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/StronglySummable/Basic.lean | MvPowerSeries.StronglySummable.weightedOrder_tendsto_top_iff | [132, 1] | [154, 66] | simp only [Set.mem_Ioi, eventually_cofinite, not_lt] | case coe
σ : Type u_1
α : Type u_2
inst✝² : DecidableEq σ
ι✝ : Type ?u.15453
inst✝¹ : Semiring α
inst✝ : Finite σ
ι : Type u_3
w : σ → ℕ
hw : ∀ (x : σ), w x ≠ 0
f : ι → MvPowerSeries σ α
hf : StronglySummable f
n : ℕ
hn : ↑n < ⊤
⊢ ∀ᶠ (x : ι) in cofinite, weightedOrder w (f x) ∈ Set.Ioi ↑n | case coe
σ : Type u_1
α : Type u_2
inst✝² : DecidableEq σ
ι✝ : Type ?u.15453
inst✝¹ : Semiring α
inst✝ : Finite σ
ι : Type u_3
w : σ → ℕ
hw : ∀ (x : σ), w x ≠ 0
f : ι → MvPowerSeries σ α
hf : StronglySummable f
n : ℕ
hn : ↑n < ⊤
⊢ {x | weightedOrder w (f x) ≤ ↑n}.Finite | Please generate a tactic in lean4 to solve the state.
STATE:
case coe
σ : Type u_1
α : Type u_2
inst✝² : DecidableEq σ
ι✝ : Type ?u.15453
inst✝¹ : Semiring α
inst✝ : Finite σ
ι : Type u_3
w : σ → ℕ
hw : ∀ (x : σ), w x ≠ 0
f : ι → MvPowerSeries σ α
hf : StronglySummable f
n : ℕ
hn : ↑n < ⊤
⊢ ∀ᶠ (x : ι) in cofinite, weig... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/StronglySummable/Basic.lean | MvPowerSeries.StronglySummable.weightedOrder_tendsto_top_iff | [132, 1] | [154, 66] | let s := {d : σ →₀ ℕ | (weight w d) ≤ n} | case coe
σ : Type u_1
α : Type u_2
inst✝² : DecidableEq σ
ι✝ : Type ?u.15453
inst✝¹ : Semiring α
inst✝ : Finite σ
ι : Type u_3
w : σ → ℕ
hw : ∀ (x : σ), w x ≠ 0
f : ι → MvPowerSeries σ α
hf : StronglySummable f
n : ℕ
hn : ↑n < ⊤
⊢ {x | weightedOrder w (f x) ≤ ↑n}.Finite | case coe
σ : Type u_1
α : Type u_2
inst✝² : DecidableEq σ
ι✝ : Type ?u.15453
inst✝¹ : Semiring α
inst✝ : Finite σ
ι : Type u_3
w : σ → ℕ
hw : ∀ (x : σ), w x ≠ 0
f : ι → MvPowerSeries σ α
hf : StronglySummable f
n : ℕ
hn : ↑n < ⊤
s : Set (σ →₀ ℕ) := {d | (weight w) d ≤ n}
⊢ {x | weightedOrder w (f x) ≤ ↑n}.Finite | Please generate a tactic in lean4 to solve the state.
STATE:
case coe
σ : Type u_1
α : Type u_2
inst✝² : DecidableEq σ
ι✝ : Type ?u.15453
inst✝¹ : Semiring α
inst✝ : Finite σ
ι : Type u_3
w : σ → ℕ
hw : ∀ (x : σ), w x ≠ 0
f : ι → MvPowerSeries σ α
hf : StronglySummable f
n : ℕ
hn : ↑n < ⊤
⊢ {x | weightedOrder w (f x) ≤... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/StronglySummable/Basic.lean | MvPowerSeries.StronglySummable.weightedOrder_tendsto_top_iff | [132, 1] | [154, 66] | suffices h_ss : {i | (f i).weightedOrder w ≤ some n} ⊆
⋃ (d : σ →₀ ℕ) (_ : d ∈ s), {i | coeff α d (f i) ≠ 0} by
exact ((finite_of_weight_le w hw n).biUnion fun d _ => hf d).subset h_ss | case coe
σ : Type u_1
α : Type u_2
inst✝² : DecidableEq σ
ι✝ : Type ?u.15453
inst✝¹ : Semiring α
inst✝ : Finite σ
ι : Type u_3
w : σ → ℕ
hw : ∀ (x : σ), w x ≠ 0
f : ι → MvPowerSeries σ α
hf : StronglySummable f
n : ℕ
hn : ↑n < ⊤
s : Set (σ →₀ ℕ) := {d | (weight w) d ≤ n}
⊢ {x | weightedOrder w (f x) ≤ ↑n}.Finite | case coe
σ : Type u_1
α : Type u_2
inst✝² : DecidableEq σ
ι✝ : Type ?u.15453
inst✝¹ : Semiring α
inst✝ : Finite σ
ι : Type u_3
w : σ → ℕ
hw : ∀ (x : σ), w x ≠ 0
f : ι → MvPowerSeries σ α
hf : StronglySummable f
n : ℕ
hn : ↑n < ⊤
s : Set (σ →₀ ℕ) := {d | (weight w) d ≤ n}
⊢ {i | weightedOrder w (f i) ≤ some n} ⊆ ⋃ d ∈ s... | Please generate a tactic in lean4 to solve the state.
STATE:
case coe
σ : Type u_1
α : Type u_2
inst✝² : DecidableEq σ
ι✝ : Type ?u.15453
inst✝¹ : Semiring α
inst✝ : Finite σ
ι : Type u_3
w : σ → ℕ
hw : ∀ (x : σ), w x ≠ 0
f : ι → MvPowerSeries σ α
hf : StronglySummable f
n : ℕ
hn : ↑n < ⊤
s : Set (σ →₀ ℕ) := {d | (weig... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/StronglySummable/Basic.lean | MvPowerSeries.StronglySummable.weightedOrder_tendsto_top_iff | [132, 1] | [154, 66] | exact ((finite_of_weight_le w hw n).biUnion fun d _ => hf d).subset h_ss | σ : Type u_1
α : Type u_2
inst✝² : DecidableEq σ
ι✝ : Type ?u.15453
inst✝¹ : Semiring α
inst✝ : Finite σ
ι : Type u_3
w : σ → ℕ
hw : ∀ (x : σ), w x ≠ 0
f : ι → MvPowerSeries σ α
hf : StronglySummable f
n : ℕ
hn : ↑n < ⊤
s : Set (σ →₀ ℕ) := {d | (weight w) d ≤ n}
h_ss : {i | weightedOrder w (f i) ≤ some n} ⊆ ⋃ d ∈ s, {i... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
σ : Type u_1
α : Type u_2
inst✝² : DecidableEq σ
ι✝ : Type ?u.15453
inst✝¹ : Semiring α
inst✝ : Finite σ
ι : Type u_3
w : σ → ℕ
hw : ∀ (x : σ), w x ≠ 0
f : ι → MvPowerSeries σ α
hf : StronglySummable f
n : ℕ
hn : ↑n < ⊤
s : Set (σ →₀ ℕ) := {d | (weight w) d ≤... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/StronglySummable/Basic.lean | MvPowerSeries.StronglySummable.weightedOrder_tendsto_top_iff | [132, 1] | [154, 66] | intro i hi | case coe
σ : Type u_1
α : Type u_2
inst✝² : DecidableEq σ
ι✝ : Type ?u.15453
inst✝¹ : Semiring α
inst✝ : Finite σ
ι : Type u_3
w : σ → ℕ
hw : ∀ (x : σ), w x ≠ 0
f : ι → MvPowerSeries σ α
hf : StronglySummable f
n : ℕ
hn : ↑n < ⊤
s : Set (σ →₀ ℕ) := {d | (weight w) d ≤ n}
⊢ {i | weightedOrder w (f i) ≤ some n} ⊆ ⋃ d ∈ s... | case coe
σ : Type u_1
α : Type u_2
inst✝² : DecidableEq σ
ι✝ : Type ?u.15453
inst✝¹ : Semiring α
inst✝ : Finite σ
ι : Type u_3
w : σ → ℕ
hw : ∀ (x : σ), w x ≠ 0
f : ι → MvPowerSeries σ α
hf : StronglySummable f
n : ℕ
hn : ↑n < ⊤
s : Set (σ →₀ ℕ) := {d | (weight w) d ≤ n}
i : ι
hi : i ∈ {i | weightedOrder w (f i) ≤ some... | Please generate a tactic in lean4 to solve the state.
STATE:
case coe
σ : Type u_1
α : Type u_2
inst✝² : DecidableEq σ
ι✝ : Type ?u.15453
inst✝¹ : Semiring α
inst✝ : Finite σ
ι : Type u_3
w : σ → ℕ
hw : ∀ (x : σ), w x ≠ 0
f : ι → MvPowerSeries σ α
hf : StronglySummable f
n : ℕ
hn : ↑n < ⊤
s : Set (σ →₀ ℕ) := {d | (weig... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/StronglySummable/Basic.lean | MvPowerSeries.StronglySummable.weightedOrder_tendsto_top_iff | [132, 1] | [154, 66] | simp only [mem_setOf_eq] at hi | case coe
σ : Type u_1
α : Type u_2
inst✝² : DecidableEq σ
ι✝ : Type ?u.15453
inst✝¹ : Semiring α
inst✝ : Finite σ
ι : Type u_3
w : σ → ℕ
hw : ∀ (x : σ), w x ≠ 0
f : ι → MvPowerSeries σ α
hf : StronglySummable f
n : ℕ
hn : ↑n < ⊤
s : Set (σ →₀ ℕ) := {d | (weight w) d ≤ n}
i : ι
hi : i ∈ {i | weightedOrder w (f i) ≤ some... | case coe
σ : Type u_1
α : Type u_2
inst✝² : DecidableEq σ
ι✝ : Type ?u.15453
inst✝¹ : Semiring α
inst✝ : Finite σ
ι : Type u_3
w : σ → ℕ
hw : ∀ (x : σ), w x ≠ 0
f : ι → MvPowerSeries σ α
hf : StronglySummable f
n : ℕ
hn : ↑n < ⊤
s : Set (σ →₀ ℕ) := {d | (weight w) d ≤ n}
i : ι
hi : weightedOrder w (f i) ≤ some n
⊢ i ∈ ... | Please generate a tactic in lean4 to solve the state.
STATE:
case coe
σ : Type u_1
α : Type u_2
inst✝² : DecidableEq σ
ι✝ : Type ?u.15453
inst✝¹ : Semiring α
inst✝ : Finite σ
ι : Type u_3
w : σ → ℕ
hw : ∀ (x : σ), w x ≠ 0
f : ι → MvPowerSeries σ α
hf : StronglySummable f
n : ℕ
hn : ↑n < ⊤
s : Set (σ →₀ ℕ) := {d | (weig... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/StronglySummable/Basic.lean | MvPowerSeries.StronglySummable.weightedOrder_tendsto_top_iff | [132, 1] | [154, 66] | simp only [mem_setOf_eq, Nat.cast_id, ne_eq, mem_iUnion, mem_setOf_eq, exists_prop] | case coe
σ : Type u_1
α : Type u_2
inst✝² : DecidableEq σ
ι✝ : Type ?u.15453
inst✝¹ : Semiring α
inst✝ : Finite σ
ι : Type u_3
w : σ → ℕ
hw : ∀ (x : σ), w x ≠ 0
f : ι → MvPowerSeries σ α
hf : StronglySummable f
n : ℕ
hn : ↑n < ⊤
s : Set (σ →₀ ℕ) := {d | (weight w) d ≤ n}
i : ι
hi : weightedOrder w (f i) ≤ some n
⊢ i ∈ ... | case coe
σ : Type u_1
α : Type u_2
inst✝² : DecidableEq σ
ι✝ : Type ?u.15453
inst✝¹ : Semiring α
inst✝ : Finite σ
ι : Type u_3
w : σ → ℕ
hw : ∀ (x : σ), w x ≠ 0
f : ι → MvPowerSeries σ α
hf : StronglySummable f
n : ℕ
hn : ↑n < ⊤
s : Set (σ →₀ ℕ) := {d | (weight w) d ≤ n}
i : ι
hi : weightedOrder w (f i) ≤ some n
⊢ ∃ i_... | Please generate a tactic in lean4 to solve the state.
STATE:
case coe
σ : Type u_1
α : Type u_2
inst✝² : DecidableEq σ
ι✝ : Type ?u.15453
inst✝¹ : Semiring α
inst✝ : Finite σ
ι : Type u_3
w : σ → ℕ
hw : ∀ (x : σ), w x ≠ 0
f : ι → MvPowerSeries σ α
hf : StronglySummable f
n : ℕ
hn : ↑n < ⊤
s : Set (σ →₀ ℕ) := {d | (weig... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/StronglySummable/Basic.lean | MvPowerSeries.StronglySummable.weightedOrder_tendsto_top_iff | [132, 1] | [154, 66] | have heq: (ENat.toNat (weightedOrder w (f i))) = weightedOrder w (f i) := by
rw [ENat.coe_toNat_eq_self]
exact ne_of_lt (lt_of_le_of_lt hi hn) | case coe
σ : Type u_1
α : Type u_2
inst✝² : DecidableEq σ
ι✝ : Type ?u.15453
inst✝¹ : Semiring α
inst✝ : Finite σ
ι : Type u_3
w : σ → ℕ
hw : ∀ (x : σ), w x ≠ 0
f : ι → MvPowerSeries σ α
hf : StronglySummable f
n : ℕ
hn : ↑n < ⊤
s : Set (σ →₀ ℕ) := {d | (weight w) d ≤ n}
i : ι
hi : weightedOrder w (f i) ≤ some n
⊢ ∃ i_... | case coe
σ : Type u_1
α : Type u_2
inst✝² : DecidableEq σ
ι✝ : Type ?u.15453
inst✝¹ : Semiring α
inst✝ : Finite σ
ι : Type u_3
w : σ → ℕ
hw : ∀ (x : σ), w x ≠ 0
f : ι → MvPowerSeries σ α
hf : StronglySummable f
n : ℕ
hn : ↑n < ⊤
s : Set (σ →₀ ℕ) := {d | (weight w) d ≤ n}
i : ι
hi : weightedOrder w (f i) ≤ some n
heq : ... | Please generate a tactic in lean4 to solve the state.
STATE:
case coe
σ : Type u_1
α : Type u_2
inst✝² : DecidableEq σ
ι✝ : Type ?u.15453
inst✝¹ : Semiring α
inst✝ : Finite σ
ι : Type u_3
w : σ → ℕ
hw : ∀ (x : σ), w x ≠ 0
f : ι → MvPowerSeries σ α
hf : StronglySummable f
n : ℕ
hn : ↑n < ⊤
s : Set (σ →₀ ℕ) := {d | (weig... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/StronglySummable/Basic.lean | MvPowerSeries.StronglySummable.weightedOrder_tendsto_top_iff | [132, 1] | [154, 66] | obtain ⟨d, hd⟩ := exists_coeff_ne_zero_of_weightedOrder w (f i) heq | case coe
σ : Type u_1
α : Type u_2
inst✝² : DecidableEq σ
ι✝ : Type ?u.15453
inst✝¹ : Semiring α
inst✝ : Finite σ
ι : Type u_3
w : σ → ℕ
hw : ∀ (x : σ), w x ≠ 0
f : ι → MvPowerSeries σ α
hf : StronglySummable f
n : ℕ
hn : ↑n < ⊤
s : Set (σ →₀ ℕ) := {d | (weight w) d ≤ n}
i : ι
hi : weightedOrder w (f i) ≤ some n
heq : ... | case coe.intro
σ : Type u_1
α : Type u_2
inst✝² : DecidableEq σ
ι✝ : Type ?u.15453
inst✝¹ : Semiring α
inst✝ : Finite σ
ι : Type u_3
w : σ → ℕ
hw : ∀ (x : σ), w x ≠ 0
f : ι → MvPowerSeries σ α
hf : StronglySummable f
n : ℕ
hn : ↑n < ⊤
s : Set (σ →₀ ℕ) := {d | (weight w) d ≤ n}
i : ι
hi : weightedOrder w (f i) ≤ some n
... | Please generate a tactic in lean4 to solve the state.
STATE:
case coe
σ : Type u_1
α : Type u_2
inst✝² : DecidableEq σ
ι✝ : Type ?u.15453
inst✝¹ : Semiring α
inst✝ : Finite σ
ι : Type u_3
w : σ → ℕ
hw : ∀ (x : σ), w x ≠ 0
f : ι → MvPowerSeries σ α
hf : StronglySummable f
n : ℕ
hn : ↑n < ⊤
s : Set (σ →₀ ℕ) := {d | (weig... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/StronglySummable/Basic.lean | MvPowerSeries.StronglySummable.weightedOrder_tendsto_top_iff | [132, 1] | [154, 66] | refine' ⟨d, _, hd.2⟩ | case coe.intro
σ : Type u_1
α : Type u_2
inst✝² : DecidableEq σ
ι✝ : Type ?u.15453
inst✝¹ : Semiring α
inst✝ : Finite σ
ι : Type u_3
w : σ → ℕ
hw : ∀ (x : σ), w x ≠ 0
f : ι → MvPowerSeries σ α
hf : StronglySummable f
n : ℕ
hn : ↑n < ⊤
s : Set (σ →₀ ℕ) := {d | (weight w) d ≤ n}
i : ι
hi : weightedOrder w (f i) ≤ some n
... | case coe.intro
σ : Type u_1
α : Type u_2
inst✝² : DecidableEq σ
ι✝ : Type ?u.15453
inst✝¹ : Semiring α
inst✝ : Finite σ
ι : Type u_3
w : σ → ℕ
hw : ∀ (x : σ), w x ≠ 0
f : ι → MvPowerSeries σ α
hf : StronglySummable f
n : ℕ
hn : ↑n < ⊤
s : Set (σ →₀ ℕ) := {d | (weight w) d ≤ n}
i : ι
hi : weightedOrder w (f i) ≤ some n
... | Please generate a tactic in lean4 to solve the state.
STATE:
case coe.intro
σ : Type u_1
α : Type u_2
inst✝² : DecidableEq σ
ι✝ : Type ?u.15453
inst✝¹ : Semiring α
inst✝ : Finite σ
ι : Type u_3
w : σ → ℕ
hw : ∀ (x : σ), w x ≠ 0
f : ι → MvPowerSeries σ α
hf : StronglySummable f
n : ℕ
hn : ↑n < ⊤
s : Set (σ →₀ ℕ) := {d |... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/StronglySummable/Basic.lean | MvPowerSeries.StronglySummable.weightedOrder_tendsto_top_iff | [132, 1] | [154, 66] | simpa [← hd.1, WithTop.some_eq_coe, Nat.cast_le] using hi | case coe.intro
σ : Type u_1
α : Type u_2
inst✝² : DecidableEq σ
ι✝ : Type ?u.15453
inst✝¹ : Semiring α
inst✝ : Finite σ
ι : Type u_3
w : σ → ℕ
hw : ∀ (x : σ), w x ≠ 0
f : ι → MvPowerSeries σ α
hf : StronglySummable f
n : ℕ
hn : ↑n < ⊤
s : Set (σ →₀ ℕ) := {d | (weight w) d ≤ n}
i : ι
hi : weightedOrder w (f i) ≤ some n
... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case coe.intro
σ : Type u_1
α : Type u_2
inst✝² : DecidableEq σ
ι✝ : Type ?u.15453
inst✝¹ : Semiring α
inst✝ : Finite σ
ι : Type u_3
w : σ → ℕ
hw : ∀ (x : σ), w x ≠ 0
f : ι → MvPowerSeries σ α
hf : StronglySummable f
n : ℕ
hn : ↑n < ⊤
s : Set (σ →₀ ℕ) := {d |... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/StronglySummable/Basic.lean | MvPowerSeries.StronglySummable.weightedOrder_tendsto_top_iff | [132, 1] | [154, 66] | rw [ENat.coe_toNat_eq_self] | σ : Type u_1
α : Type u_2
inst✝² : DecidableEq σ
ι✝ : Type ?u.15453
inst✝¹ : Semiring α
inst✝ : Finite σ
ι : Type u_3
w : σ → ℕ
hw : ∀ (x : σ), w x ≠ 0
f : ι → MvPowerSeries σ α
hf : StronglySummable f
n : ℕ
hn : ↑n < ⊤
s : Set (σ →₀ ℕ) := {d | (weight w) d ≤ n}
i : ι
hi : weightedOrder w (f i) ≤ some n
⊢ ↑(ENat.toNat ... | σ : Type u_1
α : Type u_2
inst✝² : DecidableEq σ
ι✝ : Type ?u.15453
inst✝¹ : Semiring α
inst✝ : Finite σ
ι : Type u_3
w : σ → ℕ
hw : ∀ (x : σ), w x ≠ 0
f : ι → MvPowerSeries σ α
hf : StronglySummable f
n : ℕ
hn : ↑n < ⊤
s : Set (σ →₀ ℕ) := {d | (weight w) d ≤ n}
i : ι
hi : weightedOrder w (f i) ≤ some n
⊢ weightedOrder... | Please generate a tactic in lean4 to solve the state.
STATE:
σ : Type u_1
α : Type u_2
inst✝² : DecidableEq σ
ι✝ : Type ?u.15453
inst✝¹ : Semiring α
inst✝ : Finite σ
ι : Type u_3
w : σ → ℕ
hw : ∀ (x : σ), w x ≠ 0
f : ι → MvPowerSeries σ α
hf : StronglySummable f
n : ℕ
hn : ↑n < ⊤
s : Set (σ →₀ ℕ) := {d | (weight w) d ≤... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/StronglySummable/Basic.lean | MvPowerSeries.StronglySummable.weightedOrder_tendsto_top_iff | [132, 1] | [154, 66] | exact ne_of_lt (lt_of_le_of_lt hi hn) | σ : Type u_1
α : Type u_2
inst✝² : DecidableEq σ
ι✝ : Type ?u.15453
inst✝¹ : Semiring α
inst✝ : Finite σ
ι : Type u_3
w : σ → ℕ
hw : ∀ (x : σ), w x ≠ 0
f : ι → MvPowerSeries σ α
hf : StronglySummable f
n : ℕ
hn : ↑n < ⊤
s : Set (σ →₀ ℕ) := {d | (weight w) d ≤ n}
i : ι
hi : weightedOrder w (f i) ≤ some n
⊢ weightedOrder... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
σ : Type u_1
α : Type u_2
inst✝² : DecidableEq σ
ι✝ : Type ?u.15453
inst✝¹ : Semiring α
inst✝ : Finite σ
ι : Type u_3
w : σ → ℕ
hw : ∀ (x : σ), w x ≠ 0
f : ι → MvPowerSeries σ α
hf : StronglySummable f
n : ℕ
hn : ↑n < ⊤
s : Set (σ →₀ ℕ) := {d | (weight w) d ≤... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/StronglySummable/Basic.lean | MvPowerSeries.StronglySummable.order_tendsto_top_iff | [160, 1] | [162, 44] | simp | σ : Type u_1
α : Type u_2
inst✝² : DecidableEq σ
ι : Type u_3
inst✝¹ : Semiring α
inst✝ : Finite σ
f : ι → MvPowerSeries σ α
⊢ σ → 1 ≠ 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
σ : Type u_1
α : Type u_2
inst✝² : DecidableEq σ
ι : Type u_3
inst✝¹ : Semiring α
inst✝ : Finite σ
f : ι → MvPowerSeries σ α
⊢ σ → 1 ≠ 0
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/StronglySummable/Basic.lean | MvPowerSeries.StronglySummable.not_mem_unionOfSupportOfCoeffLe_iff | [179, 1] | [183, 54] | simp only [unionOfSupportOfCoeffLe, Finset.mem_biUnion, Finset.mem_Iic, Finite.mem_toFinset,
mem_support, not_exists, ne_eq, not_and, not_not] | σ : Type u_1
α : Type u_2
inst✝² : DecidableEq σ
ι : Type u_3
inst✝¹ : Semiring α
inst✝ : DecidableEq ι
f : ι → MvPowerSeries σ α
hf : StronglySummable f
d : σ →₀ ℕ
i : ι
⊢ i ∉ hf.unionOfSupportOfCoeffLe d ↔ ∀ e ≤ d, (coeff α e) (f i) = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
σ : Type u_1
α : Type u_2
inst✝² : DecidableEq σ
ι : Type u_3
inst✝¹ : Semiring α
inst✝ : DecidableEq ι
f : ι → MvPowerSeries σ α
hf : StronglySummable f
d : σ →₀ ℕ
i : ι
⊢ i ∉ hf.unionOfSupportOfCoeffLe d ↔ ∀ e ≤ d, (coeff α e) (f i) = 0
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/StronglySummable/Basic.lean | MvPowerSeries.StronglySummable.of_subset_unionOfSupportOfCoeffLe_finite | [196, 1] | [203, 96] | suffices {I : Finset ι | I ⊆ hf.unionOfSupportOfCoeffLe d} =
(hf.unionOfSupportOfCoeffLe d).powerset by
rw [this]; apply finite_toSet | σ : Type u_1
α : Type u_2
inst✝² : DecidableEq σ
ι : Type u_3
inst✝¹ : Semiring α
inst✝ : DecidableEq ι
f : ι → MvPowerSeries σ α
hf : StronglySummable f
d : σ →₀ ℕ
⊢ {I | I ⊆ hf.unionOfSupportOfCoeffLe d}.Finite | σ : Type u_1
α : Type u_2
inst✝² : DecidableEq σ
ι : Type u_3
inst✝¹ : Semiring α
inst✝ : DecidableEq ι
f : ι → MvPowerSeries σ α
hf : StronglySummable f
d : σ →₀ ℕ
⊢ {I | I ⊆ hf.unionOfSupportOfCoeffLe d} = ↑(hf.unionOfSupportOfCoeffLe d).powerset | Please generate a tactic in lean4 to solve the state.
STATE:
σ : Type u_1
α : Type u_2
inst✝² : DecidableEq σ
ι : Type u_3
inst✝¹ : Semiring α
inst✝ : DecidableEq ι
f : ι → MvPowerSeries σ α
hf : StronglySummable f
d : σ →₀ ℕ
⊢ {I | I ⊆ hf.unionOfSupportOfCoeffLe d}.Finite
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/StronglySummable/Basic.lean | MvPowerSeries.StronglySummable.of_subset_unionOfSupportOfCoeffLe_finite | [196, 1] | [203, 96] | ext I | σ : Type u_1
α : Type u_2
inst✝² : DecidableEq σ
ι : Type u_3
inst✝¹ : Semiring α
inst✝ : DecidableEq ι
f : ι → MvPowerSeries σ α
hf : StronglySummable f
d : σ →₀ ℕ
⊢ {I | I ⊆ hf.unionOfSupportOfCoeffLe d} = ↑(hf.unionOfSupportOfCoeffLe d).powerset | case h
σ : Type u_1
α : Type u_2
inst✝² : DecidableEq σ
ι : Type u_3
inst✝¹ : Semiring α
inst✝ : DecidableEq ι
f : ι → MvPowerSeries σ α
hf : StronglySummable f
d : σ →₀ ℕ
I : Finset ι
⊢ I ∈ {I | I ⊆ hf.unionOfSupportOfCoeffLe d} ↔ I ∈ ↑(hf.unionOfSupportOfCoeffLe d).powerset | Please generate a tactic in lean4 to solve the state.
STATE:
σ : Type u_1
α : Type u_2
inst✝² : DecidableEq σ
ι : Type u_3
inst✝¹ : Semiring α
inst✝ : DecidableEq ι
f : ι → MvPowerSeries σ α
hf : StronglySummable f
d : σ →₀ ℕ
⊢ {I | I ⊆ hf.unionOfSupportOfCoeffLe d} = ↑(hf.unionOfSupportOfCoeffLe d).powerset
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/StronglySummable/Basic.lean | MvPowerSeries.StronglySummable.of_subset_unionOfSupportOfCoeffLe_finite | [196, 1] | [203, 96] | simp only [mem_setOf_eq, coe_powerset, Set.mem_preimage, mem_powerset_iff, Finset.coe_subset] | case h
σ : Type u_1
α : Type u_2
inst✝² : DecidableEq σ
ι : Type u_3
inst✝¹ : Semiring α
inst✝ : DecidableEq ι
f : ι → MvPowerSeries σ α
hf : StronglySummable f
d : σ →₀ ℕ
I : Finset ι
⊢ I ∈ {I | I ⊆ hf.unionOfSupportOfCoeffLe d} ↔ I ∈ ↑(hf.unionOfSupportOfCoeffLe d).powerset | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
σ : Type u_1
α : Type u_2
inst✝² : DecidableEq σ
ι : Type u_3
inst✝¹ : Semiring α
inst✝ : DecidableEq ι
f : ι → MvPowerSeries σ α
hf : StronglySummable f
d : σ →₀ ℕ
I : Finset ι
⊢ I ∈ {I | I ⊆ hf.unionOfSupportOfCoeffLe d} ↔ I ∈ ↑(hf.unionOfSupportOfCo... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/StronglySummable/Basic.lean | MvPowerSeries.StronglySummable.of_subset_unionOfSupportOfCoeffLe_finite | [196, 1] | [203, 96] | rw [this] | σ : Type u_1
α : Type u_2
inst✝² : DecidableEq σ
ι : Type u_3
inst✝¹ : Semiring α
inst✝ : DecidableEq ι
f : ι → MvPowerSeries σ α
hf : StronglySummable f
d : σ →₀ ℕ
this : {I | I ⊆ hf.unionOfSupportOfCoeffLe d} = ↑(hf.unionOfSupportOfCoeffLe d).powerset
⊢ {I | I ⊆ hf.unionOfSupportOfCoeffLe d}.Finite | σ : Type u_1
α : Type u_2
inst✝² : DecidableEq σ
ι : Type u_3
inst✝¹ : Semiring α
inst✝ : DecidableEq ι
f : ι → MvPowerSeries σ α
hf : StronglySummable f
d : σ →₀ ℕ
this : {I | I ⊆ hf.unionOfSupportOfCoeffLe d} = ↑(hf.unionOfSupportOfCoeffLe d).powerset
⊢ (↑(hf.unionOfSupportOfCoeffLe d).powerset).Finite | Please generate a tactic in lean4 to solve the state.
STATE:
σ : Type u_1
α : Type u_2
inst✝² : DecidableEq σ
ι : Type u_3
inst✝¹ : Semiring α
inst✝ : DecidableEq ι
f : ι → MvPowerSeries σ α
hf : StronglySummable f
d : σ →₀ ℕ
this : {I | I ⊆ hf.unionOfSupportOfCoeffLe d} = ↑(hf.unionOfSupportOfCoeffLe d).powerset
⊢ {I ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/StronglySummable/Basic.lean | MvPowerSeries.StronglySummable.of_subset_unionOfSupportOfCoeffLe_finite | [196, 1] | [203, 96] | apply finite_toSet | σ : Type u_1
α : Type u_2
inst✝² : DecidableEq σ
ι : Type u_3
inst✝¹ : Semiring α
inst✝ : DecidableEq ι
f : ι → MvPowerSeries σ α
hf : StronglySummable f
d : σ →₀ ℕ
this : {I | I ⊆ hf.unionOfSupportOfCoeffLe d} = ↑(hf.unionOfSupportOfCoeffLe d).powerset
⊢ (↑(hf.unionOfSupportOfCoeffLe d).powerset).Finite | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
σ : Type u_1
α : Type u_2
inst✝² : DecidableEq σ
ι : Type u_3
inst✝¹ : Semiring α
inst✝ : DecidableEq ι
f : ι → MvPowerSeries σ α
hf : StronglySummable f
d : σ →₀ ℕ
this : {I | I ⊆ hf.unionOfSupportOfCoeffLe d} = ↑(hf.unionOfSupportOfCoeffLe d).powerset
⊢ (↑(... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/StronglySummable/Basic.lean | MvPowerSeries.StronglySummable.support_add | [210, 1] | [220, 19] | intro d i | σ : Type u_1
α : Type u_2
inst✝² : DecidableEq σ
ι : Type u_3
inst✝¹ : Semiring α
inst✝ : DecidableEq ι
f g : ι → MvPowerSeries σ α
hf : StronglySummable f
hg : StronglySummable g
⊢ ∀ (d : σ →₀ ℕ), (support fun i => (coeff α d) ((f + g) i)) ⊆ ↑(⋯.toFinset ∪ ⋯.toFinset) | σ : Type u_1
α : Type u_2
inst✝² : DecidableEq σ
ι : Type u_3
inst✝¹ : Semiring α
inst✝ : DecidableEq ι
f g : ι → MvPowerSeries σ α
hf : StronglySummable f
hg : StronglySummable g
d : σ →₀ ℕ
i : ι
⊢ (i ∈ support fun i => (coeff α d) ((f + g) i)) → i ∈ ↑(⋯.toFinset ∪ ⋯.toFinset) | Please generate a tactic in lean4 to solve the state.
STATE:
σ : Type u_1
α : Type u_2
inst✝² : DecidableEq σ
ι : Type u_3
inst✝¹ : Semiring α
inst✝ : DecidableEq ι
f g : ι → MvPowerSeries σ α
hf : StronglySummable f
hg : StronglySummable g
⊢ ∀ (d : σ →₀ ℕ), (support fun i => (coeff α d) ((f + g) i)) ⊆ ↑(⋯.toFinset ∪ ⋯... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/StronglySummable/Basic.lean | MvPowerSeries.StronglySummable.support_add | [210, 1] | [220, 19] | simp only [Pi.add_apply, map_add, Function.mem_support, ne_eq, coe_union, Finite.coe_toFinset,
Set.mem_union] | σ : Type u_1
α : Type u_2
inst✝² : DecidableEq σ
ι : Type u_3
inst✝¹ : Semiring α
inst✝ : DecidableEq ι
f g : ι → MvPowerSeries σ α
hf : StronglySummable f
hg : StronglySummable g
d : σ →₀ ℕ
i : ι
⊢ (i ∈ support fun i => (coeff α d) ((f + g) i)) → i ∈ ↑(⋯.toFinset ∪ ⋯.toFinset) | σ : Type u_1
α : Type u_2
inst✝² : DecidableEq σ
ι : Type u_3
inst✝¹ : Semiring α
inst✝ : DecidableEq ι
f g : ι → MvPowerSeries σ α
hf : StronglySummable f
hg : StronglySummable g
d : σ →₀ ℕ
i : ι
⊢ ¬(coeff α d) (f i) + (coeff α d) (g i) = 0 → ¬(coeff α d) (f i) = 0 ∨ ¬(coeff α d) (g i) = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
σ : Type u_1
α : Type u_2
inst✝² : DecidableEq σ
ι : Type u_3
inst✝¹ : Semiring α
inst✝ : DecidableEq ι
f g : ι → MvPowerSeries σ α
hf : StronglySummable f
hg : StronglySummable g
d : σ →₀ ℕ
i : ι
⊢ (i ∈ support fun i => (coeff α d) ((f + g) i)) → i ∈ ↑(⋯.toF... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/StronglySummable/Basic.lean | MvPowerSeries.StronglySummable.support_add | [210, 1] | [220, 19] | intro h | σ : Type u_1
α : Type u_2
inst✝² : DecidableEq σ
ι : Type u_3
inst✝¹ : Semiring α
inst✝ : DecidableEq ι
f g : ι → MvPowerSeries σ α
hf : StronglySummable f
hg : StronglySummable g
d : σ →₀ ℕ
i : ι
⊢ ¬(coeff α d) (f i) + (coeff α d) (g i) = 0 → ¬(coeff α d) (f i) = 0 ∨ ¬(coeff α d) (g i) = 0 | σ : Type u_1
α : Type u_2
inst✝² : DecidableEq σ
ι : Type u_3
inst✝¹ : Semiring α
inst✝ : DecidableEq ι
f g : ι → MvPowerSeries σ α
hf : StronglySummable f
hg : StronglySummable g
d : σ →₀ ℕ
i : ι
h : ¬(coeff α d) (f i) + (coeff α d) (g i) = 0
⊢ ¬(coeff α d) (f i) = 0 ∨ ¬(coeff α d) (g i) = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
σ : Type u_1
α : Type u_2
inst✝² : DecidableEq σ
ι : Type u_3
inst✝¹ : Semiring α
inst✝ : DecidableEq ι
f g : ι → MvPowerSeries σ α
hf : StronglySummable f
hg : StronglySummable g
d : σ →₀ ℕ
i : ι
⊢ ¬(coeff α d) (f i) + (coeff α d) (g i) = 0 → ¬(coeff α d) (f... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/StronglySummable/Basic.lean | MvPowerSeries.StronglySummable.support_add | [210, 1] | [220, 19] | by_cases h₁ : coeff α d (f i) = 0 | σ : Type u_1
α : Type u_2
inst✝² : DecidableEq σ
ι : Type u_3
inst✝¹ : Semiring α
inst✝ : DecidableEq ι
f g : ι → MvPowerSeries σ α
hf : StronglySummable f
hg : StronglySummable g
d : σ →₀ ℕ
i : ι
h : ¬(coeff α d) (f i) + (coeff α d) (g i) = 0
⊢ ¬(coeff α d) (f i) = 0 ∨ ¬(coeff α d) (g i) = 0 | case pos
σ : Type u_1
α : Type u_2
inst✝² : DecidableEq σ
ι : Type u_3
inst✝¹ : Semiring α
inst✝ : DecidableEq ι
f g : ι → MvPowerSeries σ α
hf : StronglySummable f
hg : StronglySummable g
d : σ →₀ ℕ
i : ι
h : ¬(coeff α d) (f i) + (coeff α d) (g i) = 0
h₁ : (coeff α d) (f i) = 0
⊢ ¬(coeff α d) (f i) = 0 ∨ ¬(coeff α d) ... | Please generate a tactic in lean4 to solve the state.
STATE:
σ : Type u_1
α : Type u_2
inst✝² : DecidableEq σ
ι : Type u_3
inst✝¹ : Semiring α
inst✝ : DecidableEq ι
f g : ι → MvPowerSeries σ α
hf : StronglySummable f
hg : StronglySummable g
d : σ →₀ ℕ
i : ι
h : ¬(coeff α d) (f i) + (coeff α d) (g i) = 0
⊢ ¬(coeff α d) ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/StronglySummable/Basic.lean | MvPowerSeries.StronglySummable.support_add | [210, 1] | [220, 19] | right | case pos
σ : Type u_1
α : Type u_2
inst✝² : DecidableEq σ
ι : Type u_3
inst✝¹ : Semiring α
inst✝ : DecidableEq ι
f g : ι → MvPowerSeries σ α
hf : StronglySummable f
hg : StronglySummable g
d : σ →₀ ℕ
i : ι
h : ¬(coeff α d) (f i) + (coeff α d) (g i) = 0
h₁ : (coeff α d) (f i) = 0
⊢ ¬(coeff α d) (f i) = 0 ∨ ¬(coeff α d) ... | case pos.h
σ : Type u_1
α : Type u_2
inst✝² : DecidableEq σ
ι : Type u_3
inst✝¹ : Semiring α
inst✝ : DecidableEq ι
f g : ι → MvPowerSeries σ α
hf : StronglySummable f
hg : StronglySummable g
d : σ →₀ ℕ
i : ι
h : ¬(coeff α d) (f i) + (coeff α d) (g i) = 0
h₁ : (coeff α d) (f i) = 0
⊢ ¬(coeff α d) (g i) = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
σ : Type u_1
α : Type u_2
inst✝² : DecidableEq σ
ι : Type u_3
inst✝¹ : Semiring α
inst✝ : DecidableEq ι
f g : ι → MvPowerSeries σ α
hf : StronglySummable f
hg : StronglySummable g
d : σ →₀ ℕ
i : ι
h : ¬(coeff α d) (f i) + (coeff α d) (g i) = 0
h₁ : (... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/StronglySummable/Basic.lean | MvPowerSeries.StronglySummable.support_add | [210, 1] | [220, 19] | simpa [h₁] using h | case pos.h
σ : Type u_1
α : Type u_2
inst✝² : DecidableEq σ
ι : Type u_3
inst✝¹ : Semiring α
inst✝ : DecidableEq ι
f g : ι → MvPowerSeries σ α
hf : StronglySummable f
hg : StronglySummable g
d : σ →₀ ℕ
i : ι
h : ¬(coeff α d) (f i) + (coeff α d) (g i) = 0
h₁ : (coeff α d) (f i) = 0
⊢ ¬(coeff α d) (g i) = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.h
σ : Type u_1
α : Type u_2
inst✝² : DecidableEq σ
ι : Type u_3
inst✝¹ : Semiring α
inst✝ : DecidableEq ι
f g : ι → MvPowerSeries σ α
hf : StronglySummable f
hg : StronglySummable g
d : σ →₀ ℕ
i : ι
h : ¬(coeff α d) (f i) + (coeff α d) (g i) = 0
h₁ :... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/StronglySummable/Basic.lean | MvPowerSeries.StronglySummable.support_add | [210, 1] | [220, 19] | left | case neg
σ : Type u_1
α : Type u_2
inst✝² : DecidableEq σ
ι : Type u_3
inst✝¹ : Semiring α
inst✝ : DecidableEq ι
f g : ι → MvPowerSeries σ α
hf : StronglySummable f
hg : StronglySummable g
d : σ →₀ ℕ
i : ι
h : ¬(coeff α d) (f i) + (coeff α d) (g i) = 0
h₁ : ¬(coeff α d) (f i) = 0
⊢ ¬(coeff α d) (f i) = 0 ∨ ¬(coeff α d)... | case neg.h
σ : Type u_1
α : Type u_2
inst✝² : DecidableEq σ
ι : Type u_3
inst✝¹ : Semiring α
inst✝ : DecidableEq ι
f g : ι → MvPowerSeries σ α
hf : StronglySummable f
hg : StronglySummable g
d : σ →₀ ℕ
i : ι
h : ¬(coeff α d) (f i) + (coeff α d) (g i) = 0
h₁ : ¬(coeff α d) (f i) = 0
⊢ ¬(coeff α d) (f i) = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
σ : Type u_1
α : Type u_2
inst✝² : DecidableEq σ
ι : Type u_3
inst✝¹ : Semiring α
inst✝ : DecidableEq ι
f g : ι → MvPowerSeries σ α
hf : StronglySummable f
hg : StronglySummable g
d : σ →₀ ℕ
i : ι
h : ¬(coeff α d) (f i) + (coeff α d) (g i) = 0
h₁ : ¬... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/StronglySummable/Basic.lean | MvPowerSeries.StronglySummable.support_add | [210, 1] | [220, 19] | exact h₁ | case neg.h
σ : Type u_1
α : Type u_2
inst✝² : DecidableEq σ
ι : Type u_3
inst✝¹ : Semiring α
inst✝ : DecidableEq ι
f g : ι → MvPowerSeries σ α
hf : StronglySummable f
hg : StronglySummable g
d : σ →₀ ℕ
i : ι
h : ¬(coeff α d) (f i) + (coeff α d) (g i) = 0
h₁ : ¬(coeff α d) (f i) = 0
⊢ ¬(coeff α d) (f i) = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.h
σ : Type u_1
α : Type u_2
inst✝² : DecidableEq σ
ι : Type u_3
inst✝¹ : Semiring α
inst✝ : DecidableEq ι
f g : ι → MvPowerSeries σ α
hf : StronglySummable f
hg : StronglySummable g
d : σ →₀ ℕ
i : ι
h : ¬(coeff α d) (f i) + (coeff α d) (g i) = 0
h₁ :... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/StronglySummable/Basic.lean | MvPowerSeries.StronglySummable.add | [224, 1] | [226, 80] | classical exact fun d => Finite.subset (finite_toSet _) (support_add hf hg d) | σ : Type u_1
α : Type u_2
inst✝¹ : DecidableEq σ
ι : Type u_3
inst✝ : Semiring α
f g : ι → MvPowerSeries σ α
hf : StronglySummable f
hg : StronglySummable g
⊢ StronglySummable (f + g) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
σ : Type u_1
α : Type u_2
inst✝¹ : DecidableEq σ
ι : Type u_3
inst✝ : Semiring α
f g : ι → MvPowerSeries σ α
hf : StronglySummable f
hg : StronglySummable g
⊢ StronglySummable (f + g)
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/StronglySummable/Basic.lean | MvPowerSeries.StronglySummable.add | [224, 1] | [226, 80] | exact fun d => Finite.subset (finite_toSet _) (support_add hf hg d) | σ : Type u_1
α : Type u_2
inst✝¹ : DecidableEq σ
ι : Type u_3
inst✝ : Semiring α
f g : ι → MvPowerSeries σ α
hf : StronglySummable f
hg : StronglySummable g
⊢ StronglySummable (f + g) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
σ : Type u_1
α : Type u_2
inst✝¹ : DecidableEq σ
ι : Type u_3
inst✝ : Semiring α
f g : ι → MvPowerSeries σ α
hf : StronglySummable f
hg : StronglySummable g
⊢ StronglySummable (f + g)
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/StronglySummable/Basic.lean | MvPowerSeries.StronglySummable.smul | [230, 1] | [238, 45] | intro d | σ : Type u_1
α : Type u_2
inst✝¹ : DecidableEq σ
ι : Type u_3
inst✝ : Semiring α
f : ι → MvPowerSeries σ α
a : ι → α
hf : StronglySummable f
⊢ StronglySummable (a • f) | σ : Type u_1
α : Type u_2
inst✝¹ : DecidableEq σ
ι : Type u_3
inst✝ : Semiring α
f : ι → MvPowerSeries σ α
a : ι → α
hf : StronglySummable f
d : σ →₀ ℕ
⊢ (support fun i => (coeff α d) ((a • f) i)).Finite | Please generate a tactic in lean4 to solve the state.
STATE:
σ : Type u_1
α : Type u_2
inst✝¹ : DecidableEq σ
ι : Type u_3
inst✝ : Semiring α
f : ι → MvPowerSeries σ α
a : ι → α
hf : StronglySummable f
⊢ StronglySummable (a • f)
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/StronglySummable/Basic.lean | MvPowerSeries.StronglySummable.smul | [230, 1] | [238, 45] | apply Finite.subset (hf d) | σ : Type u_1
α : Type u_2
inst✝¹ : DecidableEq σ
ι : Type u_3
inst✝ : Semiring α
f : ι → MvPowerSeries σ α
a : ι → α
hf : StronglySummable f
d : σ →₀ ℕ
⊢ (support fun i => (coeff α d) ((a • f) i)).Finite | σ : Type u_1
α : Type u_2
inst✝¹ : DecidableEq σ
ι : Type u_3
inst✝ : Semiring α
f : ι → MvPowerSeries σ α
a : ι → α
hf : StronglySummable f
d : σ →₀ ℕ
⊢ (support fun i => (coeff α d) ((a • f) i)) ⊆ support fun i => (coeff α d) (f i) | Please generate a tactic in lean4 to solve the state.
STATE:
σ : Type u_1
α : Type u_2
inst✝¹ : DecidableEq σ
ι : Type u_3
inst✝ : Semiring α
f : ι → MvPowerSeries σ α
a : ι → α
hf : StronglySummable f
d : σ →₀ ℕ
⊢ (support fun i => (coeff α d) ((a • f) i)).Finite
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/StronglySummable/Basic.lean | MvPowerSeries.StronglySummable.smul | [230, 1] | [238, 45] | intro i | σ : Type u_1
α : Type u_2
inst✝¹ : DecidableEq σ
ι : Type u_3
inst✝ : Semiring α
f : ι → MvPowerSeries σ α
a : ι → α
hf : StronglySummable f
d : σ →₀ ℕ
⊢ (support fun i => (coeff α d) ((a • f) i)) ⊆ support fun i => (coeff α d) (f i) | σ : Type u_1
α : Type u_2
inst✝¹ : DecidableEq σ
ι : Type u_3
inst✝ : Semiring α
f : ι → MvPowerSeries σ α
a : ι → α
hf : StronglySummable f
d : σ →₀ ℕ
i : ι
⊢ (i ∈ support fun i => (coeff α d) ((a • f) i)) → i ∈ support fun i => (coeff α d) (f i) | Please generate a tactic in lean4 to solve the state.
STATE:
σ : Type u_1
α : Type u_2
inst✝¹ : DecidableEq σ
ι : Type u_3
inst✝ : Semiring α
f : ι → MvPowerSeries σ α
a : ι → α
hf : StronglySummable f
d : σ →₀ ℕ
⊢ (support fun i => (coeff α d) ((a • f) i)) ⊆ support fun i => (coeff α d) (f i)
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/StronglySummable/Basic.lean | MvPowerSeries.StronglySummable.smul | [230, 1] | [238, 45] | simp only [Pi.smul_apply', Pi.smul_apply, Function.mem_support, ne_eq] | σ : Type u_1
α : Type u_2
inst✝¹ : DecidableEq σ
ι : Type u_3
inst✝ : Semiring α
f : ι → MvPowerSeries σ α
a : ι → α
hf : StronglySummable f
d : σ →₀ ℕ
i : ι
⊢ (i ∈ support fun i => (coeff α d) ((a • f) i)) → i ∈ support fun i => (coeff α d) (f i) | σ : Type u_1
α : Type u_2
inst✝¹ : DecidableEq σ
ι : Type u_3
inst✝ : Semiring α
f : ι → MvPowerSeries σ α
a : ι → α
hf : StronglySummable f
d : σ →₀ ℕ
i : ι
⊢ ¬(coeff α d) (a i • f i) = 0 → ¬(coeff α d) (f i) = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
σ : Type u_1
α : Type u_2
inst✝¹ : DecidableEq σ
ι : Type u_3
inst✝ : Semiring α
f : ι → MvPowerSeries σ α
a : ι → α
hf : StronglySummable f
d : σ →₀ ℕ
i : ι
⊢ (i ∈ support fun i => (coeff α d) ((a • f) i)) → i ∈ support fun i => (coeff α d) (f i)
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/StronglySummable/Basic.lean | MvPowerSeries.StronglySummable.smul | [230, 1] | [238, 45] | intro h h' | σ : Type u_1
α : Type u_2
inst✝¹ : DecidableEq σ
ι : Type u_3
inst✝ : Semiring α
f : ι → MvPowerSeries σ α
a : ι → α
hf : StronglySummable f
d : σ →₀ ℕ
i : ι
⊢ ¬(coeff α d) (a i • f i) = 0 → ¬(coeff α d) (f i) = 0 | σ : Type u_1
α : Type u_2
inst✝¹ : DecidableEq σ
ι : Type u_3
inst✝ : Semiring α
f : ι → MvPowerSeries σ α
a : ι → α
hf : StronglySummable f
d : σ →₀ ℕ
i : ι
h : ¬(coeff α d) (a i • f i) = 0
h' : (coeff α d) (f i) = 0
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
σ : Type u_1
α : Type u_2
inst✝¹ : DecidableEq σ
ι : Type u_3
inst✝ : Semiring α
f : ι → MvPowerSeries σ α
a : ι → α
hf : StronglySummable f
d : σ →₀ ℕ
i : ι
⊢ ¬(coeff α d) (a i • f i) = 0 → ¬(coeff α d) (f i) = 0
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/StronglySummable/Basic.lean | MvPowerSeries.StronglySummable.smul | [230, 1] | [238, 45] | apply h | σ : Type u_1
α : Type u_2
inst✝¹ : DecidableEq σ
ι : Type u_3
inst✝ : Semiring α
f : ι → MvPowerSeries σ α
a : ι → α
hf : StronglySummable f
d : σ →₀ ℕ
i : ι
h : ¬(coeff α d) (a i • f i) = 0
h' : (coeff α d) (f i) = 0
⊢ False | σ : Type u_1
α : Type u_2
inst✝¹ : DecidableEq σ
ι : Type u_3
inst✝ : Semiring α
f : ι → MvPowerSeries σ α
a : ι → α
hf : StronglySummable f
d : σ →₀ ℕ
i : ι
h : ¬(coeff α d) (a i • f i) = 0
h' : (coeff α d) (f i) = 0
⊢ (coeff α d) (a i • f i) = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
σ : Type u_1
α : Type u_2
inst✝¹ : DecidableEq σ
ι : Type u_3
inst✝ : Semiring α
f : ι → MvPowerSeries σ α
a : ι → α
hf : StronglySummable f
d : σ →₀ ℕ
i : ι
h : ¬(coeff α d) (a i • f i) = 0
h' : (coeff α d) (f i) = 0
⊢ False
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/StronglySummable/Basic.lean | MvPowerSeries.StronglySummable.smul | [230, 1] | [238, 45] | rw [coeff_smul, h', MulZeroClass.mul_zero] | σ : Type u_1
α : Type u_2
inst✝¹ : DecidableEq σ
ι : Type u_3
inst✝ : Semiring α
f : ι → MvPowerSeries σ α
a : ι → α
hf : StronglySummable f
d : σ →₀ ℕ
i : ι
h : ¬(coeff α d) (a i • f i) = 0
h' : (coeff α d) (f i) = 0
⊢ (coeff α d) (a i • f i) = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
σ : Type u_1
α : Type u_2
inst✝¹ : DecidableEq σ
ι : Type u_3
inst✝ : Semiring α
f : ι → MvPowerSeries σ α
a : ι → α
hf : StronglySummable f
d : σ →₀ ℕ
i : ι
h : ¬(coeff α d) (a i • f i) = 0
h' : (coeff α d) (f i) = 0
⊢ (coeff α d) (a i • f i) = 0
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/StronglySummable/Basic.lean | MvPowerSeries.StronglySummable.support_mul | [245, 1] | [274, 13] | rintro d ⟨i, j⟩ h | σ : Type u_1
α : Type u_2
inst✝⁴ : DecidableEq σ
ι : Type u_3
inst✝³ : Semiring α
inst✝² : DecidableEq ι
inst✝¹ : DecidableEq σ
f : ι → MvPowerSeries σ α
κ : Type u_4
inst✝ : DecidableEq κ
g : κ → MvPowerSeries σ α
hf : StronglySummable f
hg : StronglySummable g
⊢ ∀ (d : σ →₀ ℕ),
(support fun i => (coeff α d) (f i.... | case mk
σ : Type u_1
α : Type u_2
inst✝⁴ : DecidableEq σ
ι : Type u_3
inst✝³ : Semiring α
inst✝² : DecidableEq ι
inst✝¹ : DecidableEq σ
f : ι → MvPowerSeries σ α
κ : Type u_4
inst✝ : DecidableEq κ
g : κ → MvPowerSeries σ α
hf : StronglySummable f
hg : StronglySummable g
d : σ →₀ ℕ
i : ι
j : κ
h : (i, j) ∈ support fun i... | Please generate a tactic in lean4 to solve the state.
STATE:
σ : Type u_1
α : Type u_2
inst✝⁴ : DecidableEq σ
ι : Type u_3
inst✝³ : Semiring α
inst✝² : DecidableEq ι
inst✝¹ : DecidableEq σ
f : ι → MvPowerSeries σ α
κ : Type u_4
inst✝ : DecidableEq κ
g : κ → MvPowerSeries σ α
hf : StronglySummable f
hg : StronglySummabl... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/StronglySummable/Basic.lean | MvPowerSeries.StronglySummable.support_mul | [245, 1] | [274, 13] | suffices ∃ p ∈ antidiagonal d, coeff α (p.fst : σ →₀ ℕ) (f i) * (coeff α p.snd) (g j) ≠ 0 by
obtain ⟨⟨b, c⟩, hbc, h'⟩ := this
simp only [mem_antidiagonal] at hbc
erw [Finset.mem_product]
simp only [Finset.mem_biUnion, mem_antidiagonal, Finite.mem_toFinset, mem_support,
ne_eq, Prod.exists]
constructor
. ... | case mk
σ : Type u_1
α : Type u_2
inst✝⁴ : DecidableEq σ
ι : Type u_3
inst✝³ : Semiring α
inst✝² : DecidableEq ι
inst✝¹ : DecidableEq σ
f : ι → MvPowerSeries σ α
κ : Type u_4
inst✝ : DecidableEq κ
g : κ → MvPowerSeries σ α
hf : StronglySummable f
hg : StronglySummable g
d : σ →₀ ℕ
i : ι
j : κ
h : (i, j) ∈ support fun i... | case mk
σ : Type u_1
α : Type u_2
inst✝⁴ : DecidableEq σ
ι : Type u_3
inst✝³ : Semiring α
inst✝² : DecidableEq ι
inst✝¹ : DecidableEq σ
f : ι → MvPowerSeries σ α
κ : Type u_4
inst✝ : DecidableEq κ
g : κ → MvPowerSeries σ α
hf : StronglySummable f
hg : StronglySummable g
d : σ →₀ ℕ
i : ι
j : κ
h : (i, j) ∈ support fun i... | Please generate a tactic in lean4 to solve the state.
STATE:
case mk
σ : Type u_1
α : Type u_2
inst✝⁴ : DecidableEq σ
ι : Type u_3
inst✝³ : Semiring α
inst✝² : DecidableEq ι
inst✝¹ : DecidableEq σ
f : ι → MvPowerSeries σ α
κ : Type u_4
inst✝ : DecidableEq κ
g : κ → MvPowerSeries σ α
hf : StronglySummable f
hg : Strongl... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/StronglySummable/Basic.lean | MvPowerSeries.StronglySummable.support_mul | [245, 1] | [274, 13] | obtain ⟨⟨b, c⟩, hbc, h'⟩ := this | σ : Type u_1
α : Type u_2
inst✝⁴ : DecidableEq σ
ι : Type u_3
inst✝³ : Semiring α
inst✝² : DecidableEq ι
inst✝¹ : DecidableEq σ
f : ι → MvPowerSeries σ α
κ : Type u_4
inst✝ : DecidableEq κ
g : κ → MvPowerSeries σ α
hf : StronglySummable f
hg : StronglySummable g
d : σ →₀ ℕ
i : ι
j : κ
h : (i, j) ∈ support fun i => (coe... | case intro.mk.intro
σ : Type u_1
α : Type u_2
inst✝⁴ : DecidableEq σ
ι : Type u_3
inst✝³ : Semiring α
inst✝² : DecidableEq ι
inst✝¹ : DecidableEq σ
f : ι → MvPowerSeries σ α
κ : Type u_4
inst✝ : DecidableEq κ
g : κ → MvPowerSeries σ α
hf : StronglySummable f
hg : StronglySummable g
d : σ →₀ ℕ
i : ι
j : κ
h : (i, j) ∈ s... | Please generate a tactic in lean4 to solve the state.
STATE:
σ : Type u_1
α : Type u_2
inst✝⁴ : DecidableEq σ
ι : Type u_3
inst✝³ : Semiring α
inst✝² : DecidableEq ι
inst✝¹ : DecidableEq σ
f : ι → MvPowerSeries σ α
κ : Type u_4
inst✝ : DecidableEq κ
g : κ → MvPowerSeries σ α
hf : StronglySummable f
hg : StronglySummabl... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/StronglySummable/Basic.lean | MvPowerSeries.StronglySummable.support_mul | [245, 1] | [274, 13] | simp only [mem_antidiagonal] at hbc | case intro.mk.intro
σ : Type u_1
α : Type u_2
inst✝⁴ : DecidableEq σ
ι : Type u_3
inst✝³ : Semiring α
inst✝² : DecidableEq ι
inst✝¹ : DecidableEq σ
f : ι → MvPowerSeries σ α
κ : Type u_4
inst✝ : DecidableEq κ
g : κ → MvPowerSeries σ α
hf : StronglySummable f
hg : StronglySummable g
d : σ →₀ ℕ
i : ι
j : κ
h : (i, j) ∈ s... | case intro.mk.intro
σ : Type u_1
α : Type u_2
inst✝⁴ : DecidableEq σ
ι : Type u_3
inst✝³ : Semiring α
inst✝² : DecidableEq ι
inst✝¹ : DecidableEq σ
f : ι → MvPowerSeries σ α
κ : Type u_4
inst✝ : DecidableEq κ
g : κ → MvPowerSeries σ α
hf : StronglySummable f
hg : StronglySummable g
d : σ →₀ ℕ
i : ι
j : κ
h : (i, j) ∈ s... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.mk.intro
σ : Type u_1
α : Type u_2
inst✝⁴ : DecidableEq σ
ι : Type u_3
inst✝³ : Semiring α
inst✝² : DecidableEq ι
inst✝¹ : DecidableEq σ
f : ι → MvPowerSeries σ α
κ : Type u_4
inst✝ : DecidableEq κ
g : κ → MvPowerSeries σ α
hf : StronglySummable f
... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/StronglySummable/Basic.lean | MvPowerSeries.StronglySummable.support_mul | [245, 1] | [274, 13] | erw [Finset.mem_product] | case intro.mk.intro
σ : Type u_1
α : Type u_2
inst✝⁴ : DecidableEq σ
ι : Type u_3
inst✝³ : Semiring α
inst✝² : DecidableEq ι
inst✝¹ : DecidableEq σ
f : ι → MvPowerSeries σ α
κ : Type u_4
inst✝ : DecidableEq κ
g : κ → MvPowerSeries σ α
hf : StronglySummable f
hg : StronglySummable g
d : σ →₀ ℕ
i : ι
j : κ
h : (i, j) ∈ s... | case intro.mk.intro
σ : Type u_1
α : Type u_2
inst✝⁴ : DecidableEq σ
ι : Type u_3
inst✝³ : Semiring α
inst✝² : DecidableEq ι
inst✝¹ : DecidableEq σ
f : ι → MvPowerSeries σ α
κ : Type u_4
inst✝ : DecidableEq κ
g : κ → MvPowerSeries σ α
hf : StronglySummable f
hg : StronglySummable g
d : σ →₀ ℕ
i : ι
j : κ
h : (i, j) ∈ s... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.mk.intro
σ : Type u_1
α : Type u_2
inst✝⁴ : DecidableEq σ
ι : Type u_3
inst✝³ : Semiring α
inst✝² : DecidableEq ι
inst✝¹ : DecidableEq σ
f : ι → MvPowerSeries σ α
κ : Type u_4
inst✝ : DecidableEq κ
g : κ → MvPowerSeries σ α
hf : StronglySummable f
... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/StronglySummable/Basic.lean | MvPowerSeries.StronglySummable.support_mul | [245, 1] | [274, 13] | simp only [Finset.mem_biUnion, mem_antidiagonal, Finite.mem_toFinset, mem_support,
ne_eq, Prod.exists] | case intro.mk.intro
σ : Type u_1
α : Type u_2
inst✝⁴ : DecidableEq σ
ι : Type u_3
inst✝³ : Semiring α
inst✝² : DecidableEq ι
inst✝¹ : DecidableEq σ
f : ι → MvPowerSeries σ α
κ : Type u_4
inst✝ : DecidableEq κ
g : κ → MvPowerSeries σ α
hf : StronglySummable f
hg : StronglySummable g
d : σ →₀ ℕ
i : ι
j : κ
h : (i, j) ∈ s... | case intro.mk.intro
σ : Type u_1
α : Type u_2
inst✝⁴ : DecidableEq σ
ι : Type u_3
inst✝³ : Semiring α
inst✝² : DecidableEq ι
inst✝¹ : DecidableEq σ
f : ι → MvPowerSeries σ α
κ : Type u_4
inst✝ : DecidableEq κ
g : κ → MvPowerSeries σ α
hf : StronglySummable f
hg : StronglySummable g
d : σ →₀ ℕ
i : ι
j : κ
h : (i, j) ∈ s... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.mk.intro
σ : Type u_1
α : Type u_2
inst✝⁴ : DecidableEq σ
ι : Type u_3
inst✝³ : Semiring α
inst✝² : DecidableEq ι
inst✝¹ : DecidableEq σ
f : ι → MvPowerSeries σ α
κ : Type u_4
inst✝ : DecidableEq κ
g : κ → MvPowerSeries σ α
hf : StronglySummable f
... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/StronglySummable/Basic.lean | MvPowerSeries.StronglySummable.support_mul | [245, 1] | [274, 13] | constructor | case intro.mk.intro
σ : Type u_1
α : Type u_2
inst✝⁴ : DecidableEq σ
ι : Type u_3
inst✝³ : Semiring α
inst✝² : DecidableEq ι
inst✝¹ : DecidableEq σ
f : ι → MvPowerSeries σ α
κ : Type u_4
inst✝ : DecidableEq κ
g : κ → MvPowerSeries σ α
hf : StronglySummable f
hg : StronglySummable g
d : σ →₀ ℕ
i : ι
j : κ
h : (i, j) ∈ s... | case intro.mk.intro.left
σ : Type u_1
α : Type u_2
inst✝⁴ : DecidableEq σ
ι : Type u_3
inst✝³ : Semiring α
inst✝² : DecidableEq ι
inst✝¹ : DecidableEq σ
f : ι → MvPowerSeries σ α
κ : Type u_4
inst✝ : DecidableEq κ
g : κ → MvPowerSeries σ α
hf : StronglySummable f
hg : StronglySummable g
d : σ →₀ ℕ
i : ι
j : κ
h : (i, j... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.mk.intro
σ : Type u_1
α : Type u_2
inst✝⁴ : DecidableEq σ
ι : Type u_3
inst✝³ : Semiring α
inst✝² : DecidableEq ι
inst✝¹ : DecidableEq σ
f : ι → MvPowerSeries σ α
κ : Type u_4
inst✝ : DecidableEq κ
g : κ → MvPowerSeries σ α
hf : StronglySummable f
... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/StronglySummable/Basic.lean | MvPowerSeries.StronglySummable.support_mul | [245, 1] | [274, 13] | . use b, c, hbc
intro h₁
apply h'
rw [h₁, MulZeroClass.zero_mul] | case intro.mk.intro.left
σ : Type u_1
α : Type u_2
inst✝⁴ : DecidableEq σ
ι : Type u_3
inst✝³ : Semiring α
inst✝² : DecidableEq ι
inst✝¹ : DecidableEq σ
f : ι → MvPowerSeries σ α
κ : Type u_4
inst✝ : DecidableEq κ
g : κ → MvPowerSeries σ α
hf : StronglySummable f
hg : StronglySummable g
d : σ →₀ ℕ
i : ι
j : κ
h : (i, j... | case intro.mk.intro.right
σ : Type u_1
α : Type u_2
inst✝⁴ : DecidableEq σ
ι : Type u_3
inst✝³ : Semiring α
inst✝² : DecidableEq ι
inst✝¹ : DecidableEq σ
f : ι → MvPowerSeries σ α
κ : Type u_4
inst✝ : DecidableEq κ
g : κ → MvPowerSeries σ α
hf : StronglySummable f
hg : StronglySummable g
d : σ →₀ ℕ
i : ι
j : κ
h : (i, ... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.mk.intro.left
σ : Type u_1
α : Type u_2
inst✝⁴ : DecidableEq σ
ι : Type u_3
inst✝³ : Semiring α
inst✝² : DecidableEq ι
inst✝¹ : DecidableEq σ
f : ι → MvPowerSeries σ α
κ : Type u_4
inst✝ : DecidableEq κ
g : κ → MvPowerSeries σ α
hf : StronglySummab... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/StronglySummable/Basic.lean | MvPowerSeries.StronglySummable.support_mul | [245, 1] | [274, 13] | . use b, c, hbc
intro h₂
apply h'
rw [h₂, MulZeroClass.mul_zero] | case intro.mk.intro.right
σ : Type u_1
α : Type u_2
inst✝⁴ : DecidableEq σ
ι : Type u_3
inst✝³ : Semiring α
inst✝² : DecidableEq ι
inst✝¹ : DecidableEq σ
f : ι → MvPowerSeries σ α
κ : Type u_4
inst✝ : DecidableEq κ
g : κ → MvPowerSeries σ α
hf : StronglySummable f
hg : StronglySummable g
d : σ →₀ ℕ
i : ι
j : κ
h : (i, ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.mk.intro.right
σ : Type u_1
α : Type u_2
inst✝⁴ : DecidableEq σ
ι : Type u_3
inst✝³ : Semiring α
inst✝² : DecidableEq ι
inst✝¹ : DecidableEq σ
f : ι → MvPowerSeries σ α
κ : Type u_4
inst✝ : DecidableEq κ
g : κ → MvPowerSeries σ α
hf : StronglySumma... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/StronglySummable/Basic.lean | MvPowerSeries.StronglySummable.support_mul | [245, 1] | [274, 13] | use b, c, hbc | case intro.mk.intro.left
σ : Type u_1
α : Type u_2
inst✝⁴ : DecidableEq σ
ι : Type u_3
inst✝³ : Semiring α
inst✝² : DecidableEq ι
inst✝¹ : DecidableEq σ
f : ι → MvPowerSeries σ α
κ : Type u_4
inst✝ : DecidableEq κ
g : κ → MvPowerSeries σ α
hf : StronglySummable f
hg : StronglySummable g
d : σ →₀ ℕ
i : ι
j : κ
h : (i, j... | case right
σ : Type u_1
α : Type u_2
inst✝⁴ : DecidableEq σ
ι : Type u_3
inst✝³ : Semiring α
inst✝² : DecidableEq ι
inst✝¹ : DecidableEq σ
f : ι → MvPowerSeries σ α
κ : Type u_4
inst✝ : DecidableEq κ
g : κ → MvPowerSeries σ α
hf : StronglySummable f
hg : StronglySummable g
d : σ →₀ ℕ
i : ι
j : κ
h : (i, j) ∈ support fu... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.mk.intro.left
σ : Type u_1
α : Type u_2
inst✝⁴ : DecidableEq σ
ι : Type u_3
inst✝³ : Semiring α
inst✝² : DecidableEq ι
inst✝¹ : DecidableEq σ
f : ι → MvPowerSeries σ α
κ : Type u_4
inst✝ : DecidableEq κ
g : κ → MvPowerSeries σ α
hf : StronglySummab... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/StronglySummable/Basic.lean | MvPowerSeries.StronglySummable.support_mul | [245, 1] | [274, 13] | intro h₁ | case right
σ : Type u_1
α : Type u_2
inst✝⁴ : DecidableEq σ
ι : Type u_3
inst✝³ : Semiring α
inst✝² : DecidableEq ι
inst✝¹ : DecidableEq σ
f : ι → MvPowerSeries σ α
κ : Type u_4
inst✝ : DecidableEq κ
g : κ → MvPowerSeries σ α
hf : StronglySummable f
hg : StronglySummable g
d : σ →₀ ℕ
i : ι
j : κ
h : (i, j) ∈ support fu... | case right
σ : Type u_1
α : Type u_2
inst✝⁴ : DecidableEq σ
ι : Type u_3
inst✝³ : Semiring α
inst✝² : DecidableEq ι
inst✝¹ : DecidableEq σ
f : ι → MvPowerSeries σ α
κ : Type u_4
inst✝ : DecidableEq κ
g : κ → MvPowerSeries σ α
hf : StronglySummable f
hg : StronglySummable g
d : σ →₀ ℕ
i : ι
j : κ
h : (i, j) ∈ support fu... | Please generate a tactic in lean4 to solve the state.
STATE:
case right
σ : Type u_1
α : Type u_2
inst✝⁴ : DecidableEq σ
ι : Type u_3
inst✝³ : Semiring α
inst✝² : DecidableEq ι
inst✝¹ : DecidableEq σ
f : ι → MvPowerSeries σ α
κ : Type u_4
inst✝ : DecidableEq κ
g : κ → MvPowerSeries σ α
hf : StronglySummable f
hg : Stro... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/StronglySummable/Basic.lean | MvPowerSeries.StronglySummable.support_mul | [245, 1] | [274, 13] | apply h' | case right
σ : Type u_1
α : Type u_2
inst✝⁴ : DecidableEq σ
ι : Type u_3
inst✝³ : Semiring α
inst✝² : DecidableEq ι
inst✝¹ : DecidableEq σ
f : ι → MvPowerSeries σ α
κ : Type u_4
inst✝ : DecidableEq κ
g : κ → MvPowerSeries σ α
hf : StronglySummable f
hg : StronglySummable g
d : σ →₀ ℕ
i : ι
j : κ
h : (i, j) ∈ support fu... | case right
σ : Type u_1
α : Type u_2
inst✝⁴ : DecidableEq σ
ι : Type u_3
inst✝³ : Semiring α
inst✝² : DecidableEq ι
inst✝¹ : DecidableEq σ
f : ι → MvPowerSeries σ α
κ : Type u_4
inst✝ : DecidableEq κ
g : κ → MvPowerSeries σ α
hf : StronglySummable f
hg : StronglySummable g
d : σ →₀ ℕ
i : ι
j : κ
h : (i, j) ∈ support fu... | Please generate a tactic in lean4 to solve the state.
STATE:
case right
σ : Type u_1
α : Type u_2
inst✝⁴ : DecidableEq σ
ι : Type u_3
inst✝³ : Semiring α
inst✝² : DecidableEq ι
inst✝¹ : DecidableEq σ
f : ι → MvPowerSeries σ α
κ : Type u_4
inst✝ : DecidableEq κ
g : κ → MvPowerSeries σ α
hf : StronglySummable f
hg : Stro... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/StronglySummable/Basic.lean | MvPowerSeries.StronglySummable.support_mul | [245, 1] | [274, 13] | rw [h₁, MulZeroClass.zero_mul] | case right
σ : Type u_1
α : Type u_2
inst✝⁴ : DecidableEq σ
ι : Type u_3
inst✝³ : Semiring α
inst✝² : DecidableEq ι
inst✝¹ : DecidableEq σ
f : ι → MvPowerSeries σ α
κ : Type u_4
inst✝ : DecidableEq κ
g : κ → MvPowerSeries σ α
hf : StronglySummable f
hg : StronglySummable g
d : σ →₀ ℕ
i : ι
j : κ
h : (i, j) ∈ support fu... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case right
σ : Type u_1
α : Type u_2
inst✝⁴ : DecidableEq σ
ι : Type u_3
inst✝³ : Semiring α
inst✝² : DecidableEq ι
inst✝¹ : DecidableEq σ
f : ι → MvPowerSeries σ α
κ : Type u_4
inst✝ : DecidableEq κ
g : κ → MvPowerSeries σ α
hf : StronglySummable f
hg : Stro... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/StronglySummable/Basic.lean | MvPowerSeries.StronglySummable.support_mul | [245, 1] | [274, 13] | use b, c, hbc | case intro.mk.intro.right
σ : Type u_1
α : Type u_2
inst✝⁴ : DecidableEq σ
ι : Type u_3
inst✝³ : Semiring α
inst✝² : DecidableEq ι
inst✝¹ : DecidableEq σ
f : ι → MvPowerSeries σ α
κ : Type u_4
inst✝ : DecidableEq κ
g : κ → MvPowerSeries σ α
hf : StronglySummable f
hg : StronglySummable g
d : σ →₀ ℕ
i : ι
j : κ
h : (i, ... | case right
σ : Type u_1
α : Type u_2
inst✝⁴ : DecidableEq σ
ι : Type u_3
inst✝³ : Semiring α
inst✝² : DecidableEq ι
inst✝¹ : DecidableEq σ
f : ι → MvPowerSeries σ α
κ : Type u_4
inst✝ : DecidableEq κ
g : κ → MvPowerSeries σ α
hf : StronglySummable f
hg : StronglySummable g
d : σ →₀ ℕ
i : ι
j : κ
h : (i, j) ∈ support fu... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.mk.intro.right
σ : Type u_1
α : Type u_2
inst✝⁴ : DecidableEq σ
ι : Type u_3
inst✝³ : Semiring α
inst✝² : DecidableEq ι
inst✝¹ : DecidableEq σ
f : ι → MvPowerSeries σ α
κ : Type u_4
inst✝ : DecidableEq κ
g : κ → MvPowerSeries σ α
hf : StronglySumma... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/StronglySummable/Basic.lean | MvPowerSeries.StronglySummable.support_mul | [245, 1] | [274, 13] | intro h₂ | case right
σ : Type u_1
α : Type u_2
inst✝⁴ : DecidableEq σ
ι : Type u_3
inst✝³ : Semiring α
inst✝² : DecidableEq ι
inst✝¹ : DecidableEq σ
f : ι → MvPowerSeries σ α
κ : Type u_4
inst✝ : DecidableEq κ
g : κ → MvPowerSeries σ α
hf : StronglySummable f
hg : StronglySummable g
d : σ →₀ ℕ
i : ι
j : κ
h : (i, j) ∈ support fu... | case right
σ : Type u_1
α : Type u_2
inst✝⁴ : DecidableEq σ
ι : Type u_3
inst✝³ : Semiring α
inst✝² : DecidableEq ι
inst✝¹ : DecidableEq σ
f : ι → MvPowerSeries σ α
κ : Type u_4
inst✝ : DecidableEq κ
g : κ → MvPowerSeries σ α
hf : StronglySummable f
hg : StronglySummable g
d : σ →₀ ℕ
i : ι
j : κ
h : (i, j) ∈ support fu... | Please generate a tactic in lean4 to solve the state.
STATE:
case right
σ : Type u_1
α : Type u_2
inst✝⁴ : DecidableEq σ
ι : Type u_3
inst✝³ : Semiring α
inst✝² : DecidableEq ι
inst✝¹ : DecidableEq σ
f : ι → MvPowerSeries σ α
κ : Type u_4
inst✝ : DecidableEq κ
g : κ → MvPowerSeries σ α
hf : StronglySummable f
hg : Stro... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/StronglySummable/Basic.lean | MvPowerSeries.StronglySummable.support_mul | [245, 1] | [274, 13] | apply h' | case right
σ : Type u_1
α : Type u_2
inst✝⁴ : DecidableEq σ
ι : Type u_3
inst✝³ : Semiring α
inst✝² : DecidableEq ι
inst✝¹ : DecidableEq σ
f : ι → MvPowerSeries σ α
κ : Type u_4
inst✝ : DecidableEq κ
g : κ → MvPowerSeries σ α
hf : StronglySummable f
hg : StronglySummable g
d : σ →₀ ℕ
i : ι
j : κ
h : (i, j) ∈ support fu... | case right
σ : Type u_1
α : Type u_2
inst✝⁴ : DecidableEq σ
ι : Type u_3
inst✝³ : Semiring α
inst✝² : DecidableEq ι
inst✝¹ : DecidableEq σ
f : ι → MvPowerSeries σ α
κ : Type u_4
inst✝ : DecidableEq κ
g : κ → MvPowerSeries σ α
hf : StronglySummable f
hg : StronglySummable g
d : σ →₀ ℕ
i : ι
j : κ
h : (i, j) ∈ support fu... | Please generate a tactic in lean4 to solve the state.
STATE:
case right
σ : Type u_1
α : Type u_2
inst✝⁴ : DecidableEq σ
ι : Type u_3
inst✝³ : Semiring α
inst✝² : DecidableEq ι
inst✝¹ : DecidableEq σ
f : ι → MvPowerSeries σ α
κ : Type u_4
inst✝ : DecidableEq κ
g : κ → MvPowerSeries σ α
hf : StronglySummable f
hg : Stro... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/StronglySummable/Basic.lean | MvPowerSeries.StronglySummable.support_mul | [245, 1] | [274, 13] | rw [h₂, MulZeroClass.mul_zero] | case right
σ : Type u_1
α : Type u_2
inst✝⁴ : DecidableEq σ
ι : Type u_3
inst✝³ : Semiring α
inst✝² : DecidableEq ι
inst✝¹ : DecidableEq σ
f : ι → MvPowerSeries σ α
κ : Type u_4
inst✝ : DecidableEq κ
g : κ → MvPowerSeries σ α
hf : StronglySummable f
hg : StronglySummable g
d : σ →₀ ℕ
i : ι
j : κ
h : (i, j) ∈ support fu... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case right
σ : Type u_1
α : Type u_2
inst✝⁴ : DecidableEq σ
ι : Type u_3
inst✝³ : Semiring α
inst✝² : DecidableEq ι
inst✝¹ : DecidableEq σ
f : ι → MvPowerSeries σ α
κ : Type u_4
inst✝ : DecidableEq κ
g : κ → MvPowerSeries σ α
hf : StronglySummable f
hg : Stro... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/StronglySummable/Basic.lean | MvPowerSeries.StronglySummable.support_mul | [245, 1] | [274, 13] | by_contra h' | case mk
σ : Type u_1
α : Type u_2
inst✝⁴ : DecidableEq σ
ι : Type u_3
inst✝³ : Semiring α
inst✝² : DecidableEq ι
inst✝¹ : DecidableEq σ
f : ι → MvPowerSeries σ α
κ : Type u_4
inst✝ : DecidableEq κ
g : κ → MvPowerSeries σ α
hf : StronglySummable f
hg : StronglySummable g
d : σ →₀ ℕ
i : ι
j : κ
h : (i, j) ∈ support fun i... | case mk
σ : Type u_1
α : Type u_2
inst✝⁴ : DecidableEq σ
ι : Type u_3
inst✝³ : Semiring α
inst✝² : DecidableEq ι
inst✝¹ : DecidableEq σ
f : ι → MvPowerSeries σ α
κ : Type u_4
inst✝ : DecidableEq κ
g : κ → MvPowerSeries σ α
hf : StronglySummable f
hg : StronglySummable g
d : σ →₀ ℕ
i : ι
j : κ
h : (i, j) ∈ support fun i... | Please generate a tactic in lean4 to solve the state.
STATE:
case mk
σ : Type u_1
α : Type u_2
inst✝⁴ : DecidableEq σ
ι : Type u_3
inst✝³ : Semiring α
inst✝² : DecidableEq ι
inst✝¹ : DecidableEq σ
f : ι → MvPowerSeries σ α
κ : Type u_4
inst✝ : DecidableEq κ
g : κ → MvPowerSeries σ α
hf : StronglySummable f
hg : Strongl... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/StronglySummable/Basic.lean | MvPowerSeries.StronglySummable.support_mul | [245, 1] | [274, 13] | refine' h (sum_eq_zero _) | case mk
σ : Type u_1
α : Type u_2
inst✝⁴ : DecidableEq σ
ι : Type u_3
inst✝³ : Semiring α
inst✝² : DecidableEq ι
inst✝¹ : DecidableEq σ
f : ι → MvPowerSeries σ α
κ : Type u_4
inst✝ : DecidableEq κ
g : κ → MvPowerSeries σ α
hf : StronglySummable f
hg : StronglySummable g
d : σ →₀ ℕ
i : ι
j : κ
h : (i, j) ∈ support fun i... | case mk
σ : Type u_1
α : Type u_2
inst✝⁴ : DecidableEq σ
ι : Type u_3
inst✝³ : Semiring α
inst✝² : DecidableEq ι
inst✝¹ : DecidableEq σ
f : ι → MvPowerSeries σ α
κ : Type u_4
inst✝ : DecidableEq κ
g : κ → MvPowerSeries σ α
hf : StronglySummable f
hg : StronglySummable g
d : σ →₀ ℕ
i : ι
j : κ
h : (i, j) ∈ support fun i... | Please generate a tactic in lean4 to solve the state.
STATE:
case mk
σ : Type u_1
α : Type u_2
inst✝⁴ : DecidableEq σ
ι : Type u_3
inst✝³ : Semiring α
inst✝² : DecidableEq ι
inst✝¹ : DecidableEq σ
f : ι → MvPowerSeries σ α
κ : Type u_4
inst✝ : DecidableEq κ
g : κ → MvPowerSeries σ α
hf : StronglySummable f
hg : Strongl... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/StronglySummable/Basic.lean | MvPowerSeries.StronglySummable.support_mul | [245, 1] | [274, 13] | push_neg at h' | case mk
σ : Type u_1
α : Type u_2
inst✝⁴ : DecidableEq σ
ι : Type u_3
inst✝³ : Semiring α
inst✝² : DecidableEq ι
inst✝¹ : DecidableEq σ
f : ι → MvPowerSeries σ α
κ : Type u_4
inst✝ : DecidableEq κ
g : κ → MvPowerSeries σ α
hf : StronglySummable f
hg : StronglySummable g
d : σ →₀ ℕ
i : ι
j : κ
h : (i, j) ∈ support fun i... | case mk
σ : Type u_1
α : Type u_2
inst✝⁴ : DecidableEq σ
ι : Type u_3
inst✝³ : Semiring α
inst✝² : DecidableEq ι
inst✝¹ : DecidableEq σ
f : ι → MvPowerSeries σ α
κ : Type u_4
inst✝ : DecidableEq κ
g : κ → MvPowerSeries σ α
hf : StronglySummable f
hg : StronglySummable g
d : σ →₀ ℕ
i : ι
j : κ
h : (i, j) ∈ support fun i... | Please generate a tactic in lean4 to solve the state.
STATE:
case mk
σ : Type u_1
α : Type u_2
inst✝⁴ : DecidableEq σ
ι : Type u_3
inst✝³ : Semiring α
inst✝² : DecidableEq ι
inst✝¹ : DecidableEq σ
f : ι → MvPowerSeries σ α
κ : Type u_4
inst✝ : DecidableEq κ
g : κ → MvPowerSeries σ α
hf : StronglySummable f
hg : Strongl... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/StronglySummable/Basic.lean | MvPowerSeries.StronglySummable.support_mul | [245, 1] | [274, 13] | simp only [mem_antidiagonal] at h' ⊢ | case mk
σ : Type u_1
α : Type u_2
inst✝⁴ : DecidableEq σ
ι : Type u_3
inst✝³ : Semiring α
inst✝² : DecidableEq ι
inst✝¹ : DecidableEq σ
f : ι → MvPowerSeries σ α
κ : Type u_4
inst✝ : DecidableEq κ
g : κ → MvPowerSeries σ α
hf : StronglySummable f
hg : StronglySummable g
d : σ →₀ ℕ
i : ι
j : κ
h : (i, j) ∈ support fun i... | case mk
σ : Type u_1
α : Type u_2
inst✝⁴ : DecidableEq σ
ι : Type u_3
inst✝³ : Semiring α
inst✝² : DecidableEq ι
inst✝¹ : DecidableEq σ
f : ι → MvPowerSeries σ α
κ : Type u_4
inst✝ : DecidableEq κ
g : κ → MvPowerSeries σ α
hf : StronglySummable f
hg : StronglySummable g
d : σ →₀ ℕ
i : ι
j : κ
h : (i, j) ∈ support fun i... | Please generate a tactic in lean4 to solve the state.
STATE:
case mk
σ : Type u_1
α : Type u_2
inst✝⁴ : DecidableEq σ
ι : Type u_3
inst✝³ : Semiring α
inst✝² : DecidableEq ι
inst✝¹ : DecidableEq σ
f : ι → MvPowerSeries σ α
κ : Type u_4
inst✝ : DecidableEq κ
g : κ → MvPowerSeries σ α
hf : StronglySummable f
hg : Strongl... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/StronglySummable/Basic.lean | MvPowerSeries.StronglySummable.support_mul | [245, 1] | [274, 13] | exact h' | case mk
σ : Type u_1
α : Type u_2
inst✝⁴ : DecidableEq σ
ι : Type u_3
inst✝³ : Semiring α
inst✝² : DecidableEq ι
inst✝¹ : DecidableEq σ
f : ι → MvPowerSeries σ α
κ : Type u_4
inst✝ : DecidableEq κ
g : κ → MvPowerSeries σ α
hf : StronglySummable f
hg : StronglySummable g
d : σ →₀ ℕ
i : ι
j : κ
h : (i, j) ∈ support fun i... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mk
σ : Type u_1
α : Type u_2
inst✝⁴ : DecidableEq σ
ι : Type u_3
inst✝³ : Semiring α
inst✝² : DecidableEq ι
inst✝¹ : DecidableEq σ
f : ι → MvPowerSeries σ α
κ : Type u_4
inst✝ : DecidableEq κ
g : κ → MvPowerSeries σ α
hf : StronglySummable f
hg : Strongl... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/StronglySummable/Basic.lean | MvPowerSeries.StronglySummable.mul | [280, 1] | [283, 80] | classical exact fun d => Finite.subset (finite_toSet _) (support_mul hf hg d) | σ : Type u_1
α : Type u_2
inst✝¹ : DecidableEq σ
ι : Type u_4
inst✝ : Semiring α
f : ι → MvPowerSeries σ α
κ : Type u_3
g : κ → MvPowerSeries σ α
hf : StronglySummable f
hg : StronglySummable g
⊢ StronglySummable fun i => f i.1 * g i.2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
σ : Type u_1
α : Type u_2
inst✝¹ : DecidableEq σ
ι : Type u_4
inst✝ : Semiring α
f : ι → MvPowerSeries σ α
κ : Type u_3
g : κ → MvPowerSeries σ α
hf : StronglySummable f
hg : StronglySummable g
⊢ StronglySummable fun i => f i.1 * g i.2
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/StronglySummable/Basic.lean | MvPowerSeries.StronglySummable.mul | [280, 1] | [283, 80] | exact fun d => Finite.subset (finite_toSet _) (support_mul hf hg d) | σ : Type u_1
α : Type u_2
inst✝¹ : DecidableEq σ
ι : Type u_4
inst✝ : Semiring α
f : ι → MvPowerSeries σ α
κ : Type u_3
g : κ → MvPowerSeries σ α
hf : StronglySummable f
hg : StronglySummable g
⊢ StronglySummable fun i => f i.1 * g i.2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
σ : Type u_1
α : Type u_2
inst✝¹ : DecidableEq σ
ι : Type u_4
inst✝ : Semiring α
f : ι → MvPowerSeries σ α
κ : Type u_3
g : κ → MvPowerSeries σ α
hf : StronglySummable f
hg : StronglySummable g
⊢ StronglySummable fun i => f i.1 * g i.2
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/StronglySummable/Basic.lean | MvPowerSeries.StronglySummable.coeff_sum | [299, 1] | [304, 86] | rw [coeff_sum_def, sum_subset (Finite.toFinset_subset.mpr hs)] | σ : Type u_1
α : Type u_2
inst✝¹ : DecidableEq σ
ι : Type u_3
inst✝ : Semiring α
f : ι → MvPowerSeries σ α
hf : StronglySummable f
d : σ →₀ ℕ
s : Finset ι
hs : (support fun i => (coeff α d) (f i)) ⊆ ↑s
⊢ (coeff α d) hf.sum = ∑ i ∈ s, (coeff α d) (f i) | σ : Type u_1
α : Type u_2
inst✝¹ : DecidableEq σ
ι : Type u_3
inst✝ : Semiring α
f : ι → MvPowerSeries σ α
hf : StronglySummable f
d : σ →₀ ℕ
s : Finset ι
hs : (support fun i => (coeff α d) (f i)) ⊆ ↑s
⊢ ∀ x ∈ s, x ∉ ⋯.toFinset → (coeff α d) (f x) = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
σ : Type u_1
α : Type u_2
inst✝¹ : DecidableEq σ
ι : Type u_3
inst✝ : Semiring α
f : ι → MvPowerSeries σ α
hf : StronglySummable f
d : σ →₀ ℕ
s : Finset ι
hs : (support fun i => (coeff α d) (f i)) ⊆ ↑s
⊢ (coeff α d) hf.sum = ∑ i ∈ s, (coeff α d) (f i)
TACTIC:... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/StronglySummable/Basic.lean | MvPowerSeries.StronglySummable.coeff_sum | [299, 1] | [304, 86] | intro i _ hi' | σ : Type u_1
α : Type u_2
inst✝¹ : DecidableEq σ
ι : Type u_3
inst✝ : Semiring α
f : ι → MvPowerSeries σ α
hf : StronglySummable f
d : σ →₀ ℕ
s : Finset ι
hs : (support fun i => (coeff α d) (f i)) ⊆ ↑s
⊢ ∀ x ∈ s, x ∉ ⋯.toFinset → (coeff α d) (f x) = 0 | σ : Type u_1
α : Type u_2
inst✝¹ : DecidableEq σ
ι : Type u_3
inst✝ : Semiring α
f : ι → MvPowerSeries σ α
hf : StronglySummable f
d : σ →₀ ℕ
s : Finset ι
hs : (support fun i => (coeff α d) (f i)) ⊆ ↑s
i : ι
a✝ : i ∈ s
hi' : i ∉ ⋯.toFinset
⊢ (coeff α d) (f i) = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
σ : Type u_1
α : Type u_2
inst✝¹ : DecidableEq σ
ι : Type u_3
inst✝ : Semiring α
f : ι → MvPowerSeries σ α
hf : StronglySummable f
d : σ →₀ ℕ
s : Finset ι
hs : (support fun i => (coeff α d) (f i)) ⊆ ↑s
⊢ ∀ x ∈ s, x ∉ ⋯.toFinset → (coeff α d) (f x) = 0
TACTIC:... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/StronglySummable/Basic.lean | MvPowerSeries.StronglySummable.coeff_sum | [299, 1] | [304, 86] | simpa only [Finite.mem_toFinset, Function.mem_support, Classical.not_not] using hi' | σ : Type u_1
α : Type u_2
inst✝¹ : DecidableEq σ
ι : Type u_3
inst✝ : Semiring α
f : ι → MvPowerSeries σ α
hf : StronglySummable f
d : σ →₀ ℕ
s : Finset ι
hs : (support fun i => (coeff α d) (f i)) ⊆ ↑s
i : ι
a✝ : i ∈ s
hi' : i ∉ ⋯.toFinset
⊢ (coeff α d) (f i) = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
σ : Type u_1
α : Type u_2
inst✝¹ : DecidableEq σ
ι : Type u_3
inst✝ : Semiring α
f : ι → MvPowerSeries σ α
hf : StronglySummable f
d : σ →₀ ℕ
s : Finset ι
hs : (support fun i => (coeff α d) (f i)) ⊆ ↑s
i : ι
a✝ : i ∈ s
hi' : i ∉ ⋯.toFinset
⊢ (coeff α d) (f i)... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/StronglySummable/Basic.lean | MvPowerSeries.StronglySummable.sum_congr | [308, 1] | [314, 57] | ext d | σ : Type u_1
α : Type u_2
inst✝¹ : DecidableEq σ
ι : Type u_3
inst✝ : Semiring α
f g : ι → MvPowerSeries σ α
hf : StronglySummable f
hg : StronglySummable g
h : f = g
⊢ hf.sum = hg.sum | case h
σ : Type u_1
α : Type u_2
inst✝¹ : DecidableEq σ
ι : Type u_3
inst✝ : Semiring α
f g : ι → MvPowerSeries σ α
hf : StronglySummable f
hg : StronglySummable g
h : f = g
d : σ →₀ ℕ
⊢ (coeff α d) hf.sum = (coeff α d) hg.sum | Please generate a tactic in lean4 to solve the state.
STATE:
σ : Type u_1
α : Type u_2
inst✝¹ : DecidableEq σ
ι : Type u_3
inst✝ : Semiring α
f g : ι → MvPowerSeries σ α
hf : StronglySummable f
hg : StronglySummable g
h : f = g
⊢ hf.sum = hg.sum
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/StronglySummable/Basic.lean | MvPowerSeries.StronglySummable.sum_congr | [308, 1] | [314, 57] | simp only [coeff_sum_def] | case h
σ : Type u_1
α : Type u_2
inst✝¹ : DecidableEq σ
ι : Type u_3
inst✝ : Semiring α
f g : ι → MvPowerSeries σ α
hf : StronglySummable f
hg : StronglySummable g
h : f = g
d : σ →₀ ℕ
⊢ (coeff α d) hf.sum = (coeff α d) hg.sum | case h
σ : Type u_1
α : Type u_2
inst✝¹ : DecidableEq σ
ι : Type u_3
inst✝ : Semiring α
f g : ι → MvPowerSeries σ α
hf : StronglySummable f
hg : StronglySummable g
h : f = g
d : σ →₀ ℕ
⊢ ∑ i ∈ ⋯.toFinset, (coeff α d) (f i) = ∑ i ∈ ⋯.toFinset, (coeff α d) (g i) | Please generate a tactic in lean4 to solve the state.
STATE:
case h
σ : Type u_1
α : Type u_2
inst✝¹ : DecidableEq σ
ι : Type u_3
inst✝ : Semiring α
f g : ι → MvPowerSeries σ α
hf : StronglySummable f
hg : StronglySummable g
h : f = g
d : σ →₀ ℕ
⊢ (coeff α d) hf.sum = (coeff α d) hg.sum
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/StronglySummable/Basic.lean | MvPowerSeries.StronglySummable.sum_congr | [308, 1] | [314, 57] | refine' Finset.sum_congr _ fun i _ => by rw [h] | case h
σ : Type u_1
α : Type u_2
inst✝¹ : DecidableEq σ
ι : Type u_3
inst✝ : Semiring α
f g : ι → MvPowerSeries σ α
hf : StronglySummable f
hg : StronglySummable g
h : f = g
d : σ →₀ ℕ
⊢ ∑ i ∈ ⋯.toFinset, (coeff α d) (f i) = ∑ i ∈ ⋯.toFinset, (coeff α d) (g i) | case h
σ : Type u_1
α : Type u_2
inst✝¹ : DecidableEq σ
ι : Type u_3
inst✝ : Semiring α
f g : ι → MvPowerSeries σ α
hf : StronglySummable f
hg : StronglySummable g
h : f = g
d : σ →₀ ℕ
⊢ ⋯.toFinset = ⋯.toFinset | Please generate a tactic in lean4 to solve the state.
STATE:
case h
σ : Type u_1
α : Type u_2
inst✝¹ : DecidableEq σ
ι : Type u_3
inst✝ : Semiring α
f g : ι → MvPowerSeries σ α
hf : StronglySummable f
hg : StronglySummable g
h : f = g
d : σ →₀ ℕ
⊢ ∑ i ∈ ⋯.toFinset, (coeff α d) (f i) = ∑ i ∈ ⋯.toFinset, (coeff α d) (g i... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/StronglySummable/Basic.lean | MvPowerSeries.StronglySummable.sum_congr | [308, 1] | [314, 57] | ext i | case h
σ : Type u_1
α : Type u_2
inst✝¹ : DecidableEq σ
ι : Type u_3
inst✝ : Semiring α
f g : ι → MvPowerSeries σ α
hf : StronglySummable f
hg : StronglySummable g
h : f = g
d : σ →₀ ℕ
⊢ ⋯.toFinset = ⋯.toFinset | case h.a
σ : Type u_1
α : Type u_2
inst✝¹ : DecidableEq σ
ι : Type u_3
inst✝ : Semiring α
f g : ι → MvPowerSeries σ α
hf : StronglySummable f
hg : StronglySummable g
h : f = g
d : σ →₀ ℕ
i : ι
⊢ i ∈ ⋯.toFinset ↔ i ∈ ⋯.toFinset | Please generate a tactic in lean4 to solve the state.
STATE:
case h
σ : Type u_1
α : Type u_2
inst✝¹ : DecidableEq σ
ι : Type u_3
inst✝ : Semiring α
f g : ι → MvPowerSeries σ α
hf : StronglySummable f
hg : StronglySummable g
h : f = g
d : σ →₀ ℕ
⊢ ⋯.toFinset = ⋯.toFinset
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/StronglySummable/Basic.lean | MvPowerSeries.StronglySummable.sum_congr | [308, 1] | [314, 57] | simp only [Finite.mem_toFinset, mem_support, ne_eq, h] | case h.a
σ : Type u_1
α : Type u_2
inst✝¹ : DecidableEq σ
ι : Type u_3
inst✝ : Semiring α
f g : ι → MvPowerSeries σ α
hf : StronglySummable f
hg : StronglySummable g
h : f = g
d : σ →₀ ℕ
i : ι
⊢ i ∈ ⋯.toFinset ↔ i ∈ ⋯.toFinset | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.a
σ : Type u_1
α : Type u_2
inst✝¹ : DecidableEq σ
ι : Type u_3
inst✝ : Semiring α
f g : ι → MvPowerSeries σ α
hf : StronglySummable f
hg : StronglySummable g
h : f = g
d : σ →₀ ℕ
i : ι
⊢ i ∈ ⋯.toFinset ↔ i ∈ ⋯.toFinset
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/StronglySummable/Basic.lean | MvPowerSeries.StronglySummable.sum_congr | [308, 1] | [314, 57] | rw [h] | σ : Type u_1
α : Type u_2
inst✝¹ : DecidableEq σ
ι : Type u_3
inst✝ : Semiring α
f g : ι → MvPowerSeries σ α
hf : StronglySummable f
hg : StronglySummable g
h : f = g
d : σ →₀ ℕ
i : ι
x✝ : i ∈ ⋯.toFinset
⊢ (coeff α d) (f i) = (coeff α d) (g i) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
σ : Type u_1
α : Type u_2
inst✝¹ : DecidableEq σ
ι : Type u_3
inst✝ : Semiring α
f g : ι → MvPowerSeries σ α
hf : StronglySummable f
hg : StronglySummable g
h : f = g
d : σ →₀ ℕ
i : ι
x✝ : i ∈ ⋯.toFinset
⊢ (coeff α d) (f i) = (coeff α d) (g i)
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/StronglySummable/Basic.lean | MvPowerSeries.StronglySummable.sum_add | [320, 1] | [329, 94] | classical
ext d
simp only [coeff_sum, Pi.add_apply, map_add]
rw [coeff_sum d (support_add hf hg d), coeff_sum d, coeff_sum d]
simp only [Pi.add_apply, map_add, Finset.union_assoc]
rw [sum_add_distrib]
all_goals
simp only [coe_union, Finite.coe_toFinset, Set.subset_union_right, Set.subset_union_left] | σ : Type u_1
α : Type u_2
inst✝¹ : DecidableEq σ
ι : Type u_3
inst✝ : Semiring α
f g : ι → MvPowerSeries σ α
hf : StronglySummable f
hg : StronglySummable g
⊢ ⋯.sum = hf.sum + hg.sum | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
σ : Type u_1
α : Type u_2
inst✝¹ : DecidableEq σ
ι : Type u_3
inst✝ : Semiring α
f g : ι → MvPowerSeries σ α
hf : StronglySummable f
hg : StronglySummable g
⊢ ⋯.sum = hf.sum + hg.sum
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/StronglySummable/Basic.lean | MvPowerSeries.StronglySummable.sum_add | [320, 1] | [329, 94] | ext d | σ : Type u_1
α : Type u_2
inst✝¹ : DecidableEq σ
ι : Type u_3
inst✝ : Semiring α
f g : ι → MvPowerSeries σ α
hf : StronglySummable f
hg : StronglySummable g
⊢ ⋯.sum = hf.sum + hg.sum | case h
σ : Type u_1
α : Type u_2
inst✝¹ : DecidableEq σ
ι : Type u_3
inst✝ : Semiring α
f g : ι → MvPowerSeries σ α
hf : StronglySummable f
hg : StronglySummable g
d : σ →₀ ℕ
⊢ (coeff α d) ⋯.sum = (coeff α d) (hf.sum + hg.sum) | Please generate a tactic in lean4 to solve the state.
STATE:
σ : Type u_1
α : Type u_2
inst✝¹ : DecidableEq σ
ι : Type u_3
inst✝ : Semiring α
f g : ι → MvPowerSeries σ α
hf : StronglySummable f
hg : StronglySummable g
⊢ ⋯.sum = hf.sum + hg.sum
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/StronglySummable/Basic.lean | MvPowerSeries.StronglySummable.sum_add | [320, 1] | [329, 94] | simp only [coeff_sum, Pi.add_apply, map_add] | case h
σ : Type u_1
α : Type u_2
inst✝¹ : DecidableEq σ
ι : Type u_3
inst✝ : Semiring α
f g : ι → MvPowerSeries σ α
hf : StronglySummable f
hg : StronglySummable g
d : σ →₀ ℕ
⊢ (coeff α d) ⋯.sum = (coeff α d) (hf.sum + hg.sum) | case h
σ : Type u_1
α : Type u_2
inst✝¹ : DecidableEq σ
ι : Type u_3
inst✝ : Semiring α
f g : ι → MvPowerSeries σ α
hf : StronglySummable f
hg : StronglySummable g
d : σ →₀ ℕ
⊢ (coeff α d) ⋯.sum = (coeff α d) hf.sum + (coeff α d) hg.sum | Please generate a tactic in lean4 to solve the state.
STATE:
case h
σ : Type u_1
α : Type u_2
inst✝¹ : DecidableEq σ
ι : Type u_3
inst✝ : Semiring α
f g : ι → MvPowerSeries σ α
hf : StronglySummable f
hg : StronglySummable g
d : σ →₀ ℕ
⊢ (coeff α d) ⋯.sum = (coeff α d) (hf.sum + hg.sum)
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/StronglySummable/Basic.lean | MvPowerSeries.StronglySummable.sum_add | [320, 1] | [329, 94] | rw [coeff_sum d (support_add hf hg d), coeff_sum d, coeff_sum d] | case h
σ : Type u_1
α : Type u_2
inst✝¹ : DecidableEq σ
ι : Type u_3
inst✝ : Semiring α
f g : ι → MvPowerSeries σ α
hf : StronglySummable f
hg : StronglySummable g
d : σ →₀ ℕ
⊢ (coeff α d) ⋯.sum = (coeff α d) hf.sum + (coeff α d) hg.sum | case h
σ : Type u_1
α : Type u_2
inst✝¹ : DecidableEq σ
ι : Type u_3
inst✝ : Semiring α
f g : ι → MvPowerSeries σ α
hf : StronglySummable f
hg : StronglySummable g
d : σ →₀ ℕ
⊢ ∑ i ∈ ⋯.toFinset ∪ ⋯.toFinset, (coeff α d) ((f + g) i) =
∑ i ∈ ?m.48819, (coeff α d) (f i) + ∑ i ∈ ?m.48941, (coeff α d) (g i)
case h
σ : ... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
σ : Type u_1
α : Type u_2
inst✝¹ : DecidableEq σ
ι : Type u_3
inst✝ : Semiring α
f g : ι → MvPowerSeries σ α
hf : StronglySummable f
hg : StronglySummable g
d : σ →₀ ℕ
⊢ (coeff α d) ⋯.sum = (coeff α d) hf.sum + (coeff α d) hg.sum
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/StronglySummable/Basic.lean | MvPowerSeries.StronglySummable.sum_add | [320, 1] | [329, 94] | simp only [Pi.add_apply, map_add, Finset.union_assoc] | case h
σ : Type u_1
α : Type u_2
inst✝¹ : DecidableEq σ
ι : Type u_3
inst✝ : Semiring α
f g : ι → MvPowerSeries σ α
hf : StronglySummable f
hg : StronglySummable g
d : σ →₀ ℕ
⊢ ∑ i ∈ ⋯.toFinset ∪ ⋯.toFinset, (coeff α d) ((f + g) i) =
∑ i ∈ ?m.48819, (coeff α d) (f i) + ∑ i ∈ ?m.48941, (coeff α d) (g i)
case h
σ : ... | case h
σ : Type u_1
α : Type u_2
inst✝¹ : DecidableEq σ
ι : Type u_3
inst✝ : Semiring α
f g : ι → MvPowerSeries σ α
hf : StronglySummable f
hg : StronglySummable g
d : σ →₀ ℕ
⊢ ∑ x ∈ ⋯.toFinset ∪ ⋯.toFinset, ((coeff α d) (f x) + (coeff α d) (g x)) =
∑ i ∈ ?m.48819, (coeff α d) (f i) + ∑ i ∈ ?m.48941, (coeff α d) (g... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
σ : Type u_1
α : Type u_2
inst✝¹ : DecidableEq σ
ι : Type u_3
inst✝ : Semiring α
f g : ι → MvPowerSeries σ α
hf : StronglySummable f
hg : StronglySummable g
d : σ →₀ ℕ
⊢ ∑ i ∈ ⋯.toFinset ∪ ⋯.toFinset, (coeff α d) ((f + g) i) =
∑ i ∈ ?m.48819, (coef... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/StronglySummable/Basic.lean | MvPowerSeries.StronglySummable.sum_add | [320, 1] | [329, 94] | rw [sum_add_distrib] | case h
σ : Type u_1
α : Type u_2
inst✝¹ : DecidableEq σ
ι : Type u_3
inst✝ : Semiring α
f g : ι → MvPowerSeries σ α
hf : StronglySummable f
hg : StronglySummable g
d : σ →₀ ℕ
⊢ ∑ x ∈ ⋯.toFinset ∪ ⋯.toFinset, ((coeff α d) (f x) + (coeff α d) (g x)) =
∑ i ∈ ?m.48819, (coeff α d) (f i) + ∑ i ∈ ?m.48941, (coeff α d) (g... | case h
σ : Type u_1
α : Type u_2
inst✝¹ : DecidableEq σ
ι : Type u_3
inst✝ : Semiring α
f g : ι → MvPowerSeries σ α
hf : StronglySummable f
hg : StronglySummable g
d : σ →₀ ℕ
⊢ (support fun i => (coeff α d) (g i)) ⊆ ↑(⋯.toFinset ∪ ⋯.toFinset)
case h
σ : Type u_1
α : Type u_2
inst✝¹ : DecidableEq σ
ι : Type u_3
inst✝ :... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
σ : Type u_1
α : Type u_2
inst✝¹ : DecidableEq σ
ι : Type u_3
inst✝ : Semiring α
f g : ι → MvPowerSeries σ α
hf : StronglySummable f
hg : StronglySummable g
d : σ →₀ ℕ
⊢ ∑ x ∈ ⋯.toFinset ∪ ⋯.toFinset, ((coeff α d) (f x) + (coeff α d) (g x)) =
∑ i ∈... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/StronglySummable/Basic.lean | MvPowerSeries.StronglySummable.sum_add | [320, 1] | [329, 94] | all_goals
simp only [coe_union, Finite.coe_toFinset, Set.subset_union_right, Set.subset_union_left] | case h
σ : Type u_1
α : Type u_2
inst✝¹ : DecidableEq σ
ι : Type u_3
inst✝ : Semiring α
f g : ι → MvPowerSeries σ α
hf : StronglySummable f
hg : StronglySummable g
d : σ →₀ ℕ
⊢ (support fun i => (coeff α d) (g i)) ⊆ ↑(⋯.toFinset ∪ ⋯.toFinset)
case h
σ : Type u_1
α : Type u_2
inst✝¹ : DecidableEq σ
ι : Type u_3
inst✝ :... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
σ : Type u_1
α : Type u_2
inst✝¹ : DecidableEq σ
ι : Type u_3
inst✝ : Semiring α
f g : ι → MvPowerSeries σ α
hf : StronglySummable f
hg : StronglySummable g
d : σ →₀ ℕ
⊢ (support fun i => (coeff α d) (g i)) ⊆ ↑(⋯.toFinset ∪ ⋯.toFinset)
case h
σ : Type... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/StronglySummable/Basic.lean | MvPowerSeries.StronglySummable.sum_add | [320, 1] | [329, 94] | simp only [coe_union, Finite.coe_toFinset, Set.subset_union_right, Set.subset_union_left] | case h
σ : Type u_1
α : Type u_2
inst✝¹ : DecidableEq σ
ι : Type u_3
inst✝ : Semiring α
f g : ι → MvPowerSeries σ α
hf : StronglySummable f
hg : StronglySummable g
d : σ →₀ ℕ
⊢ (support fun i => (coeff α d) (f i)) ⊆ ↑(⋯.toFinset ∪ ⋯.toFinset) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
σ : Type u_1
α : Type u_2
inst✝¹ : DecidableEq σ
ι : Type u_3
inst✝ : Semiring α
f g : ι → MvPowerSeries σ α
hf : StronglySummable f
hg : StronglySummable g
d : σ →₀ ℕ
⊢ (support fun i => (coeff α d) (f i)) ⊆ ↑(⋯.toFinset ∪ ⋯.toFinset)
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/StronglySummable/Basic.lean | MvPowerSeries.StronglySummable.sum_add' | [332, 1] | [341, 94] | classical
ext d
simp only [coeff_sum, Pi.add_apply, map_add]
rw [coeff_sum d (support_add hf hg d), coeff_sum d, coeff_sum d]
simp only [Pi.add_apply, map_add, Finset.union_assoc]
rw [sum_add_distrib]
all_goals
simp only [coe_union, Finite.coe_toFinset, Set.subset_union_right, Set.subset_union_left] | σ : Type u_1
α : Type u_2
inst✝¹ : DecidableEq σ
ι : Type u_3
inst✝ : Semiring α
f g : ι → MvPowerSeries σ α
hf : StronglySummable f
hg : StronglySummable g
hh : StronglySummable (f + g)
⊢ hh.sum = hf.sum + hg.sum | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
σ : Type u_1
α : Type u_2
inst✝¹ : DecidableEq σ
ι : Type u_3
inst✝ : Semiring α
f g : ι → MvPowerSeries σ α
hf : StronglySummable f
hg : StronglySummable g
hh : StronglySummable (f + g)
⊢ hh.sum = hf.sum + hg.sum
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/StronglySummable/Basic.lean | MvPowerSeries.StronglySummable.sum_add' | [332, 1] | [341, 94] | ext d | σ : Type u_1
α : Type u_2
inst✝¹ : DecidableEq σ
ι : Type u_3
inst✝ : Semiring α
f g : ι → MvPowerSeries σ α
hf : StronglySummable f
hg : StronglySummable g
hh : StronglySummable (f + g)
⊢ hh.sum = hf.sum + hg.sum | case h
σ : Type u_1
α : Type u_2
inst✝¹ : DecidableEq σ
ι : Type u_3
inst✝ : Semiring α
f g : ι → MvPowerSeries σ α
hf : StronglySummable f
hg : StronglySummable g
hh : StronglySummable (f + g)
d : σ →₀ ℕ
⊢ (coeff α d) hh.sum = (coeff α d) (hf.sum + hg.sum) | Please generate a tactic in lean4 to solve the state.
STATE:
σ : Type u_1
α : Type u_2
inst✝¹ : DecidableEq σ
ι : Type u_3
inst✝ : Semiring α
f g : ι → MvPowerSeries σ α
hf : StronglySummable f
hg : StronglySummable g
hh : StronglySummable (f + g)
⊢ hh.sum = hf.sum + hg.sum
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/StronglySummable/Basic.lean | MvPowerSeries.StronglySummable.sum_add' | [332, 1] | [341, 94] | simp only [coeff_sum, Pi.add_apply, map_add] | case h
σ : Type u_1
α : Type u_2
inst✝¹ : DecidableEq σ
ι : Type u_3
inst✝ : Semiring α
f g : ι → MvPowerSeries σ α
hf : StronglySummable f
hg : StronglySummable g
hh : StronglySummable (f + g)
d : σ →₀ ℕ
⊢ (coeff α d) hh.sum = (coeff α d) (hf.sum + hg.sum) | case h
σ : Type u_1
α : Type u_2
inst✝¹ : DecidableEq σ
ι : Type u_3
inst✝ : Semiring α
f g : ι → MvPowerSeries σ α
hf : StronglySummable f
hg : StronglySummable g
hh : StronglySummable (f + g)
d : σ →₀ ℕ
⊢ (coeff α d) hh.sum = (coeff α d) hf.sum + (coeff α d) hg.sum | Please generate a tactic in lean4 to solve the state.
STATE:
case h
σ : Type u_1
α : Type u_2
inst✝¹ : DecidableEq σ
ι : Type u_3
inst✝ : Semiring α
f g : ι → MvPowerSeries σ α
hf : StronglySummable f
hg : StronglySummable g
hh : StronglySummable (f + g)
d : σ →₀ ℕ
⊢ (coeff α d) hh.sum = (coeff α d) (hf.sum + hg.sum)
T... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/StronglySummable/Basic.lean | MvPowerSeries.StronglySummable.sum_add' | [332, 1] | [341, 94] | rw [coeff_sum d (support_add hf hg d), coeff_sum d, coeff_sum d] | case h
σ : Type u_1
α : Type u_2
inst✝¹ : DecidableEq σ
ι : Type u_3
inst✝ : Semiring α
f g : ι → MvPowerSeries σ α
hf : StronglySummable f
hg : StronglySummable g
hh : StronglySummable (f + g)
d : σ →₀ ℕ
⊢ (coeff α d) hh.sum = (coeff α d) hf.sum + (coeff α d) hg.sum | case h
σ : Type u_1
α : Type u_2
inst✝¹ : DecidableEq σ
ι : Type u_3
inst✝ : Semiring α
f g : ι → MvPowerSeries σ α
hf : StronglySummable f
hg : StronglySummable g
hh : StronglySummable (f + g)
d : σ →₀ ℕ
⊢ ∑ i ∈ ⋯.toFinset ∪ ⋯.toFinset, (coeff α d) ((f + g) i) =
∑ i ∈ ?m.52863, (coeff α d) (f i) + ∑ i ∈ ?m.52985, ... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
σ : Type u_1
α : Type u_2
inst✝¹ : DecidableEq σ
ι : Type u_3
inst✝ : Semiring α
f g : ι → MvPowerSeries σ α
hf : StronglySummable f
hg : StronglySummable g
hh : StronglySummable (f + g)
d : σ →₀ ℕ
⊢ (coeff α d) hh.sum = (coeff α d) hf.sum + (coeff α d... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/StronglySummable/Basic.lean | MvPowerSeries.StronglySummable.sum_add' | [332, 1] | [341, 94] | simp only [Pi.add_apply, map_add, Finset.union_assoc] | case h
σ : Type u_1
α : Type u_2
inst✝¹ : DecidableEq σ
ι : Type u_3
inst✝ : Semiring α
f g : ι → MvPowerSeries σ α
hf : StronglySummable f
hg : StronglySummable g
hh : StronglySummable (f + g)
d : σ →₀ ℕ
⊢ ∑ i ∈ ⋯.toFinset ∪ ⋯.toFinset, (coeff α d) ((f + g) i) =
∑ i ∈ ?m.52863, (coeff α d) (f i) + ∑ i ∈ ?m.52985, ... | case h
σ : Type u_1
α : Type u_2
inst✝¹ : DecidableEq σ
ι : Type u_3
inst✝ : Semiring α
f g : ι → MvPowerSeries σ α
hf : StronglySummable f
hg : StronglySummable g
hh : StronglySummable (f + g)
d : σ →₀ ℕ
⊢ ∑ x ∈ ⋯.toFinset ∪ ⋯.toFinset, ((coeff α d) (f x) + (coeff α d) (g x)) =
∑ i ∈ ?m.52863, (coeff α d) (f i) + ... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
σ : Type u_1
α : Type u_2
inst✝¹ : DecidableEq σ
ι : Type u_3
inst✝ : Semiring α
f g : ι → MvPowerSeries σ α
hf : StronglySummable f
hg : StronglySummable g
hh : StronglySummable (f + g)
d : σ →₀ ℕ
⊢ ∑ i ∈ ⋯.toFinset ∪ ⋯.toFinset, (coeff α d) ((f + g) ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/StronglySummable/Basic.lean | MvPowerSeries.StronglySummable.sum_add' | [332, 1] | [341, 94] | rw [sum_add_distrib] | case h
σ : Type u_1
α : Type u_2
inst✝¹ : DecidableEq σ
ι : Type u_3
inst✝ : Semiring α
f g : ι → MvPowerSeries σ α
hf : StronglySummable f
hg : StronglySummable g
hh : StronglySummable (f + g)
d : σ →₀ ℕ
⊢ ∑ x ∈ ⋯.toFinset ∪ ⋯.toFinset, ((coeff α d) (f x) + (coeff α d) (g x)) =
∑ i ∈ ?m.52863, (coeff α d) (f i) + ... | case h
σ : Type u_1
α : Type u_2
inst✝¹ : DecidableEq σ
ι : Type u_3
inst✝ : Semiring α
f g : ι → MvPowerSeries σ α
hf : StronglySummable f
hg : StronglySummable g
hh : StronglySummable (f + g)
d : σ →₀ ℕ
⊢ (support fun i => (coeff α d) (g i)) ⊆ ↑(⋯.toFinset ∪ ⋯.toFinset)
case h
σ : Type u_1
α : Type u_2
inst✝¹ : Deci... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
σ : Type u_1
α : Type u_2
inst✝¹ : DecidableEq σ
ι : Type u_3
inst✝ : Semiring α
f g : ι → MvPowerSeries σ α
hf : StronglySummable f
hg : StronglySummable g
hh : StronglySummable (f + g)
d : σ →₀ ℕ
⊢ ∑ x ∈ ⋯.toFinset ∪ ⋯.toFinset, ((coeff α d) (f x) + ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/StronglySummable/Basic.lean | MvPowerSeries.StronglySummable.sum_add' | [332, 1] | [341, 94] | all_goals
simp only [coe_union, Finite.coe_toFinset, Set.subset_union_right, Set.subset_union_left] | case h
σ : Type u_1
α : Type u_2
inst✝¹ : DecidableEq σ
ι : Type u_3
inst✝ : Semiring α
f g : ι → MvPowerSeries σ α
hf : StronglySummable f
hg : StronglySummable g
hh : StronglySummable (f + g)
d : σ →₀ ℕ
⊢ (support fun i => (coeff α d) (g i)) ⊆ ↑(⋯.toFinset ∪ ⋯.toFinset)
case h
σ : Type u_1
α : Type u_2
inst✝¹ : Deci... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
σ : Type u_1
α : Type u_2
inst✝¹ : DecidableEq σ
ι : Type u_3
inst✝ : Semiring α
f g : ι → MvPowerSeries σ α
hf : StronglySummable f
hg : StronglySummable g
hh : StronglySummable (f + g)
d : σ →₀ ℕ
⊢ (support fun i => (coeff α d) (g i)) ⊆ ↑(⋯.toFinset ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/StronglySummable/Basic.lean | MvPowerSeries.StronglySummable.sum_add' | [332, 1] | [341, 94] | simp only [coe_union, Finite.coe_toFinset, Set.subset_union_right, Set.subset_union_left] | case h
σ : Type u_1
α : Type u_2
inst✝¹ : DecidableEq σ
ι : Type u_3
inst✝ : Semiring α
f g : ι → MvPowerSeries σ α
hf : StronglySummable f
hg : StronglySummable g
hh : StronglySummable (f + g)
d : σ →₀ ℕ
⊢ (support fun i => (coeff α d) (f i)) ⊆ ↑(⋯.toFinset ∪ ⋯.toFinset) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
σ : Type u_1
α : Type u_2
inst✝¹ : DecidableEq σ
ι : Type u_3
inst✝ : Semiring α
f g : ι → MvPowerSeries σ α
hf : StronglySummable f
hg : StronglySummable g
hh : StronglySummable (f + g)
d : σ →₀ ℕ
⊢ (support fun i => (coeff α d) (f i)) ⊆ ↑(⋯.toFinset ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/StronglySummable/Basic.lean | MvPowerSeries.StronglySummable.sum_mul | [346, 1] | [360, 41] | classical
ext d
simp_rw [coeff_sum d (support_mul hf hg d), coeff_mul]
rw [sum_comm]
apply Finset.sum_congr rfl
rintro bc hbc
erw [Finset.sum_product]
rw [coeff_sum _, coeff_sum _, sum_mul_sum]
all_goals
intro x hx
apply @Finset.subset_biUnion_of_mem _ _ _ _ _ bc hbc
exact (Finite.mem_toFinset _).mpr hx | σ : Type u_1
α : Type u_2
inst✝¹ : DecidableEq σ
ι : Type u_4
inst✝ : Semiring α
f : ι → MvPowerSeries σ α
κ : Type u_3
g : κ → MvPowerSeries σ α
hf : StronglySummable f
hg : StronglySummable g
⊢ ⋯.sum = hf.sum * hg.sum | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
σ : Type u_1
α : Type u_2
inst✝¹ : DecidableEq σ
ι : Type u_4
inst✝ : Semiring α
f : ι → MvPowerSeries σ α
κ : Type u_3
g : κ → MvPowerSeries σ α
hf : StronglySummable f
hg : StronglySummable g
⊢ ⋯.sum = hf.sum * hg.sum
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/StronglySummable/Basic.lean | MvPowerSeries.StronglySummable.sum_mul | [346, 1] | [360, 41] | ext d | σ : Type u_1
α : Type u_2
inst✝¹ : DecidableEq σ
ι : Type u_4
inst✝ : Semiring α
f : ι → MvPowerSeries σ α
κ : Type u_3
g : κ → MvPowerSeries σ α
hf : StronglySummable f
hg : StronglySummable g
⊢ ⋯.sum = hf.sum * hg.sum | case h
σ : Type u_1
α : Type u_2
inst✝¹ : DecidableEq σ
ι : Type u_4
inst✝ : Semiring α
f : ι → MvPowerSeries σ α
κ : Type u_3
g : κ → MvPowerSeries σ α
hf : StronglySummable f
hg : StronglySummable g
d : σ →₀ ℕ
⊢ (coeff α d) ⋯.sum = (coeff α d) (hf.sum * hg.sum) | Please generate a tactic in lean4 to solve the state.
STATE:
σ : Type u_1
α : Type u_2
inst✝¹ : DecidableEq σ
ι : Type u_4
inst✝ : Semiring α
f : ι → MvPowerSeries σ α
κ : Type u_3
g : κ → MvPowerSeries σ α
hf : StronglySummable f
hg : StronglySummable g
⊢ ⋯.sum = hf.sum * hg.sum
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/StronglySummable/Basic.lean | MvPowerSeries.StronglySummable.sum_mul | [346, 1] | [360, 41] | simp_rw [coeff_sum d (support_mul hf hg d), coeff_mul] | case h
σ : Type u_1
α : Type u_2
inst✝¹ : DecidableEq σ
ι : Type u_4
inst✝ : Semiring α
f : ι → MvPowerSeries σ α
κ : Type u_3
g : κ → MvPowerSeries σ α
hf : StronglySummable f
hg : StronglySummable g
d : σ →₀ ℕ
⊢ (coeff α d) ⋯.sum = (coeff α d) (hf.sum * hg.sum) | case h
σ : Type u_1
α : Type u_2
inst✝¹ : DecidableEq σ
ι : Type u_4
inst✝ : Semiring α
f : ι → MvPowerSeries σ α
κ : Type u_3
g : κ → MvPowerSeries σ α
hf : StronglySummable f
hg : StronglySummable g
d : σ →₀ ℕ
⊢ ∑ x ∈ ((antidiagonal d).biUnion fun b => ⋯.toFinset).product ((antidiagonal d).biUnion fun b => ⋯.toFinset... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
σ : Type u_1
α : Type u_2
inst✝¹ : DecidableEq σ
ι : Type u_4
inst✝ : Semiring α
f : ι → MvPowerSeries σ α
κ : Type u_3
g : κ → MvPowerSeries σ α
hf : StronglySummable f
hg : StronglySummable g
d : σ →₀ ℕ
⊢ (coeff α d) ⋯.sum = (coeff α d) (hf.sum * hg.... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/StronglySummable/Basic.lean | MvPowerSeries.StronglySummable.sum_mul | [346, 1] | [360, 41] | rw [sum_comm] | case h
σ : Type u_1
α : Type u_2
inst✝¹ : DecidableEq σ
ι : Type u_4
inst✝ : Semiring α
f : ι → MvPowerSeries σ α
κ : Type u_3
g : κ → MvPowerSeries σ α
hf : StronglySummable f
hg : StronglySummable g
d : σ →₀ ℕ
⊢ ∑ x ∈ ((antidiagonal d).biUnion fun b => ⋯.toFinset).product ((antidiagonal d).biUnion fun b => ⋯.toFinset... | case h
σ : Type u_1
α : Type u_2
inst✝¹ : DecidableEq σ
ι : Type u_4
inst✝ : Semiring α
f : ι → MvPowerSeries σ α
κ : Type u_3
g : κ → MvPowerSeries σ α
hf : StronglySummable f
hg : StronglySummable g
d : σ →₀ ℕ
⊢ ∑ y ∈ antidiagonal d,
∑ x ∈ ((antidiagonal d).biUnion fun b => ⋯.toFinset).product ((antidiagonal d)... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
σ : Type u_1
α : Type u_2
inst✝¹ : DecidableEq σ
ι : Type u_4
inst✝ : Semiring α
f : ι → MvPowerSeries σ α
κ : Type u_3
g : κ → MvPowerSeries σ α
hf : StronglySummable f
hg : StronglySummable g
d : σ →₀ ℕ
⊢ ∑ x ∈ ((antidiagonal d).biUnion fun b => ⋯.to... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/StronglySummable/Basic.lean | MvPowerSeries.StronglySummable.sum_mul | [346, 1] | [360, 41] | apply Finset.sum_congr rfl | case h
σ : Type u_1
α : Type u_2
inst✝¹ : DecidableEq σ
ι : Type u_4
inst✝ : Semiring α
f : ι → MvPowerSeries σ α
κ : Type u_3
g : κ → MvPowerSeries σ α
hf : StronglySummable f
hg : StronglySummable g
d : σ →₀ ℕ
⊢ ∑ y ∈ antidiagonal d,
∑ x ∈ ((antidiagonal d).biUnion fun b => ⋯.toFinset).product ((antidiagonal d)... | case h
σ : Type u_1
α : Type u_2
inst✝¹ : DecidableEq σ
ι : Type u_4
inst✝ : Semiring α
f : ι → MvPowerSeries σ α
κ : Type u_3
g : κ → MvPowerSeries σ α
hf : StronglySummable f
hg : StronglySummable g
d : σ →₀ ℕ
⊢ ∀ x ∈ antidiagonal d,
∑ x_1 ∈ ((antidiagonal d).biUnion fun b => ⋯.toFinset).product ((antidiagonal d)... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
σ : Type u_1
α : Type u_2
inst✝¹ : DecidableEq σ
ι : Type u_4
inst✝ : Semiring α
f : ι → MvPowerSeries σ α
κ : Type u_3
g : κ → MvPowerSeries σ α
hf : StronglySummable f
hg : StronglySummable g
d : σ →₀ ℕ
⊢ ∑ y ∈ antidiagonal d,
∑ x ∈ ((antidiago... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/StronglySummable/Basic.lean | MvPowerSeries.StronglySummable.sum_mul | [346, 1] | [360, 41] | rintro bc hbc | case h
σ : Type u_1
α : Type u_2
inst✝¹ : DecidableEq σ
ι : Type u_4
inst✝ : Semiring α
f : ι → MvPowerSeries σ α
κ : Type u_3
g : κ → MvPowerSeries σ α
hf : StronglySummable f
hg : StronglySummable g
d : σ →₀ ℕ
⊢ ∀ x ∈ antidiagonal d,
∑ x_1 ∈ ((antidiagonal d).biUnion fun b => ⋯.toFinset).product ((antidiagonal d)... | case h
σ : Type u_1
α : Type u_2
inst✝¹ : DecidableEq σ
ι : Type u_4
inst✝ : Semiring α
f : ι → MvPowerSeries σ α
κ : Type u_3
g : κ → MvPowerSeries σ α
hf : StronglySummable f
hg : StronglySummable g
d : σ →₀ ℕ
bc : (σ →₀ ℕ) × (σ →₀ ℕ)
hbc : bc ∈ antidiagonal d
⊢ ∑ x ∈ ((antidiagonal d).biUnion fun b => ⋯.toFinset).pr... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
σ : Type u_1
α : Type u_2
inst✝¹ : DecidableEq σ
ι : Type u_4
inst✝ : Semiring α
f : ι → MvPowerSeries σ α
κ : Type u_3
g : κ → MvPowerSeries σ α
hf : StronglySummable f
hg : StronglySummable g
d : σ →₀ ℕ
⊢ ∀ x ∈ antidiagonal d,
∑ x_1 ∈ ((antidiago... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/StronglySummable/Basic.lean | MvPowerSeries.StronglySummable.sum_mul | [346, 1] | [360, 41] | erw [Finset.sum_product] | case h
σ : Type u_1
α : Type u_2
inst✝¹ : DecidableEq σ
ι : Type u_4
inst✝ : Semiring α
f : ι → MvPowerSeries σ α
κ : Type u_3
g : κ → MvPowerSeries σ α
hf : StronglySummable f
hg : StronglySummable g
d : σ →₀ ℕ
bc : (σ →₀ ℕ) × (σ →₀ ℕ)
hbc : bc ∈ antidiagonal d
⊢ ∑ x ∈ ((antidiagonal d).biUnion fun b => ⋯.toFinset).pr... | case h
σ : Type u_1
α : Type u_2
inst✝¹ : DecidableEq σ
ι : Type u_4
inst✝ : Semiring α
f : ι → MvPowerSeries σ α
κ : Type u_3
g : κ → MvPowerSeries σ α
hf : StronglySummable f
hg : StronglySummable g
d : σ →₀ ℕ
bc : (σ →₀ ℕ) × (σ →₀ ℕ)
hbc : bc ∈ antidiagonal d
⊢ ∑ x ∈ (antidiagonal d).biUnion fun b => ⋯.toFinset,
... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
σ : Type u_1
α : Type u_2
inst✝¹ : DecidableEq σ
ι : Type u_4
inst✝ : Semiring α
f : ι → MvPowerSeries σ α
κ : Type u_3
g : κ → MvPowerSeries σ α
hf : StronglySummable f
hg : StronglySummable g
d : σ →₀ ℕ
bc : (σ →₀ ℕ) × (σ →₀ ℕ)
hbc : bc ∈ antidiagona... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/StronglySummable/Basic.lean | MvPowerSeries.StronglySummable.sum_mul | [346, 1] | [360, 41] | rw [coeff_sum _, coeff_sum _, sum_mul_sum] | case h
σ : Type u_1
α : Type u_2
inst✝¹ : DecidableEq σ
ι : Type u_4
inst✝ : Semiring α
f : ι → MvPowerSeries σ α
κ : Type u_3
g : κ → MvPowerSeries σ α
hf : StronglySummable f
hg : StronglySummable g
d : σ →₀ ℕ
bc : (σ →₀ ℕ) × (σ →₀ ℕ)
hbc : bc ∈ antidiagonal d
⊢ ∑ x ∈ (antidiagonal d).biUnion fun b => ⋯.toFinset,
... | case h
σ : Type u_1
α : Type u_2
inst✝¹ : DecidableEq σ
ι : Type u_4
inst✝ : Semiring α
f : ι → MvPowerSeries σ α
κ : Type u_3
g : κ → MvPowerSeries σ α
hf : StronglySummable f
hg : StronglySummable g
d : σ →₀ ℕ
bc : (σ →₀ ℕ) × (σ →₀ ℕ)
hbc : bc ∈ antidiagonal d
⊢ (support fun i => (coeff α bc.2) (g i)) ⊆ ↑((antidiagon... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
σ : Type u_1
α : Type u_2
inst✝¹ : DecidableEq σ
ι : Type u_4
inst✝ : Semiring α
f : ι → MvPowerSeries σ α
κ : Type u_3
g : κ → MvPowerSeries σ α
hf : StronglySummable f
hg : StronglySummable g
d : σ →₀ ℕ
bc : (σ →₀ ℕ) × (σ →₀ ℕ)
hbc : bc ∈ antidiagona... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/StronglySummable/Basic.lean | MvPowerSeries.StronglySummable.sum_mul | [346, 1] | [360, 41] | all_goals
intro x hx
apply @Finset.subset_biUnion_of_mem _ _ _ _ _ bc hbc
exact (Finite.mem_toFinset _).mpr hx | case h
σ : Type u_1
α : Type u_2
inst✝¹ : DecidableEq σ
ι : Type u_4
inst✝ : Semiring α
f : ι → MvPowerSeries σ α
κ : Type u_3
g : κ → MvPowerSeries σ α
hf : StronglySummable f
hg : StronglySummable g
d : σ →₀ ℕ
bc : (σ →₀ ℕ) × (σ →₀ ℕ)
hbc : bc ∈ antidiagonal d
⊢ (support fun i => (coeff α bc.2) (g i)) ⊆ ↑((antidiagon... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
σ : Type u_1
α : Type u_2
inst✝¹ : DecidableEq σ
ι : Type u_4
inst✝ : Semiring α
f : ι → MvPowerSeries σ α
κ : Type u_3
g : κ → MvPowerSeries σ α
hf : StronglySummable f
hg : StronglySummable g
d : σ →₀ ℕ
bc : (σ →₀ ℕ) × (σ →₀ ℕ)
hbc : bc ∈ antidiagona... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/StronglySummable/Basic.lean | MvPowerSeries.StronglySummable.sum_mul | [346, 1] | [360, 41] | intro x hx | case h
σ : Type u_1
α : Type u_2
inst✝¹ : DecidableEq σ
ι : Type u_4
inst✝ : Semiring α
f : ι → MvPowerSeries σ α
κ : Type u_3
g : κ → MvPowerSeries σ α
hf : StronglySummable f
hg : StronglySummable g
d : σ →₀ ℕ
bc : (σ →₀ ℕ) × (σ →₀ ℕ)
hbc : bc ∈ antidiagonal d
⊢ (support fun i => (coeff α bc.1) (f i)) ⊆ ↑((antidiagon... | case h
σ : Type u_1
α : Type u_2
inst✝¹ : DecidableEq σ
ι : Type u_4
inst✝ : Semiring α
f : ι → MvPowerSeries σ α
κ : Type u_3
g : κ → MvPowerSeries σ α
hf : StronglySummable f
hg : StronglySummable g
d : σ →₀ ℕ
bc : (σ →₀ ℕ) × (σ →₀ ℕ)
hbc : bc ∈ antidiagonal d
x : ι
hx : x ∈ support fun i => (coeff α bc.1) (f i)
⊢ x ... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
σ : Type u_1
α : Type u_2
inst✝¹ : DecidableEq σ
ι : Type u_4
inst✝ : Semiring α
f : ι → MvPowerSeries σ α
κ : Type u_3
g : κ → MvPowerSeries σ α
hf : StronglySummable f
hg : StronglySummable g
d : σ →₀ ℕ
bc : (σ →₀ ℕ) × (σ →₀ ℕ)
hbc : bc ∈ antidiagona... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/StronglySummable/Basic.lean | MvPowerSeries.StronglySummable.sum_mul | [346, 1] | [360, 41] | apply @Finset.subset_biUnion_of_mem _ _ _ _ _ bc hbc | case h
σ : Type u_1
α : Type u_2
inst✝¹ : DecidableEq σ
ι : Type u_4
inst✝ : Semiring α
f : ι → MvPowerSeries σ α
κ : Type u_3
g : κ → MvPowerSeries σ α
hf : StronglySummable f
hg : StronglySummable g
d : σ →₀ ℕ
bc : (σ →₀ ℕ) × (σ →₀ ℕ)
hbc : bc ∈ antidiagonal d
x : ι
hx : x ∈ support fun i => (coeff α bc.1) (f i)
⊢ x ... | case h.a
σ : Type u_1
α : Type u_2
inst✝¹ : DecidableEq σ
ι : Type u_4
inst✝ : Semiring α
f : ι → MvPowerSeries σ α
κ : Type u_3
g : κ → MvPowerSeries σ α
hf : StronglySummable f
hg : StronglySummable g
d : σ →₀ ℕ
bc : (σ →₀ ℕ) × (σ →₀ ℕ)
hbc : bc ∈ antidiagonal d
x : ι
hx : x ∈ support fun i => (coeff α bc.1) (f i)
⊢ ... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
σ : Type u_1
α : Type u_2
inst✝¹ : DecidableEq σ
ι : Type u_4
inst✝ : Semiring α
f : ι → MvPowerSeries σ α
κ : Type u_3
g : κ → MvPowerSeries σ α
hf : StronglySummable f
hg : StronglySummable g
d : σ →₀ ℕ
bc : (σ →₀ ℕ) × (σ →₀ ℕ)
hbc : bc ∈ antidiagona... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/StronglySummable/Basic.lean | MvPowerSeries.StronglySummable.sum_mul | [346, 1] | [360, 41] | exact (Finite.mem_toFinset _).mpr hx | case h.a
σ : Type u_1
α : Type u_2
inst✝¹ : DecidableEq σ
ι : Type u_4
inst✝ : Semiring α
f : ι → MvPowerSeries σ α
κ : Type u_3
g : κ → MvPowerSeries σ α
hf : StronglySummable f
hg : StronglySummable g
d : σ →₀ ℕ
bc : (σ →₀ ℕ) × (σ →₀ ℕ)
hbc : bc ∈ antidiagonal d
x : ι
hx : x ∈ support fun i => (coeff α bc.1) (f i)
⊢ ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.a
σ : Type u_1
α : Type u_2
inst✝¹ : DecidableEq σ
ι : Type u_4
inst✝ : Semiring α
f : ι → MvPowerSeries σ α
κ : Type u_3
g : κ → MvPowerSeries σ α
hf : StronglySummable f
hg : StronglySummable g
d : σ →₀ ℕ
bc : (σ →₀ ℕ) × (σ →₀ ℕ)
hbc : bc ∈ antidiago... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/StronglySummable/Basic.lean | MvPowerSeries.StronglySummable.sum_mul' | [363, 1] | [367, 17] | rw [← sum_mul] | σ : Type u_1
α : Type u_2
inst✝¹ : DecidableEq σ
ι : Type u_4
inst✝ : Semiring α
f : ι → MvPowerSeries σ α
κ : Type u_3
g : κ → MvPowerSeries σ α
hf : StronglySummable f
hg : StronglySummable g
hh : StronglySummable fun i => f i.1 * g i.2
⊢ hh.sum = hf.sum * hg.sum | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
σ : Type u_1
α : Type u_2
inst✝¹ : DecidableEq σ
ι : Type u_4
inst✝ : Semiring α
f : ι → MvPowerSeries σ α
κ : Type u_3
g : κ → MvPowerSeries σ α
hf : StronglySummable f
hg : StronglySummable g
hh : StronglySummable fun i => f i.1 * g i.2
⊢ hh.sum = hf.sum * ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/StronglySummable/Basic.lean | MvPowerSeries.StronglySummable.of_indicator | [371, 1] | [379, 32] | intro d | σ : Type u_1
α : Type u_2
inst✝¹ : DecidableEq σ
ι : Type u_3
inst✝ : Semiring α
f : ι → MvPowerSeries σ α
hf : StronglySummable f
s : Set ι
⊢ StronglySummable fun i => s.indicator f i | σ : Type u_1
α : Type u_2
inst✝¹ : DecidableEq σ
ι : Type u_3
inst✝ : Semiring α
f : ι → MvPowerSeries σ α
hf : StronglySummable f
s : Set ι
d : σ →₀ ℕ
⊢ (support fun i => (coeff α d) ((fun i => s.indicator f i) i)).Finite | Please generate a tactic in lean4 to solve the state.
STATE:
σ : Type u_1
α : Type u_2
inst✝¹ : DecidableEq σ
ι : Type u_3
inst✝ : Semiring α
f : ι → MvPowerSeries σ α
hf : StronglySummable f
s : Set ι
⊢ StronglySummable fun i => s.indicator f i
TACTIC:
|
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