tasksource/leandojo huggingface dataset:
Theorem: Complex.hasSum_deriv_of_summable_norm
File: Mathlib/Analysis/Complex/LocallyUniformLimit.lean

tactic: rw [HasSum]
state_before: E : Type u_2
ι : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
U K : Set ℂ
z : ℂ
M r δ : ℝ
φ : Filter ι
F : ι → ℂ → E
f g : ℂ → E
u : ι → ℝ
hu : Summable u
hf : ∀ (i : ι), DifferentiableOn ℂ (F i) U
hU : IsOpen U
hF_le : ∀ (i : ι) (w : ℂ), w ∈ U → ‖F i w‖ ≤ u i
hz : z ∈ U
⊢ HasSum (fun i => deriv (F i) z) (deriv (fun w => ∑' (i : ι), F i w) z)

tactic: have hc := (tendstoUniformlyOn_tsum hu hF_le).tendstoLocallyUniformlyOn
state_before: E : Type u_2
ι : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
U K : Set ℂ
z : ℂ
M r δ : ℝ
φ : Filter ι
F : ι → ℂ → E
f g : ℂ → E
u : ι → ℝ
hu : Summable u
hf : ∀ (i : ι), DifferentiableOn ℂ (F i) U
hU : IsOpen U
hF_le : ∀ (i : ι) (w : ℂ), w ∈ U → ‖F i w‖ ≤ u i
hz : z ∈ U
⊢ Tendsto (fun s => Finset.sum s fun b => deriv (F b) z) atTop (𝓝 (deriv (fun w => ∑' (i : ι), F i w) z))

tactic: convert (hc.deriv (eventually_of_forall fun s =>
  DifferentiableOn.sum fun i _ => hf i) hU).tendsto_at hz using 1
state_before: E : Type u_2
ι : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
U K : Set ℂ
z : ℂ
M r δ : ℝ
φ : Filter ι
F : ι → ℂ → E
f g : ℂ → E
u : ι → ℝ
hu : Summable u
hf : ∀ (i : ι), DifferentiableOn ℂ (F i) U
hU : IsOpen U
hF_le : ∀ (i : ι) (w : ℂ), w ∈ U → ‖F i w‖ ≤ u i
hz : z ∈ U
hc : TendstoLocallyUniformlyOn (fun t x => Finset.sum t fun n => F n x) (fun x => ∑' (n : ι), F n x) atTop U
⊢ Tendsto (fun s => Finset.sum s fun b => deriv (F b) z) atTop (𝓝 (deriv (fun w => ∑' (i : ι), F i w) z))

tactic: ext1 s
state_before: case h.e'_3
E : Type u_2
ι : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
U K : Set ℂ
z : ℂ
M r δ : ℝ
φ : Filter ι
F : ι → ℂ → E
f g : ℂ → E
u : ι → ℝ
hu : Summable u
hf : ∀ (i : ι), DifferentiableOn ℂ (F i) U
hU : IsOpen U
hF_le : ∀ (i : ι) (w : ℂ), w ∈ U → ‖F i w‖ ≤ u i
hz : z ∈ U
hc : TendstoLocallyUniformlyOn (fun t x => Finset.sum t fun n => F n x) (fun x => ∑' (n : ι), F n x) atTop U
⊢ (fun s => Finset.sum s fun b => deriv (F b) z) = fun i => (deriv ∘ fun t x => Finset.sum t fun n => F n x) i z

tactic: exact (deriv_sum fun i _ => (hf i).differentiableAt (hU.mem_nhds hz)).symm
state_before: case h.e'_3.h
E : Type u_2
ι : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
U K : Set ℂ
z : ℂ
M r δ : ℝ
φ : Filter ι
F : ι → ℂ → E
f g : ℂ → E
u : ι → ℝ
hu : Summable u
hf : ∀ (i : ι), DifferentiableOn ℂ (F i) U
hU : IsOpen U
hF_le : ∀ (i : ι) (w : ℂ), w ∈ U → ‖F i w‖ ≤ u i
hz : z ∈ U
hc : TendstoLocallyUniformlyOn (fun t x => Finset.sum t fun n => F n x) (fun x => ∑' (n : ι), F n x) atTop U
s : Finset ι
⊢ (Finset.sum s fun b => deriv (F b) z) = (deriv ∘ fun t x => Finset.sum t fun n => F n x) s z
Theorem: CategoryTheory.MonoidalNatIso.ofComponents.inv_app
File: Mathlib/CategoryTheory/Monoidal/NaturalTransformation.lean

tactic: simp [ofComponents]
state_before: C : Type u₁
inst✝³ : Category C
inst✝² : MonoidalCategory C
D : Type u₂
inst✝¹ : Category D
inst✝ : MonoidalCategory D
F G : LaxMonoidalFunctor C D
app : (X : C) → F.obj X ≅ G.obj X
naturality : ∀ {X Y : C} (f : X ⟶ Y), F.map f ≫ (app Y).hom = (app X).hom ≫ G.map f
unit : F.ε ≫ (app (𝟙_ C)).hom = G.ε
tensor :
  ∀ (X Y : C), LaxMonoidalFunctor.μ F X Y ≫ (app (X ⊗ Y)).hom = ((app X).hom ⊗ (app Y).hom) ≫ LaxMonoidalFunctor.μ G X Y
X : C
⊢ (ofComponents app naturality unit tensor).inv.app X = (app X).inv
Theorem: NormedAddGroupHom.opNorm_zero
File: Mathlib/Analysis/Normed/Group/Hom.lean

tactic: rw [zero_mul]
state_before: V : Type ?u.313710
V₁ : Type u_1
V₂ : Type u_2
V₃ : Type ?u.313719
inst✝³ : SeminormedAddCommGroup V
inst✝² : SeminormedAddCommGroup V₁
inst✝¹ : SeminormedAddCommGroup V₂
inst✝ : SeminormedAddCommGroup V₃
f g : NormedAddGroupHom V₁ V₂
x✝ : V₁
⊢ ‖↑0 x✝‖ = 0 * ‖x✝‖

tactic: exact norm_zero
state_before: V : Type ?u.313710
V₁ : Type u_1
V₂ : Type u_2
V₃ : Type ?u.313719
inst✝³ : SeminormedAddCommGroup V
inst✝² : SeminormedAddCommGroup V₁
inst✝¹ : SeminormedAddCommGroup V₂
inst✝ : SeminormedAddCommGroup V₃
f g : NormedAddGroupHom V₁ V₂
x✝ : V₁
⊢ ‖↑0 x✝‖ = 0
Theorem: Finset.prod_erase_mul
File: Mathlib/Algebra/BigOperators/Basic.lean

tactic: rw [mul_comm, mul_prod_erase s f h]
state_before: ι : Type ?u.795937
β : Type u
α : Type v
γ : Type w
s✝ s₁ s₂ : Finset α
a✝ : α
f✝ g : α → β
inst✝¹ : CommMonoid β
inst✝ : DecidableEq α
s : Finset α
f : α → β
a : α
h : a ∈ s
⊢ (∏ x in erase s a, f x) * f a = ∏ x in s, f x
Theorem: CategoryTheory.hasTerminal_of_isSeparating
File: Mathlib/CategoryTheory/Generator.lean

tactic: haveI : Small.{v₁} 𝒢.op := small_of_injective (Set.opEquiv_self 𝒢).injective
state_before: C : Type u₁
inst✝⁴ : Category C
D : Type u₂
inst✝³ : Category D
inst✝² : WellPowered Cᵒᵖ
inst✝¹ : HasColimits C
𝒢 : Set C
inst✝ : Small ↑𝒢
h𝒢 : IsSeparating 𝒢
⊢ HasTerminal C

tactic: haveI : HasInitial Cᵒᵖ := hasInitial_of_isCoseparating ((isCoseparating_op_iff _).2 h𝒢)
state_before: C : Type u₁
inst✝⁴ : Category C
D : Type u₂
inst✝³ : Category D
inst✝² : WellPowered Cᵒᵖ
inst✝¹ : HasColimits C
𝒢 : Set C
inst✝ : Small ↑𝒢
h𝒢 : IsSeparating 𝒢
this : Small ↑(Set.op 𝒢)
⊢ HasTerminal C

tactic: exact hasTerminal_of_hasInitial_op
state_before: C : Type u₁
inst✝⁴ : Category C
D : Type u₂
inst✝³ : Category D
inst✝² : WellPowered Cᵒᵖ
inst✝¹ : HasColimits C
𝒢 : Set C
inst✝ : Small ↑𝒢
h𝒢 : IsSeparating 𝒢
this✝ : Small ↑(Set.op 𝒢)
this : HasInitial Cᵒᵖ
⊢ HasTerminal C
Theorem: MeasureTheory.OuterMeasure.f_iUnion
File: Mathlib/MeasureTheory/Measure/OuterMeasure.lean

tactic: refine' le_antisymm (m.iUnion_nat s) _
state_before: α : Type u
m : OuterMeasure α
s✝ s₁ s₂ : Set α
s : ℕ → Set α
h : ∀ (i : ℕ), IsCaratheodory m (s i)
hd : Pairwise (Disjoint on s)
⊢ ↑m (⋃ (i : ℕ), s i) = ∑' (i : ℕ), ↑m (s i)

tactic: rw [ENNReal.tsum_eq_iSup_nat]
state_before: α : Type u
m : OuterMeasure α
s✝ s₁ s₂ : Set α
s : ℕ → Set α
h : ∀ (i : ℕ), IsCaratheodory m (s i)
hd : Pairwise (Disjoint on s)
⊢ (∑' (i : ℕ), ↑m (s i)) ≤ ↑m (⋃ (i : ℕ), s i)

tactic: refine' iSup_le fun n => _
state_before: α : Type u
m : OuterMeasure α
s✝ s₁ s₂ : Set α
s : ℕ → Set α
h : ∀ (i : ℕ), IsCaratheodory m (s i)
hd : Pairwise (Disjoint on s)
⊢ (⨆ (i : ℕ), ∑ a in Finset.range i, ↑m (s a)) ≤ ↑m (⋃ (i : ℕ), s i)

tactic: have := @isCaratheodory_sum _ m _ h hd univ n
state_before: α : Type u
m : OuterMeasure α
s✝ s₁ s₂ : Set α
s : ℕ → Set α
h : ∀ (i : ℕ), IsCaratheodory m (s i)
hd : Pairwise (Disjoint on s)
n : ℕ
⊢ ∑ a in Finset.range n, ↑m (s a) ≤ ↑m (⋃ (i : ℕ), s i)

tactic: simp at this
state_before: α : Type u
m : OuterMeasure α
s✝ s₁ s₂ : Set α
s : ℕ → Set α
h : ∀ (i : ℕ), IsCaratheodory m (s i)
hd : Pairwise (Disjoint on s)
n : ℕ
this : ∑ i in Finset.range n, ↑m (univ ∩ s i) = ↑m (univ ∩ ⋃ (i : ℕ) (_ : i < n), s i)
⊢ ∑ a in Finset.range n, ↑m (s a) ≤ ↑m (⋃ (i : ℕ), s i)

tactic: simp [this]
state_before: α : Type u
m : OuterMeasure α
s✝ s₁ s₂ : Set α
s : ℕ → Set α
h : ∀ (i : ℕ), IsCaratheodory m (s i)
hd : Pairwise (Disjoint on s)
n : ℕ
this : ∑ x in Finset.range n, ↑m (s x) = ↑m (⋃ (i : ℕ) (_ : i < n), s i)
⊢ ∑ a in Finset.range n, ↑m (s a) ≤ ↑m (⋃ (i : ℕ), s i)

tactic: exact m.mono (iUnion₂_subset fun i _ => subset_iUnion _ i)
state_before: α : Type u
m : OuterMeasure α
s✝ s₁ s₂ : Set α
s : ℕ → Set α
h : ∀ (i : ℕ), IsCaratheodory m (s i)
hd : Pairwise (Disjoint on s)
n : ℕ
this : ∑ x in Finset.range n, ↑m (s x) = ↑m (⋃ (i : ℕ) (_ : i < n), s i)
⊢ ↑m (⋃ (i : ℕ) (_ : i < n), s i) ≤ ↑m (⋃ (i : ℕ), s i)
Theorem: Dfinsupp.prod_single_index
File: Mathlib/Data/Dfinsupp/Basic.lean

tactic: by_cases h : b ≠ 0
state_before: ι : Type u
γ : Type w
β : ι → Type v
β₁ : ι → Type v₁
β₂ : ι → Type v₂
dec : DecidableEq ι
inst✝² : (i : ι) → Zero (β i)
inst✝¹ : (i : ι) → (x : β i) → Decidable (x ≠ 0)
inst✝ : CommMonoid γ
i : ι
b : β i
h : (i : ι) → β i → γ
h_zero : h i 0 = 1
⊢ prod (single i b) h = h i b

tactic: simp [Dfinsupp.prod, support_single_ne_zero h]
state_before: case pos
ι : Type u
γ : Type w
β : ι → Type v
β₁ : ι → Type v₁
β₂ : ι → Type v₂
dec : DecidableEq ι
inst✝² : (i : ι) → Zero (β i)
inst✝¹ : (i : ι) → (x : β i) → Decidable (x ≠ 0)
inst✝ : CommMonoid γ
i : ι
b : β i
h✝ : (i : ι) → β i → γ
h_zero : h✝ i 0 = 1
h : b ≠ 0
⊢ prod (single i b) h✝ = h✝ i b

tactic: rw [not_not] at h
state_before: case neg
ι : Type u
γ : Type w
β : ι → Type v
β₁ : ι → Type v₁
β₂ : ι → Type v₂
dec : DecidableEq ι
inst✝² : (i : ι) → Zero (β i)
inst✝¹ : (i : ι) → (x : β i) → Decidable (x ≠ 0)
inst✝ : CommMonoid γ
i : ι
b : β i
h✝ : (i : ι) → β i → γ
h_zero : h✝ i 0 = 1
h : ¬b ≠ 0
⊢ prod (single i b) h✝ = h✝ i b

tactic: simp [h, prod_zero_index, h_zero]
state_before: case neg
ι : Type u
γ : Type w
β : ι → Type v
β₁ : ι → Type v₁
β₂ : ι → Type v₂
dec : DecidableEq ι
inst✝² : (i : ι) → Zero (β i)
inst✝¹ : (i : ι) → (x : β i) → Decidable (x ≠ 0)
inst✝ : CommMonoid γ
i : ι
b : β i
h✝ : (i : ι) → β i → γ
h_zero : h✝ i 0 = 1
h : b = 0
⊢ prod (single i b) h✝ = h✝ i b

tactic: rfl
state_before: case neg
ι : Type u
γ : Type w
β : ι → Type v
β₁ : ι → Type v₁
β₂ : ι → Type v₂
dec : DecidableEq ι
inst✝² : (i : ι) → Zero (β i)
inst✝¹ : (i : ι) → (x : β i) → Decidable (x ≠ 0)
inst✝ : CommMonoid γ
i : ι
b : β i
h✝ : (i : ι) → β i → γ
h_zero : h✝ i 0 = 1
h : b = 0
⊢ prod 0 h✝ = 1
Theorem: Set.diagonal_subset_iff
File: Mathlib/Data/Set/Prod.lean

tactic: rw [← range_diag, range_subset_iff]
state_before: α : Type u_1
s✝ t : Set α
s : Set (α × α)
⊢ diagonal α ⊆ s ↔ ∀ (x : α), (x, x) ∈ s
Theorem: MeasurableEquiv.measurable_comp_iff
File: Mathlib/MeasureTheory/MeasurableSpace.lean

tactic: have : Measurable (f ∘ (e.symm.trans e).toEquiv) := hfe.comp e.symm.measurable
state_before: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type ?u.824632
δ' : Type ?u.824635
ι : Sort uι
s t u : Set α
inst✝³ : MeasurableSpace α
inst✝² : MeasurableSpace β
inst✝¹ : MeasurableSpace γ
inst✝ : MeasurableSpace δ
f : β → γ
e : α ≃ᵐ β
hfe : Measurable (f ∘ ↑e)
⊢ Measurable f

tactic: rwa [coe_toEquiv, symm_trans_self] at this
state_before: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type ?u.824632
δ' : Type ?u.824635
ι : Sort uι
s t u : Set α
inst✝³ : MeasurableSpace α
inst✝² : MeasurableSpace β
inst✝¹ : MeasurableSpace γ
inst✝ : MeasurableSpace δ
f : β → γ
e : α ≃ᵐ β
hfe : Measurable (f ∘ ↑e)
this : Measurable (f ∘ ↑(trans (symm e) e).toEquiv)
⊢ Measurable f
Theorem: Nat.ArithmeticFunction.natCoe_mul
File: Mathlib/NumberTheory/ArithmeticFunction.lean

tactic: ext n
state_before: R : Type u_1
inst✝ : Semiring R
f g : ArithmeticFunction ℕ
⊢ ↑(f * g) = ↑f * ↑g

tactic: simp
state_before: case h
R : Type u_1
inst✝ : Semiring R
f g : ArithmeticFunction ℕ
n : ℕ
⊢ ↑↑(f * g) n = ↑(↑f * ↑g) n
Theorem: EMetric.hausdorffEdist_self
File: Mathlib/Topology/MetricSpace/HausdorffDistance.lean

tactic: simp only [hausdorffEdist_def, sup_idem, ENNReal.iSup_eq_zero]
state_before: ι : Sort ?u.35077
α : Type u
β : Type v
inst✝¹ : PseudoEMetricSpace α
inst✝ : PseudoEMetricSpace β
x y : α
s t u : Set α
Φ : α → β
⊢ hausdorffEdist s s = 0

tactic: exact fun x hx => infEdist_zero_of_mem hx
state_before: ι : Sort ?u.35077
α : Type u
β : Type v
inst✝¹ : PseudoEMetricSpace α
inst✝ : PseudoEMetricSpace β
x y : α
s t u : Set α
Φ : α → β
⊢ ∀ (i : α), i ∈ s → infEdist i s = 0
Theorem: Polynomial.reverse_natTrailingDegree
File: Mathlib/Data/Polynomial/Reverse.lean

tactic: by_cases hf : f = 0
state_before: R : Type u_1
inst✝ : Semiring R
f✝ f : R[X]
⊢ natTrailingDegree (reverse f) = 0

tactic: rw [hf, reverse_zero, natTrailingDegree_zero]
state_before: case pos
R : Type u_1
inst✝ : Semiring R
f✝ f : R[X]
hf : f = 0
⊢ natTrailingDegree (reverse f) = 0

tactic: rw [← le_zero_iff]
state_before: case neg
R : Type u_1
inst✝ : Semiring R
f✝ f : R[X]
hf : ¬f = 0
⊢ natTrailingDegree (reverse f) = 0

tactic: apply natTrailingDegree_le_of_ne_zero
state_before: case neg
R : Type u_1
inst✝ : Semiring R
f✝ f : R[X]
hf : ¬f = 0
⊢ natTrailingDegree (reverse f) ≤ 0

tactic: rw [coeff_zero_reverse]
state_before: case neg.h
R : Type u_1
inst✝ : Semiring R
f✝ f : R[X]
hf : ¬f = 0
⊢ coeff (reverse f) 0 ≠ 0

tactic: exact mt leadingCoeff_eq_zero.mp hf
state_before: case neg.h
R : Type u_1
inst✝ : Semiring R
f✝ f : R[X]
hf : ¬f = 0
⊢ leadingCoeff f ≠ 0
Theorem: dist_lineMap_right
File: Mathlib/Analysis/NormedSpace/AddTorsor.lean

tactic: rw [← dist_eq_norm', ← dist_lineMap_lineMap, lineMap_apply_one]
state_before: α : Type ?u.38369
V : Type u_3
P : Type u_1
W : Type ?u.38378
Q : Type ?u.38381
inst✝⁸ : SeminormedAddCommGroup V
inst✝⁷ : PseudoMetricSpace P
inst✝⁶ : NormedAddTorsor V P
inst✝⁵ : NormedAddCommGroup W
inst✝⁴ : MetricSpace Q
inst✝³ : NormedAddTorsor W Q
𝕜 : Type u_2
inst✝² : NormedField 𝕜
inst✝¹ : NormedSpace 𝕜 V
inst✝ : NormedSpace 𝕜 W
p₁ p₂ : P
c : 𝕜
⊢ dist (↑(lineMap p₁ p₂) c) p₂ = ‖1 - c‖ * dist p₁ p₂
Theorem: LinearIndependent.of_comp
File: Mathlib/LinearAlgebra/LinearIndependent.lean

tactic: simp_rw [← f.map_smul, ← f.map_sum, hg, f.map_zero]
state_before: ι : Type u'
ι' : Type ?u.106583
R : Type u_1
K : Type ?u.106589
M : Type u_2
M' : Type u_3
M'' : Type ?u.106598
V : Type u
V' : Type ?u.106603
v : ι → M
inst✝⁶ : Semiring R
inst✝⁵ : AddCommMonoid M
inst✝⁴ : AddCommMonoid M'
inst✝³ : AddCommMonoid M''
inst✝² : Module R M
inst✝¹ : Module R M'
inst✝ : Module R M''
a b : R
x y : M
f : M →ₗ[R] M'
hfv : LinearIndependent R (↑f ∘ v)
s : Finset ι
g : ι → R
hg : ∑ i in s, g i • v i = 0
i : ι
his : i ∈ s
⊢ ∑ i in s, g i • ↑f (v i) = 0
Theorem: Real.sinh_two_mul
File: Mathlib/Data/Complex/Exponential.lean

tactic: rw [← ofReal_inj]
state_before: x y : ℝ
⊢ sinh (2 * x) = 2 * sinh x * cosh x

tactic: simp [sinh_two_mul]
state_before: x y : ℝ
⊢ ↑(sinh (2 * x)) = ↑(2 * sinh x * cosh x)
Theorem: Finset.prod_eq_prod_diff_singleton_mul
File: Mathlib/Algebra/BigOperators/Basic.lean

tactic: rw [prod_eq_mul_prod_diff_singleton h, mul_comm]
state_before: ι : Type ?u.782719
β : Type u
α : Type v
γ : Type w
s✝ s₁ s₂ : Finset α
a : α
f✝ g : α → β
inst✝¹ : CommMonoid β
inst✝ : DecidableEq α
s : Finset α
i : α
h : i ∈ s
f : α → β
⊢ ∏ x in s, f x = (∏ x in s \ {i}, f x) * f i
Theorem: Cardinal.isRegular_aleph'_succ
File: Mathlib/SetTheory/Cardinal/Cofinality.lean

tactic: rw [aleph'_succ]
state_before: α : Type ?u.147716
r : α → α → Prop
o : Ordinal
h : ω ≤ o
⊢ IsRegular (aleph' (succ o))

tactic: exact isRegular_succ (aleph0_le_aleph'.2 h)
state_before: α : Type ?u.147716
r : α → α → Prop
o : Ordinal
h : ω ≤ o
⊢ IsRegular (succ (aleph' o))
Theorem: hasFPowerSeriesOn_cauchy_integral
File: Mathlib/MeasureTheory/Integral/CircleIntegral.lean

tactic: refine' hasSum_cauchyPowerSeries_integral hf _
state_before: E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
R : ℝ≥0
hf : CircleIntegrable f c ↑R
hR : 0 < R
y✝ : ℂ
hy : y✝ ∈ EMetric.ball 0 ↑R
⊢ HasSum (fun n => ↑(cauchyPowerSeries f c (↑R) n) fun x => y✝)
    ((2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c, ↑R), (z - (c + y✝))⁻¹ • f z)

tactic: rw [← norm_eq_abs, ← coe_nnnorm, NNReal.coe_lt_coe, ← ENNReal.coe_lt_coe]
state_before: E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
R : ℝ≥0
hf : CircleIntegrable f c ↑R
hR : 0 < R
y✝ : ℂ
hy : y✝ ∈ EMetric.ball 0 ↑R
⊢ ↑Complex.abs y✝ < ↑R

tactic: exact mem_emetric_ball_zero_iff.1 hy
state_before: E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
R : ℝ≥0
hf : CircleIntegrable f c ↑R
hR : 0 < R
y✝ : ℂ
hy : y✝ ∈ EMetric.ball 0 ↑R
⊢ ↑‖y✝‖₊ < ↑R
Theorem: HomogeneousLocalization.pow_val
File: Mathlib/RingTheory/GradedAlgebra/HomogeneousLocalization.lean

tactic: induction y using Quotient.inductionOn
state_before: ι : Type u_1
R : Type u_2
A : Type u_3
inst✝⁵ : AddCommMonoid ι
inst✝⁴ : DecidableEq ι
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
inst✝ : GradedAlgebra 𝒜
x : Submonoid A
y : HomogeneousLocalization 𝒜 x
n : ℕ
⊢ val (y ^ n) = val y ^ n

tactic: simp only [Quotient.liftOn₂'_mk, Quotient.liftOn'_mk]
state_before: case h
ι : Type u_1
R : Type u_2
A : Type u_3
inst✝⁵ : AddCommMonoid ι
inst✝⁴ : DecidableEq ι
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
inst✝ : GradedAlgebra 𝒜
x : Submonoid A
n : ℕ
a✝ : NumDenSameDeg 𝒜 x
⊢ val (Quotient.mk (Setoid.ker (embedding 𝒜 x)) a✝ ^ n) = val (Quotient.mk (Setoid.ker (embedding 𝒜 x)) a✝) ^ n

tactic: change Localization.mk _ _ = Localization.mk _ _ ^ n
state_before: case h
ι : Type u_1
R : Type u_2
A : Type u_3
inst✝⁵ : AddCommMonoid ι
inst✝⁴ : DecidableEq ι
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
inst✝ : GradedAlgebra 𝒜
x : Submonoid A
n : ℕ
a✝ : NumDenSameDeg 𝒜 x
⊢ val (Quotient.mk (Setoid.ker (embedding 𝒜 x)) a✝ ^ n) = val (Quotient.mk (Setoid.ker (embedding 𝒜 x)) a✝) ^ n

tactic: rw [Localization.mk_pow]
state_before: case h
ι : Type u_1
R : Type u_2
A : Type u_3
inst✝⁵ : AddCommMonoid ι
inst✝⁴ : DecidableEq ι
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
inst✝ : GradedAlgebra 𝒜
x : Submonoid A
n : ℕ
a✝ : NumDenSameDeg 𝒜 x
⊢ Localization.mk ↑((fun x_1 => x_1 ^ n) a✝).num
      { val := ↑((fun x_1 => x_1 ^ n) a✝).den, property := (_ : ↑((fun x_1 => x_1 ^ n) a✝).den ∈ x) } =
    Localization.mk ↑a✝.num { val := ↑a✝.den, property := (_ : ↑a✝.den ∈ x) } ^ n

tactic: dsimp only
state_before: case h
ι : Type u_1
R : Type u_2
A : Type u_3
inst✝⁵ : AddCommMonoid ι
inst✝⁴ : DecidableEq ι
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
inst✝ : GradedAlgebra 𝒜
x : Submonoid A
n : ℕ
a✝ : NumDenSameDeg 𝒜 x
⊢ Localization.mk ↑((fun x_1 => x_1 ^ n) a✝).num
      { val := ↑((fun x_1 => x_1 ^ n) a✝).den, property := (_ : ↑((fun x_1 => x_1 ^ n) a✝).den ∈ x) } =
    Localization.mk (↑a✝.num ^ n) ({ val := ↑a✝.den, property := (_ : ↑a✝.den ∈ x) } ^ n)

tactic: congr 1
state_before: case h
ι : Type u_1
R : Type u_2
A : Type u_3
inst✝⁵ : AddCommMonoid ι
inst✝⁴ : DecidableEq ι
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
inst✝ : GradedAlgebra 𝒜
x : Submonoid A
n : ℕ
a✝ : NumDenSameDeg 𝒜 x
⊢ Localization.mk ↑(a✝ ^ n).num { val := ↑(a✝ ^ n).den, property := (_ : ↑(a✝ ^ n).den ∈ x) } =
    Localization.mk (↑a✝.num ^ n) ({ val := ↑a✝.den, property := (_ : ↑a✝.den ∈ x) } ^ n)
Theorem: CategoryTheory.Limits.spanExt_inv_app_zero
File: Mathlib/CategoryTheory/Limits/Shapes/Pullbacks.lean

tactic: dsimp [spanExt]
state_before: C : Type u
inst✝¹ : Category C
D : Type u₂
inst✝ : Category D
X Y Z X' Y' Z' : C
iX : X ≅ X'
iY : Y ≅ Y'
iZ : Z ≅ Z'
f : X ⟶ Y
g : X ⟶ Z
f' : X' ⟶ Y'
g' : X' ⟶ Z'
wf : iX.hom ≫ f' = f ≫ iY.hom
wg : iX.hom ≫ g' = g ≫ iZ.hom
⊢ (spanExt iX iY iZ wf wg).inv.app WalkingSpan.zero = iX.inv
Theorem: toIocDiv_add_right'
File: Mathlib/Algebra/Order/ToIntervalMod.lean

tactic: simpa only [one_zsmul] using toIocDiv_add_zsmul' hp a b 1
state_before: α : Type u_1
inst✝ : LinearOrderedAddCommGroup α
hα : Archimedean α
p : α
hp : 0 < p
a✝ b✝ c : α
n : ℤ
a b : α
⊢ toIocDiv hp (a + p) b = toIocDiv hp a b - 1
Theorem: AffineIsometry.coe_toAffineMap
File: Mathlib/Analysis/NormedSpace/AffineIsometry.lean

tactic: rfl
state_before: 𝕜 : Type u_5
V : Type u_3
V₁ : Type ?u.24408
V₂ : Type u_4
V₃ : Type ?u.24414
V₄ : Type ?u.24417
P₁ : Type ?u.24420
P : Type u_1
P₂ : Type u_2
P₃ : Type ?u.24429
P₄ : Type ?u.24432
inst✝²⁰ : NormedField 𝕜
inst✝¹⁹ : SeminormedAddCommGroup V
inst✝¹⁸ : SeminormedAddCommGroup V₁
inst✝¹⁷ : SeminormedAddCommGroup V₂
inst✝¹⁶ : SeminormedAddCommGroup V₃
inst✝¹⁵ : SeminormedAddCommGroup V₄
inst✝¹⁴ : NormedSpace 𝕜 V
inst✝¹³ : NormedSpace 𝕜 V₁
inst✝¹² : NormedSpace 𝕜 V₂
inst✝¹¹ : NormedSpace 𝕜 V₃
inst✝¹⁰ : NormedSpace 𝕜 V₄
inst✝⁹ : PseudoMetricSpace P
inst✝⁸ : MetricSpace P₁
inst✝⁷ : PseudoMetricSpace P₂
inst✝⁶ : PseudoMetricSpace P₃
inst✝⁵ : PseudoMetricSpace P₄
inst✝⁴ : NormedAddTorsor V P
inst✝³ : NormedAddTorsor V₁ P₁
inst✝² : NormedAddTorsor V₂ P₂
inst✝¹ : NormedAddTorsor V₃ P₃
inst✝ : NormedAddTorsor V₄ P₄
f : P →ᵃⁱ[𝕜] P₂
⊢ ↑f.toAffineMap = ↑f
Theorem: Set.not_subsingleton_iff
File: Mathlib/Data/Set/Basic.lean

tactic: simp_rw [Set.Subsingleton, Set.Nontrivial, not_forall]
state_before: α : Type u
β : Type v
γ : Type w
ι : Sort x
a b : α
s s₁ s₂ t t₁ t₂ u : Set α
⊢ ¬Set.Subsingleton s ↔ Set.Nontrivial s
Theorem: IsClosed.exists_infDist_eq_dist
File: Mathlib/Topology/MetricSpace/HausdorffDistance.lean

tactic: rcases hne with ⟨z, hz⟩
state_before: ι : Sort ?u.69127
α : Type u
β : Type v
inst✝² : PseudoMetricSpace α
inst✝¹ : PseudoMetricSpace β
s t u : Set α
x✝ y : α
Φ : α → β
inst✝ : ProperSpace α
h : IsClosed s
hne : Set.Nonempty s
x : α
⊢ ∃ y, y ∈ s ∧ infDist x s = dist x y

tactic: rw [← infDist_inter_closedBall_of_mem hz]
state_before: case intro
ι : Sort ?u.69127
α : Type u
β : Type v
inst✝² : PseudoMetricSpace α
inst✝¹ : PseudoMetricSpace β
s t u : Set α
x✝ y : α
Φ : α → β
inst✝ : ProperSpace α
h : IsClosed s
x z : α
hz : z ∈ s
⊢ ∃ y, y ∈ s ∧ infDist x s = dist x y

tactic: set t := s ∩ closedBall x (dist z x)
state_before: case intro
ι : Sort ?u.69127
α : Type u
β : Type v
inst✝² : PseudoMetricSpace α
inst✝¹ : PseudoMetricSpace β
s t u : Set α
x✝ y : α
Φ : α → β
inst✝ : ProperSpace α
h : IsClosed s
x z : α
hz : z ∈ s
⊢ ∃ y, y ∈ s ∧ infDist x (s ∩ closedBall x (dist z x)) = dist x y

tactic: have htc : IsCompact t := (isCompact_closedBall x (dist z x)).inter_left h
state_before: case intro
ι : Sort ?u.69127
α : Type u
β : Type v
inst✝² : PseudoMetricSpace α
inst✝¹ : PseudoMetricSpace β
s t✝ u : Set α
x✝ y : α
Φ : α → β
inst✝ : ProperSpace α
h : IsClosed s
x z : α
hz : z ∈ s
t : Set α := s ∩ closedBall x (dist z x)
⊢ ∃ y, y ∈ s ∧ infDist x t = dist x y

tactic: have htne : t.Nonempty := ⟨z, hz, mem_closedBall.2 le_rfl⟩
state_before: case intro
ι : Sort ?u.69127
α : Type u
β : Type v
inst✝² : PseudoMetricSpace α
inst✝¹ : PseudoMetricSpace β
s t✝ u : Set α
x✝ y : α
Φ : α → β
inst✝ : ProperSpace α
h : IsClosed s
x z : α
hz : z ∈ s
t : Set α := s ∩ closedBall x (dist z x)
htc : IsCompact t
⊢ ∃ y, y ∈ s ∧ infDist x t = dist x y

tactic: obtain ⟨y, ⟨hys, -⟩, hyd⟩ : ∃ y ∈ t, infDist x t = dist x y := htc.exists_infDist_eq_dist htne x
state_before: case intro
ι : Sort ?u.69127
α : Type u
β : Type v
inst✝² : PseudoMetricSpace α
inst✝¹ : PseudoMetricSpace β
s t✝ u : Set α
x✝ y : α
Φ : α → β
inst✝ : ProperSpace α
h : IsClosed s
x z : α
hz : z ∈ s
t : Set α := s ∩ closedBall x (dist z x)
htc : IsCompact t
htne : Set.Nonempty t
⊢ ∃ y, y ∈ s ∧ infDist x t = dist x y

tactic: exact ⟨y, hys, hyd⟩
state_before: case intro.intro.intro.intro
ι : Sort ?u.69127
α : Type u
β : Type v
inst✝² : PseudoMetricSpace α
inst✝¹ : PseudoMetricSpace β
s t✝ u : Set α
x✝ y✝ : α
Φ : α → β
inst✝ : ProperSpace α
h : IsClosed s
x z : α
hz : z ∈ s
t : Set α := s ∩ closedBall x (dist z x)
htc : IsCompact t
htne : Set.Nonempty t
y : α
hyd : infDist x t = dist x y
hys : y ∈ s
⊢ ∃ y, y ∈ s ∧ infDist x t = dist x y
Theorem: Real.Angle.two_zsmul_coe_pi
File: Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean

tactic: simp [← coe_int_mul_eq_zsmul]
state_before: ⊢ 2 • ↑π = 0
Theorem: ContinuousMap.tendsto_compactOpen_iff_forall
File: Mathlib/Topology/CompactOpen.lean

tactic: rw [compactOpen_eq_sInf_induced]
state_before: α : Type u_2
β : Type u_3
γ : Type ?u.29261
inst✝² : TopologicalSpace α
inst✝¹ : TopologicalSpace β
inst✝ : TopologicalSpace γ
ι : Type u_1
l : Filter ι
F : ι → C(α, β)
f : C(α, β)
⊢ Filter.Tendsto F l (𝓝 f) ↔
    ∀ (s : Set α), IsCompact s → Filter.Tendsto (fun i => restrict s (F i)) l (𝓝 (restrict s f))

tactic: simp [nhds_iInf, nhds_induced, Filter.tendsto_comap_iff, Function.comp]
state_before: α : Type u_2
β : Type u_3
γ : Type ?u.29261
inst✝² : TopologicalSpace α
inst✝¹ : TopologicalSpace β
inst✝ : TopologicalSpace γ
ι : Type u_1
l : Filter ι
F : ι → C(α, β)
f : C(α, β)
⊢ Filter.Tendsto F l (𝓝 f) ↔
    ∀ (s : Set α), IsCompact s → Filter.Tendsto (fun i => restrict s (F i)) l (𝓝 (restrict s f))