tasksource/leandojo huggingface dataset:
Theorem: HasFDerivWithinAt.mul
File: Mathlib/Analysis/Calculus/FDeriv/Mul.lean

tactic: convert hc.mul' hd
state_before: 𝕜 : Type u_1
inst✝¹² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace 𝕜 E
F : Type ?u.715008
inst✝⁹ : NormedAddCommGroup F
inst✝⁸ : NormedSpace 𝕜 F
G : Type ?u.715103
inst✝⁷ : NormedAddCommGroup G
inst✝⁶ : NormedSpace 𝕜 G
G' : Type ?u.715198
inst✝⁵ : NormedAddCommGroup G'
inst✝⁴ : NormedSpace 𝕜 G'
f f₀ f₁ g : E → F
f' f₀' f₁' g' e : E →L[𝕜] F
x : E
s t : Set E
L L₁ L₂ : Filter E
𝔸 : Type ?u.717333
𝔸' : Type u_3
inst✝³ : NormedRing 𝔸
inst✝² : NormedCommRing 𝔸'
inst✝¹ : NormedAlgebra 𝕜 𝔸
inst✝ : NormedAlgebra 𝕜 𝔸'
a b : E → 𝔸
a' b' : E →L[𝕜] 𝔸
c d : E → 𝔸'
c' d' : E →L[𝕜] 𝔸'
hc : HasFDerivWithinAt c c' s x
hd : HasFDerivWithinAt d d' s x
⊢ HasFDerivWithinAt (fun y => c y * d y) (c x • d' + d x • c') s x

tactic: ext z
state_before: case h.e'_10.h.e'_6
𝕜 : Type u_1
inst✝¹² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace 𝕜 E
F : Type ?u.715008
inst✝⁹ : NormedAddCommGroup F
inst✝⁸ : NormedSpace 𝕜 F
G : Type ?u.715103
inst✝⁷ : NormedAddCommGroup G
inst✝⁶ : NormedSpace 𝕜 G
G' : Type ?u.715198
inst✝⁵ : NormedAddCommGroup G'
inst✝⁴ : NormedSpace 𝕜 G'
f f₀ f₁ g : E → F
f' f₀' f₁' g' e : E →L[𝕜] F
x : E
s t : Set E
L L₁ L₂ : Filter E
𝔸 : Type ?u.717333
𝔸' : Type u_3
inst✝³ : NormedRing 𝔸
inst✝² : NormedCommRing 𝔸'
inst✝¹ : NormedAlgebra 𝕜 𝔸
inst✝ : NormedAlgebra 𝕜 𝔸'
a b : E → 𝔸
a' b' : E →L[𝕜] 𝔸
c d : E → 𝔸'
c' d' : E →L[𝕜] 𝔸'
hc : HasFDerivWithinAt c c' s x
hd : HasFDerivWithinAt d d' s x
⊢ d x • c' = smulRight c' (d x)

tactic: apply mul_comm
state_before: case h.e'_10.h.e'_6.h
𝕜 : Type u_1
inst✝¹² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace 𝕜 E
F : Type ?u.715008
inst✝⁹ : NormedAddCommGroup F
inst✝⁸ : NormedSpace 𝕜 F
G : Type ?u.715103
inst✝⁷ : NormedAddCommGroup G
inst✝⁶ : NormedSpace 𝕜 G
G' : Type ?u.715198
inst✝⁵ : NormedAddCommGroup G'
inst✝⁴ : NormedSpace 𝕜 G'
f f₀ f₁ g : E → F
f' f₀' f₁' g' e : E →L[𝕜] F
x : E
s t : Set E
L L₁ L₂ : Filter E
𝔸 : Type ?u.717333
𝔸' : Type u_3
inst✝³ : NormedRing 𝔸
inst✝² : NormedCommRing 𝔸'
inst✝¹ : NormedAlgebra 𝕜 𝔸
inst✝ : NormedAlgebra 𝕜 𝔸'
a b : E → 𝔸
a' b' : E →L[𝕜] 𝔸
c d : E → 𝔸'
c' d' : E →L[𝕜] 𝔸'
hc : HasFDerivWithinAt c c' s x
hd : HasFDerivWithinAt d d' s x
z : E
⊢ ↑(d x • c') z = ↑(smulRight c' (d x)) z
Theorem: Wbtw.left_ne_right_of_ne_right
File: Mathlib/Analysis/Convex/Between.lean

tactic: rintro rfl
state_before: R : Type u_1
V : Type u_2
V' : Type ?u.162563
P : Type u_3
P' : Type ?u.162569
inst✝⁶ : OrderedRing R
inst✝⁵ : AddCommGroup V
inst✝⁴ : Module R V
inst✝³ : AddTorsor V P
inst✝² : AddCommGroup V'
inst✝¹ : Module R V'
inst✝ : AddTorsor V' P'
x y z : P
h : Wbtw R x y z
hne : y ≠ z
⊢ x ≠ z

tactic: rw [wbtw_self_iff] at h
state_before: R : Type u_1
V : Type u_2
V' : Type ?u.162563
P : Type u_3
P' : Type ?u.162569
inst✝⁶ : OrderedRing R
inst✝⁵ : AddCommGroup V
inst✝⁴ : Module R V
inst✝³ : AddTorsor V P
inst✝² : AddCommGroup V'
inst✝¹ : Module R V'
inst✝ : AddTorsor V' P'
x y : P
h : Wbtw R x y x
hne : y ≠ x
⊢ False

tactic: exact hne h
state_before: R : Type u_1
V : Type u_2
V' : Type ?u.162563
P : Type u_3
P' : Type ?u.162569
inst✝⁶ : OrderedRing R
inst✝⁵ : AddCommGroup V
inst✝⁴ : Module R V
inst✝³ : AddTorsor V P
inst✝² : AddCommGroup V'
inst✝¹ : Module R V'
inst✝ : AddTorsor V' P'
x y : P
h : y = x
hne : y ≠ x
⊢ False
Theorem: BoxIntegral.Box.disjoint_splitLower_splitUpper
File: Mathlib/Analysis/BoxIntegral/Partition/Split.lean

tactic: rw [← disjoint_withBotCoe, coe_splitLower, coe_splitUpper]
state_before: ι : Type u_1
M : Type ?u.20595
n : ℕ
I✝ : Box ι
i✝ : ι
x✝ : ℝ
y : ι → ℝ
I : Box ι
i : ι
x : ℝ
⊢ Disjoint (splitLower I i x) (splitUpper I i x)

tactic: refine' (Disjoint.inf_left' _ _).inf_right' _
state_before: ι : Type u_1
M : Type ?u.20595
n : ℕ
I✝ : Box ι
i✝ : ι
x✝ : ℝ
y : ι → ℝ
I : Box ι
i : ι
x : ℝ
⊢ Disjoint (↑I ∩ {y | y i ≤ x}) (↑I ∩ {y | x < y i})

tactic: rw [Set.disjoint_left]
state_before: ι : Type u_1
M : Type ?u.20595
n : ℕ
I✝ : Box ι
i✝ : ι
x✝ : ℝ
y : ι → ℝ
I : Box ι
i : ι
x : ℝ
⊢ Disjoint {y | y i ≤ x} {y | x < y i}

tactic: exact fun y (hle : y i ≤ x) hlt => not_lt_of_le hle hlt
state_before: ι : Type u_1
M : Type ?u.20595
n : ℕ
I✝ : Box ι
i✝ : ι
x✝ : ℝ
y : ι → ℝ
I : Box ι
i : ι
x : ℝ
⊢ ∀ ⦃a : ι → ℝ⦄, a ∈ {y | y i ≤ x} → ¬a ∈ {y | x < y i}
Theorem: FreeProduct.of_injective
File: Mathlib/GroupTheory/FreeProduct.lean

tactic: classical exact (of_leftInverse i).injective
state_before: ι : Type u_2
M : ι → Type u_1
inst✝¹ : (i : ι) → Monoid (M i)
N : Type ?u.240831
inst✝ : Monoid N
i : ι
⊢ Function.Injective ↑of

tactic: exact (of_leftInverse i).injective
state_before: ι : Type u_2
M : ι → Type u_1
inst✝¹ : (i : ι) → Monoid (M i)
N : Type ?u.240831
inst✝ : Monoid N
i : ι
⊢ Function.Injective ↑of
Theorem: MeasureTheory.Measure.count_singleton'
File: Mathlib/MeasureTheory/Measure/MeasureSpace.lean

tactic: rw [count_apply_finite' (Set.finite_singleton a) ha, Set.Finite.toFinset]
state_before: α : Type u_1
β : Type ?u.385073
γ : Type ?u.385076
δ : Type ?u.385079
ι : Type ?u.385082
R : Type ?u.385085
R' : Type ?u.385088
m0 : MeasurableSpace α
inst✝² : MeasurableSpace β
inst✝¹ : MeasurableSpace γ
μ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α
s s' t : Set α
inst✝ : MeasurableSpace α
a : α
ha : MeasurableSet {a}
⊢ ↑↑count {a} = 1

tactic: simp [@toFinset_card _ _ (Set.finite_singleton a).fintype,
  @Fintype.card_unique _ _ (Set.finite_singleton a).fintype]
state_before: α : Type u_1
β : Type ?u.385073
γ : Type ?u.385076
δ : Type ?u.385079
ι : Type ?u.385082
R : Type ?u.385085
R' : Type ?u.385088
m0 : MeasurableSpace α
inst✝² : MeasurableSpace β
inst✝¹ : MeasurableSpace γ
μ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α
s s' t : Set α
inst✝ : MeasurableSpace α
a : α
ha : MeasurableSet {a}
⊢ ↑(Finset.card (toFinset {a})) = 1
Theorem: HasFDerivAtFilter.sub_const
File: Mathlib/Analysis/Calculus/FDeriv/Add.lean

tactic: simpa only [sub_eq_add_neg] using hf.add_const (-c)
state_before: 𝕜 : Type u_1
inst✝⁸ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace 𝕜 E
F : Type u_3
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
G : Type ?u.483317
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
G' : Type ?u.483412
inst✝¹ : NormedAddCommGroup G'
inst✝ : NormedSpace 𝕜 G'
f f₀ f₁ g : E → F
f' f₀' f₁' g' e : E →L[𝕜] F
x : E
s t : Set E
L L₁ L₂ : Filter E
hf : HasFDerivAtFilter f f' x L
c : F
⊢ HasFDerivAtFilter (fun x => f x - c) f' x L
Theorem: continuous_left_toIocMod
File: Mathlib/Topology/Instances/AddCircle.lean

tactic: rw [(funext fun y => Eq.trans (by rw [neg_neg]) <| toIocMod_neg _ _ _ :
    toIocMod hp a = (fun x => p - x) ∘ toIcoMod hp (-a) ∘ Neg.neg)]
state_before: 𝕜 : Type u_1
B : Type ?u.10078
inst✝³ : LinearOrderedAddCommGroup 𝕜
inst✝² : Archimedean 𝕜
inst✝¹ : TopologicalSpace 𝕜
inst✝ : OrderTopology 𝕜
p : 𝕜
hp : 0 < p
a x : 𝕜
⊢ ContinuousWithinAt (toIocMod hp a) (Iic x) x

tactic: have : ContinuousNeg 𝕜 := TopologicalAddGroup.toContinuousNeg
state_before: 𝕜 : Type u_1
B : Type ?u.10078
inst✝³ : LinearOrderedAddCommGroup 𝕜
inst✝² : Archimedean 𝕜
inst✝¹ : TopologicalSpace 𝕜
inst✝ : OrderTopology 𝕜
p : 𝕜
hp : 0 < p
a x : 𝕜
⊢ ContinuousWithinAt ((fun x => p - x) ∘ toIcoMod hp (-a) ∘ Neg.neg) (Iic x) x

tactic: exact
  (continuous_sub_left _).continuousAt.comp_continuousWithinAt <|
    (continuous_right_toIcoMod _ _ _).comp continuous_neg.continuousWithinAt fun y => neg_le_neg
state_before: 𝕜 : Type u_1
B : Type ?u.10078
inst✝³ : LinearOrderedAddCommGroup 𝕜
inst✝² : Archimedean 𝕜
inst✝¹ : TopologicalSpace 𝕜
inst✝ : OrderTopology 𝕜
p : 𝕜
hp : 0 < p
a x : 𝕜
this : ContinuousNeg 𝕜
⊢ ContinuousWithinAt ((fun x => p - x) ∘ toIcoMod hp (-a) ∘ Neg.neg) (Iic x) x

tactic: rw [neg_neg]
state_before: 𝕜 : Type u_1
B : Type ?u.10078
inst✝³ : LinearOrderedAddCommGroup 𝕜
inst✝² : Archimedean 𝕜
inst✝¹ : TopologicalSpace 𝕜
inst✝ : OrderTopology 𝕜
p : 𝕜
hp : 0 < p
a x y : 𝕜
⊢ toIocMod hp a y = toIocMod hp a (- -y)
Theorem: Real.Angle.cos_toReal
File: Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean

tactic: conv_rhs => rw [← coe_toReal θ, cos_coe]
state_before: θ : Angle
⊢ Real.cos (toReal θ) = cos θ
Theorem: Decidable.iff_not_comm
File: Std/Logic.lean

tactic: rw [@iff_def a, @iff_def b]
state_before: a b : Prop
inst✝¹ : Decidable a
inst✝ : Decidable b
⊢ (a ↔ ¬b) ↔ (b ↔ ¬a)

tactic: exact and_congr imp_not_comm not_imp_comm
state_before: a b : Prop
inst✝¹ : Decidable a
inst✝ : Decidable b
⊢ (a → ¬b) ∧ (¬b → a) ↔ (b → ¬a) ∧ (¬a → b)
Theorem: Homeomorph.isClosed_image
File: Mathlib/Topology/Homeomorph.lean

tactic: rw [← preimage_symm, isClosed_preimage]
state_before: α : Type u_1
β : Type u_2
γ : Type ?u.64614
δ : Type ?u.64617
inst✝³ : TopologicalSpace α
inst✝² : TopologicalSpace β
inst✝¹ : TopologicalSpace γ
inst✝ : TopologicalSpace δ
h : α ≃ₜ β
s : Set α
⊢ IsClosed (↑h '' s) ↔ IsClosed s
Theorem: ContinuousMap.exists_mem_subalgebra_near_continuous_of_separatesPoints
File: Mathlib/Topology/ContinuousFunction/StoneWeierstrass.lean

tactic: obtain ⟨g, b⟩ := exists_mem_subalgebra_near_continuousMap_of_separatesPoints A w ⟨f, c⟩ ε pos
state_before: X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : CompactSpace X
A : Subalgebra ℝ C(X, ℝ)
w : Subalgebra.SeparatesPoints A
f : X → ℝ
c : Continuous f
ε : ℝ
pos : 0 < ε
⊢ ∃ g, ∀ (x : X), ‖↑↑g x - f x‖ < ε

tactic: use g
state_before: case intro
X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : CompactSpace X
A : Subalgebra ℝ C(X, ℝ)
w : Subalgebra.SeparatesPoints A
f : X → ℝ
c : Continuous f
ε : ℝ
pos : 0 < ε
g : { x // x ∈ A }
b : ‖↑g - mk f‖ < ε
⊢ ∃ g, ∀ (x : X), ‖↑↑g x - f x‖ < ε

tactic: rwa [norm_lt_iff _ pos] at b
state_before: case intro
X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : CompactSpace X
A : Subalgebra ℝ C(X, ℝ)
w : Subalgebra.SeparatesPoints A
f : X → ℝ
c : Continuous f
ε : ℝ
pos : 0 < ε
g : { x // x ∈ A }
b : ‖↑g - mk f‖ < ε
⊢ ∀ (x : X), ‖↑↑g x - f x‖ < ε
Theorem: Finset.exists_mem_insert
File: Mathlib/Data/Finset/Basic.lean

tactic: simp only [mem_insert, or_and_right, exists_or, exists_eq_left]
state_before: α : Type u_1
β : Type ?u.438273
γ : Type ?u.438276
inst✝ : DecidableEq α
a : α
s : Finset α
p : α → Prop
⊢ (∃ x, x ∈ insert a s ∧ p x) ↔ p a ∨ ∃ x, x ∈ s ∧ p x
Theorem: IsPathConnected.union
File: Mathlib/Topology/PathConnected.lean

tactic: rcases hUV with ⟨x, xU, xV⟩
state_before: X : Type u_1
Y : Type ?u.646605
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
x y z : X
ι : Type ?u.646620
F U V : Set X
hU : IsPathConnected U
hV : IsPathConnected V
hUV : Set.Nonempty (U ∩ V)
⊢ IsPathConnected (U ∪ V)

tactic: use x, Or.inl xU
state_before: case intro.intro
X : Type u_1
Y : Type ?u.646605
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
x✝ y z : X
ι : Type ?u.646620
F U V : Set X
hU : IsPathConnected U
hV : IsPathConnected V
x : X
xU : x ∈ U
xV : x ∈ V
⊢ IsPathConnected (U ∪ V)

tactic: rintro y (yU | yV)
state_before: case intro.intro
X : Type u_1
Y : Type ?u.646605
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
x✝ y z : X
ι : Type ?u.646620
F U V : Set X
hU : IsPathConnected U
hV : IsPathConnected V
x : X
xU : x ∈ U
xV : x ∈ V
⊢ ∀ {y : X}, y ∈ U ∪ V → JoinedIn (U ∪ V) x y

tactic: exact (hU.joinedIn x xU y yU).mono (subset_union_left U V)
state_before: case intro.intro.inl
X : Type u_1
Y : Type ?u.646605
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
x✝ y✝ z : X
ι : Type ?u.646620
F U V : Set X
hU : IsPathConnected U
hV : IsPathConnected V
x : X
xU : x ∈ U
xV : x ∈ V
y : X
yU : y ∈ U
⊢ JoinedIn (U ∪ V) x y

tactic: exact (hV.joinedIn x xV y yV).mono (subset_union_right U V)
state_before: case intro.intro.inr
X : Type u_1
Y : Type ?u.646605
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
x✝ y✝ z : X
ι : Type ?u.646620
F U V : Set X
hU : IsPathConnected U
hV : IsPathConnected V
x : X
xU : x ∈ U
xV : x ∈ V
y : X
yV : y ∈ V
⊢ JoinedIn (U ∪ V) x y
Theorem: inv_le_of_neg
File: Mathlib/Algebra/Order/Field/Basic.lean

tactic: rw [← inv_le_inv_of_neg hb (inv_lt_zero.2 ha), inv_inv]
state_before: ι : Type ?u.150435
α : Type u_1
β : Type ?u.150441
inst✝ : LinearOrderedField α
a b c d : α
n : ℤ
ha : a < 0
hb : b < 0
⊢ a⁻¹ ≤ b ↔ b⁻¹ ≤ a
Theorem: Real.cos_zero
File: Mathlib/Data/Complex/Exponential.lean

tactic: simp [cos]
state_before: x y : ℝ
⊢ cos 0 = 1
Theorem: Multiset.finite_toSet
File: Mathlib/Data/Set/Finite.lean

tactic: classical simpa only [← Multiset.mem_toFinset] using s.toFinset.finite_toSet
state_before: α : Type u
β : Type v
ι : Sort w
γ : Type x
s : Multiset α
⊢ Set.Finite {x | x ∈ s}

tactic: simpa only [← Multiset.mem_toFinset] using s.toFinset.finite_toSet
state_before: α : Type u
β : Type v
ι : Sort w
γ : Type x
s : Multiset α
⊢ Set.Finite {x | x ∈ s}
Theorem: List.chain'_reverse
File: Mathlib/Data/List/Chain.lean

tactic: simp only [chain'_singleton, reverse_singleton]
state_before: α : Type u
β : Type v
R r : α → α → Prop
l l₁ l₂ : List α
a✝ b a : α
⊢ Chain' R (reverse [a]) ↔ Chain' (flip R) [a]

tactic: rw [chain'_cons, reverse_cons, reverse_cons, append_assoc, cons_append, nil_append,
  chain'_split, ← reverse_cons, @chain'_reverse (b :: l), and_comm, chain'_pair, flip]
state_before: α : Type u
β : Type v
R r : α → α → Prop
l✝ l₁ l₂ : List α
a✝ b✝ a b : α
l : List α
⊢ Chain' R (reverse (a :: b :: l)) ↔ Chain' (flip R) (a :: b :: l)
Theorem: PolynomialModule.comp_single
File: Mathlib/Data/Polynomial/Module.lean

tactic: rw [comp_apply]
state_before: R : Type u_1
M : Type u_2
inst✝¹² : CommRing R
inst✝¹¹ : AddCommGroup M
inst✝¹⁰ : Module R M
I : Ideal R
S : Type ?u.629765
inst✝⁹ : CommSemiring S
inst✝⁸ : Algebra S R
inst✝⁷ : Module S M
inst✝⁶ : IsScalarTower S R M
R' : Type ?u.632417
M' : Type ?u.632420
inst✝⁵ : CommRing R'
inst✝⁴ : AddCommGroup M'
inst✝³ : Module R' M'
inst✝² : Algebra R R'
inst✝¹ : Module R M'
inst✝ : IsScalarTower R R' M'
p : R[X]
i : ℕ
m : M
⊢ ↑(comp p) (↑(single R i) m) = p ^ i • ↑(single R 0) m

tactic: erw [map_single, eval_single]
state_before: R : Type u_1
M : Type u_2
inst✝¹² : CommRing R
inst✝¹¹ : AddCommGroup M
inst✝¹⁰ : Module R M
I : Ideal R
S : Type ?u.629765
inst✝⁹ : CommSemiring S
inst✝⁸ : Algebra S R
inst✝⁷ : Module S M
inst✝⁶ : IsScalarTower S R M
R' : Type ?u.632417
M' : Type ?u.632420
inst✝⁵ : CommRing R'
inst✝⁴ : AddCommGroup M'
inst✝³ : Module R' M'
inst✝² : Algebra R R'
inst✝¹ : Module R M'
inst✝ : IsScalarTower R R' M'
p : R[X]
i : ℕ
m : M
⊢ ↑(eval p) (↑(map R[X] (lsingle R 0)) (↑(single R i) m)) = p ^ i • ↑(single R 0) m

tactic: rfl
state_before: R : Type u_1
M : Type u_2
inst✝¹² : CommRing R
inst✝¹¹ : AddCommGroup M
inst✝¹⁰ : Module R M
I : Ideal R
S : Type ?u.629765
inst✝⁹ : CommSemiring S
inst✝⁸ : Algebra S R
inst✝⁷ : Module S M
inst✝⁶ : IsScalarTower S R M
R' : Type ?u.632417
M' : Type ?u.632420
inst✝⁵ : CommRing R'
inst✝⁴ : AddCommGroup M'
inst✝³ : Module R' M'
inst✝² : Algebra R R'
inst✝¹ : Module R M'
inst✝ : IsScalarTower R R' M'
p : R[X]
i : ℕ
m : M
⊢ p ^ i • ↑(lsingle R 0) m = p ^ i • ↑(single R 0) m
Theorem: Monotone.cauchySeq_alternating_series_of_tendsto_zero
File: Mathlib/Analysis/SpecificLimits/Normed.lean

tactic: simp_rw [mul_comm]
state_before: α : Type ?u.1705167
β : Type ?u.1705170
ι : Type ?u.1705173
E : Type ?u.1705176
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
b : ℝ
f : ℕ → ℝ
z : ℕ → E
hfa : Monotone f
hf0 : Tendsto f atTop (𝓝 0)
⊢ CauchySeq fun n => ∑ i in Finset.range (n + 1), (-1) ^ i * f i

tactic: exact hfa.cauchySeq_series_mul_of_tendsto_zero_of_bounded hf0 norm_sum_neg_one_pow_le
state_before: α : Type ?u.1705167
β : Type ?u.1705170
ι : Type ?u.1705173
E : Type ?u.1705176
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
b : ℝ
f : ℕ → ℝ
z : ℕ → E
hfa : Monotone f
hf0 : Tendsto f atTop (𝓝 0)
⊢ CauchySeq fun n => ∑ x in Finset.range (n + 1), f x * (-1) ^ x
Theorem: Ordnode.BalancedSz.symm
File: Mathlib/Data/Ordmap/Ordset.lean

tactic: rw [add_comm]
state_before: α : Type ?u.7857
l r : ℕ
⊢ l + r ≤ 1 → r + l ≤ 1

tactic: exact id
state_before: α : Type ?u.7857
l r : ℕ
⊢ r + l ≤ 1 → r + l ≤ 1
Theorem: Finset.weightedVSubOfPoint_congr
File: Mathlib/LinearAlgebra/AffineSpace/Combination.lean

tactic: simp_rw [weightedVSubOfPoint_apply]
state_before: k : Type u_2
V : Type u_4
P : Type u_3
inst✝² : Ring k
inst✝¹ : AddCommGroup V
inst✝ : Module k V
S : AffineSpace V P
ι : Type u_1
s : Finset ι
ι₂ : Type ?u.22097
s₂ : Finset ι₂
w₁ w₂ : ι → k
hw : ∀ (i : ι), i ∈ s → w₁ i = w₂ i
p₁ p₂ : ι → P
hp : ∀ (i : ι), i ∈ s → p₁ i = p₂ i
b : P
⊢ ↑(weightedVSubOfPoint s p₁ b) w₁ = ↑(weightedVSubOfPoint s p₂ b) w₂

tactic: refine sum_congr rfl fun i hi => ?_
state_before: k : Type u_2
V : Type u_4
P : Type u_3
inst✝² : Ring k
inst✝¹ : AddCommGroup V
inst✝ : Module k V
S : AffineSpace V P
ι : Type u_1
s : Finset ι
ι₂ : Type ?u.22097
s₂ : Finset ι₂
w₁ w₂ : ι → k
hw : ∀ (i : ι), i ∈ s → w₁ i = w₂ i
p₁ p₂ : ι → P
hp : ∀ (i : ι), i ∈ s → p₁ i = p₂ i
b : P
⊢ ∑ i in s, w₁ i • (p₁ i -ᵥ b) = ∑ i in s, w₂ i • (p₂ i -ᵥ b)

tactic: rw [hw i hi, hp i hi]
state_before: k : Type u_2
V : Type u_4
P : Type u_3
inst✝² : Ring k
inst✝¹ : AddCommGroup V
inst✝ : Module k V
S : AffineSpace V P
ι : Type u_1
s : Finset ι
ι₂ : Type ?u.22097
s₂ : Finset ι₂
w₁ w₂ : ι → k
hw : ∀ (i : ι), i ∈ s → w₁ i = w₂ i
p₁ p₂ : ι → P
hp : ∀ (i : ι), i ∈ s → p₁ i = p₂ i
b : P
i : ι
hi : i ∈ s
⊢ w₁ i • (p₁ i -ᵥ b) = w₂ i • (p₂ i -ᵥ b)
Theorem: Cardinal.aleph0_le_add_iff
File: Mathlib/SetTheory/Cardinal/Basic.lean

tactic: simp only [← not_lt, add_lt_aleph0_iff, not_and_or]
state_before: α β : Type u
a b : Cardinal
⊢ ℵ₀ ≤ a + b ↔ ℵ₀ ≤ a ∨ ℵ₀ ≤ b
Theorem: PartENat.get_eq_iff_eq_coe
File: Mathlib/Data/Nat/PartENat.lean

tactic: rw [get_eq_iff_eq_some]
state_before: a : PartENat
ha : a.Dom
b : ℕ
⊢ Part.get a ha = b ↔ a = ↑b

tactic: rfl
state_before: a : PartENat
ha : a.Dom
b : ℕ
⊢ a = ↑b ↔ a = ↑b
Theorem: AbstractCompletion.inverse_compare
File: Mathlib/Topology/UniformSpace/AbstractCompletion.lean

tactic: have uc := pkg.uniformContinuous_compare pkg'
state_before: α : Type u_1
inst✝¹ : UniformSpace α
pkg : AbstractCompletion α
β : Type ?u.20377
inst✝ : UniformSpace β
pkg' : AbstractCompletion α
⊢ compare pkg pkg' ∘ compare pkg' pkg = id

tactic: have uc' := pkg'.uniformContinuous_compare pkg
state_before: α : Type u_1
inst✝¹ : UniformSpace α
pkg : AbstractCompletion α
β : Type ?u.20377
inst✝ : UniformSpace β
pkg' : AbstractCompletion α
uc : UniformContinuous (compare pkg pkg')
⊢ compare pkg pkg' ∘ compare pkg' pkg = id

tactic: apply pkg'.funext (uc.comp uc').continuous continuous_id
state_before: α : Type u_1
inst✝¹ : UniformSpace α
pkg : AbstractCompletion α
β : Type ?u.20377
inst✝ : UniformSpace β
pkg' : AbstractCompletion α
uc : UniformContinuous (compare pkg pkg')
uc' : UniformContinuous (compare pkg' pkg)
⊢ compare pkg pkg' ∘ compare pkg' pkg = id

tactic: intro a
state_before: α : Type u_1
inst✝¹ : UniformSpace α
pkg : AbstractCompletion α
β : Type ?u.20377
inst✝ : UniformSpace β
pkg' : AbstractCompletion α
uc : UniformContinuous (compare pkg pkg')
uc' : UniformContinuous (compare pkg' pkg)
⊢ ∀ (a : α), (compare pkg pkg' ∘ compare pkg' pkg) (coe pkg' a) = id (coe pkg' a)

tactic: rw [comp_apply, pkg'.compare_coe pkg, pkg.compare_coe pkg']
state_before: α : Type u_1
inst✝¹ : UniformSpace α
pkg : AbstractCompletion α
β : Type ?u.20377
inst✝ : UniformSpace β
pkg' : AbstractCompletion α
uc : UniformContinuous (compare pkg pkg')
uc' : UniformContinuous (compare pkg' pkg)
a : α
⊢ (compare pkg pkg' ∘ compare pkg' pkg) (coe pkg' a) = id (coe pkg' a)

tactic: rfl
state_before: α : Type u_1
inst✝¹ : UniformSpace α
pkg : AbstractCompletion α
β : Type ?u.20377
inst✝ : UniformSpace β
pkg' : AbstractCompletion α
uc : UniformContinuous (compare pkg pkg')
uc' : UniformContinuous (compare pkg' pkg)
a : α
⊢ coe pkg' a = id (coe pkg' a)
Theorem: Mathlib.Meta.NormNum.isRat_inv_neg_one
File: Mathlib/Tactic/NormNum/Basic.lean

tactic: simp [inv_neg_one]
state_before: α : Type u_1
inst✝ : DivisionRing α
⊢ (↑(Int.negOfNat 1))⁻¹ = ↑(Int.negOfNat 1)
Theorem: finsuppLequivDfinsupp_apply_apply
File: Mathlib/Data/Finsupp/ToDfinsupp.lean

tactic: simp only [@LinearEquiv.coe_coe]
state_before: ι : Type u_1
R : Type u_2
M : Type u_3
inst✝⁴ : DecidableEq ι
inst✝³ : Semiring R
inst✝² : AddCommMonoid M
inst✝¹ : (m : M) → Decidable (m ≠ 0)
inst✝ : Module R M
⊢ ↑(finsuppLequivDfinsupp R) = Finsupp.toDfinsupp

tactic: rfl
state_before: ι : Type u_1
R : Type u_2
M : Type u_3
inst✝⁴ : DecidableEq ι
inst✝³ : Semiring R
inst✝² : AddCommMonoid M
inst✝¹ : (m : M) → Decidable (m ≠ 0)
inst✝ : Module R M
⊢ ↑(finsuppLequivDfinsupp R) = Finsupp.toDfinsupp
Theorem: CategoryTheory.Adjunction.CoreHomEquiv.homEquiv_naturality_right_symm
File: Mathlib/CategoryTheory/Adjunction/Basic.lean

tactic: rw [Equiv.symm_apply_eq]
state_before: C : Type u₁
inst✝¹ : Category C
D : Type u₂
inst✝ : Category D
F : C ⥤ D
G : D ⥤ C
adj : CoreHomEquiv F G
X' X : C
Y Y' : D
f : X ⟶ G.obj Y
g : Y ⟶ Y'
⊢ ↑(homEquiv adj X Y').symm (f ≫ G.map g) = ↑(homEquiv adj X Y).symm f ≫ g

tactic: simp
state_before: C : Type u₁
inst✝¹ : Category C
D : Type u₂
inst✝ : Category D
F : C ⥤ D
G : D ⥤ C
adj : CoreHomEquiv F G
X' X : C
Y Y' : D
f : X ⟶ G.obj Y
g : Y ⟶ Y'
⊢ f ≫ G.map g = ↑(homEquiv adj X Y') (↑(homEquiv adj X Y).symm f ≫ g)
Theorem: Nat.iSup_lt_succ
File: Mathlib/Data/Nat/Lattice.lean

tactic: simp [Nat.lt_succ_iff_lt_or_eq, iSup_or, iSup_sup_eq]
state_before: α : Type u_1
inst✝ : CompleteLattice α
u : ℕ → α
n : ℕ
⊢ (⨆ (k : ℕ) (_ : k < n + 1), u k) = (⨆ (k : ℕ) (_ : k < n), u k) ⊔ u n
Theorem: Matrix.diagonal_mul_diagonal
File: Mathlib/Data/Matrix/Basic.lean

tactic: ext i j
state_before: l : Type ?u.274619
m : Type ?u.274622
n : Type u_1
o : Type ?u.274628
m' : o → Type ?u.274633
n' : o → Type ?u.274638
R : Type ?u.274641
S : Type ?u.274644
α : Type v
β : Type w
γ : Type ?u.274651
inst✝² : NonUnitalNonAssocSemiring α
inst✝¹ : Fintype n
inst✝ : DecidableEq n
d₁ d₂ : n → α
⊢ diagonal d₁ ⬝ diagonal d₂ = diagonal fun i => d₁ i * d₂ i

tactic: by_cases i = j <;>
simp [h]
state_before: case a.h
l : Type ?u.274619
m : Type ?u.274622
n : Type u_1
o : Type ?u.274628
m' : o → Type ?u.274633
n' : o → Type ?u.274638
R : Type ?u.274641
S : Type ?u.274644
α : Type v
β : Type w
γ : Type ?u.274651
inst✝² : NonUnitalNonAssocSemiring α
inst✝¹ : Fintype n
inst✝ : DecidableEq n
d₁ d₂ : n → α
i j : n
⊢ (diagonal d₁ ⬝ diagonal d₂) i j = diagonal (fun i => d₁ i * d₂ i) i j