ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Mathlib/LinearAlgebra/PiTensorProduct.lean
ι : Type u_1
ι₂ : Type u_2
ι₃ : Type u_3
R : Type u_4
inst✝¹⁰ : CommSemiring R
R₁ : Type u_5
R₂ : Type u_6
s : ι → Type u_7
inst✝⁹ : (i : ι) → AddCommMonoid (s i)
inst✝⁸ : (i : ι) → Module R (s i)
M : Type u_8
inst✝⁷ : AddCommMonoid M
inst✝⁶ : Module R M
E : Type u_9
inst✝⁵ : AddCommMonoid E
inst✝⁴ : Module R E
F : Type u_10
inst✝³ : AddCommMonoid F
t : ι → Type u_11
inst✝² : (i : ι) → AddCommMonoid (t i)
inst✝¹ : (i : ι) → Module R (t i)
inst✝ : Subsingleton ι
i₀ : ι
x : ⨂[R] (x : ι), M
this : ∀ (f : ι → M) (z : M), (fun x => z) = update f i₀ z

⊢ ∀ (r : R) (f : ι → M), ((tprod R) fun x => (lift ((ofSubsingleton R M M i₀) LinearMap.id)) (r • (tprod R) f)) = r • (tprod R) f
-- Mathlib/LinearAlgebra/PiTensorProduct.lean
ι : Type u_1
ι₂ : Type u_2
ι₃ : Type u_3
R : Type u_4
inst✝¹⁰ : CommSemiring R
R₁ : Type u_5
R₂ : Type u_6
s : ι → Type u_7
inst✝⁹ : (i : ι) → AddCommMonoid (s i)
inst✝⁸ : (i : ι) → Module R (s i)
M : Type u_8
inst✝⁷ : AddCommMonoid M
inst✝⁶ : Module R M
E : Type u_9
inst✝⁵ : AddCommMonoid E
inst✝⁴ : Module R E
F : Type u_10
inst✝³ : AddCommMonoid F
t : ι → Type u_11
inst✝² : (i : ι) → AddCommMonoid (t i)
inst✝¹ : (i : ι) → Module R (t i)
inst✝ : Subsingleton ι
i₀ : ι
x : ⨂[R] (x : ι), M
this : ∀ (f : ι → M) (z : M), (fun x => z) = update f i₀ z

⊢ ∀ (x y : ⨂[R] (i : ι), M), ((tprod R) fun x_1 => (lift ((ofSubsingleton R M M i₀) LinearMap.id)) x) = x → ((tprod R) fun x => (lift ((ofSubsingleton R M M i₀) LinearMap.id)) y) = y → ((tprod R) fun x_1 => (lift ((ofSubsingleton R M M i₀) LinearMap.id)) (x + y)) = x + y
-- Mathlib/LinearAlgebra/PiTensorProduct.lean
ι : Type u_1
ι₂ : Type u_2
ι₃ : Type u_3
R : Type u_4
inst✝¹⁰ : CommSemiring R
R₁ : Type u_5
R₂ : Type u_6
s : ι → Type u_7
inst✝⁹ : (i : ι) → AddCommMonoid (s i)
inst✝⁸ : (i : ι) → Module R (s i)
M : Type u_8
inst✝⁷ : AddCommMonoid M
inst✝⁶ : Module R M
E : Type u_9
inst✝⁵ : AddCommMonoid E
inst✝⁴ : Module R E
F : Type u_10
inst✝³ : AddCommMonoid F
t✝ : ι → Type u_11
inst✝² : (i : ι) → AddCommMonoid (t✝ i)
inst✝¹ : (i : ι) → Module R (t✝ i)
inst✝ : Subsingleton ι
i₀ : ι
t : M

⊢ { toFun := ⇑(lift ((ofSubsingleton R M M i₀) LinearMap.id)), map_add' := ⋯, map_smul' := ⋯ }.toFun ((fun m => (tprod R) fun x => m) t) = t
-- Mathlib/LinearAlgebra/PiTensorProduct.lean
ι : Type u_1
ι₂ : Type u_2
ι₃ : Type u_3
R : Type u_4
inst✝¹⁰ : CommSemiring R
R₁ : Type u_5
R₂ : Type u_6
s : ι → Type u_7
inst✝⁹ : (i : ι) → AddCommMonoid (s i)
inst✝⁸ : (i : ι) → Module R (s i)
M : Type u_8
inst✝⁷ : AddCommMonoid M
inst✝⁶ : Module R M
E : Type u_9
inst✝⁵ : AddCommMonoid E
inst✝⁴ : Module R E
F : Type u_10
inst✝³ : AddCommMonoid F
t : ι → Type u_11
inst✝² : (i : ι) → AddCommMonoid (t i)
inst✝¹ : (i : ι) → Module R (t i)
inst✝ : Subsingleton ι
i₀ : ι
r : R
x : ⨂[R] (x : ι), M

⊢ (lift ((ofSubsingleton R M M i₀) LinearMap.id)) (r • x) = (RingHom.id R) r • (lift ((ofSubsingleton R M M i₀) LinearMap.id)) x
-- .lake/packages/batteries/Batteries/Data/List/Count.lean
α : Type u_1
p q : α → Bool
l : List α
h : ∀ (x : α), x ∈ l → p x = true → q x = true

⊢ countP p l ≤ countP q l
-- Mathlib/Analysis/Calculus/Deriv/Inv.lean
𝕜 : Type u
inst✝⁴ : NontriviallyNormedField 𝕜
F : Type v
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
E : Type w
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f f₀ f₁ g : 𝕜 → F
f' f₀' f₁' g' : F
x : 𝕜
s t : Set 𝕜
L : Filter 𝕜

⊢ fderiv 𝕜 (fun x => x⁻¹) x = smulRight 1 (-(x ^ 2)⁻¹)
-- Mathlib/Data/Nat/Factorization/Defs.lean
a b m n✝ p n k : ℕ
ih : (n ^ k).factorization = k • n.factorization

⊢ (n ^ (k + 1)).factorization = (k + 1) • n.factorization
-- Mathlib/Data/Nat/Factorization/Defs.lean
a b m n p k : ℕ
ih : (0 ^ k).factorization = k • factorization 0

⊢ (0 ^ (k + 1)).factorization = (k + 1) • factorization 0
-- Mathlib/Data/Nat/Factorization/Defs.lean
a b m n✝ p n k : ℕ
ih : (n ^ k).factorization = k • n.factorization
hn : n ≠ 0

⊢ (n ^ (k + 1)).factorization = (k + 1) • n.factorization
-- Mathlib/Data/List/Infix.lean
α : Type u_1
β : Type u_2
l✝ l₁ l₂ l₃ : List α
a b : α
m✝ n✝ : ℕ
l : List α
m n : ℕ
h : m ≤ n

⊢ drop n l <+ drop m l
-- Mathlib/Data/List/Infix.lean
α : Type u_1
β : Type u_2
l✝ l₁ l₂ l₃ : List α
a b : α
m✝ n✝ : ℕ
l : List α
m n : ℕ
h : m ≤ n

⊢ drop (n - m) (drop m l) <+ drop m l
-- Mathlib/CategoryTheory/MorphismProperty/Limits.lean
C : Type u
inst✝² : Category.{v, u} C
W : MorphismProperty C
J : Type u_1
inst✝¹ : W.RespectsIso
inst✝ : HasProductsOfShape J C
hW : ∀ (X₁ X₂ : J → C) (f : (j : J) → X₁ j ⟶ X₂ j), (∀ (j : J), W (f j)) → W (Pi.map f)
X₁ X₂ : Discrete J ⥤ C
c₁ : Cone X₁
c₂ : Cone X₂
hc₁ : IsLimit c₁
hc₂ : IsLimit c₂
f : X₁ ⟶ X₂
hf : W.functorCategory (Discrete J) f
φ : (j : J) → X₁.obj { as := j } ⟶ X₂.obj { as := j } := fun j => f.app { as := j }
hf' : W (Pi.map φ)

⊢ W (hc₂.lift { pt := c₁.pt, π := c₁.π ≫ f })
-- Mathlib/CategoryTheory/MorphismProperty/Limits.lean
C : Type u
inst✝² : Category.{v, u} C
W : MorphismProperty C
J : Type u_1
inst✝¹ : W.RespectsIso
inst✝ : HasProductsOfShape J C
hW : ∀ (X₁ X₂ : J → C) (f : (j : J) → X₁ j ⟶ X₂ j), (∀ (j : J), W (f j)) → W (Pi.map f)
X₁ X₂ : Discrete J ⥤ C
c₁ : Cone X₁
c₂ : Cone X₂
hc₁ : IsLimit c₁
hc₂ : IsLimit c₂
f : X₁ ⟶ X₂
hf : W.functorCategory (Discrete J) f
φ : (j : J) → X₁.obj { as := j } ⟶ X₂.obj { as := j } := fun j => f.app { as := j }
hf' : W (Pi.map φ)

⊢ Arrow.mk (hc₂.lift { pt := c₁.pt, π := c₁.π ≫ f }) ≅ Arrow.mk (Pi.map φ)
-- Mathlib/CategoryTheory/MorphismProperty/Limits.lean
C : Type u
inst✝² : Category.{v, u} C
W : MorphismProperty C
J : Type u_1
inst✝¹ : W.RespectsIso
inst✝ : HasProductsOfShape J C
hW : ∀ (X₁ X₂ : J → C) (f : (j : J) → X₁ j ⟶ X₂ j), (∀ (j : J), W (f j)) → W (Pi.map f)
X₁ X₂ : Discrete J ⥤ C
c₁ : Cone X₁
c₂ : Cone X₂
hc₁ : IsLimit c₁
hc₂ : IsLimit c₂
f : X₁ ⟶ X₂
hf : W.functorCategory (Discrete J) f
φ : (j : J) → X₁.obj { as := j } ⟶ X₂.obj { as := j } := fun j => f.app { as := j }
hf' : W (Pi.map φ)

⊢ (hc₁.conePointUniqueUpToIso (limit.isLimit X₁) ≪≫ (Pi.isoLimit X₁).symm).hom ≫ (Arrow.mk (Pi.map φ)).hom = (Arrow.mk (hc₂.lift { pt := c₁.pt, π := c₁.π ≫ f })).hom ≫ (hc₂.conePointUniqueUpToIso (limit.isLimit X₂) ≪≫ (Pi.isoLimit X₂).symm).hom
-- Mathlib/CategoryTheory/MorphismProperty/Limits.lean
C : Type u
inst✝² : Category.{v, u} C
W : MorphismProperty C
J : Type u_1
inst✝¹ : W.RespectsIso
inst✝ : HasProductsOfShape J C
hW : ∀ (X₁ X₂ : J → C) (f : (j : J) → X₁ j ⟶ X₂ j), (∀ (j : J), W (f j)) → W (Pi.map f)
X₁ X₂ : Discrete J ⥤ C
c₁ : Cone X₁
c₂ : Cone X₂
hc₁ : IsLimit c₁
hc₂ : IsLimit c₂
f : X₁ ⟶ X₂
hf : W.functorCategory (Discrete J) f
φ : (j : J) → X₁.obj { as := j } ⟶ X₂.obj { as := j } := fun j => f.app { as := j }
hf' : W (Pi.map φ)

⊢ ∀ (j : Discrete J), ((hc₁.conePointUniqueUpToIso (limit.isLimit X₁) ≪≫ (Pi.isoLimit X₁).symm).hom ≫ (Arrow.mk (Pi.map φ)).hom) ≫ limit.π (Discrete.functor fun j => X₂.obj { as := j }) j = ((Arrow.mk (hc₂.lift { pt := c₁.pt, π := c₁.π ≫ f })).hom ≫ (hc₂.conePointUniqueUpToIso (limit.isLimit X₂) ≪≫ (Pi.isoLimit X₂).symm).hom) ≫ limit.π (Discrete.functor fun j => X₂.obj { as := j }) j
-- Mathlib/MeasureTheory/Measure/EverywherePos.lean
α : Type u_1
inst✝¹ : TopologicalSpace α
inst✝ : MeasurableSpace α
μ ν : Measure α
s k : Set α
hs : μ.IsEverywherePos s
c : ℝ≥0∞
hc : c ≠ 0
x : α
hx : x ∈ s
n : Set α
hn : n ∈ 𝓝[s] x

⊢ 0 < (c • μ) n
-- Mathlib/Algebra/Homology/Homotopy.lean
ι : Type u_1
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : Preadditive V
c : ComplexShape ι
C D E : HomologicalComplex V c
f g : C ⟶ D
h k : D ⟶ E
i k₀ : ι
hk₀ : ∀ (l : ι), ¬c.Rel k₀ l
hk₀' : ∀ (l : ι), ¬c.Rel l k₀
hom : (i j : ι) → C.X i ⟶ D.X j

⊢ C.d k₀ (c.next k₀) ≫ hom (c.next k₀) k₀ + hom k₀ (c.prev k₀) ≫ D.d (c.prev k₀) k₀ = 0
-- Mathlib/MeasureTheory/Covering/Differentiation.lean
α : Type u_1
inst✝⁵ : MetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : SecondCountableTopology α
inst✝² : BorelSpace α
inst✝¹ : IsLocallyFiniteMeasure μ
ρ : Measure α
inst✝ : IsLocallyFiniteMeasure ρ
hρ : ρ ≪ μ
p q : ℝ≥0
hpq : p < q
s : Set α := {x | ∃ c, Tendsto (fun a => ρ a / μ a) (v.filterAt x) (𝓝 c)}
o : ℕ → Set α := spanningSets (ρ + μ)
u : ℕ → Set α := fun n => s ∩ {x | v.limRatio ρ x < ↑p} ∩ o n
w : ℕ → Set α := fun n => s ∩ {x | ↑q < v.limRatio ρ x} ∩ o n
m n : ℕ

⊢ μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) = 0
-- Mathlib/MeasureTheory/Covering/Differentiation.lean
α : Type u_1
inst✝⁵ : MetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : SecondCountableTopology α
inst✝² : BorelSpace α
inst✝¹ : IsLocallyFiniteMeasure μ
ρ : Measure α
inst✝ : IsLocallyFiniteMeasure ρ
hρ : ρ ≪ μ
p q : ℝ≥0
hpq : p < q
s : Set α := {x | ∃ c, Tendsto (fun a => ρ a / μ a) (v.filterAt x) (𝓝 c)}
o : ℕ → Set α := spanningSets (ρ + μ)
u : ℕ → Set α := fun n => s ∩ {x | v.limRatio ρ x < ↑p} ∩ o n
w : ℕ → Set α := fun n => s ∩ {x | ↑q < v.limRatio ρ x} ∩ o n
m n : ℕ
I : (ρ + μ) (u m) ≠ ⊤

⊢ μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) = 0
-- Mathlib/MeasureTheory/Covering/Differentiation.lean
α : Type u_1
inst✝⁵ : MetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : SecondCountableTopology α
inst✝² : BorelSpace α
inst✝¹ : IsLocallyFiniteMeasure μ
ρ : Measure α
inst✝ : IsLocallyFiniteMeasure ρ
hρ : ρ ≪ μ
p q : ℝ≥0
hpq : p < q
s : Set α := {x | ∃ c, Tendsto (fun a => ρ a / μ a) (v.filterAt x) (𝓝 c)}
o : ℕ → Set α := spanningSets (ρ + μ)
u : ℕ → Set α := fun n => s ∩ {x | v.limRatio ρ x < ↑p} ∩ o n
w : ℕ → Set α := fun n => s ∩ {x | ↑q < v.limRatio ρ x} ∩ o n
m n : ℕ
I : (ρ + μ) (u m) ≠ ⊤
J : (ρ + μ) (w n) ≠ ⊤

⊢ μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) = 0
-- Mathlib/MeasureTheory/Covering/Differentiation.lean
α : Type u_1
inst✝⁵ : MetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : SecondCountableTopology α
inst✝² : BorelSpace α
inst✝¹ : IsLocallyFiniteMeasure μ
ρ : Measure α
inst✝ : IsLocallyFiniteMeasure ρ
hρ : ρ ≪ μ
p q : ℝ≥0
hpq : p < q
s : Set α := {x | ∃ c, Tendsto (fun a => ρ a / μ a) (v.filterAt x) (𝓝 c)}
o : ℕ → Set α := spanningSets (ρ + μ)
u : ℕ → Set α := fun n => s ∩ {x | v.limRatio ρ x < ↑p} ∩ o n
w : ℕ → Set α := fun n => s ∩ {x | ↑q < v.limRatio ρ x} ∩ o n
m n : ℕ
I : (ρ + μ) (u m) ≠ ⊤
J : (ρ + μ) (w n) ≠ ⊤
A : ρ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) ≤ ↑p * μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n))

⊢ μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) = 0
-- Mathlib/MeasureTheory/Covering/Differentiation.lean
α : Type u_1
inst✝⁵ : MetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : SecondCountableTopology α
inst✝² : BorelSpace α
inst✝¹ : IsLocallyFiniteMeasure μ
ρ : Measure α
inst✝ : IsLocallyFiniteMeasure ρ
hρ : ρ ≪ μ
p q : ℝ≥0
hpq : p < q
s : Set α := {x | ∃ c, Tendsto (fun a => ρ a / μ a) (v.filterAt x) (𝓝 c)}
o : ℕ → Set α := spanningSets (ρ + μ)
u : ℕ → Set α := fun n => s ∩ {x | v.limRatio ρ x < ↑p} ∩ o n
w : ℕ → Set α := fun n => s ∩ {x | ↑q < v.limRatio ρ x} ∩ o n
m n : ℕ
I : (ρ + μ) (u m) ≠ ⊤
J : (ρ + μ) (w n) ≠ ⊤
A : ρ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) ≤ ↑p * μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n))
B : ↑q * μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) ≤ ρ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n))

⊢ μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) = 0
-- Mathlib/MeasureTheory/Covering/Differentiation.lean
α : Type u_1
inst✝⁵ : MetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : SecondCountableTopology α
inst✝² : BorelSpace α
inst✝¹ : IsLocallyFiniteMeasure μ
ρ : Measure α
inst✝ : IsLocallyFiniteMeasure ρ
hρ : ρ ≪ μ
p q : ℝ≥0
hpq : p < q
s : Set α := {x | ∃ c, Tendsto (fun a => ρ a / μ a) (v.filterAt x) (𝓝 c)}
o : ℕ → Set α := spanningSets (ρ + μ)
u : ℕ → Set α := fun n => s ∩ {x | v.limRatio ρ x < ↑p} ∩ o n
w : ℕ → Set α := fun n => s ∩ {x | ↑q < v.limRatio ρ x} ∩ o n
m n : ℕ
I : (ρ + μ) (u m) ≠ ⊤
J : (ρ + μ) (w n) ≠ ⊤
A : ρ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) ≤ ↑p * μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n))
B : ↑q * μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) ≤ ρ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n))
h : ¬μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) = 0

⊢ False
-- Mathlib/MeasureTheory/Covering/Differentiation.lean
α : Type u_1
inst✝⁵ : MetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : SecondCountableTopology α
inst✝² : BorelSpace α
inst✝¹ : IsLocallyFiniteMeasure μ
ρ : Measure α
inst✝ : IsLocallyFiniteMeasure ρ
hρ : ρ ≪ μ
p q : ℝ≥0
hpq : p < q
s : Set α := {x | ∃ c, Tendsto (fun a => ρ a / μ a) (v.filterAt x) (𝓝 c)}
o : ℕ → Set α := spanningSets (ρ + μ)
u : ℕ → Set α := fun n => s ∩ {x | v.limRatio ρ x < ↑p} ∩ o n
w : ℕ → Set α := fun n => s ∩ {x | ↑q < v.limRatio ρ x} ∩ o n
m n : ℕ
I : (ρ + μ) (u m) ≠ ⊤
J : (ρ + μ) (w n) ≠ ⊤
A : ρ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) ≤ ↑p * μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n))
B : ↑q * μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) ≤ ρ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n))
h : ¬μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) = 0

⊢ ρ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) < ρ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n))
-- Mathlib/GroupTheory/Complement.lean
G : Type u_1
inst✝ : Group G
H K : Subgroup G
S T : Set G
hST : IsComplement S T
hHT : IsComplement (↑H) T
hSK : IsComplement S ↑K
g : G
h1 : 1 ∈ S

⊢ ↑(hST.equiv g).1 = 1 ↔ g ∈ T
-- Mathlib/Algebra/Polynomial/HasseDeriv.lean
R : Type u_1
inst✝ : Semiring R
k : ℕ
f : R[X]
r : R
hk : 0 < k

⊢ (hasseDeriv k) (C r) = 0
-- Mathlib/Order/Filter/Prod.lean
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ι : Sort u_5
s : Set α
t : Set β
f : Filter α
g : Filter β
p : α × β → Prop

⊢ (∀ᶠ (x : α × β) in f ×ˢ g, p x) ↔ ∀ᶠ (y : β × α) in g ×ˢ f, p y.swap
-- Mathlib/Order/Filter/Prod.lean
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ι : Sort u_5
s : Set α
t : Set β
f : Filter α
g : Filter β
p : α × β → Prop

⊢ (∀ᶠ (x : α × β) in map (fun p => (p.2, p.1)) (g ×ˢ f), p x) ↔ ∀ᶠ (y : β × α) in g ×ˢ f, p y.swap
-- Mathlib/Analysis/Asymptotics/Asymptotics.lean
α : Type u_1
β : Type u_2
inst✝³ : TopologicalSpace α
inst✝² : TopologicalSpace β
E : Type u_3
inst✝¹ : Norm E
F : Type u_4
inst✝ : Norm F
e : PartialHomeomorph α β
b : β
hb : b ∈ e.target
f : β → E
g : β → F

⊢ f =o[𝓝 b] g ↔ (f ∘ ↑e) =o[𝓝 (↑e.symm b)] (g ∘ ↑e)
-- Mathlib/Analysis/Asymptotics/Asymptotics.lean
α : Type u_1
β : Type u_2
inst✝³ : TopologicalSpace α
inst✝² : TopologicalSpace β
E : Type u_3
inst✝¹ : Norm E
F : Type u_4
inst✝ : Norm F
e : PartialHomeomorph α β
b : β
hb : b ∈ e.target
f : β → E
g : β → F

⊢ (∀ ⦃c : ℝ⦄, 0 < c → IsBigOWith c (𝓝 b) f g) ↔ ∀ ⦃c : ℝ⦄, 0 < c → IsBigOWith c (𝓝 (↑e.symm b)) (f ∘ ↑e) (g ∘ ↑e)
-- Mathlib/Analysis/BoxIntegral/Box/SubboxInduction.lean
ι : Type u_1
I✝ J I : Box ι
s t : Set ι
h : s ≠ t

⊢ Disjoint ↑(I.splitCenterBox s) ↑(I.splitCenterBox t)
-- Mathlib/Analysis/BoxIntegral/Box/SubboxInduction.lean
ι : Type u_1
I✝ J I : Box ι
s t : Set ι
h : s ≠ t

⊢ ↑(I.splitCenterBox s) ⊓ ↑(I.splitCenterBox t) ≤ ⊥
-- Mathlib/Analysis/BoxIntegral/Box/SubboxInduction.lean
ι : Type u_1
I✝ J I : Box ι
s t : Set ι
h : s ≠ t
y : ι → ℝ
hs : y ∈ ↑(I.splitCenterBox s)
ht : y ∈ ↑(I.splitCenterBox t)
i : ι

⊢ i ∈ s ↔ i ∈ t
-- Mathlib/Analysis/BoxIntegral/Box/SubboxInduction.lean
ι : Type u_1
I✝ J I : Box ι
s t : Set ι
h : s ≠ t
y : ι → ℝ
hs : y ∈ I ∧ ∀ (i : ι), (I.lower i + I.upper i) / 2 < y i ↔ i ∈ s
ht : y ∈ I ∧ ∀ (i : ι), (I.lower i + I.upper i) / 2 < y i ↔ i ∈ t
i : ι

⊢ i ∈ s ↔ i ∈ t
-- Mathlib/Combinatorics/SimpleGraph/Matching.lean
V : Type u_1
G G' : SimpleGraph V
M M' : G.Subgraph
v w : V
inst✝² : Fintype V
inst✝¹ : DecidableRel G.Adj
inst✝ : DecidableEq V
c : G.ConnectedComponent
hM : M.IsPerfectMatching

⊢ Even (Fintype.card ↑c.supp)
-- Mathlib/LinearAlgebra/Matrix/ToLin.lean
R : Type u_1
S : Type u_2
inst✝⁴ : CommRing R
inst✝³ : Ring S
inst✝² : Algebra R S
m : Type u_3
inst✝¹ : Fintype m
inst✝ : DecidableEq m
b : Basis m R S
x x' : S
h : (leftMulMatrix b) x = (leftMulMatrix b) x'

⊢ ((lmul R S) x) 1 = ((lmul R S) x') 1
-- Mathlib/Probability/Process/Stopping.lean
Ω : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace Ω
inst✝ : LinearOrder ι
f : Filtration ι m
τ π : Ω → ι
hτ : IsStoppingTime f τ
i : ι
this : {ω | i < τ ω} = {ω | τ ω ≤ i}ᶜ

⊢ MeasurableSet {ω | τ ω ≤ i}ᶜ
-- Mathlib/Topology/Algebra/Module/Basic.lean
R₁ : Type u_1
R₂ : Type u_2
R₃ : Type u_3
inst✝¹⁷ : Semiring R₁
inst✝¹⁶ : Semiring R₂
inst✝¹⁵ : Semiring R₃
σ₁₂ : R₁ →+* R₂
σ₂₃ : R₂ →+* R₃
σ₁₃ : R₁ →+* R₃
M₁ : Type u_4
inst✝¹⁴ : TopologicalSpace M₁
inst✝¹³ : AddCommMonoid M₁
M'₁ : Type u_5
inst✝¹² : TopologicalSpace M'₁
inst✝¹¹ : AddCommMonoid M'₁
M₂ : Type u_6
inst✝¹⁰ : TopologicalSpace M₂
inst✝⁹ : AddCommMonoid M₂
M₃ : Type u_7
inst✝⁸ : TopologicalSpace M₃
inst✝⁷ : AddCommMonoid M₃
M₄ : Type u_8
inst✝⁶ : TopologicalSpace M₄
inst✝⁵ : AddCommMonoid M₄
inst✝⁴ : Module R₁ M₁
inst✝³ : Module R₁ M'₁
inst✝² : Module R₂ M₂
inst✝¹ : Module R₃ M₃
inst✝ : ContinuousAdd M₂
ι : Type u_9
t : Finset ι
f : ι → M₁ →SL[σ₁₂] M₂

⊢ ⇑(∑ d ∈ t, f d) = ∑ d ∈ t, ⇑(f d)
-- Mathlib/CategoryTheory/Limits/HasLimits.lean
J : Type u₁
inst✝⁴ : Category.{v₁, u₁} J
K : Type u₂
inst✝³ : Category.{v₂, u₂} K
C : Type u
inst✝² : Category.{v, u} C
F✝ F : J ⥤ C
inst✝¹ : HasLimit F
G : K ⥤ C
inst✝ : HasLimit G
e : J ≌ K
w : e.functor ⋙ G ≅ F
k : K

⊢ (isoOfEquivalence e w).hom ≫ limit.π G k = limit.π F (e.inverse.obj k) ≫ w.inv.app (e.inverse.obj k) ≫ G.map (e.counit.app k)
-- Mathlib/CategoryTheory/Limits/HasLimits.lean
J : Type u₁
inst✝⁴ : Category.{v₁, u₁} J
K : Type u₂
inst✝³ : Category.{v₂, u₂} K
C : Type u
inst✝² : Category.{v, u} C
F✝ F : J ⥤ C
inst✝¹ : HasLimit F
G : K ⥤ C
inst✝ : HasLimit G
e : J ≌ K
w : e.functor ⋙ G ≅ F
k : K

⊢ (limit.isLimit G).lift ((Cones.equivalenceOfReindexing e.symm ((isoWhiskerLeft e.inverse w).symm ≪≫ e.invFunIdAssoc G)).functor.obj (limit.cone F)) ≫ limit.π G k = limit.π F (e.inverse.obj k) ≫ w.inv.app (e.inverse.obj k) ≫ G.map (e.counit.app k)
-- Mathlib/Analysis/RCLike/Basic.lean
K : Type u_1
E : Type u_2
inst✝ : RCLike K
z w : K

⊢ z < 0 ↔ re z < 0 ∧ im z = 0
-- Mathlib/Dynamics/FixedPoints/Basic.lean
α✝ : Type u
β : Type v
f✝ fa g : α✝ → α✝
x✝ y : α✝
fb : β → β
m n k : ℕ
e : Perm α✝
α : Type u_1
f : α → α
x : α

⊢ x ∈ fixedPoints f ↔ f x = x
-- Mathlib/Algebra/Lie/Weights/Killing.lean
R : Type u_1
K : Type u_2
L : Type u_3
inst✝⁹ : CommRing R
inst✝⁸ : LieRing L
inst✝⁷ : LieAlgebra R L
inst✝⁶ : Field K
inst✝⁵ : LieAlgebra K L
inst✝⁴ : FiniteDimensional K L
H : LieSubalgebra K L
inst✝³ : H.IsCartanSubalgebra
inst✝² : IsTriangularizable K (↥H) L
inst✝¹ : IsKilling K L
inst✝ : PerfectField K
α : Weight K (↥H) L

⊢ ↑(corootSpace ⇑α) = span K {(cartanEquivDual H).symm (Weight.toLinear K (↥H) L α)}
-- Mathlib/Algebra/Lie/Weights/Killing.lean
R : Type u_1
K : Type u_2
L : Type u_3
inst✝⁹ : CommRing R
inst✝⁸ : LieRing L
inst✝⁷ : LieAlgebra R L
inst✝⁶ : Field K
inst✝⁵ : LieAlgebra K L
inst✝⁴ : FiniteDimensional K L
H : LieSubalgebra K L
inst✝³ : H.IsCartanSubalgebra
inst✝² : IsTriangularizable K (↥H) L
inst✝¹ : IsKilling K L
inst✝ : PerfectField K
α : Weight K (↥H) L

⊢ ↑(corootSpace ⇑α) ≤ span K {(cartanEquivDual H).symm (Weight.toLinear K (↥H) L α)}
-- Mathlib/Algebra/Lie/Weights/Killing.lean
R : Type u_1
K : Type u_2
L : Type u_3
inst✝⁹ : CommRing R
inst✝⁸ : LieRing L
inst✝⁷ : LieAlgebra R L
inst✝⁶ : Field K
inst✝⁵ : LieAlgebra K L
inst✝⁴ : FiniteDimensional K L
H : LieSubalgebra K L
inst✝³ : H.IsCartanSubalgebra
inst✝² : IsTriangularizable K (↥H) L
inst✝¹ : IsKilling K L
inst✝ : PerfectField K
α : Weight K (↥H) L

⊢ span K {(cartanEquivDual H).symm (Weight.toLinear K (↥H) L α)} ≤ ↑(corootSpace ⇑α)
-- Mathlib/LinearAlgebra/Matrix/Gershgorin.lean
K : Type u_1
n : Type u_2
inst✝² : NormedField K
inst✝¹ : Fintype n
inst✝ : DecidableEq n
A : Matrix n n K
h : ∀ (k : n), ∑ j ∈ Finset.univ.erase k, ‖A k j‖ < ‖A k k‖

⊢ A.det ≠ 0
-- Mathlib/LinearAlgebra/Matrix/Gershgorin.lean
K : Type u_1
n : Type u_2
inst✝² : NormedField K
inst✝¹ : Fintype n
inst✝ : DecidableEq n
A : Matrix n n K
h : A.det = 0

⊢ ∃ k, ‖A k k‖ ≤ ∑ j ∈ Finset.univ.erase k, ‖A k j‖
-- Mathlib/LinearAlgebra/Matrix/Gershgorin.lean
K : Type u_1
n : Type u_2
inst✝² : NormedField K
inst✝¹ : Fintype n
inst✝ : DecidableEq n
A : Matrix n n K
h : A.det = 0

⊢ ∃ k, 0 ∈ Metric.closedBall (A k k) (∑ j ∈ Finset.univ.erase k, ‖A k j‖)
-- Mathlib/LinearAlgebra/Matrix/Gershgorin.lean
K : Type u_1
n : Type u_2
inst✝² : NormedField K
inst✝¹ : Fintype n
inst✝ : DecidableEq n
A : Matrix n n K
h : A.det = 0

⊢ Module.End.HasEigenvalue (Matrix.toLin' A) 0
-- Mathlib/LinearAlgebra/Matrix/Gershgorin.lean
K : Type u_1
n : Type u_2
inst✝² : NormedField K
inst✝¹ : Fintype n
inst✝ : DecidableEq n
A : Matrix n n K
h : A.det = 0

⊢ ⊥ ≠ LinearMap.ker (Matrix.toLin' A)
-- Mathlib/NumberTheory/DirichletCharacter/Basic.lean
R : Type u_1
inst✝ : CommMonoidWithZero R
n : ℕ
χ✝ : DirichletCharacter R n
χ : DirichletCharacter R 1
a✝ : (ZMod 1)ˣ

⊢ χ ↑a✝ = 1 ↑a✝
-- Mathlib/LinearAlgebra/TensorAlgebra/Basic.lean
R : Type u_1
inst✝² : CommSemiring R
M : Type u_2
inst✝¹ : AddCommMonoid M
inst✝ : Module R M

⊢ Disjoint (LinearMap.range (ι R)) 1
-- Mathlib/LinearAlgebra/TensorAlgebra/Basic.lean
R : Type u_1
inst✝² : CommSemiring R
M : Type u_2
inst✝¹ : AddCommMonoid M
inst✝ : Module R M

⊢ ∀ x ∈ LinearMap.range (ι R), x ∈ 1 → x = 0
-- Mathlib/LinearAlgebra/TensorAlgebra/Basic.lean
R : Type u_1
inst✝² : CommSemiring R
M : Type u_2
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
x : M
r : R
hx : (ι R) x = (Algebra.linearMap R (TensorAlgebra R M)) r

⊢ (Algebra.linearMap R (TensorAlgebra R M)) r = 0
-- Mathlib/LinearAlgebra/TensorAlgebra/Basic.lean
R : Type u_1
inst✝² : CommSemiring R
M : Type u_2
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
x : M
r : R
hx : x = 0 ∧ r = 0

⊢ (Algebra.linearMap R (TensorAlgebra R M)) r = 0
-- Mathlib/Order/Interval/Set/OrdConnected.lean
α : Type u_1
β : Type u_2
inst✝¹ : Preorder α
inst✝ : Preorder β
s✝ t s : Set α

⊢ (⇑ofDual ⁻¹' s).OrdConnected ↔ s.OrdConnected
-- Mathlib/Order/Interval/Set/OrdConnected.lean
α : Type u_1
β : Type u_2
inst✝¹ : Preorder α
inst✝ : Preorder β
s✝ t s : Set α

⊢ (∀ (x x_1 : { a // toDual a ∈ ⇑ofDual ⁻¹' s }), ⇑ofDual ⁻¹' Icc ↑x_1 ↑x ⊆ ⇑ofDual ⁻¹' s) ↔ ∀ (x x_1 : { a // a ∈ s }), Icc ↑x ↑x_1 ⊆ s
-- Mathlib/CategoryTheory/Generator.lean
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
D : Type u₂
inst✝ : Category.{v₂, u₂} D
G : Cᵒᵖ

⊢ IsCoseparator (unop G) ↔ IsSeparator G
-- Mathlib/Topology/LocallyConstant/Basic.lean
X✝ : Type u_1
Y : Type u_2
Z : Type u_3
α : Type u_4
inst✝¹ : TopologicalSpace X✝
X : Type u_5
inst✝ : TopologicalSpace X
f g : LocallyConstant X (Fin 2)
h : ⇑f ⁻¹' {0} = ⇑g ⁻¹' {0}

⊢ f = g
-- Mathlib/Topology/LocallyConstant/Basic.lean
X✝ : Type u_1
Y : Type u_2
Z : Type u_3
α : Type u_4
inst✝¹ : TopologicalSpace X✝
X : Type u_5
inst✝ : TopologicalSpace X
f g : LocallyConstant X (Fin 2)
h : ∀ (x : X), f x = 0 ↔ g x = 0

⊢ f = g
-- Mathlib/Topology/LocallyConstant/Basic.lean
X✝ : Type u_1
Y : Type u_2
Z : Type u_3
α : Type u_4
inst✝¹ : TopologicalSpace X✝
X : Type u_5
inst✝ : TopologicalSpace X
f g : LocallyConstant X (Fin 2)
h : ∀ (x : X), f x = 0 ↔ g x = 0
x : X

⊢ f x = g x
-- Mathlib/NumberTheory/Cyclotomic/PrimitiveRoots.lean
p n : ℕ+
A : Type w
B : Type z
K : Type u
L : Type v
C : Type w
inst✝⁹ : CommRing A
inst✝⁸ : CommRing B
inst✝⁷ : Algebra A B
inst✝⁶ : IsCyclotomicExtension {n} A B
inst✝⁵ : Field L
ζ✝ : L
hζ✝ : IsPrimitiveRoot ζ✝ ↑n
inst✝⁴ : Field K
inst✝³ : Algebra K L
inst✝² : Algebra K A
inst✝¹ : IsDomain A
ζ : A
inst✝ : IsCyclotomicExtension {n} K A
hζ : IsPrimitiveRoot ζ ↑n
h : Irreducible (cyclotomic (↑n) K)

⊢ minpoly K (ζ - 1) = (cyclotomic (↑n) K).comp (X + 1)