d = (monomial α d) ((coeff α d) f) d | Please generate a tactic in lean4 to solve the state.
STATE:
σ : Type u_2
α : Type u_1
inst✝¹ : Semiring α
inst✝ : TopologicalSpace α
f : MvPowerSeries σ α
d : σ →₀ ℕ
hd : ∀ (d' : σ →₀ ℕ), d' ≠ d → (monomial α d') ((coeff α d') f) d = 0
⊢ HasSum (fun i => (monomial α i) ((coeff α i) f) d) (f d)
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/Topology.lean | MvPowerSeries.hasSum_of_monomials_self | [265, 1] | [275, 27] | intro d' h | σ : Type u_2
α : Type u_1
inst✝¹ : Semiring α
inst✝ : TopologicalSpace α
f : MvPowerSeries σ α
d : σ →₀ ℕ
⊢ ∀ (d' : σ →₀ ℕ), d' ≠ d → (monomial α d') ((coeff α d') f) d = 0 | σ : Type u_2
α : Type u_1
inst✝¹ : Semiring α
inst✝ : TopologicalSpace α
f : MvPowerSeries σ α
d d' : σ →₀ ℕ
h : d' ≠ d
⊢ (monomial α d') ((coeff α d') f) d = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
σ : Type u_2
α : Type u_1
inst✝¹ : Semiring α
inst✝ : TopologicalSpace α
f : MvPowerSeries σ α
d : σ →₀ ℕ
⊢ ∀ (d' : σ →₀ ℕ), d' ≠ d → (monomial α d') ((coeff α d') f) d = 0
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/Topology.lean | MvPowerSeries.hasSum_of_monomials_self | [265, 1] | [275, 27] | change coeff α d ((monomial α d') ((coeff α d') f)) = 0 | σ : Type u_2
α : Type u_1
inst✝¹ : Semiring α
inst✝ : TopologicalSpace α
f : MvPowerSeries σ α
d d' : σ →₀ ℕ
h : d' ≠ d
⊢ (monomial α d') ((coeff α d') f) d = 0 | σ : Type u_2
α : Type u_1
inst✝¹ : Semiring α
inst✝ : TopologicalSpace α
f : MvPowerSeries σ α
d d' : σ →₀ ℕ
h : d' ≠ d
⊢ (coeff α d) ((monomial α d') ((coeff α d') f)) = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
σ : Type u_2
α : Type u_1
inst✝¹ : Semiring α
inst✝ : TopologicalSpace α
f : MvPowerSeries σ α
d d' : σ →₀ ℕ
h : d' ≠ d
⊢ (monomial α d') ((coeff α d') f) d = 0
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/Topology.lean | MvPowerSeries.hasSum_of_monomials_self | [265, 1] | [275, 27] | rw [coeff_monomial_ne (Ne.symm h)] | σ : Type u_2
α : Type u_1
inst✝¹ : Semiring α
inst✝ : TopologicalSpace α
f : MvPowerSeries σ α
d d' : σ →₀ ℕ
h : d' ≠ d
⊢ (coeff α d) ((monomial α d') ((coeff α d') f)) = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
σ : Type u_2
α : Type u_1
inst✝¹ : Semiring α
inst✝ : TopologicalSpace α
f : MvPowerSeries σ α
d d' : σ →₀ ℕ
h : d' ≠ d
⊢ (coeff α d) ((monomial α d') ((coeff α d') f)) = 0
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/Topology.lean | MvPowerSeries.hasSum_of_monomials_self | [265, 1] | [275, 27] | rw [← coeff_apply f d, ← coeff_apply (monomial α d (coeff α d f)) d, coeff_apply,
coeff_monomial_same] | case h.e'_6
σ : Type u_2
α : Type u_1
inst✝¹ : Semiring α
inst✝ : TopologicalSpace α
f : MvPowerSeries σ α
d : σ →₀ ℕ
hd : ∀ (d' : σ →₀ ℕ), d' ≠ d → (monomial α d') ((coeff α d') f) d = 0
⊢ f d = (monomial α d) ((coeff α d) f) d | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_6
σ : Type u_2
α : Type u_1
inst✝¹ : Semiring α
inst✝ : TopologicalSpace α
f : MvPowerSeries σ α
d : σ →₀ ℕ
hd : ∀ (d' : σ →₀ ℕ), d' ≠ d → (monomial α d') ((coeff α d') f) d = 0
⊢ f d = (monomial α d) ((coeff α d) f) d
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/Topology.lean | MvPowerSeries.hasSum_of_homogeneous_components_self | [283, 1] | [294, 42] | rw [Pi.hasSum] | σ : Type u_2
α : Type u_1
inst✝¹ : Semiring α
inst✝ : TopologicalSpace α
w : σ → ℕ
f : MvPowerSeries σ α
⊢ HasSum (fun p => (homogeneousComponent w p) f) f | σ : Type u_2
α : Type u_1
inst✝¹ : Semiring α
inst✝ : TopologicalSpace α
w : σ → ℕ
f : MvPowerSeries σ α
⊢ ∀ (x : σ →₀ ℕ), HasSum (fun i => (homogeneousComponent w i) f x) (f x) | Please generate a tactic in lean4 to solve the state.
STATE:
σ : Type u_2
α : Type u_1
inst✝¹ : Semiring α
inst✝ : TopologicalSpace α
w : σ → ℕ
f : MvPowerSeries σ α
⊢ HasSum (fun p => (homogeneousComponent w p) f) f
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/Topology.lean | MvPowerSeries.hasSum_of_homogeneous_components_self | [283, 1] | [294, 42] | intro d | σ : Type u_2
α : Type u_1
inst✝¹ : Semiring α
inst✝ : TopologicalSpace α
w : σ → ℕ
f : MvPowerSeries σ α
⊢ ∀ (x : σ →₀ ℕ), HasSum (fun i => (homogeneousComponent w i) f x) (f x) | σ : Type u_2
α : Type u_1
inst✝¹ : Semiring α
inst✝ : TopologicalSpace α
w : σ → ℕ
f : MvPowerSeries σ α
d : σ →₀ ℕ
⊢ HasSum (fun i => (homogeneousComponent w i) f d) (f d) | Please generate a tactic in lean4 to solve the state.
STATE:
σ : Type u_2
α : Type u_1
inst✝¹ : Semiring α
inst✝ : TopologicalSpace α
w : σ → ℕ
f : MvPowerSeries σ α
⊢ ∀ (x : σ →₀ ℕ), HasSum (fun i => (homogeneousComponent w i) f x) (f x)
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/Topology.lean | MvPowerSeries.hasSum_of_homogeneous_components_self | [283, 1] | [294, 42] | have hd : ∀ (b' : ℕ), b' ≠ (weight w) d → (homogeneousComponent w b') f d = 0 := by
intro p h
rw [← coeff_apply (homogeneousComponent w p f) d, coeff_homogeneousComponent,
if_neg (Ne.symm h)] | σ : Type u_2
α : Type u_1
inst✝¹ : Semiring α
inst✝ : TopologicalSpace α
w : σ → ℕ
f : MvPowerSeries σ α
d : σ →₀ ℕ
⊢ HasSum (fun i => (homogeneousComponent w i) f d) (f d) | σ : Type u_2
α : Type u_1
inst✝¹ : Semiring α
inst✝ : TopologicalSpace α
w : σ → ℕ
f : MvPowerSeries σ α
d : σ →₀ ℕ
hd : ∀ (b' : ℕ), b' ≠ (weight w) d → (homogeneousComponent w b') f d = 0
⊢ HasSum (fun i => (homogeneousComponent w i) f d) (f d) | Please generate a tactic in lean4 to solve the state.
STATE:
σ : Type u_2
α : Type u_1
inst✝¹ : Semiring α
inst✝ : TopologicalSpace α
w : σ → ℕ
f : MvPowerSeries σ α
d : σ →₀ ℕ
⊢ HasSum (fun i => (homogeneousComponent w i) f d) (f d)
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/Topology.lean | MvPowerSeries.hasSum_of_homogeneous_components_self | [283, 1] | [294, 42] | convert hasSum_single (weight w d) hd using 1 | σ : Type u_2
α : Type u_1
inst✝¹ : Semiring α
inst✝ : TopologicalSpace α
w : σ → ℕ
f : MvPowerSeries σ α
d : σ →₀ ℕ
hd : ∀ (b' : ℕ), b' ≠ (weight w) d → (homogeneousComponent w b') f d = 0
⊢ HasSum (fun i => (homogeneousComponent w i) f d) (f d) | case h.e'_6
σ : Type u_2
α : Type u_1
inst✝¹ : Semiring α
inst✝ : TopologicalSpace α
w : σ → ℕ
f : MvPowerSeries σ α
d : σ →₀ ℕ
hd : ∀ (b' : ℕ), b' ≠ (weight w) d → (homogeneousComponent w b') f d = 0
⊢ f d = (homogeneousComponent w ((weight w) d)) f d | Please generate a tactic in lean4 to solve the state.
STATE:
σ : Type u_2
α : Type u_1
inst✝¹ : Semiring α
inst✝ : TopologicalSpace α
w : σ → ℕ
f : MvPowerSeries σ α
d : σ →₀ ℕ
hd : ∀ (b' : ℕ), b' ≠ (weight w) d → (homogeneousComponent w b') f d = 0
⊢ HasSum (fun i => (homogeneousComponent w i) f d) (f d)
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/Topology.lean | MvPowerSeries.hasSum_of_homogeneous_components_self | [283, 1] | [294, 42] | intro p h | σ : Type u_2
α : Type u_1
inst✝¹ : Semiring α
inst✝ : TopologicalSpace α
w : σ → ℕ
f : MvPowerSeries σ α
d : σ →₀ ℕ
⊢ ∀ (b' : ℕ), b' ≠ (weight w) d → (homogeneousComponent w b') f d = 0 | σ : Type u_2
α : Type u_1
inst✝¹ : Semiring α
inst✝ : TopologicalSpace α
w : σ → ℕ
f : MvPowerSeries σ α
d : σ →₀ ℕ
p : ℕ
h : p ≠ (weight w) d
⊢ (homogeneousComponent w p) f d = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
σ : Type u_2
α : Type u_1
inst✝¹ : Semiring α
inst✝ : TopologicalSpace α
w : σ → ℕ
f : MvPowerSeries σ α
d : σ →₀ ℕ
⊢ ∀ (b' : ℕ), b' ≠ (weight w) d → (homogeneousComponent w b') f d = 0
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/Topology.lean | MvPowerSeries.hasSum_of_homogeneous_components_self | [283, 1] | [294, 42] | rw [← coeff_apply (homogeneousComponent w p f) d, coeff_homogeneousComponent,
if_neg (Ne.symm h)] | σ : Type u_2
α : Type u_1
inst✝¹ : Semiring α
inst✝ : TopologicalSpace α
w : σ → ℕ
f : MvPowerSeries σ α
d : σ →₀ ℕ
p : ℕ
h : p ≠ (weight w) d
⊢ (homogeneousComponent w p) f d = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
σ : Type u_2
α : Type u_1
inst✝¹ : Semiring α
inst✝ : TopologicalSpace α
w : σ → ℕ
f : MvPowerSeries σ α
d : σ →₀ ℕ
p : ℕ
h : p ≠ (weight w) d
⊢ (homogeneousComponent w p) f d = 0
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/Topology.lean | MvPowerSeries.hasSum_of_homogeneous_components_self | [283, 1] | [294, 42] | rw [← coeff_apply f d, ← coeff_apply (homogeneousComponent w (weight w d) f) d,
coeff_homogeneousComponent] | case h.e'_6
σ : Type u_2
α : Type u_1
inst✝¹ : Semiring α
inst✝ : TopologicalSpace α
w : σ → ℕ
f : MvPowerSeries σ α
d : σ →₀ ℕ
hd : ∀ (b' : ℕ), b' ≠ (weight w) d → (homogeneousComponent w b') f d = 0
⊢ f d = (homogeneousComponent w ((weight w) d)) f d | case h.e'_6
σ : Type u_2
α : Type u_1
inst✝¹ : Semiring α
inst✝ : TopologicalSpace α
w : σ → ℕ
f : MvPowerSeries σ α
d : σ →₀ ℕ
hd : ∀ (b' : ℕ), b' ≠ (weight w) d → (homogeneousComponent w b') f d = 0
⊢ (coeff α d) f = if (weight w) d = (weight w) d then (coeff α d) f else 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_6
σ : Type u_2
α : Type u_1
inst✝¹ : Semiring α
inst✝ : TopologicalSpace α
w : σ → ℕ
f : MvPowerSeries σ α
d : σ →₀ ℕ
hd : ∀ (b' : ℕ), b' ≠ (weight w) d → (homogeneousComponent w b') f d = 0
⊢ f d = (homogeneousComponent w ((weight w) d)) f d
TACTIC... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/Topology.lean | MvPowerSeries.hasSum_of_homogeneous_components_self | [283, 1] | [294, 42] | simp only [eq_self_iff_true, if_true] | case h.e'_6
σ : Type u_2
α : Type u_1
inst✝¹ : Semiring α
inst✝ : TopologicalSpace α
w : σ → ℕ
f : MvPowerSeries σ α
d : σ →₀ ℕ
hd : ∀ (b' : ℕ), b' ≠ (weight w) d → (homogeneousComponent w b') f d = 0
⊢ (coeff α d) f = if (weight w) d = (weight w) d then (coeff α d) f else 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_6
σ : Type u_2
α : Type u_1
inst✝¹ : Semiring α
inst✝ : TopologicalSpace α
w : σ → ℕ
f : MvPowerSeries σ α
d : σ →₀ ℕ
hd : ∀ (b' : ℕ), b' ≠ (weight w) d → (homogeneousComponent w b') f d = 0
⊢ (coeff α d) f = if (weight w) d = (weight w) d then (coe... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/Topology.lean | MvPowerSeries.as_tsum_of_homogeneous_components_self | [300, 1] | [304, 53] | haveI := t2Space σ α | σ : Type u_2
α : Type u_1
inst✝² : Semiring α
inst✝¹ : TopologicalSpace α
inst✝ : T2Space α
w : σ → ℕ
f : MvPowerSeries σ α
⊢ f = ∑' (p : ℕ), (homogeneousComponent w p) f | σ : Type u_2
α : Type u_1
inst✝² : Semiring α
inst✝¹ : TopologicalSpace α
inst✝ : T2Space α
w : σ → ℕ
f : MvPowerSeries σ α
this : T2Space (MvPowerSeries σ α)
⊢ f = ∑' (p : ℕ), (homogeneousComponent w p) f | Please generate a tactic in lean4 to solve the state.
STATE:
σ : Type u_2
α : Type u_1
inst✝² : Semiring α
inst✝¹ : TopologicalSpace α
inst✝ : T2Space α
w : σ → ℕ
f : MvPowerSeries σ α
⊢ f = ∑' (p : ℕ), (homogeneousComponent w p) f
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/Topology.lean | MvPowerSeries.as_tsum_of_homogeneous_components_self | [300, 1] | [304, 53] | exact HasSum.unique (hasSum_of_homogeneous_components_self w f)
(homogeneous_components_self_summable w f).hasSum | σ : Type u_2
α : Type u_1
inst✝² : Semiring α
inst✝¹ : TopologicalSpace α
inst✝ : T2Space α
w : σ → ℕ
f : MvPowerSeries σ α
this : T2Space (MvPowerSeries σ α)
⊢ f = ∑' (p : ℕ), (homogeneousComponent w p) f | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
σ : Type u_2
α : Type u_1
inst✝² : Semiring α
inst✝¹ : TopologicalSpace α
inst✝ : T2Space α
w : σ → ℕ
f : MvPowerSeries σ α
this : T2Space (MvPowerSeries σ α)
⊢ f = ∑' (p : ℕ), (homogeneousComponent w p) f
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | Polynomial.C_eq_smul_one | [5, 1] | [6, 25] | rw [← C_mul', mul_one] | R : Type u_1
inst✝ : Semiring R
a : R
⊢ C a = a • 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝ : Semiring R
a : R
⊢ C a = a • 1
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_factorsThrough | [35, 1] | [110, 59] | dsimp [Function.FactorsThrough] | A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ) {a : A}, a ∈ I ⊓ J → hI.dpow n a = hJ.dpow n a
n : ℕ
⊢ Function.FactorsThrough
(fun x =>
match x with
| (a, b) => ∑ k ∈ range (n + 1), hI.dpow k ↑a * hJ.dpow (n - k) ↑b)
fun x =>
mat... | A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ) {a : A}, a ∈ I ⊓ J → hI.dpow n a = hJ.dpow n a
n : ℕ
⊢ ∀ ⦃a b : ↥I × ↥J⦄,
↑a.1 + ↑a.2 = ↑b.1 + ↑b.2 →
∑ k ∈ range (n + 1), hI.dpow k ↑a.1 * hJ.dpow (n - k) ↑a.2 =
∑ k ∈ range (n + 1), hI.d... | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ) {a : A}, a ∈ I ⊓ J → hI.dpow n a = hJ.dpow n a
n : ℕ
⊢ Function.FactorsThrough
(fun x =>
match x with
| (a, b) => ∑ k ∈ range (n ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_factorsThrough | [35, 1] | [110, 59] | rintro ⟨⟨a, ha⟩, ⟨b, hb⟩⟩ ⟨⟨a', ha'⟩, ⟨b', hb'⟩⟩ H | A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ) {a : A}, a ∈ I ⊓ J → hI.dpow n a = hJ.dpow n a
n : ℕ
⊢ ∀ ⦃a b : ↥I × ↥J⦄,
↑a.1 + ↑a.2 = ↑b.1 + ↑b.2 →
∑ k ∈ range (n + 1), hI.dpow k ↑a.1 * hJ.dpow (n - k) ↑a.2 =
∑ k ∈ range (n + 1), hI.d... | case mk.mk.mk.mk.mk.mk
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ) {a : A}, a ∈ I ⊓ J → hI.dpow n a = hJ.dpow n a
n : ℕ
a : A
ha : a ∈ I
b : A
hb : b ∈ J
a' : A
ha' : a' ∈ I
b' : A
hb' : b' ∈ J
H : ↑(⟨a, ha⟩, ⟨b, hb⟩).1 + ↑(⟨a, ha⟩, ⟨b, hb⟩).2 = ↑(⟨a... | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ) {a : A}, a ∈ I ⊓ J → hI.dpow n a = hJ.dpow n a
n : ℕ
⊢ ∀ ⦃a b : ↥I × ↥J⦄,
↑a.1 + ↑a.2 = ↑b.1 + ↑b.2 →
∑ k ∈ range (n + 1), hI.dpow k ↑a... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_factorsThrough | [35, 1] | [110, 59] | dsimp only at H ⊢ | case mk.mk.mk.mk.mk.mk
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ) {a : A}, a ∈ I ⊓ J → hI.dpow n a = hJ.dpow n a
n : ℕ
a : A
ha : a ∈ I
b : A
hb : b ∈ J
a' : A
ha' : a' ∈ I
b' : A
hb' : b' ∈ J
H : ↑(⟨a, ha⟩, ⟨b, hb⟩).1 + ↑(⟨a, ha⟩, ⟨b, hb⟩).2 = ↑(⟨a... | case mk.mk.mk.mk.mk.mk
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ) {a : A}, a ∈ I ⊓ J → hI.dpow n a = hJ.dpow n a
n : ℕ
a : A
ha : a ∈ I
b : A
hb : b ∈ J
a' : A
ha' : a' ∈ I
b' : A
hb' : b' ∈ J
H : a + b = a' + b'
⊢ ∑ k ∈ range (n + 1), hI.dpow k a *... | Please generate a tactic in lean4 to solve the state.
STATE:
case mk.mk.mk.mk.mk.mk
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ) {a : A}, a ∈ I ⊓ J → hI.dpow n a = hJ.dpow n a
n : ℕ
a : A
ha : a ∈ I
b : A
hb : b ∈ J
a' : A
ha' : a' ∈ I
b' : A
hb' : b'... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_factorsThrough | [35, 1] | [110, 59] | set c := a - a' with hc | case mk.mk.mk.mk.mk.mk
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ) {a : A}, a ∈ I ⊓ J → hI.dpow n a = hJ.dpow n a
n : ℕ
a : A
ha : a ∈ I
b : A
hb : b ∈ J
a' : A
ha' : a' ∈ I
b' : A
hb' : b' ∈ J
H : a + b = a' + b'
⊢ ∑ k ∈ range (n + 1), hI.dpow k a *... | case mk.mk.mk.mk.mk.mk
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ) {a : A}, a ∈ I ⊓ J → hI.dpow n a = hJ.dpow n a
n : ℕ
a : A
ha : a ∈ I
b : A
hb : b ∈ J
a' : A
ha' : a' ∈ I
b' : A
hb' : b' ∈ J
H : a + b = a' + b'
c : A := a - a'
hc : c = a - a'
⊢ ∑ ... | Please generate a tactic in lean4 to solve the state.
STATE:
case mk.mk.mk.mk.mk.mk
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ) {a : A}, a ∈ I ⊓ J → hI.dpow n a = hJ.dpow n a
n : ℕ
a : A
ha : a ∈ I
b : A
hb : b ∈ J
a' : A
ha' : a' ∈ I
b' : A
hb' : b'... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_factorsThrough | [35, 1] | [110, 59] | have haa' : a = a' + c := by simp only [hc, add_sub_cancel] | case mk.mk.mk.mk.mk.mk
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ) {a : A}, a ∈ I ⊓ J → hI.dpow n a = hJ.dpow n a
n : ℕ
a : A
ha : a ∈ I
b : A
hb : b ∈ J
a' : A
ha' : a' ∈ I
b' : A
hb' : b' ∈ J
H : a + b = a' + b'
c : A := a - a'
hc : c = a - a'
⊢ ∑ ... | case mk.mk.mk.mk.mk.mk
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ) {a : A}, a ∈ I ⊓ J → hI.dpow n a = hJ.dpow n a
n : ℕ
a : A
ha : a ∈ I
b : A
hb : b ∈ J
a' : A
ha' : a' ∈ I
b' : A
hb' : b' ∈ J
H : a + b = a' + b'
c : A := a - a'
hc : c = a - a'
haa'... | Please generate a tactic in lean4 to solve the state.
STATE:
case mk.mk.mk.mk.mk.mk
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ) {a : A}, a ∈ I ⊓ J → hI.dpow n a = hJ.dpow n a
n : ℕ
a : A
ha : a ∈ I
b : A
hb : b ∈ J
a' : A
ha' : a' ∈ I
b' : A
hb' : b'... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_factorsThrough | [35, 1] | [110, 59] | have hbb' : b' = b + c := by
rw [← sub_eq_iff_eq_add'] at H ; rw [← H]; rw [haa']; ring | case mk.mk.mk.mk.mk.mk
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ) {a : A}, a ∈ I ⊓ J → hI.dpow n a = hJ.dpow n a
n : ℕ
a : A
ha : a ∈ I
b : A
hb : b ∈ J
a' : A
ha' : a' ∈ I
b' : A
hb' : b' ∈ J
H : a + b = a' + b'
c : A := a - a'
hc : c = a - a'
haa'... | case mk.mk.mk.mk.mk.mk
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ) {a : A}, a ∈ I ⊓ J → hI.dpow n a = hJ.dpow n a
n : ℕ
a : A
ha : a ∈ I
b : A
hb : b ∈ J
a' : A
ha' : a' ∈ I
b' : A
hb' : b' ∈ J
H : a + b = a' + b'
c : A := a - a'
hc : c = a - a'
haa'... | Please generate a tactic in lean4 to solve the state.
STATE:
case mk.mk.mk.mk.mk.mk
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ) {a : A}, a ∈ I ⊓ J → hI.dpow n a = hJ.dpow n a
n : ℕ
a : A
ha : a ∈ I
b : A
hb : b ∈ J
a' : A
ha' : a' ∈ I
b' : A
hb' : b'... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_factorsThrough | [35, 1] | [110, 59] | have hcI : c ∈ I := sub_mem ha ha' | case mk.mk.mk.mk.mk.mk
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ) {a : A}, a ∈ I ⊓ J → hI.dpow n a = hJ.dpow n a
n : ℕ
a : A
ha : a ∈ I
b : A
hb : b ∈ J
a' : A
ha' : a' ∈ I
b' : A
hb' : b' ∈ J
H : a + b = a' + b'
c : A := a - a'
hc : c = a - a'
haa'... | case mk.mk.mk.mk.mk.mk
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ) {a : A}, a ∈ I ⊓ J → hI.dpow n a = hJ.dpow n a
n : ℕ
a : A
ha : a ∈ I
b : A
hb : b ∈ J
a' : A
ha' : a' ∈ I
b' : A
hb' : b' ∈ J
H : a + b = a' + b'
c : A := a - a'
hc : c = a - a'
haa'... | Please generate a tactic in lean4 to solve the state.
STATE:
case mk.mk.mk.mk.mk.mk
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ) {a : A}, a ∈ I ⊓ J → hI.dpow n a = hJ.dpow n a
n : ℕ
a : A
ha : a ∈ I
b : A
hb : b ∈ J
a' : A
ha' : a' ∈ I
b' : A
hb' : b'... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_factorsThrough | [35, 1] | [110, 59] | have hcJ : c ∈ J := by
rw [← sub_eq_iff_eq_add'.mpr hbb']; exact sub_mem hb' hb | case mk.mk.mk.mk.mk.mk
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ) {a : A}, a ∈ I ⊓ J → hI.dpow n a = hJ.dpow n a
n : ℕ
a : A
ha : a ∈ I
b : A
hb : b ∈ J
a' : A
ha' : a' ∈ I
b' : A
hb' : b' ∈ J
H : a + b = a' + b'
c : A := a - a'
hc : c = a - a'
haa'... | case mk.mk.mk.mk.mk.mk
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ) {a : A}, a ∈ I ⊓ J → hI.dpow n a = hJ.dpow n a
n : ℕ
a : A
ha : a ∈ I
b : A
hb : b ∈ J
a' : A
ha' : a' ∈ I
b' : A
hb' : b' ∈ J
H : a + b = a' + b'
c : A := a - a'
hc : c = a - a'
haa'... | Please generate a tactic in lean4 to solve the state.
STATE:
case mk.mk.mk.mk.mk.mk
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ) {a : A}, a ∈ I ⊓ J → hI.dpow n a = hJ.dpow n a
n : ℕ
a : A
ha : a ∈ I
b : A
hb : b ∈ J
a' : A
ha' : a' ∈ I
b' : A
hb' : b'... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_factorsThrough | [35, 1] | [110, 59] | rw [haa', hbb'] | case mk.mk.mk.mk.mk.mk
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ) {a : A}, a ∈ I ⊓ J → hI.dpow n a = hJ.dpow n a
n : ℕ
a : A
ha : a ∈ I
b : A
hb : b ∈ J
a' : A
ha' : a' ∈ I
b' : A
hb' : b' ∈ J
H : a + b = a' + b'
c : A := a - a'
hc : c = a - a'
haa'... | case mk.mk.mk.mk.mk.mk
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ) {a : A}, a ∈ I ⊓ J → hI.dpow n a = hJ.dpow n a
n : ℕ
a : A
ha : a ∈ I
b : A
hb : b ∈ J
a' : A
ha' : a' ∈ I
b' : A
hb' : b' ∈ J
H : a + b = a' + b'
c : A := a - a'
hc : c = a - a'
haa'... | Please generate a tactic in lean4 to solve the state.
STATE:
case mk.mk.mk.mk.mk.mk
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ) {a : A}, a ∈ I ⊓ J → hI.dpow n a = hJ.dpow n a
n : ℕ
a : A
ha : a ∈ I
b : A
hb : b ∈ J
a' : A
ha' : a' ∈ I
b' : A
hb' : b'... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_factorsThrough | [35, 1] | [110, 59] | have Ha'c : ((Finset.range (n + 1)).sum
fun k : ℕ => hI.dpow k (a' + c) * hJ.dpow (n - k) b) =
(Finset.range (n + 1)).sum
fun k : ℕ => (Finset.range (k + 1)).sum fun l : ℕ =>
hI.dpow l a' * hJ.dpow (n - k) b * hI.dpow (k - l) c := by
apply Finset.sum_congr rfl
intro k _
rw [hI.dpow_add' k ha' ... | case mk.mk.mk.mk.mk.mk
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ) {a : A}, a ∈ I ⊓ J → hI.dpow n a = hJ.dpow n a
n : ℕ
a : A
ha : a ∈ I
b : A
hb : b ∈ J
a' : A
ha' : a' ∈ I
b' : A
hb' : b' ∈ J
H : a + b = a' + b'
c : A := a - a'
hc : c = a - a'
haa'... | case mk.mk.mk.mk.mk.mk
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ) {a : A}, a ∈ I ⊓ J → hI.dpow n a = hJ.dpow n a
n : ℕ
a : A
ha : a ∈ I
b : A
hb : b ∈ J
a' : A
ha' : a' ∈ I
b' : A
hb' : b' ∈ J
H : a + b = a' + b'
c : A := a - a'
hc : c = a - a'
haa'... | Please generate a tactic in lean4 to solve the state.
STATE:
case mk.mk.mk.mk.mk.mk
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ) {a : A}, a ∈ I ⊓ J → hI.dpow n a = hJ.dpow n a
n : ℕ
a : A
ha : a ∈ I
b : A
hb : b ∈ J
a' : A
ha' : a' ∈ I
b' : A
hb' : b'... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_factorsThrough | [35, 1] | [110, 59] | rw [Ha'c] | case mk.mk.mk.mk.mk.mk
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ) {a : A}, a ∈ I ⊓ J → hI.dpow n a = hJ.dpow n a
n : ℕ
a : A
ha : a ∈ I
b : A
hb : b ∈ J
a' : A
ha' : a' ∈ I
b' : A
hb' : b' ∈ J
H : a + b = a' + b'
c : A := a - a'
hc : c = a - a'
haa'... | case mk.mk.mk.mk.mk.mk
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ) {a : A}, a ∈ I ⊓ J → hI.dpow n a = hJ.dpow n a
n : ℕ
a : A
ha : a ∈ I
b : A
hb : b ∈ J
a' : A
ha' : a' ∈ I
b' : A
hb' : b' ∈ J
H : a + b = a' + b'
c : A := a - a'
hc : c = a - a'
haa'... | Please generate a tactic in lean4 to solve the state.
STATE:
case mk.mk.mk.mk.mk.mk
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ) {a : A}, a ∈ I ⊓ J → hI.dpow n a = hJ.dpow n a
n : ℕ
a : A
ha : a ∈ I
b : A
hb : b ∈ J
a' : A
ha' : a' ∈ I
b' : A
hb' : b'... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_factorsThrough | [35, 1] | [110, 59] | rw [Finset.sum_sigma'] | case mk.mk.mk.mk.mk.mk
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ) {a : A}, a ∈ I ⊓ J → hI.dpow n a = hJ.dpow n a
n : ℕ
a : A
ha : a ∈ I
b : A
hb : b ∈ J
a' : A
ha' : a' ∈ I
b' : A
hb' : b' ∈ J
H : a + b = a' + b'
c : A := a - a'
hc : c = a - a'
haa'... | case mk.mk.mk.mk.mk.mk
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ) {a : A}, a ∈ I ⊓ J → hI.dpow n a = hJ.dpow n a
n : ℕ
a : A
ha : a ∈ I
b : A
hb : b ∈ J
a' : A
ha' : a' ∈ I
b' : A
hb' : b' ∈ J
H : a + b = a' + b'
c : A := a - a'
hc : c = a - a'
haa'... | Please generate a tactic in lean4 to solve the state.
STATE:
case mk.mk.mk.mk.mk.mk
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ) {a : A}, a ∈ I ⊓ J → hI.dpow n a = hJ.dpow n a
n : ℕ
a : A
ha : a ∈ I
b : A
hb : b ∈ J
a' : A
ha' : a' ∈ I
b' : A
hb' : b'... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_factorsThrough | [35, 1] | [110, 59] | have Hbc : ((Finset.range (n + 1)).sum
fun k : ℕ => hI.dpow k a' * hJ.dpow (n - k) (b + c)) =
(Finset.range (n + 1)).sum
fun k : ℕ => (Finset.range (n - k + 1)).sum
fun l : ℕ => hI.dpow k a' * hJ.dpow l b * hJ.dpow (n - k - l) c := by
apply Finset.sum_congr rfl
intro k _
rw [hJ.dpow_add' (n - ... | case mk.mk.mk.mk.mk.mk
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ) {a : A}, a ∈ I ⊓ J → hI.dpow n a = hJ.dpow n a
n : ℕ
a : A
ha : a ∈ I
b : A
hb : b ∈ J
a' : A
ha' : a' ∈ I
b' : A
hb' : b' ∈ J
H : a + b = a' + b'
c : A := a - a'
hc : c = a - a'
haa'... | case mk.mk.mk.mk.mk.mk
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ) {a : A}, a ∈ I ⊓ J → hI.dpow n a = hJ.dpow n a
n : ℕ
a : A
ha : a ∈ I
b : A
hb : b ∈ J
a' : A
ha' : a' ∈ I
b' : A
hb' : b' ∈ J
H : a + b = a' + b'
c : A := a - a'
hc : c = a - a'
haa'... | Please generate a tactic in lean4 to solve the state.
STATE:
case mk.mk.mk.mk.mk.mk
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ) {a : A}, a ∈ I ⊓ J → hI.dpow n a = hJ.dpow n a
n : ℕ
a : A
ha : a ∈ I
b : A
hb : b ∈ J
a' : A
ha' : a' ∈ I
b' : A
hb' : b'... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_factorsThrough | [35, 1] | [110, 59] | rw [Hbc] | case mk.mk.mk.mk.mk.mk
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ) {a : A}, a ∈ I ⊓ J → hI.dpow n a = hJ.dpow n a
n : ℕ
a : A
ha : a ∈ I
b : A
hb : b ∈ J
a' : A
ha' : a' ∈ I
b' : A
hb' : b' ∈ J
H : a + b = a' + b'
c : A := a - a'
hc : c = a - a'
haa'... | case mk.mk.mk.mk.mk.mk
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ) {a : A}, a ∈ I ⊓ J → hI.dpow n a = hJ.dpow n a
n : ℕ
a : A
ha : a ∈ I
b : A
hb : b ∈ J
a' : A
ha' : a' ∈ I
b' : A
hb' : b' ∈ J
H : a + b = a' + b'
c : A := a - a'
hc : c = a - a'
haa'... | Please generate a tactic in lean4 to solve the state.
STATE:
case mk.mk.mk.mk.mk.mk
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ) {a : A}, a ∈ I ⊓ J → hI.dpow n a = hJ.dpow n a
n : ℕ
a : A
ha : a ∈ I
b : A
hb : b ∈ J
a' : A
ha' : a' ∈ I
b' : A
hb' : b'... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_factorsThrough | [35, 1] | [110, 59] | rw [Finset.sum_sigma'] | case mk.mk.mk.mk.mk.mk
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ) {a : A}, a ∈ I ⊓ J → hI.dpow n a = hJ.dpow n a
n : ℕ
a : A
ha : a ∈ I
b : A
hb : b ∈ J
a' : A
ha' : a' ∈ I
b' : A
hb' : b' ∈ J
H : a + b = a' + b'
c : A := a - a'
hc : c = a - a'
haa'... | case mk.mk.mk.mk.mk.mk
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ) {a : A}, a ∈ I ⊓ J → hI.dpow n a = hJ.dpow n a
n : ℕ
a : A
ha : a ∈ I
b : A
hb : b ∈ J
a' : A
ha' : a' ∈ I
b' : A
hb' : b' ∈ J
H : a + b = a' + b'
c : A := a - a'
hc : c = a - a'
haa'... | Please generate a tactic in lean4 to solve the state.
STATE:
case mk.mk.mk.mk.mk.mk
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ) {a : A}, a ∈ I ⊓ J → hI.dpow n a = hJ.dpow n a
n : ℕ
a : A
ha : a ∈ I
b : A
hb : b ∈ J
a' : A
ha' : a' ∈ I
b' : A
hb' : b'... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_factorsThrough | [35, 1] | [110, 59] | set s := (Finset.range (n + 1)).sigma fun a : ℕ => Finset.range (a + 1) with hs_def | case mk.mk.mk.mk.mk.mk
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ) {a : A}, a ∈ I ⊓ J → hI.dpow n a = hJ.dpow n a
n : ℕ
a : A
ha : a ∈ I
b : A
hb : b ∈ J
a' : A
ha' : a' ∈ I
b' : A
hb' : b' ∈ J
H : a + b = a' + b'
c : A := a - a'
hc : c = a - a'
haa'... | case mk.mk.mk.mk.mk.mk
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ) {a : A}, a ∈ I ⊓ J → hI.dpow n a = hJ.dpow n a
n : ℕ
a : A
ha : a ∈ I
b : A
hb : b ∈ J
a' : A
ha' : a' ∈ I
b' : A
hb' : b' ∈ J
H : a + b = a' + b'
c : A := a - a'
hc : c = a - a'
haa'... | Please generate a tactic in lean4 to solve the state.
STATE:
case mk.mk.mk.mk.mk.mk
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ) {a : A}, a ∈ I ⊓ J → hI.dpow n a = hJ.dpow n a
n : ℕ
a : A
ha : a ∈ I
b : A
hb : b ∈ J
a' : A
ha' : a' ∈ I
b' : A
hb' : b'... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_factorsThrough | [35, 1] | [110, 59] | set i : ∀ x : Σ _ : ℕ, ℕ, x ∈ s → Σ _ : ℕ, ℕ := fun ⟨k, m⟩ _ => ⟨m, n - k⟩ with hi_def | case mk.mk.mk.mk.mk.mk
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ) {a : A}, a ∈ I ⊓ J → hI.dpow n a = hJ.dpow n a
n : ℕ
a : A
ha : a ∈ I
b : A
hb : b ∈ J
a' : A
ha' : a' ∈ I
b' : A
hb' : b' ∈ J
H : a + b = a' + b'
c : A := a - a'
hc : c = a - a'
haa'... | case mk.mk.mk.mk.mk.mk
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ) {a : A}, a ∈ I ⊓ J → hI.dpow n a = hJ.dpow n a
n : ℕ
a : A
ha : a ∈ I
b : A
hb : b ∈ J
a' : A
ha' : a' ∈ I
b' : A
hb' : b' ∈ J
H : a + b = a' + b'
c : A := a - a'
hc : c = a - a'
haa'... | Please generate a tactic in lean4 to solve the state.
STATE:
case mk.mk.mk.mk.mk.mk
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ) {a : A}, a ∈ I ⊓ J → hI.dpow n a = hJ.dpow n a
n : ℕ
a : A
ha : a ∈ I
b : A
hb : b ∈ J
a' : A
ha' : a' ∈ I
b' : A
hb' : b'... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_factorsThrough | [35, 1] | [110, 59] | set t := (Finset.range (n + 1)).sigma fun a : ℕ => Finset.range (n - a + 1) with ht_def | case mk.mk.mk.mk.mk.mk
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ) {a : A}, a ∈ I ⊓ J → hI.dpow n a = hJ.dpow n a
n : ℕ
a : A
ha : a ∈ I
b : A
hb : b ∈ J
a' : A
ha' : a' ∈ I
b' : A
hb' : b' ∈ J
H : a + b = a' + b'
c : A := a - a'
hc : c = a - a'
haa'... | case mk.mk.mk.mk.mk.mk
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ) {a : A}, a ∈ I ⊓ J → hI.dpow n a = hJ.dpow n a
n : ℕ
a : A
ha : a ∈ I
b : A
hb : b ∈ J
a' : A
ha' : a' ∈ I
b' : A
hb' : b' ∈ J
H : a + b = a' + b'
c : A := a - a'
hc : c = a - a'
haa'... | Please generate a tactic in lean4 to solve the state.
STATE:
case mk.mk.mk.mk.mk.mk
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ) {a : A}, a ∈ I ⊓ J → hI.dpow n a = hJ.dpow n a
n : ℕ
a : A
ha : a ∈ I
b : A
hb : b ∈ J
a' : A
ha' : a' ∈ I
b' : A
hb' : b'... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_factorsThrough | [35, 1] | [110, 59] | set j : ∀ y : Σ _ : ℕ, ℕ, y ∈ t → Σ _ : ℕ, ℕ := fun ⟨k, m⟩ _ => ⟨n - m, k⟩ with hj_def | case mk.mk.mk.mk.mk.mk
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ) {a : A}, a ∈ I ⊓ J → hI.dpow n a = hJ.dpow n a
n : ℕ
a : A
ha : a ∈ I
b : A
hb : b ∈ J
a' : A
ha' : a' ∈ I
b' : A
hb' : b' ∈ J
H : a + b = a' + b'
c : A := a - a'
hc : c = a - a'
haa'... | case mk.mk.mk.mk.mk.mk
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ) {a : A}, a ∈ I ⊓ J → hI.dpow n a = hJ.dpow n a
n : ℕ
a : A
ha : a ∈ I
b : A
hb : b ∈ J
a' : A
ha' : a' ∈ I
b' : A
hb' : b' ∈ J
H : a + b = a' + b'
c : A := a - a'
hc : c = a - a'
haa'... | Please generate a tactic in lean4 to solve the state.
STATE:
case mk.mk.mk.mk.mk.mk
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ) {a : A}, a ∈ I ⊓ J → hI.dpow n a = hJ.dpow n a
n : ℕ
a : A
ha : a ∈ I
b : A
hb : b ∈ J
a' : A
ha' : a' ∈ I
b' : A
hb' : b'... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_factorsThrough | [35, 1] | [110, 59] | rw [Finset.sum_bij' i j _ _ _ _] | case mk.mk.mk.mk.mk.mk
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ) {a : A}, a ∈ I ⊓ J → hI.dpow n a = hJ.dpow n a
n : ℕ
a : A
ha : a ∈ I
b : A
hb : b ∈ J
a' : A
ha' : a' ∈ I
b' : A
hb' : b' ∈ J
H : a + b = a' + b'
c : A := a - a'
hc : c = a - a'
haa'... | case mk.mk.mk.mk.mk.mk
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ) {a : A}, a ∈ I ⊓ J → hI.dpow n a = hJ.dpow n a
n : ℕ
a : A
ha : a ∈ I
b : A
hb : b ∈ J
a' : A
ha' : a' ∈ I
b' : A
hb' : b' ∈ J
H : a + b = a' + b'
c : A := a - a'
hc : c = a - a'
haa'... | Please generate a tactic in lean4 to solve the state.
STATE:
case mk.mk.mk.mk.mk.mk
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ) {a : A}, a ∈ I ⊓ J → hI.dpow n a = hJ.dpow n a
n : ℕ
a : A
ha : a ∈ I
b : A
hb : b ∈ J
a' : A
ha' : a' ∈ I
b' : A
hb' : b'... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_factorsThrough | [35, 1] | [110, 59] | simp only [hc, add_sub_cancel] | A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ) {a : A}, a ∈ I ⊓ J → hI.dpow n a = hJ.dpow n a
n : ℕ
a : A
ha : a ∈ I
b : A
hb : b ∈ J
a' : A
ha' : a' ∈ I
b' : A
hb' : b' ∈ J
H : a + b = a' + b'
c : A := a - a'
hc : c = a - a'
⊢ a = a' + c | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ) {a : A}, a ∈ I ⊓ J → hI.dpow n a = hJ.dpow n a
n : ℕ
a : A
ha : a ∈ I
b : A
hb : b ∈ J
a' : A
ha' : a' ∈ I
b' : A
hb' : b' ∈ J
H : a + b = a' + b... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_factorsThrough | [35, 1] | [110, 59] | rw [← sub_eq_iff_eq_add'] at H | A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ) {a : A}, a ∈ I ⊓ J → hI.dpow n a = hJ.dpow n a
n : ℕ
a : A
ha : a ∈ I
b : A
hb : b ∈ J
a' : A
ha' : a' ∈ I
b' : A
hb' : b' ∈ J
H : a + b = a' + b'
c : A := a - a'
hc : c = a - a'
haa' : a = a' + c
⊢ b' = b ... | A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ) {a : A}, a ∈ I ⊓ J → hI.dpow n a = hJ.dpow n a
n : ℕ
a : A
ha : a ∈ I
b : A
hb : b ∈ J
a' : A
ha' : a' ∈ I
b' : A
hb' : b' ∈ J
H : a + b - a' = b'
c : A := a - a'
hc : c = a - a'
haa' : a = a' + c
⊢ b' = b ... | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ) {a : A}, a ∈ I ⊓ J → hI.dpow n a = hJ.dpow n a
n : ℕ
a : A
ha : a ∈ I
b : A
hb : b ∈ J
a' : A
ha' : a' ∈ I
b' : A
hb' : b' ∈ J
H : a + b = a' + b... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_factorsThrough | [35, 1] | [110, 59] | rw [← H] | A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ) {a : A}, a ∈ I ⊓ J → hI.dpow n a = hJ.dpow n a
n : ℕ
a : A
ha : a ∈ I
b : A
hb : b ∈ J
a' : A
ha' : a' ∈ I
b' : A
hb' : b' ∈ J
H : a + b - a' = b'
c : A := a - a'
hc : c = a - a'
haa' : a = a' + c
⊢ b' = b ... | A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ) {a : A}, a ∈ I ⊓ J → hI.dpow n a = hJ.dpow n a
n : ℕ
a : A
ha : a ∈ I
b : A
hb : b ∈ J
a' : A
ha' : a' ∈ I
b' : A
hb' : b' ∈ J
H : a + b - a' = b'
c : A := a - a'
hc : c = a - a'
haa' : a = a' + c
⊢ a + b -... | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ) {a : A}, a ∈ I ⊓ J → hI.dpow n a = hJ.dpow n a
n : ℕ
a : A
ha : a ∈ I
b : A
hb : b ∈ J
a' : A
ha' : a' ∈ I
b' : A
hb' : b' ∈ J
H : a + b - a' = b... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_factorsThrough | [35, 1] | [110, 59] | rw [haa'] | A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ) {a : A}, a ∈ I ⊓ J → hI.dpow n a = hJ.dpow n a
n : ℕ
a : A
ha : a ∈ I
b : A
hb : b ∈ J
a' : A
ha' : a' ∈ I
b' : A
hb' : b' ∈ J
H : a + b - a' = b'
c : A := a - a'
hc : c = a - a'
haa' : a = a' + c
⊢ a + b -... | A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ) {a : A}, a ∈ I ⊓ J → hI.dpow n a = hJ.dpow n a
n : ℕ
a : A
ha : a ∈ I
b : A
hb : b ∈ J
a' : A
ha' : a' ∈ I
b' : A
hb' : b' ∈ J
H : a + b - a' = b'
c : A := a - a'
hc : c = a - a'
haa' : a = a' + c
⊢ a' + c ... | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ) {a : A}, a ∈ I ⊓ J → hI.dpow n a = hJ.dpow n a
n : ℕ
a : A
ha : a ∈ I
b : A
hb : b ∈ J
a' : A
ha' : a' ∈ I
b' : A
hb' : b' ∈ J
H : a + b - a' = b... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_factorsThrough | [35, 1] | [110, 59] | ring | A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ) {a : A}, a ∈ I ⊓ J → hI.dpow n a = hJ.dpow n a
n : ℕ
a : A
ha : a ∈ I
b : A
hb : b ∈ J
a' : A
ha' : a' ∈ I
b' : A
hb' : b' ∈ J
H : a + b - a' = b'
c : A := a - a'
hc : c = a - a'
haa' : a = a' + c
⊢ a' + c ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ) {a : A}, a ∈ I ⊓ J → hI.dpow n a = hJ.dpow n a
n : ℕ
a : A
ha : a ∈ I
b : A
hb : b ∈ J
a' : A
ha' : a' ∈ I
b' : A
hb' : b' ∈ J
H : a + b - a' = b... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_factorsThrough | [35, 1] | [110, 59] | rw [← sub_eq_iff_eq_add'.mpr hbb'] | A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ) {a : A}, a ∈ I ⊓ J → hI.dpow n a = hJ.dpow n a
n : ℕ
a : A
ha : a ∈ I
b : A
hb : b ∈ J
a' : A
ha' : a' ∈ I
b' : A
hb' : b' ∈ J
H : a + b = a' + b'
c : A := a - a'
hc : c = a - a'
haa' : a = a' + c
hbb' : b'... | A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ) {a : A}, a ∈ I ⊓ J → hI.dpow n a = hJ.dpow n a
n : ℕ
a : A
ha : a ∈ I
b : A
hb : b ∈ J
a' : A
ha' : a' ∈ I
b' : A
hb' : b' ∈ J
H : a + b = a' + b'
c : A := a - a'
hc : c = a - a'
haa' : a = a' + c
hbb' : b'... | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ) {a : A}, a ∈ I ⊓ J → hI.dpow n a = hJ.dpow n a
n : ℕ
a : A
ha : a ∈ I
b : A
hb : b ∈ J
a' : A
ha' : a' ∈ I
b' : A
hb' : b' ∈ J
H : a + b = a' + b... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_factorsThrough | [35, 1] | [110, 59] | exact sub_mem hb' hb | A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ) {a : A}, a ∈ I ⊓ J → hI.dpow n a = hJ.dpow n a
n : ℕ
a : A
ha : a ∈ I
b : A
hb : b ∈ J
a' : A
ha' : a' ∈ I
b' : A
hb' : b' ∈ J
H : a + b = a' + b'
c : A := a - a'
hc : c = a - a'
haa' : a = a' + c
hbb' : b'... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ) {a : A}, a ∈ I ⊓ J → hI.dpow n a = hJ.dpow n a
n : ℕ
a : A
ha : a ∈ I
b : A
hb : b ∈ J
a' : A
ha' : a' ∈ I
b' : A
hb' : b' ∈ J
H : a + b = a' + b... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_factorsThrough | [35, 1] | [110, 59] | apply Finset.sum_congr rfl | A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ) {a : A}, a ∈ I ⊓ J → hI.dpow n a = hJ.dpow n a
n : ℕ
a : A
ha : a ∈ I
b : A
hb : b ∈ J
a' : A
ha' : a' ∈ I
b' : A
hb' : b' ∈ J
H : a + b = a' + b'
c : A := a - a'
hc : c = a - a'
haa' : a = a' + c
hbb' : b'... | A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ) {a : A}, a ∈ I ⊓ J → hI.dpow n a = hJ.dpow n a
n : ℕ
a : A
ha : a ∈ I
b : A
hb : b ∈ J
a' : A
ha' : a' ∈ I
b' : A
hb' : b' ∈ J
H : a + b = a' + b'
c : A := a - a'
hc : c = a - a'
haa' : a = a' + c
hbb' : b'... | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ) {a : A}, a ∈ I ⊓ J → hI.dpow n a = hJ.dpow n a
n : ℕ
a : A
ha : a ∈ I
b : A
hb : b ∈ J
a' : A
ha' : a' ∈ I
b' : A
hb' : b' ∈ J
H : a + b = a' + b... |
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